Reframing the Conceptual Change Approach in Learning and Instruction (Advances in Learning and Instruction) (Advances in Learning and Instruction) (Advances in Learning and Instruction)

RE-FRAMING THE CONCEPTUAL CHANGE APPROACH IN LEARNING AND INSTRUCTION

ADVANCES IN LEARNING AND INSTRUCTION SERIES Series Editors: K. Littleton, C. P. Constantinou, L. Mason, W.-M. Roth and R. Wegerif Further details: www.elseviersocialsciences.com www.elsevier.com/locate/series/ali Recently Published Volumes VERSCHAFFEL, DOCHY, BOEKAERTS AND VOSNIADOU Instructional Psychology: Past, Present, and Future Trends TYNJÄLÄ, VÄLIMAA AND BOULTON-LEWIS Higher Education and Working Life: Collaborations, Confrontations and Challenges ELEN AND CLARK Handling Complexity in Learning Environments HAKKARAINEN, PALONEN, PAAVOLA AND LEHTINEN Communities of Networked Expertise TUOMI-GROHN AND ENGESTROM Between School and Work DE CORTE, VERSCHAFFEL, ENTWISTLE AND VAN MERRIËNBOER Powerful Learning Environments Related journals – sample copies available online from: http://www.elsevier.com Learning and Instruction International Journal of Educational Research Computers and Education The Internet and Higher Education Early Childhood Research Quarterly

RE-FRAMING THE CONCEPTUAL CHANGE APPROACH IN LEARNING AND INSTRUCTION EDITED BY

STELLA VOSNIADOU University of Athens, Greece

ARISTIDES BALTAS National Technical University of Athens, Greece

XENIA VAMVAKOUSSI University of Athens, Greece

Published in Association with the European Association for Learning and Instruction

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ADVANCES IN LEARNING AND INSTRUCTION Series Editors: K. Littleton, C. P. Constantinou, L. Mason, W.-M. Roth and R. Wegerif Editor-in-Chief K. Littleton, Centre for Childhood Development and Learning, Open University, MK7 6AA. E-mail: [email protected]

Editorial Board P. Boscolo, University of Padova, Italy E. De Corte, University of Leuven, Belgium W.-M. Roth, University of Victoria, British Columbia, Canada Publisher’s Liasion R. Wegerif, Open University, UK

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Contents

List of Figures

xi

List of Tables

xiii

Acknowledgements

xv

Contributors

xvii

Preface

xxi

1.

The Conceptual Change Approach and its Re-Framing Stella Vosniadou

PART 1: Foundations of the Conceptual Change Approach: Kuhn’s Influence

1

17

2. The Philosophical Foundations of the Conceptual Change Approach: An Introduction Aristides Baltas

19

3. In the Wake of Thomas Kuhn’s Theory of Scientific Revolutions: The Perspective of an Historian of Science Lillian Hoddeson

25

4.

Kuhn’s Philosophical Successes? Peter Machamer

35

5.

Conceptual Change and Scientific Realism: Facing Kuhn’s Challenge Theodore Arabatzis

47

6. Background ‘Assumptions’ and the Grammar of Conceptual Change: Rescuing Kuhn by Means of Wittgenstein Aristides Baltas

63

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Contents

Commentaries

81

7.

Reflections on Conceptual Change Stathis Psillos

83

8.

Conceptual Change as Structure Change: Comment on Kuhn’s Legacy Matti Sintonen

89

PART 2: Personal Epistemology and Conceptual Change

97

9.

Personal Epistemology and Conceptual Change: An Introduction Stella Vosniadou

99

10.

Epistemological Threads in the Fabric of Conceptual Change Research P. Karen Murphy, Patricia A. Alexander, Jeffrey A. Greene and Maeghan N. Edwards

11.

Conceptions of Learning and the Experience of Understanding: Thresholds, Contextual Influences, and Knowledge Objects Noel Entwistle

123

Conceptual Change in Physics and Physics-Related Epistemological Beliefs: A Relationship under Scrutiny Christina Stathopoulou and Stella Vosniadou

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Effects of Epistemological Beliefs and Learning Text Structure on Conceptual Change Lucia Mason and Monica Gava

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12.

13.

14.

Conceptual Change Ideas: Teachers’ Views and their Instructional Practice Reinders Duit, Ari Widodo and Christoph T. Wodzinski

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197

Commentary

219

15.

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First Steps: Scholars’ Promising Movements Into a Nascent Field of Inquiry Patricia A. Alexander and Gale M. Sinatra

PART 3: Extending the Conceptual Change Approach to Mathematics Learning 16.

17.

237

Extending the Conceptual Change Approach to Mathematics Learning: An Introduction Xenia Vamvakoussi

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When We Clashed with the Real Numbers: Complexity of Conceptual Change in Number Concept Kaarina Merenluoto and Tuire Palonen

247

Contents 18.

How Many Numbers are there in a Rational Numbers Interval? Constraints, Synthetic Models and the Effect of the Number Line Xenia Vamvakoussi and Stella Vosniadou

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265

19.

Students’ Interpretations of Literal Symbols in Algebra Konstantinos P. Christou, Stella Vosniadou and Xenia Vamvakoussi

283

20.

Teaching for Conceptual Change: The Case of Infinite Sets Pessia Tsamir and Dina Tirosh

299

Commentaries

317

21.

Nurturing Conceptual Change in Mathematics Education Brian Greer and Lieven Verschaffel

319

22.

Reconceptualizing Conceptual Change Anna Sfard

329

Subject Index

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List of Figures

Figure 10.1:

Relational model of epistemological stances.

108

Figure 11.1:

Categories used in describing conceptions of knowledge and learning.

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Only 11 students, all of whom were found to hold constructivist physics-related epistemological beliefs, were found to achieve high scores in the Force and Motion Conceptual Evaluation (FMCE) instrument.

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A skeletal theoretical framework for conceptualizing the relationship between (physics-related) epistemological beliefs and physics conceptual understanding.

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Some examples of physics questions and problems used in the interviews.

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Adjusted mean scores of text-based comprehension by condition and epistemological beliefs.

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Adjusted mean of composite scores for conceptual change by condition and epistemological beliefs.

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Steps of the constructivist teaching sequences for conceptual change.

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Figure 14.2:

Design of the study.

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Figure 14.3:

Results of subcategories A-1 to A-5.

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Figure 14.4:

Results of subcategories B-1 to B-5.

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Figure 14.5:

Results of subcategories C-2 and C-3.

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Figure 14.6:

Results of subcategory D-1 to D-4.

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Figure 14.7:

Average time of each step of the CTS.

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Figure 14.8:

Analyses of the sequences of lessons.

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Figure 12.1:

Figure 12.2:

Figure 12.3: Figure 13.1: Figure 13.2: Figure 14.1:

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List of Figures

Figure 14.9: Figure 17.1:

Figure 17.2:

Figure 17.3:

Figure 20.1:

Achievement gains (Rasch parameters and residuals) for types A to D.

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A simplified model of the extensions of number domains from natural numbers to real numbers and the changes in the level of abstraction with each enlargement.

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Interviewed mathematicians in the two-dimensional field of explanations of their individual thinking and their involvement in the community of mathematics.

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A tentative “graph” of the radical conceptual changes in the extensions of the number concept and “areas” of students’ and mathematicians’ explanations.

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Vertical and Graphical Representations.

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List of Tables

Table 11.1:

Contrasting forms of understanding in revising for final examinations.

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Table 12.1:

Criteria for identifying students’ approach to learning and studying.

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Table 13.1:

Items loading on the epistemological belief dimension “Certain and simple Knowledge.”

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Frequency/percentage of responses to the open-ended questions at pretest by condition and epistemological beliefs.

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Frequency/percentage of responses to the multiple-choice questions at pre, immediate, and delayed posttests by condition and epistemological beliefs.

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Frequency/percentage of responses to the text-based comprehension questions at immediate and delayed posttests by condition and epistemological beliefs.

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Frequency/percentage of responses to the open-ended questions at immediate and delayed posttests by condition and epistemological beliefs.

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Table 14.1:

Constructivist-oriented science classrooms (COSC).

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Table 14.2:

Types of teachers’ implicit theories based on the analysis of teachers’ interviews.

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Table 17.1:

Intensity of Interviewees’ Expressions and Elaborations.

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Table 18.1:

The open-ended questionnaire (QT1).

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Table 18.2:

The forced-choice questionnaire (QT2).

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Table 18.3:

Percentage of answer types, in the total of the answers given in all six questions, as a function of the type of questionnaire and of grade.

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Table 13.2: Table 13.3:

Table 13.4:

Table 13.5:

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List of Tables

Table 18.4:

Number and percentage of students in categories formed with respect to their performance with and without the number line, as a function of grade and type of questionnaire.

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Percentage of students placed in the five categories, as a function of type of questionnaire and of grade.

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Students’ accounts of the structure of rational numbers intervals: some examples.

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Differences between the natural numbers in the context of arithmetic and literal symbols in the context of algebra.

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An example of the way in which questions were posed in the forced choice questionnaire.

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Table 19.3:

Percentages of students’ responses to the combined questions.

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Table 19.4:

Frequencies, percentages, and examples of students’ responses to each question.

293

Table 18.5: Table 18.6: Table 19.1: Table 19.2:

Acknowledgements

The present book owes its existence to the hard work of many people. We would like to make reference in particular to a set of students in the inter-disciplinary graduate programme in ‘Basic and Applied Cognitive Science’ at the University of Athens, whose help has been instrumental in the organisation of the 4th European Symposium on ‘Conceptual Change’ out of which this volume has emerged: Irini Skopeliti, Nektarios Mamalougos, Katerina Ligovanli, Kalliopi Ikospentaki, Christina Stathopoulou, Maria Koulianou, Erifylli Tsirempolou, Vassiliki Siereki, Konstantinos P. Christou, and Natassa Kyriakopoulou. We also thank all the members of the Special Interest Group on Conceptual Change of the European Association for Research on Learning and Instruction, and particularly its coordinators (at that time) Kaarina Merenluoto and Gunilla Petersson, who have contributed to the success of the 4th European Symposium. Finally we express our sincere thanks to Spyridoula Efthimiou for her superb secretarial support during the preparation of the book. The 4th European Symposium and therefore also this book would not have been possible without the financial help from the University of Athens, the Greek Ministry of Education (through its research programmes Pythagoras and Hrakleitos), the Greek Ministry of Culture, the Greek Center of Educational Research, the Commersial Bank of Greece and Gutemberg Publications. The editors Stella Vosniadou Aristides Baltas Xenia Vamvakoussi

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Contributors

Patricia A. Alexander Department of Human Development, University of Maryland, College Park, MD, USA Theodore Arabatzis Department of Philosophy and History of Science, University of Athens, Athens, Greece Aristides Baltas Faculty of Applied Mathematics and Physics, National Technical University of Athens, Athens, Greece Konstantinos P. Christou Department of Philosophy and History of Science, University of Athens, Athens, Greece Reinders Duit Institute for Science Education, University of Kiel, Kiel, Germany Maeghan N. Edwards Department of Educational Psychology, The Pennsylvania State University, PA, USA Noel Entwistle Higher and Further Education Department, University of Edinburgh, Edinburgh, UK Monica Gava Department of Developmental and Socialization Psychology (DPSS), University of Padova, Padova, Italy Jeffrey A. Greene University of Maryland, MD, USA Brian Greer 2632, NE 7th Avenue, Portland, OR, USA Lillian Hoddeson Department of History, University of Illinois at Urbana-Champaign, Urbana, IL, USA

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Peter Machamer Department of History and Philosophy of Science, University of Pittsburgh, Pittsburgh, PA, USA Lucia Mason Department of Developmental and Socialization Psychology (DPSS), University of Padova, Padova, Italy Kaarina Merenluoto Department of Teacher Education, University of Turku, Turku, Finland P. Karen Murphy Department of Educational Psychology, The Pennsylvania State University, PA, USA Tuire Palonen Faculty of Education, University of Turku, Turku, Finland Stathis Psillos Department of Philosophy and History of Science, University of Athens, Athens, Greece Matti Sintonen Department of Philosophy, University of Helsinki, Finland Anna Sfard Department of Education, University of Haifa, Haifa, Israel and Michigan State University, USA Gale M. Sinatra Department of Educational Psychology, University of Nevada, Las Vegas, NV, USA Matti Sintonen Department of Philosophy, University of Helsinki, Finland Christina Stathopoulou Department of Philosophy and History of Science, University of Athens, Athens, Greece Dina Tirosh School of Education, Tel-Aviv University, Tel-Aviv, Isreal Pessia Tsamir School of Education, Tel-Aviv University, Tel-Aviv, Israel Xenia Vamvakoussi Department of Philosophy and History of Science, University of Athens, Athens, Greece

Contributors

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Lieven Verschaffel Department of Educational Sciences, Katholic University of Leuven, Center for Instructional Psychology & Technology (CIP&T), Leuven, Belgium Stella Vosniadou Department of Philosophy and History of Science, University of Athens, Athens, Greece Ari Widodo Jurusan Pendidikan Biologi, FPMIPA UPI, Jl. Dr. Setiabudhi 229, Bandung, Indonesia Christoph T. Wodzinski Institute for Science Education, University of Kiel, Kiel, Germany

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Preface

The present volume is an outcome of the meeting of the Special Interest Group (SIG) on Conceptual Change of the European Association for Research in Learning and Instruction (EARLI), which took place in Delphi, Greece, from May 19 to May 23, 2004. This was the 4th European Symposium on Conceptual Change; the previous meetings took place in Jena (1994), Madrid (1998) and Turku (2002). This is also the third book being published as an outcome of the conferences held by the EARLI SIG. The first book was entitled New Perspectives on Conceptual Change [W. Schnotz, S. Vosniadou, & M. Carretero (Eds), Elsevier Science, 1999] and the second was entitled Reconsidering Conceptual Change: Issues in Theory and Practice [M. Limon, & L. Mason (Eds), Kluwer Academic Publishers, 2003]. The purpose of the meetings of the Conceptual Change SIG is to discuss special theoretical and methodological issues on conceptual change that are of particular interest to the SIG members. The Delphi symposium was designed to address the foundations of the conceptual change approach in the philosophy and history of science and, more specifically, the contribution of Thomas Kuhn. We were interested in examining some of the criticisms of Kuhn’s theory and in understanding how they apply to conceptual change research in learning and instruction. The present volume is the outcome of the presentations and discussions that took place at the SIG meeting. It has collected some of the papers presented by the philosophers and historians of science at the Delphi Symposium, which are particularly helpful in illuminating some of the links between philosophy and history of science and the current theoretical concerns in the field of learning and instruction. It also includes papers from invited symposia on Changes on Epistemological Beliefs and Effects of Epistemological Beliefs on Conceptual Change and Conceptual Change in Mathematics: Theoretical Issues and Educational Applications because these are particularly appropriate in the context of the re-framed conceptual change approach that we wanted to present in this book. More specifically, the book consists of three sections. The first section concerns the roots of the conceptual change approach in learning and instruction, which lie in the philosophy and history of science. The influential work of Thomas Kuhn is critically examined by philosophers and historians of science who trace aspects of Kuhnian key constructs that are still of value today. The second section examines the influence that epistemological beliefs can have on conceptual change. The third section deals with mathematics learning and teaching from a conceptual change perspective. Epistemological beliefs and

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mathematics learning and teaching are new domains of study that, we argue, can be examined in a beneficial way from a conceptual change point of view. However, this implies that the conceptual change approach should be re-defined so as to account for those aspects of learning that have been neglected before, as well as to take into consideration the particularities of a domain (mathematics learning) which differentiates it from science learning. The general introduction to the book, the introductions to the three parts(Parts I, II and III), as well as the commentaries that follow each part attempt to capture the important issues in the three areas of interest, trace the questions that need to be further investigated and strengthen the coherence among the interdisciplinary contributions. We hope that the present book will contribute, in the tradition of the Conceptual Change SIG volumes, to further advance our understanding of the problem of conceptual change in learning and its implications for instruction. The Editors Stella Vosniadou Aristides Baltas Xenia Vamvakoussi

Chapter 1

The Conceptual Change Approach and its Re-Framing Stella Vosniadou Theories of learning must provide an answer to the problem of how concepts change. Most theories do that by assuming that learning is cumulative and domain general, and that concepts change through the enrichment of prior knowledge. All of these theories find it difficult to explain both the considerable re-organization of conceptual knowledge that takes place with learning and development, and the difficulties students encounter particularly when it comes to learning some of the more advanced concepts in science and mathematics. The conceptual change approach has emerged from an effort to provide more satisfactory answers to questions regarding the radical re-organization of conceptual knowledge and the understanding of difficult science concepts. It is a constructivist approach that, unlike previous approaches, considers knowledge to be organised in domain-specific, theory-like structures, and knowledge acquisition to be characterized by theory-like changes. Although the beginning of the conceptual change approach can be traced to physics education, it is not restricted to physics but makes a larger claim about learning that transcends many domains and can apply, for example, to biology (Hatano & Inagaki, 1994), psychology (Wellman, 2002) and cognitive development in general (Carey, 1985). Recently the term ‘controversial conceptual change’ has been used to refer to the debates surrounding certain topics, such as evolution and environmental changes, or issues that are related to political and ideological differences and peace education.

The Classical Approach to Conceptual Change The foundations of the conceptual change approach lie in the attempts by philosophers and historians of science to explain theory change in science. In his well-known book The Structure of Scientific Revolutions, Thomas Kuhn (1962), following other philosophers of science, like Norwood Russell Hanson, Stephen Toulmin, Paul Feyerabend, Larry Laudan and Imre Lakatos, questioned the attempts by logical positivists and logical empiricists to treat scientific theories as sets of axioms that could be formulated in mathematical logic (known as the Received View) and the related treatment of theory change as theory reduction. According

Reframing the Conceptual Change Approach in Learning and Instruction Copyright © 2007 by Elsevier Ltd. All rights of reproduction in any form reserved. ISBN: 0-08-045355-4

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to theory reduction, a theory that enjoys a high degree of confirmation cannot ever be disconfirmed, but can only be expanded to a theory with a wider scope, or absorbed into a more inclusive and comprehensive theory (see Suppe, 1977). Kuhn argued that normal science operates within sets of shared beliefs, assumptions, commitments and practices that constitute ‘paradigms’. Discoveries emerge over time that cannot be accommodated within the existing paradigm. When these anomalies accumulate, science enters a period of crisis that is eventually resolved by a revolutionary change in paradigm. According to Kuhn, different paradigms are incommensurable. Scientific knowledge grows as we move from one paradigm to another, but it is no longer possible to imagine the results of scientific revolutions as a cumulative, linear progression. In their search for a theoretical framework to conceptualize the learning of science, some science educators turned to the philosophy and history of science as a major source of hypotheses concerning how concepts change in the process of learning science. Researchers like Viennot (1979), Driver and Easley (1978) and McCloskey (1983) realized that students bring to the science learning task alternative frameworks, preconceptions or misconceptions that are robust and difficult to extinguish. Some of them saw an analogy between theory change in science and the need for students to change their alternative frameworks and replace them with the scientific concepts instructed at school (Posner, Strike, Hewson, & Gertzog, 1982). According to Posner et al. (1982), there are four fundamental conditions that need to be fulfilled before conceptual change can happen in science education: (1) there must be dissatisfaction with existing conceptions, (2) there must be a new conception that is intelligible, (3) the new conception must appear to be plausible, and (4) the new conception should suggest the possibility of a fruitful program. This theoretical framework, which we call the ‘classsical approach’ to conceptual change, became the leading paradigm that guided research and instructional practices in science education for many years. On the basis of this ‘classical approach’, the child is like a scientist, the process of science learning is a rational process of theory replacement, conceptual change is like a gestalt shift that happens over a short period of time, and cognitive conflict is a major instructional strategy for producing conceptual change. Over the years practically all of the above-mentioned tenets of the classical approach have been seriously questioned. Some of these criticisms are similar in many respects to the criticisms voiced of Kuhn’s philosophical approach. Indeed, one of the purposes of the present volume is to examine the criticisms of the Kuhnian approach that emerged over the years in the philosophy and history of science, and evaluate their usefulness for theorizing conceptual change in learning and instruction. One important set of criticisms, coming from socio-cultural theory, has pointed out that conceptual change should not be seen as only an individual, internal, cognitive process, but also, as a social activity that takes place in a complex socio-cultural world (see Hatano, 1994; Caravita & Hallden, 1994). These researchers believe that the situational, cultural, and educational contexts should be taken into account in trying to explain how people’s concepts change. It is interesting to note that this point is actually very much in line with Kuhn’s (1962) own arguments that the notion of ‘theory’, conceived as a set of propositions, is too narrow to account for the activities of scientists and should be replaced with a ‘paradigm’. As Machamer (this volume) notes, the introduction of the notion of ‘paradigm’ by

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Kuhn shifts the emphasis from individuals’ minds to the role that the scientific community and their group commitments, shared examples, and tacit knowledge play in scientific discovery and change. Hoddeson (this volume) also points out how Kuhn’s influence helped historians of science to think about science as a social activity. We agree in part with the criticisms coming from the socio-cultural perspective and we believe that it is important to consider the important role played by socio-cultural factors in conceptual change. However, a serious limitation of radical socio-cultural (or situative) perspectives (e.g., Lave, 1996; Rogoff, 1998; Saljo, 1999; Sfard, this volume) is that they consider only the internalization or appropriation of existing cultural practices, tools, and artifacts, and do not pay adequate attention to the active role of the individual in understanding or constructing new knowledge. As Hatano (1994) aptly expresses, discussing the work of another of his Japanese colleagues (Kobayashi, 1994), ‘although understanding is a social process, it also involves much processing by an active individual mind. It is unlikely that conceptual change is induced only by social consensus. The post-change conceptual system must have not only coherence but also subjective necessity. Such a system can be built only through an individual mind’s active attempts to achieve integration and plausibility’ (p. 195). We believe that the conceptual change approach could be re-framed to account for the effects of socio-cultural factors without ignoring the contribution of the constructive individual (see also Vosniadou, in press, for a detailed discussion of this issue). Other aspects of the conceptual change approach have also been criticized. One kind of criticism centers on the nature of conceptual change and how it is achieved. For example, many researchers believe that conceptual change is a slow and gradual process and not a dramatic, gestalt shift that happens over a short period of time (Caravita & Hallden, 1994; Vosniadou, 2003; Hatano & Inagaki, 1994). Others have pointed out that science learning does not require the replacement of ‘incorrect’ with ‘correct’ conceptions, but the ability on the part of the learner to take different points of view and understand when different conceptions are appropriate depending on the context of use (e.g., Pozo, Gomez, & Sanz, 1999; Spada, 1994). Again we agree with these criticisms and we believe that it is possible to find a solution to the problem of how enrichment-kind of additive mechanisms can produce radical changes in knowledge over long periods of time. Our proposal, which will be discussed in the next section, has many similarities to the philosophical arguments made by Baltas (this volume), and provides a new perspective to the problem of incommensurability. Finally, a third kind of criticism concerns the methods of teaching for conceptual change. It has been argued that cognitive conflict is not a successful instructional strategy for producing conceptual change, as students tend to patch up local inconsistencies in a superficial way (Chinn & Brewer, 1993; Smith, diSessa, & Roschelle, 1993; Vosniadou, 1999). Motivational theorists have pointed out that the ‘classical approach’ represents ‘cold’ cognition and does not take into consideration affective and motivational factors. Effective teaching for conceptual change, they point out, should try to find ways to increase students’ motivation to change their beliefs and persuade them that conceptual change is necessary (Pintrich, Marx, & Boyle, 1993; Sinatra & Dole, 1998; Alexander, 2001). Again, we agree with the above-mentioned criticisms and believe that the conceptual change approach should utilize but cannot rely on cognitive conflict as an instructional

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strategy. In addition, attention needs to be paid to the development of motivated and intentional learners who have the metaconceptual awareness necessary in order to understand the differences between their naïve beliefs and the scientific theories to which they are exposed and are capable of using sophisticated mechanisms of hypothesis testing and deliberate belief-revision that scientists use in the process of scientific discovery (Nersessian, 1992). In view of the above-mentioned, there have been a number of attempts to re-frame the classical approach (e.g., Caravita & Hallden, 1994; Hatano, 1994) including our own (Vosniadou, 2002, 2003, in press; Vosniadou & Verschaffel, 2004). In this chapter we will try to collect all these attempts into a proposal for a more coherent, re-framed conceptual change approach to learning and instruction.

Re-Framing the Conceptual Change Approach Domain Specificity and Naïve Theories As mentioned earlier, the conceptual change approach considers conceptual change as domain-specific theory-like change. In the ‘classical approach’ the theories that needed to be changed were supposed to be students’ alternative frameworks, or misconceptions. In the re-framed approach, however, the theories that need to be changed are the naïve, intuitive, domain-specific theories constructed in early childhood, on the basis of everyday experience under the influence of lay culture. In this respect, the re-framed approach draws heavily on domain-specific approaches in cognitive developmental research (Carey, 1985; Keil, 1990; Wellman, 2002; Keil, 1992; Inagaki & Hatano, 2002). In the sections that follow we will examine in greater detail the assumptions regarding domain specificity and naïve theories. Domain specificity Most theories of learning and development, such as Piagetian and Vygotskian approaches, information processing or socio-cultural theories are domain general. They focus on principles, stages, mechanisms, strategies, etc., that are meant to characterize all aspects of development and learning. In contrast, the conceptual change approach is a domain-specific approach. It examines distinct domains of thought and attempts to describe the processes of learning and development within these domains. The idea that human cognition includes domain-specific mechanisms for learning is based on a number of independent research traditions and sets of empirical findings, some coming from animal studies (Gallistel, 1990), others based on Chomsky’s work in linguistics (Chomsky, 1988). Many cognitive developmental psychologists see domain specificity through the notion of domain-specific constraints on learning (Keil, 1981, 1990). It is argued that such constraints are needed in order to restrict the indeterminacy of experience (Goodman, 1972) and guide, amongst others, the development of language (Markman, 1989), numeric understanding (Gelman, 1990), or physical and psychological knowledge (Wellman & Gelman, 1998). There is a great deal of debate in the literature as to whether domain-specific constraints should be seen as hardwired and innate as opposed to acquired, and as having representational

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content or not (see Elman et al., 1996). Many researchers prefer to see domain-specific constraints as innate or acquired biases or preferences that do not have representational content but rather mitigate the interaction between a learning system and the environment (e.g., Keil, 1990; Inagaki & Hatano, 1991). Finally, some domain-specific approaches focus on the description of the development of expertise in different subject-matter areas, such as physics (Chi, Feltovich, & Glaser, 1981), mathematics (VanLehn, 1990; Mayer, 1985) or chess (Chase & Simon, 1973), without necessarily appealing to innate modules or constraints. The conceptual change approach can be applied to any of the above conceptualizations of domain specificity, focusing on the description and explanation of the changes that take place in the content and structure of knowledge with learning and development. Domainspecific approaches should be seen as complementary rather than contradictory to domaingeneral approaches. It is very likely that both domain-general and domain-specific mechanisms and constraints apply to development and learning (Keil, 1990). Naïve theories A basic characteristic of the conceptual change approach is the assumption that domain-specific knowledge is theory-like. The term theory-like is used to denote a relatively coherent body of domain-specific knowledge characterized by a distinct ontology and a causality that can give rise to explanation and prediction, and not an explicit, well-formed and socially shared scientific theory. For example, many cognitive developmental psychologists would agree that children are biased to notice differences in objects that move by themselves compared to those that require the push/pull of an external agent (e.g., Golinkoff, Harding, Carlson, & Sexton, 1984). This early distinction forms the basis for the differentiation of animate from inanimate objects and thus the beginnings of the distinction between naïve psychology and naïve physics. Naïve physics consists of the ontology of inanimate objects in the context of mechanical causality, while naïve psychology consists of the ontology of animate objects in the context of intentional, psychological causality. There is a great deal of experimental evidence that by the age of six children have developed a naïve theory of physics and a naïve psychology and very probably a naïve biology as well (Inagaki & Hatano, 2002). The importance of the assumption that early knowledge is organized in the form of naïve theories lies in the fact that theory-like structures are generative. As such, they make it possible for children to formulate explanations and predictions and to deal with unfamiliar problems, thus enabling them to make sense of everyday phenomena. As mentioned above, naïve theories are very different from scientific theories. They are not well formed, they are not explicit, they are not socially shared, and they are not accompanied by metaconceptual awareness. It appears that children do not fully understand that their beliefs are hypotheses that need to be tested and may be falsified, and are not very good at co-ordinating theory with evidence in order to revise their explanations (Kuhn, Amsel, & O’Loughlin, 1988; Vosniadou, 2003). How does Conceptual Change Happen? The nature of conceptual change The processes of knowledge acquisition with development can proceed either in the direction of enriching existing knowledge structures or

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towards restructuring them (Carey, 1985; Vosniadou & Brewer, 1987). Theory-like knowledge structures allow the possibility that developmental change is theory-like change and this is exactly what conceptual change is meant to be. There is a great deal of evidence that cognitive development is indeed characterized by conceptual re-organization. For example, in the domain of biology, cross-sectional developmental studies show that the biological knowledge of the 10-year old is qualitatively different from that of the 4–6-year-old child (Carey, 1985; Hatano & Inagaki, 1994), although there is disagreement as to how exactly this development proceeds. Young children explain biological phenomena within the broader framework of a naïve psychology, whereas for older children biology has become a distinct domain. Conceptual changes in the domain of biology have been described in terms of three fundamental components: first, the ontological distinctions between living/non-living and mind/body; second, the modes of inference that children employ to produce predictions regarding the behavior of biological kinds; and third, the causal-explanatory framework children employ. For example, young children employ intentional or vitalistic causality to explain biological phenomena, reason on the basis of similarity to humans, and consider plants to be non-living. In contrast, by the age of 10 most children have formed relatively well-defined biological categories from which they reason about biological phenomena, and have re-organized their concept of living things to include plants. Similar re-organizations of conceptual knowledge across early childhood years can be found amongst others, in children’s theory of mind (Wellman, 1990), theory of matter (Smith, Carey, & Wiser, 1985), and in astronomy (Vosniadou & Brewer, 1992, 1994). Our work in observational astronomy has shown that considerable qualitative changes take place in children’s concept of the earth between the ages of 4–6 and 10–12. Pre-school children consider the earth to be a stable, stationary and flat physical object located in the center of the universe. These beliefs about the earth are embedded within the larger, framework theory of physics, that is, a naïve physics. The earth is categorized as a physical object and all the beliefs that apply to physical objects in general are also applied to the earth. On the contrary, most children at the end of the elementary school think of the earth as an astronomical object, a planet, rotating around itself and revolving around the sun in a heliocentric solar system. In this process, a significant ontological shift has taken place in the concept of the earth. From a physical object for most first graders it has become a solar object for the majority of sixth grader children (Vosniadou & Skopeliti, 2005). Similar ontological shifts have been pointed out by Chi and her colleagues to occur, for example, in the case of the concept of force, light, heat, electricity etc. (Reiner, Slotta, Chi, & Resnick, 2000). The processs of conceptual change In the classical approach, theory changes with learning and development were considered the result of a rational process of theory replacement, by a thinking (like a scientist) student. Following Kuhn’s (1962) original proposal, this theory replacement was supposed to be achieved within a short period of time, like a gestalt-type switch. Again, following Kuhn, the new theory was supposed to be incommensurable to the old one. This proposal has received a great deal of criticism both from psychologists and educators. The empirical studies so far show that the process of conceptual change is slow and gradual rather than a dramatic gestalt-type shift, by learners who, unlike scientists, lack

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metaconceptual awareness of their beliefs and of the process of change or, for that matter, the need to change (Vosniadou, 2003). Furthermore, as mentioned earlier, it has become clear that conceptual change is not only an internal cognitive process but one that happens in broader situational, cultural and educational contexts, and that it is significantly influenced and facilitated by socio-cultural factors (see Hatano, 1994; also Machamer, this volume). Mechanisms of conceptual change We believe that if we consider that existing knowledge is organized in theory-like structures embedded in larger explanatory frameworks, then conceptual change can be accounted for through the use of simple, additive mechanisms, assuming, of course, that new information is coming in gradually through participation in socio-cultural activities. For example, everyday experiences with plants, such as watering plants, seeing them become bigger, or noticing that they can die, can lead children to understand that plants are similar to animals in certain properties, such as feeding, growing and dying. These similarities can eventually lead children to consider plants as living things, rather than as inanimate objects, despite the fact that they lack self-initiated movement (see Inagaki & Hatano, 1991; Vosniadou, 1999). This category change can be described as branch jumping (Thagard, 1988), or as an ontological shift (Chi, 1992). In other words, it appears that small and gradual changes in the concept of plant, through enriched experience and participation in cultural activities, together with the ability to engage in similarity-based reasoning can eventually produce changes in the larger explanatory framework within which the concept of plant was originally embedded. While this type of change can be considered as theory-like, it may be implicit and not accompanied by metaconceptual awareness. The use of bottom-up, implicit, additive mechanisms may not be very productive, however, in producing ‘instruction-induced conceptual change’. Instruction-induced conceptual change is the kind of conceptual change that requires systematic instruction in order to be achieved. Science concepts, like the concepts of force and energy, of heat and of photosynthesis, require many years of instruction in order to be understood. This is the case because scientific knowledge has developed over hundreds of years of scientific discovery into rather elaborate, counter-intuitive theories that differ in their concepts, in their structure, and in the phenomena they explain from the explanations children construct on the basis of their experience. Thus, the process of learning science appears to require children to understand a complex and counter-intuitive scientific theory, which represents an explanatory framework completely different from their naïve theories. In this process, the use of implicit, additive mechanisms is bound to produce hybrid or synthetic models (Vosniadou, in press). For example, let us take the case where a child is simply told that the earth is a sphere (and maybe shown a globe) without any further explanation. This information comes in conflict with the child’s naïve model of the earth, based on everyday experience, namely, that the earth is a flat physical object that has all the properties of physical objects, including the necessity of being supported in order not to fall ‘down’ (what has been called ‘up/down gravity’). Our studies have shown that in such situations, the use of simple, assimilatory types of mechanisms can give rise to a number of different synthetic models. One possible synthetic model is the model of a ‘dual earth’ according to which there are

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two earths: a flat, supported and stable earth on which people live, and a spherical, rotating earth, which is a ‘planet’ up in the sky. In forming this synthetic model, children simply add the information regarding the spherical earth to their existing knowledge without any other changes. Another synthetic model is that of the disc earth. According to this model, the information that the earth is spherical has been distorted to mean that the earth is round but also flat at the same time (Vosniadou & Brewer, 1992). Synthetic models are created because children revise some but not all of their beliefs about the earth that need to be changed if the scientific model is to be understood, using the bottom-up, additive mechanisms described earlier. Synthetic models happen because the children do not have explicit knowledge of their own beliefs and therefore they understand neither the contradictions between their naïve theories and the scientific explanations to which they are exposed, nor the distortions of the scientific view that they create. Such synthetic models can be observed in many subject-matter areas, from astronomy and mechanics, to history and mathematics (Gelman, 1990; Vosniadou & Brewer, 1992, 1994; Vosniadou & Ioannides, 1998; also Stafylidou & Vosniadou, 2004; Christou & Vosniadou, this volume). In mathematics, for instance, children misinterpret fractions to consist of two independent integers and order them from smaller to bigger focusing either on the nominator or on the denominator. Or, failing to understand the dense structure of rational numbers, many children think that there are no other numbers between two decimals like 0.05 and 0.06, or that there is only a finite number of numbers (e.g., 0.05, 0.051, 0.052, … 0.059, 0.06.) between them. Alternatively, they may understand that there are infinite numbers between decimals while at the same time believing that there is only a finite number of numbers between fractions (Vamvakoussi & Vosniadou, 2004, this volume). In order to avoid the construction of such synthetic models, students must (1) become aware of the inconsistencies between their naïve theories and the scientific ones, and (2) use the top-down, conscious and deliberate mechanisms for intentional learning mentioned earlier. In other words, instruction-induced conceptual change requires not only the restructuring of students’ naïve theories but also the restructuring of their modes of learning and the creation of metaconceptual awareness and intentionality (Sinatra & Pintrich, 2003; Vosniadou, 2003). Finally, it should also be added that avoidance of synthetic models can also be facilitated through the use of appropriate tools and artifacts as well as by the situational context in which the activity is taking place. The problem of incommensurability The problem of incommensurability is the claim that ‘if a scientific revolution has occurred, the meaning of terms in the former paradigm are incommensurable with (have a different meaning from) the terms of the new revolutionary paradigm’ (Machamer, this volume, p. 11). This is one of the most critical and yet most contested aspects of Kuhn’s theory. Because of the criticism he received from philosophers of science, Kuhn changed his position from one of ‘global’ incommensurability to one of ‘local’ incommensurability. The claim of local incommensurability is that only part of the meaning of the terms in the former paradigm change in the transition to the new paradigm. As Machamer (this volume) points out, this may be a move in the right direction, but it certainly weakens Kuhn’s original claim that the history of science is marked by scientific revolutions among paradigms.

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We are stopping to consider this issue here because most would admit that Kuhn’s incommensurability captures something important about learning. This ‘something’ has to do, on the one hand, with the terrible difficulty experienced by students or laymen in general in understanding the advanced concepts of current science and mathematics, and on the other with the difficulty we all experience in retrieving our old ways of thinking. While all of us were children once, it is nevertheless very difficult for us to now think like a child. Baltas (this volume) proposes a new look at Kuhn’s incommensurability that tries to save some of these intuitions. According to Baltas (this volume), paradigms include background assumptions that provide the glue, or what Wittgenstein would call the ‘grammatical hinges’ on which a theory and its concepts rest. The discovery of anomalies and inconsistencies in the process of doing normal science often necessitates a new examination of some of the background assumptions of the theory that creates this contradiction. Baltas argues that becoming aware of these assumptions, whose presence accounts for the observed contradictions, ‘opens up the grammatical space’ and allows a new paradigm/theory to occur. The above account does not claim that a new paradigm is incommensurable with the old, but, rather, that there is an asymmetry in their relation. The scientist who has made the change is capable of considering both the old and the new paradigms and understands the assumptions, which have been taken for granted in the old paradigm and caused the original anomalies. The scientist who holds on to the old assumptions, on the other hand, continues not to be able to understand how these fundamental assumptions can be questioned. Baltas’ (this volume) account of the intuition of incommensurability is consistent in part, with our account of the process of conceptual change. As mentioned earlier, our empirical work in the area of young children’s ideas about the shape of the earth and the day/night cycle, has shown that the concept of the earth is initially embedded within a naïve physics and that children apply all the presuppositions that are applicable to physical objects in general, also to the earth. Amongst those, the assumption that gravity operates up/down poses a particularly resistant obstacle to children’s understanding of the spherical shape of the earth and to cause many of the misconceptions or synthetic models that children construct. Lifting this constraint allows children to view the earth in a completely new way (see Vosniadou, 2003; Vosniadou & Brewer, 1992). Of course, this process is not under the metaconceptual control of the child who tries to understand a scientific concept, unlike the case with the scientist engaged in scientific discovery. This interpretation of the process of conceptual change is consistent with empirical results coming from many other areas of science. Just to mention one more, it appears that the assumption that force is a property of physical objects stands as an important constraint on students’ understanding of Newtonian dynamics (Ioannides & Vosniadou, 2001; Reiner et al., 2000). In addition to being consistent with the above-mentioned psychological accounts of conceptual change, Baltas’ (this volume) proposal also provides a philosophical argument to support what educational researchers have for a long time been wanting to argue for, namely, that understanding a scientific concept should not require the replacement of one theory (the incorrect) with another (the correct), but rather the ability to move on to a new, wider, broader perspective. From that point of view, conceptual change should be seen as requiring the ability to take multiple perspectives, examine different points of view and

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understand how they relate to different contexts of applicability (see, e.g., Pozo et al., 1999; Spada, 1994; Vosniadou, 1999; Vamvakoussi & Vosniadou, this volume). It would of course be wrong to assume that students have the above-mentioned abilities, that is, to take multiple perspectives and understand how they relate to different contexts of use. On the contrary, all the information we have points to the conclusion that students are not even aware of their own beliefs and especially of the hypothetical nature of their beliefs, not to mention being able to understand different points of view. Rather these should be seen as areas of future development and important aspects of an instructional approach for the fostering of conceptual change. All of that is consistent with current approaches that place a great deal of emphasis on metaconceptual awareness and intentional conceptual change, given that understanding the new paradigm is contingent upon becoming aware of the assumptions or what we call ‘entrenched presuppositions’ (Vosniadou & Brewer, 1992), that were responsible for the contradictions in the old theory and which must be denied in the new (Sinatra & Pintrich, 2003; Vosniadou, 2003). We would like to emphasize, however, that the development of metaconceptual awareness and intentional learning can be best achieved through extensive socio-cultural support (see Hatano & Inagaki, 2003). Epistemic beliefs The above considerations regarding the importance of background assumptions lead us very nicely to epistemic beliefs and their role in conceptual change. Epistemic beliefs are the beliefs individuals hold about the nature of knowledge and the process of knowing. Research has shown that personal epistemologies are not static, but changing and evolving, and there is a relatively well-established trend showing that individuals move from more absolutist and objectivist views about the nature of knowledge to views that are more relativist, constructivist, and evaluative. Up to now, research on epistemic beliefs has mainly concentrated on theoretical and methodological issues. There are different theoretical approaches on epistemic beliefs (such as the developmental approach, the multidimensional approach, the theory approach), and different kinds of instruments have been designed to measure them (e.g., quantitative vs. qualitative measures). In the present volume we look at epistemic beliefs in a new way, emphasizing their role in conceptual change and exploring some of the mechanisms that make this possible. The theoretical position we want to advance is that epistemic beliefs can have both direct and indirect influences on conceptual change. For example, beliefs in simple, stable, certain knowledge can prevent individuals from being open to new information that questions some of their basic assumptions, while on the contrary, individuals who believe that knowledge is complex, uncertain and constantly evolving may be willing to ‘open up the grammatical space’ and allow new paradigms/theories to be seriously entertained. Epistemic beliefs can influence conceptual change also in indirect ways, for example, by influencing students’ learning goals, study strategies and self-regulation. Some of these issues are addressed in the papers on epistemological beliefs and conceptual change included in the present volume. Conceptual change in mathematics The re-framed approach to conceptual change that we propose can be used to predict and explain students’ difficulties in the development of

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mathematical knowledge. It also constitutes a valuable framework for analyzing and reflecting on instructional interventions in mathematics teaching. The application of the conceptual change approach in mathematics learning and teaching is a relatively new attempt. Although there has been a great deal of research in the tradition of misconceptions during the 1970s and 1980s, the mathematics education community has been reluctant to adopt the conceptual change approach, which was developed mainly in the context of the physical sciences. This is because mathematics has been traditionally regarded as a discipline with particular characteristics that differentiate it even from its nearest neighbors, the physical sciences. Thomas Kuhn himself exempted mathematics from the pattern of scientific development and change presented in The Structure of Scientific Revolutions (see Mahoney, 1997). He did this because mathematics is based on deductive proof and not on experiment, is proven to be very tolerant to anomalies and it does not display the radical incommensurability of theory before and after revolution. Unlike science, the formulation of a new theory in mathematics usually carries mathematics to a more general level of analysis and enables a wider perspective that makes possible solutions that have been impossible to formulate before (Corry, 1993; Dauben, 1984). However, if we consider theory change in science not as replacement, as Baltas (this volume) argues, but as the ability to move to a new, wider theoretical perspective or paradigm, having understood the confining role of certain background assumptions, then the conceptual change approach can be equally well applied in science and mathematics. In fact, from a learning point of view, it appears that students are confronted with similar situations when they learn mathematics and science. As it is the case that students develop a naïve physics on the basis of everyday experience, they also develop a naïve mathematics that appears to be neurologically based (through a long process of evolution), and to consist of certain core principles or presuppositions (such as the presupposition of discreteness in the number concept) that facilitate some kinds of learning but inhibit others (Dehaene, 1998; Gelman, 2000; Lipton & Spelke, 2003). Such similarities support the argument that the conceptual change approach can be fruitfully applied in the case of mathematics learning.

Concluding Statements The re-framed approach to conceptual change is a constructivist, domain-specific approach that avoids many of the criticisms of earlier attempts to account for the process of conceptual change with learning and development. First, the focus is not on misconceptions as unitary, faulty conceptions, but on an intricate knowledge system consisting of different domain-specific areas organized in complex theory-like structures. Second, a distinction is made between naïve explanations, based on everyday experience and lay culture, and those that result from learners’ attempts to synthesize new, scientific information with existing knowledge. We consider synthetic models not to be stable misconceptions forming alternative theories, but dynamic, situated, and constantly changing representations that adapt to contextual variables or to the learner’s developing knowledge. Third, this theoretical position is a constructivist one. It can explain how new information is built on existing knowledge structures and provides a comprehensive

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framework within which meaningful and detailed predictions can be made about the knowledge acquisition process that can guide instructional interventions. It is also important to consider that the re-framed approach to conceptual change takes into consideration socio-cultural factors. It does that by considering as its primary unit of analysis the individual participating in rich socio-cultural activities, without, however, denying that knowledge can be acquired and stored in memory in some form. It also considers that teaching for conceptual change cannot be achieved through cognitive means alone but requires extensive socio-cultural support. Last, conceptual change is considered not as the replacement of an incorrect naïve theory with a correct one, but rather as an opening up of the conceptual space through increased metaconceptual awareness and epistemological sophistication, creating the possibility of entertaining different perspectives and different points of view (see Vosniadou, in press).

Acknowledgement The present work was financially supported through the program EPEAEK II in the framework of the project ‘Pythagoras — Support of University Research Groups’ with 75% from European Social Funds and 25% from National Funds.

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Vosniadou, S., & Brewer, W. F. (1992). Mental models of the earth: A study of conceptual change in childhood. Cognitive Psychology, 24, 535–585. Vosniadou, S., & Brewer, W. F. (1994). Mental models of the day/night cycle. Cognitive Science, 18, 123–183. Vosniadou, S., & Ioannides, C. (1998). From conceptual development to science education: A psychological point of view. International Journal of Science Education, 20(10), 1213–1230. Vosniadou, S., & Skopeliti, I. (2005). Developmental shifts in children’s categorization of the earth. In: B. G. Bara, L. Barsalou, & M. Bucciarelli (Eds), Proceedings of the XXVII annual conference of the cognitive science society (pp. 2325–2330). Vosniadou, S., & Verschaffel, L. (2004). Extending the conceptual change approach to mathematics learning and teaching. In: L. Verschaffel, & S. Vosniadou (Guest Editors), Conceptual change in mathematics learning and teaching (Special Issue) Learning and Instruction, 14(5), 445–451. Wellman, H. M. (1990). The child’s theory of mind. A Bradford Book. Cambridge, MA: MIT Press Wellman, H. M. (2002). Understanding the psychological world: Developing a theory of mind. In: U. Goswami (Ed.), Blackwell handbook of childhood cognitive development (pp. 167–187). Oxford, UK: Blackwell. Wellman, H. M., & Gelman, S. A. (1998). Knowledge acquisition in foundational domains. In: D. Kuhn, & R. Siegler (Eds), Cognition, perception and language. Handbook of child psychology (5th edn., Vol. 2, pp. 523–573). New York: Wiley.

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PART 1: FOUNDATIONS OF THE CONCEPTUAL CHANGE APPROACH: KUHN’S INFLUENCE

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Chapter 2

The Philosophical Foundations of the Conceptual Change Approach: An Introduction Aristides Baltas Philosophy of science situates itself by definition at the crossroads where general philosophy meets science — mostly natural science — in its results, in its structure and in its developments. It is the discipline into which general philosophical outlooks and general philosophical methods are brought to bear on the deeper understanding of science thus aiding the construction of its very self-image. Concomitantly, in a sort of inverse movement as it were, the more significant scientific achievements never fail to pose the kind of questions which provoke philosophy to respond, perhaps even at the cost of its drastic internal reorganization. This is not to say that the relations between science and philosophy of science have always been perfectly smooth and perennially harmonious. Episodes of profound misunderstanding and periods of mutual mistrust among the practitioners of the two endeavors have been witnessed in the not too distant past and will no doubt be witnessed again in the future. The long and intricate episode having been dubbed “the science wars” is a case in point.1 Nevertheless even those unhappy moments attest to one fact: philosophy of science remains constitutively open to what lies outside of it. 1

“Science wars” refers to the long and particularly heated debate, involving all kinds of media and attaining international proportions, which was sparked by an article written by the physicist Alan Sokal and published in the journal Social Text in the late 1990s. The article appeared as offering a “post-modern” interpretation of some fundamental physical theories by seemingly relying on ideas brought forth by famous French intellectuals like Derrida or Lacan. Immediately after publication, Sokal revealed that his article was a hoax and indicted Social Text for publishing it. The editors replied and a long and intricate debate ensued implicating scientists, journalists, philosophers and historians of science as well as academics of various disciplines in the humanities not only in the United States but in many other countries as well. Starting from what the ethics and the criteria for publication should be, the debate became generalized to issues concerning the differences separating and the relations holding natural and social sciences and the humanities in general, what “post-modern” thought might amount to, its influence in American universities and the reasons for this influence and so on so forth. Some of the scientists participating in the debate went as far as to argue that philosophers of science like Kuhn or even Popper had had a pernicious influence on the public understanding of science if not on the development of science itself.

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Such openness has been institutionally acknowledged since the end of World War II. Many American and European universities have set up departments and programs that have brought together philosophy of science and history of science in a way tending to turn the two into an almost single discipline. More recently, the openness in question has been further ratified by the creation of Science and Technology Studies — or Science, Technology and Society — departments or programs that add more partners to the study of science in a truly interdisciplinary spirit. In these departments or programs, philosophy of science crosses its insights and its methods not only with history of science but also with sociology of science, with the study of technology, with scientific policy studies, with engineering studies, with feminist or post-colonial approaches to such issues, with social and cultural anthropology and the list remains open. But philosophy of science has proved its open character and its outward-looking proclivities yet in another direction. This is the direction covered by some disciplines or areas of research that are not concerned directly with natural science and its development as such. Cognitive and developmental psychology, science teaching, science learning and science education in general, all disciplines concerned with the way in which real people, children and adults, acquire and change their beliefs — scientific, epistemological or everyday — have discovered in at least some areas of philosophy and history of science ideas and tools which could help them proceed with their own investigations. Conversely, significant achievements and original, productive ideas of these disciplines and areas of research have established decisively some of the more important approaches in the history and philosophy of science and have thus assisted them to articulate their own guiding ideas in close connection with what really happens when real people set out to know or are helped to come to know this or that. The works of Ronald Giere, of Paul Thagard, or of Nancy Nersessian come here readily to mind. It is almost uncontroversial that the “historicist turn” in philosophy of science, inaugurated effectively by Kuhn’s publishing the Structure of Scientific Revolutions, was the decisive factor in opening history and philosophy of science in this last direction. However, a lot of water has obviously run down the river of history and philosophy of science since then. As it was only to be expected, not all such subsequent developments could have a direct bearing on what mainly interests the present volume. What does have an important bearing on it, however, is the fact that, after a period of quasi-oblivion, the main ideas and insights of what Kuhn has bequeathed us have re-emerged forcefully on the foreground. Books, special issues of journals, many interesting articles or essays, conferences and symposia in many parts of the world attest to this fact. We should include in the list HoyningenHuene’s (1993) meticulous study, Fuller’s (2000) very controversial historio-sociological approach which, despite its valiant efforts, did not manage to lay Kuhn’s views to rest, the balanced account of Sharrock and Read (2002) as well as the collection of particularly interesting papers edited by Nickles (2003). In addition, it should be mentioned that a special issue of Philosophia Scientiae, edited by Soler (2004), has been devoted to the issue of incommensurability more or less in Kuhn’s sense of the term, while a volume covering fundamentally the same area — to be edited by Howard Sankey, Paul Hoyningen-Huene and Léna Soler and to be published by Springer — is actually at the last stages of preparation. As it is usually the case in philosophy, the historical distance covered and the new insights gained along the way have brought those who work in history and philosophy of

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science to a position where they can situate and assess which parts or which aspects of Kuhn’s work remain of lasting significance. Among them, we should include, I believe, his highlighting the role of scientific communities; his considering a paradigm as involving the theoretical dimension of science as indissolubly linked with the corresponding paradigmatic applications; his distinguishing within a unified framework how a new paradigm emerges from how it can be learnt; his connecting, at least indirectly, the incommensurability of succeeding paradigms with the obstacles one encounters in coming to understand the novel paradigm; and so on and so forth. Given what we have said just above, such significance cannot concern only history and philosophy of science narrowly conceived. It is or it can become of equal value to all the disciplines and areas of research to which philosophy and history of science have become opened by Kuhn’s work because all of them have proven capable of benefiting from the associated critical potential. In other words, the critical re-emergence in question can enhance the dialogue between history and philosophy of science, on one hand, and cognitive and developmental psychology and science education, on the other, still further. All chapters in Part I bear eloquent witness to this possibility, if not fact. Although all of them are written within the disciplinary confines of history and philosophy of science, each one of them not only throws its own particular light on this or that aspect of Kuhn’s overall approach as it is being currently discussed within that discipline but, much more importantly, it offers a set of important handles to the chapters coming next in the volume. Thus Hoddeson (this volume) sets forth how the “sociologist turn” in philosophy and history of science, sparked partly by Kuhn’s own work, eventually left behind sociology of science narrowly conceived and returned to history of science proper. This movement arrived to enlarge our understanding of the social dimension of science, the importance of which Kuhn’s work has already underscored, and to enlarge it in a way that challenged Kuhn’s own views. Thus, by taking systematically into account the social determinations of the identity of scientists (their gender, their race, the particulars of the social positioning of their discourse, etc.) these novel historiographic approaches effectively undermined the distinction between internal and external history that Kuhn had left more or less intact. This is to say that these approaches ceased to consider scientific concepts and theories as simply having a proper life of their own (internal history), radically distinguished from the evolution of social institutions and of the social relations generally wherein science functions (external history), but took the very content of scientific ideas as indebted, at least to some extent, to the social determinations in question. In this way Hoddeson offers, at least indirectly, ways and means which cognitive and developmental psychology and science education might exploit for the purpose of integrating the social element more fully into their own approaches. The “clarity and distinctness” of the major components of Kuhn’s conceptual arsenal (paradigm, puzzle solving, scientific community, incommensurability and so forth) is what the contribution of Machamer (this volume) focuses on. The ways by which these concepts have been received, criticized and elaborated upon by many philosophers of science since the Structure of Scientific Revolutions was first published, as discussed in conjunction with both their philosophical ancestry and Kuhn’s later views on the relevant issues, allow Machamer to pinpoint the insights and ideas of Kuhn’s overall approach which he considers have remained important. Among them, he highlights the relation between knowing a theory and

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being capable of applying it in novel circumstances as both aspects are tied together in Kuhn’s notion of “paradigm” (a confusing notion, as Machamer points out and as Kuhn himself later admitted), the idea of local incommensurability which Kuhn’s later work brought to the forefront and the determinative importance of countenancing science in terms of scientific communities rather than in terms of the isolated scientist. In this way Machamer’s essay helps the readers of the present volume to understand exactly where they are standing today in respect to Kuhn’s approach and to go on with their own work starting from there. Arabatzis (this volume) discusses one of the issues that have constituted a major battleground within philosophy of science at least since the 1980s, namely scientific realism. Arabatzis argues that the many views to the contrary notwithstanding, Kuhn’s approach need not imply either anti-realism or relativism. Rather, and at least with regard to the objective existence of entities like the electron, Kuhn’s views align themselves with scientific realism once they are specified and elucidated in the way Arabatzis proposes. Thus, granting that scientific concepts indeed change, he argues that at least some of them (such as “electron”) may continue referring to the same entities even after a radical paradigm change affecting profoundly their meaning has come about.2 Although this discussion might appear somewhat “esoteric” in a volume like the present one, it is my conviction that what Arabatzis has to say on the issue allows cognitive and developmental psychologists as well as science educators to lay to rest worries about the “existence of the objective world”, which inevitably crop up in their own practices, sometimes getting exacerbated by some misreadings of Kuhn’s work and by some misunderstandings of what this work in fact implies. Finally, my essay (Baltas, this volume) considers the meaning dimension of a Kuhnian paradigm change. It relies on Wittgenstein’s conception of grammar to uphold that a novel paradigm discloses and puts in question some of the grammatical conditions (which I call background “assumptions”) securing the meaning of some concepts of the old paradigm. On this basis, the essay tries to relate Kuhnian incommensurability to the grammatical challenge in question and discusses some of the associated issues arguing that Kuhn’s approach, once read in this way, can be “saved” from the charges of idealism and relativism. Moreover, that such a grammatical barrier separates succeeding paradigms explains why students are impeded in their efforts to understand the novel paradigm and thus the essay offers, at least indirectly, some handles for developing an efficient teaching strategy. Science educators as well as cognitive and developmental psychologists may thus find here some ideas intimately connected to their own work. Last but not the least, the comments of Stathis Psillos and Matti Sintonen that follow the chapters of Part I should be read carefully for a number of reasons. First, they illustrate in an exemplary fashion the kind of painstaking elucidatory work forming the daily bread of philosophy of science. Second, the points of agreement or disagreement they focus on, together with the arguments they present for the purpose, offer much room for thought to a reader eager to understand more fully the contributions these comments are making. Third, the work of Sellars, which Psillos invokes, as well as the work of Sneed and Stegmüller, which Sintonen appeals to, are, at least according to me,3 very important 2

A full presentation of the argument in conjunction with all the associated intricate historical details is to be found in Arabatzis (2006). 3 In respect to the Sneed–Stegmüller approach, I permit myself to refer to Baltas (1989).

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philosophical contributions in their own right which clarify, even if from very different angles and among many other things, what science is all about. Thus, among many other things, the way Sellars distinguishes the “scientific” from the “manifest image” (Sellars, 1991) as well as the way he treats theory change (Sellars, 1973) lays out a royal path for the serious philosophical treatment of most of the issues that Kuhn’s work has opened in direct connection with more general epistemological and metaphysical concerns. On the other hand and in the opposite direction as it were, the heavy formalism of the Sneed–Stegmüller way of conceiving the structure of mathematical physics — a way of conceiving which Kuhn himself was quite sympathetic to (Kuhn, 1976) — allows many of the same issues to become pinned down in a way assisting their clarification and their assessment. The reader of the present volume would profit greatly, I believe, if she became acquainted more fully with these not-so-well-known philosophical approaches, approaches to which Psillos and Sintonen offer us substantial, if short, introductions. To close, I can only say that the interdisciplinary dialogue that started a long time ago can go on more strongly and more productively with the help of the present volume. The reader is invited to ascertain this and, why not, to offer her own contribution.

References Arabatzis, T. (2006). Representing electrons: A biographical approach to theoretical entities. Chicago: The University of Chicago Press. Baltas, A. (1989). Louis Althusser and Joseph D. Sneed: A strange encounter in philosophy of science. In: K. Gavroglu, Y. Goudaroulis, & P. Nicolacopoulos (Eds), Imre Lakatos and theories of scientific change (pp. 269–286). Dordrecht, The Netherlands: Kluwer Academic Publishers. Fuller, S. (2000). Thomas Kuhn: A philosophical history for our times. Chicago, IL: The University of Chicago Press. Hoyningen-Huene, P. (1993). Reconstructing scientific revolutions: Thomas S. Kuhn’s philosophy of science. Chicago, IL: The University of Chicago Press. Kuhn, T. S. (1976). Theory change as structure change: Comments on the Sneed formalism. Erkenntnis, 10, 179–199. Nickles, T. (Ed.). (2003). Thomas Kuhn. Cambridge, UK: Cambridge University Press. Sellars, W. (1973). Conceptual change. In: G. Pearce, & P. Maynard (Eds), Conceptual change (pp. 77–93). Dordrecht, The Netherlands: D. Reidel. Sellars, W. (1991). Philosophy and the scientific image of man. In: W. Sellars (Ed.), Science, perception and reality (pp. 1–41). Atascadero, CA: Ridgeview Publishing Co. Sharrock, W., & Read, R. (2002). Kuhn, philosopher of the scientific revolution. Cambridge, UK: Polity Press. Soler, L. (Ed.). (2004). Le problème de l’incommensurabilité un demi-siècle après. Special Issue of Philosophia Scientiae, 8(1).

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Chapter 3

In the Wake of Thomas Kuhn’s Theory of Scientific Revolutions: The Perspective of an Historian of Science Lillian Hoddeson Thomas S. Kuhn was that special teacher who inspired me to change fields, from physics to history of science. He had made the same change himself shortly after taking his Ph.D. He had also been trained in theoretical physics but found that studying the historical development of science interested him more than practicing science. In my own case, this switch began about five years after I took my Ph.D. I remember stepping with great excitement into what I experienced as a new world, one with different assumptions about the meaning of knowledge production. Here, the data were no longer measurements but instead were traces of the past found in letters, scientific papers, reports, or even the fragile memories of individuals. Problem solving and creativity in the new domain depended on interpreting historical, rather than physical, events. I never regretted making the switch from physics to history, and I owe a special debt of gratitude to Kuhn for serving as a model of one who had the courage to break with familiar teachings and turn to new ones. Kuhn taught me many things about history that as a scientist I had not encountered. The most important was always to view events from the perspective of those who lived in the period being studied, never from the present vantage. One had to ask what was known then? What did people believe then? Why? What did the authorities preach? Was it reasonable then? What were the stakes in holding one idea rather than another? Such questions invariably will lead the historian of science outside the domain of science. Kuhn encouraged his students to look there too. For me, working with Kuhn meant returning to the status of a graduate student, even while I was on a physics teaching faculty. In those years, 1972–75, I took two seminars that Kuhn offered, both about major conceptual change in physics. One seminar focused on the scientific revolution that brought us from Newtonian mechanics to quantum mechanics; the other was on the revolution that yielded the first two laws of thermodynamics. In both seminars all the students, as well as Kuhn, read papers and books from the periods being studied in the languages in which they had been written, usually German. Their messages were

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couched in the incredibly awkward and complex formulations that scientists use when struggling to explain phenomena that the current intellectual tools cannot treat. I had already read most of Kuhn’s prior works in the history of science, including his masterpiece about the switch from the Ptolemaic to the Copernican picture in astronomy. In Kuhnian terms, one could speak of an old world giving way to a new and very differently configured one because scientists shared a new set of assumptions and agreements, what he called a new paradigm. Many scholars in different fields subsequently adapted Kuhn’s account of conceptual change, for instance to the problem of understanding how children’s models of the earth transform (Vosniadou & Brewer, 1992). This part of Kuhn’s teaching is as valuable today as it was in the years when Kuhn’s theory was still new. His well-known book, The Structure of Scientific Revolutions, became a standard text in countless fields, from history to literature to education (Kuhn, 1962). But in my field of history of science — my field and Kuhn’s — most scholars presently consider Kuhn’s Structure too limited for explaining the problems that are of most interest. Even though Kuhn moved from theoretical physics to the history of science, his approach remained close to that of a physicist, abstracting simplified patterns from sequences of events in order to seek general structures and laws. Both the strengths and the limitations of Kuhn’s theory come from this approach. If we recall the leading features of The Structure of Scientific Revolutions, we can see how much they depend on sharp conceptual oppositions and disjunctions. The basic opposition between normal and revolutionary science is the most striking. Normal science operates within the sets of shared assumptions and agreements that constitute paradigms, and by working within those constraints, scientists can dedicate their efforts to solving problems. In Kuhn’s scheme, discoveries emerge over time that cannot be accommodated within the reigning paradigm. And when these anomalies accumulate beyond a certain point, science enters a period of crisis that is eventually resolved by a revolution, a change of paradigm. Though we can trace the developments that produce these revolutions, which typically take one or more generations to work themselves out, the relation between the previous paradigm and the one that replaces it is sharply discontinuous, so that scientists working in the new paradigm inhabit a different world from that of their predecessors. This claim for the discontinuity between successive paradigms, the distinctive feature of Structure, is both subversive and enduring. It is subversive because it discredits the belief that scientific knowledge is cumulative, moving ever closer to a true and complete account of nature. Instead, Kuhn insists that different paradigms are incommensurable; scientific knowledge grows as we move from one to another, but it is no longer possible to imagine the results of scientific revolutions as a cumulative linear progression. This continues to be Kuhn’s most controversial claim; it was and is still rejected by many scientists and their allies who maintain a belief in scientific progress. It is also enduring because it gets some things right. At the level of theory, there really are revolutionary changes and incommensurable differences between earlier ways of making sense and the ways that eventually replace them — like the incommensurable pictures of burning in chemistry, where in the earlier view intangible particles called phlogista stream out of burning objects and in the later view oxygen combines chemically with the burning materials. In terms of general cultural impact, the notion of revolutionary paradigm shift is also Kuhn’s most influential achievement. It has undoubtedly been used too widely and loosely

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for describing or promoting changes that are far less radical than those he analyzed. Still, it has provided a useful model for researchers working in other fields, such as psychology and education. Such borrowings seem particularly appropriate because Kuhn borrowed, or was influenced by, psychological models such as the gestalt switch and Piaget’s stages of cognitive development. In science studies, Kuhn’s most important influence had less to do with theoretical models than with the way he helped a generation of scholars who began to think about science as a social activity that occurs in a particular period. (For disparate views on this issue, see, e.g., Keller, 1998; Pickering, 2001; Rouse, 1998). Expanding the field of study in the history of science from experiments and theories to scientific communities, and to communication within and between these communities, redirected the history of science toward considering a wider range of contributing factors and contingencies. Kuhn turned us in that direction, but he himself did not pursue it very far; his work in the history and philosophy of science remained strongly focused on the conceptual part of scientific knowledge production. Kuhn was certainly aware that there is always much more involved in scientific knowledge production than the scientific work itself. He often referred, implicitly if not explicitly, to the real-world factors that fall outside science proper, like communities, education, values, traditions, perceptions, and cognition, as well as to the material basis of phenomena. Thus in The Copernican Revolution, Kuhn (1959, pp. 14–35, 114–123) mentioned religious beliefs, scholastic education, and how astronomical observations were made. Such factors play a far less prominent role in Kuhn’s more famous account, The Structure of Scientific Revolutions, perhaps because he wanted this work to speak to a wide audience. I think he would have preferred to go past the first steps he took in considering science in social, material, and cognitive terms, as well as in intellectual terms, and build a more fully articulated theory that encompassed such aspects as gender, class, race and ethnicity, or how competitions, jealousies, ambitions, and religious beliefs enter the story of scientific knowledge production. But he could not work them rigorously or elegantly into his theory. So he left them out, perhaps hoping other scholars would figure out how to incorporate them (Keller, 1998). The fact that Kuhn did not or could not do much with the multitude of contingencies does not mean that he did not care about them. In fact, his theory’s limited ability to deal with social and cultural influences caused him considerable anguish. I came to understand this as a student in his seminar on the history of quantum mechanics during the spring of 1973. Kuhn’s plan was to retrace Werner Heisenberg’s intellectual route to his formulation of quantum mechanics. To help place ourselves into the period, we spent the first weeks of the seminar slogging through countless experimental anomalies that appeared in the work of physicists who were studying atomic phenomena in the early years of the twentieth century. Most of the problems arose from the failure to explain the masses of new data available on the atomic spectra of particular atoms. Arnold Sommerfeld’s Atombau und Spektrallinien was one of the works we plowed through, of course in the original German. The other texts included a number of confusing and abstruse journal articles by Werner Heisenberg, Wolfgang Pauli, Niels Bohr, Hendrik Kramers, John Slater, and others. For the students in the seminar, and for Kuhn, it was an exciting but extremely hard winter journey through rugged scientific terrain. For weeks we seemed to be making little noticeable progress.

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At last, about the time that spring finally came to Princeton, we encountered the first hints of a break. Reading extremely difficult German papers, we began our slow ascent to the climax of the Kuhnian narrative about quantum mechanics. We struggled as we worked through the most obscure work of all, in which Heisenberg and Kramers offered their last wheezing gasp at explaining the stationary states of an anharmonic oscillator using mysterious diagrams; it was not clear why these diagrams worked. The patchwork picture that emerged, neither classical nor quantum-mechanical, harked back to the messy adjusted Ptolemaic system before the Copernican revolution. The next step was the “turning point”, Heisenberg’s breakthrough paper in which the first version of quantum mechanics came into focus. He conceived this work in Helgoland, a barren island in the North Sea where grass cannot grow. The twenty-three year-old traveled there in June 1925 because he was suffering from a severe attack of hay fever. At this point in the seminar, Kuhn was also exhibiting some discomfort. He suddenly announced that we needed to decide “what to do about” a major paper by Paul Forman that had appeared a year earlier in the journal Historical Studies in the Physical Sciences (now Historical Studies in the Physical and Biological Sciences) (Forman, 1971). Forman, who had been one of Kuhn’s Ph.D. students, had recently worked with Kuhn in a massive effort to preserve the documents of the history of quantum mechanics and also interview leading figures in that history.1 His paper dealt with acausality and indeterminacy, key components of the new quantum mechanics picture. Forman argued, radically, that the notion of acausality arose naturally for physicists in Weimar Germany because of its uncertain and indeterminate political climate. He hypothesized that in Weimar Germany it was easier for physicists to give up the secure causal basis they had enjoyed as Newtonians. As was fairly clear to the students in the seminar, our respected teacher was experiencing some sort of a crisis of his own. Kuhn repeatedly spoke about Forman’s thesis as “terribly important”, but he could not fit its social perspective on quantum mechanics into his narrative about normal science, anomalies, revolution, and paradigm change. This failure disturbed Kuhn so much that he could not bring himself to discuss the problems raised by Forman’s paper on the Princeton campus. He called a special evening seminar to talk about it at his home. As students we spent an enjoyable evening debating whether and how Forman’s thesis could be reconciled with Kuhn’s structure, but our arguments seemed to go nowhere. Looking back on that evening, I now believe we came farther than we realized at the time, for the debate sharpened everyone’s awareness of what Kuhn’s structure could not explain. And as we went our separate ways in the years that followed, we all arrived at a richer, if messier, sense of science history. The idea that science studies must confront all the relevant factors, scientific or not, has become so widely accepted in the history of science that hardly anyone in the field argues about it now. But in the early 70s non-scientific factors posed a problem because the field was still divided into “external” and “internal” history of science. Kuhn was considered an internalist, a member of the professional sub-community that believed ideas are more important than materials, values, or social attributes. Focusing on the outside factors was somehow less reputable.

1

This project resulted in the Archives for the History of Quantum Physics available in Copenhagen, the American Institute for Physics, earlier based in New York City and now in College Park, Maryland, and some other places.

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I was among the many of Kuhn’s students who started out as internalists in the history of science. I began my career in pursuit of cases on which I could impose the Kuhnian theoretical structure. The picture seemed simply true, perhaps because it was so truly, beautifully simple. One of my first historical papers dealt with the emergence of the quantum theory of solids. As Kuhn had instructed, I read letters, scientific manuscripts, and published papers from the period, in this case 1926–1932, in the languages in which they were written. Thanks to Kuhn and his colleagues, many of the crucial archival documents could be found in the Archives for the History of Quantum Mechanics, which I examined in Copenhagen in 1976 and later in New York City. In this history, I looked for, and found, paradigms, anomalies, even revolutions. For example, Pauli’s brilliant attempt to deal with the experimental anomaly of a too weak paramagnetism in metals led directly to Sommerfeld’s transitional semiclassical electron theory of metals, and in time to Felix Bloch’s fully quantum-mechanical theory of electrons in metals. The Kuhnian structure worked here. My paper on the quantum mechanical electron theory of metals, co-authored with Gordon Baym, concluded: This development replayed the stages that the theory of the atom passed through: a classical period in severe crisis shortly after the turn of the century, a semi-classical period — the old quantum theory — during which non-classical assumptions, such as the Bohr–Sommerfeld quantization condition, were included ad hoc in the classical framework, and the revolutionary modern period, heralded by Heisenberg’s first paper on quantum mechanics. One may indeed characterize the development of the electron theory of metals as a “secondary scientific revolution”.(Hoddeson & Baym, 1980, pp. 19–20) What I gradually realized was that Kuhn’s structure fit this earliest phase in the history of the quantum theory of solids because in those years, 1926–1932, the objects of this theory were “ideal” solids, abstract models of metals and insulators.2 But as soon as physicists tried to apply this theory to real materials (sodium, copper, tin, etc.), all sorts of messy details muddied the formulation. The study of solids, as Pauli3 declared, became a “physics of dirt”, a physics, he cautioned his students, “one should not wallow in”. The problem was that each solid has its own multitude of real properties, and to incorporate all of them in the analysis would make the theory impossible to work with. An analogous dilemma was caused by the all-encompassing memory of the central character in the Jorge Luis Borges (1962) story, “Funes the Memorious.” Because Funes remembered every detail of everything he ever encountered, or even thought or dreamed of, his memory was, as he lamented, “like a garbage heap,” and this rendered him almost incapable of thought (Borges, 1962, p. 64). Just as we need to forget a good many details in order to think, the physicists who worked on the problem of extending the theory of ideal solids to real materials were forced, from about 1933, to make simplifying assumptions that would reduce the clutter of what they

2

One could make an analogous statement about why Kuhn’s theory fit the early stages of the development of quantum mechanics itself. 3 Letter to R. Peierls (July 1, 1931), cited in Hoddeson, Braun, Teichmann, and Weart (1992, p. 181, 458).

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knew and thus render it useful. And so this field became characterized by a huge number of approximations, which Pauli viewed as ugly. Many in his generation agreed that studying real solids made physics messy, just as studying how science really works has made the history of science messy. Fortunately for the progress of solid-state physics, others in the physics community (people like John Slater, John Bardeen, Frederick Seitz, Conyers Herring, and Nevill Mott) found the messy properties of real solids, including the approximations used in representing them, to be beautiful because they belonged to real, rather than ideal, solids. This group of physicists formed the first generation of physicists to refer to themselves as solid-state physicists. Studying their work occupied me for years (Hoddeson, Braun, Teichmann, & Weart, 1992, pp. 88–235). And just as a new generation of physicists was happy to work on the messy real solids, in a new solid-state physics community, which then grew into the largest subfield of modern physics now called condensed matter physics (Weart, 1992), the generation of historians of science who worked after Kuhn mostly found the messy contingent factors of real science practice rewarding to study. The price for their fuller engagement with the world was, however, abandoning the elegant Kuhnian account of successive paradigms. To follow my path in the history of science just a bit farther, I found it increasingly necessary to deal with the contingencies of real-world factors in my own work. I wrote about all kinds of influences that affected scientific change without the apparent guidance of any grand narrative — wars, patent fights, teachers, schools, families, apparatus, materials, and places. For example, when I explored the implications of the quantum theory of solids for the research at Bell Laboratories, I realized that this theory was important not only in conceptually advancing the physics, but in helping to transform the environment into one where the transistor could be invented. This happened because Bell Labs recognized its need for physicists trained in quantum mechanics, but to keep such physicists happily employed — people like William Shockley and John Bardeen — the laboratory had to offer them a considerable research freedom, at least comparable to that offered by universities. This social condition was as important to the invention of the transistor as the ghostly new paradigm of empty spaces (holes) bearing positive charge (Hoddeson, 1981a). In the transistor story, too, contingencies played a large part. The physics of semiconductors alone would probably not have led to the invention had not the United States government invested lavishly in the development of radar during World War II. This investment gave rise to contributions to the physics of silicon and germanium, which were necessary for the transistor’s invention. This wartime radar work also created an interacting network of groups from different industries and universities who were all in touch with the latest findings about semiconductors and electronics. A major Bell Labs administrator (Mervin Kelly) recognized by 1943 that these groups would surely face off against each other after the war in fierce commercial competition. As one line of defense, Bell Labs organized its semiconductor research group, which in time invented the transistor (Hoddeson, 1981b; Riordan & Hoddeson, 1997). The fact that Kuhn’s Structure does not account for the material base of science has been discussed by Andy Pickering, who explains in one example how the different apparatus used by William Fairbank and Giacomo Morpurgo in their attempts to explain the existence or non-existence of free quarks is what was really behind the Kuhnian incommensurability, the “different worlds” separating their explanations, and more generally

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between the pre-standard models pictures of the mid-1970s and the later gauge theory. Attempting to find “a fault line somewhere in the mid-1970s”, Pickering realized: The incommensurability of the old and the new physics did not seem to me much like incompatible perceptions of those funny gestalt figures: the duck and the rabbit, and so on. This incommensurability seemed to me rather to have, once more, an important material dimension. The theories and models of the old physics hung together with specific forms of apparatus and the specific material phenomena they displayed, the same went for the new physics, and these domains of instrumentation and phenomena were largely disjoint. Materially, one did the old physics differently, eliciting quite different material performances, from the new physics, and vice versa. (Pickering, 2001, p. 502) A similar case where material factors and social contingencies impart a Kuhn-like structure to a scientific development can be found in the massive effort (involving every major theoretical physicist working between 1911 and 1957) to create a first-principles theory of superconductivity. Developing the new paradigm had as much to do with post-war experiments using pure mercury isotopes, available as a result of the Manhattan Project, as it had with anomalies, crises, and paradigm shifts. As critical, perhaps, was the fact that Bardeen left Bell Labs in 1951 because he no longer experienced experimental freedom and for this reason joined a more supportive group at the University of Illinois built by Bardeen’s graduate school friend Frederick Seitz (Hoddeson & Daitch, 2002, pp. 314–329). The real problem with Kuhn’s theory of scientific revolutions is that it just is not legitimate to extract a vital artery out of the body of science and designate it as the essence of the whole business. The kinds of questions about theory change that Kuhn was drawn to are inevitably tangled up with questions that require drawing on the full range of factors that are at play in any human enterprise. The building of the atomic bomb at Los Alamos, which created all sorts of new scientific fields and problems, had as much to do with our fear of Hitler, the concerns of Jewish refugee scientists, and the material properties of amassing plutonium and uranium, as it had to do with developing and applying the underlying physics of nuclear fission (Hoddeson, Henriksen, Meade, & Westfall, 1993). And a solid explanation of the new theoretical structures emerging from the mushrooming field of particle physics after World War II has to be related to United States science policy during the Cold War, to the wartime invention of radar resonance techniques, to the expansion of graduate programs in physics, and to the development of new materials and technologies (Brown & Hoddeson, 1983). One implication of taking a more holistic look at scientific development and conceptual change is the dazzling variety of new sources we need to engage with. John Heilbron, another of Kuhn’s early Ph.D.s who worked with Kuhn on the Quantum Physics Project, suggests the rewards of drawing on the full range of sources: The industrious historian who has looked at private correspondence, government papers, foundation and university reports, newspapers, patent applications, court records, architectural monuments, scientific apparatus, motion pictures, painted neckties, and literary T-shirts — as well as the scientific literature — necessarily sees connections that the people being studied could not have known. (Heilbron, 1989, p. 48)

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To some leading historians of science, the limitations of Kuhn’s Structure appear tied with the simple fact that even after Kuhn switched fields, he continued to think like a physicist. Martin J. Klein, another physicist-turned-historian of physics, who often argued with his contemporary Thomas Kuhn, was one of the earliest to observe that it is characteristic of the physicist to want to get at the very essence of a phenomenon, to strip away all the complicating features and see as clearly and directly as he can just what is really involved. (Klein, 1972, p. 16) That is why, Klein explained (before most of us were ready to listen), physicists prize simple conceptual models while historians are drawn instead to the richness of particular moments. Klein reflected, possibly with Kuhn in mind, that this is why physicists who are interested in history often expect to be able to treat historical problems the same way — it is not surprising, but it is not justified.(Klein, 1972, p. 16). To those who regret the loss of simplicity and generalizability that has accompanied the developments that came in the wake of Kuhn, it may seem that historians of science now have no theory to replace his structure, only a welter of particulars. I think we have, if not a theory, at least a set of tacit agreements that enable our research. To me the most promising is the view that has made the old division between internal and external accounts obsolete, a view that considers all factors, whether ideas or materials, individuals or institutions, as alike in being potential actors in the stories we tell (e.g., Latour & Woolgar, 1986; Pickering, 1995). Instead of the traditional hierarchy in which ideas rule, we try to envision a democracy of all the actors, where anyone can in principle play a leading role, and where which one dominates depends both on the circumstances and how we choose to tell the story. In that vision, conceptual change becomes just one strand in the web, one dimension of social and cultural history.

Acknowledgment I thank Peter Garrett, who contributed greatly to this paper.

References Borges, J. L. (1962). Funes the memorious. In: D. Yates, & J. Irby (Eds), Labyrinths: Selected stories and other writings (pp. 59–66). New York: New Directions Publishing Corporation. Brown, L. M., & Hoddeson, L. (Eds). (1983). The birth of particle physics: Particle physics in the 1930s and 40s. New York: Cambridge University Press. Forman, P. (1971). Weimar culture, causality and quantum theory, 1918–1927: Adaptation by German physicists and mathematicians to a hostile intellectual environment. Historical Studies in the Physical Sciences, 3, 1–115.

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Heilbron, J. L. (1989). An historian’s interest in particle physics. In: L. M., Brown, M., Dresden, & L. Hoddeson (Eds), Pions to quarks: Particle physics in the 1950s (pp. 47–54). New York: Cambridge University Press. Hoddeson, L. (1981a). The entry of the quantum theory of solids into bell telephone laboratories, 1925–40: A case-study of the industrial application of fundamental science. Minerva, 18(3), 422–447. Hoddeson, L. (1981b). The discovery of the point-contact transistor. Historical Studies in the Physical Sciences, 12(1), 41–76. Hoddeson, L. H., & Baym, G. (1980). The development of the quantum mechanical electron theory of metals: 1900–28. Proceedings of the Royal Society of London A, 371, 8–22. Hoddeson, L., & Daitch, V. (2002). True genius: The life and science of John Bardeen. Washington, DC: Joseph Henry Press. Hoddeson, L., Braun, E., Teichmann, J., & Weart, S. (Eds). (1992). Out of the crystal maze: Chapters from the history of solid-state physics. New York: Oxford University Press. Hoddeson, L., Henriksen, P., Meade, R., & Westfall, C. (Eds). (1993). Critical assembly: A history of Los Alamos during the oppenheimer years, 1943–1945. New York: Cambridge University Press. Keller, E. F. (1998). Kuhn, feminism, and science? Configurations, 16, 15–19. Klein, M. J. (1972). The use and abuse of historical teaching in physics. In: S. G. Brush, & A. L King (Eds), History in the teaching of physics (pp. 12–27). Hanover, New Hampshire: University Press of New England. Kuhn, T. S. (1959). The copernican revolution: Planetary astronomy in the development of western thought (2nd ed.). New York: Vintage Books. Kuhn, T. S. (1962). The structure of scientific revolutions. Chicago: The University of Chicago Press. Latour, B., & Woolgar, S. (1986). Laboratory life: The construction of scientific facts (2nd ed.). Princeton: Princeton University Press. Pickering, A. (1995). The mangle of practice: Time, agency, and science. Chicago: The University of Chicago Press. Pickering, A. (2001). Reading the structure. Perspectives on Science, 9(4), 499–510. Riordan, M., & Hoddeson, L. (1997). Crystal fire: The birth of the information age. New York: W.W. Norton. Rouse, J. (1998). Kuhn and scientific practice. Configurations, 16, 33–50. Vosniadou, S., & Brewer, W. (1992). Mental models of the earth: A study of conceptual change in childhood. Cognitive Psychology, 24, 535–585. Weart, S. (1992). The solid community. In: L. Hoddeson, E. Braun, J. Teichmann, & S. Weart (Eds), Out of the crystal maze: Chapters from the history of solid-state physics (pp. 617–69). Oxford and New York: Oxford University Press.

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Chapter 4

Kuhn’s Philosophical Successes? Peter Machamer Introduction Thomas Kuhn’s The Structure of Scientific Revolutions was published in 1962 (Kuhn, 1962). Since its publication it has sold about 20 million copies and has been translated into 18 languages.1 This probably makes it the best-selling book on history and philosophy of science ever written. Yet, at the time of its publication, most philosophers hated the book, much to Kuhn’s dismay, and many social scientists loved it, also to his great dismay. In this chapter, I will assay some of Kuhn’s philosophical claims and critically assess what value remains. That it has been widely influential is beyond dispute. Consider a fairly recent issue of Science magazine, where the 2000 Nobel Prize winner, Arvid Carlsson (Gothenburg, Sweden) entitled his Nobel address “A Paradigm Shift in Brain Research” (Carlsson, 2001). Therein he recounted the change in basic thinking about the brain: As late as the 1950s, it was assumed that communication between nerve cells in the brain occurred by electrical impulses. A decade later, the theory of chemical transmission, which until then had been thought to occur only in the peripheral nervous system, had gained strong entrance for the central nervous system. This paradigm shift opened up an enormous new perspective in brain research, not the least by facilitating the study of brain function by means of chemical tools. Which in different ways could modify chemical signaling between nerve cells. Moreover, such tools sometimes turned out to be useful as therapeutic agents. Thus, for the first time, a variety of disorders in the central nervous system could be treated effectively. You may recall that Carlsson began his work by finding out that LSD worked on serotonin receptors and went on to work on dopamine, which led to the now standard treatment for Parkinson’s disease. But the point I wish to emphasize is that Kuhn’s “paradigm shift” has made it into the annals of the Nobel Prize winners’ account of their work.

1

These figures come from Michael Matthews in a lecture he gave at IHPST in Denver, Nov. 9, 2001.

Reframing the Conceptual Change Approach in Learning and Instruction Copyright © 2007 by Elsevier Ltd. All rights of reproduction in any form reserved. ISBN: 0-08-045355-4

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This beginning rumination takes us into the heart of our topic today: what were Kuhn’s views on the nature of science, and how have they held up almost 50 years of scrutiny?

The Nature of Kuhnian Science and Some Problems The topics I want to consider include the major philosophical concepts that have passed down into contemporary usage and which often are attributed to Kuhn’s (1962) Structure: Normal science, puzzle solving, paradigm, theory-ladenness and, of course, incommensurability. Science, Kuhn held, for the most part was normal science, as practiced by people, scientists, who had been specially trained in their field to use certain concepts and practices in order to solve scientific puzzles. So, normal science was puzzle solving. The puzzles were set by paradigms, which were basically nonpropositional sets of interconnected concepts and associated practices for applying those concepts. Paradigms directed the practicing scientists as to what things to look at (and how to conceptualize their looking) and what were the mathematical, experimental and technical procedures allowable for such explorations (e.g., what kind of equations could be used to solve which kind of problems, and what instruments would allow the scientists to measure or observe the right objects or quantities). Puzzle solving was working out the details and refining of the paradigm. Paradigm was what was learned by students of science as they were trained to be professionals in their field. They learned the proper mathematics by problem-solving exercises (exemplars) in textbooks, the proper use of instruments by apprenticeship in laboratories, and the vocabulary and preferred forms of expression by writing research reports for professional journals. It was such common training and learning by groups of people that created a scientific community of shared beliefs and practices. This emphasis on the scientific community and their group commitments, consensually shared examples, and tacit knowledge — as he called them in the 1969 Postscript to the Second Edition (Kuhn, 1970, pp. 174–210) — was one of Kuhn’s most effective lessons, not for philosophers but for historians of science, sociologists of science and science and technology studies (STS) people. The themes of science as a social activity and scientific practice are a major part of Kuhn’s legacy, but I shall not focus on them in this talk. Kuhn failed to note an interesting contrast when he introduced the idea of puzzle solving. Puzzle solving much better characterizes much of Medieval and Renaissance “science”, but there is a difference. A typical Medieval Quaestiones was nothing but a set of puzzles.2 They were compendia of various questions and an author’s proffered answers. Questions would be such as: Why do lions roar? Why do bodies fall? Can there be essential accidents? Is it possible that there are many worlds? Answers to each question would be given. But most often there were no overarching set of principles that would be invoked to make the coherence among the set of questions, neither need there be consistency

2 I remember David Lindberg pointing this out to me in the late 60s. We were discussing optics texts and how various Medieval writers tried to answer optical questions. Even within a closely constrained topic there was still no desire for consistency among answers.

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among the answers. So, in Kuhn’s terms there was no guiding paradigm. Each answer was ad hoc to a specific question. To be sure they shared some general philosophical presuppositions, basic Latinized Aristotle, but this is far from having a shared paradigm. More importantly there was no attempt at using a few “fundamental” principles and techniques to assure consistency among the various answers. Contradictory principles could be used among questions and sometimes even within discussions of the same question. Presumably, Kuhnian paradigms would not allow such ad hoc answers, except in paradigm-saving cases (as Lakatos, 1971 emphasized). Paradigms, by providing constraints that must be satisfied by all answers, were intended to ensure that the answers have a mutual consistency. So normal science is not just puzzle solving, it is principled puzzle solving in that it presupposes some fundamental principles (and practices). Yet, as noted above, a paradigm is more than a theory conceived as a set principles or laws stated in the form of a set of propositions, whether axiomatic or not. Paradigms were conceived as “vehicles” for theories, and theories were taken to provide the basic ontic commitments and behaviors of the entities of a given science. This point was picked up and developed by Laudan (1977) in his distinctions between methodological, conceptual and ontological commitments. Paradigms also were said to be like maps and contain the principles for orienting oneself in problem space, and so repeated what Toulmin (1953) had said earlier. The concept of a paradigm is no doubt the best-known legacy of Kuhn’s Structure. It also has been one of its most controversial ideas. When Shapere (1964) reviewed Structure, he spent most of his 11-page review laying out the ambiguities, inconsistencies and radical equivocations in meaning of this central term “paradigm”. Later, at a conference at the Bedford College, London in July 1965, Margaret Masterman (1970) carefully distinguished among 21 senses of the term “paradigm” and then tried to distill them all down into four basic senses.3 Kuhn (2000b) later would comment she was right. Even more telling is Kuhn’s remark: “Paradigm was a perfectly good word, until I messed it up” (2000b, p. 198). Indeed, it was; for example, “experimental paradigm” meant (and in some circles still means) an experimental design that is a field’s preferred set up used to investigate a type of problem or phenomenon. Yet this mess was just what he intended. He wanted a word that would be broader and more multifaceted than “theory”, especially where “theory” meant something like a set of law-like propositional axioms. There is no doubt that Kuhn was doing a good thing in trying to shift the locus of philosophical analysis away from axiomatic theory. The axiomatic idealization of the concept of a scientific theory usually traced its roots to Euclid and Newton. Axioms were sometimes lauded by some logical positivists (e.g., Carnap, 1939 and Reichenbach, 1958) and later by the logical empiricists and their sympathizers (including Braithwaite, 1953 and Nagel, 1961). Yet, for the most part, axiomatization had no place in the history or practice of actual science. Whoever heard of axiomatic organic chemistry or axiomatic planetary astronomy (though the latter certainly did use geometrical methods)? What I find strange is not that Kuhn wanted to liberate philosophy of science from assuming that theories could be conceived best as systems of axioms, but

3

Shapere (1971) reviewed the 2nd edition of Kuhn’s Structure and the Criticism volume.

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that any previous philosopher should have thought of the axioms as being the ideal form for scientific theories. Certainly around the turn of the 20th century the attempt of Hertz in 1894 to axiomatize mechanics (Hertz, 1956), the influence of Hilbert’s (1923) formalist mathematics and the impact of Russell and Whitehead’s (1910–1913) formal logic had their role in creating this ideal. But in fact, Newtonian or classical mechanics was not rigorously axiomatized until 1953 (McKinsey, Sugar, & Suppes, 1953); which was after the time that philosophers began to debate whether Newtonian mechanics was still really true or useful in view of its being reduced to or replaced by Einstein’s theory. In fact it is plausibly arguable that no other science has ever been axiomatized in anything approaching a rigorous way. So this being an ideal that needed attacking becomes stranger. In the same period, some years before Kuhn, Toulmin4 (1953) and Hanson (1958) brought the ideas of the late Wittgenstein into the philosophy of science.5 This questioned the idea of necessary and sufficient conditions as being the goal of analysis, while around the same time Quine (1951) attacked the analytic–synthetic distinction and later advocated a naturalized epistemology (Quine, 1959, pp. 68–90). Toulmin (1953), as we noted, held that theories were like maps, and Hanson (1958) proclaimed that theories were ways of organizing inquires into the causes by which things happened. So Kuhn publishing 4 years after Hanson was only part of the new wave in the philosophy of science. This new wave hoped to crest and crush what were taken to be the bad, old positivist traditions.6 These were heady days for those of us being trained during this time. But already the straw man, The Positivist, was beginning to emerge as the archetypal, tradition-bound, moribund bad guy. Yet Kuhn was certainly right in redrawing the parameters by which we ought to conceive science and in prodding us to move beyond theory, especially when conceived as sets of propositions. The shift in focus was to a “larger structure”, to what Lakatos (Lakatos and Musgrave 1970) later would call a “research programme”, and Laudan (1977) a “research tradition”. Paradigms extended over years, decades or even centuries, contained many nonpropositional elements, and were subject to some internal changes even while maintaining their identity as a specific paradigm. They were much more than just sets of propositions from which a scientist might draw inferences. However, this led to a large controversy over whether there were any coherent identity conditions for something’s being a paradigm. What were the criteria? Kuhn (1970) later developed the idea by talking about a disciplinary matrix that included “constellations of group commitments”, “shared examples” [“exemplars”] and tacit or intuitive knowledge about how to proceed in the doing of the science (including how to give proofs). This helped somewhat and again may be seen as right headed, but it failed to satisfy philosophically strict analytic requirements. Later Lakatos’ “hard and soft core” and Laudan’s “problems” would share similar identification difficulties. These were articulated in ways different than Kuhn but both

4 Toulmin’s (1953) Philosophy of Science was published the same year as the first edition of Wittgenstein’s (1953) Philosophical Investigations and in the same year came Quine’s (1953) From a Logical Point of View. 5 Mention should be made here too of Paul Feyerabend (1958). For a review see Machamer (1975). 6 That these bad old traditions were not so bad, and that the “new wave” was not so new and shared much with the bad old has been documented to a great length. See Alberto Coffa (1991) and Michael Friedman (1999).

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models, deep down, were completely intellectual, as opposed to Kuhn’s right-thinking emphasis on knowing as being tied to social acting and doing. Today some philosophers, like myself, might claim that science is, for the most part, a search for mechanisms and for developing research techniques by which mechanisms may be discovered and validated, wheremechanisms are socially and historically delineated and investigated. This search involves not just thoughts, but centrally experiments, procedures, productions and many types of tools. Let me stop Kuhn exposition at this point for a little philosophical excursus. I just claimed Kuhn was right to tie knowing with acting and doing. This claim is not just a pragmatist bias of mine. I would argue that tying knowledge to action goes back to Aristotle (1987a,b). Briefly, the claim is this; knowledge is not just a passive collection of sensory inputs that are then inductively worked over by the mind to provide general concepts. Positively, coming to have knowledge, or knowledge acquisition, is an active learning process on the part of the knower in the form of critical and selective integrations into multiple memory systems. Another distinct aspect of the activity comes in the recall and accessing stage when putative knowers act on the basis of their memorial (sometimes, belief) schemata and receive feedback from the environment that selectively corrects the internal memorial or schematic representations that constitute the knowledge (cf. Piaget’s accommodation and assimilation). This corrective function is necessary for intelligence (what one might call knowing well). The difference between intelligent performance and merely mechanical performance is that intelligent performance has the possibility of self-correction in virtue of being able to critically monitor its own states. Simple corrective feedback is not sufficient; a homing torpedo or a thermostat can do this. For intelligence, there must be a discrete internal representation of a goal state or an equilibrium ‘setting’ a standard for proper functioning, which may be step-wise decomposed and compared to (or regulated) and used to correct future actions (or outputs which themselves may be modified). Even in the case of procedural knowledge (pace some cognitive neuroscience theories) this must be the case, for to intelligently perform a procedure one must be able to monitor the procedural action (though this need not include an awareness of the mechanisms themselves) and correct it if it goes wrong. This is what servomechanisms and many evolutionarily wired bits of “knowledge” cannot do. Elaboration of this claim would require saying more about how goal states or norms get constructed, by evolution in certain limiting cases but more often by social training and learning. But this would take us too far beyond Kuhn and our topic. Knowledge may be conceptual or declarative, it may be discriminatory and nonconceptual, or it may be procedural. This last is often called knowing how, and it is this that Kuhn attributes to and endorses in Polyani’s idea of tacit knowledge. One has to learn to how to drive a car, properly taste a wine, develop strategies for solving jigsaw puzzles, or differentiate two changing quantities. But once one has learned, using such knowledge becomes, more or less, “automatic”. Yet in each case it is checkable and corrigible, that is correctable if the activity seems to be going, or has gone, wrong. In fact even in the putatively most passive visual cases, for learning to be successful and for knowledge to be gained, there must be an active component. Further, a third aspect, must be present for knowledge to be knowledge, it must be able to be used, at least sometimes (Machamer &

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Osbeck, 2004). This use occurs in the public domain and entails that the criteria for knowledge always have a social component.7 I might, only somewhat misleadingly, say that activity is necessary to gain knowledge, to preserve knowledge in memory, to access knowledge and to utilize knowledge. These reflections on activity and use should be a guide to making clear the educator’s key phrase, “active learning”. Let us shift our attention now to another Kuhnian conundrum. Like the logical empiricists before him, Kuhn takes up the topic of the meaning of theoretical terms. The topic arises during the discussion of what he takes to be the hallmark of scientific revolutions, incommensurability. The stark form of Kuhn’s claim would be: If a scientific revolution has occurred, then the meaning of the terms in the former paradigm are incommensurable with (i.e., have a different meaning from) the terms in the new revolutionary paradigm. I want to explore this claim about meaning for it was the cause of the most vigorous philosophical controversy that surrounded Structure. But before doing so please note one problem: Kuhn, in discussing incommensurability, slipped back into talking, like the positivists and logical empiricists, about terms and the meaning of terms. It seems we are no longer concerned with practices or activities. So I leave it to you, when one paradigm supplants another, do some or all practices change too? Does the way of solving differential equations change from Maxwell to Einstein? Did the basic way of using a microscope change after Pasteur did his revolutionary work? Of course, the answer to both questions is “Maybe yes, maybe no”. Augmenting this problem, Kuhn went on to develop the concept of “incommensurability” by conflating changes in a term’s meaning with differing theory-laden perceptions and even with changes in world views. Somehow for him these were all the same phenomenon, maybe just different aspects. But before criticizing Kuhn, let us note his influence. When the Max Plank Institute for the History of Science opened in 1989, it held a conference of which the proceedings were later published as Biographies of Scientific Objects (Daston, 1999). In this volume, Latour’s essay elaborates a “theory of ‘relative existence’”: When a phenomenon ‘definitively’ exists this does not mean that it exists forever, or independently, of all practice and discipline, but that it has been entrenched in a costly and massive institution that has to be monitored and protected with great care…. (Latour, 1999, p. 255) He says: It is impossible to pronounce the sentence ‘Ramses II died of tuberculosis’ without bringing back all the pragmatic conditions that give truth to this sentence. (Latour, 1999, p. 251) 7 It is this need for linking activity and knowledge that shows why one must read Aristotle’s remarks on epagoge (induction) in Posterior Analytics II. 19 in conjunction with what he says about active knowledge in the Nicomachean Ethics (for example at 1141B13ff.) This active link has been noted also by many others including the pragmatists (especially Dewey, 1938); it was noted by Wittgenstein (1956) in Remarks on the Foundations of Mathematics and is part of what Wittgenstein ought to have meant when he said, “look not for the meaning but for the use”.

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Latour here might mean either the use of “tuberculosis” is anachronistic, or there are only transitory ontic commitments at any time, no fact of the matter. Latour suggests we only may talk about two different objects. Pouchet and Pasteur do not define the same physical elements […] nor do they live in the same social and historical context. Each chain of convictions defines not only different links with the same elements, but different elements as well. (Latour, 1999, p. 260) This should sound familiar, for the same was said, about the Ptolemaic, Tychonic and Copernican universes, about phlogiston and oxygen chemical paradigms and about many other major changes in science. They are not just different explanations, they are different universes, different scientific objects. Latour echoes what was claimed by Kuhn (1962) 30 years before and by Hanson (1958) 4 years before that: Plus ça change, plus la meme chose. But is this claim any more plausible now with Latour than it was 40 years ago with Hanson and Kuhn (and we should not forget Feyerabend, 1962)? It is important to distinguish as Kuhn (2000a) later did, between “global” incommensurability and “local” incommensurability. The global is where one speaks about worldviews, weltaunshaungen and large-scale, social (cultural, economic political, ideological) contexts. Local refers only to small, proscribed changes that occur within some, often illdefined, identity constraints, and such changes may occur even, for example., within an individual’s concept as learning occurs. This later locution has become a preferred way of speaking with some learning psychologists (e.g., Carey & Spelke, 1996) who describe how children learn science, in terms of changing concepts that produce local incommensurabilities. For such descriptions we might as well have stayed with Festinger’s (1962) cognitive dissonance theory. Yet, surely the intuition behind local incommensurability is basically right. When anyone learns anything, some of the content of what was believed before becomes otiose, undergoes modification, and what results may well be in conflict with what had been before. So as a child learns about gravity as a force, she must relinquish some of her animistic beliefs about the cause of motion in inanimate bodies. More generally, of course, as language evolves meanings change. During the course of a human life many metaphors turn literal, extend their contexts, and domains of reference expand and contract. All these changes add new implication patterns and so a form of incommensurability. The incommensurability here is just a failure of one part of a concept to map fully onto another. The mapping relates the properties of the entities or activities and the implications licensed by the concept’s network, schema, or script. This is not a troublesome extension of the geometrical concept of “incommensurable”. The new does not fit onto the old without remainder. The new concept will have some different properties or implications that cannot be held coherently together with the old. These may be in contradiction (this was the way Baltas 1997, tried to handle incommensurability), or they may introduce new entities and activities that “redefine” the mechanism. An example of the first is allowing for a wave without requiring a medium, of the second modeling brain mechanisms not as an electrical system but as a set of chemical activities. “Chemical” refers not to different properties of the electrical objects and processes, but to different

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objects or processes. Yet both purport to describe many of the same anatomical parts and to explain the same phenomena, for example, memory or perception, by providing the neurocognitive mechanisms (as measured or operationalized by behavioral tests) by which the brain works. There is no problem of meaning or of translation in this latter type of case. One cannot translate electrical talk into chemical talk, though one may relate them in certain ways, for some chemical phenomena sometimes have or depend upon electrical properties. Global incommensurability is much more problematic. Surely Kuhn and Latour are right in pointing out that there was no such concept as tuberculosis at the time of Ramses II. So, if miraculously someone at that time had used that word, no historical actor would have understood. Tuberculosis was not an actor’s category in Ramses’ time. Or take Kuhn’s example: if an Aristotelian saw a constrained stone swinging on a string, he would not have seen a pendulum. No such pendulum category existed at that time, though the concepts of equal time and equal spaces did. But would this mean that we could not teach the historical actor, in her different time and different culture, the “new” concept? Again we must give the Greek answer: sometimes yes, sometimes no. One certainly could have taught the concept of isochrony to Aristotle. He had all the conceptual resources he needed to grasp this. What he did not have was a “context” that made time a more important variable than distance. Why such a context came about in 1603–1604 for Galileo, and not before, is a good historical question. It may be a form of the ultimate historical question: Why did this event happen now, and not at some other time? But it raises no particular philosophical difficulties except in terms of historical correlative causation. Philosophical difficulties of the kind claimed above only arise when one combines such historical claims with specific theories of meaning for example, meaning as intrinsically defined by interrelated concepts that all ‘in the head’. However, philosophically, there is no reason to accept such accounts unless there would be some evidence that makes them compelling, or, at least, plausible. The accounts assumed by Kuhn (and Feyerabend and Latour) have no such warrant. Let us consider quickly the famous case of the meaning of the term “mass”. In classical mechanics mass is conserved, in relativistic mechanics mass is not conserved (for it may be transformed into energy.) Therefore, it is claimed the term “mass” means something different in the two theories (in the two paradigms). Now there are, of course, differences. I have just noted one in the two sentences above. And this difference is not badly thought of as a difference in the meaning of the term. But to get incommensurability we also need to claim that there is no overlap in the meaning of the terms used in the different theories. Yet, to draw this conclusion one must hold that the meaning of a term is wholly and only specified by the total set of connections or implications that are allowable or licensed by the theory in which it occurs. The totality of the set is constitutive of the meaning. Partial sets of implications are not allowed. Now first of all this theory takes meaning to be an all or nothing property (Shapere 1966. remarked on this some years back). A term either keeps it meaning constant or it changes. Any change is total and complete or nonexistent. Worse, on this theory, meaning resides wholly internally in the head and has nothing to do with the world. The meaning of a term, as were, resides solely within the relations in a dictionary (or mental lexicon). Katz and Fodor (1963) actually elaborated such a theory of meaning, and it later played a

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part in Searle’s (1985) Chinese Room argument (of which Casimir Lewy of Cambridge had an earlier version). This is a preposterous theory of meaning. It is meaning without the world. Such a theory of meaning leaves out procedures of reference as having anything to do with the meaning of a term, which is most strange. What are the cognitive presuppositions of such a theory of meaning and “seeing” or conceptualizing? Simply put the cognitive theory assumed here is one where meaning is completely internal and modularly closed (like Katz and Fodor’s theory, of which Fodor still held a version as late as 1987). Now there is no doubt that conceptual schemes or categorical structures direct perceptual attention, and further there is no doubt that conceptual networks may lead to reasoning that further directs attention toward represented features. But this is far from claming that conceptual schemes are isolated cognitively, have no external relations to the world, or are not, by virtue of their parts, semimappable on the schemes of other people. Kuhn should have known of this nonindependence of meaning from Bruner & Goodman (1947) and Bruner & Postman (1949) work that had shown how value and need (as measured by external variables) affect perception and meaning, and how subjects can learn to discriminate and recognize incongruous suits and colors of playing cards when given external feedback. Both of these experiments implied the inadequacy of modularity and internalism. Further, for Kuhn, meanings must be shared by those in a paradigm and so cannot be wholly individual. A theory of the global incommensurability of meaning entails a relativistic solipsism should be taken as a reductio ad absurdum argument against any theory that implies it. Kuhn later recognized this as an implication, which is why he changed his mind and reduced his claim to local meaning incommensurability (Kuhn 2000a, 91 ff.). That Kuhn changed his mind about many things is to his great credit. He came to see many of the problems discussed above and made moves in his later works to try to address and correct them. Nonetheless, many problems remain. But the biggest by far is how could he reconcile the philosophical changes he later saw as necessary with his view of the history of science as proceeding by way of incommensurable revolutions among paradigms? If this latter cannot be defended, what point could there be in trying to modify the underlying philosophy to make it more plausible? More generally, what point could there be in talking about Kuhnian revolutions at all?

Acknowledgements This chapter is based on a lecture delivered at University of La Coruna, in Spain in March, 2002, and published as “El éxito de Kuhn, 40 años después”, in Gonzalez, W. J. (Ed.), Análisis de Thomas Kuhn: Las revoluciones científicas, Madrid: Trotta, 2004. Thirty years earlier, I took a very different line on Kuhn; see my “Understanding Scientific Change” (1975). I most happily thank Wenceslao Gonzalez for his help and able assistance on the earlier version of this chapter. Further, discussions over the years with my dear old friends and fellow curmudgeons, Aristides Baltas and J. E. McGuire, and my delightful and constant co-author Lisa Osbeck, have always been beneficial, when they have not led me astray by their wild ways. No doubt they have influenced me in ways that I dare not even recall.

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References Aristotle (1987a). Nicomachean ethics. In: J. L. Ackrill (Ed.), A new Aristotle reader. Princeton University Press. Aristotle (1987b). Posterior analytics. In: J. L. Ackrill (Ed.), A new Aristotle reader. Princeton University Press. Baltas, A. (1997). Some grammatical points about incommensurability. Paper presented at Kuhn Memorial Symposium, MIT. Braithwaite, R. (1953). Scientific explanation: A study of the function of theory, probability and law in science. London: Cambridge University Press. Bruner, J. S., & Goodman, C. D. (1947). Value and need as organizing factors in perception. Journal of Abnormal Social Psychology, 42, 33–44. Bruner, J. S., & Postman, L. (1949). On the perception of incongruity: A paradigm. Journal of Personality, 28, 206–223. Carey, S., & Spelke, E. (1996). Science and core knowledge. Philosophy of Science, 63, 515–533. Carlsson, A. (2001). A paradigm shift in brain research. Science, 294(2), 1021–1024. Carnap, R. (1939). Foundations of logic and mathematics. In: O. Neurath, C. Morris, & R. Carnap (Eds), International encyclopedia of unified science (Vol. 1(3), pp. 139–214). Chicago, IL: University of Chicago Press. Coffa, A. (1991). The semantic tradition from Kant to Carnap. Cambridge: Cambridge University Press. Daston, L. (Ed.). (1999). Biographies of scientific objects. Chicago, IL: University of Chicago Press. Dewey, J. (1938). Experience and education. London: Macmillan. Festinger, L. (1962). A theory of cognitive dissonance. Stanford, CA: Stanford University Press. Feyerabend, P. (1958). An attempt at a realistic interpretation of experience. Proceedings of the Aristotelian Society, 58, 143–170. Feyerabend, P. (1962). Explanation, reduction and empiricism. In: H. Feigl, & G. Maxwell (Eds), Scientific explanation, space and time (pp. 28–97). Minnesota Studies in the Philosophy of Science (3). Minneapolis, MN: University of Minnesota Press. Fodor, J. A. (1987). Psychosemantics. Cambridge, MA: MIT Press. Friedman, M. (1999). Reconsidering logical positivism. Cambridge: Cambridge University Press. Hanson, N. R. (1958). Patterns of discovery: An inquiry into the conceptual foundations of science. Cambridge: Cambridge University Press. Hertz, H. (1956). The principles of mechanics, presented in a new form. In: D. E. Jones, & J. T. Walley (Trans.). New York: Dover Publications (Original work published 1894). Hilbert, D. (1923). Die Logische Grundlagen der Mathematik. Mathematiche Annalen, 88, 151–165. Katz, J., & Fodor, J. (1963). The structure of a semantic theory. Language, 39, 170–210. Kuhn, T. S. (1962). The structure of scientific revolutions. Chicago, IL: University of Chicago Press. Kuhn, T. S. (1970). The structure of scientific revolutions (2nd ed.). Chicago, IL: University of Chicago Press. Kuhn, T. S. (2000a). The road since structure. In: J. Conant, & J. Haugeland (Eds), The road since structure: Philosophical essays, 1970–1993, with an autobiographical interview (pp. 90–104). Chicago, IL: University of Chicago Press. Kuhn, T. S. (2000b). A discussion with Thomas S. Kuhn. In: J. Conant, & J. Haugeland (Eds), The road since structure: Philosophical essays, 1970–1993, with an autobiographical interview (pp. 253–324). Chicago, IL: University of Chicago Press. Lakatos, I. (1971). History of science and its rational reconstruction. In: R. C. Buck, & S. Cohen (Eds), In memory of Rudolf Carnap: Proceedings of the 1970 second biennial meeting of the Philosophy of Science Association (pp. 91–135). Boston Studies in the Philosophy of Science, 8. Dordrecht, The Netherlands: Reidel.

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Lakatos, I., & Musgrave, A. (Eds). (1970). Criticism and the growth of knowledge. Cambridge: University of Cambridge Press. Latour, B. (1999). On the partial existence of existing and nonexisting objects. In: L. Daston (Ed.), Biographies of scientific objects. Cambridge: University of Chicago Press. Laudan, L. (1977). Progress and its problems. Berkeley: University of California Press. Machamer, P. (1975). Understanding scientific change. Studies in History and Philosophy of Science, 5, 373–381. Machamer, P. (2004). El éxito de Kuhn, 40 años después. In: W. J. Gonzalez (Ed.), Análisis de Thomas Kuhn: Las Revoluciones Científicas (pp. 137–153). Madrid: Trotta (in Spanish). Machamer, P., & Osbeck, L. (2003). Scientific normativity as non-epistemic: A hidden Kuhnian legacy. Social Epistemology, 17, 3–12. Machamer, P., & Osbeck, L. (2004). The social in the epistemic. In: P. Machamer, & G. Wolters (Eds), Science, values and objectivity (pp. 78–89). Pittsburgh, PA: University of Pittsburgh Press. Masterman, M. (1970). The nature of a paradigm. In: I. Lakatos, & A. Musgrave (Eds), Criticism and the growth of knowledge (pp. 59–90). Cambridge: Cambridge University Press. McKinsey, J. C. C., Sugar, A. C., & Suppes, P. (1953). Axiomatic foundations of classical particle mechanics. Journal of Rational Mechanics and Analysis, 2, 253–272. Nagel, E. (1961). The structure of science. New York: Harcourt, Brace and Co. Quine, W. v. O. (1951). Two dogmas of empiricism. Philosophical Review, 60, 20–43. Quine, W. v. O. (1953). From a logical point of view. Cambridge, MA: Harvard University Press. Quine, W. v. O. (1959). Ontological relativity and other essays. New York: Columbia University Press. Reichenbach, H. (1958). The rise of scientific philosophy. Berkeley, CA: University of California Press. Russell, B., & Whitehead, A. N. (1910–1913). Principia mathematica (Vol. 3). Cambridge: University of Cambridge Press. Searle, J. (1985). Minds, brains and science. Cambridge, MA: Harvard University Press. Shapere, D. (1964). The structure of scientific revolutions. Philosophical Review, 73, 383–394. Shapere, D. (1966). Meaning and scientific change. In: R. G. Colodny (Ed.), Mind and cosmos (pp. 41–85). Pittsburgh, PA: University of Pittsburgh Press. Shapere, D. (1971). The paradigm concept. Science, 172, 706–709. Toulmin, S. E. (1953). The philosophy of science. An introduction. London: Hutchinson’s University Library. Wittgenstein, L. (1953). Philosophical investigations (G. E. M. Anscobe, Trans.). Oxford: B. Blackwell. Wittgenstein, L. (1956). Remarks on the foundations of mathematics (G. E. M. Anscobe, Trans.). Oxford: B. Blackwell.

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Chapter 5

Conceptual Change and Scientific Realism: Facing Kuhn’s Challenge Theodore Arabatzis Introduction The impact of Thomas Kuhn’s work on history and philosophy of science has been unparalleled. His classic The Structure of Scientific Revolutions (Kuhn, 1970) has changed permanently the historiographical and philosophical landscape. Its influence on historians of science has been pervasive but, for the most part, indirect. It is a striking fact that very few historians have employed the conceptual apparatus of The Structure (i.e., “paradigm”, “normal science”, “crisis”, “revolution”, “gestalt switch”, “incommensurability”) to illuminate past scientific developments. Philosophers of science, on the other hand, have been directly affected by Kuhn’s historical philosophy of science and have debated endlessly the issues that he and his fellow traveler Paul Feyerabend raised. Those issues (e.g., the rationality of theory-choice, the reality of the ontology of science) remain at the forefront of contemporary philosophical reflection on science. One of the most far-reaching claims of Kuhn and Feyerabend concerned the nature of scientific concepts. They promoted a contextual view of concepts, according to which concepts obtain their meaning from the theoretical framework in which they are embedded. It follows that when a theoretical framework changes the concepts embedded in it change too (Feyerabend, 1962; Kuhn, 1970). Given that theory change has been quite common in history of science, conceptual change must have been a ubiquitous phenomenon. Several problematic philosophical consequences seemed to ensue from conceptual change, most notably a relativist view of theory-choice and an anti-realist stance toward the ontology of science. It was widely believed that rational choice between scientific theories could take place only against a stable, shared conceptual framework. With such a framework in place the claims of competing scientific theories could be formulated in a common language and then subjected to comparative evaluation. On the other hand, in the absence of such a framework objective theory-choice seemed impossible, since there would be no common language for formulating and comparing rival scientific theories. Furthermore, the realist requirement for a stable scientific ontology was hard to reconcile with conceptual change.

Reframing the Conceptual Change Approach in Learning and Instruction Copyright © 2007 by Elsevier Ltd. All rights of reproduction in any form reserved. ISBN: 0-08-045355-4

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If scientific concepts change, then it is not clear how they can continue to refer to the same entities, properties, or processes. This chapter focuses on the latter issue, namely the implications of conceptual change for scientific realism.1 The debate on scientific realism has taken place along several lines.2 The most important, for my purposes here, concerns the grounds that we have for believing in the reality of the unobservable entities postulated by contemporary science (atoms, electrons, photons, fields, etc.). These entities are represented by theoretical concepts. If these concepts are evolving, then what sense does it make to believe that they continue to refer to the same, and ipso facto real, entities?3 In what follows, I will discuss some of the main developments concerning this question in post-positivist philosophy of science. I will then argue that the evolution of theoretical concepts need not throw doubt, under certain conditions, on their stable identity. When theoretical concepts change they may continue to refer to the same entities, properties, or processes.

Historicizing Concepts: Kuhn’s and Feyerabend’s Anti-Realist Theses As I noted above, Kuhn and Feyerabend launched a novel and powerful attack against a realist reading of the historical development of science, that is, the view that science aims at a permanent fixed truth, of which each stage in the development of scientific knowledge is a better exemplar. (Kuhn, 1970, p. 173) On this view, the progress of science … consist[s] in ever closer specification of a single world, the actual or real one. (Kuhn, 1989, p. 24) Here I will focus on Feyerabend’s views, since they were more radical and developed in more philosophical detail than Kuhn’s early views.4 Feyerabend (1962) presented his views on conceptual change in a seminal paper. There he argued that the concept associated with a scientific term is not an intrinsic property of it, but is dependent upon the way in which the term has been incorporated into a theory. (Feyerabend, 1962, reprinted in Feyerabend, 1981, p. 74)5

1

What follows draws on material I have presented in more detail in Arabatzis (2006). I thank the University of Chicago Press for granting me permission to adapt and use portions of Chapter 9 of this book, entitled “Identifying the electron: Meaning variance and the historicity of scientific realism”. © 2006 by the University of Chicago. All rights reserved. 2 For a comprehensive analysis of this debate, from a realist point of view, see Psillos (1999). 3 cf. van Fraassen (2002, p. 56). 4 I have discussed Kuhn’s recent and more philosophically articulated views on scientific realism in Arabatzis (2001). 5 Note that Feyerabend framed the issue in terms of meanings, as opposed to concepts.

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Since scientific concepts are dependent on the theory in which they are embedded, conceptual change is a necessary consequence of theory change. Concepts “change their meaning with the progress of science”, since extension of knowledge leads to a decisive modification of the previous theories both as regards the quantitative assertions made and as regards the meanings of the main descriptive terms used. (Feyerabend, 1962, reprinted in Feyerabend, 1981, pp. 80–81) What are the implications of conceptual change for scientific realism? To use a concrete example, emphasized by both Kuhn and Feyerabend, if the physical referents of these Einsteinian concepts [space, time, and mass] are by no means identical with those of the Newtonian concepts that bear the same name (Kuhn, 1970, p. 102) then what should we conclude about the ontological status of the referents of those concepts? There are two options available to the realist. First, he or she could accept the discrepancy between Newtonian and Einsteinian concepts and argue that only the Einsteinian concepts have genuine physical referents. However, if the difference between Newtonian and Einsteinian concepts implies that the former do not have a referent then, similarly, future theoretical developments may reveal that Einsteinian terms are non-referential too. Second, the realist could deny that Kuhn and Feyerabend have provided an accurate characterization of conceptual change in science and, thus, dispute their claim that Newtonian terms do not have a referent. An argument along these lines would be that, despite the differences between Newtonian and Einsteinian concepts, there are sufficient similarities between the two sets of concepts to enable us to maintain that Newtonian terms refer to the manifestations of relativistic entities at low velocities compared with the velocity of light.6 Finally, another realist response would be to point out some difficulties in the view of concepts favored by Kuhn and Feyerabend and thus to cast doubt on their conclusions. As I mentioned above, according to that view, the concepts associated with scientific terms are theory dependent. One needs to know then what counts as part of a theory and what kinds of theory change amount to a transformation of the concepts of the theory. In this respect, as has been pointed out by Dudley Shapere, Kuhn’s and Feyerabend’s views are open to objection (see Shapere, 1966).7 “Theory” is usually understood as a systematic and articulated body of knowledge. Feyerabend on the other hand has a more inclusive conception of “theory”: The term “theory” will be used in a wide sense, including ordinary beliefs (e.g., the belief in the existence of material objects), myths (e.g., the myth of eternal recurrence), religious beliefs, etc. In short, any sufficiently general point of view concerning matter of fact will be termed a “theory”. (Feyerabend, 1965a, p. 219) 6

cf. Shapere (1991, p. 656). The critique of Feyerabend that follows is based on Shapere’s paper. The attempt, however, to improve on Feyerabend’s views so as to meet Shapere’s criticism is mine. 7

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Given this notion of “theory” and the theory dependence of concepts, it is not clear what aspects of the theoretical context are relevant to specifying a concept. Feyerabend, to the best of my knowledge, has not provided a satisfactory resolution of this difficulty. Nevertheless, this difficulty can be, at least partly, alleviated if we narrow Feyerabend’s conception of “theory” so as to exclude ordinary beliefs, myths, religious beliefs, etc. and to retain scientific theories only. In such a case a concept would be specified by the set of properties ascribed to its referent (a purported entity in nature) by the scientific theory in which the concept is embedded.8 One may object that not all the properties attributed to an entity are constitutive of the concept associated with it. The attribution of some of those properties, on such a view, is a merely factual assertion about the entity and is, therefore, irrelevant to the concept associated with it. A satisfactory rebuttal of that objection would require an extensive discussion of the analytic/synthetic distinction, a task that is beyond the scope of this chapter. Here I can only appeal to the widely accepted view, originating from Quine, that a clearcut distinction of this kind is not possible.9 Furthermore, the main realist opponent of Feyerabend, Hilary Putnam, has emphasized the impossibility of separating, in the actual use of the word, that part of the use which reflects the ‘meaning’ of the word [or, I would add, the concept associated with a word] and that part of the use which reflects deeply embedded collateral information. (Putnam, 1962, pp. 40–41) I will have more to say about this issue below. To return to my proposed modification of Feyerabend’s notion of concepts, a difficulty remains: What is the relationship between theory change and conceptual change? In other words, what kinds of modifications in a theory affect the concepts that the theory specifies? One could think of changes in a theory that would be too minor to affect the concepts embedded in it. For example, a refinement of the gravitational constant in Newton’s law of gravity would hardly change the meaning of any Newtonian terms. Feyerabend tried to alleviate this difficulty by proposing that we shall diagnose a change of meaning [conceptual change] either if a new theory entails that all concepts of the preceding theory have zero extension or if it introduces rules which cannot be interpreted as attributing specific properties to objects within already existing classes, but which change the system of classes itself. (Feyerabend, 1965b, reprinted in Feyerabend, 1981, p. 98) The former “diagnostic” procedure is far too strict. After all, one would want to diagnose a case of conceptual change if a new theory entails that some of the concepts of the preceding theory are vacuous and refer to nothing at all. 8

A similar view is held by Dudley Shapere. The only difference is that, in Shapere’s view, the concept associated with a term is identified with the set of properties attributed by the group of scientists who are using the term to the entity that the term denotes (see Shapere, 1982, p. 21). 9 For some useful reflections on this issue, see Papineau (1996).

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The latter concerns the taxonomic function of scientific theories and it presupposes the existence of unambiguous rules of classification, which can be used to collect objects, events, entities, etc. into sets. If a new theory attributes additional properties to the entities denoted by the concepts of the previous theory but preserves its taxonomic rules, one should conclude, according to Feyerabend, that no conceptual change has taken place. First, one might want to dispute his claim that the properties of an entity, to the extent that they are not related to any rules of classification, are irrelevant to the concept associated with the corresponding term. However, even if every feature associated with an entity is relevant to the specification of the corresponding concept, it would still be the case that any conceptual change brought about by the addition of new properties to a fixed set of entities would lack any anti-realist connotations. On the contrary, conceptual change of this kind would support a realist position, since it would have taken place against a stable ontological background. Furthermore, the existence of unambiguous taxonomic rules in science has been denied by Shapere (1966). Scientific classifications, in his view, are usually based on pragmatic criteria, which do not reflect any intrinsic properties of the entities classified. For example, the question Are mesons different “kinds of entities” from electrons and protons, or are they simply a different subclass of elementary particles? … can be answered either way, depending on the kind of information that is being requested for there are differences as well as similarities between electrons and mesons … [It] can be given a simple answer (“different” or “the same”) only if unwanted similarities or differences are stipulated away as inessential. (Shapere, 1966, pp. 51–52) Thus, an appropriate choice of taxonomic criteria might leave the concepts associated with the entities in question unaffected by theory change. Shapere’s criticism is based on his emphasis on the pragmatic character of scientific classifications, an emphasis that, I think, is overstated. Physics, for instance, has produced an unambiguous classification of the unobservable realm into various particles (electrons, protons, neutrons, etc.). It is true that their occasional similarities enable the classification of otherwise different entities under the same category. For example, electrons, protons, neutrons, and quarks are all classified as fermions, since they obey Fermi–Dirac statistics. But these entities, despite their occasional similarities, are clearly distinguished from one another according to their intrinsic properties (e.g., the magnitude of their charge, their mass, etc.). At least some classifications are predicated on the intrinsic properties of the entities classified and, thus, Feyerabend’s notion of conceptual change could escape Shapere’s criticism if one limited its applicability to classifications of that kind. With these qualifications, the above difficulties in the criterion of conceptual change favored by Feyerabend disappear and, thus, cannot be employed to support a realist position. An anti-realist stance is implicit in those of Feyerabend’s views that I have discussed so far. In his early work he did not adopt an explicit position on the ontological status of unobservable entities, an aspect of the realism debate with which he was not directly concerned. He was occupied, rather, with the phenomenon of meaning change and its implications for

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the then-current theories of scientific explanation and reduction. In his later writings, however, he formulated unambiguously the anti-realist perspective that was implicit in his early papers. The phenomenon of conceptual change at its most extreme (incommensurability) implies, in his view, that past scientific theories, which have been replaced by their incommensurable successors, were based on a non-existent ontology. For example, prerelativistic terms … are pretty far removed from reality (especially in view of the fact that they come from an incorrect theory implying a nonexistent ontology). (Feyerabend, 1970, p. 87) Feyerabend’s view of conceptual change, a view strikingly parallel to Kuhn’s, presented a serious challenge to scientific realists. It seemed that the instability of scientific concepts had as an immediate corollary the collapse of a realist attitude toward their referents. Given the then-prevalent belief (inspired by Frege) that a concept is specified by a set of conditions that are necessary and sufficient for its correct application, the slightest change in those conditions (conceptual change) would imply that the concept as previously used was vacuous, that is, it referred to nothing at all. Hence Kuhn’s recent remark that “the history of science is the history of developing vacuity” (Kuhn, 1989, p. 32). Furthermore, even if one rejects the ‘necessary and sufficient conditions’ view of concepts, as most philosophers (and, as far as I know, all psychologists) nowadays do, a problem remains. If a concept evolves over time it is not clear how it can refer to the same entity.10 Why is it, for example, that Faraday’s field concept and Einstein’s field concept have the same counterpart in nature? (cf. Nersessian, 1984). If they do not, as Kuhn and Feyerabend argued, then one is forced to admit that there are no entities in nature with the properties of Faraday’s field. Only the Einsteinian field concept may have a genuine referent. And even that possibility may be unlikely. Consider the following alarming scenario, due to Hilary Putnam: What if all the theoretical entities postulated by one generation (molecules, genes, etc., as well as electrons) invariably do not exist from the standpoint of later science? … One reason this is a serious worry is that eventually the following meta-induction becomes overwhelmingly compelling: just as no term used in science of more than fifty (or whatever) years ago referred, so it will turn out that no term used now refers (except maybe observation terms, if there are such). (Putnam, 1984, pp. 145–146) Thus, Putnam saw clearly the, prima facie, anti-realist implications of conceptual change. Let me now turn to his significant attempt to evade those implications.

Putnam on Conceptual Change: A Realist Way Out? In the early 1970s Putnam tried to disentangle concepts from their referents by suggesting that the latter are independent and essential constituents of the former. That is, a concept 10

Remember that the requirement for a stable ontology is crucial for realism.

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is determined, to a large extent, by its referent and not vice versa. Whatever changes the other components of a concept may undergo the “referential” component will remain unaffected. Putnam was not particularly concerned with theoretical concepts denoting unobservable entities. Following Kripke, whose ideas were developed in the context of modal logic, he was occupied, rather, with natural kind concepts in general. The purpose of his arguments was to show that the referents of natural kind concepts have not been affected by the development of scientific knowledge. Even though the concept [of a natural kind, e.g., fish] is not exactly correct (as a description of the corresponding natural kind)…that does not make it a fiction. …The concept is continually changing as a result of the impact of scientific discoveries, but that does not mean that it ceases to correspond to the same natural kind … . (Putnam, 1973, p. 197) I have two comments here. First, I do not think that one can exclude the possibility that future changes in the concept of, for example, fish will affect its reference. The concept may change so that organisms that are now classified as fish will cease to be thus classified. Second, the example of “fish”, or any other observable natural kind concept, is not very helpful when it comes to questions of scientific realism. Even if the reference of that concept changed, the existence of the individual organisms that we classify as fish would not be questioned. And that is because the existence of those organisms is established on grounds independent of our system of classification. That is not the case, however, when the natural kind concept refers to unobservable entities. Besides the fact that several such concepts have been abandoned,11 even when they have been retained, the lack of independent, physical access to the entities denoted by those concepts makes problematic the claim that, in case of conceptual change, their referents remain stable. Whatever the merit of Putnam’s arguments for the referential stability of observable natural kind concepts, I will argue that, when it comes to theoretical concepts, which denote unobservable entities, his theory of reference does not remove the threat that conceptual change poses to scientific realism.12 Putnam’s view of concepts was motivated by a basic realist intuition, namely “that there are successive scientific theories about the same things: about heat, about electricity, about electrons, and so forth”. For instance, Bohr would have been referring to electrons when he used the word “electron”, notwithstanding the fact that some of his beliefs about electrons were mistaken, and we are referring to those same particles notwithstanding the fact that some of our beliefs — even beliefs included in our scientific “definition” of the term “electron” — may very likely turn out to be equally mistaken. (Putnam, 1973, p. 197)

11

For a very long list of concepts that have been abandoned, see Laudan (1981). Several critics of Putnam have also argued, for different reasons than I will suggest below, that his theory does not work for theoretical concepts (see Enç, 1976; Kroon, 1985; Nola, 1980). 12

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To flesh out this realist intuition Putnam had to reject the ‘necessary and sufficient conditions’ view of concepts. In his view, membership in the extension of a term is not determined by a set of necessary and sufficient conditions, which constitute the concept associated with that term. Rather the reference of a term (concept) is fixed through our causal interaction with the world: things which are given existentially … help to fix reference. Actual things, whatever their description, which have played a certain causal role in our acquisition and use of terms determine what the term refers to. (Putnam, 1983, p. 73) Thus, the reference of a concept is an independent component of it and is not affected by (even drastic) changes the other components might undergo.13 One of the main arguments that Putnam directed against the necessary and sufficient conditions view of concepts was the following thought experiment.14 Consider our familiar concept of water. The reference of the concept of water, on the necessary and sufficient conditions view, is determined by the phenomenological properties we attribute to water (transparent, tasteless, thirst-quenching, etc.). Putnam, however, pointed out that the discovery of a substance with the same phenomenological properties as our familiar water but with a totally different structure (XYZ) is conceivable. In that case the reference of our concept of water would not include the novel substance (XYZ). Thus, the “stereotypical” features of an entity or substance, despite being part of the concept associated with it, do not determine the concept’s reference. It follows that the evolution of a concept need not affect the stability of its reference. Putnam’s rejection of the necessary and sufficient conditions view neutralizes, to some extent, the anti-realist implications of conceptual change. If a concept could refer to the same entity(ies) despite the fact that it has evolved, anti-realism would not be an inescapable consequence of the instability of scientific concepts. I put the qualifier “to some extent” because a belief in the existence of, say, electrons would make sense only if a core of the concept of the electron (i.e., some of our beliefs about electrons) has remained stable since the initial proposal of the electron hypothesis. Cataclysmic changes of scientific concepts, which would amount to the abandonment of every belief about the corresponding entities, would be indeed a threat to scientific realism. Barring such radical cases, Putnam’s rejection of the necessary and sufficient conditions view of concepts should be an indispensable ingredient of any realist position. Nevertheless, Putnam’s success in disentangling the reference of a concept from the rest of it does not extend to cases where a concept refers to an unobservable entity (or a class of

13

Here it is worth pointing out that Putnam’s views on reference have remained intact, even though his more recent views on scientific realism depart considerably from the realist intuitions that motivated his earlier attempts to articulate a theory of meaning. In a recent discussion of this issue, for example, he insists that “the reference of the terms ‘water,’ ‘leopard,’ ‘gold,’ and so forth is partly fixed by the substances and organisms themselves… [while] the ‘meaning’ of these terms is open to indefinite future scientific discovery” (Putnam, 1990, pp. 109–110). 14 See Putnam (1975a, pp. 223ff).

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such entities). In order for the referent of a concept to be an essential and stable constituent of the concept in question, as Putnam’s theory requires, it should be possible to identify the referent without relying on the other constituents of the concept. To that effect, one should have independent, physical access to the entities denoted by a concept, independent of (most of) our beliefs about them. This independent access guarantees, extreme skepticism excluded, that those entities are real. However, in the case of unobservable entities no such direct epistemic access exists. All that we have is a set of indirectly confirmed beliefs about those entities and it is not clear that we can identify them independently of those beliefs.15 Furthermore, since the problem under examination is the ontological status of unobservable entities, it goes without saying that a theory like Putnam’s, which presupposes their existence, cannot be employed without circularity to support a realist position. My evaluation of Putnam’s theory as a potential realist tool was based on the assumption that unobservable entities are beyond direct empirical access. This assumption has been forcefully challenged by Ian Hacking, whose views on scientific realism are the subject of the next section.

Hacking’s Entity Realism Hacking (1983) has advanced a realist position, which is based on a close examination of experimental practice. A satisfactory resolution of the problem of scientific realism would be possible, Hacking claims, only if we stopped being preoccupied with scientific theorizing and shifted instead our focus of analysis toward experimentation. Such a shift of emphasis would be enough to make us all realists with respect to some unobservable entities, but would not weaken our anti-realist convictions with respect to the theories that postulate those entities. This peculiar mix of realism about entities and anti-realism about theories follows from two central aspects of experimental practice. On the one hand, the manipulation of unobservable entities in the laboratory provides sufficient grounds for believing in their existence. On the other hand, the fact that experimentalists use, according to the purpose in hand, a variety of sometimes incompatible theoretical models of those entities, generates strong doubts that any of those models accurately represent reality. All these models, however, have some aspects in common, namely a core of statements about the causal properties of the corresponding entities, properties which we have come to know by manipulating those entities in various experimental contexts. One can (should) be a realist about this common core, which, however, does not deserve to be called a “theory”. Hacking’s entity realism can be summarized in his aptly chosen slogan: “If you spray [e.g.] electrons then they are real” (Hacking, 1983, p. 24). Notwithstanding the charm of Hacking’s slogan, it fails to impress philosophers in the empiricist tradition. Van Fraassen, for instance, when asked to evaluate Hacking’s argument, responded in the following way: “If they are 15

I will suggest below a recipe for identifying the reference of theoretical concepts denoting unobservable entities that would allow one to include reference as a component of a concept. However, it is not applicable to entities that have not been (and perhaps cannot be) subject to experimental investigation (e.g., black holes). Furthermore, even where it is applicable, the referential stability of a concept requires historical investigation and should not be taken for granted prior to such an investigation.

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real then you spray them”.16 Van Fraassen does not imply, of course, that everything real can be sprayed. Rather, his point is that one can use the expression “spraying of electrons” (as the best available description of a given experimental situation) without committing oneself to believing in the existence of electrons. The coherence of his position stems from the possibility that a new theory, which would not include electrons in its ontology, could adequately account for the experimental situation that Hacking, following contemporary experimentalists, describes as “spraying of electrons”. It is conceivable that the process now described as “spraying of electrons” might be re-described in terms of an alternative theory based on a different ontology. Another objection has been raised against Hacking’s claim that manipulability is a necessary and sufficient criterion for establishing the existence of an unobservable entity, namely that it does not do justice to actual scientific practice (see Morrison, 1990). In particular, his exclusive emphasis on manipulability fails to capture the variety of evaluative criteria that are employed within scientific practice for demonstrating the reality of an unobservable entity.17 The most important of those criteria seems to be the empirical adequacy of the theory that postulates the entity in question. Even though manipulability is one of the criteria that scientists often employ, it fails to carry conviction in all contexts. Furthermore, scientists need to have a fairly definite idea of what it is that they are manipulating before invoking manipulability as a demonstrative principle. In many experimental contexts we know that we are manipulating something without knowing what it is that we are manipulating. Manipulability, however, was supposed to provide adequate grounds for the existence of, for example, electrons and not merely for the existence of an “I know-not-what” something. Thus, Hacking’s manipulability criterion fails as a descriptive account of the plurality of principles that scientists employ in the construction of “existence proofs” for unobservable entities. It also fails as a normative account, since even within actual scientific practice manipulability does not (and should not) provide adequate reasons for belief in the entities that are supposedly manipulated. A further problematic aspect of Hacking’s realist position is associated with his “home truths” (low-level generalizations) about, for example, electrons that we supposedly know independently of any high-level theory. Hacking does not specify what kind of electron properties he has in mind but one could guess that his “home truths” would include wellknown causal properties of electrons, like their charge, mass, and spin, which enable us to manipulate them in order to investigate other less well known aspects of nature. It is difficult to see, however, how one could isolate the concepts denoting those properties from the background theory in which they are embedded. To use a concrete example, it is difficult to obtain an understanding of “charge” independently of any high-level theory, especially in view of the fact that “charge” meant different things for different scientists.

16

Personal communication. One could deny that Hacking’s criterion needs to capture the richness of scientific practice, by interpreting it as a normative as opposed to a descriptive criterion. However, Hacking himself stresses the descriptive (as well as the normative) dimensions of his criterion. For him, the main reason that scientists themselves believe in the reality of some unobservable entities is their ability to manipulate those entities: “The vast majority of experimental physicists are realists about some theoretical entities, namely the ones they use. I claim that they cannot help being so” (Hacking, 1983, p. 262). 17

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Hacking could argue that all those different and incompatible conceptions of charge had certain aspects in common, that is, the causal properties that we have associated with electric charge all along (e.g., the ability of charges to attract, or repel each other). But even those properties may vary across theoretical frameworks. In classical electromagnetic theories, for instance, one of the causal properties of a charged particle was that it emitted radiation when performing accelerated motion. In the old quantum theory of the atom, on the other hand, charged particles in accelerated motion (electrons orbiting around the nucleus) did not radiate. Hacking’s attempt to isolate the causal properties of electrons from any background theory was motivated by the plurality of incompatible models about electrons. However, it turns out that the causal properties themselves have been interpreted via several, incompatible theories. He could, of course, search again for a common core shared by those theories, but it is not clear that there would be an ending point to this process. A closely associated problem is Hacking’s selective realism. Let us grant for the sake of argument that belief in the existence of the “well-known causal properties” of some entity does not depend on how we interpret those properties theoretically. As I already mentioned, in his view, one should maintain a realist position with respect to those properties, but, nevertheless, should not commit oneself to believing that the theories involved in the interpretation of the relevant experiments are true.18 However, as was pointed out by Duhem, experimental results falsify (or, I would add, confirm) whole chunks of knowledge consisting of high-level theory(ies), an understanding of the instruments involved in the given experimental context, statements about initial conditions, etc. In view of this very plausible holistic epistemology, which Hacking has not by any means discredited, his selective realism does not make much sense. To put it another way, it is far from clear why our ability to make sense of certain experimental situations confers a privileged status on the entities supposedly involved in the experiments but, nevertheless, cannot be employed as an argument for a realist stance toward the theory(ies) involved in the interpretation of those experiments. The following example would suffice to illustrate my point: The manipulation of electrons requires knowledge of their behavior in various experimental contexts (e.g., in a cloud chamber). To control that behavior one has to know, in addition to the causal properties of electrons, several background theories (electromagnetism, mechanics, etc.), which predict how an entity with those properties would behave in a given context. It is those background theories along with Hacking’s “home-truths” that enable intervention. Thus, if our ability to intervene is an argument for realism, that realism should be of a very broad kind covering entities and theories alike. Having discussed some significant positions that have been advanced with regard to the implications of conceptual change for scientific realism, I will now put forward my own approach to this problem.

How to Rescue Scientific Realism from Conceptual Change To present my own view on conceptual change and scientific realism, it is necessary to adopt a particular view of scientific concepts. In what follows, I take a concept to be the 18

Perhaps one should be a realist about the phenomenological theories of the instruments involved.

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set of features that are ascribed to its putative referent, by the theory in which the concept is embedded. In case that no such systematic theory has been developed, a concept would consist of the characteristics attributed to its counterpart in nature, by the group of scientists who are using it. These characteristics consist of two interrelated kinds: the properties of the entity in question and the laws obeyed by it. The former are usually specified via the latter. For instance, what charge is depends on the laws obeyed by charged bodies, for example, the laws of classical electromagnetism. Conversely, the laws obeyed by an entity depend on its properties. For example, the electrons qua charged particles are supposed to obey Coulomb’s law. As I already mentioned, one may find objectionable the suggestion that all the characteristics ascribed by a theory to an entity are part of the corresponding concept. In particular, one may call for a distinction between the concept-constitutive and the merely factual aspects of the information concerning the referent of a concept. I think, however, that such a distinction cannot be drawn in a satisfactory way. To begin with, no distinction of this kind is inherent in the scientific representation of an entity. For example, electrons are represented as subatomic particles, with a specific amount of charge, a specific rest mass, spin, etc. There is nothing in this representation, however, that explicitly distinguishes some of these properties as constitutive of the concept of the electron. Could one impose such a distinction and on what grounds? I can think of two alternatives, none of which seems adequate. First, one could argue that only those properties that are relevant to the classification of an entity are part of the corresponding concept. For example, only the subset of the properties of electrons that enables us to distinguish them from other elementary particles should be considered part of the concept of the electron. This recipe has the undesirable consequence that a concept may change if the domain of classification is enriched. Consider, again, the case of electrons. Until the early 1910s there was no need to distinguish them from other charged particles, because they were the only particles postulated. When the concept of the proton was introduced, however, the electron’s negative charge, or the small value of its mass became crucial for distinguishing it from a proton. Should we say that with the postulation of protons the concept of the electron changed? Would someone who learned the concept before the proposal of protons learn something different from someone who learned about electrons after that proposal? Second, one could argue that only those properties that are used for identifying the reference of a term belong to the associated concept. These are properties, as I will suggest below, which enable the identification of an entity in an experimental situation. In the case of electrons, their charge-to-mass ratio was crucial for deciding whether various phenomena (cathode rays, Zeeman effect, etc.) were their observable manifestations. Other properties, for example, the size of electrons, were not significant in that respect (see Arabatzis, 2006). The problem with this suggestion is that the properties employed for identifying the reference of a term may not suffice to convey the associated concept. The concept of the electron, for instance, is not exhausted by the electron’s charge-to-mass ratio. One has to know much more than that to know what an electron is. The view of concepts I have suggested differs both from the necessary and sufficient conditions view and from Putnam’s notion. It differs from the former in that the properties associated with an entity and the laws it is supposed to obey are neither necessary nor sufficient conditions that any such entity should fulfill. The advancement of science may very

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well eliminate several of those characteristics, reveal further properties of the entity in question, or alter the laws it is supposed to obey without thereby affecting the reference of the corresponding concept. The view of concepts I advocate also departs from Putnam’s in that it does not require the referent of a term to be a stable and independent component of the associated concept. Dropping this requirement has two advantages. First, no problem arises when a concept denotes unobservable entities with no experimental manifestations, whose reference, therefore, cannot be identified independently of a description. Second, the stability of a concept’s reference is left open and, as I will argue below, becomes an issue that has to be settled through historical work. Given the above view of concepts a realist position is compatible with the phenomenon of conceptual change to the extent that a core of the evolving concept in question has survived changes in theoretical perspective.19 Thus, histories of scientific concepts can play a seminal role in evaluating the tenability of a realist attitude toward the corresponding entities. On the one hand, a realist stance toward a particular entity would be discredited if the historical record revealed that the corresponding concept has undergone a radical transformation, which has affected even the most central features previously associated with that entity. On the other hand, if historical analysis shows that a core of the concept has remained unaffected by changes in theoretical perspective, this would be an argument (albeit not a conclusive one) for maintaining a realist position with respect to the entity under examination. Such a position would involve the inductively grounded belief that the core of properties attributed to the entity will survive changes in high-level theory. One might want to challenge the view that the survival of a conceptual core enables a realist perspective to get off the ground. In particular, one might ask for the specific criteria that privilege this set of beliefs (the “core”) over the rest of a concept that proved highly unstable. I will not attempt to respond to this line of criticism, because I think that there is another way out for the aspiring realist. In particular, there is a way to identify the reference of a theoretical concept, denoting an unobservable entity, without relying heavily on that entity’s description. Theoretical concepts are usually introduced and articulated to make sense of experimentally obtained phenomena. The concept of the electron, for instance, was initially developed to account for the behavior of cathode rays in a vacuum tube and for the magnetic splitting of spectral lines (see Arabatzis, 2006). When presented with such a concept, one has to identify the experimental situations that are taken to manifest the presence of (are causally attributed to) its referent, the corresponding unobservable entity. Thus, the experimental situations associated with a concept provide a way to track its referent. Putnam has anticipated this proposal by suggesting that we can identify a magnitude (e.g., charge) by, for example, singling it out as the magnitude which is causally responsible for certain effects (Putnam, 1975b, p. ix)

19

A similar suggestion has been defended by Putnam (1974, p. 275). For some other options that are open to the realist, see below. cf. also Psillos (1999, p. 295).

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However, he takes for granted that the reference of scientific concepts does not change over time. In my view, on the other hand, the realist intuition that (mature) science has developed against a stable ontological background has to pass the test of historical scrutiny. It is an open question whether the reference of, say, the concept of charge has remained stable. Having provided a criterion for identifying reference, the stability of the reference of a specific concept over time should be investigated historically. Historical research, thus, obtains an essential role for evaluating the tenability of a realist attitude toward the ontology of science. Furthermore, Putnam does not address a question that, I think, is important: What binds certain effects together as manifestations of a single entity? (cf. Kuhn, 1979, p. 411). Different effects can be attributed to the same entity only if they have some qualitative or quantitative features in common. First, they may share some common qualitative feature, which indicates that a single entity is “behind” all of them. For example, it was known since the 17th century that the processes of combustion, calcination (oxidation), and respiration took place only in the presence of atmospheric air. That fact was explained by the hypothesis that air was necessary for absorbing phlogiston, the entity that was given off in all those processes. Second, from the quantitative features of an experimental situation it may be possible to infer the value of some property of the hidden entity involved in it.20 If inferences from different situations converge to the same results, then this agreement binds them together as effects of the same entity. Furthermore, the value in question provides a way of identifying the entity in novel experimental situations. For example, at the turn of the 19th century several experimental phenomena (the Zeeman effect, cathode rays, -rays, etc.) were attributed to the presence and action of hypothetical charged particles. It was not a priori evident that the particles involved in all those different phenomena were the same. Physicists were led to that conclusion when it turned out that the particles that were responsible for the Zeeman effect had approximately the same charge-to-mass ratio with the particles which constituted cathode rays. From that point on, the charge-to-mass ratio of the electron functioned as a criterion for identifying it in novel experimental situations (for details see Arabatzis, 2006). To be a realist about a particular entity one has to establish the referential continuity of the corresponding concept. That is, to show that over the history of the concept the experimental situations that were assumed to be the observable manifestations of its referent have remained stable or, at least, exhibited a cumulative development (i.e., the previous set of situations associated with the concept was a subset of the current set). This expansion may happen in three ways. First, new situations may get attributed to a familiar entity, without any modification in the theoretical description of that entity. In that case the corresponding concept will remain unaffected. Second, the accommodation of new situations as manifestations of a familiar entity may require the attribution of additional properties to that entity, which, moreover, do not conflict with its previously established properties. In that case, the concept will expand in a cumulative way and its reference will remain the same. Third, the accommodation of novel situations may require the attribution of novel 20

This type of “backward” inference is sometimes called “deduction from the phenomena”. I think that this expression is misleading, because these deductions always involve additional theoretical assumptions about, for instance, the entity’s properties or the laws it obeys. See the very enlightening discussion in Worrall (2000).

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properties to the entity, which, however, are incompatible with some of its previously accepted properties. The latter will, in the process, get rejected. In that case, the concept will change in a non-cumulative fashion. Despite that change, the concept may still refer to the same entity, provided that the experimental situations previously associated with the concept continue to be attributed to its referent. As long as that “family” of experimental situations remains invariant (or expands in a cumulative fashion) a realist stance toward the entity involved in them is not threatened by changes in that entity’s description. Thus, the question whether realism is a defensible approach vis-à-vis a certain entity requires a historical reconstruction of the evolution of the corresponding concept (which requires considering the experimental situations that track its reference). Unless the historical and philosophical community undertakes successfully the enormous historiographical task of showing the referential continuity of most scientific concepts, any realist position will make sense only with respect to specific entities (local realism).

Acknowledgment I thank Aristides Baltas, Stathis Psillos, Matti Sintonen, and Stella Vosniadou for their helpful comments.

References Arabatzis, T. (2001). Can a historian of science be a scientific realist? Philosophy of Science, 68, supplement, S531–S541. Arabatzis, T. (2006). Representing electrons: A biographical approach to theoretical entities. Chicago, IL: The University of Chicago Press. Enç, B. (1976). Reference of theoretical terms. Nous, 10, 261–282. Feyerabend, P. (1962). Explanation, reduction and empiricism. In: H. Feigl, & G. Maxwell (Eds), Scientific explanation, space and time (pp. 28–97). Minnesota Studies in the Philosophy of Science (Vol. 3). Minneapolis, MN: University of Minnesota Press. Feyerabend, P. K. (1965a). Problems of empiricism. In: R. G. Colodny (Ed.), Beyond the edge of certainty (pp. 145–260). Englewood Cliffs, NJ: Prentice-Hall. Feyerabend, P. K. (1965b). On the ‘meaning’ of scientific terms. Journal of Philosophy, 62 (10), 266–274. Feyerabend, P. K. (1970). Against method: Outline of an anarchistic theory of knowledge. In: M. Radner, & S. Winokur (Eds), Analysis of theories and models of physics and psychology (pp. 17–130). Minnesota Studies in the Philosophy of Science (Vol. 4). Minneapolis, MN: University of Minnesota Press. Feyerabend, P. K. (1981). Realism, rationalism and scientific method: Philosophical papers (Vol. 1). Cambridge: Cambridge University Press. Hacking, I. (1983). Representing and intervening: Introductory topics in the philosophy of natural science. Cambridge: Cambridge University Press. Kroon, F. W. (1985). Theoretical terms and the causal view of reference. Australasian Journal of Philosophy, 63 (2), 143–166. Kuhn, T. S. (1970). The structure of scientific revolutions (2nd ed.). Chicago, IL: The University of Chicago Press.

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Kuhn, T. S. (1979). Metaphor in science. In: A. Ortony (Ed.), Metaphor and thought (pp. 409–419). Cambridge: Cambridge University Press. Kuhn, T. S. (1989). Possible worlds in history of science. In: S. Allén (Ed.), Possible worlds in humanities, arts and sciences (pp. 9–32). Berlin: Walter de Gruyter. Laudan, L. (1981). A confutation of convergent realism. Philosophy of Science, 48, 19–49. Morrison, M. (1990). Theory, intervention and realism. Synthese, 82, 1–22. Nersessian, N. J. (1984). Faraday to Einstein: Constructing meaning in scientific theories. Dordrecht, The Netherlands: Martinus Nijhoff. Nola, R. (1980). Fixing the reference of theoretical terms. Philosophy of Science, 47, 505–531. Papineau, D. (1996). Theory-dependent terms. Philosophy of Science, 63, 1–20. Psillos, S. (1999). Scientific realism: How science tracks truth. London: Routledge. Putnam, H. (1962). The analytic and the synthetic. In: H. Putnam, Philosophical papers (Vol. 2, pp. 33–69). Cambridge: Cambridge University Press. Putnam, H. (1973). Explanation and reference. In: H. Putnam, Philosophical papers (Vol. 2, pp. 196–214). Cambridge: Cambridge University Press. Putnam, H. (1974). Language and reality, In: H. Putnam, Philosophical papers (Vol. 2, pp. 272–290). Cambridge: Cambridge University Press. Putnam, H. (1975a). The meaning of ‘meaning’. In: H. Putnam, Philosophical papers (Vol. 2, pp. 215–271). Cambridge: Cambridge University Press. Putnam, H. (1975b). Mind, language and reality: Philosophical papers (Vol. 2). Cambridge: Cambridge University Press. Putnam, H. (1983). Reference and truth. In: H. Putnam, Realism and reason: Philosophical papers (Vol. 3, pp. 69–86). Cambridge: Cambridge University Press. Putnam, H. (1984). What is realism? In: J. Leplin (Ed.), Scientific realism (pp. 140–153). Berkeley, CA: University of California Press. Putnam, H. (1990). Realism with a human face. Cambridge: Harvard University Press. Shapere, D. (1966). Meaning and scientific change. In: R. G. Colodny (Ed.), Mind & cosmos: Essays in contemporary science and philosophy (pp. 41–85). Pittsburgh, PA: University of Pittsburgh Press. Shapere, D. (1982). Reason, reference, and the quest for knowledge. Philosophy of Science, 49, 1–23. Shapere, D. (1991). Leplin on essentialism. Philosophy of Science, 58, 655–677. Van Fraassen, B. C. (2002). The empirical stance. New Haven, CT: Yale University Press. Worrall, J. (2000). The scope, limits, and distinctiveness of the method of ‘deduction from the phenomena’: Some lessons from Newton’s ‘demonstrations’ in optics. British Journal for the Philosophy of Science, 51, 45–80.

Chapter 6

Background ‘Assumptions’ and the Grammar of Conceptual Change: Rescuing Kuhn by Means of Wittgenstein Aristides Baltas Introduction It has been variously noticed but mostly not discussed at the length required that no scientific concept and no scientific theory can work only at the explicit level. Since no inquiry can start from the zero point of an absolute beginning, all concepts and all theories, even the most formalized or highly counterintuitive, cannot but relate to some aspects of common wisdom and hence cannot but rely on an amorphous plethora of assessments, appraisals, commitments, connotations, prejudgements and so forth that are taken for granted one way or another. That they are taken for granted means that the process of inquiry has not come to dispute their established roles and hence it has allowed them to continue performing their work more or less silently, inexpressibly, hiding their function from view. Following standard terminology, we can say that such assessments, appraisals, commitments etc. form the inquiry’s background. The ‘elements’ of such background have been variously called: among other things, they have been named “idola” by Bacon, “hidden lemmas” by Lakatos, “natural interpretations” by Feyerabend, “ontological commitments” by Laudan, ingredients of “practical ideology” by Althusser, pieces of “tacit knowledge” by Polanyi, or “historically changeable Kantian categories” by Kuhn. Let us give them a name in our turn and call them, for reasons that will appear in a moment, background ‘assumptions’ with quotes. The present investigation starts by endorsing Kuhn’s insight that the unavoidable existence of such background ‘assumptions’ offers an additional argument as to why the notion of paradigm — in the sense of “disciplinary matrix” — should replace the traditional notion of theory: a notion richer than “theory” is required if we are to take into account factors instrumental for the functioning of scientific concepts but not involved in the explicit level. We can usher Wittgenstein into the picture if we try to move a step further in our efforts to understand the exact role played by such ‘assumptions’. In §4.002 of the Tractatus LogicoPhilosophicus, Wittgenstein (1986) talks of the “enormously complicated silent adjustments

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(Abmachungen)” at work in our understanding of everyday language. And in his On Certainty (Wittgenstein, 1969) he returns, I take it, to the same idea talking now of the “hinges” that “have to stay put” for the door of understanding to move (pp. 341–343). Without entering into a Wittgenstein exegesis, I just take the background ‘assumptions’ in question as simply another name for either these “silent adjustments” or those “hinges”. This is to say that I consider them as mutely adjusting the usage of the concepts at work in any given language game so as to assure their legitimate sense. In other words, these background ‘assumptions’ are the “hinges” sustaining the concept’s manifest meaning by noiselessly dictating the conditions under which it makes sense; it is they that make up the latent grammatical conditions of the concept’s meaningfulness in the corresponding context of its use. What follows is an attempt to draw the implications of this idea for conceptual change.1 It will be shown that Kuhn’s overall account can, with some qualifications, be rescued from the charges of idealism and relativism while some implications for education will also be drawn.

Background ‘Assumptions’ and Paradigm Change Since on most views the meaning of a concept derives from the propositions engaging it, it is to propositions we have to turn in order to assess concretely the role of background ‘assumptions’. Thus some such ‘assumptions’ may render a proposition’s truth perfectly obvious while hiding themselves and the role they are silently playing in doing this behind the glare, so to speak, of that very obviousness. For example, on some such ‘assumptions’, the earth cannot but be immobile for anybody possessing her senses. On the other hand, some other background ‘assumptions’ may lie well concealed beneath our taking another proposition as totally inconceivable. For example, again on some background ‘assumptions’, no action at a distance can be conceived for how can bodies react instantaneously to the presence of another faraway body when, in addition, there is nothing in between? The development of classical mechanics has unearthed the ‘assumptions’ lying silently beneath the above propositions. And it has done that in a way making both this old obviousness and that old inconceivability appear today as astonishingly short sighted to almost everybody, that is, to all those having successfully gone through an elementary physics course. Teachers of physics are well aware of this, for overcoming the obstacles to such changes in understanding provides their daily bread. Moreover, such ‘assumptions’ may also work under a tautology, a contradiction, a deductive consequence or an analytic truth, mutely assuring the logical status of the propositions concerned while again hiding themselves and their role from view. Putnam’s (2000) example is that somebody can be either naked or not naked but only on the silent background ‘assumption’ that she is not wearing a net. It is well known that Lakatos’s (1976) “hidden lemmas” mess up rigorous deductive chains while it is analytically the case that the earth cannot be a planet, if the concept “planet” is indeed defined as referring to celestial bodies 1

What follows is borrowed heavily from Baltas (2004). In that paper I focused on radical scientific discovery, while here I shift the emphasis toward the concomitant radical conceptual change. In addition, I try to clarify some formulations and forward a few remarks intending to open the discussion along different paths.

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moving around the earth. All kinds of examples can be added, implicating everyday or highly elaborated concepts. As Wittgenstein (1969) has shown in his On Certainty, the ‘assumptions’ in question are not proper assumptions — hence the quotes. They are neither a priori and indubitable nor a posteriori and open to doubt. In normal circumstances,2 the role they are playing remains veiled and they cannot be moved around in the space of justifications. This is another way of saying that they are the “hinges” that have to stay put if the inquiry is to proceed with its questions, its reasons and its doubts. And it is in this sense that they constitute the latent quasi-logical, that is, precisely, the grammatical, conditions allowing the concepts they sustain to have the meaning they do. In forming the latent grammatical conditions allowing the concepts involved in a scientific inquiry to make sense, these ‘assumptions’ underwrite the “natural interpretation” (Feyerabend’s way of referring to their function) of the corresponding conceptual system and, by means of it, everything this interpretation determines as, for instance, the sense in which the corresponding experimental transactions and their results are to be taken. Thus they are involved in all the pictures, the analogies, the metaphors etc., which scientists are based on in order to understand the concepts their own work produces, and to communicate their results. In this way they assure the overall grammatical consistency of the process of inquiry, channeling it away from the ungrammatical. Given this, we can take an additional step that Wittgenstein, I believe, fully licences although he does not explicitly take (since he is not particularly concerned with conceptual change) and states that while channeling the investigation away from the ungrammatical, the very same background ‘assumptions’ form the meaning bounds of the concepts implicated thus surreptitiously closing the corresponding horizon. On this basis, a paradigm change can be understood as centred on the disclosure of such an ‘assumption’ (or a set thereof) and to the effects of such a disclosure. The pattern is as follows: First, a process of inquiry stumbles on a Kuhnian “anomaly” which, in the most severe cases, is a total deadlock or contradiction — what confronts the scientists implicated makes no sense to them and hence they find themselves confronting a complete impasse. On the point above, we can say that, one, such a deadlock or contradiction surfaces when the grammatical bounds of the concepts involved tend to be trespassed and that, two, it is the background ‘assumptions’ implicated that silently determine these grammatical bounds, since their function is precisely to steer the process of inquiry clear of the ungrammatical. Hence it is they that silently determine what is a deadlock or contradiction in the given context. Given this, the second step of paradigm change amounts to the illumination of the background ‘assumption’ at issue and to its emergence from out of the background. Such a disclosure transgresses the bounds in question, amounting thus to a leap into the ungrammatical for whatever concerns the old paradigm, and retraces accordingly — both widens and modifies — the grammatical space available to the inquiry. This becomes the grammatical space underwriting the novel paradigm. Thus an episode of paradigm change turns an ‘assumption’ that, as we have said, is formlessly taken along as a matter of course and to which, accordingly, questions could

2

For reasons to appear later, “normal” here should be taken as pointing toward Kuhn’s “normal science”.

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not be addressed (i.e., an ‘assumption’ with quotes) into a distinct proposition that can be doubted and thence conceptually and experimentally examined (i.e., a proper assumption without quotes). This proposition thus becomes open to rejection, revision, justification and so forth.3 It is these new possibilities for the development of the process of inquiry that activate the novel grammatical space opened by the disclosure in question and it is in this way that the novel paradigm becomes articulated. Thereby, the conceptual system concerned is itself not only enriched but also transformed substantially.4 We have a case of conceptual change. We should stress that the disclosure of a background ‘assumption’ accomplishes when it happens to two distinct things. On the one hand, it clears, as we have just said, novel grammatical space. As the horizon of inquiry is no longer closed by the dumb existence of this ‘assumption’, new avenues of research are opened, new questions are asked and new answers are given, the ones, precisely, that deploy the novel paradigm. On the other hand, such a disclosure creates a novel vantage point from where the preceding state of the investigation can be looked at anew. This is a vantage point whereby the anomalies whose unaccountable existence had induced the disclosure in the first place appear as misconstruals due to the work that the disclosed ‘assumption’ had been silently performing. Hence reasons can be adduced post hoc for the change that has occurred, reasons that appear as correcting previous inadvertences or oversights and hence reasons that tend to assess this change as invariably progressive. The spontaneous whiggism of practising scientists, rehearsed at length in the literature, is based precisely on this.

Examples of Conceptual Change Although interesting debates have been taking place on whether the ideas of conceptual change, paradigm change and revolution can apply to mathematics,5 with Wittgenstein apparently having laid the issue to rest in advance for it would have been misleadingly formulated,6 I believe that examples from mathematics do help in clarifying what is here at stake. Accordingly, my first example will be Cantor’s theory of the infinite, understood as a deep conceptual change concerning enumeration. We know that Cantor’s theory of sets is the result of his efforts to harness mathematically the infinite while the necessary ground was opened by Cantor’s famous definition: a set is of infinite cardinality (Cantor’s novel concept whose meaning widens and modifies what the old concept of enumeration used to cover) if and only if its elements can be put into one-to-one correspondence with the elements of one of its proper subsets. What 3

And not just to a simple dismissal and replacement, as would have happened, for example, in the ideological change involved in a religious conversion. This distinction opens various important issues, which I cannot enter into here. 4 No scientific concept can perform its function in isolation from the other concepts (Baltas, 1988). Hence background ‘assumptions’ form the invisible grammatical ‘glue’ allowing the conceptual system to be ‘naturally’ understood (see Baltas, 2004). 5 An interesting volume on the issue is Gillies (1992). 6 For Wittgenstein’s overall conception of mathematics, see Shanker (1987). The author, I take it, leaves Wittgenstein’s would-be position on paradigm change in mathematics interestingly inconclusive.

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relates Cantor’s definition to what we have been saying is the fact that our author, before offering his definition, was up against a total deadlock, which, in hindsight, we can name “the anomaly of enumeration”. On the one hand, since the set, say, of even numbers is a proper subset of the set of natural numbers, it obviously contains ‘fewer’ elements while, obviously again, the two contain an ‘equal’ number of elements, since these can be put into one-to-one correspondence. What Cantor did in order to escape the anomaly can then be considered very ‘simple’: he did not try to resolve or in any way overcome the contradiction but merely turned the very deadlock into a definition! Let us try spelling this out. The deadlock Cantor encountered was determined as that by some background ‘assumption’ working under the time-honoured conception of the extremely close connection (if not identity) between the part/whole and the smaller/greater relations. Within the then emerging language of set theory, the first relation is defined like this: a “proper subset” of a given set is one included in that set without exhausting it; a non-empty remainder is left. Bringing in the second relation, we can then hold that the set, say, of even numbers, is smaller, and cannot be but smaller, than the set of natural numbers, since the latter includes the odd numbers as well. On the other hand, within the same language and again by definition, sets have elements, that is, they are the sets of their elements. And another time-honoured idea is that the even numbers can be put into one-to-one correspondence with the natural numbers: the even number 2n corresponds one-to-one to the natural number n. This is exactly what gives rise to the “anomaly of enumeration”, for then the set of even numbers has the same number of elements as the set of natural numbers although a moment ago we demonstrated that the former is smaller than the latter — How else could “smaller” be cashed in here, if it is only elements that sets can have? — and hence it cannot but have fewer elements! Cantor’s change of paradigm, the deep conceptual change he effected, amounts then to his accepting this contradiction and go on defining infinite sets as precisely those for which both of the contradiction’s horns hold simultaneously. The well-entrenched background ‘assumption’ disclosed through this move is that always and without exception, analytically so to speak, a collection of items that is smaller than another, one forming a proper part of its corresponding whole, cannot but include fewer items. Cantor’s definition opens up the grammatical space where it is not necessarily the case that the relation “properly included in” coincides with the relation “has fewer elements than”: for finite sets the two relations are equivalent but for infinite sets they are not. Giving room to this distinction — a distinction literally unthinkable before — sets up this novel grammatical space wherein the old contradiction becomes automatically resolved. The articulation of the new paradigm of enumeration, that is, Cantor’s theory of cardinality, is nothing but the exploitation of this novel grammatical space through the deployment of that distinction’s far-reaching consequences. A radical conceptual change in mathematics has thus occurred. We have to note, however, that after all is said and done, after the radical novelty of Cantor’s proposal has become absorbed and assimilated into the common wisdom of logicians and mathematicians, his move can appear only as simple and natural: Why expect that the infinite should obey the same “laws” as the finite? Post festum, but only post festum, the question is, of course, totally justified in its disarming candidness: Why indeed? My second example proposes a summary reconstruction of Bohr’s theory of the atom as a decisive step toward the formulation of quantum mechanics. Schematically speaking,

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Bohr’s revolutionary proposal may be reconstructed as follows. For reasons too complex to enter into here, Bohr could not, or would not, question either the electromagnetic theory of his time or the planetary model of the atom. But the two together demonstratively lead to an unstable atom: the accelerated motion of the revolving electrons makes them radiate and thus forces them to fall onto the nucleus. If we deem unstable atoms unacceptable at any price, we have again an anomaly in the form of a contradiction. Bohr’s overcoming the anomaly can then be considered as similar in structure to Cantor’s. He refuses succumbing to the contradiction but goes forward by accepting both its horns in the form of a positive definition of an entirely new physical concept: atoms in the “ground state” do not radiate and hence they are stable. But they do not radiate and are stable only by fiat, that is, merely because they are defined in precisely this way by Bohr.7 Since, now, the trajectories of the electrons in this newly defined state do not obey the classical electromagnetic laws, these can be no ‘real’ trajectories’, whatever this might mean or imply. In other words, Bohr’s new definition opens up the novel grammatical space capable of hosting the new distinction his definition comes up with, namely that between moving ‘classical’ bodies, which do emit radiation when following particular trajectories, and newly defined “moving quantum bodies” that do not. Distinguishing in this way ‘classical’ from quantum motion unavoidably puts under fire the fundamental concept of “trajectory”, leading thereby, grammatically if not historically, to the follow-up distinction between ‘classical’ trajectories, where position and velocity (or momentum) can be determined simultaneously with arbitrary precision, and quantum ‘trajectories’ where they cannot. Given that “trajectory” is, by definition, the sequence in time of the determined positions and velocities (the tangents of the corresponding curve) of a moving body, this further distinction is tantamount to the disclosure and the subsequent questioning of a quasi-elemental background ‘assumption’: it is not necessarily the case, as it had been silently ‘assumed’ up to then, that the trajectories of all physical bodies do possess this, their defining, property. Although this makes quantum ‘trajectories’ impossible to strictly represent in space, cloud chambers and other experimental artefacts demonstrate that the concept should not be altogether jettisoned and hence that classical and quantum concepts bear very complex relations to one another. Concurrently, the disclosure of this ‘assumption’ opens the grammatical connection between quantum ‘trajectories’ and the particle/wave duality thus placing the articulation of the new paradigm of quantum mechanics well on its track. The radical conceptual change associated with quantum mechanics finds here one of its roots. But again, after Bohr’s initial move and its various grammatical implications have become assimilated into the common wisdom of at least the practising physicists, the kind of candid question we encountered in the case of Cantor inevitably surfaces: Why expect that the exceedingly small should obey the laws governing the medium-sized bodies of our everyday experience? The whole practice of teaching introductory quantum mechanics is founded on the apparent naturalness of precisely this question. 7

Evidently, this simple picture does not intend to exhaust Bohr’s logical, conceptual or experimental motivations for proposing his model of the atom; it does not seek to trace the history of this proposal, be it even in such a summary fashion; nor does it aim at reconstructing Bohr’s way of thinking on the issue. It only wants to suggest that paradigm change and conceptual change are concomitant with deep grammatical changes, the understanding of which may help us coming to grips with the philosophical issues involved.

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My last example concerns the other major revolution in the physics of the 20th century, namely relativity theory. It is well known that Maxwell’s electromagnetic theory achieved, among other things, the unification of electricity, magnetism and optics. One of the more salient features of that achievement was the prediction of electromagnetic waves, the existence of which was later confirmed by Hertz. Given that, classically speaking, the concept “wave” is defined as the propagation of a medium’s disturbances, within the grammatical space supporting this definition — that is, within the grammatical space available to classical physics — the concept “wave” is analytically related to the existence of a material medium. Therefore such a medium, called “the ether” after a long tradition, could not but exist and it was the physicists’ duty to pin down its properties. However, all efforts to such effect failed systematically, the ether’s character becoming thereby all the more elusive and mysterious. A crisis situation thus settled in wherein the ether started to appear as the source of a whole set of anomalies. It was in this context that Einstein published his “On the Electromagnetics of Moving Bodies”, introducing the special theory of relativity. Speaking very roughly, the theory of relativity can be seen ex post facto as resulting from the disclosure of some of the background ‘assumptions’ silently underlying the classical concepts of motion and the concomitant postulation — again a case of overcoming an anomaly through a novel definition — of a new kind of entity, different from both particles and media, the electromagnetic field.8 The disclosure of such ‘assumptions’ opens the grammatical space permitting the existence of ‘non-classical’ waves that can propagate without any material medium being disturbed, a space wherein the ether thus becomes indeed “superfluous” (Einstein’s word). And this implies that the anomalies associated with its purported existence disappear for good. But, on the other hand, since such ‘nonclassical’ waves propagate in vacuum, that is, in absence of the material medium whose disturbance they are, amounting thus to a ‘disturbance’ of nothing that ‘propagates’ in itself (!), the disclosure in question transforms radically the whole classical conception of motion. It is this conceptual change that forms the basis for the deployment of the new paradigm of relativistic physics. And once again, and always post festum, we can repeat with both the practising physicists and the teachers of modern physics that follow in their wake: Why expect that the exceedingly fast should obey the laws governing the kinds of motion we encounter daily? We see that in this example as well the spontaneous whiggism of both physicists and teachers of physics finds room to thrive.

Rescuing Kuhn In the above sections we have been using the term “conceptual change” rather loosely. However, much of the criticism against Kuhn has revolved around the exact meaning of this term, especially as regards concepts that keep the same name across paradigm change (e.g., classical and relativistic mass). We are therefore obliged to ask: What exactly do we 8

This is not historically accurate in the details. Remaining however close to the historical facts, it permits us to bring out the grammatical structure of Einstein’s breakthrough.

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mean when we say that such concepts change? Is there a kind of continuity that can justify their keeping the same name or, as Kuhn has mostly been taken to imply, does their incommensurability make their meanings totally unconnected and hence their keeping the same name merely a matter of convention? Or is the very idea of meaning change and hence of incommensurability a non-issue? To try answering, we should have a close look on how what we have been saying bears on Kuhn’s ideas on paradigm change. We will attest then that the main charges against Kuhn’s account can be dispelled. Idealism First, from the psychological point of view, the disclosure of a background ‘assumption’ does amount, as Kuhn would have it, to a “Eureka!” experience inducing a gestalt shift that makes one see the world under a new light. Given that the main ingredients of paradigms cannot but be ideas, Kuhn’s maintaining that people residing in different paradigms “live in different worlds” seems to imply that it is ideas in general that determine the world and hence are primary in respect to it. As this is the core thesis of idealism, Kuhn has been charged, apparently with good reasons, for offering an idealist account of science. However, on a closer look, this charge is totally misguided. For, as given above, Kuhn’s account of paradigm change can only imply that it is the world, not only as fully independent of ideas but also as absolutely primary in respect to them, that can be the only agency responsible for anomalies, that is for the deadlocks or contradictions on which paradigms stumble. This is to say that it is only the world that can display the inadequacy of a conceptual system, the fact that such a system does not possess the resources for coping with the relevant deadlocks or contradictions, and hence for accounting for the world in the corresponding respect. It is only the world, as fully independent of the ideas articulating the various paradigms, which, by resisting them, can force us into accepting that those ideas fall short of it and thence are subordinate to it and fully reliant upon it, for better or for worse. Ideas are at peace with the world, allowing scientific, or even naïve, realism to appear as compelling, only as long as no dire anomalies surface to exert their full pressure.9 It follows that Kuhn’s position on paradigm change not only cannot be charged as idealist but, on the contrary, that it is a position vindicating with a vengeance the sovereignty of the world in respect to ideas, that is, the absolute primacy that practically all forms of naturalism demand of it. Communication Breakdown To go on, let us look at the phenomenon Kuhn calls “communication breakdown”. On the present view, communication between scientists invariably breaks down in the relevant respect when these reside in succeeding paradigms. Why this is the case should by now be obvious. The scientist residing in the old paradigm has not undergone the pertinent “Eureka!” experience and still holds fast to the corresponding background ‘assumption’, while remaining unaware that a questionable assumption is involved in the first place. 9

Such resistance of the world to the theories or paradigms trying to account for its various aspects should form, at least according to me, the starting point for a viable realist position, a position which, for reasons related to the above argument, I call “negative realism”. I have tried to explore this idea in Baltas (1997). See, however, Psillos’s (2002) interesting and still unanswered objections.

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Given the way we explicated the function of such an ‘assumption’, it follows that continuing to hold fast to it implies that the scientist implicated cannot understand how it could be possibly questioned. Such questioning is literally unthinkable from the old vantage point, as it would amount, for example, to the destruction of an analytic relation between concepts, that is, of a relation that is inconceivable otherwise. Accordingly, the scientist still residing in the old paradigm can literally not understand the effects of the disclosure in question and hence neither the concepts the colleague residing in the new paradigm is using nor much of her way of talking. It should be clear, nevertheless, that no kind of simple-mindedness is involved here. To say it provocatively, Lorentz’s, say, continuing to admit till the end that he never came to understand relativity theory may even make him more ‘rational’ than Einstein. The old man had precise, well-articulated reasons — logical, conceptual, empirical and historical — for clinging, even unawares, to the background ‘assumptions’ assuring the meaning of the old concepts while his young colleague only made a senseless leap into the ungrammatical. That Lorentz never underwent the appropriate “Eureka!” experience in no way demonstrates some kind of intellectual inferiority. In any case, the communication breakdown under consideration is no global breakdown. Although the grammatical change involved cannot but rebound to a greater or lesser extent throughout the old conceptual system (since the relevant concepts are closely interconnected), the two parties continue sharing the enormous grammatical space of their common language within which the blind spots of their communication can be circumscribed, even if very hazily. For example, both parties can agree that explaining phenomena is what they are after while remaining completely at odds, not only on what such explanations should exactly amount to but also on which are the phenomena to be explained.10 We should add that the communication situation is not totally hopeless even in respect to such ‘dialogues’. Since the disclosure of a background ‘assumption’ creates, as we have said, a novel vantage point whereby one can see what prevented this new way of looking before, one is armed with important rhetorical ammunition.11 Although such ammunition cannot compel logically those holding the old paradigm to undergo the pertinent “Eureka!” experience, it can serve to surround, so to speak, the corresponding background ‘assumption’ so as to force those opponents to realize, at least indirectly and in the long run, that they are unwittingly harbouring unwarranted biases. Our pedagogical practices when we are teaching counterintuitive theories do bear an eloquent witness to the need for exerting such violence. Asymmetry The next point we should note is that the relation between the two succeeding paradigms is not symmetrical, and this is in many senses. For one, it is not grammatically and hence epistemically indifferent within which paradigm one resides. Since the grammatical space available to the new paradigm includes the possibility of examining (of negating, of modifying, of accepting etc.) an additional assumption unavailable to the old, namely the one

10

The communication gap between Galileo and the Aristotelians, or even that between Frege and Hilbert on what a mathematical definition should amount to, are cases in point. 11 See, for example, Pera (1994).

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resulting from the disclosure of the ‘assumption’ that had been silently taken for granted, it is a grammatical space objectively (i.e., independently of the relevant beliefs and convictions) wider than that available to the old paradigm. It is crucial to underline, however, that the assessment of an objectively greater width is possible only from the vantage point determined by the new paradigm. Those clinging to the old continue to be unwittingly constrained by the background ‘assumption’ in question, to remain blind to the possibility of its interpellation and, therefore, incapable of surveying the greater width attained. Accordingly, if we keep Kuhn’s metaphor of the “gestalt switch”, we should note that, unlike the duck/rabbit case and at least thus far in our discussion, this is not a switch that allows one to go back and forth. It is a semiconducting switch, directing from the old paradigm to the new through the relevant “Eureka!” experience.12 This is to elaborate the asymmetry we noted: if we continue residing within the old paradigm, we are blocked in our understanding at least parts of what our interlocutors are talking about. In fact we even consider what they are telling us as nonsensical, since ungrammatical. If we reside within the new paradigm, we can understand both what our interlocutors are talking about and why they are talking this way. We are not blocked because, to go one step further, the grammatical space available to us can accommodate an explicit interpretation in our terms of the old conceptual system as a whole, an interpretation concomitant with the reinterpretation of the empirical phenomena involved. In the case of post-Galilean physics, where a mathematical structure always underpins the physical conceptual system thus rendering concepts and conceptual relations more precise, the interpretation in question is based on an explicit, although imperfect, translation of at least a crucial part of the old conceptual system in terms of the new.13 The case of mathematics where no empirical phenomena are involved should be treated separately and will not concern us in the remainder of the chapter. What I mean can be spelled out as follows. In the passage from the old paradigm to the new, what first14 opens up is the grammatical space tied to the particular concepts located at the core of the deadlock or contradiction on which the old paradigm stumbled in the first place. This opening up blows up and changes dramatically the meaning of these concepts. Some of them may become fully discarded (as phlogiston or caloric) while some others acquire a radically new meaning, even as their names, for reasons we will see in a moment, remain the same. The novel grammatical space is such that three closely connected things become possible within it.

12

Hence the asymmetry in question should not be identified with Kuhn’s “incommensurability”. The latter concerns the relation between the languages of the two paradigms and does allow a movement back and forth as, say, between English and French. Thus “incommensurability” can be experienced only from the vantage point of the novel paradigm and, in this sense, it presupposes the asymmetry we are discussing (see Baltas, 2004). 13 Two remarks are in order: 1. My running together “translation” and “interpretation” here against Kuhn’s (2000) careful distinction of the two terms is allowed by the asymmetry we are discussing while Kuhn has in mind only his “incommensurability”. See note above as well as Baltas (2004). 2. My formulation seems to imply that, to the extent that phenomena become ‘merely’ reinterpreted across paradigm change, they, pace Kuhn, remain per se invariant. I will try arguing below why this implication need not follow although the kind of continuity suggested by my formulation seems to me inescapable. 14 Not necessarily in the temporal or historical sense. But what we are concerned with here is only the reconstruction of episodes of paradigm change, not an accurate historical account of their occurrence.

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First, one can understand on its basis the role the disclosed ‘assumption’ had been playing in assuring the coherence and self-consistency of the old conceptual system. Thereby that old system can be interpreted in a way making clear the reasons for both its successes and its failures in accounting for the empirical phenomena in its domain. Second and concurrently, the novel grammatical space can host a reinterpretation in the new language of the empirical phenomena that had been countenanced, and as they had been countenanced, by the old. The relevant parts and aspects of nature are being understood now in terms of the new conceptual system and of the novel grammatical space supporting it. Third, the new conceptual system can account successfully for at least some aspects of some of the phenomena (as reinterpreted) that were at the heart of the deadlock or contradiction from which the whole process started, that is, those that the old conceptual system found impossible to handle — as well as for at least some aspects of at least some phenomena (as reinterpreted) that had been considered up to then as being successfully accounted for by the old conceptual system. Continuity Given this, let us turn to the crucial question of whether some kind of continuity can be said to exist across paradigm change and of how this can be fleshed out. If, as we have said, the new paradigm resolves the deadlock or contradiction on which the old paradigm had stumbled, the question is exactly what is resolved in this way and, concomitantly, what relation the two succeeding paradigms might entertain with each other through this resolution. To answer, we have to distinguish the relation between concepts at the two ends of the passage in question from the corresponding relation between phenomena. Let us start from concepts. We have said that paradigm change occurs when a new positive definition is proposed that trespasses the grammatical bounds of some old concept(s), instituting thereby the grammatical space underlying the new paradigm. As our examples have shown, the newly defined concept (cardinal number, quantum trajectory, electromagnetic field) can host a distinction between itself and the interpretation/translation in terms of the novel conceptual system of the concept(s) whose meaning bounds have been thus trespassed. If this were not the case, the newly defined concept would not bear at all on the anomaly at issue; it would be irrelevant and thus remain perfectly idle in respect to it. On the other hand, that the old concept(s) have to be interpreted/translated for becoming thus accommodated is a consequence of their impossibility of overstepping these bounds. It follows that this interpretation/translation, in being precisely an interpretation/translation, retains the memory of this impossibility. But the fact that the positive definition in question institutes a novel grammatical space, wherein the ensuing new concept functions fully grammatically in tandem with the interpretation/translation of the old, makes the old ungrammaticality disappear from sight and represses its memory.15 The survival of this 15

It follows that this ungrammaticality becomes eventually forgotten, fully covered under what the resultant grammatical space starts compelling everyone to take for granted. Thus, in the long run, the achievements of the superseded paradigm appear retrospectively as trivial, as obviously mistaken or as fully incomprehensible. It is then up to the historian of science to reveal the revolutionary character of the corresponding ungrammatical leap and to dispense intellectual justice.

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impossibility, even merely in the guise of a repressed memory, keeps the new conceptual system linked to the old, thus rendering it continuous with it in that precise sense. Going now to the phenomena implicated, we can say that, repressing the memory of this impossibility goes hand in hand with the reinterpretation of these phenomena by the new conceptual system in terms of the novel grammatical space. But in what way do we have here a reinterpretation of the phenomena exhibiting the anomaly? If we do not raise the issue we cannot but be silently assuming that these are merely reinterpreted and thus that they retain as such their identity, which inevitably begs the question. To answer, we repeat that, on our account, the appearance of a deadlock or contradiction in respect to some phenomena means only that the world manifests its resistance to the conceptual system intending to capture them. This is to say, first, that it is the force of this resistance that the ungrammatical move into the new paradigm has to overcome and, second, that the success of this move appeases this resistance by making the new conceptual system conform to the world in the relevant respect. But this can only mean that ‘something’ of the world, ‘something’ of the materiality of the phenomena concerned, is carried invariant across paradigm change.16 For, appeasing such resistance and making the new conceptual system conform to the world can only mean that this new system manages to capture adequately the very same ‘something’ that resisted its capture by the old conceptual system. In other words, the same ‘something’ is being conceptualized in terms of an inadequate system at one end of the relevant passage and in terms of a correspondingly adequate one at the other end.17 The novel grammatical space is itself stamped by the persistence of this ‘something’: the fact that what is at issue is an impossibility having concerned the phenomena exhibiting the anomaly, an impossibility; moreover, that continues to survive — even only as a repressed memory — within the grammatical space having overcome the resistance they had exhibited marks precisely the elusive passage of the same ‘something’ across the ungrammatical leap. We can add that the fact that this ‘something’ remains invariant is faithfully reflected in the pertinent “Eureka!” experience, for this is an experience that cannot engage but a single thing at both its ends: after having undergone it, we understand exactly what we were incapable of understanding before. In one word, it is the invariance of this ‘something’ across paradigm change that allows maintaining that the new conceptual system indeed reinterprets the phenomena exhibiting the Kuhnian anomaly. It is precisely this that makes the new paradigm, not merely different from the old, but the one that succeeds it. 16

This is ‘something’ that any brand of naturalism would minimally require, if the world is to be independent of ideas and primary in respect to them. For an argument why this ‘something’ need not amount to a Kantian “thing in itself”, see Baltas (1997). 17 I should note that no “convergence to the truth” is implied by such a passage from the inadequate to the adequate. As we will explicate below, a kind of progress is involved, but it amounts only to a relation between two grammatical spaces while the world remains, so to speak, gravely silent in the background, permanently refusing to pronounce itself otherwise than through resisting inadequate conceptual systems. This is to say that, as concerns the world, the kind of progress involved is ‘negative’: the new conceptual system merely shows that the world in the relevant respect is not in fact as the old system takes it to be. And that is that. The novel grammatical space inevitably carries background ‘assumptions’ of its own that another, perhaps much more radical, paradigm change may eventually disclose, showing thereby that the new conceptual system too is grossly inadequate. For more details see below as well as Baltas (1997).

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The grounds of this continuity provide also the reasons why it is not only historically but also grammatically required that the names of the old concepts be preserved in the novel grammatical space as far as its continuity with the old calls for: these names constitute the necessary grammatical reminder of this continuity, that is, of its inescapability. We have to acknowledge, however, that such continuity through interpretation/translation and reinterpretation makes the phenomena captured by the old conceptual system appear as ‘at bottom’ identical with their reinterpretation in the new system. And such a form of continuity is close enough to a continuity of reference to almost compel us, together with most practising scientists, to say that the two paradigms talk about the same phenomena tout court. Yet on the above this cannot be strictly correct. If, as we have said, the world manifests itself only through its resistance to our conceptual systems, then the phenomena we are considering cannot provide a firm hold to the relation of reference, at least as that relation has been standardly discussed. Viewed from another angle, this is to say — as Kuhn too would have it — that no phenomena can be given to our experience and to our perception independently of the concepts capturing them,18 for, to use received terminology, they are always, in a sense, theory-laden.19 In our terms, this simply means that the ‘something’ of their materiality that remains invariant across paradigm change cannot be captured independently of some conceptual system or other, with all the corresponding background ‘assumptions’ inevitably entering into play and the attendant threat of an anomaly always lurking in the shadows. The above entail that the continuity of succeeding paradigms through interpretation/ translation and reinterpretation is always a vital feature of paradigm change, accomplishing at least two purposes. First, such continuity is essential for justifying the use of the old conceptual system when translatable counterparts exist or when the corresponding limit conditions are satisfied:20 after relativity theory and/or quantum mechanics became the reigning paradigms, classical mechanics can be used without many scientific or philosophical qualms (which does not mean straightforwardly or unproblematically), thanks to the continuity assuring the corresponding interpretation/translation and reinterpretation. Second, the same continuity allows assessing the role the disclosed ‘assumption’ had been playing within the old conceptual system and hence accounting for the successes and failures of the old paradigm as well as for the coming to being of its successor. As specifically in the case of physics, this translation saves — sanctions21 — as limiting cases the translatable, precisely, concepts and conceptual relations of the old paradigm, its existence permits us to understand how that paradigm could have been successful in the relevant respects despite the role being played by the ‘assumption’ in question. Concurrently, that not all parts of the old conceptual system are saved through this translation22 permits us to

18

As McDowell (1994) phrases it, the “conceptual goes all the way down”. For an interesting way to distinguish the conceptual from the theory-laden that is consonant with the present account and on the basis of which the continuity we are talking about can perhaps be spelled out more fully, see Pagondiotis (2004). 20 For details see Baltas (2004). 21 The use of this term intends an allusion to Bachelard’s (1975) “histoire sanctionnée”. It lies outside the scope of the present paper to develop the proximity (and the differences) between the approaches of Bachelard and of Kuhn. 22 Bachelard’s (1975) “histoire périmée” finds its correspondent here. 19

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understand the ways in which the horizon of inquiry had been closed by the silent work performed by the same ‘assumption’. In particular, we can understand why the new paradigm could not have come into being without the ungrammatical leap represented by the novel definition, a kind of leap; however, that preserves the continuity between the two paradigms precisely through the interpretation/translation and reinterpretation at issue. Surreptitiously shifting this logical “without” to a temporal “before” obliterates such imperfection in translation as well as other telling differences between two succeeding paradigms, and makes us perceive the old paradigm tout court as either totally discardable or as just a limiting case of the new. As we have been repeating, this misconception lies at the roots of the ways we tend to teach modern science while simultaneously feeding the spontaneous whiggism of practising scientists. Thus, such continuity spells out the ways in which the grammatical space attached to the new paradigm is indeed objectively wider than that attached to the old. Kuhn’s alleged relativism can be offset at precisely this point. Relativism First of all it is imperative to emphasize that attaining this objectively greater width is an irreversible achievement. Once a new grammatical possibility becomes available through the disclosure of a background ‘assumption’ and once the corresponding implications have become domesticated through the successes of the new paradigm, scientific reason cannot, grammatically if not historically, wipe out this possibility, forget its existence, and act as if it were not there. Forsaking the “Eureka!” experience, retracing the steps of paradigm change and making the disclosed assumption re-enter the amorphous background supporting grammatically the old conceptual system is obviously impossible. This is to say that the route for regaining the lost innocence, the route leading back to conceiving the old concepts strictly the way they were being conceived before the disclosure, is blocked. At least part of the grammatical glue assuring the coherence and self-consistency of the old conceptual system has been found out and, to that extent, it has lost for good the corresponding gripping power. The two succeeding paradigms are asymmetrical also in the sense that the old paradigm has become definitively superseded. It follows that no room is left for an extra-paradigmatic vantage point, that is, a point from where one could impartially assess the relative merits and demerits of paradigms, biased by none. As Kuhn would have it, nothing at all can be conceived outside a paradigm and, as McDowell (1994) would formulate it, no view “from sideways on” can ever be available: we can reason, only in terms of a paradigm, on the appropriate more general understanding of the term. But if we always find ourselves within a paradigm without the possibility of acceding to extra-paradigmatic neutral ground, it seems to follow that a paradigm is as good as any other. And if that is indeed the case, rationality receives a lethal blow while all kinds of relativist positions, from social constructivism to various forms of alleged ‘post-modernism’, find a privileged soil to thrive on: if rationality cannot be saved even within science, then it cannot be saved anywhere and hence we are free to choose the paradigm that suits best our conventions, our interests, or even our whims. However, this is a non sequitur and the corresponding charge is, once again, totally misguided. Once we accept that we cannot access God’s standpoint, where no latent

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grammatical conditions need exist and all background ‘assumptions’ without exception lay bare to the gazing, no peril whatsoever to rationality can be forthcoming and no kind of relativism can be implied from our account. The change from a paradigm to its successor is as rational a procedure as any in human thought and action. The catchword is what we have been repeating almost ad nauseam above: there is an inherent asymmetry between succeeding paradigms in the sense that the grammatical space available to the new paradigm is objectively wider than that available to the old. That the new paradigm supersedes definitively the old means that, after having undergone the “Eureka!” experience, we necessarily reside within the novel grammatical space with no possibility of going back and hence that no real choice between the two paradigms can be at issue.23 In that precise sense, objectivity need not imply neutrality in respect to paradigms and the attendant impartiality; their tie may appear unbreakable only from the point of view of God. Scientific Progress But if that is the case, then our account appears as open to the opposite charge: Do we endorse cumulative progress? The answer is no, for that the old paradigm has become definitively superseded does not imply that the new paradigm can do better than its predecessor in respect to all the (reinterpretations of the) empirical issues that the old had confronted, successfully or not, on its own. After the ungrammatical leap represented by the novel definition and after the initial empirical successes securing the irreversible character of the expansion in grammatical space, the new paradigm may indeed solve (reinterpretations of) some outstanding old puzzles, it may dissolve and completely discard through the corresponding interpretation/ translation of some others, or it may stumble in its efforts to solve (the reinterpretations of) some of the puzzles that the old paradigm had successfully tackled in its own terms. It may even prove the case that, apart from the inaugurating initial successes, no substantial claim of the new paradigm can survive future empirical trials or that the inaugurating successes themselves need to be reinterpreted. A new paradigm change may be called for, a paradigm change that will disclose additional background ‘assumptions’, those that will appear, ex post facto and from the resulting novel point of view, as responsible for the trouble this new paradigm had been encountering. This implies that there is no paradigm-free way to assess the successes and failures of paradigms so as to come up with the conclusion that the new paradigm performs unqualifiedly better than the old. For the same reason as above, namely that God’s standpoint is unavailable, no neutral scale can exist, as the idea of cumulative progress would have it, on which to place such successes and failures, count them impartially and draw the balance. Nevertheless, progress does occur. As we have been repeating, the novel grammatical space is objectively

23

Theory comparison, theory choice, incommensurability etc. can become issues only after the novel paradigm has been established and from its own vantage point (see Baltas, 2004).

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wider and the old paradigm has become definitively superseded. Progress concomitant with paradigm change amounts ‘merely’ to this.

Concluding Remarks: Implications for Education We have said above that the definition instituting the grammatical space of the new paradigm can be perceived only as senseless from the vantage point of the old paradigm. This is inescapable and most cases of the relevant communication breakdown rehearsed in the literature are due precisely to this. How then, can we teach the conceptual ingredients of the new paradigm to those still clinging to the old? How can we bridge the corresponding gap in understanding? The answer that follows from the above is simple: we can only start from openly acknowledging that what we are about to teach to our students cannot but sound senseless to them and go on by judiciously employing appropriate pictures and metaphors that do not hide but, on the contrary, bring out the nonsensical, precisely, character of the basic concepts at issue. Here is an example. Stephen Weinberg, Nobel laureate in physics and grand master of exposition, in addressing himself (Weinberg, 1977) to those who do not know relativity theory, ‘defines’ the electromagnetic field as “a taut membrane without the membrane”. This is a self-destructing and hence nonsensical ‘definition’, openly admitting that the concept of the electromagnetic field can make no sense at all from the point of view of classical physics. At the same time, however, this ‘definition’ proposes a kind of ‘picture’ intending to ‘depict’ the only kind of ‘meaning’ the concept can have in the context of the old paradigm. Indeed, from the vantage point of classical physics, the electromagnetic field can be conceived, as we have said, only as the disturbances of nothing that propagate in themselves, that is, only in terms of this or some equivalent nonsensical ‘picture’ or metaphor. We seem therefore entitled to generalize: teaching effectively the ingredients of a new paradigm cannot but be based on the astute use of nonsense. And this brings me to my last remark. Volumes of inconclusive discussions have been devoted to the conception of nonsense that Wittgenstein (1986) promotes in the Tractatus Logico-Philosophicus. Although this is certainly not the place to launch yet another line of approach to the issue, the above allows me to forward, very tentatively, the following. If we take the famous penultimate paragraph of that work (§ 6.54: “My propositions are elucidatory in this way: he who understands me finally recognizes them as senseless...”) to mean basically that Wittgenstein uses nonsense as a basic tool for making his views understood by his intended interlocutors, then nonsense does amount for him too to a kind of pedagogical device. And this seems to imply that paradigm change need not be that foreign to him either. If this simple observation can be developed into an account that holds water, then it would perhaps be worthwhile trying to read the Tractatus Logico-Philosophicus retrospectively on the basis of the Kuhnian arsenal as we strived to clarify it here.24 Some more light might be shed then on both Wittgenstein’s and Kuhn’s ideas, ideas that remain important for they continue to inform our views of language, of science and of the world. 24

See in this respect Baltas (2005) in Greek, with an English draft available on request.

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References Bachelard, G. (1975). Le nouvel esprit scientifique. Paris: PUF. Baltas, A. (1988). On the structure of physics as a science. In: D. Batens, & J. P. van Bendegem (Eds), Theory and experiment, recent insights and new perspectives on their relation (pp. 207–226). Dordrecht, The Netherlands: Reidel Publishing Co. Baltas, A. (1997). Constraints and resistance: Stating a case for negative realism. In: E. Agazzi (Ed.), Realism and quantum physics (pp. 74–96). Amsterdam-Atlanta, GA: Rodopi. Baltas, A. (2004). On the grammatical aspects of radical scientific discovery. Philosophia Scientia, 8(1), 169–201. Baltas, A. (2005). On paradigms in philosophy: Showing, elucidation, nonsense, philosophy, silence, ethics in Wittgenstein’s Tractatus”. In: K. Ierodiakonou (Ed.), The use of paradigms in philosophy. Athens: Ekkremes (in Greek). Gillies, D. (Ed.). (1992). Revolutions in mathematics. Oxford: The Clarendon Press. Kuhn, T. S. (2000). Commensurability, compatibility, communicability. In: J. Conant, & J. Haugeland (Eds), Thomas S. Kuhn, the road since structure (pp. 33–57). Chicago, IL: Chicago University Press. Lakatos, I. (1976). Proofs and refutations. Cambridge, UK: Cambridge University Press. McDowell, J. (1994). Mind and world. Cambridge, MA: Harvard University Press. Pagondiotis, K. (2004). Can the perceptual be conceptual and non-theory-laden?. In: A. Raftopoulos (Ed.), Cognitive penetrability of perception: An interdisciplinary approach. New York: Nova Scotia Publishing Co. Pera, M. (1994). The discourses of science. Chicago, IL: Chicago University Press. Psillos, S. (2002). What is the positive content of negative realism. Neusis, 11, 85–96 (in Greek). Putnam, H. (2000). Rethinking mathematical necessity. In: A. Crary, & R. Read (Eds), The new Wittgenstein (pp. 218–231). London: Routledge. Shanker, S. G. (1987). Wittgenstein and the turning point in the philosophy of mathematics. Albany, New York State: The State University of New York Press. Weinberg, S. (1977). The search for unity: Notes for a history of quantum field theory. Daedalus, 107, 17–35. Wittgenstein, L. (1969). In: G. E. M. Anscombe, & D. Paul (Trans.), On certainty. Oxford: Blackwell. Wittgenstein, L. (1986). In: C. K. Ogden (Trans.) Tractatus Logico-Philosophicus. London: Routledge and Kegan Paul.

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COMMENTARIES

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Chapter 7

Reflections on Conceptual Change Stathis Psillos When discussed among philosophers of science, the issue of conceptual change brought in its tow incommensurability. Kuhn and Feyerabend both suggested that in the transition from the old to the new paradigm (or theory, or conceptual scheme) there is a deep conceptual asymmetry. Hoddeson (this volume) narrates elegantly the shock that the community felt and explains the ways that several members of it dealt with the postKuhnian trauma. In Structure, Kuhn made notoriously extravagant claims about the proponents of competing paradigms practicing their trades in different worlds, or being transported to different worlds etc. When the dust settled, Kuhn drew a distinction between global and local incommensurability and defended a version of the latter. Machamer (this volume) discusses this issue in his piece and makes what I take it to be the exactly right point about global incommensurability: it entails relativistic solipsism (which, I add, a moment’s reflection shows that it is either incoherent or absurd); hence, any theory that entails global incommensurability is reduced to absurdity. What then of local incommensurability? In his mature thought, Kuhn argued that it occurs when the competing theories have locally different taxonomies of natural kinds (what Kuhn called lexical structures). It amounts to a claim of local untranslatability, due to a mismatch between the lexical taxonomies associated with the two theories. Kuhn, actually, was very careful in characterising the situations in which incommensurability arises. The obvious objection to his claim is that mismatches in the lexical structures may well be there and yet, the lexica might be sufficiently similar to enable setting up at least rough-and-ready correspondences among their nodes. To this he replied that not any “old difference” yields incommensurability; rather, there must be differences that test the compatibility of nodes in the lexical structures. He talks, for instance, about the no-overlap principle which captures the old Lockean thought that there are gaps or chasms in nature: the members of two natural kinds that are not related to each other as genus and species are disjoint. Incommensurability, then, arises when lexical structure A classifies entity x under kind K and structure B classifies entity x under the disjoint kind L; or when lexical structure A classifies entity x under kind K and structure B classifies entity y — but not x — under the kind K.

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Machamer (this volume) thinks Kuhn is basically right about local incommensurability. But I do not see why this should give us pause. As stated, the thesis of local incommensurability merely recapitulates the claim that there is conceptual change. To talk of conceptual change as opposed to conceptual replacement presupposes that something remains unaltered while something else changes. Conceptual replacement does occur but gives no rise to incommensurability. Conceptual change, on the other hand, reassigns referents (extensions) to new kinds or reshuffles the referents (extensions) of old kinds. But this would be a problem only if concepts were identified extensionally. Since they are not, there is no problem. The real issue, I take it, concerns the similarities and differences in the conceptual roles assigned to them. These can offer criteria for conceptual similarity and difference. To clarify this point I will appeal to Sellars (1973). He defended a kind of nominalism about concepts. He denied that concepts are abstract entities (types or universals) to which thought is somehow related. Concepts are not the kind of thing one is related to by having certain thoughts — e.g., the concept DOG is not the kind of thing one is related to when one has dog-thoughts. He took the line that concepts are (or are related to) dispositions and abilities (hence, concepts are not things of any sort), viz., dispositions to think thoughts of certain things — e.g., to have the concept DOG is to have the ability to think thoughts about dogs, and in particular to think thoughts to the effect that such and such thing is a dog. Sellars conjoined this dispositional account of concepts with a functionalist account of thought: thoughts have the content they do because they function in a certain way and two thoughts with the same function have the same content. If that is what is important about thoughts (as Sellars says, apart from their functional role, thoughts are neurophysiological processes), concepts — being constituents of thoughts — can be identified functionally as well. Identical are those concepts that function in exactly the same way. This move opens up the space of comparing concepts according to their functional similarity. How a concept functions will depend on its connection to other concepts (that is, to its conceptual role) as well as the rules (formal or material, as Sellars would have it) that determine its contribution to a conceptual framework. Similarity of function is then the means to build bridges among concepts — even if these concepts are not identical (despite the fact that they might have the same names.) For instance, one could say that the function of a pitcher in a baseball game is similar to the function of a bowler in a game of cricket. (So: the concepts PITCHER and BOWLER are similar — this is illuminating, I take it, since we can now focus on where they differ. Pitching is primarily defensive — it aims to prevent the other team from scoring runs — while bowling is offensive — it aims to remove the batsman). Similarly, (mutatis mutandis) one could say that the function of relativistic mass in special relativity is similar to the function of classical mass in classical physics. All this presupposes that the functional role of a concept is not determined in a holistic and undifferentiated way by the framework in which it belongs. Arabatzis (this volume) disagrees. He defends conceptual holism (that is, the view that a concept is identified holistically by means of all properties possessed by, and all law-like connections that characterise, the entities to which it applies). But conceptual holism cannot explain the robustness (or invariance) that many concepts possess, viz., that they do not change very easily, and certainly they do not change when small alterations take place in their conditions of application. Besides, conceptual holism cannot explain what the possession of a concept consists

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in since it cannot consist in the possession of a full theory. The sensible thing to say is that not everything (in a conceptual scheme) is constitutive of its concepts and not all belief changes constitute concept changes. That a line should be drawn seems imperative. Arabatzis seems to be saying this: concepts are individuated holistically, but there can be conceptual stability provided there are independent ways to identify what the concept stands for as the same again. These independent ways are tied to experimental procedures in which a stable referent for a concept is identified. Therefore, Arabatzis makes experiments the locus of conceptual stability. But there seems to be a problem with this. Let us take theories to be (or to fix the) modes of presentations (senses) of concepts and let us take the experimental procedures to fix the reference of a concept (the type of entity for which the concept stands for). How are these related to each other? It seems that, on Arabatzis’ story, these two semantic vehicles move parallel to each other and it is a fortunate coincidence that they, if at all, cross each other’s path. There is no reason to think that the referent identified in a series of stable experiments is the referent of a concept whose content is identified holistically by a theory unless the theory informs the experiments and the experiments expand the theory. Here again, if the whole of the theory informs the experiment, when the theory changes, the experiment changes too and the referent that has been identified experimentally changes too. I do not doubt that experiments offer useful ways to identify entities; the issue is that they have to identify an entity as the referent of a concept. If this concept is individuated (or is introduced) in the process of the experiment, well and good. If this concept is individuated by a theory, we need an extra assumption that it is this concept’s referent that is identified by the experiment. With this in mind, let us move back to Sellars (1973). I take him to have argued that it is precisely because some concepts are similar, that is, they play a similar functional role in their respective frameworks, and that not all parts of the framework are equally responsible for determining the content of concepts. But how can we classify concepts as being similar? Sellars’ deepest thought in this connection was that it is a mistake to think that there is just one single concept of X. Conceptual change (not replacement) occurs when we move from one concept of X to another concept of X and this, of course, is compatible with the thought that these concepts of X are relevantly similar to each other (hence, there is conceptual continuity of the sort that allows for conceptual change). To illustrate this, Sellars considers the following cases: (A) ISOSCELES TRIANGLE vs. SCALENE TRIANGLE (B) EUCLIDEAN TRIANGLE vs. RIEMANNIAN TRIANGLE In case (A), the two concepts operate within a certain (Euclidean) framework and are species of a genus. Using type-hierarchies, we could say that EUCLIDEAN TRIANGLE is a supertype, having ISOSCELES TRIANGLE and SCALENE TRIANGLE as subtypes. These two subtypes are similar in specific respects (since they inherit the properties of their common supertype) and different in others (in virtue of which they are classified in different subtypes). In case (B), the two concepts operate within different frameworks. Using type-hierarchies, we could say that both EUCLIDEAN TRIANGLE and RIEMANNIAN TRIANGLE are subtypes of a supertype TRIANGLE. What differentiates the subtypes is the fact that EUCLIDEAN TRIANGLE is governed by the axioms of

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Euclidean geometry, while RIEMANNIAN TRIANGLE is governed by the axioms of Riemannian geometry. It is obvious that ISOSCELES TRIANGLE and SCALENE TRIANGLE are more similar to each other than are EUCLIDEAN TRIANGLE and RIEMANNIAN TRIANGLE. But there is no reason to think that both EUCLIDEAN TRIANGLE and RIEMANNIAN TRIANGLE are not TRIANGLE concepts. There is, as Sellars (1973, p. 90) put it, a generic functioning of TRIANGLE “which abstracts from the specific differences between Euclidean and Riemannian geometries”. Sellars’ point is that it is wrong to think of these two concepts as single concepts. Rather, the expressions “Euclidean” and “Riemannian” qualify the concept TRIANGLE. (So, strictly speaking, we should write: Euclidean TRIANGLE and Riemannian TRIANGLE.) The Euclidean TRIANGLE and the Riemannian TRIANGLE are both varieties of TRIANGLE; they function in a similar way in their respective frameworks and they function in a way that TRIANGLE (more abstractly specified) functions. In On Certainty, Wittgenstein (1969) claimed that some propositions play the role of hinges that hold together a language-game (or a conceptual scheme). He did not think, however, that hinges share something deep in common in virtue of which they function as hinges. They are not analytic truths, nor a priori truths. They are rather heterogeneous collections of propositions (including arithmetical truths, straight empirical propositions and others). What hinges share in common is that they function in a certain way, viz., within certain language games, they cannot be doubted without revealing some conceptual shortcoming (cf. p. 137). Their functioning this way, according to Wittgenstein, is the result of an act (or a deed, as he put it): the act that has to do with that “we just can’t investigate everything, and for that reason we are forced to rest content with assumption”. (p. 343) Baltas (this volume) finds in this Wittgensteinian story a way to understand what happens during a revolution — and in particular a way to understand a conceptual change. What he calls background “assumptions” are hinges, with the difference that they are hidden — they lie in the background of concepts and determine their conditions of their application. According to Baltas, background “assumptions” determine the logical grammar of a concept. His idea is that a revolution (a paradigm change) consists in a leap into the ungrammatical (from the point of view of the superseded paradigm) while it also consists in bringing to the foreground and challenging the background assumptions of the concepts of the old paradigm (from the point of view of the new paradigm). This is an interesting asymmetrical relation: the new paradigm opens up a new grammatical (hence conceptual) space that was not possibly available from within the old paradigm. The emergence of a new paradigm, according to Baltas, turns a background assumption into an ordinary proposition, and in particular, one that can be subjected to doubt, empirical examination, revision or rejection. By challenging this assumption, the new paradigm redefines an old concept by removing a presupposition for its application. How are the new and the old concepts related to each other? As Baltas himself notes, there must be some continuity (similarity) between the two concepts; otherwise, it will be unclear that they relate to the same phenomena and in particular, that the conceptual change was prompted (at least partly) as a response to an empirical anomaly that the old

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paradigm faced. Now, Baltas seems to be saying three things in response to the question above. First, there is continuity because the non-grammatical character of the new concept vis-à-vis the old one is repressed in memory — scientists, that is, forget or choose to forget that there was a leap into the ungrammatical. Second, in the transition from the old to the new paradigm, some worldly item (whatever constituted the anomaly that brought the old paradigm to crisis) remains invariant. Third, the retention of the same name for the old and the new concepts constitutes “the necessary grammatical reminder of this [conceptual] continuity”. However, even if we were to grant these three points, they would locate the continuity at the wrong place. The locus of conceptual continuity should be similarities among the concepts, i.e., among their content (their functions, or their conceptual texture, as Sellars (1973) would put it). Anything else, desirable though it may be, is an extrinsic characteristic. Baltas’ first point makes conceptual continuity a matter of (fortunate) psychological contrivance; his second point makes it a matter of referential stability; and his third point, makes conceptual continuity a matter of naming. It is worth noting that Sellars’ account sketched above would be a better candidate for capturing Baltas’ insight. If we think of not just one but of many concepts of X (e.g., classical WAVE; non-classical WAVE etc.), Baltas’ idea of changes in the background assumptions can be part of the story that explains how these concepts are similar to, and different from, each other. What more should we expect from a theory of conceptual change?

Acknowledgements Many thanks to my student Milena Ivanova, who made me focus my ideas on some key issues.

References Sellars, W. (1973). Conceptual change. In: G. Pearce, & P. Maynard (Eds), Conceptual change (pp. 77–93). Dordrecht, The Netherlands: D. Reidel. Wittgenstein, L. (1969). On certainty (G. E. M. Anscombe, & D. Paul, Trans.). Oxford: Blackwell.

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Chapter 8

Conceptual Change as Structure Change: Comment on Kuhn’s Legacy Matti Sintonen Thomas Kuhn is best known as a chief critic of the once-dominant Received View of scientific theories and growth of knowledge. The four chapters by Hoddeson, Baltas, Arabatzis, and Machamer give a clear and philosophically sophisticated account of Kuhn’s contribution to the new historicist understanding of science that challenged and in part replaced this Received View. The little I wish to add concerns the way Kuhn conceived scientific theories, how scientific terms acquired their meaning, how theory change and conceptual change could be elaborated, as well as the potential implications of these views for learning and instruction. The best way to appreciate Kuhn’s contribution is to see it against the problems of the Received View. In this view scientific theories were conceived as interpreted (and ideally axiomatized) logical structures, where the theoretical (unobservable) and non-theoretical (observable) terms and concepts were connected by laws. Theories, therefore, were sets of sentences or statements, and once so interpreted they were typically considered true (or often “strictly speaking” false) claims about the world. Although proponents of the view did not think there could be any logic of discovery, there was a logic of justification: experiments and observations on one hand and logic and reason on the other hand force scientists to abandon false theories and hence to pave the way towards Truth. Kuhn came to doubt the credentials of the Received View already in the 1940s. Together with a growing number of scientists and philosophers, he felt that it was more of an exercise in logic than a description of what actually went on in science. One shortcoming, germane to our topic, was that the view had virtually nothing to say about how theories were learned and taught, and its account of how the theoretical (non-observational) concepts acquired their meaning for individuals and the communities was dead wrong. Reflecting on Hempel’s (1957) canonical work on concept formation and its influence on his own views, Kuhn (1993) summarized his departure from the Received View as follows. First, theoretical terms and along with them concepts are not learned the way logicians interpret so-far uninterpreted symbols. Rather, they are learned in use. Second, these uses

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are manifested in paradigmatic examples of “nature’s behaviour”. Third, there must be several such exemplars for the meaning to catch on; and, finally, when the process is complete, the language or concept learner has acquired not only meanings but also, inseparably, generalizations about nature. (Kuhn, 1993, p. 312)1 As Arabatzis (this volume) nicely shows, the idea that language and world, meanings and beliefs, are learned simultaneously owes to Willard Van Quine’s theory of meaning. Quine had challenged the dogmas of the Received View but his proposal, adopted by Kuhn, made a commitment to thorough conceptual contextualism or even holism: whenever there is a change in the theoretical assumptions the crucial concepts alter in meaning. But if theory change always involves conceptual change, it is difficult to escape relativist and antirealist conclusions concerning the ontology of science. All manners of unpalatable or absurd philosophical and empirical consequences would follow. Arabatzis argues that the dynamic view of growth of knowledge championed by Kuhn in particular does not have these drastic consequences. Holism in this extreme form flies in the face of scientists’ intuition and common sense, and I am not sure that even its weaker forms can or need be rescued. The reason is that the way Kuhn contrasted the Received View and his own relied heavily on the sort of linguistic view that is currently receding. Instead of trying to formulate a theory of reference suitable for radical conceptual change (some variety of causal theory might work), I propose to focus on a more mundane description of radical theory changes as a change in the structure of conceptual systems, that is, paradigms and theories. Kuhn’s reflections explain some of the ambiguities of “paradigms” and “theory” noted by Hoddeson and Machamer (this volume). A paradigm narrowly conceived is an exemplary problem solution, whereas a broad reading yields a “disciplinary matrix” or “a constellation of group commitments”. As Machamer puts it, paradigms in this broader sense are mainly non-propositional sets of interconnected concepts, coupled with shared guidelines for application. They provide a communal touchstone for what problems are worth focusing on, and what kinds of conceptual equipment can be used in the pursuit of solutions. Paradigms in this broad sense include not only exemplars as one ingredient, but also the symbolic generalizations and the (cognitive) values of the group or community. Also, theories can be taken in two different senses: as more specific, precisely formulated “principles or laws”; and as complex conceptual structures that include precisely formulated laws (and other ingredients). Second, a theory (in either sense) includes as a crucial element a set of intended applications, what a theory is “about”. To see how Kuhn’s

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Kuhn is here referring to Carl Hempel’s (1957). It is of some interest to note that this is the positivists’ series of publications called Encyclopedia of Unified Science, the very same series in which Kuhn’s Structure of Scientific Theories was first published. Note also that Kuhn’s ties to the positivist views are not accidental; his manner of phrasing the problems of meaning and concept formation owes to Hempel’s account. Kuhn came to disagree with it, but his problem was the same. It is now conceivable that the scene in history and philosophy of science has changed so that the problem is no longer amongst the most pressing ones — and, therefore, also that our way of seeing the issue has changed.

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account departs company from the Received View, let us call, following the so-called structuralist explication, the precisely formulated theories theory-elements and big theories theory-nets. Theory-nets are sets of interconnected elements, the sort of “larger structures” or “research programmes” alluded to by Machamer (this volume).2 Now, theory-elements (and hence nets) have a dual structure, consisting of a set of intended applications and a theory-core, most notably the laws that specify the models of the theory. In the Received View the laws were assumed to have a universal scope, whereas in Kuhn they are explicitly tied to particular types of structures described in a non-theoretical vocabulary. It is important for this view that the intended applications are not given as explicit lists but rather through exemplars and similarity relations: anything that is sufficiently similar to a paradigmatic exemplar falls into the scope of the theory-element. Theories in mature scientific fields, therefore, are not isolated individuals but parts of a larger, even global structure. For our purposes the most interesting type of structure is a paradigmatic theory-net with one fundamental theory-core, specified in terms of the deepest or most entrenched laws and principles. The basic laws give rise to hierarchy-specific theory-elements that zero in on more limited classes of applications by making more specific claims. Normal science, Kuhnian puzzle solving, then amounts to adding such applications (special laws) to the already existing structure. Initially, a net might contain just a few well-established (sufficiently confirmed) applications, although the claim made might be extremely strong: for example, that not just gravitational phenomena (falling bodies) but pendulums, oscillators, and even electromagnetic phenomena can be captured under the fundamental laws by hitting upon suitable special laws. A theory-evolution represents such historical development, and as Machamer says, it can span over decades or even centuries. In a comment on Sneed’s (1971) account of theory dynamics Kuhn distinguished between these two types of change of structure, one evolutionary and the other revolutionary: “Particularly striking to me, of course, is that its manner of doing so appears to demand the existence of (at last) two quite distinct sorts of alteration of time”, one in which the fundamental theory-core remains the same and is extended to cover new applications, the other one in which a fundamental core is abandoned for another. The allegiance of a scientist are to the core and not the special laws that govern particular applications. Two men can “share belief in a core and in certain of its exemplary applications… even though their beliefs about its possible expansion differ widely.” (Kuhn, 1976, p. 187) It is this basic core, also, which is immune to refutation during normal scientific evolution. To get a better handle on revolutionary conceptual change we need to return to the duality of cores and intended applications. Very roughly, what happens in revolutionary change is that the structure of the concepts which enable classification of phenomena as similar

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I shall skip all formal details and only present the fundamental ideas needed to appreciate its virtues in explicating a conceptual change. The view was first formulated by Sneed (1971) and it was developed by Stegmüller (1973). The resultant collaboration led into what was known as the structuralist or non-statement view. In private conversation Kuhn confessed that the view had become too technical to his tastes (“I am too old to start learning category theory”, he said to me in 1986). Nevertheless, he wrote, Stegmüller has understood his work “better than any other philosopher who has made more than passing reference to it.” (Kuhn, 1976, p. 179)

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changes. Put metaphorically, scientific theories (and paradigms in the wider sense) are something that provide a community of scientists with well-defined puzzles, food for thought, as well as means for solving these puzzles, that is, the cutlery. Kuhn writes of Sneed: “For him as for me, the adequate specification of a theory must include specification of some set of exemplary applications”. (Kuhn, 1976, p. 182) What falls within the jurisdiction of a theory is determined through their similarity with these exemplars, not by giving a list or by specifying necessary and sufficient conditions. And if this is how they are specified, and how the concepts entering the theory get their identities, there can be subtle individual or communal differences, arising from background and previous training, in the way a theory is understood. That similarity measures are the way theories are specified is in line with Baltas’s (this volume) Wittgensteinian suggestion. Behind all explicit claims and theory formulations (theories in the more restricted sense), there always are background assumptions or hinge propositions in Wittgenstein’s sense (although to what extent they are propositional is an issue). These implicit assumptions, variously described by different scholars, delineate the horizon of possible explicit questions and answers. They therefore comprise the bounds of sense and amount to the grammar of conceptual change. Eventually, when an anomaly blows up the bounds of sense, questions formerly inconceivable begin to shape. Clearly, such a revolutionary disclosure corresponds to overthrowing the fundamental theoryelement and, therefore, bringing to end normal-scientific evolution. Baltas suggests that this Wittgenstein-inspired view by Kuhn does not lead to idealism or relativism, nor does it jeopardise rationality and scientific progress — and here Arabatzis agrees. The notion that exemplars, paradigms in the narrow sense, are an aspiring student’s entering wedge into science, and that paradigms in the narrow and broad sense specify a grammar of possible questions, has implications for the manner in which would-be members of a scientific profession become card-carrying members. To become a Newtonian is to learn to see the applications through the fundamental Newtonian laws, and hence to group falling bodies, planets and comets, pendulums, and oscillators as similar. This does not come out as natural but may need overcoming of initial or “natural” groupings. Furthermore, this has implications for the way sciences are learnt and taught, and how what is learnt at some stage directs attention to where to go next: “Need I emphasize”, Kuhn wrote, “that learning a theory is learning successive applications in some appropriate order and that using it is designing still others?” The failure of the Received View was, therefore, not just in its purely static account of theories — confinement to snapshots — but also a failure to address how the crucial concepts of a theory acquire content in and through use, scientific practice. Note also that the community of card-carrying members of a paradigm need not be unanimous when concepts are extended to cover new instances. Despite the communally shared exemplars they allow leeway in judgments on future cases. Despite seemingly smooth communication within practitioners of a field there may be underlying differences in the way phenomena are grouped. I have focused on (mature) scientific theories as hierarchically organized theory-nets that evolve by budding more specific theories because it gives a visual picture of theory change at different levels of generality. Theories also are, in this view, historically evolving particulars.

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Furthermore, although the picture does not solve the problem of reference, it throws some light on the problem of meaning and conceptual change, for adding new special laws (e.g., force laws in physics) brings about conceptual change. To illustrate this, and to conclude this comment, I shall point out the relevance of structure change for cognitive science in two cases: Giere’s (1994) account of the cognitive structure of scientific theories and Thagard’s (1992) views on conceptual revolutions. The aim is to give an idea of how Kuhn’s account could be elaborated (for a thorough and insightful account of Kuhn and cognitive science, see Nersessian, 2003). Giere (1994) favors the semantic or model-based approach to scientific theories, as has been briefly outlined above. He is interested in canvassing a naturalistic approach to science that explicitly incorporates human agents. Consequently, he draws on what cognitive psychologists have had to say about the nature of concepts and categories. Very briefly, Giere sides with those who object to the classical view of categories as sets of necessary and sufficient conditions. There is, he suggests, a close connection between categories and theories which can be brought out to the open by following the pioneering work by Jerome Bruner in the 1950s and Eleanor Rosch and others in the 1970s. In Rosch’s studies on language acquisition the key hypothesis, confirmed in different experimental setups and across different cultures, was that concepts are structured or graded. Following work on the essentially culture-independent or near-universal color terms Rosch extended the idea to artifacts and what philosophers call natural kinds. A prototype of a chair has four legs, but more peripheral (shall we say, degenerate) instances can be included, with allowance on increased time of recognition. Within the category “bird” there are prototypic or indeed exemplary species (robins) but, with some imagination and time, ostriches (no wings) and even bats (very degenerate birds, on second thought not really birds at all) can be counted in. In these studies on categorization, there was also another interesting finding, viz., that grading can vary horizontally (intracategory grading) or vertically (intercategory grading). Now, if we would have to guess what level, for example, in a vertical category is more fundamental, one could easily go for the most general one as the basic category. For instance, all lassies are collies, collies are dogs, dogs are mammals, mammals are animals and animals are living things. So the category of “living things” would be the most fundamental one. However, Rosch (1978) argued that in this hierarchy “dog” is basic and dogs are the easiest to identify and group together; categories higher than that exhibit less similarity, and the gain in similarity when moving down to its subcategories is insignificant. The notion of similarity does heavy-duty work here, and so do those of a prototype or exemplar. Although these notions are far from being unproblematic, there is support for the view that similarity to prototypes or exemplars is fundamental to categorization. Moreover, exemplars serve as entering wedges into categories and hence conceptual networks. Giere’s (1994) point in venturing into the realm of cognitive psychology is that just as natural and artificial kinds exhibit graded structures, so do scientific theories. Theories are systems of models that have both a vertical and a horizontal structure. His example is, not surprisingly, the structure of classical mechanics. The most fundamental level of description, from the point of view of physical theory, would be that of the three Newtonian laws, which employs the notion of force but imposes no particular form in it (this would be the basic laws in a theory-net in the sense specified above). The next level splits these models into two groups depending on whether the total energy of the system studied is constant in

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time or not. On the next level down the hierarchy, different types of forces impose additional structure. Finally, on levels IV and V we get down to familiar descriptions of classes of models and their visual representations: free fall and inclined planes as types of rectilinear motion with constant force, pendulums and springs as types of harmonic motion, as well as circular and elliptical orbits as types of orbital motion. These categories can then be given more fine-graded content still. Now, from the point of view of an abstract mind wedded to viewing all these classes of models as models of classical mechanics the fundamental level is level I, that of Newton’s three laws. However, Giere argues that level IV and the visual models are basic in the Roschian sense. Physics is taught by giving examples and exercises in terms of this level, but once they have been utilized as ladders to climb up to the abstract level they can be kicked away. To become an expert is to learn to see the intended applications of classical mechanics through and by help of the higher level mathematical equations, indeed Kuhn’s symbolic generalizations. And Giere indeed surmises that most of Kuhn’s (1962) exemplary problem solutions do correspond to Roschian basic-level models. This view also tallies nicely with that advanced by Thagard on conceptual revolutions. Thagard writes that an understanding of conceptual revolutions requires much more than a view of the nature of isolated concepts. We need to see how concepts can fit together into conceptual systems and what is involved in the replacement of such systems. (1992, p. 30) This Thagard seeks to provide. He gives an account of conceptual systems in which concepts are organized into hierarchies by the help of various types of links, such as kind, instance, rule, property, and part links. This makes it possible to grade changes with respect to their degree of radicalness in terms of adding or deleting nodes and links, and of adding or deleting rules. As Thagard sums up, changes in kind-relations and part-relations usually involve a restructuring of conceptual systems that is qualitatively different from mere addition or deletion of nodes and links. (Thagard, 1992, p. 32)

References Giere, R. (1994). The cognitive structure of scientific theories. Philosophy of Science, 61, 276–296. Hempel, C. (1957). Fundamentals of concept formation in empirical science. Chicago, IL: University of Chicago Press. Kuhn, T. (1962). The structure of scientific revolutions. Chicago, IL: University of Chicago Press. Kuhn, T. (1976). Theory-change as structure-change: Comments on the Sneed formalism. Erkenntnis, 10, 179–199. Kuhn, T. (1993). Afterwords. In: P. Horwich (Ed.), World changes. Thomas Kuhn and the nature of science. Cambridge: The MIT Press. Nersessian, N. (2003). Kuhn, conceptual change and cognitive science. In: T. Nickles (Ed.), Thomas Kuhn. Cambridge: University of Cambridge Press.

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Rosch, E. (1978). Principles of categorization. In: E. Rosch, & B. Lloyd (Eds), Cognition and categorization. Hillsdale, NJ: Erlbaum. Sneed, J. D. (1971). The logical structure of mathematical physics. Dordrecht, The Netherlands: D. Reidel. Stegmüller, W. (1973). Theorienstrukturen und theoriendynamik. Berlin: Springer. Thagard, P. (1992). Conceptual revolutions. Princeton, NJ: Princeton University Press.

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PART 2: PERSONAL EPISTEMOLOGY AND CONCEPTUAL CHANGE

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Chapter 9

Personal Epistemology and Conceptual Change: An Introduction Stella Vosniadou

Definitional Issues Epistemology is the branch of philosophy that studies the nature, source, and limits of knowledge. Psychologists are not interested in epistemology per se but rather in understanding individuals’ beliefs about knowledge. The term ‘personal epistemology’ is used in the psychological literature to characterize the beliefs of individuals about the nature of knowledge and the processes of knowing. The interest is in finding out how these beliefs differ from individual to individual, how they change with development and learning, and how they influence learning and studying in the subject-matter areas. The terms epistemic and epistemological beliefs are also used in the literature. In this volume, Murphy et al. and Alexander and Sinatra propose, following Kitchener (2002), to use the term epistemic to denote individuals’ beliefs about knowledge, and epistemological to denote beliefs about the study of knowledge. According to this definition, individuals, such as students and researchers, have epistemic beliefs. Researchers investigating the beliefs of individuals about knowledge are, however, engaged in research on epistemological beliefs. These terms are not used in the same way by all the researchers in the field, including the authors of the chapters in the present volume. Furthermore, as Stathopoulou and Vosniadou (this volume) note, there is a set of additional terms, such as epistemic cognition, epistemological perspectives, epistemological theories, epistemological resources, etc., which different researchers use, reflecting the diversity of the theoretical approaches to the construct of interest. Most of the researchers recognize the lack of terminological and sometimes conceptual clarity in this field of research, which, as Alexander and Sinatra note in their commentary, may nevertheless be justified on the grounds that it is relatively new and emergent; in fact, hardly more than about 10 years of age.

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Epistemological Beliefs Research There is considerable disagreement as to how it is best to conceptualize the nature and development of epistemological beliefs, although practically all researchers agree that personal epistemologies change and evolve with learning and development. Some researchers use quantitative instruments assuming that epistemological beliefs can be conceptualized as consisting of distinct dimensions each of which represents a kind of continuum between extremes (e.g., Schommer-Aikins, 2002). Others use qualitative measures, such as interviews, considering personal epistemologies as more coherent sets of beliefs evolving through hierarchically organized sequence of positions (Perry, 1998), stages (King & Kitchener, 1994) or levels (Kuhn, 1991). Regardless of the particular theoretical framework adopted, personal epistemologies are usually seen as changing through development and learning, following an ‘upward’ movement from dualistic/absolutist and objectivist views, to relativist and subjectivist views and finally to contextual, constructivist, and evaluative perspectives of knowledge and knowing (Hofer, 2002). This issue needs, however, to be investigated further as there may be important cross-cultural differences in the development of personal epistemology.

The Relationship between Personal Epistemology and Conceptual Change In the present volume we are not interested in investigating epistemological beliefs per se but in understanding the role they play in conceptual change. Research has shown that a constructivist epistemology is positively related to skills and attitudes important for learning, such as critical thinking, self-regulation, cognitive flexibility, the ability to communicate ideas and to learn from collaboration. Previous studies have also demonstrated the relationship between a constructivist epistemology and text comprehension. Ryan (1984) found that students who are relativists in their beliefs about knowledge, i.e., who perceive knowledge as context dependent, are more successful in comprehension monitoring and tend to use high-level comprehension strategies. On the contrary, students who are dualists, i.e., who perceive knowledge as factual, right or wrong, who are more likely to study for recall of facts from texts. Naïve beliefs concerning the structure of knowledge — beliefs concerning knowledge as an accumulation of discrete, concrete, knowable facts — have been found to be related to poor text comprehension in the social and physical sciences, and to affect the comprehension and related problem solving of statistical text in a negative way (Schommer, 1990; Schommer, Crouse, & Rhodes, 1992). Naïve beliefs concerning the stability/certainty of knowledge, i.e., seeing knowledge as unchanging, have a negative effect on the interpretation of controversial evidence (Kardash & Scholes, 1996) and tentative text. Students who believe that knowledge is absolute and certain do not change their previous views easily and tend to draw inappropriate absolute conclusions from tentative text. (Schommer, 1990). The theoretical position that we would like to advance is that epistemological beliefs can have both a direct and an indirect influence on conceptual change. For example, beliefs

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in simple, stable, and certain knowledge can prevent individuals from being open to new information that questions some of their basic assumptions, while on the contrary, individuals who believe that knowledge is complex, uncertain, and constantly evolving may be willing to ‘open up the grammatical space’ and allow new paradigms/theories to be seriously entertained. Epistemological beliefs can influence conceptual change also in indirect ways, for example, by influencing students’ learning goals, study strategies, and selfregulation. Some of these issues are addressed in the papers on epistemological beliefs and conceptual change included in the present volume.

The Chapters in the Present Volume The chapter by Murphy et al. examines how psychology researchers, implicitly or explicitly, convey epistemological beliefs in their conceptual change research. This is a new and interesting topic of research. Murphy et al. justifiably claim that sometimes the researchers involved in studying the epistemic beliefs of individuals have not examined their own epistemological stances. The attempt to bring some insights and knowledge from the philosophical study of epistemology to bear on the psychological studies of epistemological beliefs research is also worthwhile. Nevertheless, a great deal of caution must be exercised with respect to the kinds of conclusions that can be drawn from these studies. More specifically, there is a danger here to attribute to researchers the epistemic beliefs that the researchers attribute to the participants of their studies, as, for example, attributing to Vosniadou and Brewer (1992) the foundational epistemic beliefs they attributed to their subjects. Furthermore researchers are likely to have epistemic beliefs that cut across a number of the epistemic stances described in the paper, whereas any single empirical study must by necessity concentrate on only a few variables. Finally there may be important differences between the studies that attempt to provide information about students’ epistemic beliefs as compared to interventions that attempt to change these beliefs. In interventions, usually more attention is paid to sociocultural factors. Entwistle (this volume) reviews a range of studies that show that beliefs about knowledge and learning go through similar qualitative changes over time and that they eventually become integrated and affect the approaches to studying adopted by students. This chapter takes a perspective on conceptual development and change that derives mainly from research in teaching and learning at the university level, but it is also based on ideas about the nature of concepts, epistemological development, and contextual influences on the emergence and use of conceptions. The author concludes that only through a deep approach to studying, an approach that links intention with learning processes, can students achieve the kinds of conceptual understanding demanded by university degree courses. The chapter by Stathopoulou and Vosniadou follows as a nice continuation of the Entwistle chapter. It is designed to further explore the relationship between personal epistemology and conceptual change in physics. The authors present results suggesting that a constructivist epistemology may guide the adoption of a deep approach to studying, which may in turn facilitate physics understanding. Previous research by Stathopoulou and

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Vosniadou (in press) demonstrated that secondary school students with constructivist physics epistemologies had better understanding of Newtonian dynamics compared to their classmates with less constructivist physics epistemologies. In the present study, indepth interviews, think-alouds, and observations during problem-solving of 10 students selected from the Stathopoulou and Vosniadou (2006) study were analyzed in terms of their approaches to physics learning and studying. Five of these students were found to hold constructivist physics epistemologies as well as a high degree of physics understanding, whereas the remaining five did not have a constructivist physics epistemology and were low in physics understanding. The interviews showed that the five students with the constructivist epistemologies and high conceptual understanding of physics adopted a deep approach to studying and learning, as described by Entwistle (this volume). In contrast, the remaining five students who did not have a constructivist personal epistemology and were far from having achieved conceptual change in physics, adopted a superficial approach to learning and studying. The chapter by Mason and Gava also reports a study that investigated the relationship between epistemological beliefs and learning from a refutational text. The results showed that eighth graders with more advanced personal epistemologies (who believed that knowledge is complex and uncertain) were able to profit more from reading a refutational text compared to students with less advanced personal epistemologies. Duit and his colleagues present data from a recent video study on the practice of introductory physics instruction in Germany. The analysis of these data showed that there is a large gap between teachers’ beliefs about the nature of knowledge and how to teach it and the actual use of such ideas in instructional practice. Teachers’ thinking about instruction was oriented mostly towards the content to be taught. Most of the teachers were not informed about recent conceptual change ideas or about the influence of epistemological beliefs on students’ understanding. In their commentary, Alexander and Sinatra raise a number of important theoretical and methodological issues regarding epistemological beliefs research and the relationship between individual’s epistemic beliefs and conceptual change. One important issue is the lack of theoretical and terminological clarity. The discussants ask for greater explicitness of definition of the terms used by researchers so that the field can eventually achieve more solid conceptual and lexical foundations. They also discuss extensively the complexity of the constructs involved and of their interrelationships, urging researchers to be more explicit about the theoretical models they propose and to design studies that can further develop and test these models. Finally, the importance of conducting more long-term developmental and classroom-intervention studies using carefully designed measures of epistemic beliefs and appropriate statistical procedures in order to be able to better understand the interrelationship between personal epistemology and conceptual change is discussed.

Acknowledgments The present work was financially supported through the program EPEAEK II in the framework of the project ‘‘Pythagoras — Support of University Research Groups” with 75% from European Social Funds and 25% from National Funds.

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References Hofer, B. K. (2002). Personal epistemology as a psychological and educational construct: An introduction. In: B. K. Hofer, & P. R. Pintrich (Eds), Personal epistemology: The psychology of beliefs about knowledge and knowing (pp. 3–14). Mahwah, NJ: Erlbaum. Kardash, C. M., & Scholes, R. J. (1996). Effects of preexisting beliefs, epistemological beliefs and need for cognition on interpretation of controversial issues. Journal of Educational Psychology, 88, 260–271. King, P. M., & Kitchener, K. S. (1994). Developing reflective judgement: Understanding and promoting intellectual growth and critical thinking in adolescents and adults. San Francisco, CA: Jossey-Bass. Kitchener, R. (2002). Folk epistemology: An introduction. New Ideas in Psychology, 20, 89–105. Kuhn, D. (1991). The skills of argument. Cambridge: Cambridge University Press. Perry, W. C. Jr. (1998). Forms of intellectual and ethical development in the college years: A scheme. San Francisco, CA: Jossey-Bass. Ryan, M. P. (1984). Monitoring text comprehension: Individual differences in epistemological standards. Journal of Educational Psychology, 76(2), 248–258. Schommer, M. (1990). Effects of beliefs about the nature of knowledge on comprehension. Journal of Educational Psychology, 82, 498–504. Schommer-Aikins, M. (2002). An evolving theoretical framework for an epistemological belief system. In: B. K. Hofer, & P. R. Pintrich (Eds), Personal epistemology: The psychology of beliefs about knowledge and knowing (pp. 103–118). Mahwah, NJ: Erlbaum. Schommer, M., Crouse, A., & Rhodes, N. (1992). Epistemological beliefs and mathematical text comprehension: Believing it is simple does not make it so. Journal of Educational Psychology, 84(4), 435–443. Stathopoulou, C., & Vosniadou, S. (in press). Exploring the relationship between physics-related epistemological beliefs and physics understanding. Contemporary Educational Psychology. Vosniadou, S., & Brewer, W. F. (1992). Mental models of the earth: A study of conceptual change in childhood. Cognitive Psychology, 24, 535–585.

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Chapter 10

Epistemological Threads in the Fabric of Conceptual Change Research P. Karen Murphy, Patricia A. Alexander, Jeffrey A. Greene and Maeghan N. Edwards Never as much as today have I thought that philosophy was indispensable to respond to the most urgent questions of society…Never have we had such a need for the philosophical memory. (Derrida, 2002) Throughout this volume, the body of theory and research on conceptual change is discussed and debated by international scholars who have helped to shape this very domain of study. Among those renowned scholars are individuals invested in understanding the place of epistemic beliefs in the realization of conceptual change. What we hope to contribute to this dialogue is a preliminary, yet critical, exploration of the philosophical threads related to epistemology found in the fabric of conceptual change research. For this initial epistemological foray into conceptual change research, we have chosen to examine how psychological researchers implicitly or explicitly infuse epistemic tenets into their conceptual change studies. This theoretical analysis unfolded in several phases. First, we began with a review of how philosophers convey the nature of knowledge. Second, based on that review, we selected several prominent epistemological frameworks (e.g., foundationalism and reliabilism), and delineated their defining features (e.g., stances regarding knowledge, substantive conditions, and justification of knowledge). Next, we constructed prototypic conceptual change studies representing selected epistemic belief frameworks. Essentially those prototypes were examples of how the philosophical frameworks might manifest in conceptual change research. Finally, we searched the literature on conceptual change for evidence of implicit and explicit epistemic assumptions evident in the design and execution of research. That evidence is shared here in the form of selected conceptual change studies that can be distinguished by their differential treatment of knowledge justification. For clarity we have taken pains to distinguish between epistemic and epistemological beliefs. The former are beliefs about knowledge, whereas the latter are beliefs about the study of knowledge. Kitchener (2002) has made a strong argument for maintaining this distinction. Students most likely have epistemic beliefs, beliefs about knowledge. Reframing the Conceptual Change Approach in Learning and Instruction Copyright © 2007 by Elsevier Ltd. All rights of reproduction in any form reserved. ISBN: 0-08-045355-4

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Many researchers have epistemic beliefs as well, such as what means of justification are privileged over others. Finally, those looking at the implicit or explicit assumptions regarding the study of knowledge are engaging in research on epistemological beliefs. We consider this epistemological delving into the conceptual change literature worthwhile for several reasons. First, there would appear to be an inherent relation between epistemic beliefs and conceptual change. That is, if one is to investigate the nature and process of change in conceptual understanding, it would seem that some judgments as to the nature or source of knowledge must be assumed by those conducting this research. Second, we are of the mind that all psychological research, including the conceptual change literature, would benefit from rediscovering its philosophical roots (Murphy, 2003). Much of contemporary educational research seems almost devoid of an awareness of the foundational literature in philosophy from which it was birthed. As Derrida (2002) described, we need a philosophical memory, especially when trying to examine some of the more urgent questions of society, such as conceptual change. A caveat, of course, is that although philosophical memory and even current philosophical thinking can inform theory and interpretation, it cannot replace empirical evidence. Such empirical evidence is vital in understanding the nature of the conceptual change process. This is particularly true in the fields of cognitive and educational psychology due to the epistemic importance of empirical evidence as a form of justification (Southerland, Sinatra, & Matthews, 2001). Third, and most relevant to this volume, we have found ourselves enticed by the burgeoning psychological literature on epistemic beliefs. In many ways, the psychological literature on epistemic beliefs functions independent of substantive influence from philosophy. Like others, we posit that this independence is due, in part, to variations in the central questions guiding these disciplines. That is, philosophers are intrigued by what it means to know, whereas psychologists are more concerned with how one comes to know (Pollock & Cruz, 1999). Despite the subtle variations in focus, we maintain that there is much to be learned from the philosophical annuls pertaining to the theoretical study of epistemology. This sojourn into the conceptual change literature will cast a discerning eye on the epistemic aspects of studies, including our own, populating the pages of psychological journals. Reciprocally, by bringing epistemic beliefs into the limelight, we may furnish conceptual change researchers with a new lens for examining their work.

Framing the Epistemology Literature Epistemology is a core area of inquiry within philosophy (Moser, 1995). Consequently, the philosophical literature pertaining to epistemology and its central tenets is both vast and varied. Nonetheless, there are defining questions that characterize the philosophical literature on epistemology. To our understanding, those defining questions are: What is the nature of knowledge; what is the source of knowledge; and what are the limits of knowledge (Bonjour & Sosa, 2003; Pollock & Cruz, 1999; Moser, 1995)? This characterization of epistemology does not include skepticism, the belief that it is impossible to “know” anything. While this is an interesting philosophical argument, in terms of our focus on conceptual change, it is of little merit. An epistemological skeptic would not only denounce our attempts to understand knowledge, but also question the rationale for

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facilitating conceptual change. The skeptic would ask how any ethical human being could assert that another’s concepts were in need of changing, when knowledge is not possible. Indeed, many modern philosophers see skepticism as a useful tool for testing the robustness of epistemological stances, but do not give it serious consideration as an actual explanation of knowledge or knowing (Pollock & Cruz, 1999; Quinton, 2001; Williams, 2001). Given the skeptics’ refutation of both the content and the analytic lens of this chapter, we set aside the skeptical question as intriguing but tangential to our purposes. Thus, although the three questions about knowledge have relevance for educational research, understanding the nature of knowledge is particularly important in the study of conceptual change. As such, we set ourselves the task of clarifying this defining question. Our approach to this challenging exercise was multilayered. We first turned to three well-respected encyclopedias on philosophy: the Routledge Encyclopedia of Philosophy (Craig, 2004), the Cambridge Dictionary of Philosophy (Moser, 1995), and the Stanford Encyclopedia of Philosophy (Bradie & Harms, 2004; Feldman, 2001; Fumerton, 2000; Goldman, 2001; Greco, 2002; Klein, 2001; Kvanvig, 2003; Steup, 2001; Young, 2001). These volumes are frequently required readings for entry-level courses in philosophy, and served as clear and concise summaries of the boundless library of classic and contemporary writings on epistemology. These encyclopedias proved useful to us in clarifying and extending our prior knowledge of epistemology, and in identifying other sources that functioned as yet another layer of essential discourse to be analyzed and synthesized. For example, in reading the Stanford Encyclopedia we were directed to Ernest Sosa’s two edited volumes on Knowledge and Justification. These tiers of sources were invaluable in conceptualizing eight principal stances in epistemology: 1. Foundationalism, which holds that all knowledge is derived through ascent from basic beliefs internal to the knower; 2. Coherentism, which sees all of one’s beliefs as mutually reinforcing and thereby justified as knowledge only in their mutual coherence; 3. Direct Realism, which states that justification cannot be solely a function of beliefs about perceptions but a function of the perception themselves; 4. Probabilism, which assesses the cognitive mechanisms that produce beliefs in terms of their ability to produce knowledge with a high likelihood of being true; 5. Reliabilism, which judges the veracity of knowledge based on the cognitive mechanisms’ reliability in producing true beliefs; 6. Social epistemology, which focuses on social practices and their influences on one’s beliefs about knowledge; 7. Evolutionary epistemology, which emphasizes the process of natural selection in the formation and maintenance of justified true belief; and 8. Virtue epistemology, which focuses on the character of the knower rather than individuals’ beliefs or collections of beliefs. Next, we narrowed our focus to four epistemological frames that appear most pertinent to psychological research, in general, and the conceptual change literature, in particular: foundationalism; coherentism; reliabilism; and social epistemology. We chose to narrow our focus for two reasons. First, there are simply too many epistemological frames studied in philosophy, and a cogent presentation of them would not be possible within one

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DOXASTIC STANCES

Foundationalism Coherentism

NORMATIVE STANCES (Internalist Theories)

Direct Realism Virtue Epistemology NONDOXASTIC STANCES

Probabilism Reliabilism Social Epistemology

NATURALIST STANCES (Externalist Theories)

Evolutionary Skepticism

Figure 10.1: Relational model of epistemological stances. manuscript. In addition, some of the epistemological frames seemed less relevant to academic learning or are less well developed in the philosophical literature. In reviewing literatures on each of these frames as well as literature comparing the frames (e.g., Moser, 1995; Sosa, 1994), it is important to establish a priori that despite the fact that we discuss each of the frames separately, we do not view the frames as conceptually or theoretically mutually exclusive. From a conceptual or theoretical perspective, there are overlaps in nature of justification (e.g., doxastic vs. non-doxastic) and the source of justification (e.g., internal or external). Such similarities or overlaps are illustrated in Figure 10.1. From an operational perspective, it would seem that contextual and social factors could strongly influence the predominant epistemic frame enacted at any particular time.1 For example, an individual might enact a reliabilist frame in regard to their scientific research while holding to a foundationalist perspective in their personal religious life.

An Epistemological Primer Prior to exploring the aforementioned epistemological frameworks, we briefly discuss the philosophical analysis of knowledge. Like psychologists (e.g., Alexander, Schallert, & Hare, 1991), philosophers distinguish between many kinds of knowledge including empirical (a posteriori), nonempirical (a priori), knowledge by acquaintance, knowledge by description, and “how to” knowledge. The kinds of knowledge with which psychologists are most familiar would be “how to” knowledge (i.e., procedural knowledge) and knowledge by description (i.e., propositional or declarative knowledge). It is propositional knowledge that is typically the crux of epistemological inquiry; that is, what does it mean to say that an individual knows something to be the case? 1

The issue of contextual and social influences on the enactment of epistemic beliefs was raised by Stella Vosniadou during revisions of the chapter.

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Formal logic is the means by which epistemological theory is understood and tested, and those propositions can be expressed in the form of syllogisms of the kind (DeRose, 2004): 1. P is true, 2. S believes that P, and 3. S is justified in believing that P. This standard analysis of knowledge claims that truth is necessary for knowledge, but it is not sufficient. If one is to know that p, not only must it be true that p: one must also believe that p, at least implicitly. (Sosa, 1994, p. xi) In essence, the nature of knowledge can be broken down into three distinct, yet related, conditions: the belief condition, the truth condition, and the justification condition. In terms of the belief condition, most epistemologists agree that knowledge requires belief but belief does not require knowledge (Moser & vander Nat, 2003). One knows something only if one believes it, but not all things that are believed are in fact knowledge. Knowledge also requires truth and justification. You know that all college professors are brilliant only if it is true that all college professors are brilliant. If there is one dull college professor, you do not know that all college professors are brilliant. Knowledge thus has a truth requirement. (Moser & vander Nat, 2003, p. 6) Most epistemologists agree that even justified belief is not knowledge unless it is, in fact, true. One might be epistemically responsible in the means by which one justifies a belief, but still be incorrect. In this case, the believer is not faulted, but he or she does not retain the status of having knowledge. It would be expected that the believer would acknowledge such when shown conclusive contradictory evidence. The belief and truth conditions are fairly standard aspects of most epistemologies. Most epistemological stances vary in terms of how they treat the justification condition. The first differentiation concerns whether or not those stances posit that beliefs about perceptions are enough to justify knowledge. Doxastic views contend that the only viable and sufficient means of justification derive from the beliefs that one holds; specifically, beliefs about perceptions (Figure 10.1). Non-doxastic views agree that perception beliefs are important, but insufficient for justification. Instead, non-doxastic views require an examination of the cognitive processes from which knowledge derives. Within the doxastic tradition, there are two fundamental schools (i.e., foundationalism and coherentism). Foundationalists assert that there are certain self-evident foundational beliefs that require no further justification. It is from these beliefs that other knowledge claims are built through the process of epistemic ascent. For example, sense data about the behavior of clouds might lead to conclusions about the mechanics of wind. Coherentism agrees that knowledge claims are built from beliefs and perceptions, but denies that there are certain foundational claims; rather all beliefs must be

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judged correct through an evaluation of their coherence with other beliefs. Coherent systems are knowledge, incoherent systems are not. A coherent system would include a set of perceptions that are non-contradictory such as seeing, smelling, and tasting the presence of a piece of cake. Despite their differences, both foundationalism and coherentism rely on internal sources of justification. As such, one might expect a certain degree of overlap between these frameworks when evidenced in research and practice. For example, it seems reasonable that a researcher or teacher could value both the power of students’ experiential beliefs as well as the coherence among those beliefs. However, doxastic epistemological theories are subject to a serious criticism in the form of the Gettier problem (Moser, 1995; Pollock & Cruz, 1999; Quinton, 2001). Prior to Gettier (1963), all epistemology was doxastic, claiming that knowledge was justified true belief. However, Gettier demonstrated that these conditions were not sufficient. For example, we can imagine a woman named Barbara who has good reason to believe a nonetheless false proposition that John owns a radio (i.e., she saw John carrying a radio). Barbara also chooses to guess where another person named Mike is, such as in Florida, knowing that she has no reason to believe it to be true. Coincidentally, Mike is in Florida. Now, Barbara is justified in believing the disjunction “either John owns a radio or Mike is in Florida.” Here, it is obvious that while this statement is true, we would not say that Barbara knows either that John owns a radio or that Mike is in Florida. This and other challenges to doxastic theories, while seemingly easy to resolve, have been a persistent and tenacious problem. Some epistemologists have suggested that the Gettier problem illustrates that the justification, truth, and belief conditions are not sufficient for knowledge. Their solution has been to institute various other kinds of additional conditions beyond justified true belief. Epistemic frames advancing conditions for knowledge beyond justified true belief are referred to as non-doxastic. Non-doxastic views divide into two main schools based upon how they choose to bolster the justification condition (i.e., internalism and externalism), while foundationalism and coherentism rely on beliefs only. Internalists judge the validity of cognitive processes based upon preestablished epistemic norms that are employed to judge the viability of a particular cognitive process to serve as justification. For example, reasoning would serve as justification for knowledge, but guessing would not be an acceptable form of justification. Externalists determine the validity of cognitive processes based on external metrics. Naturalists, for example, believe that cognitive processes should be compared to scientific theories and data to determine their validity. Among the naturalists, reliabilists assess cognitive processes based upon how effectively they predict events or data in the real world, thus they employ an external source or criterion for justification of cognitive processing. Social epistemologists also assess the predictive reliability of cognitive processes, but do so through the testimony of established authorities and beliefs institutionalized within a particular culture or community of practice.

Overlaying Epistemological Stances on Conceptual Change Studies Given this epistemological primer, we turn to four epistemologies prevalent, yet implicit and unexamined, in the conceptual change literature. As a means of introducing these four,

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we have constructed prototypic studies of conceptual change and describe why their features coincide with the defining characteristics of each. To make the distinctions between these studies more transparent, we will hold certain elements consistent. Specifically, our conceptual problem across the four hypothetical studies will be Galileo’s Principle of Falling Bodies and our participants will be eighth graders of normal cognitive ability representing diverse sociocultural backgrounds. Foundationalism Because of the importance of students’ self-justifying, core beliefs, and the role that perception plays in foundationalism, we would want to initiate this study by pre-assessing students’ beliefs relative to the Principle in some detail. Moreover, we would want to create an intervention that explicitly confronts students’ unjustified beliefs through sensory experience. Thus, in the pre-assessment, we might have students make initial predictions about the outcome of dropping two objects of different sizes and weights, and then require them to explain those predictions; exposing their self-justifying core beliefs. For example, students may be asked to watch a simulation that depicts a bowling ball and a small rubber ball being dropped from a tower. Before the balls hit the ground, the simulation is stopped. Students would then be asked to describe what would happen and why. This process would be repeated with objects varying along perceptual dimensions. In all cases, the goal would be for students to render a prediction and to explain that prediction. Once the pre-assessment is complete and students’ misperceptions are exposed, an intervention would be initiated that would be multisensory in nature. That is, we would present a number of predictive events that convey the Principle in a way that draws separately on sensory data. For example, as in the pre-assessment task, students might be asked to watch a simulation similar to that viewed earlier and witness the outcome in slow motion. Likewise, the students might close their eyes and listen intently to what they hear when a water-filled balloon and a glass are similarly dropped from a set height. As before, the outcomes would be discussed in terms of any initial self-justifying beliefs. Following a delay, students would again be shown the pre-assessment simulation and predict the outcome. The explanations for their predictions would provide evidence of conceptual change relative to the target Principle. Coherentism What differentiates coherentism from foundationalism is the lack of foundational beliefs; thus, any intervention designed to change a concept or mental model must instead address multiple beliefs in an attempt to invalidate the coherence of the system. Consequently, rather than rely solely on the previous simulation to alter students’ conceptions of the Principle, we might employ a structured inquiry during the simulation to uncover multiple misperceptions within the network and then provide evidence contrary to those misperceptions. For example, because mass, velocity, and acceleration are variables intertwined within the understanding of the Principle, we might design interventions to correct misperceptions in any of these areas. Thus, in the simulation we might first attempt to isolate the effect of velocity. Students would then collect data relative to

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velocity during the interventions. Similarly, we could run specific simulations for which mass or acceleration is the key variable. Moreover, because we are concerned about the relations among the beliefs that are part of this system, we would also seek to address the seeming inconsistencies across these key concepts and their relations to each other by requiring students to explain the effect of mass in relation to velocity, or acceleration in association with mass and velocity. The coherentist assumption would be that a “tipping point” would be reached where the contrary evidence would outweigh the coherence of the misperceived system, thus allowing for instantiation of new beliefs. Reliabilism A reliabilist would approach conceptual change differently. Since reliabilism is an externalist theory, one necessary criterion for justification would be the quality of the cognitive processes used in substantiating belief, where the validity of the cognitive processes would be based on external metrics (e.g., comparisons to scientific data). Therefore, a reliabilist intervention could depend upon experimental data that are beyond human perception (e.g., reasoning) to illuminate students’ misperceptions so long as the cognitive processing effectively predicts events or data in the real world. For example, scientific theories could be gleaned from the subject’s misperceptions and empirically tested. When those theories fail to reliably produce accurate predictions, the experimenter could facilitate conceptual change by presenting principles or accurate scientific theories to the students. The modified or newly formed theories could then be experimentally tested. When the results demonstrate the reliability of the newly presented theory, students could accrete that theory through cognitive processing. This cognitive processing of the new theory could be enough to foster conceptual change in the reliabilists’ view. Thus, the reliabilist has the ability to address conceptual change through perceptual data and scientific hypothesis testing that is not directly observable. Whether a student would find this convincing is a separate matter, but the externalist epistemology allows for greater variety in the ways conceptual change could be invoked. In essence, while doxastic theories are “I’ll believe it when I see it” theories, non-doxastic theories, such as reliabilism, allow for the possibility that more abstract evidence may be convincing. Social Epistemology Social epistemologists not only acknowledge the importance of the natural sciences in the justification of beliefs, but also consider the influence of social and cultural forces as equally privileged mechanisms of justification. Therefore, in planning a prototypic study of conceptual change from this perspective, several features would likely become important. For one, in the process of exposing students’ pre-existing beliefs, we would not only want to determine what students’ hold to be true about the Principle of Falling Bodies, but we would also need to ascertain the source of those beliefs. Moreover, in the pre-assessment, we would want to learn what sources of evidence (e.g., known experts or written text) students deem more convincing or authoritative. It might be that the experimenter is not seen as a valid source of new information, and therefore, another method should be used to promote conceptual change.

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In addition, while we could use the types of simulation described under reliabilism, we would incorporate more cooperative or collaborative group discussions to introduce alternative evidential sources that students might perceive as more convincing. Further, when it comes to the post-test, we would be interested in the changes in understanding that students manifest about the Principle, and concerned with any demonstrated changes in their sources of evidence or shifts in the perceived credibility of sources.

Prototypic Conceptual Change Investigations Given the extensiveness of the conceptual change literature and in light of our stated intention to forward a preliminary analysis, we set out to narrow the research base to those articles likely to display epistemological threads. Thus, we first conducted a broad search employing several electronic databases including PsycINFO, ERIC, Philosopher’s Index, and Education Abstracts. Our initial parameters were reference to conceptual change in title, abstract, or in key concepts, and a publication or release date of 1999 or later. We also conducted a search by author for the past 10 years, including the names of six researchers known for their work both in conceptual change and epistemology. Those authors were Stella Vosniadou, Clark Chinn, William Brewer, Lucia Mason, Gale Sinatra, and Janice Dole. This process resulted in an initial pool of almost 400 works. To narrow this pool, we established six key inclusion criteria. Specifically, we decided to focus on those works that: (a) mentioned conceptual change in title or in abstract; (b) were empirical in nature (i.e., no research summaries or theoretical analyses); (c) could be fully retrieved from the archived source (i.e., no summary of conference proceedings or dissertation abstracts); (d) were not solely about teacher conceptual change; (e) incorporated some learning outcome; and (f) were published in English (our linguistic limitations at cause). By applying these criteria, we succeeded in narrowing the pool to 90 works. Of these, we selected 40 of the most promising pieces for subsequent analysis. In particular, we chose those works that were written by well-known conceptual change researchers and were published in leading general educational research journals (e.g., Cognition and Instruction, Journal of Educational Psychology, or Contemporary Educational Psychology). Our goal was to locate studies seemingly well matched to the epistemological frameworks we identified. Ultimately, for this chapter, we engaged in detailed analyses of six articles that afforded us an opportunity to view the conceptual change research through an epistemological lens. Those six studies are: Vosniadou and Brewer (1994), Chinn and Brewer (2001), Nussbaum and Sinatra (2003), Mason (1996), Qian and Alvermann (1995), and Hofer (2004). We acknowledge that the categorizations posed herein are highly inferential, since none of the authors freely exposed their epistemic leanings for our benefit. In essence, our goal was to look for evidence of similarities between the aforementioned philosophical, epistemological stances and the views of selected researchers in a particular study. Also, in all the studies perused, the researchers’ lens was focused on the manifestations of students’ epistemic cognition and not on their own epistemic beliefs. Still, we considered the goals, design, methodology, and data sources of these studies to be vestiges of the researchers’

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epistemic views. As such, there is the possibility that a given researcher might emphasize or expose different epistemic beliefs in one study than another. Work by William Brewer is a good case in point. Specifically, in the pages that follow, we categorized his work with Vosniadou (i.e., Vosniadou & Brewer, 1994) as evidencing threads of foundationalism, whereas we interpret his work with Chinn as more closely aligned with coherentism (Chinn & Brewer, 2001). In both cases, however, we see evidence of a focus on beliefs internal to the knower (i.e., doxastic). Vosniadou and Brewer (1994) Stella Vosniadou and William Brewer (1994) offer an analysis of first-, third-, and fifthgrade children’s understandings of the day/night cycle (e.g., disappearance of the sun during the night, disappearance of the stars during the day, or apparent movement of the moon). Their results revealed that children used a collection of well-defined mental models to explain those various phenomena. The mental models were empirically accurate, consistent, and parsimoniously explained. Younger children constructed intuitive mental models based on beliefs about prior experiences and situational factors, whereas older children constructed mental models that synthesized aspects of culturally accepted scientific information with prior knowledge, and a few students constructed culturally accepted scientific mental models as evidenced in their verbal protocols and pictorial representations. Traces of foundationalism can be seen in Vosniadou and Brewer’s theoretical framework. For example, the researchers assumed that children possessed beliefs on the basis of their interpretations of their everyday experiences. Children are not blank slates when they are first exposed to the culturally accepted, scientific views, but bring to the acquisition task some initial knowledge about the physical world that appears based on interpretations of everyday life. (p. 124) Moreover, they posited that children’s beliefs have a hierarchical structure in which some beliefs are foundational (i.e., presuppositions), while other beliefs can be conceptualized as second-order. Presuppositions may be innate or empirically acquired constraints which are present from early infancy and which guide the way children interpret their observations and the information they receive from the culture to construct knowledge structures. (p. 124) Within Vosniadou and Brewer’s frame those presuppositions and second-order beliefs served as constraints that must be exposed, and potentially lifted, during the conceptual change process. Given the strength of some presuppositions and second-order beliefs, students will sometimes form alternative models that incorporate aspects of their presuppositions. Indeed, Vosniadou and Brewer have termed such models synthetic because they are formed precisely because presuppositions cannot be removed or

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replaced through exposure to cultural information during instruction about scientific phenomena. Conceptual change is seen as the product of the gradual lifting of constraints, as presuppositions, beliefs, and mental models are added, eliminated, or revised during the knowledge acquisition process. (p. 124) The purpose of this particular study was to document mental models of the day/night cycle of various aged children, and to compare the variations in those models to the hypothesized transition from intuitive models to scientific models. We also see evidence of coherence theory in this study. For instance, the researchers were interested in the coherence between children’s explanations of various individual phenomena (e.g., sun, moon, stars) and their relations to the overarching process of the day/night cycle. Whether foundationalism or coherence theory, it is clear that Vosniadou and Brewer’s mental models framework is doxastic in nature. This understanding is confirmed by the presence of the Gettier problem in the results. That is, the children seemed to offer what they perceived as justified beliefs. But the beliefs were false because they were based on inaccurate perceptions gleaned from everyday experiences. Children believed …from their observations that the day/night cycle is causally related to the appearance and disappearance of both the sun and the moon. The erroneous belief that the moon is causally implicated in the day/night cycle appeared to be present in the synthetic models of the oldest children in our sample. (p. 171) The researchers go on to state that they do not understand why these beliefs persisted. The Gettier problem seems to shed light on this situation. These students were justifying their beliefs based on false or coincidentally true beliefs, and philosophically speaking, such instances should not be considered knowledge. Chinn and Brewer (2001) The purpose of this research was to test a theory (i.e., models-of-data) of how students evaluate data. The models-of-data theory is intended to provide an account of both how students evaluate data and of how they design research studies such as experiments. Chinn and Brewer (2001) hoped to categorize the types of reasons given to explain why students accept or reject aspects of data. To investigate students’ reasons, these researchers asked 168 undergraduate students to read one of the two theoretical texts on either mass extinction or dinosaur metabolism. After reading, students rated the extent to which they believed the theory. After rating the initial theory, students read about data either supporting or refuting the original theory. Then, students rated their beliefs in the data and explained their beliefs. Finally, students offered ratings of the consistency between the theory and the data, as well as explanations of their data. Chinn and Brewer found that students’ ratings and explanations generally upheld the models-of-data theory. That is, these researchers found that when

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evaluating data, individuals created a cognitive model that integrated features of the data with a theoretical interpretation of the data. In essence, the strength of Chinn and Brewer’s present work is its modeling of how people interpret data and evidence. What is important for our purposes are Chinn and Brewer’s theoretical assumptions in their models-of-data, as well as their interpretations of the findings. What we saw in both their theoretical framework and their interpretation of the outcomes was evidence of a coherence approach to epistemic cognition. Like coherence theory, Chinn and Brewer’s model focuses on the relation between beliefs and their place in a comprehensive system of knowledge. [W]e propose that when people evaluate data, they construct a cognitive model of the data according to the perspective of the person who is reporting the data....The individual seeks to undermine one or more of the links in the model, often by seeking alternative causes that individuate particular links. If the individual succeeds in identifying a plausible alternative cause for an event, the data can be discounted. (p. 337) This notion of examining the links between various beliefs is at the heart of coherence theory and Chinn and Brewer’s research findings. A model of data is evaluated by examining the plausibility of the links in the model. (p. 334) Nussbaum and Sinatra (2003) Argumentation was used as a means to encourage cognitive engagement in Nussbaum and Sinatra’s (2003) study of conceptual change. Participants were asked to guess the path of an object dropped from various moving contexts such as a plane or a conveyor belt. Participants who did not predict a Newtonian-path for the object were prompted to argue for “another option” that was in fact the scientifically correct one. The authors found that subjects who argued for the scientific option were more likely to revise their initial guess to one more scientific as compared to a control group. This change in conception was hypothesized to be due to the degree of cognitive engagement involved in the argumentation intervention. The Nussbaum and Sinatra (2003) article demonstrates a reliance on not only naturalistic empiricism, but also the power of individuals’ ability to critique their own cognitive processes. The authors discussed a commitment to fostering scientific thinking, conducting experiments, and resolving discrepancies between points of view and data. The researchers’ focus on science places them within a naturalistic epistemology. Additional evidence of a naturalistic approach to epistemic cognition can be found in the fact that during the multiple iterations of the falling object simulation, participants were asked to evaluate the reliability of their underlying cognitive activities. Thus, we believe the Nussbaum and Sinatra research was derived from a reliabilist epistemology. What is significant in this categorization is that it appeared that the authors believed conceptions change through introspection and examination of the internal logic of one’s conceptions. Yet, Nussbaum

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and Sinatra attributed conceptual change to argumentation, as opposed to the kinds of cognitive dissonance that such argumentation most likely invokes in the subject who initially guessed incorrectly. In essence, they are describing the manner in which analysis of cognitive processing is encouraged, but they do not mention the actual analysis itself. Mason (1996) In the context of a larger two-year study, Mason (1996) attempted to explicate students’ processes in making justified scientific claims in a group discussion. Students in five fifthgrade classrooms discussed different environmental education units in both whole-class and small-group settings in order to construct new scientific knowledge. Mason identified students’ scientific claims and justifications for those claims in their group discourse. After categorizing students’ appeals to personal experiences, accepted scientific principles, analogies, generalizations, and other relevant data, Mason concluded that the students in the discussions were able to learn from each other by negotiating and sharing meanings. We hold that this study and its underlying theoretical framework offer threads of social epistemology. This framework differs from reliabilism in that the justifications formed by the students do not come solely from the reliability of their own cognitive processes. Instead, Mason posited that knowledge could be constructed with and through others, with or without an appeal to one’s own cognitive processes. Mason specifically expressed this social epistemic framework by stating that not only can students build knowledge with each other, but that they can also use the same social interactions with their peers as justifications for their new conceptions. When a shared idea passes from mind to mind, co-constructed knowledge becomes stronger and deeper than the knowledge transmitted in traditional science classrooms. (p. 427) Hofer (2004) and Qian and Alvermann (1995) Both Hofer’s (2004) and Qian and Alvermann’s (1995) studies highlight differences between philosophical and psychological definitions of epistemology. Hofer (2004) performed exploratory qualitative analyses with students in two different chemistry courses. In each course, student interviews and class observations provided data used to analyze the underlying epistemic assumptions fostered by instruction. Hofer’s goal was to “examine how beliefs about knowledge and knowing are communicated in the classroom and how they are situated in classroom interaction” (Hofer, 2004, p. 135). Ultimately, she hoped to describe the evolution of students’ epistemic understanding in a contextualized learning environment. Hofer categorized the epistemic beliefs into dimensions including the certainty of knowledge, the simplicity of knowledge, the source of knowledge, and the justification for knowing, based upon Schommer-Aikins’ work (e.g., 2002). Thus, Hofer asserted that she was able to substantiate the existence of dimensions and to explicate how they were instantiated in the classroom and internalized by students. One of Hofer’s main points was that students inferred epistemic assumptions of instructors and milieus, and in so doing, acquired learning in ways that were contextualized and situated.

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Qian and Alvermann (1995) examined the relation between learned helplessness, epistemic beliefs, and conceptual change within the domain of science. A group of 212 high school students were classified according to their level of prior knowledge about Newton’s Theory of Motion. The authors examined how learned helplessness and epistemic beliefs differentially impacted students who varied in their level of prior knowledge. Refutational text on Newton’s Theory was employed as the intervention, and students were given a subsequent post-test to measure conceptual change. Qian and Alvermann (1995) found three epistemological (as they termed them) belief factors: quick learning, the simplicity and certainty of knowledge, and the innate nature of learning ability. The researchers found little evidence of any relation between learned helplessness and epistemic beliefs, and the only factors significantly associated with conceptual change were simple/certain knowledge and quick learning factors, both of which were negatively related. In prior knowledge groups, significant gains in conceptual change were realized but there was no interaction with epistemological beliefs or learned helplessness. Overall, Qian and Alvermann found support for the three latent factors that they termed epistemological beliefs, and they found that refutational text facilitated conceptual change among learners. We would submit that both of these studies have more psychological than philosophical views of epistemic cognition. Pollock and Cruz (1999) discussed a fundamental difference between philosophical and psychological analyses of epistemology. While philosophers are concerned primarily with the question “What is it to know?” psychologists are interested in the range of ways in which humans acquire and justify their knowing. Therefore, psychological analyses of epistemic cognition should capture the variability people display. For example, while the psychological literature on personal epistemology discusses student assumptions about knowledge, such as the limits and simplicity of knowledge (Hofer, 2004; Qian & Alvermann, 1995), the philosophical literature seems unconcerned with such issues (Pollock & Cruz, 1999; Sosa, 1994).

Concluding Thoughts and Future Directions As we stated at the outset, we engaged in this process of tracing the threads of epistemology in conceptual change research with an enthusiasm that can only be born from naïveté. Had we realized the truly daunting nature of the task we had set for ourselves, we may not have been willing to initiate the pursuit. Nonetheless, we have personally been well served by the endeavor, learning much about the ideas and issues that reside at the juncture of epistemology and conceptual change. We also believe that our analysis can prove informative to those invested in understanding conceptual change or sparking conceptual growth in others. First, among the insights we derived from this venture is the realization that within the conceptual change literature, the epistemic threads are often so buried as to be unidentifiable. It might be argued that conceptual change researchers are not required to make their epistemic views public or even to acknowledge the vast literature in epistemology or some may maintain that interpretations of their views are likely dependent on the nature of the selected study. Yet, we maintain our initial argument that those seeking to change knowledge or

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beliefs in others must themselves possess a tacit, if not explicit, conception as to what it means to know or some grounded notions of what constitutes knowledge. Second, while we retain our belief that conceptual change researchers would benefit from rediscovering their epistemological roots, we acknowledge that this wading into the philosophical waters might be tantamount to an arctic dip. For those who are unfamiliar with contemporary philosophical writings or who have not revisited classic works, dealing with the terminology and discourse of this literature could prove a shock to the system. As with any complex field, philosophers have their own lexicons and discursive patterns that seem strange and uninviting to the casual reader. We would welcome those rare souls who are equally at home in both the psychological and philosophical literatures, and who are able to weave the theories and research of one community into the fabric of another. Perhaps in this way, our desire to see deeper and richer threads of epistemological thought in the conceptual change research might be realized. Finally, as suggested, there appears to be two rather distinct streams of epistemic inquiry that only sporadically intertwine. On the one hand, there is philosophical epistemology that utilizes formal logic and that coheres around the three basic questions of what is knowledge, particularly the justification condition: what are its sources; and what are its limits; which are primarily questions about the nature of knowledge? On the other hand, there is psychological epistemology that seems more invested in the acquisition of knowledge or, in the case of conceptual change research, in changing knowledge. The inherent nature of knowledge is either taken as a given or considered tangential to the agenda at hand. Thus, we focused here on the justification condition as it is the most relevant to the conceptual change literature. However, researchers would do well to survey other conditions and explore how their assumptions about knowledge influence their work. Even given these conclusions, we are left with the question as to whether or not most psychologists are aware of their underlying epistemic stances. If they are not aware, then the authors themselves are unsure how their stance might manifest in their conceptual change work. It could be that researchers posit a different epistemic stance for themselves than they do for those they study. For example, most conceptual change researchers, in our view, espouse a naturalist epistemology as the most correct. This implicit assumption could manifest in two different but equally questionable ways. First, it could be the case that a naturalist researcher sees naïve student conceptualizations (e.g., Vosniadou and Brewer, 1994’s findings regarding children’s belief that the sun goes to sleep behind a mountain at night) as an instance of a “less-developed” epistemic stance, such as foundationalism. Such beliefs about the sophistication of epistemic stances and implicit epistemologies have been raised by philosophers such as Pollock and Cruz (1999) in their suggestions regarding the critiques of doxastic theories using the Gettier problem, and non-doxastics’ superiority in their ability to deal with such challenges. Thus, a psychological researcher with this set of implicit epistemic assumptions would see conceptual change development as an analogue to development in the philosophical epistemology literature itself, moving from doxastic to non-doxastic theories. However, a second way in which a psychologist’s implicit epistemological stance could manifest would be in interpreting all data from that point of view. In this example, a naturalist researcher, using a reliabilist method of establishing justification, would characterize the Vosniadou and Brewer (1994) example not as an instance of foundationalist

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thinking but rather as an improper cognitive mechanism lacking sufficient reliability in testing. There are two empirical questions here. First, do researchers assume that people can hold different epistemic stances, and second, how do students justify their knowledge, via simple perception, cognitive mechanisms, or another way? Obviously, the answers to these questions would have clear ramifications for the manner in which conceptual change was studied and what kinds of interventions would be appropriate to foster change. There is an obvious danger in attempting to categorize the epistemological underpinnings of researchers’ work. A more valid and collaborative next step would be to interview selected researchers to expose qualitatively their epistemic and epistemological beliefs. Not only would this help us understand the topology of the conceptual change landscape, but it might also afford us insights into the kinds of stances held by experts in the domain of psychology. Likewise, a more comprehensive review of the 90 articles we initially identified would give us further evidence of the variability of epistemic and epistemological assumptions within the psychological literature. It could be that individual studies have varying degrees of each epistemology discussed herein. To these future studies, we would add a review of the major models of conceptual change as a means of juxtaposing these researchers’ theoretical premises with the conceptualizations and operationalizations that emerge in their empirical studies. Our goal would be to subject the conceptual change research to a test of internal consistency. There may be those who would appreciate our efforts to delve into the philosophical literature and seek to weave its elements into the fabric of conceptual change research, but others might perceive this as a futile undertaking with little to contribute to the real problems of learning and teaching. To the contrary, the eloquent words of Derrida (2002) that opened this chapter are a poignant reminder that we cannot afford to lose our philosophical memory. Rather than be esoteric or divorced from significant educational questions of the day, philosophy is a voice that needs to be heeded. When it comes to conceptual change that voice reminds us that questions of belief, truth, and justification cannot be set aside when we elect to expose the understanding of others and clearly not when we feel empowered to alter those understandings. Without question, there is much more to learn about the place of epistemology in conceptual change theory and research. We have done little more here than open the door on a complex and controversial realm of investigation, and we welcome the questions, concerns, and insights of those who have long been invested in this program of research or who share our passion for the building a philosophical memory that can inform and guide.

References Alexander, P. A., Schallert, D. L., & Hare, V. C. (1991). Coming to terms: How researchers in learning and literacy talk about knowledge. Review of Educational Research, 61, 315–343. Bonjour, L., & Sosa, E. (2003). Epistemic justification: Internalism vs. externalism, foundations vs. virtues. Malden, MA: Blackwell Publishing. Bradie, M., & Harms, W. (2004). Evolutionary epistemology. In: E. N. Zalta (Ed.), Stanford encyclopedia of philosophy. Retrieved April 2, 2004, from http://plato.stanford.edu/archives/win2003/bradie

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Chinn, C. A., & Brewer, W. F. (2001). Models of data: A theory of how people evaluate data. Cognition and Instruction, 19(3), 323–393. Craig, E. (Ed.). (2004). Routledge encyclopedia of philosophy. London: Routledge. DeRose, K. (2004). Sosa, safety, sensitivity, and skeptical hypotheses. In: J. Greco (Ed.), Ernest sosa and his critics (pp. 22–41). Malden, MA: Blackwell Publishing. Derrida, J. (2002). Culture et dépendences—spécial Jacques Derrida, presented by F.-O. Giesbert, with the participation of E. Levy, C. Pépin, D. Schnick, & S. Werba (France 3 Television, May 2002). D. Egéa-Kuehne (Trans.). Feldman, R. (2001). Naturalized epistemology. In: E. N. Zalta (Ed.), Stanford encyclopedia of philosophy. Retrieved April 2, 2004, from http://plato.stanford.edu/archives/win2003/feldman Fumerton, R. (2000). Foundationalist theories of epistemic justification. In: E. N. Zalta (Ed.), Stanford encyclopedia of philosophy. Retrieved April 2, 2004, from http://plato.stanford.edu/ archives/win2003/fumerton Gettier, E. (1963). Is justified true belief knowledge? Analysis, 23, 121–123. Goldman, A. (2001). Social epistemology. In: E. N. Zalta (Ed.), Stanford encyclopedia of philosophy. Retrieved April 2, 2004, from http://plato.stanford.edu/archives/win2003/goldman Greco, J. (2002). Virtue epistemology. In: E. N. Zalta (Ed.), Stanford encyclopedia of philosophy. Retrieved April 2, 2004, from http://plato.stanford.edu/archives/win2003/greco Hofer, B. K. (2004). Exploring the dimensions of personal epistemology in differing classroom contexts: Student interpretations during the first year of college. Contemporary Educational Psychology, 29, 129–163. Kitchener, R. (2002). Folk epistemology: An introduction. New Ideas in Psychology, 20, 89–105. Klein, P. (2001). Skepticism. In: E. N. Zalta (Ed.), Stanford encyclopedia of philosophy. Retrieved April 2, 2004, from http://plato.stanford.edu/archives/win2003/klein. Kvanvig, J. (2003). Coherentist theories of justification. In: E. N. Zalta (Ed.), Stanford encyclopedia of philosophy. Retrieved April 2, 2004, from http://plato.stanford.edu/archives/win2003/kvanvig Mason, L. (1996). An analysis of children’s construction of new knowledge through their use of reasoning and arguing in classroom discussions. International Journal of Qualitative Studies in Education, 9(3), 411–433. Moser, P. K. (1995). Epistemology. In: R. Audi (Ed.), Cambridge dictionary of philosophy (pp. 233–238). New York: Cambridge University Press. Moser, P. K., & vander Nat, A. (2003). Human knowledge: Classical and contemporary approaches (3rd ed.). New York: Oxford University Press. Murphy, P. K. (2003). The philosophy in thee: Tracing philosophical influences in educational psychology. Educational Psychologist, 38, 137–146. Nussbaum, E. M., & Sinatra, G. M. (2003). Argument and conceptual engagement. Contemporary Educational Psychology, 28, 384–395. Pollock, J. L., & Cruz, J. (1999). Contemporary Theories of Knowledge (2nd ed.). New York: Rowman & Littlefield. Qian, G., & Alvermann, D. (1995). Role of epistemological beliefs and learned helplessness in secondary school students’ learning science concepts from text. Journal of Educational Psychology, 87(2), 282–292. Quinton, A. (2001). The rise, fall, and rise of epistemology. In: A. O’Hear (Ed.), Philosophy at the New Millennium (pp. 61–72). New York: Cambridge University Press. Schommer-Aikins, M. (2002). An evolving theoretical framework for an epistemological belief system. In: B. K. Hofer, & P. R. Pintrich (Eds), Personal epistemology: The psychology of beliefs about knowledge and knowing (pp. 103–118). Mahwah, NJ: Erlbaum. Sosa, E. (1994). Knowledge and justification (Vol. 1). Brookfield, VT: Dartmouth Publishing Company Ltd.

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Southerland, S. A., Sinatra, G. M., & Matthews, M. R. (2001). Belief, knowledge, and science education. Educational Psychology Review, 13(4), 325–351. Steup, M. (2001). The analysis of knowledge. In: E. N. Zalta (Ed.), Stanford encyclopedia of philosophy. Retrieved April 2, 2004, from http://plato.stanford.edu/archives/win2003/steup Vosniadou, S., & Brewer, W. F. (1994). Mental models of the day/night cycle. Cognitive Science, 18, 123–183. Williams, M. (2001). Problems of knowledge: A critical introduction to epistemology. New York: Oxford University Press. Young, J. O. (2001). The coherence theory of truth. In: E. N. Zalta (Ed.), Stanford encyclopedia of philosophy. Retrieved April 2, 2004, from http://plato.stanford.edu/archives/win2003/young

Chapter 11

Conceptions of Learning and the Experience of Understanding: Thresholds, Contextual Influences, and Knowledge Objects Noel Entwistle Introduction This chapter takes a perspective on conceptual development and change that derives mainly from research into teaching and learning at university level, but is also based on ideas about the nature of concepts, epistemological development, and contextual influences on the emergence and use of conceptions. An important distinction has to be drawn between concepts that are used to describe the physical world and those which describe human behaviour. The latter not only carry with them descriptions of a particular way of thinking about a construct, but also have implications for action. The starting point of this chapter is a set of overlapping ideas about conceptions of knowledge and learning. In both, there is a transition in the development where a marked qualitative change takes place. At this stage, ways of thinking about aspects of knowledge and processes of learning become integrated to create a conception that can affect the approaches to studying adopted by students and the level of understanding they reach in their courses. Whether changes in behaviour actually do flow from conceptual development, however, depends on the context within which students are operating and the particular circumstances facing them. Crucial for students is their intention in approaching learning tasks. Only through a deep approach, which links intention with learning processes, can the kinds of conceptual understanding that degree courses demand be reached. This chapter explores students’ experiences in developing academic understanding to provide a counterpoint to the research into conceptual development. The various elements within the chapter are finally brought together to show parallels in the development of conceptions and of understanding, and in the ways in which context influences the expression of conceptions in approaches and actions.

Reframing the Conceptual Change Approach in Learning and Instruction Copyright © 2007 by Elsevier Ltd. All rights of reproduction in any form reserved. ISBN: 0-08-045355-4

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Concepts and Conceptions A necessary starting point is to look at the sense in which the terms concept and conception are being used here. “Concept” is most frequently used to describe a grouping of objects or behaviours with the same defining features that has become recognised through research or widespread usage. The same term has also been used in the literature on conceptual change to indicate individuals’ different ways of thinking about a particular grouping. Here, for clarity, these individual variations are referred to as “conceptions”. The early work on conceptual development, based on memory research, described the importance of building up an extensive set of clearly defined concepts that are well differentiated from similar ones (Ausubel, Novak, & Hanesian, 1978). And it also suggested the existence of hierarchical concept “trees” in which narrower concepts are subsumed within broader ones, rather like a zoological classification system (Baddeley, 1976). While there was good evidence for some such logical mental organisation of concepts, the effect of everyday experience and individual differences in conceptualisation make the structures much less straightforward. Conceptions evolve through increasing knowledge and experience as they bring together additional aspects of the concept, moving closer, in most cases, to commonly agreed definitions and also becoming more inclusive. Physical concepts help us understand the world around us and also to use those understandings to our advantage in controlling and utilising its materials and properties. Social science concepts help us to make sense of our personal and social worlds, and the conceptions that people develop also have a use — to provide guidelines for action. Kelly (1955) was one of the first psychologists to draw attention to the idiosyncratic nature of such conceptions. He argued that people develop their own personal constructs as a way of making sense of their own perceptions of the world and of social behaviour. In trying to understand initially puzzling phenomena, Kelly suggested that individuals build up their own personal constructs from their experiences. A personal construct was seen as a device for construing or interpreting perceptions that helps not only in understanding other people’s behaviour but also in controlling the person’s own behaviour. Kelly saw these personal constructs as being continuously modified through subsequent experiences and, as they related to specific contexts, they were seen as explaining only a limited set of occurrences. These ideas paved the way for a more contextualised view of the way conceptions develop.

Contextualisation of Conceptions Educational research in the development of concepts in the “hard” sciences initially developed from memory research and therefore concentrated on helping students to replace naïve concepts with scientifically accurate ones which have greater explanatory potential. The more advanced conceptions were expected to replace those derived from everyday experience, with the change sometimes occurring in response to the “correct” concept being offered. But increasingly it is being recognised that conceptual change can be a slow

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process, as in the change from novice to expert that involves both the restructuring of existing conceptions and an awareness of a broader view. In trying to understand what is going on, it is important to make a distinction between the process of conceptual change and the end result of conceptual change … With increases in age and with expertise, we not only have a restructured system, in the sense of developing radically different representations of reality that were not available before; … [we also acquire] a more flexible system, a system that makes it easier to take different perspectives and different points of view … [Of course], it is difficult to understand other points of view if you do not even recognise what your own point of view is … Increased awareness of one’s own beliefs and presuppositions … [or metaconceptual awareness] … is a necessary step in the process of conceptual change. (Vosniadou, 1999, pp. 11–12) Recent research has also been showing that conceptions are developed in relation to certain situations and demands, and for distinct purposes, and so have to be seen as contextualised (Halldén, 1999; Halldén et al., 2002). Halldén (1999) has argued that conceptions can be contextualised in a variety of ways. Some have their roots in everyday experience — a situational context. Such conceptions, developed in everyday interactions with, say, parents or peers, will often appear naïve or incomplete when compared with those required for sophisticated academic understandings of phenomena. Halldén has also pointed to contextualisation within the “more embracing ideas” (1999, p. 64) of a cognitive context, with educational institutions providing such contexts in terms of generally agreed concepts, but this context also includes the conceptions built up by the individual learner. In academic debate, linkages between concepts are generally made explicit, but in the learning experiences of students, influenced by their own personal cognitive context, the connections often remain tacit or implicit, and may not even be recognised. In developing their conceptions, students experience the limitations suggested by Meno’s paradox: an adequate conception often requires a student to create links with a broader conceptualisation of the topic, which has to be built from the very ideas that the student has yet to understand. In practice, the initial grasp of abstract concepts will often be incomplete or inaccurate, but it is improved by revisiting the ideas several times and conceptualising them in increasingly complete and complex ways. The final form of contextualisation identified by Halldén is the cultural context or the form of discourse in which the concept is discussed (whether everyday or academic), with development being an inevitably gradual and recursive process through multiple and varying experiences. Learning is not to be looked upon as a linear process in which we first learn ‘facts’ … and then try to understand these facts … Rather learning is to be regarded as a simultaneous processing of these levels where the learner is continuously oscillating between [them] … In the beginning … both the understanding of the meaning of facts and the theoretical understanding are vague … When we are trying to learn something entirely new, our point of

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Halldén goes on to distinguish three distinct processes within conceptual change. Early research into the development of scientific conceptions focused on the first of these — replacing naïve conceptions with “approved” versions through direct teaching or reading. The second process involves providing an entirely new conception or conceptual framework, but in a way that enables students to reinterpret their existing knowledge through the lens of this new way of thinking. In this way, the new conception may have transformative properties, leading to more inclusive, and more complex, representations. The final process involves the independent development of a new conceptualisation from people’s own experiences and knowledge, which enrich their “repertoire of conceptualisations” (Halldén 1999, p. 56). These repertoires will include naïve conceptions acquired early in life, others developed to make sense of subsequent everyday experience, and those which are personal variants of the formally defined concepts introduced in educational contexts. Particular events and circumstances will trigger conceptions acquired in equivalent contexts, as an automatic response. Subsequent reflection, or the reactions of others, may cause an alternative conception to be brought to the fore. Such reflection can be seen in Halldén’s research into young children’s conceptions of the earth, which suggests that an initial flat-earth conception changes towards a more accurate representation, in response to subsequent questions from the interviewer (Halldén et al., 2002). The sudden switches in conception observed in this study indicate that several conceptions had been previously established that could be triggered by specific questions. If we accept that people acquire and retain repertoires of conceptions that have been developed out of particular sets of experiences within specific contexts, and that these are brought into play by similar experiences, contexts and situations, and for specific purposes, then we should arrive at a more complete picture of conceptual development. In higher education, all three of Halldén’s processes of conceptual change play a part. Students are given concepts, some of which can be understood immediately through prior knowledge and experience, although the more abstract or counter-intuitive concepts often cause difficulty before a satisfactory individual conception can be achieved — satisfactory, that is, to both the learner and the teacher. Eventually, students are expected to develop their own distinctive conceptions of the discipline and to describe and justify their understandings within a well-validated academic discourse. In a recent project investigating the influence of teaching–learning environments on student learning (ETL project, 2005), we have identified a category of concepts that appear to have a clear transformative property. Meyer and Land (2002, 2006) have labelled these as threshold concepts, with the notion of “opportunity cost” in economics being a particularly clear example. A threshold concept can be considered as akin to a portal, opening up a new and previously inaccessible way of thinking about something. It represents

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a transformed way of understanding, or interpreting, or viewing something without which the learner cannot progress. As a consequence of comprehending a threshold concept, there may thus be a transformed internal view of subject matter, subject landscape, or even world view. (Meyer & Land, 2003, p. 1) This notion of threshold concepts that open up a way of thinking is valuable in making sense of students’ difficulties in many subject areas, but it is particularly important in considering certain concepts in the social sciences. Conceptions developed out of experience from social situations lead to the formation of personal constructs that are used both to make sense of other people’s behaviour and to guide one’s own. Thus, if there are threshold concepts in relation to academic learning, they are likely to affect the way in which students go about their everyday studying. We shall come to this shortly but, first, some explanation is needed of the distinctive research approaches that have been used to identify conceptions relating to student learning.

Research Approaches to Investigating Student Learning The earliest systematic work on students’ ways of studying used Likert-type attitude scales to assess differences in study methods and attitudes to teaching (Brown & Holtzman, 1966). More recent research in Europe and Australasia has used a combination of interviews and inventories to identify concepts and categories that provide clear ways of describing variations in the ways in which students learn and study. This had its origins in the work of a research group in Gothenburg in the 1970s led by Ference Marton, which introduced a distinctive qualitative research approach that was subsequently developed into phenomenography (Marton & Booth, 1997). The main method involved interviewing students about their ways of learning, but a distinctive aspect of the interviewing took students, stage by stage, into deeper and deeper reflections on the processes of learning and studying involved. The subsequent analysis of the transcripts involved not just establishing categories describing the main similarities and differences in students’ responses, but also systematically exploring the relationships between those categories, and clarifying their meaning. Subsequent work on phenomenography identified the main conceptions held by students of a variety of key scientific concepts, showing how incomplete conceptions contained elements of the whole, while others represented a historical way of conceiving the idea (Marton & Booth, 1997). This research displays a cross-section of the differing stages in conceptual development reached by the students interviewed and draws attention to the importance of looking critically at the meaning of the categories, a process which often identifies several types of relationship between them. One problem with the way in which phenomenography developed was the tendency to focus on conceptions without reference to the context in which they were formed, and this has been a particular focus of criticism (Säljö, 1997). Other work using the same general approach has sought to retain the link with context, using a variety of approaches that involve what Svensson (1997a) has described as contextual analysis, while a different

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technique — intentional analysis developed by Halldén — has also been used in research in student learning (Scheja, 2002). Interview studies, in their varying forms, have played a major part in establishing the concepts described in this chapter, with their meaning and validity having been explored further through many surveys in several countries. These surveys have used inventories to refine the definition of the categories and to determine the statistical relationships that exist between them (see, for example, Entwistle & Ramsden, 1983; Biggs, 1987; Vermunt, 1998; Meyer, 2002). The interplay between qualitative and quantitative methods has proved a powerful way of developing a better understanding of student learning and how it is influenced by teaching.

Concepts Describing Students’ Ways of Thinking About Knowledge and Learning Conceptions of Knowledge In research into the conceptual development in the “hard” sciences, the “target” concepts are always clearly defined and students’ conceptions can be seen as moving towards those targets. The qualitative differences found in conceptions within the social sciences are less easy to interpret, as the meaning of the concept may well be contested by theorists offering differing definitions. The notion of “knowledge” is certainly affected by such controversy with post-modernists confronting positivists over relativism. Such varying interpretations can also be seen in the variation of conceptions among university students, but there is also evidence of progression towards more complete conceptualisation. Perry (1970) conducted a seminal investigation into changes in the ways students at Harvard and Ratcliffe Colleges thought about knowledge, which seems to link mainly with the third of Halldén’s processes of conceptual development. Perry identified a series of positions or stages through which students progressed as they gradually acquired the ways of thinking characteristic of academic discourse. Although staff provide the “raw material” to encourage changes in conceptions of knowledge, few courses examine the nature of subject knowledge in a way that allows students to progress epistemologically through direct teaching. Rather, a realisation of the existence of alternative conceptions of knowledge dawns only slowly as the ways of thinking within academic discourse are added to their conceptual repertoires. Perry’s interviews enabled him to distinguish dualistic thinking — expecting a single “right” answer — from relativistic reasoning in which students are able to accept uncertainty in knowledge and use evidence and logic to test alternative explanations. Perry traced the initially hesitant move from dualistic to relativistic thinking which, once established, could lead on to a firm, evidentially based commitment to a particular interpretation or perspective. The commitment among the students Perry interviewed was associated with a tolerant view of competing interpretations, and also moral feelings about taking up justifiable stances. The top half of Figure 11.1 summarises the steps in this epistemological progress and draws attention to the transformative characteristics of the emergence of relativistic thinking. For Perry (1988), this stage represented a pivotal position that allows students to see

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Conceptions of knowledge Relativism

Dualism Knowledge as absolute, provided by authorities

Multiple perspectives opinions of equal value

Awareness of knowledge as provisional

Evidence used to reason among alternatives

Commitment to a personal, reasoned perspective

Pivotal position

Recognising differing forms of knowledge and learning processes

Expanding awareness through a broader, integrative conception

Changing as a person sense of identity

Threshold Acquiring factual information

Memorising what has to be learned

Applying and using knowledge

Understanding what has been learned

Reproducing

Seeing things in a different way

Seeking meaning

Conceptions of learning

Figure 11.1: Categories used in describing conceptions of knowledge and learning.

knowledge in a crucially different way, and so becomes a threshold which opens up the subject in important ways. (Relativism) has taken us over a watershed, a critical traverse in our Pilgrim’s Progress … In crossing the ridge of the divide, … (students) see before (them) a perspective in which the relation of learner to knowledge is radically transformed. In this new context, ‘Authority,’ formerly a source and dispenser of all knowing, (becomes) … a resource, a mentor, a model … (Students) are no longer receptacles but the primary agents responsible for their own learning … As students speak from this new perspective, they speak more reflectively. And yet the underlying theme continues: the learner’s evolution of what it means to know. (Perry, 1988, p. 156) The recognition of both knowledge and values as provisional can bring with it a feeling of insecurity, which may inhibit further development for a while, or lead to retreat into the “safer” territory provided by the apparent certainty of dualism. If students manage to move on, they develop not just an understanding of the academic discourse and ways of reasoning used in the discipline they are studying, but also the greater tolerance of other interpretations of evidence that Vosniadou (1999) sees as a more general characteristic of conceptual growth. Subsequent research has largely confirmed the developmental trend identified by Perry (Hofer & Pintrich, 2002), but with some important qualifications suggesting gender differences (Clinchy, 2002), as well as suggestions that the trend is not unidimensional. Schommer-Aikins (2002) has found several factors within her measures of epistemological development that follow rather different trajectories, although some of these seem to overlap with the descriptions of students’ conceptions of learning (Pintrich, 2002).

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There are certain aspects of Perry’s scheme that deserve further consideration. First, the recognition that students can regress in their epistemology suggests that the less developed conceptions of knowledge are not entirely replaced by relativistic ways of thinking, and so we have further evidence of co-existence of conceptions and the alternation between conceptions due to contextual variations. Second, if we examine the nature of the categories there is some confusion. Perry found good evidence of progression towards more sophisticated views of knowledge, but his descriptions of relativism bring together two rather different aspects — philosophical views on the nature of knowledge and students’ gradual recognition of how evidence is used in their discipline to justify conclusions or theories. The extent to which students will perceive relativism in its philosophical sense will depend markedly on the subject area: in the sciences, competing theories can be tested against some generally agreed measurements, while in the humanities the contested nature of knowledge and its relativism becomes self-evident, and so is more likely to be recognised by the students. It seems likely, therefore, that students in different subject areas will be seeing relativism in importantly different ways. Conceptions of Learning Säljö (1979) investigated different conceptions of learning by interviewing adults with differing levels of educational experience. He found a series of categories that differentiated broadly between learning as “reproducing knowledge” and learning as “seeking meaning”, a distinction which parallels the changes in students’ ways of thinking about knowledge. The earlier categories describe learning simply in terms of acquiring factual information, with learning being equated with memorising what has to be learned, and reproducing it as required. A further step recognises the way learning depends on using knowledge in future situations, before reaching a crucial transformation in ways of thinking about learning. The emphasis shifts from seeing learning as the accumulation and use of unrelated pieces of information at the behest of others, to viewing it as a process in which understanding is developed and knowledge takes on a personal meaning. Here, a threshold is again reached and the landscape of learning changes, as relativism in relation to learning processes is appreciated. The final step in Säljö’s progression describes learning as seeing things in a different way (as with the threshold concepts mentioned previously) and implies the use of a more sophisticated conceptualisation of the material, with existing representations being reorganised into forms of understanding that fit reality more closely and are more satisfying to the learner. This set of categories, shown in the bottom half of Figure 11.1, describes qualitative differences that were empirically related to the length and type of education received, and so appear to be developmental. In a subsequent study, Säljö (1982) found that adults who think about learning in a more sophisticated way recognise a variety of different learning processes, and understand that effective learning depends on finding the most appropriate ways of tackling a particular task for a specific purpose, in a given context. This awareness can also lead to a more active engagement with learning as students monitor their progress towards understanding. Until students have grasped the importance and value of developing their own individual conceptions and understandings of academic material, they are unlikely to appreciate the implications of contrasting learning processes. The recognition

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of learning as developing personal meaning, rather than just acquiring other people’s knowledge, thus creates what can be seen as an emergent property of the more advanced conceptions — sometimes referred to as metalearning — which parallels the notion of “metaconceptual awareness”, mentioned in the quote from Vosniadou, earlier. Here we have not just a potential repertoire of conceptions, but also an advanced conception that includes a recognition of how best to marshal the different learning processes to achieve academic goals. We also have two processes — rote learning and meaningful learning (Ausubel et al., 1978), which are different in kind, thus creating, as with Perry’s categories, a progression which is not straightforward, being influenced by experience and circumstances. Students initially rely on rote learning in a routine way, and only gradually recognise that other kinds of learning are more appropriate for the tasks they meet in higher education. Although staff will generally expect students to be already well aware of the kinds of learning they should be using, that is often not the case. If we take the parallels with Perry’s students, the metacognitive awareness of multiple ways of learning can be expected to emerge, along with the recognition of the academic discourse and use of evidence appropriate to their specific discipline, only in the latter stages of a degree course. Other research on conceptions of learning (Marton, Dall’Alba, & Beaty, 1993) suggested a sixth category — changing as a person — which indicates a distinct form of learning that goes beyond the cognitive aspects reported by Säljö. It links the development of conceptions of learning to a changing sense of identity and so introduces the broader sense of learning also seen in Perry’s final category.

Approaches to Learning and Studying As we have seen, the more sophisticated conceptions of both knowledge and learning are transformative, leading to greater metacognitive awareness. It would be strange, then, if such changes in ways of thinking about knowledge and learning did not have marked effects on how students study, or on the quality of learning achieved. Later on we shall see that this does happen, although not in a straightforward way. First, however, we need to explore how students go about their everyday study activities. In the literature, there are differing sets of terms and constructs describing these differences from various theoretical perspectives, but with a good deal in common (Entwistle & McCune, 2004). For simplicity, we will use just one set here, one that is based on the work of Marton and his colleagues. The construct of approach to learning was derived from interviews with students who had been asked to read a short academic article that contained evidence used to build up an argument and reach a conclusion (Marton & Säljö, 1976). The students were told that they would be asked questions on the article afterwards. First, they were asked to summarise the general meaning of the article and then to explain how they had gone about their reading. Analysis of the first question indicated substantial differences in the level of understanding of the author’s meaning. The second question was analysed to try to provide reasons for these different levels of outcome and the researchers found a major distinction in the focus of attention students reported. Some students had adopted what came to be called a surface approach: they concentrated on the words and information, trying to spot

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questions likely to be asked and then memorising the answers. Their focus was thus on the information itself, rather than on its meaning or the relationship between evidence and conclusions. Other students had been trying to understand the meaning of the article for themselves, which involved paying careful attention to how well the evidence supported the conclusions — a deep approach. This contrast can be seen in comments of two students in later interviews at Lancaster. In reading the article, I was looking out mainly for facts and examples. I read the article more carefully than I usually would, taking notes, knowing that I was to answer questions about it. I thought the questions would be about the facts in the article … This did influence the way I read; I tried to memorise names and figures quoted, etc. I tried to concentrate — too hard — therefore my attention seemed to be on ‘concentration’ rather than on reading, thinking, interpreting and remembering — something I find happening all the time I’m reading text-books. I read more slowly than usual, knowing I’d have to answer questions, but I didn’t speculate on what sort of questions they’d be. I was looking for the argument and whatever points were used to illustrate it. I could not avoid relating the article to other things I’d read, past experience, and associations, etc. My feelings about the issues raised made me hope [the author] would present a more convincing argument than he did, so that I could formulate and adapt my ideas more closely, according to the reaction I felt to his argument. (Entwistle, 1988, pp. 77–78) Svensson (1977) analysed the Gothenburg interviews independently and saw an equivalent distinction, but with a different emphasis. He described the difference as atomistic and holistic, contrasting students who were merely noting and ordering parts of the text with those who were “integrating parts by use of some organising principle” (Svensson, 1997b, p. 65). The atomistic focus brings with it the use of rote learning, while a deep, holistic approach depends on making connections between ideas. Developing understanding often involves an interplay between inductive and deductive thinking, considering the patterns of relationships in relation to the whole topic and examining the detailed evidence and logic. Here we see parallels with relativistic reasoning and the more integrative conceptions of learning. The metacognitive awareness of the uses of contrasting learning processes is also a feature of deep approaches; students intending to understand may choose to supplement inductive and deductive thinking with rote-learning, where that is a prerequisite for understanding. The deep approach thus can include the memorising of detail or terminology, although its focus will still be on the development of personal understanding. Subsequent studies developed the contrast between deep and surface approaches by asking students about their experiences of everyday studying (Entwistle & Ramsden, 1983). These investigations confirmed the validity and importance of the distinction across a wide range of educational contexts and cultures (Richardson, 2000), but there has been less agreement about the extent to which approaches to learning are consistent across contexts. The evidence on the contextualisation of conceptions, and Marton’s own ideas about

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the nature of approaches to learning, indicate that approaches must be variable. As everyone has the basic mental processes used in deep and surface approaches, there has to be an element of choice in deciding how learning tasks are tackled. Previous success in using one or other approach will, however, lead to its subsequent use, and can build towards its more habitual use. To the extent that the learning required in different courses, or by different academic staff, is similar, a relatively consistent approach can be expected, but where the contexts are perceived to be different, then the approaches will vary accordingly (Ramsden, 1997). We thus have to accept elements of both consistency and variability in approaches to learning (Thomas & Bain, 1984). The element of consistency means that students can be asked about their typical approaches using inventories, with the deep/surface dichotomy generally emerging from the factor analysis of students’ responses (Biggs, 1993; Entwistle & McCune, 2004; Richardson, 2000). Questionnaire surveys have repeatedly shown overall relationships between approaches and students’ experiences of instructional methods, which imply an important element of stability (ETL project, 2005), but interview studies have drawn attention to individual reactions to the same experiences of teaching, and also the effect of this specific context in determining whether an established approach is changed. In a study by McCune (2000), for example, a student described how a tutor’s encouragement of a critical approach in reading had affected her approach. That’s really made me think a lot more about what I’m actually reading … I think that’s helped me with my psychology as well, because I have to keep remembering … that they’re only ideas and psychologists’ views on something, and it’s not actually dead set … It makes me less trusting of what I’m reading, but in a way it makes me feel more independent in my work, … less like I’m being taken in by what they’re writing, if I actually think, ‘Why are they saying (that)?’ (Entwistle, McCune, & Walker, 2001, p. 121) From other comments in this study, it appears that students have to be both alert and “ready” if they are to take advantage of a tutor’s encouragement to use a deep approach (McCune, 2000, 2004). There also has to be sufficient incentive to change established habits. Having a conception of learning that embraces personal understanding is usually associated with an intention to adopt a deep approach, but the necessary learning processes may not always flow from that intention. There must be adequate prior knowledge, but the approach adopted also depends on the reasons for taking the course and feelings about the tutor. Context, circumstance and feelings all seem to affect whether a conception is fully expressed in action, or whether the intention to use a specific approach actually leads to the actions necessary to carry it through successfully (Calder, 1989). A similar conclusion can also be drawn from research into the conceptions of teaching held by academic staff. Over the last decade, a series of interview studies have described differences in the conceptions that academic staff have of teaching, with a main dichotomy emerging that contrasts teaching seen as the process of transmitting knowledge with teaching envisaged as the encouragement of conceptual development (Kember, 1998). At one pole, the focus is on the knowledge that the student is expected to learn, viewed entirely from the perspective of the member of staff. The opposite conception sees teaching as the

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process of encouraging the student to acquire a new way of looking at the world and includes descriptions of how best to explain the ideas to students in ways that open out the subject for them. This contrast can be illustrated in extracts from two lecturers interviewed as part of a study by Prosser, Trigwell, & Taylor (1994). I put great emphasis on behavioural objectives and making sure that I cover the syllabus thoroughly. I also think it is important that the students take away a good set of notes: it gives them confidence if they have those notes to study from and also gives them a clear idea of what they have to know in order to pass the exam. And I’ll help them by essentially writing the notes for them … In my teaching, I also try to make the students realise the importance of being accurate, for example in the way they write up lab reports … (and) I also give a lot of (tests) to make sure they know their stuff. Lectures provide a presence that a book doesn’t, and you can utilise that in the lecture by a directly engaging question. The lecturer can actually engage (the student) in that question, in a much more interactive mode … I suppose I’m saying that the function of the lecture is to bring inquiry to life … [In my teaching], I’m constantly challenging the students … to think something through for themselves. I think that generates a certain sense of intrigue and understanding is related to that. (And) in my preparation I actually have to create this (situation) every time, rather than just remember (the content). (The lecture) is a conversation in which there’s active listening involved. (Prosser, personal communication) Van Driel, Verloop, Van Werven, and Dekkers (1997) also found these two extremes in the views of the lecturers they interviewed, with staff who emphasised the transmission of knowledge also blaming students for poor performance and low levels of motivation. But they also described an intermediate category in which staff saw the importance of making the students actively involved in their own learning, while the staff retained responsibility for introducing and controlling those opportunities. There is some indication that an advanced conception integrates different forms of knowledge about the subject, teaching, and student learning (Entwistle & Walker, 2002) and also brings together the earlier approaches and actions into a flexible whole which can be varied according to circumstances, as explained in the following reflective comment from a physics lecturer. In the early days, it was a matter of being prepared, presenting content confidently and accurately, and being in control. Some days it worked well and others it was a struggle, and I never knew quite why — it must have been the students or perhaps the weather that day. Later, as I developed more mastery, I could teach in a more conversational style, maintaining a sense of theatre, creating and taking opportunities to engage students’ interest and thinking. Examples, demonstrations and questions can be chosen to maximise such engagement and wherever possible to elucidate and challenge students’ preconceptions. The experience of teaching now, from my point

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of view, is more akin to a masterful jazz musician improvising and interacting with partners, allowing the instrument itself to speak, to express and inspire — rather than having to clumsily pluck or blow to force a predictable outcome. (Entwistle & Walker, 2002, pp. 29–30) In interviews about teaching, staff often move interchangeably in their comments from expressions of general conceptions to descriptions of specific approaches and actions (Samuelowicz & Bain, 2001). Unless the questions are phrased in ways that focus clearly on one aspect, the intermixture of conceptions, approaches, and actions creates confusion in the analysis. Beyond the immediate confusion, however, a pattern emerges. Staff who have reached a sophisticated conception of teaching are more likely to show metacognitive awareness of the links between teaching and learning, and to develop flexible ways of teaching that are adopted to the needs of both the subject and the students. But an advanced conception does not also lead to the equivalent actions. There is accumulating evidence that lecturers are more likely to make explicit use of their conceptions when planning courses and thinking broadly about teaching (McAlpine, Weston, Berthiaume, & FairbankRoch , 2006) than when preparing their day-to-day teaching (Eley, 2006). It thus appears that conceptions of either learning or teaching become conscious and explicitly used only in certain circumstances and for particular purposes. In a recent study across five disciplines, we have found many instances in which staff have been prevented from teaching in what they would see as better ways by lack of time or resources, or by the competing pressures from research and administration (ETL, 2005). An undeveloped conception strongly limits the approaches and actions that follow from it, but a more sophisticated conception will not always lead to equivalent approaches and actions. It depends on context and circumstances. As we have seen, correlational studies of conceptions and approaches emphasise consistent relationships between conceptions and approaches, but interview studies bring out individual variations in reactions, indicating how relative stability in approaches can disguise important elements of variability and change. Because the evidence contains both these elements, either one can be highlighted, depending on the expectations and theoretical orientation of the researcher, but both remain part of the phenomenon being investigated (Entwistle, McCune & Walker 2001). Bringing together all these findings with those from other studies that have linked conceptions of learning with study strategies (Lonka & Lindblom-Ylanne, 1996; Vermetten, 1999), a chain of influence in terms of student characteristics can be traced from the more stable individual differences and prior knowledge, through conceptions of learning, to approaches to studying and levels of understanding (Entwistle, 2007). We come now to variations in the development of academic understanding, which seem to parallel the processes involved in conceptual growth, but on a broader canvas.

The Nature of Academic Understanding Although understanding is generally cited as being one of the main aims of higher education, with conceptual understanding being specifically mentioned, rather little research has

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been conducted into its nature or the way in which it is experienced by students. In psychology, understanding is often described in relation to studies of problem solving (Nickerson, 1985), but that is a special case. Academic understanding itself has only recently become a major focus in educational research, and then mainly at school level (Wiske, 1998; Newton, 2000; Entwistle & Smith, 2002). Much of the research on student learning has taken the meaning of understanding rather for granted, but there has been a series of small-scale studies investigating how understanding is experienced by final-year students as they prepare for final examinations (Entwistle & Entwistle, 1992, 2003). In the interviews, students were taken deeply into their experiences through successive invitations either to describe more fully or to explain their approaches. They were also asked specifically how they had experienced understanding. At its heart was a feeling of satisfaction, but its expression ranged from the sudden “aha” (as confusion on a particular topic was replaced by insight) to a less dramatic feeling coming from the ability to follow a lecture or from an emerging appreciation of the nature of the discipline itself. This feeling involved a recognition of the meaning and significance of the material learned and, on occasions, was explicitly linked to previous personal or professional experience. The feeling of understanding also included a sense of coherence and connectedness — “things clicking into place” or “locking into a pattern” — conveying an implication of completeness and closure. Nevertheless, students recognised that their understanding might well develop further, that understanding involved no more than provisional wholeness. Their current understanding was felt to be adequate for the present, but might well be modified and extended in the future. This almost paradoxical combination of completeness and potential for further development does seem to be central to the concept of understanding. Interview comments suggested that, even close to the completion of their degree, some of the students had not actively sought conceptual understanding, while several were so concerned about “passing” that they talked mainly about how they would meet the perceived exam requirements. These marked differences suggested that there was a hierarchy of forms of understanding (Table 11.1), which could be seen from students’ comments on their revision strategies. The initial analysis produced five categories of description that

Table 11.1: Contrasting forms of understanding in revising for final examinations. Students’ understandings were found to differ in three main ways 1. Breadth of understanding 2. Depth or level of understanding 3. Structure used to organise the material being learned a. imposing little or no structure on the facts learned b. relying exclusively on the lecturer’s structures c. producing prepared answers to previous years’ questions d. adapting own understanding to expected question types e. using own well-justified conception of the topic and subject area

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indicated the different ways in which students had prepared for the exam, but looking at the relationships between the categories suggested underlying differences that were more salient. The form of understanding depended on its breadth — the amount of material that the students sought to integrate within their understanding. It also reflected its depth — the amount of effort put into deepening understanding, leading to different levels of understanding. Perhaps most importantly, the form of understanding reflected the extent to which there was a coherent structure shaping the understanding, and whether this structure had been provided by the teacher or created through the student’s own active consideration of the topic. As the categories shown in Table 11.1 were derived from students’ descriptions of their understanding, and processes of reaching and using it, these seem to represent conceptions of the type of understanding students believed they would have to demonstrate. The apparent progression is towards greater self-reliance in constructing a personal understanding, rather than relying on the content and structure provided, and it also represents a conception broadening from a focus on individual topics to be revised prior to the exam, towards a conception of the subject as a whole and its underlying discourse. There are echoes here of both Halldén’s third process within conceptual development, showing independent development of a new conceptualisation, and Perry’s relativism in the sense of recognising and using the academic discourse. This development towards a coherent, idiosyncratic structure developed by students to encapsulate their understanding led to a further exploration of the data, focusing on what had been said about constructing the personal structures. It soon became clear that students who had put substantial effort into developing their own structure as part of the revision process, often experienced their understandings almost as entities that could be brought to mind, “viewed” in some sense, reconsidered and reshaped to fit new requirements, without fundamentally changing what had been understood. These entities were described as knowledge objects (Entwistle & Marton, 1994; Entwistle, 1998). (It should be noted that the term has been used in a rather different sense by some other researchers.) The main characteristics of knowledge objects came from their being seen “virtually as a picture”, and with components within them which could be used to pull in related information, as required. The details were not in the picture but could be accessed from it, while the structure offered the students a logical path through which to develop an argument in the essays they were required to write in demonstrating their understanding. Given the abstract nature of the idea of a “knowledge object”, more extensive quotes have been included here to clarify its meaning. The comments relate mainly to the use of summary notes in preparing for the exams and how the understandings were used during them. I can see that (part of my summary notes) virtually as a picture, and I can review it and bring in more facts about each part … Looking at a particular part of the diagram sort of triggers off other thoughts. I find schematics, in flow-diagrams and the like, very useful because a schematic acts a bit like a syllabus; it tells you what you should know, without actually telling you what it is … The basic diagram is on paper, but details about the diagram are added on later by myself in my head … The facts are stored separately, and the schematic is like an index, I suppose ….

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Students’ structured summary notes were often recalled as images in the exams, with the summary notes often structured to form a visual mnemonic, but the feeling of understanding did not always relate directly to imagery. Some students only visualised the structure of their notes during the examinations when they were in difficulty. For them, understanding was tapped directly, with visualisation being “available”, but not necessarily used. (In exams), I just clear my mind and something comes … You know it’s visual in some ways, but it’s also just there without necessarily being visual … (It’s not as if) you remember a page, and the page is locked in your memory. What I’m saying is that the ideas are locked in your memory and they display as a page when you’re thinking about it, but not necessarily when you’re putting it down … You can sort of by-pass the conscious perception of your memory: it may not be a visual memory, but (sometimes) it may have to be perceived as a visual memory … I think, in a stress situation like an examination, you don’t actually (have to) reach for it, it comes out automatically. That may show that it’s not actually a visual memory, as such, but a visual expression of ‘central memory’. (Entwistle & Entwistle, 2003, p. 38) Although knowledge objects have been described here in only one context — preparation for examinations — they have also been detected in students’ descriptions of writing coursework essays (term papers), but only when the students engaged actively and personally with the topic (Entwistle, 1995). Evidence of the potential longevity of knowledge objects has also been found — in one instance, over a period of several years, with visual imagery playing an important part in the retained structure. There is also anecdotal evidence to suggest that staff regularly make use of knowledge objects in preparing papers (or a chapter like this). Staff also have an awareness of knowledge objects as they are giving lectures, although evidence for this remains impressionistic (Entwistle, 1998). There is apparent similarity between a “knowledge object” and what has previously been called a “schema” (see Westen, 1999), but the knowledge object involves a sense of personal attachment to the way the understanding has been constructed, and also describes a particular way of bringing together ideas and evidence to justify an understanding within

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an academic discourse. It is thus a form of contextualised schema that can also be seen in terms of neurological descriptions of mental processes. Mind is a stream of conscious and subconscious experience … It is at root the coded representation of sensory impressions and the memory and imagination of sensory impressions … Long-term memory recalls specific events… It also re-creates not just moving images and sound but meaning in the form of linked concepts simultaneously experienced … By spreading activation, the conscious mind summons information from the store of long-term memory … and holds it for a brief period in short-term memory. During this time it processes the information, … while scenarios arising from the information compete for dominance … As the scenarios of consciousness fly by, driven by stimuli and drawing upon memories of prior scenarios, they are weighted and modified by emotion … which animates and focuses mental activity … What we call meaning is the linkage among neural networks created by spreading activation that enlarges imagery and engages emotion. (Wilson, 1998, pp. 119, 121, 122, 123, 126)

Conclusion This series of examples of conceptual development and experiences of understanding has helped to clarify the nature of advanced conceptions of psychological constructs, and how conceptions relate to approaches and actions. There is evidence of stability in the use of a particular conception, approach, or understanding, but also indications that people have repertoires that are built up within different contexts, triggered by specific contexts, and used for differing purposes. It is not at all clear just how many conceptions are likely to co-exist. Phenomenographic analyses have generally produced between three and seven distinguishable categories, although with no suggestion that these can co-exist within any individual. The tension between the idea of stable conceptions stored in long-term memory, and the evidence for contextualisation of conceptions, leads to intriguing questions about the relationships between differing conceptions within the memory, and how each is triggered. The descriptions of students using knowledge objects provides a window into the processes of concept formation, as the development of understanding becomes a more conscious process that can be reported. In the interviews at university level, conceptions and understandings were described as if they had become firmly established in memory, ready to be brought into play confidently and quickly in exams. But the knowledge objects, although well constructed, were not in a fixed form. They were perceived to be flexible and students talked about adapting them to differing demands and purposes. It appears that they provided a generic logical path to explain a topic within an exam, but one that could be modified to take account of the demands of the question and the anticipated audience of examiners. We suggested earlier that some social science concepts guide actions, and from the evidence presented it seems that the best-developed conceptions create emergent properties, seen for example in threshold concepts that open up the subject, and in the way

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metacognitive awareness of studying arises from an advanced conception of learning. But the path from advanced conceptions of knowledge and learning, to intentions to understand, and on to the actions that fulfil that intention within approaches to studying, is strongly affected by the context. It is influenced by the situation the student experiences in the classroom, the relationship with the teacher, the perception of the task, and the reward system in terms of grade criteria (Entwistle, 2000, 2007). Research into the conceptions of teaching held by staff also points to an advanced conception that develops over time into a metacognitive view of the links between subject knowledge, pedagogical knowledge and the needs of students. Whether or not the conception is expressed in equivalent actions, however, again depends on circumstances and context. It appears that an advanced conception of conceptual development must now be seen in a more complex way than was envisaged in the earlier studies based on cognitive psychology, while the ideas from social constructivism on their own also fail to capture the whole process. We need a theory that brings together cognitive constructivism with current ideas about the influence of context and evidence for the co-existence of rather different conceptions related to varying situational contexts. From a review of the current literature, such a theory is still no more than a gleam in the eye of the author.

Acknowledgements I am grateful to Ola Halldén for lengthy discussions about the nature of conceptions on a number of occasions, but in particular on the journeys between Athens and Delphi. I also found comments made by Patricia Alexander at the symposium very helpful in improving Figure 11.1. Above all, I am grateful to colleagues who have collaborated in the research on student learning over the years and, in particular, Velda McCune, Hilary Tait, and Paul Walker who were involved in the work on conceptions and approaches, and Dorothy and Abigail Entwistle who were involved with developing the ideas on knowledge objects.

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Marton, F., & Säljö, R. (1976). On qualitative differences in learning – I: Outcome and process. British Journal of Educational Psychology, 46, 4–11. Marton, F., Dall’Alba, G., & Beaty, E. (1993). Conceptions of learning. International Journal of Educational Research, 19, 277–300. McAlpine, L., Weston, C., Berthiaume, D., & Fairbank-Roch, G. (2006). How do professors explain their thinking when planning and teaching? Higher Education, 51, 125–155. McCune, V. (2000). The development of first-year students’ approaches to studying. Unpublished Ph.D. thesis. University of Edinburgh, Edinburgh. McCune, V. (2004). Development of first-year students’ conceptions of essay writing. Higher Education, 47, 257–282. Meyer, J. H. F. (2002). An overview of the development and application of the Reflections on Learning Inventory (RoLI). Durham, NC: University of Durham, Centre for Learning, Teaching and Research in Higher Education. Meyer, J. H. F., & Land, R. (2002). Threshold concepts and troublesome knowledge: Linkages to ways of thinking and practising within the disciplines. In: C. Rust (Ed.), Improving student learning: Improving student learning theory and practice – 10 years on (pp. 412–424). Oxford: OxfordBrookes University, Centre for Staff and Learning Development. Meyer, J. H. F., & Land, R. (2006). (Eds.) Overcoming barriers to student understanding: Threshold concepts and troublesome knowledge. London: Routledge. Newton, D. P. (2000). Teaching for understanding. London: Routledge Falmer. Nickerson, R. S. (1985). Understanding understanding. American Journal of Education, 93, 201–239. Perry, W. G. (1970). Forms of intellectual and ethical development in the college years: A scheme. New York: Holt, Rinehart and Winston. Perry, W. G. (1988). Different worlds in the same classroom. In: P. Ramsden (Ed.), Improving learning: New perspectives (pp. 145–161). London: Kogan Page. Pintrich, P. R. (2002). Future challenges and directions for theory and research on personal epistemology. In: B. K. Hofer, & P. R. Pintrich (Eds), Personal epistemologies: The psychology of beliefs about knowledge and knowing (pp. 389–414). Mahwah, NJ: Lawrence Erlbaum. Prosser, M., Trigwell, K., & Taylor, P. (1994). A phenomenographic study of academics’ conceptions of science learning and teaching. Learning and Instruction, 4, 217–232. Ramsden, P. (1997). The context of learning in academic departments. In: F. Marton, D. J. Hounsell, & N. J. Entwistle (Eds), The experience of learning (2nd ed.) (pp. 198–216). Edinburgh: Scottish Academic Press (now available at http://www.tla.ed.ac.uk/resources/EoL.html). Richardson, J. T. E. (2000). Researching student learning: Approaches to studying in campus-based and distance education. Buckingham: SRHE/Open University Press. Säljö, R. (1979). Learning in the learner’s perspective – I: Some Common-Sense conceptions. (Report 76). Gothenburg: University of Gothenburg, Department of Education. Säljö, R. (1982). Learning and understanding: A study of differences in constructing meaning from a text. Gothenburg: Acta Universitatis Gothoburgensis. Säljö, R. (1997). Talk as data and practice: A critical look at phenomenographic inquiry and the appeal to experience. Higher Education Research and Development, 16, 173–190. Samuelowicz, K., & Bain, J. D. (2001). Revisiting academics’ beliefs about teaching and learning. Higher Education, 41, 299–325. Scheja, M. (2002). Contextualising studies in higher education: First-year experiences of studying and learning in engineering. Ph.D. thesis. Department of Education, Stockholm University, Stockholm. Schommer-Aikins, M. (2002). An evolving theoretical framework for an ‘epistemological belief system’. In: B. K. Hofer, & P. R. Pintrich (Eds), Personal epistemologies: The psychology of beliefs about knowledge and knowing (pp. 103–118). Mahwah, NJ: Lawrence Erlbaum.

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Svensson, L. (1977). On qualitative differences in learning – III: Study skill and learning. British Journal of Educational Psychology, 47, 233–243. Svensson, L. (1997a). Theoretical foundations of phenomenography. Higher Education Research and Development, 16, 159–171. Svensson, L. (1997b). Skill in learning and organising knowledge. In: F. Marton, D. J. Hounsell, & N. J. Entwistle (Eds), The experience of learning (2nd ed.) (pp. 59–71). Edinburgh: Scottish Academic Press (now available at http://www.tla.ed.ac.uk/resources/EoL.html). Thomas, P. R., & Bain, J. D. (1984). Contextual dependence of learning approaches: The effects of assessments. Human Learning, 3, 227–240. Van Driel, J. H., Verloop, N., Van Werven, H. I., & Dekkers, H. (1997). Teachers’ craft knowledge and curriculum innovation in higher engineering education. Higher Education, 34, 105–122. Vermetten, Y. (1999) Consistency and variability of student learning in higher education. Doctoral dissertation. Brabant: Catholic University of Brabant. Vermunt, J. D. (1998). The regulation of constructive learning processes. British Journal of Educational Psychology, 68, 149–171. Vosniadou, S. (1999). Conceptual change research: The state of the art and future directions. In: W. Schnotz, S. Vosniadou, & M. Carretero (Eds), New perspectives on conceptual change (pp. 3–13). Oxford: Pergamon. Westen, D. (1999). Psychology: Mind, brain and culture (2nd ed.). Chichester: Wiley. Wilson, E. O. (1998). Consilience. London: Little, Brown & Co. Wiske, M. S. (Ed.). (1998). Teaching for understanding: Linking research with practice. San Francisco, CA: Jossey-Bass.

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Chapter 12

Conceptual Change in Physics and Physics-Related Epistemological Beliefs: A Relationship under Scrutiny Christina Stathopoulou and Stella Vosniadou

Introduction There is a diversity of theoretical and conceptual approaches to the construct of ‘epistemological beliefs.’ This is reflected by the fact that the construct is assigned different terms in the literature, such as personal epistemology, epistemic beliefs, ways of knowing, epistemological perspectives, epistemological reflection, epistemological thinking, epistemological theories, epistemological resources, etc. This diversity indicates that what we call ‘epistemological beliefs’1 may not be the same in all relevant studies, or at least, that the boundaries of the construct may differ (Hofer & Pintrich, 1997; Pintrich, 2002). As Pintrich points out, the key issue concerns what should be considered as the core or essence of personal epistemology and what should be left out of the definition or considered as related but distinct constructs. (Pintrich, 2002, p. 390) The present chapter draws on the more conventional definition of epistemology, as the study concerning “the nature and limits of claims to know” (Harre, 2002), and suggests that personal epistemological beliefs cluster in two general areas: the nature of knowledge, including structure and stability of knowledge, and the nature of the process of knowing, including source and justification of knowing (see also Hofer & Pintrich, 1997). We consider personal epistemological beliefs as individually held theory-like structures, namely, systems of beliefs that are nonetheless interconnected (see also Hofer & Pintrich, 1997).

1

We have decided to keep the term epistemological beliefs and, alternatively, personal epistemology since this is widely used by researchers in the field, until there is some consensus on a new terminology.

Reframing the Conceptual Change Approach in Learning and Instruction Copyright © 2007 by Elsevier Ltd. All rights of reproduction in any form reserved. ISBN: 0-08-045355-4

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This theoretical position is in line with the conceptual change approach adopted here, and suggests that personal epistemology forms initially a narrow but relatively coherent set of beliefs regarding the nature of knowledge and the process of knowing, which is based on the limited range of children’s initial experiences and information they receive from lay culture. This set of beliefs becomes gradually more differentiated as experience and/or cultural information accumulates, gradually changing some beliefs, but not others, and connecting them to different contexts of use. It should be mentioned that the term theorylike is used to denote an explanatory structure that can generate explanations and predictions, but which, unlike a scientific theory, is not explicit, well formed or socially shared. It is assumed that most individuals are not metaconceptually aware of their epistemological beliefs. The theory approach to epistemological beliefs adopted in the present study can be seen as a bridge between the developmental approach (according to which epistemological beliefs form a rather coherent, developmental structure that does not allow for within-stage of development variation (e.g., Baxter Magolda, 1992; Belenky, Clinchy, Golberger, & Tarule, 1986; King & Kitchener, 1994; Kuhn, 1991; Perry, 1998)), and the multidimensional approach (according to which personal epistemology is a system of rather orthogonal, uncoordinated dimensions, that are more or less independent, developing not necessarily in synchrony (e.g., Schommer, 1990, 1994; Schommer, Crouse, & Rhodes, 1992)). Conceptualizing epistemological beliefs as theory-like structures helps us understand better how they can be acquired and changed, how they can influence individuals’ learning in areas such as physics, and how it is possible to have different epistemological beliefs in different disciplines (Buehl, Alexander, & Murphy, 2002), since it allows for general and domain-specific beliefs to co-exist in an interconnected network (see also Hofer, 2000; Hofer & Pintrich, 1997), which is nonetheless contextually bound. Regardless of the conceptual and theoretical approach to the construct, changes in epistemological thinking have always been conceptualized as involving an ‘upward’ movement from dualistic/absolutist and objectivist views to more and more relativist, subjectivist, contextual, constructivist, and evaluative perspectives of knowledge and knowing (Hofer, 2002; Hofer & Pintrich, 1997; Pintrich, 2002). In this context, physics-related epistemological beliefs have been seen to change in a variety of specific ways. Songer and Linn (1991) focus on what they call ‘productive science views.’ According to their suggestion, scientific knowledge is a dynamic, socially constructed set of ideas, that progresses through either evolutionary or even revolutionary changes in perspective and, therefore, it is controversial, particularly in periods preceding discoveries, and is relevant to the lives of individuals and societies. Smith, Maclin, Houghton, and Hennessey (2000), following Carey and Smith (1993), describe the preferred, constructivist, epistemology of science as an epistemology in which students are aware of the central role of ideas in the knowledge acquisition process and of how ideas are developed and revised through a process of conjecture, argument and test. (Smith et al., 2000, p. 350) Driver, Leach, Millar, and Scott (1996) think that the more complex and presumably more mature epistemological views are characterized by ‘model-based reasoning.’

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According to this view, students understand that scientific inquiry involves evaluation of theories in light of new evidence, and that there may be multiple explanatory models, involving theoretical entities that cannot be observed. Also, description is clearly differentiated from explanation. Finally, other researchers emphasize that students’ views about the nature of scientific knowledge and knowing are strongly influenced by context (e.g., Elby & Hammer, 2001; Leach, Millar, Ryder, & Sere, 2000). We will use the term ‘constructivist epistemology’ to refer to a set of epistemological beliefs that have more or less the contextual, constructivist, and evaluative characteristics described above, having, however, in mind, that the context-dependence of epistemological beliefs challenges the across-contexts generalizations about the nature of knowledge and knowing. A constructivist personal epistemology has been found positively related to skills and attitudes important for learning. Previous studies have demonstrated the relationship of the construct with comprehension, learning, academic performance, and conceptual change. For example Ryan (1984), following and extending the work of Perry (1998), investigated the effect of epistemological development on comprehension and metacomprehension. He found that students who are relativists in their beliefs about knowledge, i.e., who perceive knowledge as context dependent, are more successful in comprehension monitoring and tend to use high-level comprehension strategies, as opposed to dualists (who perceive knowledge as factual, right or wrong). The latter are more likely to study for recall of facts from texts. Beliefs concerning the structure of knowledge, that is, considering knowledge as an accumulation of discrete, concrete, knowable facts, have been found to be related to poor text comprehension in such areas as the social and physical sciences. They have also been found to affect the comprehension and related problem solving of statistical text in a negative way (Schommer, 1990; Schommer et al., 1992). Beliefs concerning the stability/ certainty of knowledge, that is, viewing knowledge as unchanging, attaining/approximating absolute truth, have been found to negatively affect the interpretation of controversial evidence (Kardash & Scholes, 1996) and tentative text (Schommer, 1990).

Epistemological Beliefs and Physics Understanding There is some empirical evidence that supports the position that a constructivist physics epistemology facilitates physics understanding. Songer and Linn (1991) found that secondary students who viewed science as a dynamic process of developing and changing ideas, and also considered interpretation and integration of ideas as strategies that facilitate learning, were more likely to understand concepts in thermodynamics and to integrate them around scientific principles, than students who viewed science as an accumulation of true and unchanging facts. Qian and Alverman (1995) investigated the influence of secondary students’ epistemological beliefs on learning counter-intuitive science concepts from a refutational text (Newton’s first law vs. the ‘impetus’ theory). Their results showed that epistemological beliefs were strongly related to what they call ‘conceptual change learning’ regarding projectile motion. More specifically, the students who believed in simple-certain knowledge and quick learning were less likely to change their positions on projectile motion after reading an expository refutational text, compared to the students ‘who viewed knowledge as complex and evolving’ and learning as a time-consuming process.

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In our previous work we investigated the relationship between Greek secondary school students’ physics-related epistemological beliefs and physics understanding (Stathopoulou & Vosniadou, 2006). The beliefs of 394 Greek 10th-grade students were measured through a specially designed paper and pencil questionnaire, the Greek Epistemological Belief Evaluation instrument for Physics (GEBEP). The results revealed four dimensions underlying students’ beliefs: structure of knowledge, construction & stability of knowledge, attainability of truth, and source of knowledge. In a subsequent study we selected 38 students with the highest scores in all the dimensions underlying the GEBEP (the constructivist epistemological beliefs group) and 38 students with the lowest scores (the lessconstructivist epistemological beliefs group). These 76 students were administered a reliable instrument for measuring their conceptual understanding of Newtonian dynamics, the Force and Motion Conceptual Evaluation instrument (FMCE) (Thornton & Sokoloff, 1998). The design of the FMCE is based on results received by the extant science education research on students’ ‘misconceptions.’ FMCE has been administered to many thousands of secondary and university students in the US. It has also been used extensively in our research lab with secondary and university physics students in Greece. Thus, substantial information is available that allows us to know, on the basis of the students’ performance on this test, when they can be said to have understood Newton’s three laws, in other words, to have achieved conceptual change regarding force and motion. The results showed that beliefs regarding the nature of physics knowledge and the process of knowing are related to conceptual change in physics. More specifically, students’ epistemological beliefs concerning the structure as well as the construction and stability of physics knowledge were found to predict high scores in the FMCE, suggesting a deep understanding of Newtonian dynamics. Furthermore, the results showed that only the students in the constructivist epistemological beliefs group achieved a deep conceptual understanding of Newtonian dynamics. As shown in Figure 12.1, only 11 students out of

70

Number of students

60

27

50 E.B. Group

40 38

30

Less-constructivist Epistemological Beliefs (N=38)

20 10 0

11

0-21 22-43 Physics Conceptual Understanding on the basis of the FMCE score

Constructivist Epistemological Beliefs (N=38)

Figure 12.1: Only 11 students, all of whom were found to hold constructivist physics-related epistemological beliefs, were found to achieve high scores in the Force and Motion Conceptual Evaluation (FMCE) instrument.

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the 76 were found to have high scores in the FMCE and all of these students had constructivist physics-related epistemological beliefs. We have interpreted these results to suggest that a constructivist physics epistemology may be a necessary, although not sufficient, condition for conceptual change in Newtonian dynamics (Stathopoulou & Vosniadou, 2006). In a third study, we studied three 10th-grade students in the context of a Computer Supported Collaborative Learning (CSCL) environment2 implemented in the instruction of Newtonian dynamics (Mol, Stathopoulou, Kollias, & Vosniadou, 2003). These three students had strong non-constructivist beliefs regarding the source of knowing in physics, which limited their ability to a more effective use of the CSCL environment. For example, they believed that they should rely on the authority of the teacher to tell them which was the correct answer and, as a result, they had great difficulty resolving inconsistencies and disagreements by themselves through argumentation. Their epistemological beliefs prevented them from relying on themselves and developing the more reflective approach necessary in order to resolve, in a principled way, differences of opinion regarding physics knowledge. Why may Epistemological Beliefs Facilitate Conceptual Change? In their influential model of conceptual change as a rational process, Posner, Strike, Hewson, and Gertzog (1982), suggested four conditions3 for a successful conceptual change to take place in the learner’s conceptual ecology. The term conceptual ecology was used to describe the learner’s existing interrelated networks of concepts that influence the selection of a new concept playing a central and organizing role in thought. Among these, personally held ‘epistemological commitments,’ namely, assumptions or views concerning the nature of knowledge and knowing were considered as playing an important role. In a ‘revisionist’ approach of the initial overtly rational model of conceptual change, Strike and Posner suggested that “motives and goals and their institutional and social sources need to be considered” (1992, p. 162) as well, in attempting to describe a learner’s evolving conceptual ecology and understand the construct’s impact on conceptual understanding. The need to incorporate variables of motivational and affective character, such as personal beliefs and attitudes,4 into models of conceptual change is stressed by cognitive/developmental psychologists who also go beyond an approach that emphasizes the overtly rational nature of conceptual change (e.g., Pintrich, Marx, & Boyle, 1993; Pintrich, 1999; Dole & Sinatra, 1998; Gregoire, 2003; Sinatra, 2005). Hofer and Pintrich (1997) suggested that epistemological beliefs functioning as implicit theories interacting with the educational context can influence academic achievement indirectly by affecting goal orientation. In other words, epistemological beliefs can give rise to certain types of learning goals, such as mastery, performance, and completion goals, which in turn, can function as guides for cognitive and metacognitive strategy use. 2

The research was performed as part of the European Project ITCOLE (http://www.euro-cscl.org/site/itcole ) That is, dissatisfaction with the current concept and also the ineligibility, plausibility, and fruitfulness of the new concept. 4 For example, beliefs about the nature of knowledge, and knowing, beliefs about learning, about the role of self as learner, goal orientation, motivation to engage in academic tasks, interest/values. 3

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The conceptual change approach adopted in this study considers that, in many cases, the construction of a scientific concept, such as the concept of force, requires students to radically reorganize prior knowledge (Carey, 1985, 1992; Vosniadou, 1999, 2002, 2003; Vosniadou, in press). This is the case, because by the time they are exposed to systematic science instruction, students have already constructed an initial naïve physics based on their everyday observations and cultural experiences. This naïve physics is very different from the scientific theories to which they are exposed at school and can stand in the way of learning physics. We explain the phenomenon of ‘misconceptions’ observed in science classrooms, at all levels of education, as resulting from students’ attempts to add the new, to-be-acquired information to an incompatible knowledge base, thus forming synthetic models (Ioannides & Vosniadou, 2002; Vosniadou, 1999, 2002). We believe that conceptual change is a slow and gradual process that not only involves cognitive factors but is also influenced by motivational and affective variables, such as personal beliefs and attitudes, as well as by the physical and social/cultural environment (Dole & Sinatra, 1998; Pintrich, 1999; Sinatra, 2005; Vosniadou, 2002, 2003). According to our theoretical position, physics-related epistemological beliefs can influence the knowledge acquisition process directly or indirectly, just like ontological presuppositions and other beliefs of a motivational and affective character can do. They can influence both the kind of new information that is picked up from the physical and sociocultural context and the way in which this information is interpreted (Vosniadou, 1994, 2002, 2003; Vosniadou & Brewer, 1994). For example, beliefs in simple and/or certain knowledge may affect the learning process directly by focusing students’ attention on factual information, while beliefs in complex and/or evolving knowledge may cause students to focus more on patterns of relationships and their change over time. Figure 12.2 presents a skeletal theoretical framework for conceptualizing the relationship between physics-related epistemological beliefs and physics conceptual understanding.

Physical and Socio-Cultural Context

PresuppositionsCommitments-Beliefs

New Cultural & Observational Information

Ontological Epistemological Motivational

Cognitive& Metaconceptual Processes Strategy use Metaconceptual awareness Self-Regulation

Conceptual Understanding in Physics

Figure 12.2: A skeletal theoretical framework for conceptualizing the relationship between (physics-related) epistemological beliefs and physics conceptual understanding.

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Epistemological, ontological, and motivational beliefs may also affect students’ achievement indirectly, by influencing students’ learning goals, study strategies, and selfregulation, as discussed earlier. For example, the belief in simple knowledge may lead to the selection of rehearsal strategies to strengthen memorization and recall of piecemeal factual information. The skeletal framework presented in Figure 12.2 is based on prior work by Vosniadou regarding conceptual change in physical science (e.g., Vosniadou, 1994, 2003), and also takes into consideration some current suggestions about the role of motivational and affective as well as social/cultural variables (Pintrich, 1999; Pintrich et al., 1993; Strike & Posner, 1992).

A Case Study: How do Study Strategies Intervene? In the pages that follow we present preliminary results from a case study which attempted, through in-depth interviews, think-alouds, and observations, to understand the indirect effect of physics-related epistemological beliefs on physics understanding. The in-depth interviews were administered to 10 students, who were selected out of the 76 students in the Stathopoulou and Vosniadou (2006) study that investigated the relationship between physics-related epistemological beliefs and physics understanding. These 10 students were selected on the basis of their scores in the GEBEP, as well as in the FMCE. Five of the students had high scores in all dimensions underlying the GEBEP and also high scores in the FMCE, thus comprising the group of students who were considered to hold constructivist physics epistemologies and to have achieved an in-depth understanding of physics. In contrast, the remaining five had low scores in all dimensions underlying the GEBEP and also low scores in the FMCE, thus comprising the group of students considered to hold lessconstructivist physics epistemologies and to have poor/superficial physics understanding. It was hypothesized that the approach to learning and studying adopted by the students, that is, deep vs. superficial (e.g., Entwistle, this volume; Entwistle, Tait, & McCune, 2000), and the related selection of study strategies, may intervene in the relationship between personal physics-related epistemological beliefs and conceptual change in physics. More specifically, we hypothesized that a constructivist physics personal epistemology is more likely to guide students to the adoption of a deep approach to learning and studying, and therefore, to facilitate physics understanding, than a less-constructivist epistemology. Of course, the relationship between personal epistemology and physics understanding is likely to be a reciprocal one. As students develop a deeper understanding in physics their personal physics-related epistemologies would be likely to change. On the basis of these hypotheses, the five students who were found to score high on both the GEBEP and the FMCE were expected to adopt a deep approach to studying, as opposed to the remaining five students, with low scores on both the GEBEP and the FMCE, who were expected to adopt a superficial approach to studying. Following Entwistle (e.g., Entwistle, this volume; Entwistle et al., 2000), a deep approach to learning and studying involves goals of personal making of meaning, and accordingly, deep strategy use, such as integration of ideas, looking for patterns and underlying principles, examining in detail evidence and logic, and monitoring of understanding. It also involves metaconceptual awareness, that is, awareness of one’s own beliefs. A deep

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approach to learning and studying may also involve some performance goals, such as time management and organized studying that do not contradict the goal of personal making of meaning. In contrast, a superficial approach to learning is characterized by performance orientation, lack of purpose, and superficial strategy use, such as memorization of facts, and syllabus boundness. It is also characterized by lack of metaconceptual awareness.

Method Participants Ten 10th-grade Greek students participated in this study, six of whom were boys and four were girls. As mentioned earlier, these students were selected from a pool of students that participated in the Stathopoulou and Vosniadou (in press) study, on the basis of their scores in the GEBEP and the FMCE. Five of the students were selected because they had achieved high scores in all dimensions underlying the GEBEP and also high scores in the FMCE, whereas the remaining five were selected because they had achieved low scores in all dimensions underlying the GEBEP and also low scores in the FMCE. Materials and Procedure The interview started with a discussion with each participating student about the nature of physics knowledge, the role of teachers, textbooks, peers and the self in learning physics, as well as about the role of experience and experiment in the justification of knowing. This was done in order to re-examine the participants’ physics-related epistemologies and to reveal aspects of students’ approaches to physics learning and studying. The discussion also concerned the interviewees’ attitudes towards physics and their study goals and strategies. The second part of the interviews involved discussion, think-alouds, and observation as each student was faced with tasks, such as answering questions regarding particular dynamics-related situations, and solving problems in the area of Newtonian dynamics. We wanted in this way to re-examine the depth of students’ conceptual understanding of dynamics, but most importantly, to investigate in a context-sensitive way students’ approaches to learning and studying and particularly their strategies in the context of a problem-solving task. Some examples of the physics questions and problems that were used during the interviews are shown in Figure 12.3. We selected the particular questions and problems because they targeted some well-known ‘misconceptions’ of students, such as the ‘impetus misconception.’ Each student was interviewed for about two hours. Data regarding students’ grades in school physics were also collected.

Results The analysis of the interviews gave results consistent with those received by the instruments that were earlier used to measure epistemological beliefs and physics understanding

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Identify the forces acting on the ball.

i. Identify the forces acting on the sliding down girl. ii. In case that the boy on the top of the slide had pushed the girl instantly, do you believe that the same, as previously, forces would act on her, as she slides down the slide? Explain. iii. Consider the slide as a frictionless inclined plane of given length and angle. If you also know the girl’s mass and weight can you find her speed at the bottom of the slide? Explain.

An object with a mass of 2 Kg moves with a speed of 3 Km /h. Can you find the net external force on the object so that it keeps moving in the same direction with the speed of 3 km/h? If yes, find it. If not, what other information should you be given?

Figure 12.3: Some examples of physics questions and problems used in the interviews. (i.e., the GEBEP and the FMCE respectively). More specifically, all five students who were selected because they scored high in both the GEBEP and the FMCE, were indeed found to hold constructivist physics epistemologies and to have achieved conceptual change in Newtonian dynamics, in contrast to the remaining five students who were selected for their low scores in the GEBEP and the FMCE. Further analysis of the interview-data was done for the purpose of identifying patterns of responses pointing to what may be described as deep approach to studying, as opposed to superficial approach to studying. Preliminary results showed, as expected, that all five students who were found to hold constructivist physics epistemologies, and also to have achieved conceptual change in physics (in the area of Newtonian dynamics), had adopted what may be considered a deep approach to learning and studying. The remaining five students showed evidence of adopting

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a superficial approach to learning and studying. The criteria for identifying students’ approaches were defined, as mentioned earlier, on the basis of Entwistle’s descriptions of a deep as opposed to a superficial approach to learning and studying (Entwistle, this volume; also, Entwistle et al., 2000) and are presented in Table 12.1. They were categorized around three dimensions of learning and studying: goals, strategies, and metaconceptual awareness. A deep approach to learning and studying involves goals of personal making of meaning, and deep strategy use, such as integration of ideas. It also involves awareness of one’s own beliefs and thus, a good sense of understanding or failure to understand. A superficial approach involves performance goals, lack of purpose, and superficial strategy use, such as rote learning. We also assume that this approach is accompanied by low awareness of one’s own beliefs. To illustrate the differences in approaches to learning and studying adopted by students with qualitatively different physics-related epistemological beliefs, two students, John and Michael, were selected as examples. Below we present ‘in greater details’ results from the interviews of these two students. A closer look at John and Michael The two students who served as examples in the present study, John and Michael, were very different in terms of their personal physics epistemologies and their conceptual understanding of dynamics, but both had very high grades in school physics. Comparable performance in school physics was important for selecting the two students because controlling for this variable has the potential to help us understand better the influence of physics-related epistemological beliefs on physics understanding. John had scored high on all the factors underlying the GEBEP and also impressively high (40 out of 43) on the FMCE, while Michael had scored low on all the factors underlying the GEBEP and also very low (6 out of 43) on the FMCE. In what follows, we present excerpts from the transcribed interviews with the two students, in order to clarify the differences in their approaches to learning and studying. We will focus on the three criteria used to identify students’ approach to learning and studying, described in Table 12.1.

Table 12.1: Criteria for identifying students’ approach to learning and studying. Approaches to learning and studying

Criteria Goals

Strategy use

Deep approach

Meaning-making

Integrating ideas

Superficial approach

Performance orientation or lack of purpose

Rote learning

Metaconceptual awareness Being aware of one’s own beliefs Not being aware of one’s own beliefs

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Goals: Meaning Making vs. Performance Orientation John appears to seek personal making of meaning, by trying to work things out for himself, to relate them to what he already knows. John: Physics is relatively difficult for the average student because (…) you have to learn things, not by rote, but to learn, to put them in your mind. You need time to consolidate things… Interviewer: What do you mean by saying “to put them in your mind” and “to consolidate things”? John: I mean to relate them, that is, you don’t need to have [separate] laws for force and motion, and later other for energy and other for momentum. If you don’t relate them you can’t learn them really good. His meaning seeking is demonstrated by the fact that he monitors his performance and takes corrective steps, when he realizes failures in understanding: John: [Sometimes] you realize that something is missing, or that you haven’t really understood [something]. You have to go back, I mean, I don’t know what most students do, but I will go back, because if I haven’t understood, I can’t go on. Michael, on the other hand, appears to be performance-oriented in the sense that he worries not really about understanding for himself, but rather, about doing what, he thinks, the physics teacher expects of him. This seems to be his major concern and having the teacher’s approval is what he is after. This is rather clear in the following excerpt: Michael: When the physics teacher asks something, regardless of whom he asks, I say the answer to myself and if I was quicker in answering [than the other student], and the answer was correct, I believe that it is OK. I’m sure that I know that. Interviewer: Is this enough to be sure that everything is OK? Is there anything else [about your learning] to worry about? Michael: (laughing) No, no, no, and if the teacher tells you, which is not usual, “you’re doing well, keep working” then, I believe, that you are at a good level. Strategies: Integrating Ideas vs. Rote Learning John appears to adopt the deep strategy use such as integration of ideas on the basis of some organizing principles. He also admits that he finds it rather intriguing to try to relate ideas and to play around with ideas. John: Our mind can remember a lot of things, but you cannot overload it, so, you can keep in your mind some basic formulas that you can easily store in your memory (…) You can relate different parts of a theory.

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Christina Stathopoulou and Stella Vosniadou Thus you can relate formulas with other things that may help you to reproduce any formula needed (…) Formulas derive from theory. It’s easier to remember the formulas if you understand the theory. (…) [You know that linear velocity in uniform circular motion is equal to 2r/ T rather than 2T/r] because you remember from theory that linear velocity equals the arc length divided by the [corresponding] time. You know that the length of a circle is 2r and the corresponding time equals the period T, thus…

And later, when he was asked to identify the forces exerted on a girl sliding down a slide, and to find her speed at the bottom of the slide, he said: John: (having drawn the forces, in the scientifically accepted way, without difficulty) Right… her speed. Let me see, I can work either with kinematic equations or with energy (He starts working with kinematics, suggesting that there is friction between the slide and the girl). Interviewer: Suggest the friction is negligible. John: Then I would work with energy. Interviewer: I see, [in that case] you would prefer a different way. John: I don’t mind actually, both ways are good. Also, in another instance: John: [You may engage yourself in solving a problem that is not an assignment and looks strange or intriguing, because] it is the curiosity to deal with something that may not come to an end but you want simply to play around with this, to try this out. In contrast, Michael appears to adopt superficial strategy use, such as rehearsal and memorization: Michael: I have my own way, I mean with some formulas that are difficult [to remember] I find some key words and I keep them in mind so that when is needed I can remember exactly what it was about. Or by writing formulas, if you write them many times, I believe that your hand is getting used [to writing them] and at the moment you must write them it is easier, they come back in memory. Later, answering a question concerning what he does when he simply cannot remember a formula; for example, how he can decide whether the linear velocity in uniform circular motion is equal to 2r/T or to 2T/r he said: Michael: I believe that if you have learned a formula well, from my personal experience, because I’m good with formulas and I can remember them, hum I’m familiarized…, but I believe that if my mind gets stuck,

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the student will try to recall it; if he doesn’t, he will try to solve the problem and (…) he will get such a result that he will understand that something’s wrong. Interviewer: This is what you do? Michael: I am sure that I will write the formulas [right], because I know them, I mean, it is so easy to write them if you have learnt them (…) I have no trouble with formulas. I write the formula and everything goes right, the problem is easily solved (…). The problem goes itself, for me it is something that you can learn, what to do first, second, third, and the problem is solved like that. I mean it’s so easy. In another instance, Michael was given the mass and the speed of a moving object and was asked to find the net force on the object in case it kept moving in the same direction with the same speed. Michael: I usually have more data given. I believe that I should have been given a force F so that I could find the resultant force in the x direction, to find in the vertical direction that the weight equals the [normal force] N and then to find the friction, classical methodology. Something is missing. I believe that I should be given the F.

Being Metaconceptually Aware vs. Not Being Metaconceptually Aware Another important aspect of a deep approach to learning and studying is awareness of one’s own beliefs. Being aware of one’s own beliefs, means being also able to recognize different points of view. John is aware of his own beliefs and of the change in his beliefs. For example, he knows that he has abandoned the ‘impetus misconception,’ that is, that there must be always a force in the direction of the movement (McCloskey, 1983; Gunstone & Watts, 1985; Vosniadou, 2002; Ioannides & Vosniadou, 2002). Thus, when John is asked to identify the forces on a ball that was hit by a football player he says: John: Now there is no force from the football player, it is away from him, the only force is its weight. Yes there is no force [from the football player], there is a velocity that was acquired when it was in contact with him, when he exerted a force on the ball. Interviewer: Is this a spontaneous response? I mean, are there any cases when you are about to say that there is also an acquired force and then you say “no there isn’t [such a force]”? John: Yes, I haven’t got away [from this belief] completely, but when I think about it I understand that there isn’t such a force, since there is no contact [with the football player]. I mean with a first thought, I may be confused, but with a second, no I don’t. Michael, on the other hand, is not aware of his own beliefs and their development. The following dialogue, when he was also asked to identify the forces on a ball that was

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headed by a football player is revealing, as it shows that he is not aware of his failure to understand: Michael: This body has definitely a weight and we can see its direction and put another force F (he draws a force F in the direction of the ball’s movement) Interviewer: I can see why you say “there is definitely its weight” it is like saying that the ball is in the field of gravity, but why do you think there is also this force F? Michael: We can see that he hits the ball with his head and now the ball is free [to move] Interviewer: Yes, the ball is not in contact with his head isn’t it? Michael: (crossing out the force F) Yes it has left his head. Interviewer: I see that you cross out the force F. Michael: Yes, because I made a mistake, there is not such a force; there is its weight and a backward force. We can say that it is the reaction from the air that slows down the ball. Interviewer: If we consider this force negligible Michael, how does it move? Michael: It slows down, ah we said [it is] negligible, hum it falls freely? Interviewer: It moves downwards, vertically? Michael: Let’s say this is the ground, right? It will move…I forgot, I don’t remember how it moves. Later, when Michael was faced with the problem concerning the girl sliding down a slide, he drew a force in the direction of the movement and when he came to a dead end due to the lack of information about such a force, he wondered whether he should cross out that force. In short, John, the student with a constructivist physics-related epistemology, and the one showing evidence of in-depth physics understanding, adopted a deep approach to learning and studying as exemplified by his tendency to seek personal meaningmaking, to actively monitor his understanding, to integrate ideas and also, to be aware of his own beliefs and of changes in his beliefs. The opposite, that is, the adoption of a superficial approach to learning and studying, was found for Michael, the student with a less-constructivist physics epistemology who showed evidence of poor conceptual physics understanding. Michael appeared to be performance-oriented, to rely on memorization of what has to be learned and rote learning, not to be concerned about integrating ideas and finally, not to be aware of his own beliefs.

Discussion The study presented in this chapter provided preliminary evidence that the adopted approach to learning and the consequent selection of study strategies may intervene in the relationship between epistemological beliefs and conceptual change. Of the 10 students investigated, the five who held constructivist physics epistemologies and had achieved

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conceptual change in dynamics, adopted a deep approach to learning and studying. They were all oriented to personal meaning-making through the selection of deep study strategies. This process paralleled an increasing metaconceptual awareness. In contrast, the remaining five students who had less-constructivist physics epistemologies and were far from having achieved deep conceptual understanding of dynamics, showed evidence of a superficial approach to learning and studying. They were performance-oriented and preferred the selection of superficial study strategies, while no evidence of substantial metaconceptual awareness was found. The examples of John and Michael helped us to better understand these differences. The findings of the present study are in line with Entwistle’s (this volume) suggestion that the development of conceptions of knowledge (dualistic vs. relativistic) parallels the development of conceptions of learning (reproducing vs. seeking meaning), and that this is a process of increasing metacognitive awareness. They also agree with the hypotheses emerging from the suggested theoretical framework according to which epistemological beliefs may either facilitate or constrain the knowledge acquisition process directly, through guiding attention to certain information and through influencing intentions regarding knowledge construction and revision, as well as indirectly through certain mediating cognitive, metacognitive, and/or motivational factors, such as goal orientation and study strategies (e.g., Vosniadou, 1994, 2003, in press; Sinatra & Pintrich, 2003; Mason, 2003; Pintrich, 1999; Dole & Sinatra, 1998; Sinatra, 2005). Students’ belief that physics knowledge is a piecemeal collection of factual information, rather than a complex system of interrelated concepts, may make them more likely to ‘early foreclose’ (Kruglanski, 1989) their critical thinking in the process of learning (Pintrich et al., 1993; Pintrich, 1999; Qian & Alverman, 1995). It is also possible that students with such a predisposition may be precluded from deep information processing and from developing useful metacognitive skills. They would thus be expected to use superficial strategies (e.g., memorization of facts and formulas) instead of deeper strategies (e.g., organization and integration on the basis of principles). Such strategies lead to inert knowledge and prevent successful transfer. At the metacognitive level, they would be expected not to be aware of the need for information management and evaluation of learning outcomes. In any case, a significant impact on conceptual change in physics would be expected. Schommer et al. (1992) also suggested that belief in simple, as opposed to complex, knowledge may influence both students’ selection of cognitive strategies during the process of learning and their standards for judging learning outcomes. With regard to the finding that students’ beliefs in unchanging physics knowledge are also related to a superficial physics understanding, it could be argued that this may be the case because such a belief may constrain the evaluation and filter the interpretation of tentative and controversial information that does not concur with existing knowledge (e.g., Schommer–Aikins, 2002; Qian & Alverman, 1995). Students, who believe that physics knowledge does not change, may prefer to avoid ‘threatening’ new information, rather than examining and changing their existing conceptions. Similar explanations have been provided in the literature to account for the way students respond to anomalous data in conditions of cognitive conflict (e.g., Chinn & Brewer, 1993; Chinn & Malhotra, 2002; Mason, 2000). Finally, the results of our studies also suggest that epistemological beliefs maybe better predictors of conceptual change in physics than grades in physics (Stathopoulou &

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Vosniadou, in press). As we saw, the two students who served as examples in the present study, John and Michael, were very different in terms of their personal physics epistemologies and their conceptual understanding of Newtonian dynamics, but both had very high grades in school physics. Previous studies with Greek 1st-year university physics students have also shown that about half of them could not answer in the scientifically accepted way all the FMCE questions concerning the three Newton’s laws of motion, despite the fact that they took demanding entrance examinations and were selected on the basis of their high grades in school physics (Mol, Stathopoulou, & Vosniadou, 2004). This may be the case, because high grades can result not only from in-depth physics understanding but also from such factors as efficient use of rules, formalistic and algorithmic approach to problem-solving, adaptation to the teacher’s preferred techniques, rote learning, or what could be called a ‘strategic approach’ to learning and studying (Entwistle et al., 2000). Thus, it seems reasonable to suggest that a student with a deep conceptual understanding in physics would be expected to have high grades in school physics but the opposite may not necessarily be the case. To conclude, it appears that epistemological beliefs influence conceptual change in a variety of different ways. Understanding these ways involves more than an approach to conceptual change as an overtly rational process. The conflict between what is already known and the new, to-be-acquired information creates a learning situation in which affective and motivational variables can play an important role (Dole & Sinatra, 1998; Gregoire, 2003; Pintrich et al., 1993; Pintrich, 1999; Sinatra, 2005; Vosniadou, 2003). The relationship between epistemological beliefs and conceptual change is likely to be, to some extent, a reciprocal one (Pintrich, 2002). Since personally held beliefs about knowledge and knowing (in physics) are themselves subject to change, it is rather reasonable to suggest that deeper understanding (in physics) may provide feedback that influences epistemological beliefs. The exact processes through which epistemological beliefs change is not addressed in this study, but it is definitely an issue that needs careful investigation. A number of researchers emphasize the importance of constructivist instruction in facilitating the development of personal epistemologies (e.g., Bell & Linn, 2002; Carey & Smith, 1993, Roth & Roychoudhury, 1994; Smith et al., 2000). As noted earlier, conceptualizing epistemological beliefs as theory-like structures can help us understand better the mechanisms of their change by drawing on various cognitive mechanisms, as well as on motivational and affective variables that are involved in conceptual change models (e.g., Dole & Sinatra, 1998; Gregoire, 2003; Pintrich, 1999; Pintrich et al., 1993; Sinatra, 2005; Vosniadou, 1994, 2003).

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Mason, L. (2003). Personal epistemologies and intentional conceptual change. In: G. M. Sinatra, & P. R. Pintrich (Eds), Intentional conceptual change (pp. 199–236). Mahwah, NJ: Lawrence Erlbaum Associates. McCloskey, M. (1983). Naive theories of motion. In: D. Gentner, & A. L. Stevens (Eds), Mental models. Hillsdale, NJ: Erlbaum. Mol, A., Stathopoulou, C., Kollias, V. P., Vosniadou, S. (2003, July). Gradual learning of science in a CSCL environment and the quest of epistemologically sophisticated learners. Paper presented at the 3rd IEEE International Conference on Advanced Learning Technologies (ICALT’03), Athens, Greece. Mol, A., Stathopoulou, C., & Vosniadou, S. (2004). Consistency versus fragmentation in Mechanics. In: S. Vosniadou, C. Stathopoulou, X. Vamvakoussi, & N. Mamalougos (Eds), Abstracts of the 4th European symposium of the European association for research on learning and instruction on conceptual change: Philosophical, historical, psychological and educational approaches (pp. 133–135). Athens: Gutenberg Press. Perry, W. C., Jr. (1998). Forms of intellectual and ethical development in the college years: A scheme. San Francisco, CA: Jossey-Bass. Pintrich, P. R. (1999). Motivational beliefs as resources for and constrains on conceptual change. In: W. Schnotz, S. Vosniadou, & M. Carretero (Eds), New perspectives on conceptual change (pp. 33–50). Oxford: Elsevier. Pintrich, P. R. (2002). Future challenges and directions for theory and research on personal epistemologies. In: B. K. Hofer, & P. R. Pintrich (Eds), Personal epistemology: The psychology of beliefs about knowledge and knowing (pp. 103–118). Mahwah, NJ: Lawrence Erlbaum Associates. Pintrich, P. R., Marx, R. W., & Boyle, R. A. (1993). Beyond cold conceptual change: The role of motivational beliefs and classroom contextual factors in the process of conceptual change. Review of Educational Research, 63(2), 167–199. Posner, G. J., Strike, K. A., Hewson, P. W., & Gertzog, W. A. (1982). Accommodation of a scientific conception. Toward a theory of conceptual change. Science Education, 66(2), 211–227. Qian, G., & Alvermann, D. (1995). Role of epistemological beliefs and learned helplessness in secondary school students’ learning science concepts from text. Journal of Educational Psychology, 87(2) 282–292. Roth, W. M., & Roychoudhury, A. (1994). Physics students’ epistemologies and views about knowing and learning. Journal of Research in Science Teaching, 31 (1), 5–30. Ryan, M. P. (1984). Monitoring text comprehension: Individual differences in epistemological standards. Journal of Educational Psychology, 76(2), 248–258. Schommer, M. (1990). Effects of beliefs about the nature of knowledge on comprehension. Journal of Educational Psychology, 82, 498–504. Schommer, M. (1994). An emerging conceptualization of epistemological beliefs and their role in learning. In: R. Garner, & P. Alexander (Eds), Beliefs about text and instruction with text (pp. 25–40). Hillsdale, NJ: Erlbaum. Schommer–Aikins, M. (2002). An evolving theoretical framework for an epistemological belief system. In: B. K. Hofer, & P. R. Pintrich (Eds), Personal epistemology: The psychology of beliefs about knowledge and knowing (pp. 103–118). Mahwah, NJ: Lawrence Erlbaum Associates. Schommer, M., Crouse, A., & Rhodes, N. (1992). Epistemological beliefs and mathematical text comprehension: Believing it is simple does not make it so. Journal of Educational Psychology, 84(4), 435–443. Sinatra, G. M. (2005). The “warming trend” in conceptual change research: The legacy of Paul Pintrich. Educational Psychologist, 40(2), 107–115.

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Chapter 13

Effects of Epistemological Beliefs and Learning Text Structure on Conceptual Change Lucia Mason and Monica Gava Introduction Classroom learning often implies the re-organization of students’ existing knowledge structures, that is, conceptual change. Research in this field is carried out by cognitive, developmental, and educational psychologists, as well as by educators, especially in science domains, who are interested in the learning processes that are activated when disciplinary contents are taught in the classroom. Different models have been elaborated to explain why students hold alternative conceptions and have difficulty revising them (e.g., Chi, Slotta, & de Leeuw, 1994; Dole & Sinatra, 1998; Vosniadou, 1994). An important issue that arises from conceptual change research is that several factors are involved in the complex and dynamic process of knowledge revision. These include not only learners’ cognitive, motivational, and affective characteristics, but also social, instructional, and contextual aspects of learning situations (Limón & Mason, 2002; Schnotz, Vosniadou, & Carretero, 1999; Sinatra, 2005; Sinatra & Pintrich, 2003a). This study aimed at extending current understanding about the role of two of these variables, one related to the learners, that is their epistemological beliefs, and the other related to the material to be understood, that is the structure of the text to be learned. These two factors have received attention only relatively recently and their effects have been examined separately (Qian & Pain, 2002). On the one hand, it has emerged that beliefs about the nature of knowledge and knowing can constrain or facilitate revision of existing knowledge (e.g., Qian & Alvermann, 2000). On the other hand, it has been shown that the use of innovative learning texts, such as “refutational” ones, can be effective in knowledge restructuring processes (e.g., Guzzetti, Snyder, Glass, & Gamas, 1993). Since most school learning is the outcome of studying texts, the design of effective learning texts is a crucial question. Issues from the two different lines of investigation have been taken into account in the present study by focusing on the effects on conceptual change of epistemological beliefs and the design of refutational texts.

Reframing the Conceptual Change Approach in Learning and Instruction Copyright © 2007 by Elsevier Ltd. All rights of reproduction in any form reserved. ISBN: 0-08-045355-4

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Epistemological Beliefs and Conceptual Change When confronted with new knowledge to be learned, students do not just activate their preexisting knowledge about the topic but also their beliefs about knowledge itself. Beliefs about knowledge and knowing, namely epistemological beliefs, are individuals’ convictions about the nature, organization, and source of knowledge, its truth value and justification criteria of assertions (Hofer & Pintrich, 1997, 2002). Schommer (1990) proposed a conceptualization of personal epistemology as a set of more or less independent belief dimensions. Two dimensions regard the nature of knowledge: belief in simple (isolated and unambiguous items of information) or complex (set of interrelated concepts) knowledge and belief in certain (absolute and stable) or uncertain (continuously evolving) knowledge. Two dimensions regard the nature of learning: belief in quick (fast process) or gradual (slow process) learning and belief in a fixed (unmodifiable) or malleable (improvable) ability to learn. The inclusion of beliefs about learning was questioned by researchers (Hofer & Pintrich, 1997) who propose limiting personal epistemology to the nature of knowledge only. However, they recognize that beliefs about knowledge are strictly connected to beliefs about learning. A systematic research program on the relationship between epistemological beliefs and learning has been carried out by Schommer (e.g., 1990, 1993) through her 63-item Epistemological Questionnaire. Among other findings, for instance, she showed that the more students believe in certain knowledge and quick learning, the more they write inappropriate absolute conclusions to a text involving hypothetical material. Of particular interest to the present study are investigations about the effects of students’ epistemological beliefs on conceptual change, which have all been conducted using revised versions of Schommer’s instrument. In the study by Qian and Alvermann (1995), students with less sophisticated beliefs were less likely to abandon their naïve conceptions of motion after reading a text on the Newtonian theory, while students with more advanced beliefs generated greater change in their conceptual structures. Windschitl and Andre (1998) investigated the relationship between epistemological beliefs and knowledge revision with respect to an innovative, constructivist or a traditional, objectivist learning environment. Students in the former produced significantly greater conceptual change regarding the topic of the human cardiovascular system than students in the latter. In addition, students with more sophisticated beliefs about the nature and acquisition of knowledge changed their own conceptions more in the constructivist, rather than the traditional learning environment. Mason (2000, 2001) also investigated the effects of beliefs about the nature of knowledge in relation to the acceptance of anomalous data on theory change involving two controversial topics, one scientific and the other historical. For the first topic in particular, students who believed in the changing nature of knowledge were more likely to accept evidence conflicting with their prior conceptions and, consequently, to change them. In the study by Southerland and Sinatra (2003), not only epistemological beliefs but also cognitive dispositions (e.g., the disposition to engage in effortful thinking, open-minded thinking and not to identify with one’s beliefs but rather to weigh up new evidence) were examined in relation to the acceptance of three scientific topics varying in degree of controversy, that is, human evolution, animal evolution, and photosynthesis. It was found that epistemological beliefs and cognitive dispositions predicted acceptance of scientific knowledge only for the

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most controversial topic, human evolution. Students with more sophisticated views about knowledge and a greater disposition toward critical and open-minded thinking were more likely to accept human evolution than students who have less advanced epistemological beliefs and are less disposed toward critical and flexible thinking. To give an account of the influence of personal epistemology on processes of conceptual change, Mason (2003) appealed to the key notion of intentionality in learning (Sinatra & Pintrich, 2003b). She proposed that beliefs about knowledge and knowing may or may not guide students toward the goal of learning through knowledge revision, starting from recognizing a problem of knowledge, such as lack of knowledge or contradictory information, in their conceptual structures. Only advanced epistemological beliefs, that is, beliefs in complex, hypothetical, and evolving knowledge, are conducive to that recognition, which requires that students be engaged in learning as problem solving to be able to intentionally produce changes in knowledge. The result of this process, sustained by a sophisticated epistemological view, is a knowledge goal that implies the revision of prior conceptions and higher levels of understanding.

Refutational Texts and Conceptual Change A number of studies in reading education research have investigated the effects of text characteristics, such as coherence and the relationship between coherence and prior knowledge, on learning complex disciplinary content (e.g., Beck, McKeown, Sinatra, & Loxterman, 1991; Boscolo & Mason, 2003; McNamara, Kintsch, Songer, & Kintsch, 1996). Of particular interest to the present study is the design of refutational texts, that is, texts that directly state students’ alternative conceptions about a topic, refute them, and present the scientific conceptions as viable alternatives (e.g., Alvermann & Hynd, 1989; Hynd, 2003). The design of refutational texts to promote conceptual change has taken into account Kintsch’s (1988) model of text comprehension, which distinguishes two levels of comprehension: the textbase and situation model. The textbase is a propositional representation of the information expressed in a text, which is formed by extracting semantic information from a text at both local (microstructure) and global (macrostructure) levels, and coding it into a coherent network of propositions. The situation model is the result of integrating the reader’s prior knowledge with the information provided by the text. Learning from text is more than forming a good textbase. It implies a situation model, which is a mental representation of the text topic linked to the reader’s knowledge base. Refutational texts have been proposed as powerful means of helping students construct understanding at the situational model level. Not only do they activate the readers’ alternative prior knowledge but they also make explicit the inconsistencies between this knowledge and the new knowledge to be learned. Refutational texts were found to be more effective than traditional, straightforward informational texts in engendering conceptual change about physics contents. This was especially so for less-skilled readers and also when various other tools are used to facilitate knowledge revision (Alvermann & Hynd, 1989; Alvermann, Hynd, & Quin, 1997; Guzzetti et al., 1993; Hynd, McWhorter, Phares, & Suttles, 1994). The greater effectiveness of a refutational text, compared with a traditional expository text about energy was also found in a study with elementary school students,

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although it did not explicitly address pre- and post-reading conceptions (Diakidoy, Kendeou, & Ioannides, 2003). Mikkilä-Erdman (2002) has also documented the advantages, in terms of conceptual change, of a refutational text about photosynthesis over a typical textbook text. Finally, it has emerged that a refutational text promotes greater change in epistemological beliefs about mathematics in preservice teachers than a traditional text (Gregoire-Gill, Ashton, & Algina, 2004). Hynd (2003) has interpreted the benefit of refutational texts by reference to the four conditions for conceptual change, highlighted in the well-known article by Posner and associates (Posner, Strike, Hewson, & Gertzog, 1982). These conditions are satisfied by refutational texts because they elicit dissatisfaction with the reader’s current conceptions, explain the scientific concept clearly and, in depth, make it plausible through believable examples, and, finally, show the usefulness of the explanatory value of the new concept. The power of refutational texts has also been explained by reference to the characteristics that anomalous data should have, as underlined by Chinn and Brewer (1993, 1998). They should be credible, unambiguous, and in multiple forms. Furthermore, Hynd (2003) referred to social psychology literature and explained the power of refutational texts also in terms of central, not peripheral, text information processing because readers are actively engaged in processing the text ideas they perceive as personally relevant.

Research Questions and Hypotheses Relying on the findings from the two lines of research mentioned above, this study aimed at examining how students’ epistemological beliefs, and the structure of a learning text, affect revision of their alternative conceptions about natural selection and biological evolution. The variable of students’ reading comprehension skill was also considered as covariate in the statistical analyses. Although participants with reading difficulties were not considered in the analysis, the reading skills of the other students, who made up regular heterogeneous classes, varied. Three research questions guided the study: 1. Is students’ text-based comprehension at the immediate and delayed posttests affected by epistemological beliefs, text structure, and/or their interaction? 2. Is students’ conceptual change about natural selection and biological evolution, as reflected in changes in their explanations from pre to immediate and delayed tests, affected by epistemological beliefs, text structure, and/or their interaction? 3. Is students’ metaconceptual awareness of the changes in their own conceptual structures at the immediate posttest affected by epistemological beliefs, text structure, and/ or their interaction? For research question 1, we hypothesized that no significant differences would emerge between participants for text comprehension at the textbase level in relation to epistemological beliefs and text structure. According to the model of text comprehension proposed by Kintsch (1988), retention-comprehension questions should ascertain participants’ comprehension at the level of textbase, that is, the propositional representation of the information expressed in the text. Since the refutational and traditional texts did not

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differentiate in terms of information, participants would construct the same textbase representation, that is, a coherent network of propositions, in the two conditions. For research question 2, we hypothesized that beliefs about the nature of knowledge and text structure would make a difference immediately after reading the text and at a delayed posttest. Students with more advanced epistemological beliefs would change their alternative conceptions more than students with less advanced epistemological beliefs, and these changes would be more persistent. Conceptual change would imply text comprehension at the situation model level in Kintsch’s terms (1988), that is, the reader’s prior knowledge being integrated with information given in the text. Participants with more advanced beliefs about the nature of knowledge would be helped in perceiving a knowledge problem in their conceptual structures and in working toward solving it. We hypothesized that the text structure would also affect students’ revision of knowledge. By directly stating and challenging their alternative conceptions, the refutational text, more than a traditional text, would create or refine students’ metaconceptual awareness about their own representations as well as the scientific representations. This is an essential condition for conceptual change (Vosniadou, 2003; Wiser & Amin, 2001). The former can be perceived as limited while the latter as having more explanatory value. In addition, we wondered whether an interaction between the two variables would emerge and, if so, whether the refutational text could contribute to compensating less mature beliefs about the nature of knowledge in the change of explanations about biological evolution. Compared with a traditional text, a text that activates and challenges students’ alternative conceptions could create awareness of a conflict between their own conceptions and scientific ones in students who believe less in the complex and uncertain nature of knowledge. For research question 3, we hypothesized that both epistemological beliefs and text structure would affect metaconceptual awareness about changes in students’ conceptual structures, as both factors would help students to be aware of the differences between their prior conceptions and the scientific ones. In this case too we wondered whether the interaction between the two variables could also affect this performance by producing a compensation effect between reading a refutational text and less advanced epistemological beliefs. The refutational text may facilitate students who believe less in the complex and uncertain nature of knowledge to be metaconceptually aware that their initial conceptions changed as they were considered inappropriate after the text reading.

Method Participants First phase. In the first phase, the epistemological beliefs of 559 eighth graders, attending public middle schools in a province of northeastern Italy, were measured. There were 282 girls and 277 boys. All were native speakers of Italian, Caucasian, and shared a homogeneous middle-class social background. Second phase. In the second phase, 147 of the 559 students, who made up seven classes, were involved in the study. Of these, 10 participants were found to have reading difficulties (see below) on administration of a standardized reading comprehension text. In agreement with their teachers, they took part in the study, but their production was not considered for

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statistical analyses. In addition, 12 students were not present at the posttest. There were therefore 125 participants who provided data for both the pre and immediate posttests (65 girls and 60 boys). Of these 125 students, 15 were absent on the day of the delayed posttest. Therefore, 110 participants (58 girls and 52 boys) underwent the pre, post, and delayed posttest design. Participants’ scores in the epistemological dimension of Certain and simple knowledge (see the Results section) were dichotomized on the basis of the median and were used to create two mutually exclusive groups: one made up of students with more advanced beliefs about the nature of knowledge (n ⫽ 48; 25 in the experimental condition and 23 in the control condition) and the other of students with less advanced beliefs (n ⫽ 62; 29 in the experimental condition and 33 in the control condition). Conditions Each class was randomly assigned to one of two conditions: experimental and control. Students in the experimental condition read a refutational text about natural selection and biological evolution. In the control condition, the students read a traditional text about the topic (see next section). Prereading Tasks Epistemological beliefs. A shorter version of Schommer’s Epistemological Questionnaire was administered to measure students’ beliefs about the nature and acquisition of knowledge. This is a reduced and adapted version in Italian, which had already been used in a previous study (Lando, 2003). It comprises 36 items to be rated on a 5-point Likert-type scale (1 ⫽ totally disagree; 5 ⫽ totally agree). The items refer to the four epistemological dimensions proposed by Schommer (1990). Nine items regard each of the four dimensions: simplicity vs. complexity of knowledge, certainty vs. uncertainty of knowledge, fixed vs. malleable ability to learn, and quick vs. gradual learning. Prior knowledge. Participants’ preexisting conceptions about natural selection and biological evolution were ascertained by means of five open-ended and four multiple-choice questions. All questions were of a “generative” nature (Vosniadou, 1994), which introduced a problem to be solved by the construction of an appropriate representation of natural selection. The open-ended questions asked students to formulate an explanation of the problem, while the multiple-choice questions asked them to choose one of the four explanations. Some of these questions were devised by taking into account questions asked in previous studies (e.g., Bandiera, 1991; Bishop & Anderson, 1990; Demastes-Southerland, Good, & Peebles, 1995; Lawson & Thompson, 1988; Lawson & Weser, 1990) and the grade level of our participants, who were younger than those involved in previous research. Other questions, formulated by the authors, were new. The reliability of open-ended and multiple-choice questions at pretest were .68 and .76, respectively. For each participant a composite score for the explanations was calculated by summing the scores for answers to the open-ended questions and multiple-choice questions. Reading comprehension. Participants were individually administered an Italian standardized test for eighth grade to measure their expository text reading comprehension skills (Cornoldi & Colpo, 1995).

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Reading Text The effects of learning text structure on conceptual change were examined by means of two versions of the same text about natural selection and biological evolution according to Darwin’s theory. In the control condition students were given a text taken from a common science textbook (Braccini, Bosco, & Durante, 1998). In the experimental condition, students were given a refutational text, which was prepared by modifying the structure of the traditional text used in the control condition. Four parts were added and no other modifications were made. Each of these four parts was written to activate, by directly stating and challenging, a specific alternative conception held by the participants, which had already been identified in previous studies on the topic (e.g., Bandiera, 1991; Bishop & Anderson, 1990; Demastes-Southerland et al., 1995; Ferrari, & Chi, 1998; Lawson & Thompson, 1988). Each part added to the refutational text was also written in keeping with the conceptual continuity and text flow (Beck et al., 1991). Text coherence was modified as little as possible with the insertion of each part. The first part added considered the question of the origin of a new species and aimed at refuting the creationist perspective. The second underlined the importance of random intraspecies variability and aimed at undermining the alternative conception that all exemplars of a single species are the same. The third examined the concept of “struggle for survival” underlining that it should not be considered in physical terms. The fourth part added challenged the naïve conception of the individual’s adaptation to the environment based on the need and/or will to change. This alternative conception is confuted to introduce the theory of adaptation of the species to the environment. In Appendix an excerpt from the traditional text and another from the refutational are presented. Postreading Tasks Text-based comprehension. Six open questions were asked to assess participants’ comprehension at the level of textbase. The answers to these questions required information that was explicitly stated in the text. Conceptual change as change in explanations. The same generative open-ended and multiple-choice questions used to ascertain participants’ prior knowledge were asked again at both the immediate and the delayed posttests to examine any changes in participants’ explanations and the stability of these changes. Answers to these questions should ascertain text comprehension at the level of situation model (Kintsch, 1988), based on the integration between prior knowledge and knowledge provided by the text. At the immediate posttest, reliability of open-ended and multiple-choice generative questions were .78 and .82, respectively. Metaconceptual awareness. Two questions were asked to examine participants’ metaconceptual awareness of the changes in their conceptual structures. They were first asked, “Do you think that you have changed any conception after reading the text?” If they answered positively, they were asked, “What was your conception before reading and what is it now?”

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Coding Answers to the open-ended questions, for measuring comprehension at textbase level, were scored 0–2 according to their degree of correctness and completeness (Table 13.4). All answers were coded by two independent judges. Inter-rater agreement was 93%. Disagreements were resolved in conference through discussion in the presence of a third judge. Answers to the generative open-ended questions asked at pretest and immediate and delayed posttests to measure knowledge revision were scored 0–3 according to their degree of correctness and completeness (Tables 13.2 and 13.5). All answers were coded by the same two independent judges. Inter-rater agreement was 94%. Disagreements were resolved in conference through discussion in the presence of a third judge. Answers to the multiple-choice questions asked at pretest and immediate and delayed posttests to measure knowledge revision were scored 0–2. Of the four choices, two were wrong, one correct but generic, and one correct and complete (Table 13.3). Answers to the questions about metaconceptual awareness were assigned 1 point for mentioning each of the referred concepts. For instance, the answer “I believed that the species were always the same since the beginning, now I know that they evolved” scored 1. The answer “Before I thought like Lamarcke, that is, species acquire new features through the use of their organs and then these features are transmitted. Now I think that speciation is based on a mutation” scored 2. All answers were coded by the same two independent judges. Inter-rater agreement was 98%. Disagreements were resolved in conference through discussion in the presence of a third judge.

Table 13.1: Items loading on the epistemological belief dimension “Certain and simple Knowledge.” Item Almost everything one reads is believable If scientists try hard enough, they can find the truth to almost everything The most important aspect of scientific work is precise measurement and careful work Scientists can ultimately arrive at the truth Most words have one clear meaning Truth is unchanging It is a waste of time to work on problems that have no chance of arriving at a clear-cut and unambiguous answer The best thing about a science course is that most problems have only one right answer Today’s facts may be tomorrow’s fiction If you try to combine new ideas from a textbook with what you already know, you will get mixed up

Loading .488 .548 .677 .603 .701 .567 .424 .501 .432 .457

Table 13.2: Frequency/percentage of responses to the open-ended questions at pretest by condition and epistemological beliefs. Questions

Response categories

Condition/epistemological beliefs Experimental Less advanced

Q2. The first cheetahs used to run at a maximum speed of only 20 km/h. Nowadays cheetahs run 60 km/h when they are hunting prey. In your opinion, how can the evolution of their running capacity be explained?

Not pertinent/unclear (0) They will become dark as they need to be dark (0) Dark butterflies will survive (1) There will be more dark butterflies than the light colored as they reproduce (2) Dark butterflies can camouflage themselves, so they adapt and

Less advanced

2 (8%) 4 (16%) 6 (24%) 8 (32%)

1 (3%) 5 (17%) 3 (10%) 13 (45%)

3 (9%) 7 (21%) 6 (18%) 13 (39%)

1 (4%) 5 (22%) 4 (17%) 9 (39%)

5 (20%)

7 (24%)

4 (12%)

4 (17%)

4 (16%) 12 (48%) 7 (28%)

3 (10%) 16 (55%) 8 (28%)

6 (18%) 20 (61%) 6 (18%)

2 (9%) 12 (52%) 9 (39%)

2 (8%)

1 (3%)

–

–

–

–

1 (3%)

More advanced

–

–

1 (3%)

–

–

6 (24%) 12 (48%)

6 (21%) 7 (59%)

5 (15%) 21 (64%)

4 (17%) 16 (70%)

5 (20%) 2 (8%)

5 (17%) 1 (3%)

4 (12%) 2 (6%)

3 (13%) –

–

–

1 (3%)

–

(Continued)

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Q3. In UK there is a species of butterfly called ‘Biston Betularia’. There are two types of this butterfly. One type has light-colored wings, the other dark-colored wings. Most butterflies have lightcolored wings and land on birch trees, which have light-

Not pertinent/unclear (0)a Yes, strong muscles are transmitted (0) No, the man changed over time (1) No, the man was not born with strong muscles and cannot transmit them (2) No, strong muscles are not part of genetic inheritance and cannot be transmitted (3) Not pertinent/unclear (0) Running increases running capacity (0) Need or will to run faster increases running capacity (0) It depends on the struggle for survival (1) It depends on the struggle for survival: The fastest cheetahs were favored (2) Natural selection favored survival of the most adaptable, those that were fast, the winners in the struggle for survival (3)

More advanced

Effects of Epistemological Beliefs and Learning Text Structure

Q1. If a man has developed strong muscles because of having done a lot of physical exercise, will his children be born with stronger muscles?

Control

Table 13.2: (Continued) Response categories

Condition/epistemological beliefs

Less advanced

Q4. The arctic fox lives best at very low temperatures. It has a thick coat, which, of course, is very important for survival. What do you think is the o rigin of this thick coat?

Q5. Bacteria that infest the human body often cause disease. Antibiotics are used to kill them. However, some of these bacteria manage to resist antibiotics. In your opinion, in 20 years, for instance, will current antibiotics still be effective against bacteria?

Control

More advanced

Less advanced

More advanced

reproduce, while the light colored are destined to become extinct (3)

Not pertinent/unclear (0) Arctic foxes were created with a thick coat (0) Arctic foxes needed to have a thicker coat because of the low temperature (0) A mutation was the cause of the thick coat (1) Not pertinent/unclear (0) No, bacteria will get used to antibiotics (0) No, bacteria will build on their own resistance to survive (0) No, bacteria are resistant (1) No, fewer and fewer bacteria will be killed by antibiotics, the others will become resistant (2) No, if they are resistant, they reproduce and the new bacteria will also be resistant (3)

1 (4%) 11 (44%)

3 (10%) 0 (34%)

5 (15%) 14 (42%)

2 (9%) 11 (48%)

12 (48%)

15 (52%)

14 (42%)

10 (43%)

1 (4%)

1 (3%)

–

–

18 (72%) –

16 (55%) 4 (14%)

24 (73%) 4 (12%)

13 (57%) 3 (13%)

5 (20%)

7 (27%)

4 (12%)

5 (22%)

2 (8%) –

–

– 1 (3%)

1 (3%)

– –

1 (3%)

2 (9%) –

–

aScoring (0) means that no points were attributed to the answer, (1) indicates a correct but incomplete answer, (2) indicates a correct and more elaborated answer, and (3) indicates a correct and complete answer.

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Experimental

colored bark. In this way are camouflaged and are not they eaten by birds. What would happen in the long run to the butterflies’ wings if the bark of the birch became darker because of pollution?

174

Questions

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175

Procedure Data gathering took place in three sessions. The topic of natural selection and biological evolution had not yet been dealt with in any of the classes. In the first session (pretest) the order of tasks was: Epistemological Questionnaire, pretest generative open-ended and multiple-choice questions, and reading comprehension test. The first session took about 2 hours. The second session took place about a week after the first. Participants in the control condition were asked to study the traditional text, while participants in the experimental condition were asked to study the refutational text. After reading the text (immediate posttest), participants were asked to answer open-ended and multiple-choice generative questions, and metaconceptual questions based on the text. This session also took about 2 hours. The third session (delayed posttest) took place 2 months after the posttest and lasted about 1.5 hours. Participants were asked all the same questions again.

Results Epistemological Beliefs The initial 559 participants’ item ratings for each of the 36 items of the reduced and adapted version of Schommer’s Epistemological Questionnaire were used as a variable in the factor analysis. To be consistent with Schommer (1990), a principal-axis factoring with varimax rotation was used for an exploratory factor analysis. A three-factor model emerged with all eigenvalues ⬎1.00 as cut-off. The three factors accounted for 45.3% of the variance after rotation and were given descriptive titles on the basis of high-loading items. The three factors are: “Externally controlled learning” (
= .90), “Quick learning” (
= .84), and “Certain and simple knowledge” (
= .82). Weighted factor scores were calculated for each belief dimension. Only scores for Certain and simple knowledge, that is, the factor which was most relevant to the present study, were used for subsequent analyses. It should be pointed out that also in Qian and Alvermann’s (1995) study using Schommer’s revised 32-item questionnaire, the dimension “Simple and certain knowledge” emerged. Table 13.1 shows the items that loaded ⬎.40 in the belief dimension of Certain and simple knowledge. Reading Comprehension Skills According to the scores in the standardized test of reading comprehension skills, each participant could be considered as having high-, intermediate-, or low-level skills. Prior Knowledge Answers to the open-ended and multiple-choice questions revealed that the typical alternative conceptions, already identified in previous studies, were common among participants (Tables 13.2 and 13.3). Overall, it can be said that students held naïve conceptions of an organism’s adaptation to the environment when they explained the biological evolution

176

Questions

Response categories

Condition/epistemological beliefs Pretest Experimental

Immediate posttest

Control

Experimental

Control

Delayed Posttest Experimental

Control

Less More Less More Less More advanced advanced advanced advanced advanced advanced

Less More Less More advanced advanced advanced advanced

Less advanced

More advanced

–

1 (3%) 1 (3%) 1 (4%) –

4 (14%)

1 (3%) 3 (13%) 4 (16%) –

5 (15%) 1 (4%)

8 (32%)

4 (14%)

9 (27%)

6 (26%)

5 (20%)

4 (14%)

11 (33%)

17 (68%)

24 (83%)

23 (70%)

16 (70%)

20 21 (80%) (72%)

Q1 A lady with brown Incorrect hair, married to a (choice c man who also has and d) (0)a brown hair, has Correct, been dyeing her generic hair blond since (choice a) she was 18 years (1) old. When she has Correct and a child, will this complete child have brown or (choice b) blond hair? Why? (2)

10 (43%)

21 10 (64%) (43%)

21 (84%)

5 (17%)

27 (82%)

7 (30%)

–

24 (83%)

1 (3%)

15 (65%)

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Table 13.3: Frequency/percentage of responses to the multiple-choice questions at pre, immediate, and delayed posttests by condition and epistemological beliefs.

Q2

Q3 Why did geese develop palmate feet?

Incorrect (choice b and c) (0) Correct, generic (choice d) (1) Correct and complete (choice a) (2)

19 18 (76%) (62%)

21 (64%)

14 (61%)

11 5 (44%) (17%)

18 12 (55%) (52%)

14 (56%)

10 (34%)

20 (61%)

13 (57%)

3 4 (12%) (14%)

7 (21%)

6 (26%)

5 4 (20%) (14%)

4 3 (12%) (13%)

2 (8%)

–

2 (6%)

2 (9%)

3 7 (12%) (24%)

5 (15%)

3 (13%)

9 (36%)

20 (69%)

11 (33%)

8 (35%)

9 (36%)

19 (66%)

11 (33%)

8 (35%)

21 24 (84%) (83%)

29 (88%)

18 (78%)

15 (60%)

8 (28%)

24 (73%)

14 (61%)

17 (68%)

12 (41%)

25 (76%)

16 (70%)

1 (4%) 3 (10%) –

–

3 (12%) 4 (14%)

1 (3%) 1 (4%)

2 (8%)

5 (17%) 2 (6%)

–

3 2 (12%) (7%)

5 (22%)

7 (28%)

8 (24%)

6 (24%)

12 (41%)

7 (30%)

4 (12%)

17 (59%)

8 (35%)

6 (18%)

(Continued)

Effects of Epistemological Beliefs and Learning Text Structure

Many years ago Incorrect when insecticides (choice a started to be used, and c) (0) they were very Correct, effective for generic killing flies and (choice d) mosquitoes. Now, (1) after 40 years, Correct and when the same complete insecticides are (choice b) sprayed, much (2) fewer mosquitoes are killed. How can you explain this phenomenon?

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Response categories

Condition/epistemological beliefs Pretest Experimental

Q4 If a population of geese were forced to live in an environment where water to swim in was not available, what would happen?

Incorrect (choice a and b) (0) Correct, generic (choice c) (1) Correct and complete (choice d) (2)

Immediate posttest

Control

Experimental

Control

Delayed Posttest Experimental

Less More Less More Less More advanced advanced advanced advanced advanced advanced

Less More Less More advanced advanced advanced advanced

17 25 (68%) (86%)

27 (82%)

14 18 (61%) (72%)

14 (48%)

23 (70%)

14 (61%)

18 (72%)

2 (8%)

2 (6%)

1 (4%)

–

2 (7%)

4 (12%)

1 (4%)

4 (12%)

8 (35%)

7 (28%)

13 (45%)

6 (18%)

8 (35%)

2 (7%)

6 2 (24%) (7%)

Control Less advanced

More advanced

17 (59%)

22 (67%)

15 (65%)

1 (4%)

5 (17%)

3 (9%)

–

6 (24%)

7 (24%)

8 (24%)

8 (35%)

aScoring (0) was attributed to the wrong choice, (1) was attributed to the correct but generic choice, and (2) was attributed to the correct choice.

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Questions

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Table 13.3: (Continued)

Effects of Epistemological Beliefs and Learning Text Structure

179

mentioned in the questions. According to these conceptions, adaptation is due to the need or will to survive, or to the use or exercise of a feature. The creationist conception was detected only in the explanations regarding the question of the coat of the artic fox, which asked explicitly about its origin. Overall, answers to the multiple-choice generative questions at pretest substantially confirmed the presence of naïve conceptions about adaptation to the environment. Preliminary analyses through independent-sample t-tests revealed that the students in the two conditions, experimental and control, did not differ significantly for prior knowledge, t (123) ⫽ .98, p ⫽ .32, reading comprehension skills, t (123) ⫽ .61, p ⫽ .54 or epistemological beliefs, t (123) = .99, p = .32. Text-Based Comprehension A repeated-measure ANCOVA, with epistemological beliefs (more and less advanced) and condition (experimental, control) as between-subject variables, time (immediate posttest and delayed posttest) as the within-subject variable, and reading comprehension as the covariate, was carried out. It revealed the main effect of testing time, F (1, 105) ⫽ 11.83, p = .001, as well as the interaction time ⫻ condition, F (1, 105) ⫽ 4.70, p ⬍ .05. Two months after studying the text, students’ overall scores decreased, but the scores of participants who had read the refutational text decreased significantly less. In this case, the text structure affected the stability of comprehension at the textbase level (Table 13.4). No differences related to epistemological beliefs emerged. Figure 13.1 shows mean scores for text-based comprehension at the immediate and delayed posttests.

Immediate posttest Delayed posttest 9 8 7 6 5 4 3 2 1 0 Less advanced

More advanced

Experimental

Less advanced

More advanced

Control

Figure 13.1: Adjusted mean scores of text-based comprehension by condition and epistemological beliefs.

Response categories

Condition/epistemological beliefs Immediate posttest Experimental

Q1. What should The parents (0)a the variability Chance mutations of individual or “mixing” of characteristics parents’ be attributed characteristics (1) to? Chance mutations and “mixing” of parents’ characteristics (2) Q2. What does “struggle for survival” mean?

Fight between animals (0) Competition for food (1) Competition for food and territory, to avoid predators, to overcome diseases (2)

Delayed posttest

Control

Experimental

Control

Less advanced

More advanced

Less advanced

More advanced

Less advanced

More advanced

Less advanced

More advanced

18 (72%) 4 (16%)

15 (52%) 2 (7%)

19 (58%) 4 (12%)

16 (70%) 2 (9%)

23 (92%) 2 (8%)

25 (86%) 4 (14%)

25 (76%) 8 (24%)

18 (78%) 5 (22%)

3 (12%)

12 (41%)

10 (30%)

5 (22%)

–

–

–

–

9 (36%)

4 (14%)

9 (27%)

8 (35%)

19 (76%)

19 (66%)

27 (82%)

16 (70%)

10 (40%)

12 (41%)

8 (24%)

4 (17%)

6 (24%)

10 (34%)

5 (15%)

5 (22%)

6 (24%)

13 (45%)

16 (48%)

11 (48%)

–

1 (3%)

2 (9%)

–

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Questions

180

Table 13.4: Frequency/percentage of responses to the text-based comprehension questions at immediate and delayed posttests by condition and epistemological beliefs.

Q3. Which The strongest individuals survive and are more the weakest advantaged die (0) and which are Animals destined to die with good in the struggle characteristics for survival? for the environment survive, the others die (1) Animals with the characteristics fitting the environment survive and have offspring, the others do not live long (2)

9 (36%)

4 (14%)

12 (36%)

4 (17%)

16 (64%)

12 (41%)

7 (28%)

11 (38%)

7 (21)

3 (13%)

8 (32%)

7 (24%)

9 (36%)

14 (48%)

14 (42%)

16 (70%)

1 (4%)

10 (34%)

1 (3%)

2 (9%)

Q4. What favors natural selection?

6 (24%)

3 (10%)

6 (18%)

5 (22%)

19 (76%)

17 (59%)

32 (97%)

23 (100%)

3 (12%)

2 (7%)

2 (6%)

1 (4%)

1 (4%)

5 (17%)

–

–

16 (64%)

24 (83%)

25 (76%)

17 (74%)

5 (20%)

7 (24%)

–

1 (3%)

18 (78%)

3 (13%)

–

181

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Effects of Epistemological Beliefs and Learning Text Structure

The strongest or the most clever animals (0) Survival of the animals that can adapt to the environment (1) Survival of the fittest who transmit their characteristics to offspring (2)

32 (97%)

Response categories

Condition/epistemological beliefs Immediate posttest Experimental

Q5. What are mutations?

Q6. What is speciation?

Delayed posttest

Control

Experimental

Control

Less advanced

More advanced

Less advanced

More advanced

Less advanced

More advanced

Less advanced

More advanced

2 (8%)

4 (14%)

8 (24%)

2 (9%)

6 (24%)

10 (34%)

16 (48%)

10 (43%)

6 (24%)

2 (7%)

1 (3%)

5 (22%)

10 (40%)

12 (41%)

12 (36%)

8 (35%)

17 (68%)

23 (79%)

24 (73%)

16 (70%)

9 (36%)

7 (24%)

5 (15%)

5 (22%)

2 (8%) 15 (60%)

2 (7%) 7 (24%)

1 (3%) 19 (58%)

2 (9%) 9 (39%)

18 (72%) 6 (24%)

19 (66%) 8 (28%)

29 (88%) 2 (6%)

19 (82%) 2 (9%)

8 (32%)

20 (69%)

13 (39%)

12 (52%)

1 (4%)

2 (7%)

2 (6%)

2 (9%)

Changes in animals’ appearance (0) Variation in genetic inheritance (1) Positive or negative chance variations in genetic inheritance (2) Reproduction (0) Origin of a new species (1) Origin of a new species when mutations accumulate over generations (2)

aScoring (0) means that no points were attributed to the answer, (1) indicates a correct but incomplete answer, (2) indicates a correct and more elaborated answer, and (3) indicates a correct and complete answer.

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Questions

182

Table 13.4: (Continued)

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183

Conceptual Change A repeated-measure ANCOVA, with epistemological beliefs (less and more advanced) and condition (experimental, control) as between-subject variables, conceptual knowledge at the three different times (composite scores for explanations at pretest, immediate, and delayed posttests) as the within-subject variable, and reading comprehension as the covariate, was carried out. From this analysis the main effects of testing time, F (2, 104) = 3.16, p <.05 and the interactions time ⫻ epistemological beliefs, F (2, 104) = 3.75, p < .05, time ⫻ condition, F (2, 104) ⫽ 17.50, p < .001, and time ⫻ epistemological beliefs ⫻ condition, F (2, 104) ⫽ 5.72, p ⫽ .01, emerged. The covariate also correlated significantly with the three scores, F (2, 104) ⫽ 9.93, p ⫽ .001. Overall, participants’ scientific knowledge about natural selection and biological evolution increased from the pre to the immediate posttest. However, participants’ scores decreased slightly from the immediate to the delayed posttest. Regarding epistemological beliefs, students with more advanced convictions about the nature of knowledge progressed conceptually from the pre to the immediate posttest much more than those with less advanced convictions. As far as the type of text is concerned, students who received information from the refutational text were facilitated in conceptual change much more than students who read the traditional text (Tables 13.3 and 13.5). From the interaction testing time ⫻ beliefs ⫻ condition, it emerged that participants who changed their alternative prior conceptions more at the immediate posttest were those with more advanced epistemological convictions and who read the refutational text. Moreover, students with less advanced epistemological beliefs who read the refutational text outscored those at the same level of epistemic cognition who read the traditional text. Furthermore, students with the highest outcomes at the immediate posttest were also those whose scores decreased more at the delayed posttest. However, they still significantly outscored those who progressed less at the immediate posttest and whose scores did not significantly decrease at the delayed posttest. Figure 13.2 shows means of composite scores for generative questions at pre, immediate, and delayed posttests. It is worth noting that the covariate prior knowledge correlates with conceptual change measures. Conceptual Change in Explanations: Examples To illustrate cases of conceptual change due to text reading, some examples of explanations that show a revision of personal conceptions from pre to immediate posttest follow. For the question about cheetahs’ increased running speed, the pretest explanation given by a student in the experimental condition (signed as P6) expressed a naïve preconception about adaptation, which was then successfully revised. The explanation given at the delayed posttest referred to the crucial ideas of intraspecies variation and natural selection. Pretest: The ability to run faster developed over time, over some decades in fact, so muscles and bones changed allowing cheetahs to increase their speed. Cheetahs’ running faster and faster influenced the speed. (P6, exp.)

184

Questions

Response categories

Condition/epistemological beliefs Immediate posttest Experimental Less advanced

Q1 If a man has developed strong muscles because of having done a lot of physical exercise, will his children be born with stronger muscles?

Not pertinent/ unclear(0)a Yes, strong muscles are transmitted (0) No, the man changed over time (1) No, the man was not born with strong muscles and cannot transmit them (2) No, strong muscles are not part of genetic inheritance and cannot be transmitted (3)

Delayed posttest

Control

More Less More advanced advanced advanced

Experimental Less advanced

More advanced

Control Less More advanced advanced

1 (4%)

1 (3%)

5 (15%)

3 (13%)

3 (12%)

2 (7%)

4 (12%)

4 (17%)

3 (12%)

2 (7%)

8 (24%)

9 (39%)

3 (12%)

3 (10%)

5 (15%)

6 (26%)

2 (8%)

2 (7%)

7 (21%)

3 (13%)

2 (8%)

4 (14%)

8 (24%)

2 (9%)

8 (32%)

5 (17%)

9 (27%)

6 (26%)

7 (28%)

7 (24%)

11 (33%)

9 (39%)

11 (44%)

19 (66%)

4 (12%)

2 (9%)

10 (40%)

13 (45%)

5 (15%)

2 (9%)

Lucia Mason and Monica Gava

Table 13.5: Frequency/percentage of responses to the open-ended questions at immediate and delayed posttests by condition and epistemological beliefs.

Q3 In UK there is a species of butterfly called ‘Biston Betularia.’

Not pertinent/ unclear (0) Running increases running capacity (0) Need or will to run faster increases running capacity (0) It depends on the struggle for survival (1) It depends on the struggle for survival: The fastest cheetahs were favored (2) Natural selection favored survival of the most adaptable, those that were fast, the winners in the struggle for survival (3) Not pertinent/ unclear (0) They will become dark as they need to be dark (0)

1 (4%) 7 (28%)

– 5 (17%)

1 (3%) 11 (33%)

– 8 (35%)

1 (4%) 8 (32%)

– 2 (7%)

9 (36%)

7 (24%)

14 (42%)

12 (52%)

10 (40%)

12 (41%)

4 (16%)

5 (17%)

4 (12%)

1 (4%)

2 (8%)

4 (14%)

4 (12%)

2 (8%)

1 (3%)

2 (6%)

1 (4%)

3 (10%)

1 (3%)

–

2 (8%)

11 (38%)

1 (3%)

2 (9%)

3 (12%)

8 (28%)

3 (9%)

–

5 (20%)

3 (10%)

11 (33%)

4 (17%)

7 (28%)

5 (17%)

11 (33%)

12 (48%)

10 (34%)

18 (55%)

13 (57%)

8 (32%)

7 (24%)

13 (39%) 12 (52%)

–

2 (6%) 8 (24%)

1 (4%) 6 (26%)

15 (45%) 14 (61%)

2 (9%)

5 (22%)

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Q2 The first cheetahs used to run at a maximum speed of only 20 km/h. Nowadays cheetahs run 60 km/h when they are hunting prey. In your opinion, how can the evolution of their running capacity be explained?

Response categories

Condition/epistemological beliefs Immediate posttest Experimental Less advanced

There are two types of this butterfly. One type has lightcolored wings, the other dark colored wings. Most butterflies reproduce, while have lightcolored wings and land on birch trees, which have lightcolored bark. In this way they are camouflaged and are not eaten by birds. What would happen in the long run to the butterflies’ wings if the bark of the birch became darker because of pollution?

Dark butterflies will survive (1) There will be more dark butterflies than as they reproduce (2) Dark butterflies can camouflage themselves,so they adapt and the light colored are destined to become extinct (3)

Delayed posttest

Control

More Less More advanced advanced advanced

Experimental Less advanced

More advanced

Control Less More advanced advanced

5 (20%)

3 (10%)

3 (9%)

4 (17%)

2 (8%)

8 (28%)

5 (15%)

1 (4%)

3 (10%)

1 (3%)

1 (4%)

6 (24%)

6 (21%)

4 (12%) –

2 (8%)

10 (34%)

1 (4%)

2 (8%)

3 (10%)

–

–

5 (22%)

1 (4%)

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Questions

186

Table 13.5: (Continued)

Not pertinent/ unclear (0) Arctic foxes were created with a thick coat (0) Arctic foxes needed to have a thicker coat because of the low temperature (0) A mutation was the cause of the thick coat (1) A mutation favored the adaptation of foxes with a thick coat (2) By chance mutation foxes with thick coats adapted, survived and reproduced (3)

–

1 (3%)

1 (4%)

1 (4%)

1 (3%)

2 (6%)

–

12 (48%)

9 (31%)

20 (61%)

14 (61%)

15 (60%)

13 (45%)

8 (32%)

3 (10%)

8 (24%)

7 (30%)

8 (32%)

3 (10%)

10 (30%)

2 (8%)

7 (24%)

1 (3%)

1 (40%)

–

1 (3%)

2 (6%)

–

–

1 (3%)

–

5 (17%)

1 (3%)

–

1 (4%)

6 (21%)

2 (6%)

–

13 (52%)

11 (38%)

17 (52%) 12 (52%)

–

–

–

–

–

1 (4%)

9 (31%)

3 (9%)

–

14 (56%)

15 (52%)

19 (58%)

14 (61%)

5 (20%)

6 (21%)

9 (27%)

6 (26%)

2 (8%)

1 (3%)

2 (6%)

3 (13%)

8 (32%)

8 (28%)

16 (48%) 16 (70%)

13 (39%)

7 (30%)

– 8 (35%)

187

Q5 Bacteria that infest Not pertinent/ the human body unclear (0) often cause disease. No, bacteria will Antibiotics are used get used to to kill them. antibiotics (0) However, some of No, bacteria will these bacteria build on their manage to resist own resistance antibiotics. In your to survive (0)

2 (8%)

Effects of Epistemological Beliefs and Learning Text Structure

Q4 The arctic fox lives best at very low temperatures. It has a thick coat, which, of course, is very important for survival. What do you think is the origin of this thick coat?

(Continued)

Response categories

Condition/epistemological beliefs Immediate posttest Experimental Less advanced

opinion, in 20 years, No, bacteria for instance, are resistant (1) will current No, fewer and fewer antibiotics still be bacteria will be effective against killed by bacteria? antibiotics, the others will become resistant (2) No, if they are resistant, they reproduce and the new bacteria will also be resistant (3)

Delayed posttest

Control

More Less More advanced advanced advanced

Experimental Less advanced

More advanced

Control Less More advanced advanced

2 (8%)

–

2 (6%)

–

1 (4%)

2 (7%)

1 (3%)

2 (8%)

–

2 (6%)

–

3 (12%)

1 (3%)

^1 (3%)

–

1 (3%)

–

7 (24%)

1 (3%)

–

2 (8%)

7 (24%)

–

3 (13%)

aScoring (0) means that no points were attributed to the answer, (1) indicates a correct but incomplete answer, (2) indicates a correct and more elaborated answer, and (3) indicates a correct and complete answer.

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Questions

188

Table 13.5: (Continued )

Effects of Epistemological Beliefs and Learning Text Structure

189

Pretest Immediate posttest Delayed posttest 14 12 10 8 6 4 2 0 Less advanced

More advanced

Experimental

Less advanced

More advanced

Control

Figure 13.2: Adjusted mean of composite scores for conceptual change by condition and epistemological beliefs. Delayed posttest: The first cheetahs had different characteristics. Some ran faster than others, thus this advantageous characteristic allowed faster cheetahs to prevail over the slower ones, and over time the slow cheetahs became extinct. (P6, exp.) For the question about the origin of the thick coat of the arctic fox, a change in explanation from pre to the posttest is evident in the following answers given by another student in the experimental condition (signed as P92). Her naïve preconception was based on the idea that the need for survival leads to adaptation to the environment. The new conception she constructed appealed to the notions of intraspecies variation and natural selection. She understood that the latter favors only the species members with the most suitable characteristics. Pretest: I think that the arctic fox, in the beginning, was a fox with a normal coat but over time its coat changed because of the very low temperatures. It therefore became thicker and thicker, allowing it to survive. (P92, exp.) Delayed posttest: These foxes with a thicker coat are the descendants of the first foxes that inhabited the earth. They were actually “privileged” because they could

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Lucia Mason and Monica Gava adapt to living at low temperatures, while at the same time there were foxes with less thick coats. Foxes with a less thick coat disappeared, while the “privileged” survived and had offspring that had offspring. Thus, after millions of years the entire species has adapted to the environment. (P92, exp.)

Metaconceptual Awareness An ANCOVA with epistemological beliefs and condition as the between-subject factors, reading comprehension as the covariate, and the immediate posttest score for metaconceptual awareness as the outcome measure revealed the effects of both independent variables, F (1, 105) ⫽ 5.58, p ⬍ .05, and F (1, 105) ⫽ 8.89, p ⬍ .01, respectively. Students with more advanced beliefs about the nature of knowledge (M ⫽ .69, SE ⫽ .10) showed greater metaconceptual awareness of the changes in their conceptual structures after text reading than students with less advanced beliefs (M ⫽ .34, SE ⫽ .10). Participants who read the refutational text (M ⫽ .73, SE ⫽ .10) were also metaconceptually more aware than students who read the traditional text (M ⫽ .29, SE ⫽ .10). No interaction between the two variables emerged. The following are examples of the different degrees of participant awareness of the change in their personal conceptions, having read the text, as expressed at the immediate posttest: Before, I thought that cheetahs had developed a skeleton suitable to the environment and that foxes had developed their coats to adapt to the environment. Now I know that it was only possible for members of the species to survive if they had a suitable skeleton or coat. (P8, exp.) Before, I thought animals changed because they tried to adapt to the environment so that they would not die. I have learned now that their characteristics changed because of natural selection not because of wanting to adapt to the environment. Through natural selection only those who have some advantages for survival in fact survive. The example of giraffes in the text led me to think a lot about the evolution of some animals. (P58, exp.) These examples illustrate the students’ metacognitive ability to think about their past and current thinking, although at increasing levels of awareness.

Discussion and Implications This study aimed at extending current understanding of the role of two variables that affect conceptual change, which had been investigated separately, that is, epistemological beliefs and the structure of the text to be read. The former is a learner’s variable that reflects a more or less advanced position about the nature of knowledge. In this study, it reflects convictions about knowledge as simple and certain vs. complex and continuously evolving. The latter variable is a factor regarding instructional material that

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reflects the type of text from which new information can be constructed. In this study two types of text were used: an ordinary expository science text whose primary function was to give new information, and a refutational text that not only gives new, correct information but also explicitly states and refutes alternative conceptions by presenting the scientific conceptions as viable alternatives. Our first research question asked if participants’ comprehension at the textbase level would be affected by epistemological beliefs, text structure, and/or their interaction. Findings partially confirm our hypothesis, since epistemological beliefs did not affect this level of text comprehension but text structure did. Students’ scores decreased at the delayed posttest and the decrease was less evident for students in the experimental condition, who read the refutational text. The second research question asked if students’ conceptual change about the topic — reflected in explanations given in answer to open-ended questions and in the choice of explanations for the multiple-choice questions — would be affected by epistemological beliefs and text structure. We also wondered if reading a refutational text could compensate less advanced epistemological understanding. Findings partially confirm our hypothesis and show that, overall, students’ scientific knowledge of the topic improved, but those who believed more in complex and uncertain knowledge generated more conceptual change than those who believed more in simple and certain knowledge. Similarly, students who read the refutational text changed their preconceptions more than students who read the traditional text. Our data on the effects of beliefs about knowledge and text structure confirm the findings of previous studies on these two variables, which were investigated separately. On the one hand, all studies that addressed the relationship between epistemological beliefs (measured by the revised version of Schommer’s questionnaire) and conceptual change found that less advanced epistemological beliefs, that is belief in knowledge as absolute, simple, stable, and transmitted by authority, are associated with lower performances in knowledge revision. Conversely, more sophisticated beliefs, that is belief in knowledge as complex, uncertain, and derived from reason, are associated with higher performances (e.g., Mason, 2000; Qian & Alvermann, 1995; Sinatra, Southerland, McConaughy, & Demastes, 2003, Windschitl & Andre, 1998). As previously proposed (Mason, 2003), epistemological beliefs mediate cognitive functions in the process of intentional conceptual change (Sinatra & Pintrich, 2003a). They may or may not guide students toward the goal of learning through knowledge revision, starting from recognizing a problem of knowledge, such as lack of knowledge or contradictory information, in their conceptual structures. Only beliefs in complex, hypothetical, and evolving knowledge are conducive to that recognition, which requires that students be engaged in learning as problem solving to be able to intentionally produce changes in knowledge (Mason, 2003). On the other hand, it emerges from almost all studies on the effects of refutational texts that they are a powerful means of promoting understanding of complex scientific knowledge at the situation model level (e.g., Alvermann & Hynd, 1989; Guzzetti et al., 1993; Diakidoy et al., 2003). This positive effect may be explained in terms of the creation or refinement of metacognitive awareness of one’s current conceptions and the new conceptions to be learned. While the former are introduced as limited and then refuted, the latter are introduced as viable representations, that is, intelligible, credible, and fruitful.

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The power of refutational texts can also be explained in terms of deeper text processing stimulated by a comparison between the readers’ preconceptions and the new conceptions to be learned. The interaction between testing time, epistemological beliefs, and text type is particularly interesting. A powerful combination of the personal and instructional variables to support conceptual change emerged. Students with higher levels of epistemological thinking are those who are better able to take advantage of reading a text that explicitly states and refutes their preconceptions. A compensation effect between less mature beliefs about the nature of knowledge and refutational text also emerged since students who believed less in the complexity and uncertainty of knowledge and read the refutational text outscored students at the same level of epistemological thinking who read the traditional text. The correlation between reading comprehension skills and conceptual change measures leads us to reflect, once again, on the importance of these skills in academic learning. Improving students’ reading comprehension is, in any case, an essential way to sustain their conceptual learning from a text. The third research question asked if at posttest, participants’ metaconceptual awareness of the changes in their cognitive structures would also be affected by epistemological beliefs and text structure. We also wondered if reading a refutational text could compensate less advanced epistemological understanding. Findings add to the literature that both factors help students think about their thinking, as hypothesized. To believe more that knowledge is complex and uncertain enhances metaconceptual awareness. Reading a refutational text that states and refutes alternative preconceptions also creates or refines awareness of the changes in one’s cognitive structures. These data are in line with Vosniadou’s (1994, 2003) recommendation for the creation of a powerful learning environment for conceptual change as well as with Wiser and Amin’s (2001) positive effects of metaconceptual teaching about thermal phenomena. To some extent, our outcomes also confirm the data from a previous study on the understanding of evolution by Lawson and Worsnop (1992), who pointed out that reflective reasoning was a strong predictor of learning about the topic. No interaction between the personal and instructional variables emerged for metaconceptual awareness. We should underline that in this study we did not examine students’ religious beliefs, as in other research in science education (e.g., Dagher & BouJaoude, 1997; DemastesSoutherland et al., 1995). Since the topic was animal, and not human, evolution, we did not take into account the consideration that this variable may mediate acceptance of scientific knowledge, which may be understood, but not believed as truth. More recently, Sinatra and her associates (2003) have showed not only the role of beliefs about knowledge but also cognitive dispositions, such as the disposition to engage in effortful and open-minded thinking, and not to identify with one’s beliefs but rather to weigh up new evidence in the understanding and acceptance of human evolution. Furthermore, it should be pointed out that we investigated only short-term conceptual change. Learning was only stimulated through the reading of one text. On one hand, this is in line with what takes place in regular classroom settings, at least in Italy, from the last years of primary school. New knowledge comes mainly from text reading, especially when abstract concepts are acquired, so that observation and experimentation can be scarcely

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applied to learning, such as in the case of natural selection and biological evolution. On the other hand, we know that reading a refutational text accompanied by group or pair discussion can increase the positive effects of the text itself (Hynd, Qian, Ridgeway, & Pickle, 1991). Further research is needed to examine the role and contribution of argumentation, promoted by social-cognitive interaction, in the construction of more advanced explanations in different teaching conditions. From the educational point of view, our findings lead us to draw two main implications for teaching for conceptual change. First of all, since students’ representations about knowledge affect the process of knowledge construction and reconstruction, it is important to ascertain their epistemological beliefs. To acquire knowledge is the main, or one of the main, reasons for attending school. What do students think knowledge is and where does it come from? (Kuhn, 2000). It is also crucial to implement educational interventions aimed at changing convictions that are not conducive to deepening conceptual understanding (e.g., Conley, Pintrich, Vekiri, & Harrison, 2004). The way in which scientific knowledge is presented can itself foster or hinder the refinement of thinking about knowledge and knowing. The second implication regards the instructional material to be studied. Adequate space should be given, especially for complex topics, to texts that explicitly demonstrate the limitations of alternative conceptions and the fruitfulness of intelligible and credible ones. This type of text should be considered not only as a source of scientific knowledge but also as a tool for promoting the refinement of students’ metacognitive awareness. A careful and critical analysis of the text content and a successful integration of the new information into conceptual structures are related to students’ awareness of the differences between their own, and scientific conceptions. All this means supporting students’ deep understanding, that is, a knowledge construction process which is not fictitious or pseudo, as happens when new knowledge is ignored, distorted, or remains inert.

Appendix Excerpt from the Traditional Expository Text Individuals of the same species are not all exactly the same. There is a great variability in characteristics, which are partially inheritable, that is, they can be transmitted to the offspring. Although Darwin considered the variability in characteristics of the exemplars of a species a fundamental aspect of his theory, he did not know how it originated. Nowadays, this variability is attributed to mutations in genetic inheritance, which are totally a matter of chance, and to a “mixing” of the parents’ characteristics that occurs through reproduction. Excerpt from the Refutational Text Variability is fundamental in Darwin’s theory. Why is variability so important? Imagine that animals of the same species are exactly the same and perfectly adapted to a given

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environment. Imagine that that environment changes. No animal is adaptable to the new environment; they all die and therefore the species ends. Conversely, if one member of the species is different from the rest, it is more likely that, by chance, it has some characteristics that are appropriate for the new environment. This “lucky” animal survives, leaves offspring behind, and the species it belongs to continues. Individuals of the same species are therefore different. Try and look at a baby. The baby only looks like its father and mother in part. This is due to a “mixing” of the parents’ characteristics during reproduction. Variability is also due to mutations of the genetic inheritance, which are totally a matter of chance.

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Dole, J. A., & Sinatra, G. M. (1998). Reconceptualizing change in the cognitive construction of knowledge. Educational Psychologist, 33(2/3), 109–128. Ferrari, M., & Chi, M. T. H. (1998). The nature of naive explanations of natural selection. International Journal of Science Education, 20, 1231–1256. Gregoire-Gill, M., Ashton, P. T., & Algina, J. (2004). Changing preservice teachers’ epistemological beliefs about teaching and learning in mathematics: An intervention study. Contemporary Educational Psychology, 29, 164–185. Guzzetti, B. J., Snyder, T. E., Glass, G. V., & Gamas, W. S. (1993). Promoting conceptual change in science: A comparative metaanalysis of instructional interventions from reading education and science education. Reading Research Quarterly, 28, 117–159. Hofer, B. K., & Pintrich, P. R. (1997). The development of epistemological theories: Beliefs about knowledge and knowing and their relation to learning. Review of Educational Research, 67, 88–140. Hofer, B. K., & Pintrich, P. R. (Eds). (2002). Personal epistemology: The psychology of beliefs about knowledge and knowing. Mahwah, NJ: Laurence Erlbaum Associates. Hynd, C. (2003). Conceptual change in response to persuasive messages. In: G. M. Sinatra, & P. R. Pintrich (Eds), Intentional conceptual change (pp. 291–315). Mahwah, NJ: Lawrence Erlbaum Associates. Hynd, C., McWhorter, Y., Phares, V., & Suttles, W. (1994). The role of instructional variables in conceptual change in high school physics topics. Journal of Research in Science Teaching, 31, 933–946. Hynd, C., Qian, G., Ridgeway, V., & Pickle, M. (1991). Promoting conceptual change with science texts and discussion. Journal of Reading, 34, 596–601. Kintsch, W. (1988). The use of knowledge in discourse comprehension: A construction-integration model. Psychological Review, 95, 163–182. Kuhn, D. (2000). Theory of mind, metacognition, and reasoning: A life-span perspective. In: P. Mitchell, & K. J. Riggs (Eds), Children’s reasoning and the mind (pp. 301–326). Hove, UK: Psychology Press. Lando, M. (2003). Credenze epistemologiche generali e dominio-specifiche [Domain-general and domain-specific epistemological beliefs]. Unpublished degree dissertation. University of Padua, Italy. Lawson, A. E., & Thompson, L. D. (1988). Formal reasoning ability and misconceptions concerning genetics and natural selection. Journal of Research in Science Teaching, 25, 73–746. Lawson, A. E., & Weser, J. (1990). The rejection of nonscientific beliefs about life: Effects of instruction and reasoning skills. Journal of Research in Science Teaching, 27, 589–606. Lawson, A. E., & Worsnop, W. A. (1992). Learning about evolution and rejecting a belief in special creation: Effects of reflective reasoning skill, prior knowledge, prior belief and religious commitment. Journal of Research in Science Teaching, 29, 143–166. Limón, M., & Mason, L. (Eds). (2002). Reconsidering conceptual change. Issues in theory and practice. Dordrecht, The Netherlands: Kluwer. Mason, L. (2000). Role of anomalous data and epistemological beliefs in middle students’ theory change on two controversial topics. European Journal of Psychology of Education, 15, 329–346. Mason, L. (2001). Responses to anomalous data on controversial topics and theory change. Learning and Instruction, 11, 453–483. Mason, L. (2003). Personal epistemologies and intentional conceptual change. In: G. M. Sinatra, & P. R. Pintrich (Eds), Intentional conceptual change (pp. 199–236). Mahwah, NJ: Lawrence Erlbaum Associates. McNamara, D. S., Kintsch, E., Songer, N. B., & Kintsch, W. (1996). Are good texts always better? Interactions of text coherence, background knowledge, and levels of understanding in learning from text. Cognition and Instruction, 14, 1–43.

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Mikkilä-Erdmann, M. (2002). Science learning through text: The effect of text design and text comprehension skills on conceptual change. In: M. Limón, & L. Mason (Eds), Reconsidering conceptual change. Issues in theory and practice (pp. 337–356). Dordrecht, The Netherlands: Kluwer. Posner, G. J., Strike, K. A., Hewson, P. W., & Gertzog, W. A. (1982). Accommodation of a scientific conception: Toward a theory of conceptual change. Science Education, 66, 211–227. Qian, G., & Alvermann, D. (1995). Role of epistemological beliefs and learned helplessness in secondary school students’ learning science concepts from text. Journal of Educational Psychology, 87, 282–292. Qian, G., & Alvermann, D. E. (2000). The relationship between epistemological beliefs and conceptual change learning. Reading & Writing Quarterly, 16, 59–74. Qian, G., & Pan, J. (2002). A comparison of epistemological beliefs and learning from science text between American and Chinese high school students. In: B. K. Hofer, & P. R. Pintrich (Eds), Personal epistemology. The psychology of beliefs about knowledge and knowing (pp. 365–385). Mahwah, NJ: Lawrence Erlbaum Associates. Schnotz, W., Vosniadou, S., & Carretero, M. (Eds). (1999). New perspectives on conceptual change. Amsterdam: Pergamon. Schommer, M. (1990). Effects of beliefs about the nature of knowledge on comprehension. Journal of Educational Psychology, 82, 498–504. Schommer, M. (1993). Epistemological development and academic performance among secondary students. Journal of Educational Psychology, 85, 406–411. Sinatra, G. M. (2005). The “warming trend” in conceptual change research: The legacy of Paul R. Pintrich. Educational Psychologist, 40, 107–115. Sinatra, G. M., & Pintrich, P. R. (Eds). (2003a). Intentional conceptual change. Mahwah, NJ: Lawrence Erlbaum Associates. Sinatra, G. M., & Pintrich, P. R. (Eds). (2003b). The role of intentions in conceptual change learning. In: G. M. Sinatra, & P. R. Pintrich (Eds), Intentional conceptual change (pp. 1–18). Mahwah, NJ: Lawrence Erlbaum Associates. Sinatra, G. M., Southerland, S. A., McConaughy, F., & Demastes, J. (2003). Intentions and beliefs in students’ understanding and acceptance of biological evolution. Journal of Research in Science Teaching, 40, 510–528. Southerland, S. A., & Sinatra, G. M. (2003). Learning about biological evolution: A special case of intentional conceptual change. In: G. M. Sinatra, & P. R. Pintrich (Eds), Intentional conceptual change (pp. 317–345). Mahwah, NJ: Lawrence Erlbaum Associates. Vosniadou, S. (1994). Capturing and modeling the process of conceptual change. Learning and Instruction, 4, 45–69. Vosniadou, S. (2003). Exploring the relationships between conceptual change and intentional learning. In: G. M. Sinatra, & P. R. Pintrich (Eds), Intentional conceptual change (pp. 377–406). Mahwah, NJ: Lawrence Erlbaum Associates. Windschitl, M., & Andre, T. (1998). Using computer simulations to enhance conceptual change: The roles of constructivist instruction and student epistemological beliefs. Journal of Research in Science Teaching, 35, 145–160. Wiser, M., & Amin, T. (2001). “Is heat hot?” Inducing conceptual change by integrating everyday and scientific perspectives on thermal phenomena. Learning and Instruction, 11, 331–355.

Chapter 14

Conceptual Change Ideas: Teachers’ Views and their Instructional Practice Reinders Duit, Ari Widodo and Christoph T. Wodzinski

Objectives Conceptual change views have played a key role in constructivist-oriented attempts to improve science teaching and learning since the 1980s (Duit & Treagust, 1998, 2003; Mason, 2001; Schnotz, Vosniadou, & Carretero, 1999). Whereas constructivist ideas provide the epistemological frame for conceptualizing teaching and learning processes, the term conceptual change denotes that learning science includes major reconstruction of the already existing conceptual structures (Duit, 1999). Conceptual change views play a major part in recent programs to improve the quality of science instruction (Beeth et al., 2003). It seems, however, that constructivist ideas and findings of research on conceptual change are not familiar to most science teachers. This lack of teachers’ familiarity with the recent state of research on efficient teaching and learning is seen as the key barrier towards improving science teaching and learning (Anderson & Helms, 2001). Teachers’ epistemological beliefs about teaching and learning as well as their views about efficient instruction have to undergo major conceptual changes. Fostering these changes has played a significant role in approaches of teacher professional development (West & Staub, 2003). The study presented here does not address the issue of changing teachers’ views. It is investigating whether a sample of German physics teachers is familiar with conceptual change ideas, on the one hand, and whether their instruction meets key characteristics of such approaches, on the other. Subsequent studies drawing on these results attempt to develop teachers’ epistemological beliefs and views about instruction as well as their instructional behavior. An analysis of the literature has resulted in two sets of categories denoting key issues of conceptual change ideas. The first category system, COSC (constructivist-oriented science classrooms), is designed to identify the appearance of key characteristics of constructivist learning environments (Phillips, 2000). The second category system, CTS

Reframing the Conceptual Change Approach in Learning and Instruction Copyright © 2007 by Elsevier Ltd. All rights of reproduction in any form reserved. ISBN: 0-08-045355-4

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(constructivist teaching sequences), allows one to investigate whether the phase structure of instruction follows suggestions for constructivist teaching sequences in the literature (e.g., Biemans, Deel, & Simons, 2001; Driver, 1989).1 Additional data on teachers’ thinking about teaching and learning science are available from teachers’ interviews.2 The study aims at investigating the appearance of key characteristics of constructivist learning environments and constructivist teaching sequences in ordinary science classrooms. It is not the intention to use constructivism to judge teachers’ teaching, rather the main objectives of the study are to explore the appearance of constructivist ideas in ordinary classroom practice, to identify teachers’ difficulties to implement constructivist ideas, and to explore the possibilities of using constructivist ideas to improve practice. The study presented is part of the first phase of a larger project that aims at investigating physics teachers’ and students’ scripts of introductory physics instruction.3 The study shares major features of the TIMS Video Study on mathematics instruction in the USA, Japan, and Germany (Stigler, Gonzales, Kawanaka, Knoll, & Serrano, 1999) and of the more recent TIMSS-R video studies on science instruction (Roth et al., 2001).

Constructivist-Oriented Science Classrooms Constructivist principles of teaching and learning (e.g., Driver, 1989; Duit & Treagust, 1998, 2003; Matthews, 2000; Phillips, 2000; Watts, 1994) provide the theoretical framework for the development of the coding system to identify key characteristics of constructivist informed learning environments. Constructivism suggests that first, knowledge is human construction. From the constructivist point of view, knowledge is not an objective representation of the world; rather it is a human construct. Natural objects or phenomena are themselves “objective” and “real” but the observations and interpretations of them are affected by the subjective interpretation schemes of the observer. Second, knowledge is constructed within certain social and material contexts and, consequently, it is affected by issues such as ideologies, religion, politics, economics, human interests, and by the particular material features of the learning environment. Third, knowledge is tentative. Our knowledge about the world is not a mere copy of the reality outside but it is our tentative construction about it. Scientific truth is not absolute but it is relative and may change over time.

1

Ari Widodo developed the coding instruments COSC and CTS. For more details of the instruments and the results of his study see Widodo (2004). The Coding Manuals are also available from the first author of the present chapter ([email protected]). 2

Christoph T. Wodzinski (née Müller) was responsible for developing methods to analyze the interviews and for interpreting the results (see Müller, 2004) 3

The team of the first phase of this project comprises: Manfred Prenzel, Tina Seidel, Reinders Duit, Manfred Lehrke, Rolf Rimmele, Christoph T. Müller, Maike Tesch, Lena Meyer, Inger Marie Dalehefte and Ari Widodo. The project is part of the priority program “BIQUA - The Quality of School: Studying Students’ Learning in Math and Science and Their Cross-Curricular Competencies Depending on In-School and Out-of-School Contexts” sponsored by the German Science Foundation that includes a total of 23 projects.

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Since constructivism includes views of what knowledge is and how knowledge is acquired, principles emerge from both perspectives that need to be taken into account in developing criteria for “constructivist learning environments.” Science learning should provide students with opportunities to experience science as a body of knowledge and as a knowledge generation process (Duschl & Gitomer, 1991). Students should learn facts, laws, and theories and how science works and knowledge is developed. It suggests that principles of the nature of science (McComas, 1998; OECD PISA, 1999) should also be considered in teaching science. Key characteristics of constructivist-oriented learning environments have been provided in a number of research projects. In developing our COSC instrument we draw especially on the two versions of CLES (constructivist learning environment scale; Aldridge, Fraser, Taylor, & Chen, 2000; Fraser, 1998; Taylor, Fraser, & Fisher, 1997; Taylor & Fraser, 1991) and STAM (secondary teacher analysis matrix —science version; Gallagher & Parker, 1995). Furthermore, criteria for constructivist learning environments developed by Labudde (2000) as well as Tenenbaum, Naidu, Jegede, and Austin (2001) are taken into account. The COSC consists of five categories and each is developed further into three to six subcategories (Table 14.1). The first category, “Facilitating Knowledge Constructions,” represents that knowledge is seen as human construction, that learners have pre-instructional conceptions, that learning is an active process of knowledge construction, and that learning is a change in learners’ conceptions, that is, a conceptual change. It identifies the extent to which students’ prior knowledge and conceptual change strategies are explored and employed to facilitate students’ knowledge constructions. The second category, “The Relevance and the Meaningfulness of the Learning Experience,” represents the view that knowledge construction is embedded within a particular social and material context that may support or hamper conceptual change, that is, students’ construction processes. This category identifies the extent to which students’ learning needs are addressed and how resources are utilized to provide relevant and meaningful learning experiences to the students. The third category, “Social Interactions,” focuses on the issue that knowledge is socially constructed. It identifies the extent to which students are given opportunities to socially interact with each other and the teacher, in different forms of social organizations. Different forms of social organizations provide different learning experiences to the students. Social interaction is also essential for conceptual change since students need to be informed about the ideas of other students and realize that their point of view may not be shared by others. The fourth category, “Fostering Students to be Independent Learners,” represents constructivist views that learners are purposive and ultimately responsible for their own learning. It identifies the extent to which students are given some freedom to organize their own learning and are fostered to be independent learners (see Sinatra and Pintrich (2003) on the importance of independent learning for intentional conceptual change). The fifth category, “Science, Scientific Knowledge, and Scientists,” represents the constructivist views that science knowledge is human construction and that science knowledge is tentative (see McComas, 1998). It identifies the extent to which science lessons provide the students with opportunities to experience how scientific knowledge is constantly being developed and revised.

Table 14.1: Constructivist-oriented science classrooms (COSC). C. Social interactions

1. Making the students aware of the status of their learning within the whole subject 2. Exploring students’ prior knowledge or ideas 3. Exploring students’ ways of thinking

1. Exploring students’ interests, attitudes, and feelings

1. Student–student interactions

4. Providing thinkingprovoking problems

4. Using resources from everyday life

5. Addressing students’ conceptions

5. Discussing applications of the concepts learned

a. Using evolutionary ways b. Using evolutionary ways

2. Addressing students’ learning needs 3. Addressing real-life events, phenomena, or examples

D. Fostering students to be independent learners

1. Providing the students with some freedom to organize their own learning a. Simple interactions 2. Encouraging the among the students students to rethink their own ideas b. Students exchange 3. Encouraging the ideas with other students to be selfstudents regulative and reflective

2. Student–teacher interactions a. Simple interactions between students and the teacher b. Students exchange ideas with the teacher 3. Social organization of the class a. Individual setting b. Group setting c. Classroom setting

4. Taking into account students’ critical voices

E. Science, scientific knowledge, and scientists 1. Acknowledging the tentativeness of science 2. Acknowledging differences in theories or views 3. The roles of observation and evidence, hypotheses, theories, and laws in science 4. Acknowledging differences in the ways to do science 5. Acknowledging the limitations of science explanations

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Constructivist Teaching Sequences Major characteristics of the previously described COSC instrument are the way teachers address students’ conceptions when developing physics concepts, and whether students’ interests, attitudes, and needs are explicitly taken into account, in short, whether a learning environment is provided that persistently supports students’ construction processes, that is, fosters conceptual change. In order to allow a more fine-graded investigation of teachers’ attempts to guide students from their pre-instructional conceptions to the physics concepts, we have developed an additional coding system. The sequencing of instruction observed is analyzed from the perspective of constructivist teaching and learning sequences which have played a significant role in conceptual change approaches. For constructing the “reference model” for steps of constructivist teaching sequences in Figure 14.1, we draw on major approaches in the literature (Biemans & Simons, 1999; Cosgrove & Osborne, 1985; Driver, 1989; Hodson, 1993; Lawson, Abraham, & Renner, 1989; Nunez-Oviedo, Clement, & Rea-Ramirez, 2002). According to our model, constructivist teaching progresses in a spiral sequence of five steps: 1. Introduction: It identifies efforts to prepare students for the topic, to promote students’ readiness, and to generate students’ interests in the lesson. 2. Exploring students’ prior knowledge: It identifies teachers’ efforts to explore students’ prior knowledge related to the topic. 3. Restructuring students’ conceptions: It identifies attempts to facilitate conceptual change. 4. Applying the newly constructed ideas: It identifies attempts to apply the concepts learned to other contexts or to real life. 5. Reviewing the new ideas: It identifies attempts to encourage students to compare the newly achieved and the previous conceptions.

12 Introduction

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Figure 14.1: Steps of the constructivist teaching sequences for conceptual change.

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Design of the Study Thirteen classes of schools in two German states participating in a national quality development program comprise the sample (13 teachers; 344 grades 7 and 8 students). Two three-lessons sequences on introduction to the electric circuit and the force concept were video-documented for each teacher (Figure 14.2). Major data sources are the videos of the lessons. Further data sources are student questionnaires. In the beginning and at the end of the school year, a questionnaire was used to investigate the development of students’ affective variables (e.g., interests, self-concept, and meta-cognitive views) and achievement (regarding the two topics video-documented). After each lesson, video-documented brief questionnaires — especially on students’ views of instructional activities and learning motivation during instruction — were provided. On the teacher side, there was a brief questionnaire on their views of learning in the beginning of the school year. Teachers were also asked to provide drafts of the planned instruction on the two topics documented. After videotaping the second topic, teachers were interviewed about various facets of teaching and learning physics. They were also asked to comment on three video-documented episodes from their instruction. Various quantitative and qualitative data are available. Accordingly, quantitative and qualitative research methods are employed (see for a similar approach the “video-survey methodology” of TIMSS-R Video; Roth et al., 2001). Two video cameras were used. One of them followed the actions of the teachers, the other documented the whole class. Every lesson was fully transcribed and coded in various ways (Prenzel, Duit, Euler, Lehrke, & Seidel, 2001); for a summary Duit, Müller, Tesch, & Widodo, 2004) by using the software “Videograph” (Rimmele, 2003) developed by one of our team members.4 For each of the coding systems used the interrater-reliabilty was carefully examined. In the following, we refer only to the coding systems COSC and CTS September 2000

July 2001

Video Topic 1 3 periods

Q1

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Qca

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TI Q 2

Q1/Q2 : Students' questionnaires on interest and assessment of knowledge Qca : Students' questionnaires with regard to the videotapedlessons TI : Teachers' interviews withregardtothe videotapedlessons

Figure 14.2: Design of the study. 4

Available from: http://www.ipn.uni-kiel.de/projekte/video/videostu.htm

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presented above. For them, on average two independent coders agreed in more than 90% of the decisions. The COSC and the CTS are designed as time-based category systems. In this study the coding time unit is 10 seconds. The COSC is designed as a two-option category system. In coding practice, after observing a 10-second unit, coders decide whether the unit meets the criteria for the related subcategories. When the unit meets the criteria, it is coded “yes” and when it does not meet the criteria it is coded “no.” The CTS is designed as a five-option category system.

Results: Learning Environments of the Lessons (COSC) In general, the results from the COSC show that the lessons meet the characteristics of constructivist-oriented science classrooms only to a certain extent. Most subcategories were observed for less than 10 minutes. A number of characteristics were not observed at all, namely subcategories D-3 and D-4 as well as all subcategories of category E. Facilitating Knowledge Constructions

Observed(in minutes)

Two important characteristics for facilitating students to construct knowledge, namely making the students aware of the status of their learning and exploring students’ ways of thinking, were very seldom observed, while the other three subcategories were observed more often (Figure 14.3). Regarding the issue of making students aware of the status of their learning, a number of teachers in the interview claimed that they do not provide a “preview” of the lesson because they do not like to lose moments of surprise. Although exploring students’ prior knowledge was observed relatively often, frequently little was obtained from these attempts. Usually teachers asked the students to think about what they already know about the topic in question, sometimes requesting them to write down what they think. However, the questions asked by the teachers were usually very general and the identified prior knowledge not specific enough to allow the 10 8 6 4 2 0 A-1

A-2

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A-1: Making students aware of the status of their learning. A-2: Exploring students’ prior knowledge or ideas. A-3: Exploring students’ ways of thinking. A-4: Providing thinking-provoking problems. A-5: Addressing students' conceptions.

Figure 14.3: Results of subcategories A-1 to A-5.

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teachers to detect students’ pre-conceptions. Additionally, after exploring students’ prior knowledge, often no further actions were taken to address it. It seems that the teachers did not have clear ideas of how to make use of students’ prior knowledge. This finding is similar to Hewson, Tabachnick, Zeichner, and Lemberger’s (1999) results that prospective teachers explored students’ prior knowledge, but only few were able to use it to plan their teaching. At first sight, it is rather pleasing that a substantial amount of time (in average about 20%) was spent to address students’ conceptions — which is a key issue in constructivist conceptual change approaches. Only seldom observed were the use of analogies and cognitive conflicts. “Evolutionary” strategies (drawing on step by step development of ideas) were more frequent than “revolutionary” strategies (drawing on sudden insight facilitated by cognitive conflicts). As will be discussed later (see the results of the CTS instrument), the use of conceptual change strategies as proposed in the literature (Duit, 1999) was rather infrequent. This finding is not surprising taking into account that the interviews revealed that most teachers were not well informed about the literature on conceptual change (Müller, 2004). The most frequent kind of addressing students’ conceptions was embedded in the dominant “questioning–developing” mode of classroom discourse. The teacher opened the discussion with a question and in a series of further questions the science explanation was developed. This attempt to actively include students in the development of the concept in question was often reduced to a dyadic question–answer game in which the teacher asked a question, a student gave an answer, and the teacher only took the answers on board that led to the intended solution. Briefly summarized, from the perspective of conceptual change strategies, there were rather limited opportunities for the students to become aware of their ideas and in which way they differed from the science view. There was further rather limited support for students to construct the new science view. Relevance and Meaningfulness of The Learning Experience As shown in Figure 14.4, lessons seldom addressed issues related to students’ interests and learning needs (B-1, B-2). However, the lessons were quite often situated within students’ everyday life as they frequently addressed real-life phenomena or real-life examples (B-3) and used resources from everyday life (B-4). These results suggest that there were significant efforts by the teachers to provide learning experiences that were relevant to students’ everyday life experiences. Although it was less frequently observed than subcategories B-3 and B-4, application of knowledge (B-5) was also given certain attention. Included were applications addressing issues of real life. Social Interactions Simple interactions were more common than intensive interactions that include exchange of ideas5 (Figure 14.5). Simple interaction denotes primarily the common dyadic style

5

Due to the difficulties to code student–student interactions, no results for subcategories C-1a and C-1b are available.

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where the teacher asks a question, a student gives an answer, the teacher asks the next question and so forth. Intensive interactions concern classroom discourses in which students have a voice and that include true exchanges of ideas. Further, students had only limited opportunities to work in groups and to work individually. In some lessons only classwork but no student individual or student group work occurred. Since lessons were generally rather classwork-oriented and less individual-oriented, it was unlikely that lessons catered to students’ individual learning needs and speeds. Fostering Students to be Independent Learners

Observed (in minute)

As shown in Figure 14.6, teachers put remarkable efforts on fostering students’ independence (see D-1). However, other aspects required for students to be independent learners (especially D-3 and D-4) were seldom observed. For example, one of the important features of being independent learners is that students should be able to learn how to learn, how to control and to monitor their own learning. Unfortunately, these issues were not well addressed by the teachers. It was rarely observed that students’ critical voices were taken into account. Taylor et al. (1997) claim that it is an essential feature of efficient teaching and 20 15 10 5 0 B-1

B-2

B-3

B-4

B-5

subcategory

B-1: Exploring students’ interests, attitudes, and feelings. B-2: Addressing students’ learning needs. B-3: Addressing real-life events, phenomena, or examples. B-4: Using resources from everyday life. B-5: Discussing applications of the concepts learned.

Observed (in minute)

Figure 14.4: Results of subcategories B-1 to B-5. 40 30 20 10 0 C-2a

C-2b

C-3a

C-3b

subcategory

C-2a: Simple interactions between students and the teacher C-2b: Students exchange ideas with the teacher C-3a: Individual setting C-3b: Group setting C-3c: Classroom setting.

Figure 14.5: Results of subcategories C-2 and C-3.

C-3c

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12 10 8 6 4 2 0 D-1

D-2

D-3

D-4

subcategory

D-1: Providing students with some freedom to organise their own learning. D-2: Encouraging students to re-think their own ideas. D-3: Encouraging students to be self-regulative and reflective. D-4: Taking into account students’ critical voices.

Figure 14.6: Results of subcategory D-1 to D-4.

learning that students have a voice in instruction, that is, that students’ contributions to classroom discourse are taken seriously by the teacher. Hanrahan (1998) argued that this may only be expected if students are explicitly encouraged to do that. Science, Scientific Knowledge, and Scientists The subcategories denoting views of the nature of science could not be observed. In other words, they did not play any role in the introductory physics instruction video-documented. In the interviews, some teachers argued that it is too early for such considerations at introductory level. But it became also evident that many teachers were not very familiar with views of the nature of science addressed by the category E.

Results: Constructivist Teaching Sequences (CTS) As presented in Figure 14.7, all five steps were observed in the lessons. However, this does not mean that each lesson always included all five steps. The sum of the length of all steps reveals that in only 21 minutes of the time, steps of the constructivist teaching sequences could be observed. This suggests that about 50% of teaching time was spent for other issues, such as addressing discipline, taking notes, or doing activities that do not clearly refer to any of the five steps. Analyses of the individual lessons show that many lessons had not progressed in steps as suggested by the constructivist teaching sequences. To provide examples, Figure 14.8 displays the appearance of the steps for two teachers’ introductory lessons of the topic “electric circuit.” A number of lessons did not show complete steps. T-7’s lesson in Figure 14.8, for instance, did not include application and review. Further, a number of lessons did not progress in steps as suggested by the constructivist teaching sequences. T-1’s lesson in Figure 14.8 to a certain extent meets the CTS. It began with a short introduction, followed by extended explorations of students’ prior knowledge. By the end of the lesson a short restructuring was done followed by application and review. In contrast, T-7’s lesson included only introduction, exploration, and restructuring.

Length (in minute)

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6 4 2 0 1

2

3

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Figure 14.7: Average time of each step of the CTS.

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Figure 14.8: Analyses of the sequences of lessons. Analyses of the sequences reveal the following findings. First, in 26 lessons (38%), one or more steps were missing. The fourth step (applying the conceptions) and the fifth step (reviewing) were the two most frequently missing steps. Second, in many lessons one or more steps were skipped or they progressed in unclear sequences.

Patterns of Instruction and the Development of Achievement and Interests In order to investigate relations between certain patterns of instruction and the development of student cognitive and affective variables within the school year of videotaping (Figure 14.2), correlations between the frequencies of certain codings, on the one hand,

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and the development of student achievement and affective variables (like interests) measured by a pre- and a post-questionnaire, on the other, were calculated. Due to the small number of classes participating in the study these measures have to be interpreted with great care. At best they provide preliminary hypotheses that have to be further investigated in subsequent studies. For the coding system COSC there are two subcategories where substantially high correlations (we used partial correlations) with student achievement occurred — for the length of time students are challenged with thinking-provoking problems (A-4) and the time spent in addressing students’ conceptions (A-5). These results are in accordance with findings from other studies (e.g., Ditton, 2002) that constructivist learning environments may promote students’ learning. There are also somewhat weak indications that embedding instruction in everyday life contexts is linked to advanced achievement. Interestingly, there is no substantial correlation between achievement gains and subcategory A-2 “Exploring students’ prior knowledge or ideas.” As argued above, the limited scope of the teachers’ exploration attempts may be responsible for this result, which at first sight appears to be in contradiction to arguments in the constructivist literature that exploring students’ ideas is essential. Substantial correlations between certain codings (denoting certain instructional patterns) and the development of affective variables could not be observed (for a detailed discussion see Widodo, 2004). For the coding system CTS there is a significant correlation between the total time in which steps of the CTS were observed for the 13 teachers and student achievements gains (Widodo, 2004). This finding appears to indicate that the five coded steps (Figure 14.1) denote the essential phases of effective instruction. This result is in accordance with findings from other studies that orienting instruction at constructivist teaching sequences promotes students’ learning (e.g., Biemans et al., 2001).

Teacher Interviews In-depth interviews with each of the 13 teachers provide the data for investigating teachers’ epistemological beliefs about teaching and learning as well as their views about efficient instruction. The study presented here also explores the match between teachers’ beliefs and views and their actual classroom behavior. Further attention is given to the relations between teachers’ views and teaching behavior on the one hand, and the development of student achievement and affective variables on the other. The first part of the interviews explored teachers’ ideas about the key issues of the study, namely: • • • • •

the role of experiments, the significance of various science processes, views of the nature of science, their preferred aims, and their epistemological beliefs of teaching and learning.

In the second part of the interview teachers were presented with three video sequences from their instruction: (1) introduction into one of the topics taught, (2) a typical whole

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class activity, and (3) a typical experiment. The semistructured interviews lasted between 60 and 90 minutes. They were carried out by the first author. They were audio-documented and fully transcribed. The interviews were analyzed from two different but complementary perspectives. First, a qualitative analysis drawing on various theoretical perspectives (like theoretical frames on the role of experiments and processes or constructivist views) was carried out. This analysis was based on the “qualitative content analysis” as provided by Mayring (2000). This analysis proceeds in three steps. As a first step, the utterances documented in the transcripts are paraphrased in such a way that key features of the meaning become apparent. As a second step, these key features are thematically organized and summarized as “prototypical statements.” These statements — as a third and last step — are systematically categorized. This analysis resulted in a brief summary description of teachers’ views concerning the above-mentioned key issues. Second, a system of categories was developed to carry out a turn-to-turn coding of the interviews. These categories were oriented at the categories used for coding the lessons in order to allow an analysis of what teachers think and what they do in classrooms. Due to the small sample of 13 teachers, statistical analysis can only be used to generate hypothetical information on eventually relevant interdependencies. For this attempt, a cluster analysis was conducted on the quantitative data of the coding process. It resulted in four types of teachers’ implicit theories. These types matched the results of the qualitative analysis. Teachers’ implicit theories were compared with their actual (videotaped) classroom practice. Finally, relations between the types of implicit theories on the one hand and student achievement gains over one school year and the development of students’ affective variables on the other were investigated.

Findings on Teachers’ Views of Teaching and Learning Physics Teachers’ Familiarity with the Recent Literature on Teaching and Learning Science and Their Views of Learning Science Most teachers are not well informed about key ideas of conceptual change research, neither of constructivist theories nor of conceptions of teaching and learning in general. Their views of their students’ learning usually do not meet the state of recent theories of teaching and learning. In general, most teachers appear to lack an explicit view of learning. Some of the teachers hold implicit theories, that contain some “intuitive” constructivist aspects, for instance, they want to be learning counselors, and they are aware of the importance of students’ cognitive activity and the interpreting character of students’ observations and understanding. However, we also find teachers who characterize themselves as mediators of facts and information and who are not aware of students’ interpretational frames and the role of students’ pre-instructional conceptions. They mostly think that giving good instructions is a guarantee for successful learning. Only one teacher was well informed about current theories of teaching and learning, namely constructivist theories. Ironically, the instruction of this teacher was the least constructivist.

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The analyses of the interviews also revealed that the following way of thinking about instruction is predominating. Major concerns are considerations on the science topic in question. Considerations on students’ learning capabilities and difficulties are often marginal or nearly totally missing. These findings clearly have to do with the previously mentioned lack of explicit views about students’ learning.

Types of Teachers’ Implicit Theories The cluster analysis of quantitative coding of the teachers’ interviews resulted in four types of teachers’ implicit theories (Table 14.2). Table 14.2: Types of teachers’ implicit theories based on the analysis of teachers’ interviews. Cognitive apprenticeship

Cognitive self-effort

Learning viewed as stepwise supported development in argumentative discourse

Topic-oriented learning opportunities should be provided — but support is not explicitly mentioned

Mistakes are learning opportunities Conceptual Growth

Type C View themselves primarily as physicists Aim of instruction: introduction into the physics view, including understanding concepts, principle and processes

Conceptual Change

Type A View themselves as teachers

Type B View themselves as pushing forward, to overcome learning resistance Clear flow of instruction (order, precision) Aim of instruction: Incite interest to understand everyday life issues Type D View themselves as providers of learning opportunities

Teaching oriented to student understanding; provide students with chances to find out themselves and to think

Expect/miss interest & student initiatives

Aims: modeling, understanding everyday life

Students shall find out themselves Instruction open but not substantially structured

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The succession in which the clusters occurred in the analysis was type A, B, C, and D. These types could be described by certain characteristic items included in the cluster analysis. Based on these items, two theoretically founded dimensions were constructed. Each type of teachers’ implicit theories could be allocated to one of the cells in this twodimensional framework. For the first dimension, one pole represents the idea that learning can be viewed as stepwise supported development in argumentative discourse, while the other pole represents the idea that teaching means to provide topic-oriented learning opportunities, whereby the need to support the use of the learning opportunities is not explicitly mentioned. The poles of this dimension could be characterized by “cognitive apprenticeship” and “cognitive self-effort.” For the second dimension, one pole represents the idea that teaching means to provide opportunities to change students’ scientific concepts, while the other pole represents the idea that teaching physics means to present physical phenomena and to mediate the structure of the topic. Here, the poles of the dimension could be called “conceptual change” and “conceptual growth.” Taking these two dimensions together, we can identify four types of teachers as described below. Statistical analysis reveals substantial differences between the classes allocated to the types of implicit theories concerning student achievement gains over one school year. Due to the small sample of 13 teachers, the following results have to be interpreted with caution. They may be viewed as hypotheses only. Between the type of teacher (being a member of one type of implicit theories) and students’ achievement gains (pre–post, i.e., from the beginning to the end of the school year) we found a positive and significant correlation (r ⫽ 0.697, p ⫽ 0.008 for Rasch parameters; r ⫽ 0.755, p ⫽ 0.003 for residuals; Figure 14.9). Between the type of teacher and students’ affective variables we found no significant correlations. Nevertheless, we can observe substantial differences in the development of interest, self-concept, and self-evaluated competence between students of the different types of teachers. 2.5

10

achievement gains (Rasch)

2

8 6 4

1.5

2 0 -2

1

-4 -6

0.5

achievement gains (residual)

achievement gains (Rasch) achievement gains (residual)

-8 0

-10 type A

type B

type C

type D

Figure 14.9: Achievement gains (Rasch parameters and residuals) for types A to D.

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Regarding achievement gains, the average gain for classes of the “cognitive apprenticeship” teachers is significantly higher than for the “cognitive self-effort” teachers (Figure 14.9). There is also a somewhat marginal difference between the “conceptual growth” and “conceptual change” poles of the “cognitive apprenticeship” dimension with the “conceptual change” teachers being ahead. From our point of view, these results indicate that to foster students’ learning in the sense of constructing their own knowledge it is necessary not only to provide opportunities for students to think and work independently but to structure learning processes. Students seem to need stepwise guidance in the sense of cognitive apprenticeship to connect new knowledge with existing structures. The role of the teacher in this process should be a kind of learning counselor, not only somebody providing learning opportunities without telling students how to use them. The results also show a strong relationship between teachers’ perspectives on issues of instruction and their own role definition. Therefore, developing teachers’ content-specific pedagogical knowledge is necessary not only to foster their methodological competencies but also to develop a new view of their role as teachers.

Teachers’ Views in a Nutshell Seen from the constructivist conceptual change perspective underlying the present chapter, teachers’ thinking about teaching and learning physics meets characteristics of the constructivist view only marginally. Most teachers are not well informed about recent views of teaching and learning science, they do not hold explicit views of student learning, and their ways of thinking about instruction is rather topic-oriented only. However, results also show that a number of teachers implicitly hold views that are in accordance with constructivist ideas. And if they implicitly hold such views the achievement gains appear to be better as compared to teachers who do not hold them.

Summary of Findings Findings of the Whole Video-Study The findings presented in this chapter are part of a larger set of studies within the first phase of the IPN Videostudy Physics. It turned out (Duit et al., 2004; Prenzel et al., 2002; Seidel et al., 2002) that the physics instruction of our 13 teachers is strongly teacher dominated. The teachers prefer a script in which a straight path to a certain concept or to the results of a certain experiment is used. Our data also show that there is substantial variance between the 13 teachers participating. Therefore, we observe 13 scripts that are individual and characteristic of the single teachers. There is also a broad range of teaching methods — although teacher-dominated methods prevail. We see, for instance, also teaching and learning methods that allow students some freedom to follow their own ideas. However, certain narrowness is to be seen in the scripts of most teachers. This becomes

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apparent, for instance, in the dominating use of the questioning– developing approach. Rather seldom does it happen that the dyadic dialogue between the teacher and students is widened to a discourse including interactions between students. There are also several examples in our data that the questioning–answering game turns out to be rather inefficient. This is, for instance, the case when teachers try to develop characteristics of physics concepts. Often teachers ask questions students cannot answer at that time. Hence, they start a guessing game. The teachers’ views about “good” physics teaching and learning as revealed by the teacher interviews also show a rich repertoire of thinking patterns about instruction, on the one hand, and a certain narrowness, on the other. Many teachers hold elaborate ideas about their way of teaching. However, considerations about the content in question predominate planning. Reflections about students’ perspectives and their role in the learning process play a comparably minor role. It turned out that most teachers do not hold elaborate views about teaching and learning. Further, most teachers are not informed about research findings on teaching and learning physics. Briefly summarized, two general orientations of instruction may be distinguished: 1. Instructional: Oriented to physics; focus on physics concepts; learning viewed as knowledge transmission. 2. Constructivist: Focus is on student learning, in particular on the conditions necessary to support learning; learning viewed as student construction. The instructional orientation predominates teaching behavior and teachers’ beliefs. There is a large gap between the kind of thinking about efficient teaching and learning physics, as discussed in the research-based literature, and the thinking of the teachers in our study and their instructional practices. Findings Concerning Constructivist-oriented Science Instruction As presented in the previous sections, the instructional patterns observed meet patterns characteristic of constructivist-oriented science instruction only to a limited extent. Most teachers’ views of teaching and learning science are also quite far from constructivist ideas. Furthermore, most teachers are not informed about publications addressing constructivist views. This also holds for publications written for teachers in teachers’ journals. It is even true for publications in which the role of pre-instructional (alternative) conceptions (such as conceptions on the electric circuit or the force concept) are discussed — although a substantial number of such publications appeared since the 1980s in Germany. If instructional patterns and teachers’ views to a certain extent meet constructivist ideas, achievement seems to be better. Two major facets are included in what is called constructivist here. First, learning should be initiated with students’ conceptions that might be in contrast to the physics views and need to be developed as described in the constructivist literature on conceptual change. Second, it is not sufficient to provide learning opportunities but learning has to be continuously supported, for example, by means of cognitive activation.

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Significance of the Findings Owing to the small and particularly composed sample, the results presented here may only be viewed as preliminary hypotheses. In the second phase of the IPN video study we will investigate whether they also hold for the larger and somewhat more representative sample of 50 teachers. A video study in Switzerland which is a sister project of the IPN study (Labudde, Gerber, & Knierim, 2003) provides an additional chance to further develop the hypotheses of the first phase. Concerning research methods, the first phase has confirmed that video analysis is a rather powerful means to investigate instructional practice. This is especially true for a design that is not restricted to just documenting videos (which was the case for the TIMS Video Study in mathematics; Stigler et al., 1999). The set of student and teacher questionnaires as well as interviews used, on the one hand, and coding the video-documented lessons on the other, have provided deep insights into the physics classroom that go far beyond results that may be gained by the single research methods (cf. Roth et al., 2001). The interplay of quantitative and qualitative interpretation modes has proven rather valuable.

Conclusions Our studies have shown that teachers’ views about their students’ learning are limited — if the current state of research in teaching and learning is taken as reference. It further became apparent that teachers’ thinking about instruction is predominantly oriented to the content in question. It appears that teacher professional development needs to address these two issues. It seems that teachers’ epistemological beliefs and their thinking about instruction, on the one hand, and their instructional practice, on the other, have to be developed “in parallel.” Recent approaches of teacher professional development take this into account. West and Staub (2003), for instance, developed an approach called “contentfocused coaching.” They attempt to make teachers familiar (a) with the recent state of research knowledge about teaching and learning and (b) with the view of a fundamental interplay of all variables of instructional planning (content, aims, teaching methods, and media as well as students’ views). In a number of studies these changes of teachers’ epistemological beliefs and views are explicitly modeled as conceptual changes (Beeth & Hewson, 1999; Duit & Treagust, 2003). In the third phase of the IPN Video Study we will investigate in which way instructional videos may support guiding teachers from their views and practices towards overcoming the above limitation of their primarily content-oriented thinking. In a project to improve physics instruction in Germany (Physics in Context; Duit, Euler, Friege, Komorek, & Mikelskis-Seifert, 2003) development of teachers’ thinking about instruction is a key issue. A final remark concerns the significance of research on teaching and learning on improving practice. The state of theory building on conceptual change has become more and more sophisticated and the teaching and learning strategies developed have become more and more complex during the past 25 years. These developments are, of course, necessary in order to address the complex phenomena of teaching and learning (science)

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more and more adequately. However, the gap between what is necessary from the researcher perspective and what may be set into practice by “normal” teachers has also increased more and more. In other words, there is the paradox that in order to adequately address teaching and learning processes, research alienates the teachers and hence widens the “theory–practice” gap. The major message of the present chapter is that it is necessary to close the gap between theory and practice at least to a certain extent. What research on conceptual change has to offer classroom practice is not set into actual practice to a substantial extent. Of course, teacher development programs are essential in order to change teachers’ views of teaching and learning and their practice. However, it also appears to be necessary to “simplify” instructional theories and conceptual change strategies in such a way that they may become part of teachers’ routines.

Acknowledgments Ari Widodo is a lecturer at the Indonesian University of Education in Bandung. His Ph.D. studies at the IPN were supported by a scholarship of DAAD (German Academic Exchange Service). Christoph T. Wodzinski’s work was made possible by a grant from DFG — Deutsche Forschungsgemeinschaft (German Science Foundation). We are also grateful for the contributions of the other team members of the IPN Video Study Physics in which the work presented here is embedded (see footnote 1 above).

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COMMENTARY

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Chapter 15

First Steps: Scholars’ Promising Movements Into a Nascent Field of Inquiry Patricia A. Alexander and Gale M. Sinatra It is likely that most readers of this volume have had the touching experience of watching infants as they first struggle to walk unaided. They teeter on uncertain feet and legs and move ever so hesitantly, but with dogged determination. The frequent stumbles and falls, which are part and parcel of this experience, can prove quite frustrating to the children and worrisome to those bearing witness. Yet, these setbacks or impediments are only temporary and are critical lessons in balance and mobility for the children. Before they can conquer the art of walking, young children must come to understand what their bodies can and cannot do and how those bodies must work in concert with their physical surroundings. Those first steps are truly developmental milestones that ultimately signal a new-found freedom for toddlers. Once bound to a limited physical space or dependent on others for mobility, young children find themselves able to step out into a world that had been virtually inaccessible to them. As we read the collection of chapters in this section, we appreciated that we were also bearing witness to first steps — those uncertain, tentative, but inspiring initial movements that are a milestone to subsequent development. However, we do not see ourselves in the role of doting family members who have already mastered this physical feat and are merely watching the hesitant and unsophisticated movements of those less mature than we. On the contrary, as we share our reflections and recommendations on the writings by Entwistle, Murphy, Alexander, Greene, and Edwards, Mason and Gava, Duit, Widodo, and Wodzinski, and Stathopoulou and Vosniadou (this volume), we see ourselves as literally on comparable footing. Because the nature and boundaries between epistemology, conceptual change, and learning are just coming into focus, we are, in reality, partners in this development and, thus, trying out our own theoretical and empirical legs. The advantage we have in this commentary is that we have the luxury of looking across these diverse and intriguing chapters, able to identify recurring patterns that warrant care and attention if they are to foster further movement and that hint at the next steps that might be taken. Specifically, we have organized this analysis into two sections. In the first section, we outline five themes that emerged from the five chapters — themes that represent areas of concern for those taking initial steps to define this relatively new field of inquiry. Three of

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those themes deal with the clarity, the boundaries and contexts, and the complexity of the constructs under investigation. What do we mean by epistemological beliefs or conceptual change, for example, and when do the studies dealing with epistemology actually cross into the territory associated with ontology? Moreover, how are researchers to be true to the complex nature of epistemology, conceptual change, or learning without being overwhelmed by that complexity? We also explore questions of development and interrelatedness of epistemology, conceptual change, and learning. Here the concern is change over time both within and across the literatures on epistemology and conceptual change. For instance, how can researchers address the changing sophistication of individuals’ knowledge and beliefs over time, especially as they grow in expertise within academic domains? Further, to what degree do transformations in one of these areas (e.g., epistemic beliefs) contribute to developmental shifts in the others (e.g., conceptual change)? We will also examine if current developmental notions, as represented in these chapters, can be judged as indicative of universal transformations, rather than reflections of Western cultures or postindustrial societies? Finally, beyond the questions of how epistemology, conceptual change, and learning are related to each other, we want to explore the degree to which these phenomena are directly or indirectly related to other educational and psychological factors, including learner motivation and pedagogical practices. One question to be considered here is the degree to which formal schooling or the classroom contexts shape students’ epistemic beliefs or their conceptions or the degree to which students’ own personalities, goals, and motivations become the defining forces in the formation of such beliefs and conceptions. In the second section of this chapter, we revisit these five themes of conceptual clarity, boundaries and contexts, complexity, development, and interrelatedness. However, the goal in this section is to pose next steps. What can these scholars and others committed to the nascent field of inquiry (ourselves included) do to ensure that we move more steadily and assuredly into studies of the interplay of learning, epistemology, and conceptual change? How can current researchers lay the appropriate groundwork so that others can follow with greater ease and confidence?

Assessing the First Steps It is never easy to conduct research on human learning and development. Humans are notoriously willful creatures and reluctant participants in psychological probings and proddings. It is even more precarious to attempt studies that go beyond the study of directly observable and tractable phenomena (e.g., salivating animals) and instead seek to infer relations (to say nothing of causality) between constructs (e.g., knowledge and beliefs) that only occasionally rise to the surface of consciousness. This is certainly the reality faced by the brave authors of these intriguing and provocative chapters. Even as we laud Entwistle, Murphy and colleagues, Mason and Gava, Duit and colleagues, and Stathopoulou and Vosniadou (this volume) for their creative and fearless forays into difficult and less-charted terrain, we want to provide some cautions or warnings that might help them from faltering or stumbling in their subsequent investigations.

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Conceptual Clarity Readers of the aforementioned chapters will be confronted with an array of terminology central to this emergent and interdisciplinary field of inquiry. Not surprisingly, that lexicon includes words and phrases about epistemology (e.g., epistemological beliefs, epistemic beliefs, and personal epistemology) and conceptual change (e.g., concepts, conceptual knowledge, and knowledge restructuring), as well as learning (e.g., constructivism). As we have stated, researchers like the scholars contributing to this volume are already pursuing rather elusive constructs that are notoriously hard to uncover and validate. It would seem advisable, therefore, for researchers exploring such constructs as knowledge, beliefs, and learning to take great care to establish the conceptual and operational parameters for their key constructs. Regrettably, this is not always the case within this growing literature. Rather, the literature exploring the relations between epistemology, conceptual change, and learning is often plagued by terminology that is undefined, poorly specified, or variably defined. This is a broad concern that the authors of these chapters rightfully acknowledge. In certain instances, this conceptual wobbliness seems attributable to the newness of the research. For example apart from the work of Perry (1970), the psychological research on epistemology is very much in its infancy, having really come into its own in the last decade. Consequently, it is not surprising that researchers in this area are still seeking solid ground with regard to labels and definitions they use. These conceptual growing pains are quite apparent in discussions over what exactly researchers are studying when they ask questions about the knowledge and knowing. In this volume, both Mason and Gava and Stathopoulou and Vosniadou use the terms epistemological beliefs and personal epistemology to signify the type of beliefs they are examining. Even while selecting these particular labels, Stathopoulou and Vosniadou acknowledge the conceptual unsteadiness they are confronting: There is a diversity of theoretical and conceptual approaches to the construct of ‘epistemological beliefs’. This is reflected by the fact that the construct is assigned different terms in the literature such as, personal epistemology, epistemic beliefs, ways of knowing, epistemological perspectives, epistemological reflection epistemological thinking, epistemological theories, epistemological resources etc. This diversity indicates that what we call ‘epistemological beliefs’ may not be the same in all relevant studies, or, at least, that the boundaries of the construct may differ. (Hofer & Pintrich, 1997; Pintrich, 2002) (this volume: p. 145) Murphy et al. raise the same conceptual issue. Yet, their decision was to draw a sharp distinction between epistemological beliefs and epistemic beliefs — their preferred term — and to avoid the notion of personal epistemology entirely. As these authors stated: For clarity we have taken pains to distinguish between epistemic and epistemological beliefs. The former are beliefs about knowledge, whereas the latter are beliefs about the study of knowledge. Kitchener (2002) has made a strong argument for maintaining this distinction. Students most likely

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Patricia A. Alexander and Gale M. Sinatra have epistemic beliefs, beliefs about knowledge. Many researchers have epistemic beliefs as well, such as what means of justification are privileged over others. Finally, those looking at the implicit or explicit assumptions regarding the study of knowledge are engaging in research on epistemological beliefs. (this volume: p. 105–106)

Although such conceptual wrangling may be worrisome, we feel that, in the case of epistemological versus epistemic versus personal epistemological beliefs, it bodes well for this emergent field. Specifically, it suggests that this literature is maturing to the point of dealing with complicated but fundamental quandaries that have previously been overlooked or ignored. Whether this conceptual maturation continues or leads to greater consistency with this psychological research in knowledge and knowing remains to be seen. In direct contrast to the epistemological/epistemic beliefs debate, some of this conceptual teetering may reflect overconfidence or undue comfort with often-used words within more mature lines of inquiry. Entwistle (this volume) makes this point when he offers an explicit definition of concepts — a word for which a definition is often assumed. ‘Concept’ is most frequently used to describe a grouping of objects or behaviours with the same defining features that has become recognised through research or widespread usage. The same term has also been used in the literature on conceptual change to indicate individuals’ different ways of thinking about a particular grouping. Here, for clarity, these individual variations are referred to as ‘conceptions’. (this volume: p. 124) Nowhere is this problem of definition through assumption more apparent than in researchers’ treatment of the three words most at the heart of this program of research — knowledge, beliefs, and learning. Researchers like Entwistle and Mason and Gava (this volume) convey a sensitivity to the labels applied to psychological studies of individuals’ perceptions of knowledge and beliefs. Nonetheless, as the chapter by Murphy et al. indicates, researchers may be slow to reveal the definitions of knowledge and beliefs that guide the design and interpretation of their investigations, even if such definitions are well articulated in their minds. We see the conceptual definition of these constructs, along with an explication of learning, to be important to anchor subsequent investigations, especially given that scholars in this realm are expressly engaged in the study of epistemic beliefs. One problematic term, which is at the heart of the study by Duit et al. (this volume), is constructivism. Despite the fact that Duit et al. portray the notion of constructivism and constructivist teaching in rather tidy language, it is far from a tidy construct. As Bereiter (1994), Phillips (1995) and others have argued, there are many faces to constructivism. Moreover, a constructivistic orientation toward learning (i.e., knowledge is constructed) should not be confused with attempts to convert such a learning theory into pedagogical practice. In essence, if all knowledge is, in actuality, constructed, then the mode of instructional delivery would be inconsequential. Whether teachers lecture or engage students in active learning, those students would still form their knowledge via construction. In addition, the effectiveness of constructivist pedagogy has recently been called into question (Kirschner, Sweller, &

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Clark, 2006), further complicating the issue. Thus, adding constructivism to the discussion of epistemological beliefs and conceptual change may only serve to exacerbate the problem of conceptual clarity in this nascent literature. Boundaries and Contexts Perhaps the unsteadiness we perceive in the terms that mark this emergent field of inquiry is indicative of an even deeper issue; the nature of the constructs under study and the conceptual boundaries between them. In effect, it is one thing to differentially label the same construct, but it is another to use the same label to signify varied constructs. Within these five informative chapters, for example, we see scholars struggling to transfer particular philosophical constructs (e.g., epistemology) into the realms of psychology or pedagogy. In so doing, these researchers must confront the possibility that they are now engaged in the study of fundamentally altered concepts (Alexander, 2006). Part of the struggle in terminology use captured in the epistemological versus epistemic beliefs debate reflects this awareness. However, what researchers in this realm may be overlooking is that they have not only stepped across the philosophical threshold into psychology, but may have also transitioned from epistemology to ontology. Specifically, when we look at the questions researchers are asking about knowledge, many of them move beyond questions about the source or justification of knowledge and more directly target questions about “what is knowledge.” When “nature” is the central concern, researchers may have crossed the boundary from epistemology (“what is known”) into ontology (“what is”). This is an argument recently outlined by Murphy et al.: We would suggest, however, that the psychological literature might benefit from the stricter definitions of philosophy, separating epistemological concepts, such as justification, from more ontological constructs, such as the relations among objects of knowledge or characteristics of knowledge (e.g., certainty or simplicity). It would also help coordinate the two disciplines to emphasize the personal aspects of epistemology in the psychological literature, to clearly identify the differences between the two fields. (this volume: p. 119) We concur with the authors of these five chapters that it is important to ascertain what students, teachers, and researchers believe knowledge and concepts to be. However, as with Murphy et al. (this volume) and others (Alexander, 2006), we feel it is essential to accurately mark this as more ontological than epistemological territory. Until we have a more definitive sense of what researchers and their study participants mean by knowledge, knowing, or beliefs, then it is more difficult to interpret their responses to questions about the source, certainty, or justification of knowledge or malleability of beliefs (Alexander & Dochy, 1995; Sinatra & Kardash, 2004). In a way, we see this association between ontology and epistemology as analogous to questions of validity and reliability in educational measurement. Researchers may well be able to establish that their factor structures or categorizations of questionnaire or interview data are reliable, as the current researchers have done. However, until these scholars can

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demonstrate that the measurements are also psychometrically (i.e., ontologically) valid, results remain suspect. Thus, until those of us committed to the investigation of epistemological beliefs, conceptual change, and learning have dealt more directly and clearly with ontological and epistemological boundaries can we have confidence in the insights and recommendations forwarded. This overarching concern over crossing conceptual boundaries is not restricted solely to the broad epistemological and ontological terrains, but it pertains as well to the nature of knowledge in general, as opposed to the nature of scientific knowledge in particular. Notably, in the Mason and Gava, Duit et al., and Stathopoulou and Vosniadou chapters (and indirectly in the Murphy et al. chapter), researchers are primarily focused on individuals’ knowledge and beliefs within the domain of science. Mason and Gava (this volume) ask students to process refutational texts about natural selection and biological evolution and Duit et al. (this volume) study constructivist science teaching, while participants in Stathopoulou and Vosniadou (this volume) are working within the domain of physics. Consequently, it is important to ascertain whether these scholars are exploring the nature, sources, certainty, or justification of knowledge per se or the nature of science or beliefs about its associated knowledge. As Sinatra and colleagues have argued (Sinatra, Southerland, McConaughy, & Demastes, 2003; Southerland, Sinatra, & Mathews, 2001), there is a fine but critical distinction between views on the nature of science and epistemic beliefs relative to this domain and general beliefs about knowledge and knowing. Specifically, those holding to the domain-specific perspective on epistemological beliefs would contend that individuals may view scientific knowledge differently than knowledge associated with other domains (e.g., history or reading). Moreover, tensions between the physical world, which is at the heart of science, and the nonphysical or metaphysical realms aligned with other domains (e.g., history or religion, respectively) could well drive different world or ontological views that translate into varied judgments about the source or justification of knowledge. What these issues related to conceptual boundaries suggest is that those studying the nature and interplay of epistemology, conceptual change, and learning must be more definitive about the territory in which they are presently moving if subsequent development is to be ensured. For example, how does one talk about the major reconstruction of scientific understandings (Duit et al., this volume), if one holds to the belief that all such knowledge is personally constructed? How does one reconcile personally constructed knowledge with the “scientific body of knowledge” or call for the learning of “facts, laws, and theories,” as Duit et al. have (this volume: p. 199) without sounding oxymoronic. Complexity Clarity and contextualizing are just two of the themes to arise from the chapters by Entwistle, Murphy et al., Mason and Gava, Duit et al., and Stathopoulou and Vosniadou. One of the most paradoxical themes pertains to the fundamental complexity of the core constructs under study. In essence, what these initial explorations have made apparent is that each of the focal constructs under investigation — epistemic beliefs, conceptual change, and learning — is subtle in form, intangible in nature, and empirically complicated. The difficulty experienced in constructing a valid and reliable measure of epistemic beliefs is just one indicator of this problem.

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Indeed, the literature is replete with studies that involve the creation or validation of a questionnaire or interview protocol about epistemic beliefs (or epistemological beliefs or personal epistemology; Buehl & Alexander, 2005; Hofer & Pintrich, 2002; King & Kitchener, 1994; Schommer, 1990; Schraw, Bendixen, & Dunkle, 2002; Wood & Kardash, 2002). In the current volume, Mason and Gava opted to measure students’ general beliefs about knowledge and knowing by using a “reduced and adapted” version of Schommer’s (1990) often-used Epistemological Questionnaire. Duit and colleagues (this volume) relied upon the Constructivist-Oriented Science Classrooms (COSC) instrument to categorize teachers’ instructional behaviors. By comparison, Stathopoulou and Vosniadou (this volume) used a cultural- and domain-specific measure of epistemic beliefs, the Greek Epistemological Beliefs Evaluation Instrument for Physics (GEBEP). What these two studies demonstrate is that variability in the use of domain-general or domain-specific tools exists, and efforts to craft psychometrically sound measures continue. But measurement of epistemic beliefs is only one small piece of the complex puzzle in the literature that considers such beliefs in relation to conceptual change or learning. Here beliefs about knowledge and knowing must be assessed in relation to other equally complex constructs, most notably conceptual change, as well as characteristics of respondents. Even though the issue of measure interdependence is not often considered, it seems feasible to assume that such interdependence warrants reflection. That is to say, there is an understanding that items appearing on the same questionnaire or test do not function completely independently. The content or focus of one item can affect thoughts and responses to subsequent items. Similarly, it is reasonable to assume that the researchers’ decisions to incorporate multiple measurements tools (e.g., epistemology questionnaire, prior knowledge test, or refutational texts) in the same study can have a cumulative effect not evidenced when a single empirical test is used. There is also the continuing question for researchers as to how to disentangle potentially significant learner characteristics within the study of epistemological or epistemic beliefs. Trying to understand the relation between school experiences and students’ beliefs about knowledge and knowing (Entwistle, this volume) or between researchers’ mental models and the way they approach the study of conceptual change (Murphy et al., this volume) has the characteristics of the classic chicken–egg problem. What came first and what affected what? In this fledgling domain of research, we are understandably far from answering that elemental question. Development The unsure footing researchers face on their nascent forays into this conceptual terrain is made more difficult by the fact that the road they travel is itself under construction. That is, the constructs under investigation — epistemic beliefs, conceptual change, knowledge, and learning — are themselves ever developing in learners. Still more challenging, these constructs likely develop in concert with one another, affecting how each develops as the learner matures. As an example, research suggests that students who view knowledge as fixed and unchanging are less likely to change their perspective than those who view knowledge as evolving (Qian & Alvermann, 1995, 2000, Windschitl & Andre, 1998). Developmentally (through both maturation and educational experience) students become increasingly aware of their epistemic beliefs. As students acquire the capabilities to be reflective about their

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knowledge and beliefs, and teachers and activities bring students’ views to the foreground, students are afforded the opportunity to expand or restructure their knowledge. Entwistle describes how Vosniadou (1999) and others have portrayed this interactive developmental process whereby with age and with expertise, [learners acquire] a more flexible system, a system that makes it easier to take different perspectives and different points of view. (Vosniadou, 1999, pp. 11–12) (this volume: p. 125) Similarly, Mason and Gava (this volume) illustrate the effects of students’ emerging metaconceptual awareness. It is this greater developed facility for metaconceptual awareness that affords epistemic belief change, which further bootstraps conceptual knowledge development. Evidence for this effect can be seen in the Mason and Gava chapter. Specifically, these authors found that students who viewed knowledge as constructed, showed greater metacognitive awareness, which likely contributed to their greater degree of conceptual change. The view of conceptual development portrayed in these chapters, particularly Entwistle’s analysis, reflects the complex developmental interplay among these constructs and also echoes the view of knowledge development portrayed in the Model of Domain Learning or MDL (Alexander, 1997, 2005; Alexander, Sperl, Buehl, Fives, & Chiu, 2004). The MDL is a useful framework for understanding how the development of knowledge/expertise interacts with the development of beliefs about knowledge. As one acquires a richer knowledge base, and moves toward expertise in a domain, one becomes able to see the complexity of ideas and understand multiple sides of an issue. This appreciation for the complex and evolving nature of knowledge promotes opportunities for epistemic change (Sinatra, 2005). The MDL also emphasizes that development occurs within a domain of knowledge. As we see in the Duit et al. and Stanthopoulou and Vosniadou chapters (this volume), characteristics of the domain affect how knowledge within that domain develops. For example, according to Stanthopoulou and Vosniadou’s findings, infants come prepared to acquire knowledge about the physical world. Physics knowledge is further developed and refined through experience with everyday objects and events. Thus, beliefs about the nature of knowledge in the domain of physics are likely strongly held and deeply entrenched. Overall, these findings highlight the need for researchers to take steps toward understanding knowledge and belief development within different domains of learning. As we discuss the development of epistemic beliefs, we purposefully avoid the characterization of development as moving from a “naïve” or absolutist view toward a more “sophisticated” view of knowledge as constructed and interpreted. Stanthopoulou and Vosniadou (this volume) take up the issue of what constitutes a sophisticated personal epistemology, preferring instead the nomenclature of constructivist epistemology. Some have argued that epistemic development does proceed toward a view of knowledge as constructed and interpretive (Bendixen & Rule, 2004; Duit et al., this volume). However, evidence is emerging which suggests that such a view of knowledge may be more reflective of a Western or academic orientation toward knowledge and knowing. For instance, Karabenick and Moosa (2005) recently demonstrated that a group of Middle Eastern (Omani) college students were more likely to believe in simple and certain knowledge and

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to regard authorities as the source of knowledge than were their U.S. counterparts. In addition, Weinstock (2005), in a study of ethnic and gender differences in epistemological understanding demonstrated that 7th and 9th grade Bedouin students (particularly boys) were more likely to be absolutist in orientation than Jewish students of the same grade level, suggesting epistemological beliefs are both developmental and culturally embedded. Certainly, it could be argued that the polemics of radio talk shows and 24-hour news broadcasts in the United States reflect a view of complex issues of politics and morality as dualistic and simplistic if not dogmatic and absolutist. Thus, the valuing of complex and nuanced views of knowledge may be unique to those in the academy who may not share the beliefs about knowledge and knowing prevalent within the broader cultural community to which they belong. This observation harkens back to Perry’s (1970) conclusions drawn from the study of Harvard undergraduates. Interrelatedness Yet, continued development relative to learning, knowledge, and conceptual change entails more than simple step-by-step progress in these individual areas. As the first steps in human locomotion is just a prelude to an array of complex physical routines, so too should the independent developments in learning, knowledge, and conceptual change foreshadow coordinated and choreographed study of these particular constructs in conjunction with significant learner and social/contextual variables. That is, as we come to understand how epistemic beliefs interact with other learner characteristics in specific situational and instructional contexts, we can begin to design learning environments and instructional strategies to support conceptual development/change. Entwistle (this volume) explains that we are attempting to describe a very complex system. Like all complex systems, this involves identifying the important components and describing how those components interact and affect the overall functioning of the system. However, as Entwistle points out, there is one important distinguishing aspect of the system we hope to describe — the knowledge construction system. That is, it is an intentional system. This means that while we are trying to understand the interrelatedness of the constructs in the learning system, the learner has the capacity to compound this complexity. Several of the chapters in this section describe this capacity of the learner as engaging in “metalearning” or “deep learning” (Duit et al., this volume; Entwistle, this volume, Stanthopoulou & Vosniadou, this volume). For instance, Entwistle explains that learners are engaging in “metalearning” when they …think about learning in a more sophisticated way [and] recognize a variety of different learning processes, and understand that effective learning depends on finding the most appropriate ways of tackling a particular task for a specific purpose in a given situation, in a given context. This awareness can also lead to a more active engagement with learning as students monitor their process towards understanding. (this volume: p. 130) The awareness, reflectivity, and self-regulated aspects of learning portrayed in these chapters is characteristic of intentional learners (Bereiter & Scardamalia, 1989) who are

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capable of changing their knowledge and beliefs through the successful orchestration of the learning situation. Sinatra and Pintrich (2003, p. 6) described this process as intentional conceptual change or the “goal-directed and conscious initiation and regulation of cognitive, metacognitive, and motivational processes to bring about a change in knowledge.” As described in these chapters, intentional, reflective learners have many choices to make when confronted with a new conception. They have the ability to choose which learning strategies to employ, which knowledge and beliefs to bring to bear, which learning goals to apply, and ultimately which conception to adopt. When multiple options for learning are available to the student, Seigler (1996) has argued that learners make adaptive choices among those options resulting in the more successful strategies prevailing. Metalearning (i.e., deep learning or intentional learning) involves the opportunity to make many choices with important consequences for the learner. Learners may make a controlled intentional analysis of the new conception and chose whether it has adaptive utility for them. This suggests that, ultimately, it is the learner who must manage the interconnections among the components in the complex learning system. Therefore, instructional strategies that afford students the opportunity to make productive choices are more likely to facilitate an orchestration of the components of the learning process that results in knowledge restructuring and long-lasting conceptual change. Particular pedagogical practices and instructional techniques such as refutational texts (see Mason & Gava, this volume), as well as argumentation, discussion, and experimentation, are more likely to give students the opportunity to make choices that support epistemic belief and knowledge change (Alexander, Fives Buehl, & Mulhern, 2002; Fives, Alexander, & Buehl, 2001; Murphy & Alexander, 2004; Sinatra & Kardash, 2004). These pedagogical approaches allow students to see both sides of an issue, weigh issues and arguments, examine their own beliefs and knowledge, and actively engage in knowledge construction and/or restructuring. The choice to explore and manage the interconnectivity of these constructs is not the sole purview of the learner. As Duit et al. (this volume), Entwistle (this volume), and Murphy et al. (this volume) explain, the teacher and the researcher must also examine their views of epistemic beliefs, conceptual change, and learning. These self-reflections are necessary if teachers are to orchestrate the components of the learning process in the classroom to facilitate knowledge development in their students. They are also requisite for researchers — ourselves included — so that we may take interconnectivity into account and design research programs to facilitate the construction of our knowledge base in this domain.

Proposing Next Steps From the outset of this commentary, we have extolled the intellectual insights and scholarly daring of the authors contributing to this section. We have expressed our appreciation for the critical, groundbreaking steps taken by these contributing authors, even as they teetered across relatively uncharted terrain at the juncture of epistemology, conceptual change, and learning. Nonetheless, like proud parents participating in these empirical developments, we want to see even more movement and are willing to provide whatever coaxing, enticement, or support it might take to stimulate increased activity in this area.

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Toward that end, we visit anew our five initial themes: conceptual clarity, construct boundaries, complexity, development, and interrelatedness. We consider several issues that cross these themes and that seem to merit further analysis or investigation. We do not intend to be comprehensive or exhaustive in this proposal. Rather, it is our hope that the possibilities we forward might spark the thinking and actions of others who are similarly invested in the relations between epistemic beliefs, conceptual change, and learning. Introspection and Explication The theme of conceptual clarity in particular suggests that subsequent movement in research on epistemic beliefs, conceptual change, and learning is dependent upon increased lexical and semantic specificity. As researchers frame their investigations, forge their empirical tools, interact with participants, and interpret their data, they must take certain precautionary steps. First, they should reflect on the meanings they are ascribing to their words, including such basic concepts as knowledge, knowing, or beliefs. The chapter by Murphy and colleagues reminds us that we, as educational researchers, may be far quicker to analyze the meanings that others attach to terms like knowledge or beliefs than we are to consider our own epistemological or ontological frames. Such introspection cannot hurt and may well give researchers a new perspective on these fundamental constructs and their study of them. Second, concurrent with the introspection, researchers should incorporate more explicit definitions of underlying constructs and concepts in their writings and presentations. It has been documented in the literature that researchers often fail to provide explicit definitions and elect instead to define constructs implicitly or referentially (Alexander, Schallert, & Hare, 1991; Murphy & Alexander, 2000). That is to say, they may explain a particular construct (e.g., teacher quality or reading) by describing some of its outcomes, salient attributes, or underlying processes (e.g., higher student performance or phonological awareness). Alternatively, authors may refer to key studies or leading researchers associated with the target constructs perhaps under the assumption that those references or names will suffice as grounding for the central constructs. However, we feel that the nascent, multidimensional field represented in these five chapters is better served by the systematic and explicit definition of terms, beginning with words like knowledge, beliefs, concept, conceptual change, or learning upon which this inquiry is framed. We expect that there will be debates and controversies as researchers work to conceptually define these core constructs. But, once the dust settles, we believe that a more solid foundation will have been laid for future decades of theory and research. Integration and Modeling Other themes we identified pertained to conceptual boundaries, complexity, and interrelatedness. Progress in all three of these areas seems predicated on more than semantic specificity and explication, however. It rests on researchers’ ability to conceptualize how those constructs operate collectively, as well as how they relate to other variables or function within varying contexts. One means of concretizing these conceptualizations is through theoretical models that specify the nature or weight of interactions between and

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among variables or that demonstrate how the relations between and among variables change over time or across contexts. Recently, a number of scholars including Bendixen and Rule (2004), Dole and Sinatra (1998), Gill et al. (2004), and Murphy (2005) have attempted to model the process by which beliefs (or knowledge) undergo transformation. Although these models are still in development or under test, they share certain correspondences that can be catalysts for future study. In addition, the intricacy and complexity of the dimensions and processes portrayed in these models are unavoidable. And yet, none of these models do more than capture some aspect of knowledge or belief change. Upon examination, the aforementioned models also suggest that affective concerns may play more significant roles in the changing of beliefs than in knowledge change. Still, these proposed models differ markedly in the specific components they contain or exclude and the paths or causal chains they predict. For instance, across the previously cited models, there is a distinction in the degree to which doubt, uncertainty, or dissonance is considered an essential trigger for belief or knowledge change. In general, we look forward to the maturation of these emergent models and to the focus they bring to central constructs and their interrelations. Instructional Interventions and Developmental Research For all the hypotheses and predictions the current chapters and emergent models contribute to the literature on the nature of knowledge and belief change, there remains a paucity of intervention and longitudinal studies. Consequently, understandings about relations between epistemic beliefs, conceptual change, and learning are still based largely on correlational data. What causes what and what happens over time are questions that such correlational data cannot adequately answer. Without question, therefore, there is a compelling need to expand the empirical base in this field. That expansion should entail both classroom-based studies that explore the effects of direct interventions (such as suggested by the Mason and Gava, Duit et al., and the Stathopoulou and Vosniadou chapters, this volume) and long-term studies that address developmental questions. As discussed, however, these intervention and developmental studies cannot be parochial in perspective, ignoring the possibility of cultural influences on students’ changing beliefs and knowledge. Instead, it is important that the empirical programs of research we are envisioning embrace sociocultural factors and take place in varied cultural settings. On this latter point, the current volume which represents a truly international perspective on conceptual change and epistemology is to be praised. In our opinion, a similar international orientation to classroom and developmental studies of epistemic beliefs, conceptual change, and learning should be aggressively pursued. Measurement Tools and Statistical Procedures Even a cursory examination of the literature in epistemic beliefs, conceptual change, and learning brings serious measurement and statistical concerns to light — concerns that must be confronted if future progress is to be made. For example, so much of what we have come to know about epistemic beliefs rests almost exclusively on self-report data.

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It will take a very creative mind to find alternative means of uncovering individuals’ beliefs about knowledge and knowing that move beyond self-reports or at a minimum corroborate them. Moreover, there has been continuing efforts to construct knowledge and beliefs measures that result in reliable and valid data. This concern is especially true for measures of epistemic beliefs that do not always produce data that match hypothesized factor structures or that may compound ontology issues with epistemological ones (Alexander, 2006). Combined with the search for more effective and informative measures to epistemic beliefs, conceptual change, and learning, researchers must continue their quest for statistical procedures that permit more specific and accurate representation of the interplay between these key constructs. Here again, the concern arises that current statistical procedures simply may not be able to do justice to the complexity or developmental nature of epistemic beliefs, conceptual change, and learning. Therefore, even as we call for more enriched and elaborate models of the change process, we acknowledge that existing models may already over-reach our statistical capabilities (Kulikowich & Young, 2001). As a community, we have tended to rely too heavily on data-reduction procedures that are part of the latent-variable modeling family (e.g., factor analysis or structural equation modeling) that cannot readily or effectively accommodate the extensive number of constructs and relations that must be depicted in our theoretical and conceptual models. As researchers have argued (Alexander, Murphy, & Kulikowich, in press; Kulikowich & Young, 2001), viable statistical alternatives likely exist. However, those alternatives can only serve us if our tasks and interventions undergo dramatic transformations as well. Consequently, future understandings about the relations between epistemic beliefs, conceptual change, and learning will require the direct involvement of those with quantitative and qualitative expertise who can assist in the identification or development of appropriate statistical tools that effectively complement experimental tasks and interventions.

Concluding Thoughts The nature of first steps is such that they represent precarious and rudimentary acts that are simultaneously significant and foundational; so it is with the findings, implications, and recommendations outlined by the authors of these preceding chapters. Entwistle, Murphy and colleagues, Mason and Gava, Duit and colleagues, and Stathopoulou and Vosniadou are scholars who stand in the vanguard of exciting research that seeks to bring the literatures on epistemic beliefs, conceptual change, and learning into alignment. We recognize that future progress in this multidimensional arena may not be certain or steady and that those attempting these movements may occasionally stumble or fall along the way. Nonetheless, we feel that we are bearing witness to a meaningful event in this nascent field of inquiry and look forward to whatever unfolds in the next decade of theory and research. Our underlying hope is that indeed the findings, implications, and recommendations presented in the aforementioned chapters are truly first steps in what promises to be a long and fruitful journey.

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Vosniadou, S. (1999). Conceptual change research: State of the art and future directions. In: W. Schnotz, S. Vosniadou, & M. Carretero (Eds), New perspectives on conceptual change. Oxford, UK: Elsevier. Weinstock, M. (2005). Grade level, gender, and ethnic differences in epistemological understanding ithin domains. In: F. Hearle, & L. Bendixen (Co-chairs), Current research in children’s personal epistemology: Implications for developmental aspects of learning and instruction. Symposium presented at the Meeting of the European Association for Research in Learning and Instruction, Nicosia, Cyprus. Windschitl, M., & Andre, T. (1998). Using computer simulations to enhance conceptual change: The roles of constructivist instruction and student epistemological beliefs. Journal of Research in Science Teaching, 35, 145–160. Wood, P., & Kardash, C. (2002). Critical elements in the design and analysis of studies of epistemology. In: B. K. Hofer, & P. R. Pintrich (Eds), Personal epistemology: The psychology of beliefs about knowledge and knowing (pp. 261–275). Mahwah, NJ: Lawrence Erlbaum Associates.

PART 3: EXTENDING THE CONCEPTUAL CHANGE APPROACH TO MATHEMATICS LEARNING

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Chapter 16

Extending the Conceptual Change Approach to Mathematics Learning: An Introduction Xenia Vamvakoussi This part Extending the Conceptual Change Approach to Mathematics Learning comprises four chapters and two commentaries, which attempt to show the relevance of the conceptual change approach for the learning and teaching of mathematics. In the first chapter, Merenluoto and Palonen present the results of interviews with seven professional mathematicians reflecting on their efforts to understand various aspects of the number concept, from their privileged position as members of the community of mathematicians. The following two chapters by Vamvakoussi and Vosniadou, and Christou, Vosniadou and Vamvakoussi, attempt to show that the conceptual change approach can make meaningful predictions regarding students’ difficulties to understand the mathematical concept of rational number and the use of literal symbols in algebra. In the fourth chapter, Tsamir and Tirosh describe a number of teaching interventions designed to facilitate students’ understanding of certain properties of infinite sets, in light of conceptual change instruction design principles. The commentators look at the chapters from different points of view. Brian Greer and Lieven Verschaffel argue in favor of a conceptual change perspective, both in the context of the historical evolution of mathematics as a discipline and in the context of mathematics education. On the contrary, Anna Sfard attempts to critically examine the assumptions and findings of the first four chapters by reinterpreting them through the lenses of a different theoretical framework, which she describes as communicational. In their attempt to extend the conceptual change approach in the learning and teaching of mathematics all the chapters of this part address, either implicitly or explicitly, certain issues of importance, which we will try to highlight in this short introduction.

Can the Conceptual Change Approach Bring Anything New to Research on Mathematics Education? We believe that there is at least one new and important issue that the conceptual change approach brings to mathematics education and this concerns the role of prior knowledge in mathematics learning and teaching. Reframing the Conceptual Change Approach in Learning and Instruction Copyright © 2007 by Elsevier Ltd. All rights of reproduction in any form reserved. ISBN: 0-08-045355-4

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When it comes to mathematics as a school subject, the importance of prior knowledge in further learning is widely recognized. Indeed, it is reported (e.g., Grossman & Stodolsky, 1995) that mathematics educators consider their subject to be highly sequential, more than science or social sciences educators. This implies that students cannot understand new mathematical content if they have not mastered what has been already taught — which of course seems as a reasonable thing to believe. However, it reflects a limited view of the role of prior knowledge in learning: Prior knowledge is considered to be a base upon which further learning is built and it may constrain learning when it is either inadequate or even absent. This view is compatible with the belief that learning is additive and may prevent mathematics teachers, as well as designers of mathematics curricula, from taking a conceptual restructuring perspective on the processes of mathematics learning. As Resnick (2006) points out, it has taken some time for mathematics education researchers to realize that initial mathematical understandings may not always be supportive of further learning, and that in some cases they may in fact inhibit further learning. She makes an example out of the Cognitively Guided Instruction (CGI), a well-documented approach to mathematics education developed by Carpenter and his colleagues (Carpenter & Moser, 1984; Carpenter, Fennema, & Peterson, 1987), which encouraged and trained mathematics teachers to build instruction on children’s intuitive ideas. Although CGI had strong results for primary-level teachers and their students, its effect was not that strong for older students, a fact that has been attributed to the lack of an adequate body of research evidence related to students’ ideas on more advanced mathematical concepts. According to Resnick, this explanation was based on the implicit assumption that learning of elementary and more advanced concepts would not be fundamentally different — the possibility that knowledge based on intuition would become deeply entrenched and stand in the way of further learning has not been anticipated. On the other hand, it would be unfair to say that mathematics education researchers have not noticed the fact that in some cases prior knowledge can inhibit new learning. Fischbein (1987), for example, was one of the first to notice that intuitive beliefs may be an important contributor to students’ systematic errors in mathematics, a fact also noted by Vergnaud (1989) and Sfard (1987). Brousseau’s theory of epistemological obstacles (1997) shares some similarities with the conceptual change approach. Still, as pointed out by Greer (2004), such theoretical approaches have not found their way into the actual mathematics teaching practices. To summarize, although there have been attempts in mathematics education research to take a wider perspective on the role of prior knowledge in further learning, it seems that both research and practice can profit by a well-articulated constructivist framework that predicts and explains a wide range of difficulties that students may meet and the kinds of misconceptions that should be expected; also, as Greer and Verschaffel (this volume) argue, the particular value of the conceptual change framework lies in the fact that it proposes a number of specific and testable principles (see Vosniadou, Ioannides, Dimitrakopoulou, & Papademetriou, 2001) that can guide instruction and research on instruction. This particular characteristic is in line with what De Corte (2004) describes as a major goal of current and future research on the learning and teaching mathematics.

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Can the Conceptual Change Approach be Applied to Mathematics Learning? The attempt to apply the conceptual change theoretical framework in mathematics learning is relatively recent (Vosniadou & Verschaffel, 2004). The conceptual change framework has its roots in the philosophy and history of science and has been initially applied to explain students’ misconceptions mainly in learning science. The question whether this framework can be applied fruitfully in mathematics learning is a plausible one; it might be the case that patterns of change in the course of mathematics history growth, as well as in mathematics learning, are not similar to science and science learning. As Greer and Verschaffel (this volume) argue, although several distinctions can be made between the characters of the changes that occur in science and in mathematics, there are several changes in mathematics that merit the title “revolutionary”, in the sense that prior “background assumptions” (Baltas, this volume) are discarded or that a significant part of the ‘older’ mathematics will come to be replaced or dramatically augmented by concepts and techniques that visibly change the vocabulary and grammar of mathematics. (Dauben, 1992, p. 80, quoted by Greer & Verschaffel, this volume) This issue is also discussed in the chapters authored by Merenluoto and Palonen, and Vamvakoussi and Vosniadou. Vamvakoussi and Vosniadou (this volume) draw on the argument that, from a historical point of view, the conceptual change perspective is applicable to the development of mathematical concepts rather than theories (see also Vamvakoussi & Vosniadou, 2004). They refer to the number concept to show that in the process of its development the meaning of the term “number”, as well as the ontological status of its referents, has undergone considerable changes. Merenluoto and Palonen (this volume) also refer to the concept of number, to point out that progress in mathematics is far from being linear and smooth. They focus on the controversies and discussions that lie behind the number concept to highlight the social character of mathematical knowledge construction. In educational context, it appears that not only it is expected from students to master under instruction parts of the body of mathematical knowledge that took millennia to develop (Greer, 2006; Greer & Verschaffel, this volume), but also the discussions, controversies, and shared conventions within the mathematics community are concealed (Merenluoto & Palonen, this volume). Tsamir and Tirosh (this volume) argue that it may be profitable for students to be informed of mathematicians’ difficulties with counterintuitive concepts. This is in line with a conceptual change instruction principle, stating that it is important in instruction to distinguish new information that is consistent with prior knowledge from new information that runs contrary to prior knowledge. (Vosniadou et al., 2001, p. 393)

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Tsamir and Tirosh presented their students with two excerpts of statements made by well-known mathematicians who acknowledged the difficulties involved in the comparison of infinite sets and stressed that it is legitimate to wonder about the nature of infinity, in an effective attempt to address students’ entrenched presuppositions about enumeration. On the other hand, learning within the re-framed conceptual change framework does not require the replacement of “incorrect” with “correct”’ conceptions, but the ability on the part of the learner to take different points of view and understand when different conceptions are appropriate depending on the context of use (Vosniadou, this volume). Merenluoto and Palonen’s (this volume) participants, being expert mathematicians, describe nicely how their thinking about numbers differs with respect to the context and how they are able to move flexibly from one context to the other. In educational context, Tsamir and Tirosh (this volume) aim at supporting their students develop this ability in relation to the comparison of sets: They report that helping students to become aware of the different techniques they use to compare finite sets and select the appropriate technique to use in the case of infinite sets proved to be gainful.

How is Mathematics Learning Different from Science Learning? As argued in the introduction of this volume, within the re-framed conceptual change framework, mathematics learning shares many similarities with science learning: As it is the case that students develop a naïve physics on the basis of everyday experience, they also develop a naïve mathematics which appears to be neurologically based (through a long process of evolution), and to consist of certain core principles or presuppositions (such as the presupposition of discreteness in the number concept) that facilitate some kinds of learning but inhibit others (Dehaene, 1998; Gelman, 2000; Lipton & Spelke, 2003). However, the role of empirical facts is not the same in science and mathematics (Tsamir & Tirosh, this volume). There are arguably more opportunities to have real life experiences of gravity and motion, than of irrational numbers. As pointed out by Greer (2004) and Greer and Verschaffel (this volume), most mathematical concepts are introduced in the course of formal instruction. The question arises, is the conceptual change approach applicable only in the case of elementary concepts that can be directly related to experience? An immediate answer can be given through the chapters authored by Christou et al., Tsamir and Tirosh, Vamvakoussi and Vosniadou, which refer to students at secondary school and at university level. They show that initial understandings of natural numbers and enumeration, which are bound to be directly grounded in experience, interfere with the understanding of more advanced mathematical content, such as rational numbers, literal symbols in algebra, and the comparison of infinite sets. However, there are also studies conducted within the conceptual change framework dealing with concepts that are introduced through instruction and are not related — at least not directly — to experience. For instance, Biza and Zachariades (2006) investigated first-year university mathematics students’ understanding of the tangent line on a curve. They showed that students generate synthetic models of the tangent line, reflecting the specific properties of the circle tangent line that they have been initially introduced to in high school.

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Another thing that differentiates mathematics learning from science learning is the abstract nature of its objects of study and the extensive use of symbols. For a mathematically versed person, it may be easy to consider 3, –3.14, , 2/3, and –7i + 6 as objects that can be called numbers (Greer & Verschaffel, this volume), and to treat them as such. But for the mathematics novice, the same task is extremely complicated: One has to disregard superficial differences (like differences in notation) and deeper differences (like differences in use), to conceptualize 2/3 as an object, rather than as a process. In addition, Markovitz and Sowder (1991) observe that a major difficulty faced by learners in mathematics is the use of different symbols to represent the same idea and similar-looking symbols to represent different ideas. For instance, 2/4, 3/6, 0.50 all refer to the number “one half”. On the other hand, one uses the symbol minus (–) to denote an operation (subtraction) and to signify “negativity” (e.g. Vlassis, 2004). To make meaningful predictions related to mathematics learning, the conceptual change framework should account for the difficulties stemming out of these particular characteristics. In this volume, Christou et al. argue that that prior knowledge of numbers, especially natural numbers, in the context of arithmetic, interferes with the interpretation of literal symbols. They present evidence showing that students tend to interpret literal symbols as standing mainly for natural numbers. They also show that students tend to interpet what they call the phenomenal sign of an algebraic expression (such as the negative sign in ‘–x’) on the basis of their knowledge of the actual sign of numbers in the context of arithmetic. Vamvakoussi and Vosniadou (this volume) point out that interpreting rational number notation is a major difficulty for students. They present evidence that students tend to interpret different symbolic representations (fractional, decimal) of a rational number to stand for different numbers, which results in thinking of fractions and decimals as two different, unrelated sets of numbers. They suggest that students’ initial explanatory frameworks of number are usually simply enriched with “new kinds of numbers”, i.e. fractions and decimals. They appeal to learning by mere enrichment to explain students’ generation of synthetic models of rational numbers intervals. Such findings and interpretations are consistent with the conceptual change approach. Of course, there is much to be done to explore further the advantages and limitations of this approach to mathematics learning.

What can We Gain from a Conceptual Change Approach to Mathematics Learning and Teaching? Tirosh and Tsamir (2004) argue that mathematics education and conceptual change research both can gain from each other. In this volume, Tsamir and Tirosh systematically evaluate certain principles for instruction aiming at conceptual change through reflecting on the design of successful learning environments. They refer to a number of their studies aiming at teaching effectively the comparison of infinite sets. It appears that the choices they made, based on their rich experience and their sensitivity about the role of intuitive beliefs in learning, are highly compatible with the conceptual change instruction design principles. One could expect that it would be helpful for mathematics educators to be aware of these principles when starting to design a mathematics learning environment.

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In addition, Vosniadou and Vamvakoussi (2006) argue that the conceptual change perspective could be helpful in the design of mathematics curricula. In Greer’s (2006) words, it is essential to take a long-term perspective in the design of mathematics instruction, which involves careful sampling from the valid range of examples, the anticipation of expansions of meaning, the search for bridging devices when such expansions are necessary, and the open discussion with students of why conceptual change is necessary, and often difficult, which takes them deep into historical and philosophical analyses. (Greer, 2006, p. 178)

How is the Conceptual Change Approach Related to Other Approaches to Learning and Instruction? The application of the conceptual change approach in mathematics may also reveal the limitations of this framework — after all, no theory can possibly explain all phenomena of learning, as Sfard (this volume) suggests. This is why the claim made in the introduction, namely that the conceptual change approach is complementary and not incompatible with other approaches, such as the situated cognition approach (Vosniadou, in press; Vosniadou & Vamvakousi, 2006), has to be further substantiated. This claim has both theoretical and educational implications. In the case of mathematics learning, it is hard to deny that learning how to count has a social and communicational character. Before children construct a principled understanding of counting (Gelman, 2000), they must make several attempts to assign to the counting ritual the same meaning as their parents do. It takes a lot of effort and it is presumably an effort initially invested on communication (Sfard & Lavie, 2005). However, there are differences between the conceptual change approach and the communicational framework briefly presented by Sfard (this volume) with respect to their theoretical assumptions. In particular, within the communicational framework, learning is viewed as change in discourse (see, for example, Sfard & Lavie, 2005), whereas from a conceptual change perspective, learning is accomplished through participation in social activities and exhibited in changes in discourse, but also involves changes in assumed mental structures. Not committing oneself to account for mental structures can probably result in frameworks in which less theoretical constructs need to be developed, thus avoiding the use of inadequately defined terms described by Sfard (this volume). However, we believe that theoretical frameworks that take into consideration the cognitive component of learning have greater explanatory power (for a thorough discussion see Vosniadou, in press). Within the conceptual change framework, the importance of communication is fully recognized, especially in relation to education. Vosniadou et al. (2001) propose that learning environments that facilitate group discussion and the verbal expression of ideas make possible for students to express their internal representations of phenomena and compare them with those of the others. In a more general fashion, learning environments of this kind help students increase their metaconceptual awareness of their thinking, which is an important component of intentional learning (Vosniadou, in press, 2003). In support of this view, Tsamir and Tirosh (this volume) claim that the implementation of this principle had

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positive effect on their students’ learning about the comparison of infinite sets. However, further research is needed to examine the educational implications of the conceptual change approach in mathematics learning and teaching.

Acknowledgments The present work was financially supported through the program EPEAEK II in the framework of the project “Pythagoras — Support of University Research Groups” with 75% from European Social Funds and 25% from National Funds.

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Resnick, L. B. (2006). The dilemma of mathematical intuition in learning. In: J. Novotná, H. Moraová, M. Krátká, & N .Stehlíková (Eds), Proceedings of the 30th conference of the international group for the psychology of mathematics education (Vol. 1, pp. 173–175). Prague: PME. Sfard, A. (1987). Mathematical practices, anomalies and classroom communication problems. In: P. Ernest (Ed.), Constructing mathematical lnowledge: Epistemology and mathematics education (Studies in Mathematics Education Series, Vol. 4). London: The Falmer press. Sfard, A., & Lavie, I. (2005). Why cannot children see as the same what grown ups cannot see as different? Early numerical thinking revisited. Cognition and Instruction, 23(2), 237–309. Tirosh, D., & Tsamir, P. (2004). What can mathematics education gain from the conceptual change approach? And what can the conceptual change approach gain from its application to mathematics education? Learning and Instruction, 14, 535–540. Vamvakoussi, X., & Vosniadou, S. (2004). Understanding the structure of the set of rational numbers: A conceptual change approach. Learning and Instruction, 14, 453–467. Vergnaud, G. (1989). L’obstacle des nombres négatifs et l’introduction à l’algèbre. In: N. Bednarz, & C. Garnier (Eds), Construction des Savoirs (pp. 76–83). Ottawa: Agence d’ARC. Verschaffel, L., & Vosniadou, S. (Guest Eds.). (2004). The conceptual change approach to mathematics learning and teaching. Learning and Instruction (Special Issue), 14, 445–548. Vlassis, J. (2004). Making sense of the minus sign or becoming flexible in ‘negativity’. Learning and Instruction, 14, 469–484. Vosniadou, S. (2003). Exploring the relationships between conceptual change and intentional learning. In: G. M. Sinatra, & P. R. Pintrich (Eds), Intentional conceptual change (pp. 377–406). Mahwah, NJ: Lawrence Erlbaum Associates. Vosniadou, S. (in press). The cognitive-situative divide and the problem of conceptual change. Special Issue of Educational Psychologist. Vosniadou, S., Ioannides, C., Dimitrakopoulou, A., & Papademetriou, E. (2001). Designing learning environments to promote conceptual change in science. Learning and Instruction, 11, 381–419. Vosniadou, S., & Vamvakoussi, X. (2006). Examining mathematics learning from a conceptual change point of view: Implications for the design of learning environments. In: L. Verschaffel, F. Dochy, M. Boekaerts, & S. Vosniadou, (Eds), Instructional psychology: Past, present and future trends. Sixteen essays in honour of Erik De Corte (pp. 55–70). Oxford: Elsevier press.

Chapter 17

When We Clashed with the Real Numbers: Complexity of Conceptual Change in Number Concept Kaarina Merenluoto and Tuire Palonen Introduction Empirical research on conceptual change in the learning of science has been extensive over the last two decades (e.g. Carey, 1985; Carey & Spelke, 1994; Chi, Slotta, & de Leeuw, 1994; Hatano & Inagaki, 1998; Ioannides & Vosniadou, 2002; KarmiloffSmith, 1995; Posner, Strike, Hewson, & Gertzog, 1982; Vosniadou, 1994). The main findings have been that, in situations where learners’ prior knowledge is incompatible with the scientific concepts accepted and defined by the communities of scientists, students are prone to make systematic errors or have misconceptions. Many of these misconceptions are synthetic models (cf. Vosniadou, 1994; 1999), indicating that students are assimilating new information into prior knowledge instead of making radical changes. The results of several teaching experiments to facilitate conceptual change indicate that radical conceptual change is an extremely difficult and complicated process, and that prior conceptions seem to be quite resistant to teaching efforts (e.g. Duit, Roth, Komorek, & Wilbers, 2001; Mikkilä-Erdmann, 2002; Trumper, 1997; Wiser & Amin, 2001). In addition is the social and cultural nature of the learning environments and communities in this process (Kelly & Green, 1998; Posner et al., 1982; Strike & Posner, 1982). Although there has been a great deal of research on conceptual change in the learning of science during recent decades, there is not much research from this perspective in mathematics learning. This is because, in traditional contexts, mathematics is considered as a hierarchical body of knowledge, where new discoveries are not treated as changes but as enlargements (Boyer, 1959), and in which the previous ones are included as substructures. Hence, in mathematics culture, changes are not treated as revolutions as they are in the development of science concepts (cf. Kuhn, 1970). Recent empirical studies in mathematics learning from a conceptual change perspective (e.g. Lehtinen,

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Merenluoto, & Kasanen, 1997; Merenluoto & Lehtinen, 2002; Stafylidou & Vosniadou, 2004; Vamvakoussi & Vosniadou, 2004) indicate that the extensions in the number concept actually require a radical revision of prior knowledge structures: a conceptual change. The natural numbers and their discrete nature have a high intuitive acceptance attached to them as being self-evident, self-justifiable or self-explanatory. Therefore, students in the school context seem to have a tendency towards overconfidence (Fischbein, 1987), and towards relying on the logic of natural numbers in their reasoning. This leads to systematic and resistant misconceptions in problems where rational or real numbers are used (e.g. De Corte & Verschaffel, 1996). Even at the higher levels of education, the majority of Finnish students was not able to explain how they think about numbers, and was unaware of the fundamental conceptual differences between the number domains (Merenluoto & Lehtinen, 2002, 2004). In order to better understand this difficulty in the school context, we analyze retrospective interviews in which mathematicians explain their history in learning real numbers as well as their informal, metaphorical descriptions of their thinking about numbers. We also examine how the shared practices in the mathematics community and the mathematicians’ involvement in the mathematics culture interact with explanations of individual learning and thinking about real numbers. By the term school context, we mean the class community, which does not necessarily provide the respective conceptual tools with which to share mathematical understanding. By the term mathematics culture, we mean the knowledge, a practice of mathematicians, which is mediated by the historical development of mathematics, as it has been co-constructed over two thousand years. There, the conceptual tools (like common language, acceptance of conventions, awareness of inconsistencies and ways to deal with them) are developed through discussions, scientific debate and argumentations. By the term mathematics community, we mean the society or colloquium in which these jointly developed conceptual tools are used by practicing mathematicians. In the following pages, we first provide a brief historical background regarding the development of the concept of real numbers and then explain some of the foundational problems related to these definitions.

Historical Background of Real Numbers The notion of real numbers is one of the most complex and profound concepts in mathematics, and it has a long and complicated history of development. Behind this long development period lays the dichotomy between discrete and continuous quantities that was faced already in Antiquity. The so-called exhaustion method of Archimedes dealt with some aspects of this dichotomy. This method was a primitive limiting process to deal with infinite small quantities, which stated: If from the greater there be subtracted a magnitude greater than its half, and from that which is left a magnitude greater than its half, and this process be repeated continually, there will be left some magnitude which will be less than the lesser magnitude set out. (Boyer, 1959, p. 33)

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The concept of limit was actually embedded in this process, but because of the problems with infinitesimals — infinitely small quantities — the efforts to use numbers in order to handle continuity led to paradoxes that disturbed mathematicians for two thousand years (Boyer, 1959; Dantzig, 1954; Kline, 1980). Finally, in the latter part of the 19th century, the continuous and discrete quantities were integrated into the abstract theory of real numbers. Mathematicians have different kinds of rigorous constructs for these numbers, the most familiar ones being those where the real numbers are constructed as limits of Cauchy’s sequence1 or as Dedekind’s cuts2 in the number line. From the point of view of conceptual change, it is important to notice that the long development period of the theory of real numbers indicates the radical nature of the change, which was required when the initial thinking of numbers as discrete objects was integrated with the knowledge belonging ontologically to another domain of continuous objects (Boyer, 1959; Kline, 1980; Rudin, 1953). This integration also required a radical change in the discussion and communication in the socially shared practices of mathematicians. After this change, mathematical reasoning on continuity was no longer based on intuitions of physical reality, but on logically consistent arguments. In addition, during the long period of development, which preceded the formation of this rigorous theory, mathematicians like Newton tried to justify the concept of limit with the existence of continuity, for example, using the “ultimate ratios in which the quantities vanish” (Boyer, 1959, p. 198). Finally, during the 19th century, in the modern theory of real numbers, the concept of continuity is based on the existence of limit, that is, on the abstract density of numbers. The hierarchical system of extensions in the number concept, constructed after the formation of the theory of real numbers at the end of the 19th century, is typical of mathematics. This hierarchical system is a product of successive abstractions, in which the numbers from the previous domain are used to build the more advanced set of numbers. Then the more advanced domain creates a new frame of reference, from which mapping to the previous domain is formed (see Landau, 1960). The numbers in the previous domain are then treated as a subset of the more advanced ones (Figure 17.1). Thus, for example, the natural numbers are treated as a subset of rational numbers, and rational numbers as a subset of real numbers. In Figure 17.1, the set of real numbers is, for clarification, presented as a set model. However, for the mathematicians it is more common to use the number line as a representation of the infinite set of real numbers (the continuum). According to the continuum hypothesis, only real numbers, including the transcendental numbers (like  and e), form the continuum of numbers. This hypothesis, proposed by Cantor, is based on cardinalities of infinite sets, and it claims that any infinite set of real numbers contains either exactly as many elements as there are natural numbers or as many as there are real numbers. Thus, it states that there is no other transfinite number between the cardinality of natural and the

1 A number l is called the limit of an infinite sequence u1, u2, u3,... if for any positive number  ⬎ 0 we can find a positive number N depending on  such that 冷un ⫺ l冷 ⬍  for all integers n ⬎ N. 2 Dedekind’s “cut” means a cut in the number line; thus, for example, when a cut is defined by the number 兹2苶, it represents all the numbers that are smaller than 兹苶2 on the number line.

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Real numbers

Rational numbes Integers Natural numbers

no smallest”

numbers like π and

2

”

0, 1, 2,… ”

no biggest”

numbers like 1 , 37 , 4 43

....-3, -2, -1,

no next”

”

”

continuum of real numbers”

Figure 17.1: A simplified model of the extensions of number domains from natural numbers to real numbers and the changes in the level of abstraction with each enlargement.

cardinality of real numbers. This hypothesis, however, led to serious logical problems among mathematicians. Finally in 1930, Gödel proved that there will always be statements within an axiomatic system which can be neither proved nor disproved on the basis of these axioms. Later, in 1953, Cohen showed the independence of the continuum hypothesis from the other axioms of set theory. This implies that it is impossible to prove that the continuum hypothesis is either true or false from the usual axioms of set theory. In the mathematics culture, these cognitive conflicts were finally handled by mutual agreements and shared discussions which opened up the possibility of choice between different set theories, and a way to avoid disturbing conflicts. These socially negotiated contracts, rules and examples, which have been constructed through discussions among mathematicians, are behind the modern concepts of limit and continuity also used in the school context. According to several studies, these concepts are difficult also for students today (e.g. Cornu, 1991; Fishbein, Jehiam, & Cohen, 1995; Merenluoto & Lehtinen, 2002; Sierpinska, 1987; Szydlik, 2000; Williams, 1991). We argue that in the development of students from novices to experts (Patel & Groen, 1991) there is a connection between the individual knowledge acquisition processes and participation in the culture of expertise (Sfard, 1998). The conceptual tools developed, like real numbers, are jointly formulated during the history of the development of the concept. Through these kinds of processes, the experiences and practices of a community are transformed into external and communicable items. In the literature, this has been called reification. Reification means treating an abstraction or thought as if it were a concrete and real object (cf. Sfard, 1991, 1994). Reifications represent shared narratives, forms, symbols, common language, acceptance of conventions, awareness of inconsistencies and ways to deal with them, all of which support the members’ activity and interaction. The concept of internalization refers, in turn, to the social cultural approach according to which the interaction

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between different community members fosters the conceptual growth of the newcomers (Hakkarainen, Palonen, Paavola, & Lehtinen, 2004; Galperin, 1966; Vygotsky, 1978).

Problems in Bridging Individual and Community-Level Changes In the school context, students do not have access to the shared narratives of the mathematics culture. The historical conflicts concerning real numbers are usually not taught explicitly; in fact, it is possible that they are not even mentioned to the students who are struggling to understand these concepts. It is important to notice that at the upper secondary level, the teaching of limit and continuity, the core concepts of real numbers, usually occurs in a very “novel” domain for the students: they have not discussed concepts, such as continuity, in mathematics before. What they have experienced about continuity and limits comes from their everyday experience, where the limit is considered as something that restricts or stops the continuity. Accordingly, their thinking on continuity is mediated by metaphors in which continuity is related to the continuity of time, motion or direction. In fact, the general idea of the continuity of motion is one of the basic features of the physical world, and even infants grasp it at some intuitive level (e.g. Spelke, 1991). This idea is vague, dynamically and instinctively understood in the context of motion or time, but not with numbers (Dantzig, 1954; Cornu, 1991). Further, everyday thinking of continuity is even strengthened in the school practice of teaching where the continuity of function is often described as the result of a continuous motion: “the pencil never leaves the paper”. Thus, students try to use their prior knowledge and metaphors from their everyday experiences to interpret these mathematical concepts. In our previous survey (Merenluoto & Lehtinen, 2002), less than 5% of the 538 upper secondary level students showed any indications of a more radical conceptual change in their explanations of limit and continuity. The majority of students described continuity as something that “does not stop” or “what you can draw without lifting your pen” or is in “one piece”. The limit in students’ explanations represented something that stops the continuity: “it is the value, where you can not go further, a stop-sign” or they described it as approaching something: “…It approaches a value”, “the graphs approach the point but do not hit it”, “the value the function approaches but never reaches” (see also Cornu, 1991). The answers of students suggested that they did not recognize the crucial underlying similarity between the density of numbers on the number line and the continuity of a function based on the concept of limit. In the present study, our aim is to show that “doing” mathematics in the school context is very different from “doing” mathematics within the mathematics community. One of the important differences between students and mathematicians is that the latter are able to meaningfully participate in the mathematics community. For this purpose, we analyze retrospective interviews in which mathematicians narrated their learning history of real numbers and provided informal, metaphorical descriptions of their thinking about numbers. We also examined how the shared practices in the community of mathematicians and their involvement in the culture of mathematics interact with their explanations of individual learning and thinking of real numbers.

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Method Participants Ten Finnish university mathematicians, chosen at random, were contacted by phone, and seven of them agreed to participate in the study. In Finland, most of the mathematicians in universities are male, and in this group there were six men and one woman. All participants were working at the same university, except for the female participant. Their expertise in mathematics was in the domains of functional analysis, analytic number theory, algebraic number theory, applied mathematics, coding theory and automata theory. Five of them had recently been teaching courses on analysis, where the concept of real numbers was taught. We refer to the participants using the arbitrary names: Jim, Mathias, Jonas, Eric, Carl, Thomas and Lea, in order to keep the interviewees’ identity in the strictest confidence. Eric, Thomas, Carl and Jim were professors in mathematics while Jonas, Mathias and Lea were lecturers. Procedure The interviews were conducted by the authors of this chapter. The conditions in which the individual interviews were held were informal and relaxed; the interview took place in the interviewees’ offices at the university. In these semistructured interviews, the mathematicians were asked to explain their personal learning history of real numbers, their informal thinking of numbers and the basis of their certainty in using real numbers. Each interview lasted about 1–1.5 hours, and was tape-recorded and transcribed. Analysis of Data The analysis of the transcriptions was based only on what the participants spontaneously said during these interviews without making generalizations about their thinking. Two main criteria were used in the qualitative content analysis: (1) how the mathematicians elaborated their individual meta-level thinking about real numbers (their acquisition of knowledge, conceptual change), and (2) how they spontaneously described their process of initiation into the mathematics community in relation to their understanding of real numbers (their introductory experience, their references to the historical development, the mathematics culture, their descriptions of their individual certainty in relation to the logical problems in the history of real numbers). In addition, a frequency analysis was used to measure the intensity of these expressions and elaborations, as follows: We calculated (1) how many times the interviewee spontaneously referred to the above-mentioned two main criteria during the interview, and (2) how many words they used in these spontaneous expressions at a time. Thus, for example, during the interview, Carl spontaneously explained his informal thinking of numbers 15 times during the interview using altogether 303 words in these expressions (Table 17.1).

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Table 17.1: Intensity of Interviewees’ Expressions and Elaborations. Frequency of Spontaneous Expressions of Elaboration Number History Eric Thomas Carl Jonas Mathias Jim Lea

18 16 15 13 8 9 5

Count of Words in the Expressions of Elaboration Altogether

References to

Number History

References to

9 8 5 2 1 3 0

424 507 303 265 405 192 177

490 218 84 65 13 107 0

In addition to the detailed frequency analysis, two independent researchers analyzed the data and arranged the interviewees in order (from one to seven) according to the abovementioned frequency criteria. Cohen’s Kappa was calculated and had a value of .619, while the Spearman’s rank correlations were .857 for the individual elaboration of thinking and .955 for references to mathematics culture.

Results and Discussion We begin by presenting some of the interviewees’ typical learning experiences, and then we proceed to describe how they elaborated their initiation into the mathematics community in relation to their understanding of real numbers, and building their own personal certainty in relation to the logical problems in the culture of mathematics. We continue by describing how the mathematicians described everyday thinking of numbers and their informal and metaphorical thinking of real numbers. Finally, we present the relation between the spontaneous references to the culture of mathematics, and the elaboration of informal thinking of real numbers suggested by the data. Memories from Typical Learning Experiences While all the interviewed mathematicians clearly liked mathematics, their spontaneous references to interest in and enjoyment of mathematics varied. Lea simply mentioned that during her school years she understood mathematics and it was “fun”. Jim, on the other hand, showed his enthusiasm for his experiences and made several references to the beauty of mathematics. He and Eric both mentioned that mathematics is like music and if taught like music, it brings joy to large groups of people.

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When explaining their learning history, the mathematicians explained how during their school years, the density of numbers was easily understood intuitively by thinking of the density of numbers on the number line. Eric and Thomas spontaneously explained their deliberate strategies to learn calculus. Eric read university-level mathematical books already in his early teens. Besides thinking formally, he also tried to find informal ways to explain things to himself. Thomas, in turn, made a visit to the USA as a teenager, where the concepts of calculus were taught in such a way that he found the derivative and the use of the rigorous ␦– definition3 — fascinating. This discovery paved the way for him “to understand the mathematical truths”. When looking back on their school years, these mathematicians did not mention any particular difficulties in learning the numbers during their school years; instead they excitedly referred to their fondness for mathematics. Initiation into the Mathematics Culture: A Clash with the Real Numbers There are two traditional ways to introduce real numbers in university mathematics: the axiomatic approach and the construction approach. In the axiomatic approach, the axioms of real numbers are presented to the students and all the following results with real numbers are then deducted from these axioms. According to the construction approach, the real numbers are constructed from rational numbers in vertical abstractions and, as a result, the real numbers are defined as limits of Cauchy’s sequences or as Dedekind’s cuts. Thomas spontaneously mentioned at five separate times during the interview that knowing the rules of proof gave him a pleasant feeling of confidence. When explaining his encounter with real numbers at university, he told how the rigorous axiomatic approach gave him a feeling of comfort. However, in spite of that, he reported how he clashed with the real numbers. Carl described this experience as follows: Quite a knock-out it was at the beginning [—], for several weeks I did not understand a thing. But for him “this difficulty required a few weeks of thinking” and then came the “flash of understanding” after which it was easy for him. Lea told how she understood notions like the “supremum”,4 but in the beginning she did not understand their relation to real numbers. For Mathias, the construction of real numbers from rational numbers was still mysterious: The system of extension; it was such a collision then [—] it was difficult [— ], definitely difficult [—], when it was explained [—] I did not understand it [—] I was so puzzled.. 3

The formal procedure used in dealing with the limit concept. There, for example, the number l is the limit of f(x) as x approaches x0 if for any positive number  (however small) we can find some positive number  such that 冷f(x) ⫺ l冷 ⬍  whenever 0⬍ 冷x ⫺ x0冷 ⬍ . 4 “Supremum” means the “least upper bound” that is the very thing that differentiates the real numbers from the rational ones; then, for example, 兹苶2 being an irrational number is the “least upper bound” for all the numbers smaller than 兹苶2. Thus, in the domain of rational numbers, there is no “least upper bound” for respective numbers.

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In fact, in his recall of past memories, he mentioned the name “Dedekind” five times, when speaking of something that was really complicated and difficult: … the professor explained Dedekind’s cut; and the whole autumn semester I was wondering, what on earth is this about [—], I did not understand anything there, maybe somebody did, but I did not. It was only later that the understanding came. Jim and Jonas, who did not use words like “clash” or “collision”, did, however, refer to a conceptual leap in their learning of real numbers. Jonas explained that he was very excited about the axiomatic approach: “I did not know that it was possible to do it that way!”. But then he added: “But it was so difficult, at first I dif not understand”. Jim explained how he taught the construction of real numbers to new students: “I really constructed them from rational numbers” but then he admitted that, pedagogically, it was not a very good choice because it was too difficult. Thus, the memories from learning mathematics during the school years were pleasant ones for the interviewed mathematicians. It was not until they faced the abstract construction of real numbers during the early stages of their initiation into the mathematics community, that they experienced an essential change in their “doing” of mathematics. However, because of their great interest in mathematics, this change, “a clash”, led to a deeper level of understanding of real numbers. Formal and Informal Thinking of Real Numbers Given their wide content knowledge, the mathematicians were able to see the whole picture of numbers. Therefore, the concept of real numbers for them seemed to represent the whole construction of the theory of real numbers, such as abstract constructions, dynamics of delta–epsilon processes, Cauchy’s sequences and the logical problems with the continuum hypothesis. Carl explained that speaking about numbers and counting, there is no point of bothering your brains so that all the real numbers are there: in mathematics we strive cognitive economy. Lea explained how if some concept has several features, you are aware of them the whole time, but if in some situation, they are not needed, it is pointless to carry them around with you all the time. Matthias vividly explained his thinking of the construction of real numbers as being still mysterious. He explained that for him real numbers were decimal numbers to begin with, and only when he is unable to prove something, then “I need to think more deeply about the constructions”. But in everyday thinking “I do not need the system”.

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Thus, they made a clear difference between their formal and informal thinking of numbers. All of them referred to their understanding of the formal representations of real numbers, but most of them explained that “nobody really thinks that way”. In fact, the thought of using real numbers in everyday thinking was even slightly amusing for them. The majority of mathematicians spontaneously and emphatically claimed that they do not use real numbers in their everyday dealing with numbers. For them, all everyday numbers are rational numbers, and the real numbers are only for theoretical purposes. Everyday numbers are [—] always approximations of real numbers, [—] the real numbers are theoretical objects only for the use of mathematicians. (Eric) When I think of buying milk… I do not think about the price on the number line (laughing). (Thomas) … I do not need irrational numbers when chopping wood at my summer house (laughing). (Jonas) When explaining their informal, meta-level thinking of real numbers, the mathematicians used efficient metaphors, based on the underlying general principles in the abstraction of density. Jim and Jonas referred to the geometrical abstraction of real numbers as the density of numbers on the number line. Thomas, who had recently been teaching a course on so-called hyper-real numbers, described his metaphor for these numbers “as softened atoms padded with little infinitesimals” (referring to the nonstandard analysis). Eric’s description was the most elaborate: I think of the real numbers in a naïve way from a topological viewpoint using the concept of neighbourhood, where it is not so much the question about the point itself but its neighbourhood– so that every real number defines a systemic neighbourhood where it can be infinitesimally small and every number has its own neighbourhood. I like this kind of thinking more than trying to think about a point.5 You cannot imagine one point, whereas it is possible to think about its neighbourhoods. Hence one real number is a systemic neighbourhood. (Eric) The mathematicians expressed multiple informal representations of these numbers, and were able to flexibly shift from one representation to the other. In addition, they were metaconceptually aware of their efficient metaphors in thinking of these numbers. They were also very aware of the difference between formal and informal representations of these numbers, and able to flexibly shift between them. Constructing Independent Personal Certainty in Relation to the Logical Problems in the Culture of Mathematics During the interviews, the mathematicians were asked to explain their personal certainty related to the concept of real numbers. Most of the interviewees then described it in 5

By a point he means the mathematical point, which has no dimensions.

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relation to the logical problems in the mathematics culture. Four of the interviewed mathematicians clearly described their certainty about real numbers on two different levels. On the one hand, they were aware of the logical problems and puzzlement in the historical background, and of the agreements made to deal with these uncertainties within the community of mathematics. For example, Thomas explained:

…But of course here there are all kinds of logical problems and when you are aware of them then you know that even thinking about the number line is not that self-evident. There are all kinds of questions that can be found like this hypothesis of continuum. And referring to the agreements to deal with this problem, he added, “you just have to make a choice of what kind of set theory you want to select”. On the other hand, they all made a very distinct difference regarding their personal certainty. For example, Carl explained that: There are two levels to this certainty: when I think of numbers the way I use them in everyday life, or as a mathematician, there is no kind of uncertainty. But if you go into the deep levels of logic, there were great revolutions, when all these modern paradoxes in mathematics were found. But then he continued by explaining his individual certainty in relation to the problems in the culture of mathematics: But they have no influence on the normal mathematics, although there are properties which are very surprising, but I do not let those paradoxes bother me or disturb my good night’s sleep. Correspondingly, Jonas explained that “this logical part of the theory of real numbers, the theory is so powerful that I can not be absolutely sure if there are logical inconsistencies in it… but I have a very high subjective confidence in it. It is a true theory. Thus, the mathematicians spontaneously referred to the mathematics culture, especially when explaining their personal certainty of real numbers in relation to the logical problems. The mathematicians who made most of the spontaneous references also used more words in elaborating these explanations (Table 17.1). According to a frequency analysis we placed interviewed mathematicians in a twodimensional field (Figure 17.2). One dimension (the vertical axis) represents the number of spontaneous references to the culture of mathematics and the other (the horizontal axis) the frequencies of their elaboration of informal thinking of numbers. There, Lea and Mathias only referred to their personal (un)certainty in dealing with real numbers (Figure 17.2). Lea also admitted that she had never intentionally thought about the difference between rational and real numbers, because “they are just numbers”.

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Figure 17.2: Interviewed mathematicians in the two-dimensional field of explanations of their individual thinking and their involvement in the community of mathematics. Lea and Mathias, who gave rather scanty explanations of their thinking of numbers and still showed some uncertainties in their own explanations, did not spontaneously refer to the culture of mathematics during the interview, whereas the others did. What is also worth noting is that Mathias, Jonas and Jim explained their thinking of real numbers through the geometric representation of real numbers as points on the number line. Eric and Thomas were the only ones who also spontaneously referred to their epistemology of mathematics as a living and changing system of knowledge, which is not an empirical science. Students may think of mathematics as a complete structure where nothing can be added [—] but in fact, it is a continuously living and changing system. Although you may think that today’s mathematics is ready and complete, after a while you find that again something new has been created. (Eric) Mathematics is not an empirical science… yes, its conventions make sense, but they are, however, only conventions and logical selections of what we need to do… like, which of the set theories is to be chosen if we accept the continuum hypothesis. (Thomas) When the mathematicians explained their processes of entering into the community of mathematicians almost all of them used words like “clash” and “knock-out” in their descriptions of their initiation into the construction of real numbers. Thus, most of them remembered how learning to understand the notion of real numbers, especially the construction of those numbers, was something special and required them to adopt a new kind

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of abstract thinking. In their individual construction processes, most of them also clearly defined their individual certainty about these numbers in relation to the logical problems in the culture of mathematics, of which they were aware. There were clear differences in the frequencies of spontaneous references to the mathematics culture between the interviewed mathematicians (Figure 17.2). Thomas, Eric, Carl and Jim, who made most of the comments, were all publishing professors in mathematics with experience of international discussions within mathematics community. Compared to them, Lea, Matthias and Jonas made clearly fewer references to the mathematics culture. The fact that the latter were lecturers in mathematics suggests that prospective mathematicians gradually gain access to the culture of mathematics.

Summary and Conclusions In this chapter we have analyzed retrospective interviews in which mathematicians narrated their learning history of real numbers and elaborated on their informal, metaphorical thinking of numbers. Our main argument is that prospective mathematicians’ initiation into the mathematics community involves initiation into the mathematics culture, whereas students in the school context are introduced only to pieces of content knowledge. The results suggest major differences between mathematicians (the experts in the present study) and students — the novices in our previous study (Merenluoto & Lehtinen, 2002) — a difference that is of course self-evident. However, there are some important differences that need to be pointed out in order to understand the problems students have in trying to understand these difficult concepts. In the first instance, the experts’ content knowledge is considerably broader than novices’, which allows them to see the whole picture, to entertain multiple representations, and to shift flexibly from one to the other. The mathematicians’ informal descriptions of numbers, their spontaneous references to the number line and efficient use of these metaphors for density of numbers and for neighborhoods suggest a flexible strategy with active use of the number line, and conscious attention paid to the underlying fundamental principles of real numbers explained in the theories of expertise (e.g. Chi, Glaser, & Farr, 1988; Chi, Feltovich, & Glaser, 1981). What is also related to advanced mathematical thinking was their flexible changes from one representation to another (e.g. Dreyfus, 1991), and the fact that they paid primary attention to the relations between mathematical objects (e.g. Hadamard, 1945; Liljedahl, 2004); all these indicated that they had constructed a systemic environment of numbers: a reification (cf. Rudin, 1953; Landau, 1960). The experts were also metaconceptually aware of their thinking and capable of making a distinction between their everyday and formal thinking of real numbers. Most of them spontaneously stated their relation to the mathematics culture. On the other hand, the novices in the school context gave no indications of this kind of distinction. In fact, the students did not understand even the basic differences between the different number domains. It is especially important to pay attention to the difference between students’ explanations of limit and continuous objects presented in our previous research (Merenluoto & Lehtinen, 2002, 2004), and the explanations given by the mathematicians (Figure 17.3).

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Figure 17.3: A tentative “graph” of the radical conceptual changes in the extensions of the number concept and “areas” of students’ and mathematicians’ explanations. The metaphor is one of the basic mechanisms behind conceptualization (Sfard, 1998), and different metaphors may lead to different ways of thinking. When examining the differences between these metaphors, it is possible to understand why the conceptual change from discrete numbers to the density of numbers does not usually or easily occur during school mathematics learning. In the teaching practices in the school context, the problems inherent in the extensions of its number concept are often implicitly passed over. It is also possible that many teachers do not understand these concepts very well either. In our endeavors to facilitate conceptual change in the number concept, we need to give students some access of the shared practices in the mathematics community. We could, for example, explain the efficient metaphors the mathematicians use in explaining their thinking of real numbers, thus paying attention to the density of numbers as a precondition for existence of the limit. Further, we could discuss these and present them side by side to discuss the difference between the everyday metaphors and the metaphors the mathematicians use (see also Wiser & Amin, 2001). The results of the present study suggest that for those who do not have problems with rational numbers, and who want to enter into the mathematics community, the essential conceptual leap in the extensions of the number concept seems to be in the process of constructing real numbers out of rational numbers. It is also important to notice that for the majority of students, the novices, at the heart of conceptual change is the conceptual difference between natural numbers compared to rational and real numbers: the difference between discrete and continuous quantities (e.g. Vamvakoussi & Vosniadou, this volume). This difference is embedded in the rules of calculation of these numbers, which are different from the rules when working with natural numbers. It is important to notice that for

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the majority of students in Finland, even in the upper levels of education, the most problematic conceptual leap in the constriction of the number concept is the extension from natural numbers to rational numbers. For these students, understanding rational numbers seems to be clearly constrained by their foremost practice to try to remember the rules without understanding (Merenluoto, 2005). Cultural knowledge can scaffold individuals’ conceptual development and guide new members, such as novice students, toward a deeper understanding of the difficult abstractions in mathematics. Learning the language of mathematics and its way to communicate requires not only a series of individual flashes of insight, but also adopting the socially negotiated rules and shared representations, descriptions, examples and scripts, all co-constructed within the mathematics community. Some of these can also be constructed through discussions in the school context.

References Boyer, C. B. (1959). The history of calculus and its conceptual development. New York: Dover Publications. Carey, S. (1985). Conceptual change in childhood. Cambridge, MA: MIT Press. Carey, S., & Spelke, E. (1994). Domain-specific knowledge and conceptual change. In: L. A. Hirsfeld, & S. A. Gelman (Eds), Mapping the mind. Domain specificy in cognition and culture (pp. 169–200). Cambridge, MA: Cambridge University Press. Chi, M. T. H., Feltovich, P., & Glaser, R. C. (1981). Categorization and representation of physics problems by experts and novices. Cognitive Science, 5, 121–125. Chi, M. T. H., Glaser, R., & Farr, M. F. (1988). The nature of expertise. Hillsdale, NJ: Lawrence Erlbaum. Chi, M. T. H., Slotta, J. D., & de Leeuw, N. (1994). From things to process: A theory of conceptual change for learning science concepts. Learning and Instruction, 4, 27–43. Cornu, B. (1991). Limits. In: D. Tall (Ed.), Advanced mathematical thinking (pp. 153–166). Dordrecht, The Netherlands: Kluwer. Dantzig, T. (1954). Number, the language of science. New York: The Free Press. De Corte, E., & Verschaffel, L. (1996). An empirical test of the impact of primitive intuitive models of operations on solving word problems with a multiplicative structure. Learning and Instruction, 6, 219–242. Dreyfus, T. (1991). Advanced mathematical thinking processes. In: D. Tall (Ed.), Advanced mathematical thinking (pp. 25–41). Dordrecht, The Netherlands: Kluwer. Duit, R., Roth, W. M., Komorek, M., & Wilbers, J. (2001). Fostering conceptual change by analogies — between Scylla and Charybdis. Learning and Instruction, 11, 283–304. Fischbein, E. (1987). Intuition in science and mathematics. An educational approach. Dordrecht, The Netherlands: D. Reidel. Fishbein, E., Jehiam, R., & Cohen, D. (1995). The concept of irrational numbers in high-school students and prospective teachers. Educational Studies in Mathematics, 29, 29–44. Galperin, P. Y. (1966). On the notion of internalization. Voprosy Psikhologii, 12, 25–32. Hadamard, J. (1945). The psychology of invention in the mathematics field. A mathematicians mind. Princeton, NJ: Princeton University Press. Hakkarainen, K., Palonen, T., Paavola, S., & Lehtinen, E. (2004). Communities of networked expertise. Professional and educational perspectives. Oxford: Elsevier.

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Hatano, G., & Inagaki, K. (1998). Qualitative changes in intuitive biology. European Journal of Psychology of Education, 12, 111–130. Ioannides, C., & Vosniadou, S. (2002). The changing meanings of force. Cognitive Science Quarterly, 2(1), 5–62. Karmiloff-Smith, A. (1995). Beyond modularity: A developmental perspective on cognitive science. Cambridge, MA: MIT Press. Kelly, G. J., & Green, J. (1998). The social nature of knowing: Toward a sociocultural perspective on conceptual change and knowledge construction. In: B. Guzzetti, & C. Hynd (Eds), Perspectives on conceptual change. Multiple ways to understand knowing and learning in a complex world (pp. 145–182). Mahwah, NJ: Lawrence Erlbaum Associates. Kline, M. (1980). Mathematics. The loss of certainty. New York: Oxford University Press. Kuhn, T. S. (1970). The structure of scientific revolutions. Chicago, IL: University of Chicago Press. Landau, E. (1960). Foundations of analysis. The arithmetic of whole, rational, irrational and complex numbers. New York: Chelsea Publishing. Lehtinen, E., Merenluoto, K., & Kasanen, E. (1997). Conceptual change in mathematics: From rational to (un)real numbers. European Journal of Psychology of Education, 12(2), 131–145. Liljedahl, P. (2004). Mathematical discovery: Hadamard resurrected. In: M. J. Høines, & A. B. Fuglestad (Eds), Proceedings of the 28th conference of the international group for the psychology of mathematics education (Vol. 4, pp. 249–256). Bergen: PME. Merenluoto, K. (2005). Restrained by beliefs in algorithms and discrete numbers: Student teachers solution strategies in problem using rational numbers. In: E. Pehkonen (Ed.), Problem solving in mathematics education. Proceedings of the Promath 2004 meeting (pp. 115–125). Lahti, Finland: University of Helsinki. Merenluoto, K., & Lehtinen, E. (2002). Conceptual change in mathematics: Understanding the real numbers. In: M. Limón, & L. Mason (Eds), Reconsidering conceptual change. Issues in theory and practice (pp. 233–258). Dordrecht, The Netherlands: Kluwer. Merenluoto, K., & Lehtinen, E. (2004). Number concept and conceptual change: Towards a systemic model of the processes of change. Learning and Instruction, 14, 519–536. Mikkilä-Erdmann, M. (2002). Textbook text as a tool for promoting conceptual change in science. Doctoral dissertation. University of Turku, Annales Universitateis Turkuensis B 249. Patel, V. L., & Groen, G. J. (1991). The general and specific nature of medical expertise: A critical look. In: K. A. Ericsson, & J. Smith (Eds), Toward a general theory of expertise: Prospects and limits (pp. 93–125). Cambridge: Cambridge University Press. Posner, G., Strike, K., Hewson, P., & Gertzog, W. (1982). Accommodation of a scientific conception: Toward a theory of conceptual change. Science Education, 66, 211–227. Rudin, W. (1953). Principles of mathematical analysis. London: McGraw-Hill. Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22(1), 1–6. Sfard, A. (1994). Reification as a birth of a metaphor. For the Learning of Mathematics, 14(1), 44–55. Sfard, A. (1998). On two metaphors on learning and the dangers of choosing just one. Educational Researcher, 27(2), 4–13. Sierpinska, A. (1987). Humanities students and epistemological obstacles related to limits. Educational Studies in Mathematics, 18, 371–397. Spelke, S. E. (1991). Physical knowledge in infancy: Reflections on Piaget’s theory. In: S. Carey, & R. Gelman (Eds), The epigenesis of mind: Essays on biology and cognition (pp. 133–170). Hillsadale, NJ: Erlbaum. Stafylidou, S., & Vosniadou, S. (2004). The development of students’ understanding of the numerical value of fractions. Learning and Instruction, 14, 503–518.

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Strike, K., & Posner, G. (1982). Conceptual change in science teaching. European Journal of Science Education, 4, 231–240. Szydlik, J. E. (2000). Mathematical beliefs and conceptual understanding of the limit of a function. Journal for Research in Mathematics Education, 31(3), 258–276. Trumper, R. (1997). Applying conceptual conflict strategies in the learning of the energy concept. Research in Science & Technological Education, 15, 5–19. Vamvakoussi, X., & Vosniadou, S. (2004). Understanding the structure of the set of rational numbers: A conceptual change approach. Learning and Instruction, 14, 453–468. Vosniadou, S. (1994). Capturing and modelling the process of conceptual change. Learning and Instruction, 4, 45–69. Vosniadou, S. (1999). Conceptual change research: State of art and future directions. In: W. Schnotz, S. Vosniadou, & M. Carretero (Eds), New perspectives on conceptual change (pp. 3–14). Oxford: Elsevier. Vygotsky, L. S. (1978). Mind in society: The development of higher psychological processes. Cambridge, MA: Harvard University Press. Williams, S. R. (1991). Models of limit held by college students. Journal for Research in Mathematics Education, 22(3), 219–236. Wiser, M., & Amin, T. (2001). “Is heat hot?” Inducing conceptual change by integrating everyday and scientific perspectives on thermal phenomena. Learning and Instruction, 11, 331–356.

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Chapter 18

How Many Numbers are there in a Rational Numbers Interval? Constraints, Synthetic Models and the Effect of the Number Line Xenia Vamvakoussi and Stella Vosniadou The Set of Rational Numbers: An Expansion of the Natural Numbers Set? A rational number is a number that can be expressed as the ratio of two integers, like 1/2, ⫺2/3 etc. Alternatively, a rational number can be expressed either as a simple decimal or as a recurring decimal, like 0.23, ⫺0.4, 0.3454545…, ⫺0.333…, etc. It follows that all natural numbers are included in the rational numbers set, since 2, for instance, can be represented as 2/1 or 2.0. The same holds for any negative whole number, since ⫺3, for instance, can be represented as ⫺3/1 or ⫺3.0. The rational numbers set is closed under subtraction and division, in the sense that the difference and the quotient of any rational number are rational numbers as well. This is a property that does not hold within the natural numbers set: For instance, the outcomes of 3 ⫺ 5 or 3:5 are not natural numbers. The need for closure under subtraction and division offers a plausible explanation for the need to expand the set of natural numbers to the set of rational numbers. This expansion can be described linearly in the following way: The natural numbers set is expanded to the set of integers to comprise the negative whole numbers. The new set is closed under subtraction, e.g., 3 ⫺ 5 ⫽ ⫺2 is an integer number. The set of integers is then expanded to the set of rational numbers; the new set is closed under subtraction and division, e.g., 3 ⫺ 5, 3:5 are rational numbers. This characterization of the shift from natural to rational numbers may represent a good way to make a plausible summary of the historical development of the number concept. However, it does not accurately reflect the historical process. More importantly, it conceals the fact that the historical development of the rational number concept was far from being linear and smooth. There were harsh debates in the history of mathematics about the kind of mathematical constructs that can be considered as numbers and the kinds of operations that are legitimate (see also Merenluoto & Palonen, this volume). In a more general fashion, the shift from natural to rational numbers involved changes in the status and meaning

Reframing the Conceptual Change Approach in Learning and Instruction Copyright © 2007 by Elsevier Ltd. All rights of reproduction in any form reserved. ISBN: 0-08-045355-4

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of the term “number” that cannot be accounted for in terms of the mere expansion of the natural number concept. For example, to accept negative numbers as numbers meant that the conceptualization of numbers as answers to the questions “How many?” and “How much?” for discrete and continuous quantities, respectively, had to be radically changed (Dunmore, 1992). From a different perspective, the shift from accepting that 3 ⫺ 5 is a legitimate operation to accepting that 3 ⫺ 5 is a mathematical object can be viewed as an ontological shift during which a process now becomes a member of the category of objects (Sfard, 1991). The same holds for the shift in the conceptualization of the ratios of integers from relations between numbers to numbers. What we are trying to say here is that the change from natural to rational numbers cannot be accounted for as a mere expansion of the natural numbers set, in the sense that there is more involved than just “adding” new numbers to it. Along with the changes mentioned above, the two sets also have radically different structures: The set of natural numbers is discrete, i.e., between two successive natural numbers there is no other natural number, whereas the set of rational numbers is dense, i.e., between any two, non-equal rational numbers, there are infinitely many numbers. In this chapter, we investigate secondary and upper secondary students’ understanding of this particular property of the rational numbers set, namely density. Understanding density is closely related to the development of the rational number concept. We argue that students’ initial explanatory frameworks for number are tied around natural numbers and their basic properties, such as discreteness. When students are introduced to other kinds of numbers, like fractions and decimals, they have to deal with constructs that took millennia to develop. Although students do not have to recapitulate the historical growth of these mathematical ideas, they still face the challenging task to construct meaning for these new mathematical objects called “numbers” and, in particular, their notation. Moreover, while moving to a wider explanatory framework for number, students have to realize the confining role of the presuppositions pertaining to their initial framework, which are bound to interfere in the interpretation of new information about rational numbers. We argue that the shift to wider explanatory frameworks for number cannot be accomplished by a mere enrichment of the initial frameworks for number, but requires conceptual change. That is, it requires changes in the structure of the number concept, in the fundamental presuppositions within which it is embedded, as well as change in the contexts of its use.

Initial Explanatory Frameworks for Number It has been argued that even before instruction children form a principled understanding of number that has its roots in the act of counting (Gelman, 2000; Gelman & Gallistel, 1978). During the first years of instruction, this initial understanding of number supports children to reason about natural numbers and to learn about their properties, to build strategies in relation to natural numbers operations, etc. For instance, the counting algorithm supports children to build the successor principle (e.g., that given a specific natural number, one can always find its next), which in turn may help them to infer that there are infinitely many natural numbers (Hartnett & Gelman, 1998). Addition and subtraction are

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conceptualized in terms of counting [see, for example, Resnick (1989) for a report of counting-based strategies], multiplication is viewed as repeated addition (e.g., Fischbein, Deri, Nello, & Marino,, 1985), and numbers can be ordered by means of their position on the count list. In terms of the conceptual change theoretical framework that we adopt (Vosniadou, this volume; Vosniadou & Verschaffel, 2004), this complex network of relations and operations constitutes young children’s initial explanatory framework of number (see also Smith, Solomon, & Carey, 2005). The conceptual change framework predicts that when children will encounter numbers with different properties than natural numbers, their initial explanatory framework for number will stand in the way of further learning. Indeed, there is a great deal of evidence showing that students at various levels of instruction make use of their knowledge of natural number to conceptualize rational numbers and make sense of decimal and fraction notation, often resulting in making systematic errors in ordering, operations, and notation of rational numbers (e.g., Fischbein et al., 1985; Moskal & Magone, 2000; Resnick et al., 1989; Stafylidou & Vosniadou, 2004; Yujing & Yong-Di, 2005). Many researchers attribute these difficulties to the constraints students’ prior knowledge about natural numbers imposes on the development of the rational number concept. Such findings and interpretations are compatible with the conceptual change approach and constitute evidence of the confining role of students’ initial explanatory frameworks for number. In this chapter, we will test the predictive power of this framework in relation to students’ understanding of the dense structure of the rational numbers set, which has not been extensively investigated so far, as pointed out also by Smith et al. (2005).

Understanding the Structure of the Set of Rational Numbers: A Conceptual Change Approach To understand the structure of the rational numbers set, students must (a) realize that discreteness is a property of natural numbers which is not preserved in the rational numbers set, and (b) be aware of the fact that the rational numbers set consists of elements that can be represented either as fractions or as decimals, yet remain invariant under different symbolic representations. We assume that the discreteness of numbers is a fundamental presupposition (Vosniadou, 1994, 2001) of the initial explanatory framework of number, within which “orderability” is not differentiated from “nextness” (Greer, 2004). Research in the area of mathematics education has provided evidence that the idea of discreteness of numbers is a barrier to understanding the dense structure of the rational and real numbers set for students at various levels of instruction (Malara, 2001; Merenluoto & Lehtinen, 2002, 2004; Neumann, 1998) as well as for prospective teachers (Tirosh, Fischbein, Graeber, & Wilson, 1999). Understanding that a rational number can be represented in multiple ways, but that its different symbolic representations refer to the same mathematical entity, requires the ability to move flexibly among different symbolic representations of rational numbers. This task, although easy for a mathematically versed person, is very challenging for the novices in mathematics. As Markovits and Sowder (1991) argue, one of the major difficulties faced

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by learners in mathematics is the use of different symbols to represent the same ideas. In the case of rational numbers notation, there is evidence to suggest that students interpret different symbolic representations to refer to different mathematical entities. This phenomenon has been noted by Khoury and Zazkis (1994) and reported by O’Connor (2001) as a fact noticed by mathematics teachers. In our previous work (Vamvakoussi & Vosniadou, 2004), we found evidence that students treated different symbolic representations as if they were different numbers in their attempts to describe the structure of rational numbers intervals. For example, during an interview, a 9th grader stated that there are infinitely many numbers between 3/8 and 5/8 and she mentioned a number of fractions, all of which were equivalent to 4/8, such as 4.0/8, 8/16, etc., indicating her belief that these symbols represented different numbers. Another student said that there is no other number between 3/8 and 5/8 and he explained that 4/8 can be simplified to 1/2, and 1/2 is not between 3/8 and 5/8. This student, although he “knew” that 1/2 is equivalent to 4/8, failed to assign to it the properties of 4/8. In this sense, he considered it to be a different mathematical object. The belief that different symbolic representations stand for different numbers may have another implication regarding students’ thinking about the structure of the rational numbers set. We suggest that students may consider fractions and decimals to be different, unrelated “sets” of numbers. This assumption is compatible with evidence coming from research in various domains showing that novices tend to categorize objects on the basis of superficial characteristics (e.g., Chi, Feltovich, & Glaser, 1981). In the case of rational numbers, this tendency may be enhanced by the fact that there are considerable differences between the operations, as well as between the ordering of decimals and fractions. Thinking of the rational numbers set as consisting of different, unrelated sets of numbers is bound to constrain students’ understanding of its dense structure. For instance, Neumann (1998) reports that 7th graders had difficulties accepting that there could be a fraction between 0.3 and 0.6. In a previous study, Vamvakoussi and Vosniadou (2004) found that students tend to think differently for fractions than for decimals with respect to their structure. For instance, a student stated that there is a finite number of numbers between two given decimals, but infinitely many numbers between two given fractions. To summarize, children’s initial explanatory frameworks of number are tied around natural numbers. In elementary school, children are introduced to new “kinds” of numbers, i.e., fractions and decimals and later on to negative numbers as well. Accepting that these novel constructs are full-fledged numbers is a major conceptual shift per se (see, for example, Sfard, 1991). But let us focus on the expansion of numbers, from a learner’s point of view. To master the rational number concept (at the level required in school context), students must move from their initial to a wider explanatory framework for number, realizing that certain presuppositions that held before, such as the discreteness of numbers, are only valid within specific contexts. In developing this wider framework, it does not suffice to accept that the term “number” refers equally well to natural numbers, fractions, and decimals; one must also realize that fractions and decimals (simple and recurring),1 despite

1

Here and in the following we will use the term “decimal number” with reference only to the decimal numbers that belong to the rational numbers set.

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their differences in notation, ordering, operations, and contexts of use, are alternative representations of rational numbers and not different kinds of numbers. This requires the ability on the part of the learner to take different perspectives on fractions and decimals, according to the context of use, while being able to think of them as interchangeable representations of the same mathematical entities, namely rational numbers. According to the conceptual change framework, the construction of this wider explanatory framework for number, accompanied by the deposition of natural number of its privileged position and the development of different perspectives with respect to rational number notation, cannot but be difficult, gradual, and time-consuming; moreover, in the process, misconceptions will appear. In particular, it is predicted that students are bound to generate synthetic models of the structure of the set of rational numbers, which are caused when learners assimilate aspects of the new, incompatible information in their existing knowledge. While examining the development of the number concept, it is essential to take into consideration the artifacts and tools available in school context to support rational and real number reasoning. Indeed, there is a general agreement that external representations have the potential to facilitate students to grasp mathematical ideas. With respect to numbers, students are presented with external representations, such as Venn diagrams and the number line. In particular, the number line, being itself continuous, appears to be a good external representation to convey the idea of density of numbers. However, some mathematics educators argue that the meaning of a mathematical idea is not necessarily carried by a more concrete representation (e.g., Clements & McMillen, 1996). This view has its counterpart in the conceptual change literature, where there is an ongoing discussion about the effect of external representations, such as the globe or the map, on students’ reasoning and understanding — in this case, about the shape of the Earth (Ivarsson, Schoultz, & Säljö, 2002; Schoultz, Säljö, & Wyndhamn, 2001; Vosniadou, Skopeliti, & Ikospentaki, 2005). According to Vosniadou and her colleagues, the effect of external representations should be analyzed within a constructivist framework, where the external representations are themselves interpreted on the basis of students’ prior knowledge. Among the aims of this study was to investigate the effect of the number line on students’ reasoning about the density of rational numbers.

The Present Study In the present study we investigated 9th and 11th graders’ understanding of the dense structure of the rational numbers set and the effect of the number line on their reasoning. Understanding density is not an explicit goal of the Greek secondary school mathematics curriculum. Still, students by grade 8 have already been taught everything they are supposed to know about real numbers, including ordering, turning a decimal into a fraction and vice versa. They also have dealt with tasks that implicitly refer to density. For instance, in the 7th grade book, students are asked to find a number between two fractions with common denominators and successive numerators. Eighth graders are explicitly taught how to approximate 兹苶2 with two rational numbers. Moreover, in grade 10, students are introduced, through Venn diagrams, to the interrelations between the various subsets of the real numbers. They also review everything they are supposed to know about real numbers, including properties, operations, and ordering. From grade 8 and on, students use the real number line to represent real numbers. The mathematics

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teachers in the schools where this study was conducted told us that they had mentioned to the students on several occasions that there are infinitely many numbers in an interval. We assumed that the idea of discreteness, being a fundamental presupposition of initial explanatory frameworks of number, is a major barrier on students’ understanding of the dense structure of the set of rational numbers. We also assumed that new knowledge of fractions and decimals is used by students in their attempts to describe rational number intervals, but it is constrained by their belief that different symbolic representations refer to different numbers, and its implications that were discussed in the introduction. More specifically, we expected that: • The performance of 11th graders would be better than 9th graders’. Still, the answer that there is a finite number of numbers in a rational number interval would appear frequently in both age groups. • The presence of the number line would have a helpful but limited effect on students’ performance in tasks related to density. • Students would perform better in forced-choice items, than in open-ended items. • Understanding the structure of the set of rational numbers would be a slow and gradual process, and not an “all or nothing” situation: We assumed that there would be intermediate levels of understanding and that students at these levels would form synthetic models of the structure of rational numbers intervals. We expected these models to reflect the presupposition of discreteness and the belief that different symbolic representations stand for different numbers.

Method Participants The participants of this study were 301 students, of which 164 were 9th graders and 137 were 11th graders (approximate age 15 and 17 years, respectively). They came from five different schools in the Athens area. Almost half of our participants were girls. Materials We designed two types of questionnaires: an open-ended (QT1) and a forced-choice one (QT2). Both types of questionnaires had two parts, each consisting of three questions. One part made use of the number line while the other did not. Tables 18.1 and 18.2 show the items in QT1 and QT2, respectively. All questions had a common, general form — indicated as GF on the tables — which focuses on the number of numbers in a given rational numbers interval.2

2

The number of numbers in an interval is not the only aspect of the property of density; there is also the question of whether there is a “next number” to a given rational number. However, we chose to focus on this particular aspect of the property, which we considered to be appropriate for our participants’ age and level of instruction.

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Table 18.1: The open-ended questionnaire (QT1). Items without the number line GF

Are there any numbers that are greater than a and, at the same time, less than b? If yes, define how many/which these numbers are If no, explain why

Q1

a ⫽ 0.005, b ⫽ 0.006

Q2

a ⫽ 3/8, b ⫽ 5/8

Q3

a ⫽ 0.001, b ⫽ 0.001

Items with the number line (Number line present and number a already placed on the number line) GF

Place b on the number line Are there any numbers that are greater than a and, at the same time, less than b? If yes, define how many/which these numbers are If no, explain why

Q4

a ⫽ 0.01, b ⫽ 0.02

Q5

a ⫽ 1/3, b ⫽ 2/3

Q6

a ⫽ 0.01, b ⫽ 0.1

Procedure The students in each class were equally divided between the QT1 and the QT2 condition. Half of them received the items with the number line first (NL1), whereas the other half received first the items without the number line (NL2). In both cases, the first part of the questionnaire was withdrawn, before the second part was administered. Students had one school hour (about 45 minutes) to answer all six questions.

Results Students’ responses to each of the six questions were marked as “Finite,” “Undefined,” and “Infinite.” “Finite” refers to the response type “There is finite number of numbers,” while “Infinite” refers to the response type “There are infinitely many numbers.” The answer “Undefined” appeared only in the open-ended questionnaires when a student selected the answer “Yes” to claim that there are numbers in a given interval but did not give any further information regarding the number of numbers. Table 18.3 shows the percentage of every type of answer, in the total of the answers given in all six questions, as a function of grade [9th (GR1), 11th (GR2)] and type of questionnaire (QT1, QT2).

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Table 18.2: The forced choice questionnaire (QT2). Items without the number line GF How many numbers are there, that are greater than a and, at the same time, less than b? i. There is no such number ii. There are the following numbers: iii. There are infinitely many decimals/ fractions iv. There are infinitely many numbers: simple decimals, decimals with infinitely many decimal digits, fractions, square roots v. None of the above. I believe that … Q1

a ⫽ 0.005, b ⫽ 0.006 ii. 0.0051, 0.0052, 0.0053, 0.0054, 0.0055, 0.0056, 0.0057, 0.0058, 0.0059

Q2

a ⫽ 3/8, b ⫽ 5/8 i*. There is only one number, namely 4/8 ii*. There are infinitely many fractions, all equivalent to 4/8, e.g., 8/16

Q3

a ⫽ 0.001, b ⫽ 0.01 ii. a. 0.002, 0.003, 0.004, 0.005, 0.006, 0.007, 0.008, 0.009 ii. b. 0.0011, 0.0012, 0.0013,…, 0.0097, 0.0098, 0.0099

Items with the number line (Number line present and number a already placed on the number line) GF

Place b on the number line. How many numbers are there, that are greater than a and, at the same time, less than b? i. There is no such number ii. There are the following numbers: iii. There are infinitely many decimals (or fractions) iv. There are infinitely many numbers: simple decimals, decimals with infinitely many decimal digits, fractions, square roots v. None of the above. I believe that …

Q4

a ⫽ 0.01, b ⫽ 0.02 ii. 0.11, 0.12, 0.13, 0.14, 0.15, 0.16, 0.17, 0.18, 0.19

Q5

a ⫽ 1/3, b ⫽ 2/3 ii.

Q6

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 ᎏ, ᎏ, ᎏ, ᎏ, ᎏ, ᎏ, ᎏ, ᎏ, ᎏ 3 3 3 3 3 3 3 3 3

a ⫽ 0.01, b ⫽ 0.1 ii. a. 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08, 0.09 ii. b. 0.011, 0.012, 0.013,..., 0.020, 0.021,..., 0.099

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Table 18.3: Percentage of answer types, in the total of the answers given in all six questions, as a function of the type of questionnaire and of grade. Open-Ended (QT1)

Finite Infinite Undefined No answer

Forced-Choice (QT2)

9th grade, (GR1) N ⫽ 498 (83 ⫻ 6)

11th grade (GR2), N ⫽ 396 (66 ⫻ 6)

9th grade (GR1) N ⫽ 486 (81 ⫻ 6)

11th grade (GR2) N ⫽ 426 (71 ⫻ 6)

65% 11.9% 20.4% 2.8%

41.7% 26% 28.2% 4%

52% 42.7% – 5.3%

34.5% 61.3% – 4.2%

The Presupposition of Discreteness As shown in Table 18.3, the answer “Finite” was the dominant answer for 9th graders, in QT1 and QT2. The same holds for 11th graders, but only in QT1. In QT2, 11th graders gave the “Infinite” answer more often. Still, about one-third of the answers were “Finite.” This result shows that the presupposition of discreteness was strong in the 9th graders and remained strong in the 11th graders too. Differences by Age In order to calculate each student’s overall performance in QT2, we scored the “Finite” and “Infinite” answers as 1 and 2, respectively, and calculated the sum of the scores in all six questions. In QT1, we scored the “Finite,” “Undefined,” and “Infinite” answers as 1, 2, and 3, respectively, and calculated the sum of the scores in all six questions. We scored the answer “Undefined” higher than the “Finite” and lower than the “Infinite” answer, respectively, assuming that: • a student who could not define the number of numbers in an interval, yet took their existence for granted, should be scored higher than a student who answered that there are no numbers in the same interval. • a student who was able to characterize the number of numbers in an interval as “infinitely many” should be scored higher than a student who did not use this expression. Under both conditions (QT1, QT2), either when a student gave no answer or when he gave an irrelevant answer, s/he was scored with 0. We compared 9th and 11th graders’ performance in QT1 by performing a Mann–Whitney test, which showed that 11th graders’ overall performance was significantly better than 9th graders’ (z ⫽ ⫺2.862, p ⬍ .05). A second Mann–Whitney test showed that 11th graders’ overall performance was significantly better than 9th graders’ (z ⫽ ⫺2.537, p ⬍ .05) in QT2 as well.

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Effects of the Number Line Examining the influence of the number line, we first tested for possible order effects. We compared the performance of students who received the items with the number line first, to the performance of those who received first the items without the number line (NL1 vs. NL2). We performed four independent Mann–Whitney tests (GRi ⫻ QTj, i, j ⫽ 1, 2) that showed no significant order effect. As a result, the order of presentation of the number line was not taken into consideration in subsequent statistical analyses. In order to test for possible effects of the presence of the number line on students’ performance, we performed four independent Wilcoxon signed ranks tests (GRi ⫻ QTj, i, j ⫽ 1, 2). We found no significant difference in 11th graders’ performance in the presence of the number line either in QT1 or in QT2. However, the presence of the number line improved significantly the performance of 9th graders, in both QT1 (z ⫽ ⫺2.467, p ⬍ .05) and QT2 (z ⫻ ⫺3.395, p ⫽ .001). In order to examine further the effect of the number line on students’ performance, we separated the students into three categories: those who scored higher in the items with the number line, those who scored lower in these items, and the remaining students who scored the same in the items with and without the number line. Table 18.4 shows the number and percentage of students in each category for each grade and type of questionnaire. As can be seen in Table 18.4, there are a considerable number of students whose performance was not affected at all by the presence of the number line. Moreover, the number of 11th graders who scored higher in the questions with the number line is close to the number of those who scored lower in these questions. Taking a closer look at our data we find that in QT1, only 3 of the 21 9th graders whose overall performance was better for questions with the number line had actually moved from “Finite” to “Infinite” answers; in addition, only one of them gave the “Infinite” answer consistently in all questions with the number line. In QT2, 20 of the 31 9th graders who performed better for questions with the number line moved from “Finite” to “Infinite” answers. Only seven of these students gave Table 18.4: Number and percentage of students in categories formed with respect to their performance with and without the number line, as a function of grade and type of questionnaire. Open-Ended (QT1) Performance

9th grade (GR1), N = 83

11th grade (GR2) N = 66

Better with the number line Worse with the number line No effect

21 (25.3%) 6 (7.2%) 56 (67.5%)

16 (24.2%) 20 (30.3%) 30 (45.5%)

Forced-Choice (QT2) 9th grade (GR1), N = 81 31 (38.3%) 8 (9.9%) 42 (51.8%)

11th grade (GR2), N = 71 16 (22.5%) 10 (14.1%) 45 (63.4%)

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the “Infinite” answer consistently for all questions with the number line. In conclusion, it appears that the presence of the number line had a limited effect on students’ performance. Effect of Questionnaire Type We categorized all students into three groups on the basis of their responses to all six questions. More specifically, we placed in the first group students who answered only “Finite.” In the third group, we placed students who answered only “Infinite.” In the second, intermediate group, we placed all remaining students. The first, second, and third group will be referred to as “Finiteness,” “Mixed,” and “Infinity” category, respectively. A  2 test showed that the effect of the questionnaire type was statistically significant (df(2), p ⫽ .001). More specifically, students who filled in the open-ended questionnaires were found more often in the category “Finiteness” and less often in the other two categories, as compared to students who filled in the forced-choice questionnaires. Intermediate Levels of Understanding In our attempts to examine intermediate levels of understanding of the structure of the rational numbers set, we refined our categorization further and identified two subcategories for the category “Finiteness”: Discreteness and Refined Discreteness, both in QT1 and in QT2. In the Discreteness category we placed students who considered the given numbers to be successive, e.g., 0.005 to be immediately next to 0.006, in all six questions. In the Refined Discreteness category, we placed students who did not consider the given numbers to be successive, in at least one out of six questions. These students still answered that there is a finite number of numbers in the interval, e.g., 0.0051, 0.0052, …, 0.0059 between 0.005 and 0.006. In the case of forced-choice questionnaires, we were able to define two sub-categories for the category “Infinity”: Constrained Density and Density. Students who answered that there are infinitely many numbers of the same symbolic representation in the given interval, for at least one out of the six questions, were placed in the “Constrained Density” category. Students in this category were reluctant to accept, for example, that there may be decimals between two fractions or vice versa. In the “Density” category, we placed the students who answered that there are infinitely many numbers, regardless of their symbolic representation, in all six questions. Almost half of the students of the “Infinity” category were placed in the “Constrained Density” category. Table 18.5 summarizes the percentage of students placed in the Discreteness, Refined Discreteness, Mixed, Constrained Density, and Density categories, for both grades and types of questionnaire. As shown in Table 18.5, there were three intermediate levels of understanding, between the naïve level of Discreteness and the more sophisticated level of Density. Students in the Refined Discreteness category were constrained by the presupposition of discreteness. Students in the Constrained Density category had overcome the barrier of discreteness, yet, they were constrained by their belief that different symbolic representations stand for different numbers, in the sense that they tend to group numbers according to their symbolic representation. Finally, students in the Mixed category, who answered that there are infinitely

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Table 18.5: Percentage of students placed in the five categories, as a function of type of questionnaire and of grade. Category

Discreteness Refined discreteness Mixed Constrained density Density

Open-Ended Questionnaire

Forced-Choice Questionnaire

9th grade (N ⫽ 83)

11th grade (N ⫽ 66)

9th grade (N ⫽ 81)

11th grade (N ⫽ 71)

30% 16.9%

12.1% 7.6%

4.9% 30.9%

4.2% 16.9%

43.4% 9.6%

66.7% 13.6%

40.7% 12.3%

42.3% 15.5%

11.1%

21.1%

many numbers in some, but not all questions, are still constrained by the idea of discreteness and are also affected by the symbolic representation of numbers. Indeed, more than half (53.9%) of the students in the “Mixed” category referred consistently to intervals with different structure, according to the symbolic representation of the first and the last number. For instance, some answered that there are infinitely many numbers between two decimals but a finite number of numbers between two fractions, or vice versa. The belief that different symbolic representations refer to different numbers was expressed also in a more explicit way. More specifically, in the case of forced-choice questionnaires, 12 out of 81 (15%) 11th graders and 14 out of 71 (19.70%) 9th graders answered that there are infinitely many fractions between 3/8 and 5/8, all equivalent to 4/8. We also noted that in the case of open-ended questionnaires, 8% of the students in the Discreteness category (total 149 students), answered that there is no number between 3/8 and 5/8, probably because they simplified 4/8 to 1/2 and decided that it is not between 3/8 and 5/8.

Discussion Our results agree with previous findings showing that the idea of discreteness is a barrier to the understanding of the dense structure of the rational numbers set (Malara, 2001; Merenluoto & Lehtinen, 2002, 2004; Neumann, 1998; Tirosh et al., 1999). They also agree with findings showing that realizing that rational numbers remain invariant under different symbolic representations is a major conceptual difficulty for students (e.g., Khoury & Zazkis, 1994). These results are in full agreement with the predictions made within the conceptual change framework adopted. Understanding density appears to be a slow and gradual process, in the course of which students are bound to generate synthetic models of the rational numbers intervals. Indeed, with the exception of students in the Density category, our participants gave alternative

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accounts of the structure of the given intervals. These accounts reflected the enrichment of an initial explanatory framework of numbers with information about fractions and decimals. However, within this wider explanatory framework, these new “kinds of numbers” were considered as different, unrelated sorts of numbers; in addition, the presupposition of discreteness appeared to play a confining role. In this sense, the accounts generated by our participants can be interpreted as synthetic models of the structure of the rational numbers intervals. In Table 18.6 there is a more detailed description of the synthetic models generated by our participants.

Table 18.6: Students’ accounts of the structure of rational numbers intervals: some examples. Category

Description

Example

Discreteness

Intervals that preserve the discrete structure of natural numbers The initial numbers are considered successive Intervals that preserve the discrete structure of natural numbers The initial numbers are not considered successive Different structure, according to the symbolic representation of the first and last numbers

(0.005–0.006)

Refined discreteness

Within the mixed category

Constrained density

Density

Intervals that contain “infinitely many” equivalent numbers Intervals that contain infinitely many numbers of the same symbolic representation Any interval contains infinitely many numbers, regardless of their symbolic representations

(1/3–2/3) (0.0051, 0.0052, 0.0053,…,0.0059, 0.006) (3.0/8, 3.1/8,…, 4.0/8,…, 5/8) (Decimal, infinitely many decimals, decimal) and (fraction, finite number of fractions, fraction). Or vice versa (3/8, 4/8, 4.0/8, 8/16,…, 5/8) (Decimal, infinitely many decimals, decimal) or (fractions, infinitely many fractions,fractions)

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Students in the Discreteness category treated the given numbers as if they were successive, thus preserving the discrete structure of the set of natural numbers. Students in the Refined Discreteness category also preserved the discrete structure of natural numbers. However, they made use of new knowledge about rational numbers to give a more sophisticated answer. For instance, they appealed to the fact that adding one decimal digit to 0.005, one gets a number that is bigger than 0.005 and still smaller than 0.006, to conclude that there are (a finite number of) numbers between 0.005 and 0.006. Yet, their commitment to the idea of discreteness prevented them from using this knowledge efficiently and infer that there can be even more numbers in the given interval. Students who completed the forced-choice questionnaires were placed more often in the Refined Discreteness category than in the Discreteness category. Apparently, while these students were facilitated by the presence of more sophisticated answers, they were not able to overcome the presupposition of discreteness and chose the answer “Infinite.” This result, along with the finding that students gave more sophisticated answers in QT1, as compared to QT2, is in accordance with findings from the conceptual change research, which show that students perform better in forced-choice tasks (Vosniadou, Skopeliti, & Ikospentaki,, 2004). Students in the Constrained Density category were able to answer that there are infinitely many numbers in all given intervals, but were reluctant to accept that these numbers can have various symbolic representations. This type of synthetic model reveals students’ disposition to group numbers according to their symbolic representation, which reflects the belief that different symbolic representations stand for different numbers. Among the students in the Mixed Category, those who answered differently, according to the symbolic representation of the first and last number of the interval, generated synthetic models of the structure of rational numbers intervals, for example discrete structure between fractions and dense structure between decimals. We suggest that these students were also constrained by their belief that different symbolic representations refer to different numbers, with different properties. This suggestion is based on findings from our previous work (Vamvakoussi & Vosniadou, 2004). For example, a 9th grader who participated in that study claimed that there are infinitely many numbers between 3/8 and 5/8, yet there is a finite number of numbers between 0.001 and 0.01. When asked to consider both his answers, he explained that if one turns the same decimals into fractions, one can find more, infinitely many numbers between them. The belief that the different symbolic representations of a number refer to different numbers is explicitly expressed in the case of the students who answered that there are “infinitely many” numbers between 3/8 and 5/8, all being fractions equivalent to 4/8. We believe that the same belief drove some students to answer that there is no other number between these fractions. Based on the findings of a previous study (Vamvakoussi & Vosniadou, 2004) we suggest that these students converted 4/8 into 1/2, and then assumed that 1/2 is not between 3/8 and 5/8, based on the difference between the symbolic representations. If this is the case, these students, although they may know that 1/2 is equivalent to 4/8, fail to assign to it the properties of 4/8. In this sense, they consider it to be a different mathematical entity. Finally, our findings with respect to the number line support the view that the meaning of a mathematical idea is not necessarily carried by the mere presence of a more concrete

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representation (e.g., Clements & McMillen, 1996). Our results indicate that the number line, despite being continuous, does not facilitate considerably students’ reasoning about the infinity of numbers in a rational number interval. This is because the number line improved significantly only the performance of 9th graders, and not of the 11th graders. Second, even in this case, the 9th graders who scored higher in the presence of the number line did not answer consistently that there are infinitely many numbers when the number line was present; in fact, some of them did not change any of their answers from “Finite” to “Infinite.” Third, the order of appearance of the number did not have a significant effect on students’ performance. It is important to note that the effect of the number line was not always positive, especially with respect to 11th graders. This result, although not statistically significant, is interesting and somewhat surprising: Taking a second look at the questions with the number line we noticed that they might be considered easier than the questions without the number line. Indeed, the decimals involved in the first case have fewer decimal digits than those that appear in the second case. Although further research is needed to clarify this issue, we believe that this result might be a side effect of what Greek students are taught in upper secondary school about the line in the context of Euclidean geometry. More specifically, in 10th grade, students are introduced to the notion of a line as a set of points, which is different from the “holistic” line (Nunez & Lakoff, 1998) and may convey the idea that lines are discrete in nature. If the number line is interpreted on the basis of students’ thinking about the geometrical line, as Vosniadou et al. (2005) would suggest, then this could account to some extent for the decrease of 11th graders’ performance in the presence of the number line. The findings of this study support the hypothesis that understanding the dense structure of the rational numbers set requires a number of changes in students’ explanatory frameworks for number. These changes involve objectifying rational numbers, conceptualizing natural numbers as a subset of rational numbers, and assigning proper meaning to fractional and decimal notation, which involves abandoning the belief that different symbolic representations refer to different numbers; most importantly, students must stop thinking about rational numbers in terms of natural numbers and their properties. These changes cannot be accomplished by mere enrichment of the initial explanatory frameworks of number. Our view is very similar to the view of Smith et al. (2005), who argue that the shift in younger children’s thinking about numbers is explained in terms of conceptual change. We fully agree that “conceptual changes involve differentiations and coalescences such that the extension of a concept and its relations to other concepts are qualitatively different after the change than before it.” (Smith et al., 2005, p. 133) However, we do not believe that these differentiations and coalescences “commit the child to concepts that would be incoherent in each side of the divide” (Smith et al., 2005, p. 133). We want to stress that we do not see the antecedent and subsequent states as “incommensurable,” in the sense that after the change children think of numbers in terms of rational numbers. Rather, we argue for the ability on part of the learner to move flexibly between natural and rational number reasoning, according to the context of use. Learners who have not made

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the transition to the wider explanatory framework for number are clearly not in a position to entertain both perspectives. However, as it has been nicely illustrated in Merenluoto and Palonen (this volume), mathematically versed persons not only do they show this ability, but they are also fully aware of their different modes of thinking about numbers. Conceptual change in the number concept cannot but be gradual and time consuming and they become even more difficult if the design of instruction builds on the presupposition that learning is accomplished through additive mechanisms. To give an example, in the Greek 8th grade mathematics textbook, which is based on a procedure-oriented curriculum, the set of rational numbers is presented in the following way: “All the numbers that we know, namely, natural numbers, decimals, and fractions, together with the respective negative numbers, constitute the set of rational numbers.” This “definition” certainly builds on students’ prior knowledge about numbers, but at the same time, enhances students’ initial tendency to group numbers on the basis of their symbolic representations and presents the rational numbers set as consisting of different, unrelated sorts of numbers; in a more general fashion, it presents the rational numbers set as an expansion of the natural numbers set, in the course of which “new” numbers are being added to it. The fact that students are explicitly taught how to turn a decimal into a fraction and vice versa does not necessarily support them in accomplishing the corresponding conceptual knowledge, as suggested by the results of our study. The rational number concept is notoriously difficult for students to develop and this fact has been amply demonstrated in numerous studies. We suggest that the conceptual change approach could make a useful theoretical tool in the attempt to synthesize such widespread findings. In a more general fashion, it might be useful to look at mathematics curricula through the lenses of the conceptual change approach and reconsider the sequence of appearance of certain mathematical concepts (see also Vosniadou & Vamvakoussi, 2006). According to Resnick (2006), the answer to the question “Is there a ‘best’ developmental sequence for teaching mathematical concepts that will maximize positive effects of prior mathematical learning and minimize interference?” is yet to come. Resnick also points out that students should not be the only targets of conceptual change interventions — teachers should also be taken into consideration. The first step towards this direction would probably be to inform teachers about the issue of conceptual change and stress the fact that an expansion of a concept from a mathematics point of view may not correspond to an enrichment of the prior knowledge of the learner.

Acknowledgments The present study was funded through the program EPEAEK II in the framework of the project “Pythagoras — Support of University Research Groups” with 75% contribution from European Social Funds and 25% contribution from National Funds.

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Chapter 19

Students’ Interpretations of Literal Symbols in Algebra Konstantinos P. Christou, Stella Vosniadou and Xenia Vamvakoussi Large-scale research both in Great Britain (Hart, 1981) and in the United States (Carpenter, Corbitt, Kepner, Lindquist, & Reys, 1981) has shown that students have great difficulties in understanding algebra. Many students mention specifically the use of literal symbols as the origin of their difficulties, saying that they understood mathematics until literal symbols appeared (Sackur, 1995). In the present study, the conceptual change approach will be used as an explanatory framework for better understanding students’ difficulties in interpreting the use of literal symbols as variables in algebra.

The Conceptual Change Approach The conceptual change approach focuses on knowledge acquisition in specific domains and describes learning as a process that sometimes requires the significant reorganization of existing knowledge structures and not just their enrichment (Vosniadou, 1999; Vosniadou & Brewer, 1992). According to this approach, by the time formal education starts, students have already constructed a commonsense understanding of the world based on their everyday experiences and the influence of the culture. This prior knowledge can stand in the way of acquiring new information when the new learning content is incompatible with what is already known. In these cases, the acquisition of new information requires conceptual change. Conceptual change is more difficult than learning that can be accomplished through enrichment. When students use additive mechanisms to assimilate incompatible information to what they already know (enrichment) they produce a type of misconception, which can be explained as “synthetic models” (Vosniadou, this volume). Recently the conceptual change approach has been applied to the learning and teaching of mathematics (see Verschaffel & Vosniadou, 2004) with most of the relevant studies focusing on the development of the number concept. Prior research has indicated that students’ principle understanding of numbers is grounded on the natural numbers

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(Gelman, 2000). Many students even in secondary education tend to project their number concept grounded on natural numbers onto a non-natural number input. This seems to be one of the reasons why misconceptions and difficulties appear when numbers other than natural, such as rational numbers, are introduced in the mathematics curriculum. For example, many of the errors students make in the case of fractions can be interpreted to be caused by the application of properties of natural numbers to fractions (e.g., Gelman, 2000; Stafylidou & Vosniadou, 2004). Vamvakoussi and Vosniadou (2004) argue that the presupposition of discreteness, which is a property that characterizes the natural numbers, constrains students’ understanding of density, which is a property of rational numbers. Other research also indicates that prior knowledge of natural numbers hinders students’ understanding of the properties of rational numbers (Resnick et al., 1989). Students’ prior experience based on calculating only with natural numbers is considered to be responsible for students’ belief that “multiplication always makes the number larger”. This belief in turn hinders students’ understanding of calculations when real numbers are involved (Fischbein, Deri, Nello, & Marino, 1985). The purpose of the present study is to examine students’ difficulties to interpret the use of literal symbols in algebra. Literal symbols are used in many ways in algebra: They are used to stand for mathematical objects such as functions, matrixes, etc., but they are mostly used to represent the concept of variable. A variable is a mathematical entity that can be used to represent any number within a range of numbers and can stand on its own right in the algebraic formal system. We hypothesized that students’ prior knowledge about the way numbers are used in the context of arithmetic is likely to affect their interpretation of the use of literal symbols in algebra. Findings from previous research are consistent with this hypothesis. In the next session we discuss some of the most important relevant findings.

Research on Students’ Interpretations of the Use of Literal Symbols in Algebra Previous studies have demonstrated a series of misconceptions students have in relation to the use of literal symbols in algebra. For example, students often view literal symbols as labels for objects, i.e., they think that ‘D’ stands for David, ‘h’ for height, or they believe that ‘y’ — in the task “add 3 to 5y” — refers to anything with a ‘y’ like a yacht, a yoghurt or a yam. Alternatively, when students think of literal symbols as numbers they usually believe that they stand for a specific number only (Collis, 1975; Booth, 1984; Knuth, Alibali, McNeil, Weinberg, & Stephens, 2005; Kuchemann, 1978, 1981; Stacey & MacGregor, 1997). These misconceptions appear to be quite strong and difficult to change. Booth (1982, 1984) designed a teaching experiment specifically to encourage the acquisition of the interpretation of the literal symbol as generalized number, but found that students still faced great difficulties, even when specific instructions were given to them. Another difficulty students appear to have with the use of literal symbols in algebra is known as ‘the lack of closure’ error (Collis, 1975), which refer to students’ unwillingness to accept algebraic expressions that contain literal symbols as final answers.

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When presented with tasks such as “write the number which is twice as big as x”, students are unwilling to accept “2x” as a final answer (Booth, 1984; Firth, 1975), but insist on replacing it with a specific number. Based on such findings Booth (1984) suggested that students consider mathematics to be an empirical subject, which requires the production of numerical answers only. The ‘concatenation problem’ presents another difficulty that students have with the use of literal symbols in algebra. In arithmetic, concatenation denotes implicit addition, as both in place-value numeration, e.g., 27 equals 20 plus 7, and in mixed-numeral notation, e.g., 2 ½ means 2 plus ½ (Matz, 1980). On the contrary, in algebra, concatenation denotes multiplication; 2a means 2 times a. When asked to substitute 2 for a in 3a, students tend to interpret concatenation as it is used in arithmetic responding with 32. It is only when asked specifically to respond as ‘in algebra’ that some students reply with ‘3 times 2’ (Chalouh & Herscovics, 1988). Students’ tendency to think that literal symbols can stand for objects, names of objects, or specific numbers only was originally explained in terms of the Piagetian theory. In other words, it was argued that the students had not yet reached the stage of formal operations and, therefore, they were not capable of acquiring the concept of variable (e.g., Collis, 1975; Kuchemann, 1978, 1981). Other researchers noticed that there might be an interaction between students’ knowledge of arithmetic and their attempts to learn new content in algebra. For example, Booth (1984, 1988) suggested that students’ difficulties in algebra may be partly due to their problematic knowledge in arithmetic. As the students construct their algebraic notions on the basis of their experience in arithmetic, erroneous arithmetical knowledge can be transferred in algebra. On the contrary, Matz (1980) argued that students’ difficulties with algebra are not necessarily due to problematic knowledge of arithmetic, but rather reflect inappropriate use of the properties of arithmetic to interpret a new field in mathematics. A similar, albeit more elaborated account than that of Matz’s was offered by Kieran (1992). According to Kieran, most of the problems students have in their introduction to algebra arise because of the shift to a set of conventions different from those used in arithmetic. For example, in arithmetic, letters can be used as labels: ‘m’ can be used to denote meters, monkeys, etc., and 12 m can mean 12 meters, that is, 12 times 1 meter. But in algebra, 12 m can mean 12 times the number of meters. Algebra uses many of the symbols used in arithmetic, such as the equal sign or the addition and subtraction signs, but in different ways. For example, the equal sign is used in arithmetic as the ‘enter command’ for the result of a calculation to be announced. Kieran (1981) argues that the belief of novices in algebra that the equal sign is a “do something signal” rather than a symbol of the equivalence between the left and right sides of an equation is demonstrated by their initial reluctance to accept statements such as 4 ⫹ 3 ⫽ 6 ⫹ 1. Both Matz and Kieran suggest that students’ prior experience with arithmetic and the fact that symbols have different roles in arithmetic compared to algebra can explain some of students’ difficulties in algebra. This view is highly compatible with the conceptual change approach. The conceptual change approach provides an explanatory framework that can account for the previous findings and also make new and meaningful predictions regarding students’ difficulties with the use of literal symbols in algebra.

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The Conceptual Change Approach and the Use of Literal Symbols in Algebra When students are introduced to algebra, they face the difficult task of (a) assigning meaning to new symbols and (b) assigning new meaning to old symbols, which were used in the context of arithmetic. According to Resnick (1987), when students connect an algebraic expression1 with the ‘world of numbers’, they give a referential meaning to the algebraic expression which can affect their performance in various mathematical tasks. We suggest that a number of misconceptions that students have with the use of literal symbols could be explained to result from the inappropriate transfer of prior knowledge about numbers in the context of arithmetic, for the interpretation of literal symbols in algebra. Because students’ explanatory frameworks of number are initially tied around natural numbers (Gelman & Gallistel, 1978; Gelman, 2000), we will start by making explicit, in Table 19.1, some of the important differences between the use of natural numbers in the context of arithmetic and the use of literal symbols as variables in the context of algebra. Table 19.1: Differences between the natural numbers in the context of arithmetic and literal symbols in the context of algebra. Natural Numbers in arithmetic

Literal symbols as variables in algebra

Form Sign

1, 2, 3, … Actual sign (positive)

a, b, x, y, … Phenomenal sign (positive or negative)

Symbolic Representation

Each number in the natural number set has a unique symbolic representation — different symbols correspond to different numbers.

Each literal symbol corresponds to a range of real numbers — different symbols could stand for the same number.

Ordering —Density

Natural numbers can be ordered by means of their position on the count list. There is always a successor or a preceding number. There are no numbers between two subsequent numbers.

Literal symbols in algebra cannot be ordered by means of their position on the alphabet. There is no such thing as a successor or preceding literal symbols.

Relationship to the unit

The unit is the smallest natural number.

There is no smallest number that can be substituted for a variable, unless otherwise specified.

1 An algebraic expression is a mathematical object that contains one or more variables together with other symbols, such as numbers or signs.

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As we can see in Table 19.1, natural numbers are expressed as a series of digits (1, 2, 37, etc.), whereas variables are expressed as letters of the alphabet (a, b, x, y, etc.). Natural numbers are positive numbers and as such they have no sign attached to them. On the other hand, variables may have a phenomenal sign, which is the intuitively obvious sign that the variable appears to have as a superficial characteristic of its form. Mathematically speaking, the variable does not have a specific sign of its own. The values of a variable are determined when specific numbers are substituted for the literal symbol. Variables can stand for either positive or negative numbers independently of the phenomenal sign that is attached to them. For example, the variable which is represented by literal symbol ‘x’ can stand for positive and also for negative numbers; ‘⫺x’ can stand for positive or negative numbers as well [this happens because ⫺(⫺3)⫽3]. In the natural number set, every natural number has a unique symbolic representation and different symbols represent different numbers. For example, the symbol of the natural number ‘2’ stands only for the number ‘2’. On the other hand, in the algebraic notation, a literal symbol could stand for a range of real numbers. For example, the literal symbol ‘x’ could stand for any real number such as 2, 4.555, ½, etc., and different literal symbols could stand for the same number. The arithmetical value of a literal symbol is determined by the number that it represents and for this reason literal symbols cannot be ordered without reference to the numbers they stand for. For the same reason, a literal symbol does not have a lowest arithmetical value unless the range of numbers that it represents has a lower limit. Because literal symbols can stand for a range of real numbers they do not have any special relation with the unit. The conceptual change perspective predicts that incompatibility between the use of literal symbols in algebra and students’ prior knowledge about numbers, in particular natural numbers, could result in errors. These errors could be explained to originate in students’ tendency to use their prior experience with numbers in the context of arithmetic to interpret literal symbols in algebra. Findings of prior studies on students’ difficulties with the use of literal symbols in algebra are consistent with this view. For example, some students believe that when the literal symbol changes, then the value that it represents also changes (Booth, 1984; Kuchemann, 1981; Wagner, 1981). These students explain that ‘x ⫹ y ⫹ z’ can never equal ‘x ⫹ p ⫹ z’ because ‘different literal symbols must correspond to different numbers’. They are unwilling to accept that different symbols could stand for the same value. However, this belief is applicable to natural numbers, where each number has a unique symbolic representation and where different symbols stand for different numbers. There is also evidence that some students associate literal symbols with natural numbers, in the sense that they respond as if there is a correspondence between the linear ordering of the alphabet and that of the natural numbers system (Booth, 1984; Stacey & MacGregor, 1997; Wagner, 1981). For example, they tend to assign the numerical value 8 to the literal symbol ‘h’ (used to represent a boy’s height), because ‘h’ is the eighth letter in the alphabet. Or they say that 10 ⫹ h ⫽ 18, and 10 ⫹ h ⫽ R, because the tenth letter after ‘h’ in the alphabet is ‘R’. In this study, we hypothesized that prior experience with numbers, in particular with natural numbers, would result in a strong tendency on the part of the students to interpret literal symbols as standing mostly for natural numbers. We also hypothesized that it would

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be difficult for students to understand that variables have a phenomenal sign, which is not the actual sign of the values they represent. This hypothesis is based on the fact that in the context of arithmetic, no sign implies positive value. This is a characteristic of natural numbers that holds for all positive numbers. When students are introduced to negative numbers they learn that the presence of the negative sign means ‘negative value’. We thus predicted that students would tend to interpret ‘x’ to stand for positive numbers and ‘⫺x’ to stand for negative numbers, a phenomenon also noted by researchers such as Chiarugi, Fracassina, and Furinghetti (1990) and Vlassis (2004). These hypotheses were investigated in a series of empirical studies. Previous work by Christou and Vosniadou (2005) investigated some of the above-mentioned hypotheses. They gave 8th- and 9th-grade students a questionnaire (Questionnaire A, QR/A), which asked them to write down the numbers they thought could be assigned to algebraic expressions such as ‘a’, ‘⫺b’, ‘4g’, ‘a/b’, ‘d+d+d’, etc. The results showed that only about one-third of the students gave the mathematically correct response, namely that ‘all numbers can be assigned to each algebraic expression’. When asked, for example, to write down numbers they thought could be assigned to the algebraic expression ‘a’, 66% of the students responded only with natural numbers. Natural numbers were mostly used in the remaining questions as well. In most of their responses students substituted only natural numbers for the literal symbols of the given algebraic expression and maintained the form of the algebraic expression: fraction-like in the case of ‘a/b’, multiples of 4 in the case of ‘4g’, natural numbers larger than 3 in the case of ‘k + 3’, etc. When asked to write down numbers that can be assigned to ‘⫺b’, 72% of the students responded only with negative whole numbers (⫺1, ⫺2, ⫺3, etc.). Again, we interpreted these responses to reflect students’ tendency to substitute only natural numbers for the literal symbol ‘b’ and to maintain the phenomenal negative sign of the given algebraic expression ‘⫺b’. Very few students answered the question by providing numbers other than natural numbers, such as decimals, fractions, negatives, or real numbers. Students in this questionnaire were affected by the phenomenal sign of the algebraic expressions in the sense that they maintained it when they substituted numbers for the literal symbols. The majority of the students assigned only positive numbers to the positive-like algebraic expressions and negative numbers to the negative-like algebraic expression ‘⫺b’. It could, therefore, be objected that the students responded with the first numbers that came to their mind, in full knowledge that their answer was correct, since all values can be assigned to any algebraic expression. However, these responses differ in important ways from the responses expected from a mathematically sophisticated participant. In order to further explore this possible hypothesis, we designed a second open-ended questionnaire (Questionnaire B, QR/B) in which the students were asked to write down numbers that they thought could not be assigned to a set of algebraic expressions. The set of algebraic expressions used was the same as the one used in QR/A. Unlike QR/A, in QR/B there is only one correct response — namely, that “all numbers can be assigned to each algebraic expression” or that “there are no such numbers that cannot be assigned to each algebraic expression”. The results obtained in QR/B showed that again only about one-third of the students gave this mathematically correct response. About half of the students said that negative

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whole numbers (⫺1, ⫺2, ⫺3, etc.) could not be assigned to the algebraic expression ‘a’ and that natural numbers (1, 2, 3, etc.) could not be assigned to ‘⫺b’. Similarly in the remaining algebraic expressions, ‘4g’, ‘a/b’, ‘d⫹d⫹d’, and ‘k ⫹ 3’, the students tended to respond by replacing literal symbols only with negative whole numbers, while maintaining the form of the algebraic expression. For example, the majority of the students gave numbers such as (⫺1) ⫹ (⫺1) ⫹ (⫺1), or (⫺2) ⫹ (⫺2) ⫹ (⫺2), as numbers that could not be assigned to the algebraic expression ‘d⫹d⫹d’, and numbers such as (⫺2)/(⫺3), and (⫺3)/(⫺4) as numbers that could not be assigned to the fraction-like algebraic expression ‘a/b’. We suggest that these students interpreted the literal symbols to stand only for natural numbers and thus they thought that the additive inverses of natural numbers (the negative whole numbers) could not be substituted for the literal symbols. In QR/B students appeared again to be sensitive to the phenomenal sign of the algebraic expressions which they seem to have interpreted as the actual sign of the values that they represented. The majority of the students responded to the questions by using numbers with the opposite sign of the phenomenal sign of the algebraic expressions used. They said that negative numbers could not be assigned to the positive-like algebraic expressions and that positive numbers could not be assigned to the negative-like algebraic expression ‘⫺b’. In order to further examine students’ tendencies to maintain or change the phenomenal sign of the given algebraic expression as a function of the questionnaire type, students’ responses were subjected to a one-way ANOVA. Responses that maintained the phenomenal sign were marked as 1, responses that changed the phenomenal sign were marked as 2, and mathematically correct responses were marked as 3. The results showed main effects for questionnaire type [F(1, 281) ⫽ 6.126, p ⬍ 0.05], which were due to the fact that students maintained the phenomenal sign in QR/A but changed it in QR/B. We interpreted students’ sensitivity to the phenomenal sign of the algebraic expression to be intricately related to their belief that literal symbols in algebra stand only for natural numbers. Students think of ‘⫺7x’, for example, as always negative and ‘7x’ as always positive because they tend to think of the literal symbol ‘x’ as only standing for natural numbers. In another ANOVA we examined the effect of students’ tendency to substitute only natural numbers vs non-natural numbers for the literal symbols themselves, independently of the sign of the algebraic expression, in the two questionnaires. Responses that substituted literal symbols only with natural numbers were marked as 1, responses that used nonnatural numbers were marked as 2, and mathematically correct responses were marked as 3. The results showed no significant differences between the two questionnaires. In both questionnaires, students tended to substitute mostly natural numbers for the literal symbols themselves and appeared unwilling to also present any fractions, decimal numbers, or real numbers under any condition. This finding was consistent with the theoretical hypothesis of the research, namely, that students tend to consider literal symbols in algebra to stand for natural numbers only. A possible criticism of our experiment could be that the students provided natural numbers not because they thought that these are the only ‘correct’ substitutions for the literal symbols, but because these are the most common numbers, used both at school and in everyday situations. In school mathematics, natural numbers are used in most of the pro blems students are asked to solve and the solution to these problems, most of the time,

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involves natural numbers only. Thus, it could be argued that students responded with natural numbers because they thought that this was what they should do and not because they did not know that it is possible to substitute the literal symbols with non-natural numbers. In order to further explore this possibility we designed another study in which we used a forced-choice questionnaire. The advantage of a forced-choice questionnaire, in comparison to the open-ended ones used earlier, is that it presents students with specific alternatives that can include both natural and non-natural numbers. It can thus provide a more rigorous test of the hypothesis that students interpret literal symbols in algebra to stand only for natural numbers.

The Present Study In this study, we constructed a forced-choice questionnaire (Questionnaire C, QR/C) that presented students with a set of specific alternatives for the same algebraic expressions used in Questionnaires A and B described earlier. These alternatives included both natural and non-natural numbers such as negative integers, positive and negative fractions, and positive and negative decimals. The correct response, namely that ‘all numbers can be assigned’, was one of the alternatives. The students were asked to choose the alternatives that they thought could not be assigned to the given algebraic expression. We used the negative substitution form (could not be assigned) because it is only in this condition that we can say with certainty that only the mathematically correct response applies. In the positive substitution condition all responses can be considered to be technically correct. If students indeed interpret literal symbols to stand only for natural numbers, they should exclude some numbers from the given set, such as fractions, decimals, etc., depending on the form of the given algebraic expression. For example, given the algebraic expression ‘a/b’, they should think that only positive fractions could be assigned to it, and thus that whole numbers or even decimal numbers could not be assigned. Alternatively, we would expect that in the case of ‘⫺b’, they would tend to exclude all the positive numbers of the given set of alternatives.

Method Participants Thirty-four children participated in this study. There were 8th and 9th graders (mean age 14.5 years old) from two middle-class high schools in Athens. All of them completed the forced-choice questionnaire (QR/C). Materials QR/C consisted of the following six algebraic expressions: Q1: a, Q2: ⫺b, Q3: 4g, Q4: a/b, Q5: d⫹d⫹d, and Q6: k ⫹ 3. For each algebraic expression, the students were asked

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to choose from a given set of alternative numbers those they thought could not be assigned to them. The set of alternatives consisted of 11 number choices, which included positive and negative fractions, positive and negative decimals, positive and negative integers. The twelfth alternative was always the correct response, namely, that all numbers can be assigned to each algebraic expression. An example is shown in Table 19.2. Procedure The following instructions were given to the students: “In algebra, we use literal symbols (such as a, b, x, y, etc.) mostly to stand for numbers. In this questionnaire we use such symbols. Read the following questions carefully. If you think there are some numbers among the given alternatives that cannot be assigned to the given algebraic expressions, please place a circle around them. You may choose more than one numbers if you wish”. Students completed the questionnaire in the presence of one of the experimenters and their mathematics teacher in their classroom.

Results Based on the findings from our previous studies, we created three main categories of responses namely ‘NN/1’, ‘NN/2’, and ‘Phenomenal sign’. The category ‘NN/1’ attempted to capture all responses that reflected the belief that literal symbols stand for natural numbers only (NN belief). For example, students who chose all numbers from the given alternatives except the positive fractions for the algebraic expression ‘a/b’, would be placed in this category. The category ‘NN/2’ captured responses that included some but not all alternatives predicted by the NN belief. For example, in the case of ‘a/b’, students could choose all the integers of the given set of alternatives. These responses would be placed in this category. In the ‘Phenomenal sign’ category we placed student’s responses that included all numbers with the opposite sign from the phenomenal one. For example, in this category we placed Table 19.2: An example of the way in which questions were posed in the forced choice questionnaire. Are there some numbers among the following alternatives that you think cannot be assigned to 4g? 2 i) ⫺ ᎏ 3

a) 6

e) 6.74

b) 2

5 f) ᎏ 7

c) ⫺0.25

g) 8

k) 2.333

d) –3

h) 4

l) No, all numbers can be assigned to it.

j) 8

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students’ responses only with negative numbers in the case of ‘a/b’, or only with positive numbers in the case of ‘⫺b’. There were three additional categories of responses: ‘Non-systematic’, ‘No response’, and ‘Correct’. The ‘Non-systematic’ category was used for all non-systematic responses. The ‘No response’ category included null responses, and the ‘Correct’ category represented the correct alternative. All responses were categorized by one of the experimenters, and a second rater scored half of the responses using the same criteria. The agreement of the categorization was 96%. All disagreements were discussed until consensus was achieved. Tables 19.3, and 19.4 in more detail, presented the frequencies and the percentages of each category of responses. Table 19.3 presents the total percentage of students’ responses to all the questions for each response category. Only 18.6% of the students’ responses represented the ‘correct response despite the fact that it was an explicit alternative in all questions. One-third of students’ responses (30.3%) were affected by the NN belief in the strict (22%, NN/1) or in a more differentiated way (8.3%, NN/2). One-fourth of students’ responses (25.4%) were affected by the phenomenal sign of the algebraic expressions. There was a large percentage of responses in the non-systematic category (16.1%) that could be explained by the complexity and counterintuitiveness of the questions in QR/C, the fact that they were expressed in the negative form, and, finally, the forced choice nature of the questionnaire. Previous studies have also shown that non-systematic responses appear more frequently in forced-choice questionnaires (see Vosniadou, Skopeliti, & Ikospentaki, 2004) than in open-ended ones. Table 19.4 presents in greater detail the frequencies and percentages for students’ responses in each category for the 6 algebraic expressions, together with examples of the type of numbers/responses for each category. In the case of the algebraic expression ‘a’, only about one-third of the students (29.4%) responded by selecting the correct response, namely that ‘all values can be assigned to it’. The majority of the responses reflected the belief that the literal symbol ‘a’ stands only for positive numbers (38.2%). Another 20.5% of the responses indicated that numbers other than natural numbers could not be assigned to ‘a’. In the remaining algebraic expressions, such as ‘4g’, ‘k ⫹ 3’, or ‘a/b’, students’ responses appeared to be slightly different. The majority of the students were affected

Table 19.3: Percentages of students’ responses to the combined questions. Questionnaire C (choose, from the given set of numbers, those that you think cannot be assigned to the given algebraic expressions) Categories

Correct

NN/1

NN/2

Phenomenal Sign

Nonsystematic

No Response

Mean percentage

18.6%

22% 30.3%

8.3%

25.4%

16.1%

9.3%

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more by the NN belief than by the phenomenal sign. For example, in the case of ‘a/b’, 32.3% of the responses were placed in the category NN/1, and 11.7% in the category ‘Phenomenal sign’. In the remaining algebraic expressions the results were similar with the exception of ‘⫺b’. In this case, the results were quite different because of the presence of the negative sign. About half of the students’ responses (52.9%) indicated that students interpreted this expression as standing for negative numbers only. Table 19.4: Frequencies, percentages, and examples of students’ responses to each question. Questionnaire C (choose, from the given set of numbers, those that you think cannot be assigned to the given algebraic expressions) Questions

Correct

NN/1

10 (29.4%)

All but NN 7 (20.5%)

Decimals, Negatives fractions 3 13 (8.8%) (38.2%) Natural numbers

Positives

6 (17.6%)

All but negative wholes 2 (5.9%)

2 (5.9%)

18 (52.9%)

7 (20.5%)

All but NN 9 (26.5%)

Q1: a

Q2: ⫺b

Q3: 4g

NN/2

Phenomenal Sign

5 (14.7%)

Integers

Negatives

6 (17.6%)

All but positive fractions 11 (32.3%)

2 (5.9%)

4 (11.7%)

Fractions Negatives

4 (11.7%)

All but NN 10 (29.4%)

Fractions Negatives

5 (14.7%)

All but NN 6 (17.6%)

Q5: d⫹d⫹d

Q6:  + 3

1 (2.9%)

⫺

5 (14.7%)

1 (2.9%)

10 (29.4%)

3 (8.8%)

5 (14.7%)

6 (17.6%)

6 (17.6%)

5 (14.7%)

6 (17.6%)

4 (11.7%)

Negatives ⫺

a Q4: ᎏ b

NonNo systematic Response

3 (8.8%)

7 (20.5%)

6 (17.6%)

6 (17.6%)

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Discussion The results from the present study are consistent with the findings from our previous study (Christou & Vosniadou, 2005), and support the hypothesis of the conceptual change approach that students’ interpretation of the use of literal symbols in algebra is strongly influenced by their experience with numbers, in particular natural numbers, in the context of arithmetic. The conclusion was based on two sources of evidence: first, students tended to substitute only natural numbers for the literal symbols of the algebraic expressions, and second, students interpreted the phenomenal sign of the algebraic expressions as the sign of the numbers that they represent. This was the case, despite the fact that students were taught in school that each literal symbol corresponds to any real number. Students interpreted algebraic expressions such as ‘k ⫹ 3’ or ‘d ⫹ d ⫹ d’ to stand mostly for positive numbers and believed that negative numbers cannot be assigned to them. The greatest influence of the phenomenal sign appeared in the case of the negativelike algebraic expression ‘⫺b’, which the majority of the students interpreted as standing for negative numbers only. This finding is consistent with results from Vlassis’ (2004) research with polynomials, where students appeared to consider the minus sign at the beginning of a polynomial as the sign of a negative number. We interpret students’ belief that the phenomenal sign of the algebraic expression is the actual one of the numbers that it represents to originate from prior experience with arithmetic. In the context of arithmetic, numbers with “no sign” means numbers with “positive value”, whereas numbers with “negative sign” means numbers with “negative value”. The transfer of this knowledge in the area of algebra causes a misconception, which is strong even in the case of the older students and constrains the acquisition of more advanced mathematical concepts such as, for example, the absolute value of a number. For any real number a, the absolute value of a, denoted |a|, is always a positive number, so |a| is equal to a, if a ⱖ 0 or to ⫺a, if a ⬍ 0. As the students are affected by the phenomenal sign of the algebraic expressions they do not think of ‘–a’ as a symbol that could possibly stand for a positive number when ‘a’ stands for a negative one (see Chiarugi et al., 1990). In order for students to understand that the phenomenal sign of an algebraic expression is not the sign of the numbers that it represents, they need to reorganize their prior knowledge about numbers as shaped in the context of arithmetic. Furthermore, students’ prior experience with numbers in the context of arithmetic constrains their understanding of the generalized nature of a literal symbol, i.e., as a variable that stands for any real number. The present findings agree with the previous research which shows that the initial understanding of number as natural number may hinder the acquisition of more advanced mathematical concepts, as in the case of fractions, rational numbers, or algebraic rules, etc. (Gelman, 2000; Resnick et al., 1989; Stafylidou & Vosniadou, 2004; Vamvakoussi & Vosniadou, 2004). Resnick (1987) has argued that algebraic expressions can take their meaning from their position in the formal system of algebra. Over and above, there is also a referential meaning, which algebraic expressions take either from the situations in which relations among quantities and actions upon quantities play a role or from its connection with the ‘world of numbers’. She noted that student’s capability to assign a referential meaning to the algebraic expressions affects their performance in algebraic transformation tasks. Focusing on

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the referential meaning that can be assigned to the algebraic expressions with its linkage to the world of numbers, we found that many students tend to think that literal symbols represent only natural numbers and as a consequence they have a very restricted range of numbers from which the algebraic expressions take their referential meaning. This affects students’ performance in many mathematical tasks such as, for example, when students have to estimate the sign or the value of an algebraic expression in situations where the monotony of a function is tested, or in the case of a radicand. We believe that we have provided some evidence that the conceptual change theoretical framework can help us systematize students’ errors in interpreting the use of literal symbols in algebra. The results of the present study can also provide useful information for the design of curricula and for instruction. It is important for teachers of algebra as well as for the curriculum designers to be familiar with students’ beliefs and the possible reasons for their mistakes when they use literal symbols in algebra, as well as in other domains for example physics, chemistry, etc. (see, e.g., Heck, 2001; Sherin, 2001). Greer (1994, 2005) has suggested various devices for expanding arithmetic operations beyond natural numbers. He proposed to give students mathematical tasks, which use non-natural numbers as factors, such as the equation 2.67x2 – 3.86x – 12.23 = 0, as this could help them extend their conceptual fields beyond the natural numbers. Some researchers investigate the implications of introducing algebraic thinking in elementary school (Carraher, Schliemann, & Brizuela, 2001). Of course, more detailed empirical research is needed to further investigate students’ difficulties and the effect of specific instructional innovations before introducing them in schools.

Acknowledgments The research was supported by a grant from the Program ‘‘Pythagoras — Support of University Research Groups” (EPEAEK II), 75% from European Social Funds and 25% from National Funds.

References Booth, L. R. (1982). Developing a teaching module in beginning algebra, Proceedings of the 6th conference of the international group for the psychology of mathematics education, Antwerp (pp. 280–285). Booth, L. R. (1984). Algebra: Children’s strategies and errors. Windsor, Berkshire: NFER-Nelson. Booth, L. R. (1988). Children’s difficulties in beginning algebra. In: The ideas of algebra, K-I2, 1988 NCTM Yearbook (pp. 20–32). Reston, Virginia: National Council of Teachers of Mathematics. Carpenter, T. P., Corbitt, M. K., Kepner, H. S. Jr., Lindquist, M. M., & Reys, R. E. (1981). Results from the second mathematics assessment of the national assessment of educational progress. Reston, VA: National Council of Teachers of Mathematics. Carraher, D., Schliemann, A., & Brizuela, B. (2001). Can young students operate on unknowns? In: M. van den Heuvel-Panhuizen (Ed.), Proceedings of the 25th conference of the international group for the psychology of mathematics education, Utrecht University, Utrecht (Vol. 1, pp. 130–140).

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Chalouh, L., & Herscovics, N. (1988). Teaching algebraic expressions in a meaningful way. In: A. F. Coxford (Eds), The Ideas of Algebra, K-12 (1988 Yearbook) (pp. 33–42). Reston, VA: National Council of Teachers of Mathematics. Chiarugi, I., Fracassina, G., & Furinghetti, F. (1990). Learning difficulties behind the notion of absolute value. Proceedings of the 13th conference of the International Group for the Psychology of Mathematics Education, PME, Mexico (Vol 3, pp. 231–238). Christou, K. P., & Vosniadou, S. (2005). How students interpret literal symbols in algebra: A conceptual change approach. In: B. G. Bara, L. Barsalou, & M. Bucciarelli (Eds), Proceedings of the cognitive science 2005 conference Stressa, Italy (pp. 453–458). Collis, K. F. (1975). The development of formal reasoning. Newcastle, Australia: University of Newcastle. Firth, D. E. (1975). A Study of Rule Dependence in Elementary Algebra. Unpublished master’s thesis. University of Nottingham, England. Fischbein, E., Deri, M., Nello, M., & Marino, M. (1985). The role of implicit models in solving problems in multiplication and division. Journal of Research in Mathematics Education, 16, 3–17. Gelman, R. (2000). The epigenesis of mathematical thinking. Journal of Applied Developmental Psychology, 21, 27–37. Gelman, R., & Gallistel, C. R. (1978). The child’s understanding of number. Cambridge, MA: Harvard University Press. Greer, B. (1994). Extending the meaning of multiplication and division. In: G. Harel, & J. Confrey (Eds), The development of multiplicative reasoning in the learning of mathematics (pp. 61–85). Albany, NY: SUNY Press. Greer, B. (2005). Designing for conceptual change. In: J. Novotná, H. Moraová, M. Krátká, & N. Stehlíková (Eds), Proceedings of the 30th conference of the International Group for the Psychology of Mathematics Education, PME, Prague. (Vol. 1, pp. 171–172). Hart, K. M. (1981). Children’s Understanding of Mathematics: 11–16. London: John Murray. Heck, A. (2001). Variables in computer algebra, mathematics, and science. International Journal of Computers in Mathematics Education, 8(3), 195–221. Kieran, C. (1981). Concepts associated with the equal symbol. Educational Studies in Mathematics, 12, 317–326. Kieran, C. (1992). The learning and teaching of school algebra. In: D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 390–419). New York: Macmillan. Knuth, E. J., Alibali, M. W., McNeil, N. M., Weinberg, A., & Stephens, A. C. (2005). Middleschool understanding of core algebraic concepts: Equivalence and variable. International Reviews on Mathematical Education, 37(1), 68–76. Kuchemann, D. (1978). Children’s understanding of numerical variables. Mathematics in School, 7(4), 23–26. Kuchemann, D. (1981). Algebra. In: K. Hart (Ed.), Children’s understanding of mathematics: 11–16 (pp. 102–119). London: John Murray. Matz, M. (1980). Building a metaphoric theory of mathematical thought. Journal of Mathematical Behavior, 3(1), 93–166. Resnick, L. B. (1987). Understanding algebra. In: J. A. Sloboda, & D. Rogers (Eds), Cognitive process in mathematics (pp. 169–203). Oxford: Clarendon Press. Resnick, L. B., Nesher, P., Leonard, F., Magone, M., Omanson, S., & Peled, I. (1989). Conceptual bases of arithmetic errors: The case of decimal fractions. Journal for Research in Mathematics Education, 20, 8–27. Sackur, C. (1995). Blind Calculators in algebra: Write false interviews. In: E. Cohors-Fresenborh (Ed.), Proceedings of the first European Research Conference on Mathematics Education (ERCME ’95) (pp. 82–85). Osnabruck, Germany.

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Sherin, B. L. (2001). How students understands physics equations. Cognition and Instruction, 19(4), 476–541. Stacey, K., & MacGregor, M. (1997). Ideas about symbolism that students bring to algebra. The Mathematics Teacher, 90(2), 110–113. Stafylidou, S., & Vosniadou, S. (2004). Students’ understanding of the numerical value of fractions: A conceptual change approach. Learning and Instruction, 14, 503–518. Vamvakoussi, X., & Vosniadou, S. (2004). Understanding the structure of the set of rational numbers: A conceptual change approach. Learning and Instruction, 14, 453–467. Verschaffel, L., & Vosniadou, S. (Guest Eds). (2004). The conceptual change approach to mathematics learning and teaching. Special Issue of Learning and Instruction, 14. Vlassis, J. (2004). Making sense of the minus sign or becoming flexible in ‘negativity’. Learning and Instruction, 14, 469–484. Vosniadou, S. (1999). Conceptual change research: State of the art and future directions. In: W. Schnotz, S. Vosniadou, & M. Carretero (Eds), New Perspectives on Conceptual Change (pp. 3–13). Amsterdam: Elsevier. Vosniadou, S., & Brewer, W. F. (1992). Mental models of the earth: A study of conceptual change in childhood. Cognitive Psychology, 24, 535–585. Vosniadou, S., Skopeliti, I., & Ikospentaki, K. (2004). Modes of knowing and ways of reasoning in elementary astronomy. Cognitive Development, 19, 203–222. Wagner, S. (1981). Conservation of equation and function under transformations of variables. Journal for Research in Mathematics Education, 12, 107–118.

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Chapter 20

Teaching for Conceptual Change: The Case of Infinite Sets Pessia Tsamir and Dina Tirosh

Introduction The conceptual change approach has been widely used to interpret students’ solutions to science problems in a series of developmental studies referring to science education (e.g., Carey, 1985; Posner, Strike, Hewson, & Gertzog, 1982; Vosniadou & Brewer, 1992; Wiser & Amin, 2001). This approach was mainly used to explain knowledge acquisition in specific domains, while describing the importance of reorganization of existing knowledge structures in some processes of learning. Vosniadou, Ioannides, Dimitrakopoulou, and Papademetriou (2001) argued that intuitive knowledge of the physical world is necessary for functioning in the world, and that learners tend to develop various beliefs and presuppositions related to scientific topics. However, scientific explanations of the physical world often run counter to fundamental principles of intuitive knowledge, which are confirmed by our everyday experience. Consequently, in the process of learning, new information interferes with prior knowledge, resulting in the construction of synthetic models (or misconceptions), and this shows that knowledge acquisition is a gradual process during which existing knowledge structures are revised slowly. Similarly, when studying mathematics, in the course of accumulating mathematical knowledge, the student goes through successive processes of generalization, while also experiencing the extension of various mathematical systems. For instance, the concept of number, a central concept in mathematics, is introduced very early on in primary school via the system of natural numbers. Then, gradual transitions occur beginning with integers, through rational numbers, irrational numbers and real numbers, and concluding with the system of complex numbers presented in the upper grades of high school. The move from one number system to a wider one preserves some numerical characteristics, adds some others, while yet others are lost. For example, the transition from natural numbers to integers enables one to solve a problem like 5–7 (closure under subtraction). Yet, at the same

Reframing the Conceptual Change Approach in Learning and Instruction Copyright © 2007 by Elsevier Ltd. All rights of reproduction in any form reserved. ISBN: 0-08-045355-4

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time it becomes impossible to generalize that subtraction “always makes smaller” and the system no longer has a smallest number. Awareness of the changes caused by the enlargement of mathematical systems and the ability to identify the variant and invariant elements under a specific transition are important factors in the growth of mathematical knowledge. However, research findings clearly indicate that learners tend to attribute all properties of a specific domain of numbers to a more general one (e.g., regarding rational numbers: Greer, 1992; Hart, 1981; Klein & Tirosh, 1997; regarding decimals: Moloney & Stacey, 1996; Putt, 1995; regarding negative numbers: Hefendehl, 1991; Streefland, 1996; regarding irrational numbers: Fischbein, Jehiam, & Cohen, 1995). This general description suggests that the conceptual change framework, much like other theories that relate to intuitive knowledge (e.g., Fischbein, 1987; Tall & Vinner, 1981), may be suitable for analyzing the development of students’ mathematical knowledge. Indeed, in recent years, several researchers have attempted to explore the promises of this theoretical framework to mathematics learning and teaching. A special issue devoted to the conceptual change approach to mathematics learning and teaching was recently published in Learning and Instruction (Vosniadou & Verschaffel, 2004). Most studies explored the conceptual changes involved in the transition from one number system to a wider one (e.g., Merenluoto & Lehtinen, 2004; Stafylidou & Vosniadou, 2004; Vlassis, 2004; Vamvakoussi & Vosniadou, 2004; see also, Hartnett & Gelman, 1998). In this chapter we discuss the applicability of the conceptual change approach to the learning and teaching of the Cantorian Set Theory. We focus on one major aspect of this theory: the equivalency of infinite sets. The terms “comparing infinite sets,”1 “comparing infinite quantities,” and “determining the equivalency of infinite sets” are used interchangeably to account for the comparison of the cardinalities of these sets. The Cantorian Set Theory is the most commonly used theory of infinity today. Yet students face great difficulties in acquiring various properties of the equivalency of infinite sets (Borasi, 1985; Duval, 1983; Fischbein, Tirosh, & Hess, 1979; Lakoff & Nunez, 2000; Tall, 1980, 1992, 2001; Tirosh, 1991; Tsamir, 1999). It was reported that when asked to compare the numbers of elements in two infinite sets students at different grade levels used methods that are adequate only for the comparison of the number of elements in finite sets. For example, students expect that the number of elements in a set, which is the union of two distinct, non empty sets, is larger than that of the number of elements in each of these sets. This, however, is true for finite sets, but not for infinite ones. It seems evident from the related research findings and also from the historical development of the Cantorian Set Theory that the acquisition of various aspects of the theory in general and the equivalency of infinite sets, in particular, necessitates radical reconstruction. In the course of the last twenty years we designed and evaluated several methods of teaching the Cantorian Set Theory (Tirosh, 1991; Tirosh, Fischbein, & Dor, 1985; Tirosh & Tsamir, 1996; Tsamir, 1999, 2003a; Tsamir & Tirosh, 1999). These instructional practices were inspired by Fischbein’s (1987) theory of intuition in mathematics and science. The major principles that guided the development of these instructional practices (as described, for instance, in Tirosh, 1991) were: identifying the intuitive criteria students use

1

The term “comparing infinite sets” is an abbreviation to “comparing the number of elements in two infinite sets”

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to compare infinite quantities; raising students’ awareness of inconsistencies in their own thinking; discussing the origins of students’ intuitions about infinity; progressing from finite to infinite sets; stressing that it is legitimate to wonder about infinity; emphasizing the relativity of mathematics; and strengthening students’ confidence in the new definitions. We evaluated the impact of traditional courses with little or no emphasis on students’ intuitive tendencies to overgeneralize from finite to infinite sets, and of courses that were developed in line with the principles listed above, on high-school students and on prospective mathematics teachers’ intuitive and formal knowledge of Cantorian Set Theory. Our findings in the different studies indicate that instruction that implemented these principles led to the promotion of the reconstruction of knowledge structures (Tirosh, 1991; Tsamir, 1999). These interventions promoted learners’ awareness of the differences between finite and infinite systems, and of the contradictions that result from interchangeably applying different criteria when comparing infinite sets. Looking at these instructional interventions through different lenses could provide additional insights into their pros and cons. In this chapter we show that the instructional design principles deriving from the conceptual change approach (as presented by Vosniadou et al., 2001) offer a valuable framework for analyzing and reflecting on instructional interventions in mathematics. More specifically, we focus on the mathematical notion of equivalency of infinite sets, using the instructional design recommendations of the conceptual change approach to analyze and reflect on related learning environments.

Using the Conceptual Change Approach to Reflect on Learning Environments: The Case of Infinite Sets In 2001, Vosniadou, Ioannides, Dimitrakopoulou and Papademetriou described eight recommendations for designing learning environments to promote conceptual change in science. In the following section we show that these principles could be applied for reflecting on learning environments that aimed at promoting learners’ grasp of the equivalency of infinite sets. For this purpose we revisit examples from instructional interventions that we developed, reflecting on them in light of Vosniadou et al.’s recommendations. Each of the following subsections is dedicated to one of the recommendations. Taking into Consideration Students’ Prior Knowledge The realization that students do not come to school as empty vessels but have representations, beliefs and presuppositions about the way the physical world operates that are difficult to change has important implications for the design of science instruction. (Vosniadou et al., 2001, p. 392) Vosniadou et al. emphasize the need, when designing instruction, to consider the ways in which students see the physical world. They mention two sources where this information might be found: the accumulated data on students’ conceptions and ways of thinking,

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published in science education literature, and the examination of students’ ideas prior to instruction. This recommendation indicates the importance of a body of knowledge regarding students’ conceptions, and their ways of thinking about concepts whose development could necessitate radical reconstruction. In the case of equivalency of infinite sets such a body of knowledge exists. Research in the field of mathematics education indicates that the transition from the comparison of finite sets to the comparison of infinite sets (i.e., the comparison of the number of elements in these sets) is problematic for many students (e.g., Borasi, 1985; Duval, 1983; Falk, Gassner, Ben Zoor, & Ben Simon, 1986; Fischbein et al., 1979; Martin & Wheeler, 1987; Sierpinska, 1987; Tall, 1980). When asked to compare the number of elements in two infinite sets, students used different methods in their comparisons, inevitably leading to contradictions. For instance, when asked to compare the number of elements in the set of natural numbers and the set of positive, even numbers, students used inclusion, 1:1 correspondence, and single infinity as their bases for comparison. The use of inclusion (i.e., justifying the claim that two sets consist of different numbers of elements by stating that a proper subset has less elements than the set itself) led them to conclude that the infinite sets have an unequal number of elements. When using 1:1 correspondence (i.e., justifying the claim that two sets have the same number of elements by pairing each element of one set with a unique element of the other with no “leftovers”), or single infinity (i.e., all infinite sets have the same number of elements — infinity) they concluded that the number of elements was equal. Students often accepted these two contradictory solutions to the same problem as both valid (e.g., Tirosh & Tsamir, 1996). In our studies, we found that consulting the research in mathematics education on learners’ conceptions of the equivalency of infinite sets is imperative for instruction related to this complex concept. Our decision to start from reflecting on strategies that are often implicitly used to compare the number of elements in finite sets, and only then to move to infinite sets was based on the consistent observation, in various studies, that students tend to apply methods applicable only for finite sets, to infinite ones. Some examples will be presented later on. Consulting the accumulated data on students’ conceptions and ways of thinking, as described in the mathematics education literature, is one component of prior knowledge that Vosniadou et al. (2001) recommended to consider when designing learning environment. The other component of prior knowledge that Vosniadou et al. (2001) discussed — the examination of the ideas held by the students prior to instruction — was also found to be important. For example, in a pretest conducted with 11th graders enrolled in the same school, significant differences were found between the methods that students from two classes used for comparing infinite sets. In one class, almost all students argued that “all infinite sets have the same number of elements — infinity,” while in another class most students tended to use inclusion considerations. Our search for an explanation of this unexpected diversity led us to examine their curriculum. It was found that at the time of the pretest, the first class was learning about limits, addressing infinity as “a single sized entity,” while the others were not. Such differences were taken into account when designing the learning environments for each of these classes (Tsamir, 1990).

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Facilitating Metaconceptual Awareness To help students increase their metaconceptual awareness it is necessary to create learning environments that make it possible for them to express their representations and beliefs. This can be done in environments that facilitate group discussion and the verbal expression of ideas. It is also important to create learning environments that make it possible for students to express their internal representations of phenomena, to compare them with those of others. (Vosniadou et al., 2001, p. 392) Vosniadou et al. provide an example of the design and implementation of a learning environment to teach mechanics to 5th graders. The class was divided into small groups in which the students were first asked to write individual answers in their notebooks, then to discuss the written answers with their group peers, and finally, to agree on a “group response” to be presented in a class discussion. During class discussion, students were invited to challenge the answers of their own and other groups’ representatives. The authors commented that there were often unresolved disagreements and an experiment was conducted in the class, using everyday materials, to provide an objective answer to the problem being investigated. Similar instructional sequences were included in our intervention studies. Students were often first asked to individually describe in writing the methods they used to determine whether two sets have the same number of elements. Then, they were asked to discuss their solutions in small groups in an attempt to convince their peers, before reporting on their solutions to the entire class. During the class discussion, various opinions were raised and discussed. For example, in the following activity, we aimed at promoting prospective teachers’ awareness of the different criteria that they used for the comparison of finite sets. The development of this activity was based on our findings, which students tend to argue, that counting is the only strategy for comparing finite sets (the activity is also described in Tsamir, 1999). Comparing finite sets. The activity consisted of three stages (Appendix A). Stage 1 aimed at exposing the prospective teachers’ tendency to claim that counting the number of elements in each finite set and comparing the two resulting numbers is the only method for comparing sets. For this purpose two disjoint, finite sets were introduced, both with small numbers of elements. Stage 2 aimed at raising the prospective teachers’ awareness of various methods that they actually use for the comparison of the number of elements in two finite sets. Here, five problems were presented, each intended to elicit a different method for the comparison of finite sets: Problem 1 — 1:1 correspondence; Problem 2 — counting; Problem 3 — inclusion; Problem 4 — intervals (comparing distances between “consecutive elements” in the two sets) and 1:1 correspondence; and Problem 5 — intervals. Stage 3 had several aims, including: (a) to practice reflection upon the methods used to compare the number of elements in the sets; (b) to conclude that comparisons are possible even when the elements cannot be counted; (c) to realize that counting, even when applicable, is not always

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the best method for comparing; and (d) to realize that counting is actually creating 1:1 correspondence between a set and a specific subset of the natural numbers. In our study, we addressed mathematics prospective teachers for secondary school, and indeed, all participants individually solved the comparison in Stage 1 by counting. When asked to reach a generalized conclusion, during small-group discussions and later on in class discussion, they responded that counting is the only way to compare finite sets. Like in Stage 1, the problems in Stage 2 were first solved individually. These problems could be solved by various methods. The participants did in fact use all four available methods to solve the problems. They listed these methods in response to the first assignment in the reflection-stage (Stage 3) (counting, mainly in Problem 2; 1:1 correspondence, mainly in Problem 1; inclusion, mainly in Problem 3; and intervals, mainly in Problems 4 and 5). When all the methods used were presented in a single table, two findings emerged: (a) for each comparison, each prospective teacher used a single method; (b) 1:1 correspondence was used by several prospective teachers to solve some problems. The class discussion emphasized that: (1) the number of elements in two finite sets could be compared even when counting was impossible (see Problem 1 and Problem 5). (2) Even where counting was applicable, it was not necessarily the preferable method for attaining the solution (see Problems 3 and 4). (3) Counting is basically pairing, that is, creating 1:1 correspondence between the elements of a given set and a subset of the natural numbers. The combination of individual work, of small group work, and of class discussions was crucial in raising students’ awareness of the various explicit as well as implicit methods they themselves used to compare finite sets. Addressing Students’ Entrenched Presuppositions It is important in instruction to distinguish new information that is consistent with prior knowledge from new information that runs contrary to prior knowledge. (Vosniadou et al., 2001, p. 393) Vosniadou et al. further emphasize that when new information is consistent with prior knowledge, it is most likely to be understood even if it is presented as a fact without any further explication. However, when the new information runs contrary to the existing conceptual structures, simply presenting the new information as a fact may not be adequate. Vosniadou et al. noted that in science instruction counterintuitive information is often introduced as a fact. In the case of the equivalency of infinite sets, some of the criteria for comparing the number of elements in two sets could be used only for finite sets (i.e., counting), other methods could, in principle, be used for both finite and infinite sets (e.g., 1:1 correspondence and inclusion). However, some of the latter methods are considered inadequate for comparing the number of elements in two infinite sets, in the framework of Cantorian set theory (e.g., inclusion). In fact, here only one criterion is applicable to both finite and infinite sets, namely, 1:1 correspondence. Actually, Cantor used this criterion for defining the equivalency of two sets, either finite or infinite. We have noted before that students regarded counting as the criterion for comparing sets, and that they had only finite sets in mind when expressing this view. However, when

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extending the system, relating both to finite and infinite sets, obviously, this criterion cannot be used for the new types of sets. In our instruction we led the students to acknowledge the limitations of the “counting” criterion. The participants themselves noted that this criterion could not be used for comparing the number of elements in the new type of sets. This raised their awareness of the need to choose another, more suitable criterion. Thus, they became aware of the need to re-examine each of the criteria that had already been discussed with reference to finite sets. In addressing students’ entrenched presuppositions, we stressed that it is both natural and legitimate to wonder about the nature and characteristics of infinite sets. For this purpose we extensively related to statements of mathematicians who acknowledged the difficulties involved in this topic. For instance, we presented the students with the following two excerpts: the first showing that even Galileo acknowledged the puzzling aspect of infinity, and the second, by Hann, discussing the difficulties related to the equivalency between a set (natural numbers) and its proper subset (positive, even numbers). Here a difficulty presents itself which appears to me insoluble. Since it is clear that we may have one line segment longer than another, each containing an infinite number of points, we are forced to admit that, within one and the same class, we may have something greater than infinity, because the infinity of points in the long line segment is greater than the infinity of points in the short line segment. This assigning to an infinite quantity a value greater than infinity is quite beyond my comprehension. (Galileo, 1954/1638, p. 31) An infinite set is equivalent to at least one of its proper subsets. If we look for examples of enumerable infinite sets we arrive immediately at highly surprising results. The set of all positive even numbers is an enumerable infinite set and has the same cardinal number as the set of all the natural numbers, though we would be inclined to think that there are fewer even numbers than natural numbers. (Hann, 1956, p. 1604) Motivation for Conceptual Change Students often do not see the reason to change their beliefs and presupposition because they provide good explanations of their everyday experiences, function adequately in the everyday world, and are tied to years of confirmation. (Vosniadou et al., 2001, p. 393) Vosniadou et al. (2001) relate to two major aspects in the task of persuading students to invest the substantial effort required to re-examine their initial explanations, and to become literate. The first aspect is creating an environment that will motivate the desired changes and the second is to ensure that experiences embedded in the environment are selected carefully so that they are theoretically relevant. Vosniadou et al. further explain that “theoretically relevant” means addressing

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We designed various activities aimed at increasing students’ motivation and interest in the comparison of sets. In these activities students experienced surprise when realizing that they were unaware of several adequate and inadequate criteria that they had been using in various situations. Here, we describe one such activity. Inclusions? 1:1 Correspondence? Intervals? The activity unfolded in three stages (Appendix B). Stage 1 aimed at (a) increasing prospective teachers’ awareness of the validity of all four methods (counting, 1:1 correspondence, inclusion, and intervals) for comparing finite sets and (b) increasing prospective teachers’ awareness of the extent to which each of the methods is applicable in a given problem. Stage 2 aimed at (a) discussing the role of consistency in mathematical systems and (b) demonstrating that 1:1 correspondence is the only method that is always applicable. At Stage 3 the students were encouraged to draw the final conclusions for finite sets, namely that all four criteria can be used to compare the number of elements in two finite sets but that 1:1 correspondences is the most general criterion. Here, much like in the activity previously described under Stages 1 and 2, the participants individually solved the problems. It was interesting for them to discover that the comparison of the number of elements in two finite sets can be conducted by using counting, inclusion, intervals, and 1:1 correspondence without violating the consistency of the theory. The students were quite surprised to realize that 1:1 correspondence is a method that enabled them to compare the number of elements in various sets, in all these problems. This, in itself, created some motivation to examine the usefulness of the various methods that they intuitively used to compare the number of elements in infinite sets. Cognitive Conflict Conceptual change is a difficult process that involves the reorganization of not just one misconception or one belief, but an interrelated system of beliefs and presupposition that takes a long time to be accomplished. Thus, conceptual change cannot but be a gradual affair, which requires the use of many different instructional interventions of which cognitive conflict is only one. (Vosniadou et al., 2001, p. 394) Vosniadou et al. (2001) claimed that cognitive conflict should be used in science instruction with caution. They showed examples of ways of using cognitive conflict, where the conflict was triggered by making children realize that their explanatory framework could not explain some empirical results. In the case of comparing infinite sets, we extensively used the cognitive conflict approach to increase students’ realization that “something is wrong” and in need of reconstruction. We also used other approaches, such as analogy (Tsamir, 2003).

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We shall here present two related activities; one was used with 10th graders and the other with prospective teachers. Even numbers and multiples of four: Inclusion vs. single infinity. The activity was used with 10th graders. It aimed at encouraging participants to discuss the consequences of relying only on intuitive criteria when comparing the number of elements in two infinite sets. In this activity the class was divided into teams of four; each student was first asked to provide his/her own intuitive answer to the question: Two sets are given: N = {2, 4, 6, 8, 10, …}, M = {4, 8, 12, 16, 20, ….} Is the number of elements in set M equal to the number of elements in set N? Explain your answer. Each team was then instructed to discuss its answers and to reach mutual agreement about the correct response. Notably, when applying the question “Are these two equal” to the context of infinite sets it has no formal meaning for the students, because the term “equal” was not yet defined in the context of infinite sets. However, all the students provided intuitive answers to this question, based on the meaning that they attributed to the term “equal”. We found that in each team, some students claimed that set M has less elements, because it is included in set N, while other students in the same team claimed that both sets are infinite and therefore they both have an infinite number of elements. The participating students noted that both answers seemed reasonable and that they were unable to decide which of them was correct. They further expressed the feeling that something was wrong. Natural numbers and multiples of 100: Inclusion, 1:1 correspondence, and single infinity. This activity was used with prospective teachers. It aimed at promoting their awareness that the application of different methods when comparing infinite sets leads to contradictory answers. Participants were first asked to individually solve several comparisons of infinite sets tasks. For instance, Given B = {1, 2, 3, 4, 5, 6, 7, …}, P = {100, 200, 300, 400, 500, 600, 700, …} The number of elements in sets P and S is equal/not equal. Explain … We chose the tasks so that they would trigger the use of three different criteria (inclusion, 1:1 correspondence, and single infinity). Indeed, most students used one method for each problem; however, they commonly used more than one method in the comparisons conducted in the different problems. The prospective teachers were then asked, once more, to individually solve the same problems. This time, however, the instruction was “Try to apply all the three methods to each problem.” Here is an example: Given B = {1, 2, 3, 4, 5, 6, 7, …}, P = {100, 200, 300, 400, 500, 600, 700, …} Is ‘1:1’ correspondence applicable? Yes/No If your answer is Yes — Use this method to solve the problem Is the number of elements in set B equal to the number of elements in set P? Yes/No Is ‘inclusion’ applicable? Yes/No If your answer is Yes — Use this method to solve the problem

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Is the number of elements in set B equal to the number of elements in set P? Yes/No Is ‘all infinities are equal’ applicable? Yes/No If your answer is Yes — Use this method to solve the problem Is the number of elements in set B equal to the number of elements in set P? Yes/No The participants, while performing this activity, realized that the application of the three methods led to contradictory solutions. This created a state of confusion and raised questions (For example: Why did it work with finite sets? What should be done in the case of infinite sets?). Participants were then asked to reflect in groups on their responses. The specific guiding questions were: 1. Is it OK to alternatively use these methods for the comparison of infinite sets? Why? 2. In your opinion, which (if any) of the various methods for comparing infinite sets is preferable? Why? In the class discussion, after each group presented its approach, the participants reached the conclusion that at least two of the above-mentioned methods are valid for the comparison of infinite sets, but only if used exclusively to compare all pairs of infinite sets. Otherwise, essential consistency is violated. Consequently, the choice of a method for the comparison of all infinite sets should be made in advance, and this method must then be used exclusively. A question that naturally arose was: Is there a preferable method? that is, one that provides more conclusive answers. Participants found that: (1) all infinities are equal, that is, all sets should have the same number of elements, and therefore there is no reason for comparison; (2) inclusion was only occasionally applicable; (3) 1:1 correspondence was found to enable the comparison of infinite sets both when such a relationship existed (indicating “equality”) and when it could be proved that there is no such correspondence (indicating “inequality”). Moreover, this method was also applicable for comparing finite sets. Thus, 1:1 correspondence appeared to be the most applicable method both for finite sets and for infinite sets. This assignment, before formally studying the theorems of the Cantorian Set Theory, pointed to the possibility of using various methods when comparing given sets as a property lost in the transition from finite to infinite sets; another property that was lost in this transition is: “The whole is always greater than its part.” The possible equivalency of a set and its proper subset is a characteristic gained in this transition. Providing Models and External Representations Models and external representations can be used to clarify aspects of a scientific explanation that are not apparent when the explanation is given in a linguistic or mathematical ways. The visual qualities of a model are useful in making an explanation better understood and more easily memorized. (Vosniadou et al., 2001, p. 394)

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Vosniadou et al. showed how in the learning environments that they created to teach mechanics to 5th grade students they used measurements of forces, symbolic representations of force and energy, and a friction model to clarify aspects of the scientific explanations that were not apparent when the explanation was given in a linguistic or mathematical way. In the case of comparison of infinite sets, we presented 16–18-year-old students and prospective teachers with different representations of problems that required the comparison of the number of elements in pairs of infinite sets. For example, we used four representations of the task asking to compare the number of natural numbers with that of the natural numbers larger than 2, namely, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …} and {3, 4, 5, 6, 7, 8, 9, 10, …}. Horizontal representation — the sets are presented beside each other. A = {1, 2, 3, 4, 5, …}, B = {3, 4, 5, 6, 7, …} Vertical representation — the sets are presented one below the other. A = {1, 2, 3, 4, 5, …} B = {3, 4, 5, 6, 7, …} Explicit representation — the sets are presented one below the other so that it is visually easy to match each element of one set with one element of the other set. Here the pairs are 1↔1+2, 2↔2+2, 3↔3+2, 4↔4+2, 5↔5+2 etc. A = {1 , 2 , 3 , 4 , 5 , …} B = {1+2, 2+2, 3+2, 4+2, 5+2, …} Geometrical representation — the elements of the given sets are related to geometrical figures that trigger the matching of each element of one set with one element of the other set (Figure 20.1). For example, participants were asked to consider the two sets of numbers, and the set of trapezoids: The upper base of each trapezoid is shorter by 2 cm than the bottom base. The first upper base is 1 cm long and each following segment is 1 cm longer than the previous one. The first bottom base is 3 cm long and each following bottom base segment is 1 cm longer than the previous one. Set A presents the numbers that express the lengths (in cm) of the upper bases; and set B presents the numbers that express the lengths (in cm) of the bottom bases in the trapezoids. A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …} B = {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, …}

1cm

{

3cm

2cm ,

4cm

3cm ,

5cm

4cm ,

6cm

, ...

}

Figure 20.1: Vertical and Graphical Representations.

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For each representation, students were asked to judge whether the numbers of elements in Sets A and B were equal, and to explain their answers. It was found that the horizontal representation encouraged more “natural numbers” responses, usually justified by inclusion arguments, to the effect, for instance, that “set A consists of all the elements that are included in set B in addition to two extra elements. Thus, there are two more elements in set A than in set B.” The other representations evoked “the same number of elements” responses, accompanied by various types of justifications. The vertical representation triggered “single infinity” justifications, claiming that “all infinite sets have the same number of elements, so the presented sets are also equal.” The explicit and geometrical representations triggered one-to-one correspondence considerations, pointing to a way to pair matching elements. Hence, students reached contradictory responses to the different representations of the same task; for instance, “equal number of elements” to the geometrical and explicit representations, and “unequal number of elements” to the horizontal representation. We would like to emphasize that two representations, the explicit and the geometrical, led to responses consistent with Cantorian Set Theory, in terms of both the judgment and the method used. These representations were used to design anchoring tasks in instructional interventions (e.g., Tsamir, 2003b; Tsamir & Tirosh, 1999). Breadth of Coverage of the Curriculum It may be more profitable to design curricula that focus on the deep exploration and understanding of a few, key concepts in one subject matter area than curricula that cover a great deal of material in a superficial way. (Vosniadou et al., 2001, p. 391) Vosniadou et al. suggest focusing on a limited number of key concepts and giving students enough time and adequate learning environments to achieve a qualitative understanding of these concepts. Clearly, the concepts: “equal,” “unequal,” and “equivalent” are key concepts in mathematics. They are widely used in various mathematical domains including arithmetic, algebra, trigonometry, analysis, and geometry. In fact, it is hard to imagine a mathematical domain where these notions play no significant part. Hence, devoting time and attention to designing learning environments that address these notions in various ways, and evaluating them, should be a major aim of mathematics learning and teaching. Order of Acquisition of the Concept Involved The relational structure in which the concepts of a domain are acquired needs to be taken into consideration. (Vosniadou et al., 2001, p. 391) Vosniadou et al. explained that in the design and implementation of learning environments a great deal of attention should be paid to the sequence in which the concepts were introduced. This should be done to avoid the formation of new misconceptions and to overcome existing ones. In designing instructional interventions Vosniadou et al. (2001)

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attempted to consider possible synthetic models that could be constructed by children on the basis of their prior knowledge, and to use this analysis as basic criteria for the design of the order of the concepts to be taught. When considering the sequencing of instruction in the case of the equivalency of infinite sets, clearly, one needs to think about the order of the presentation of the relevant concepts (e.g., set, finite sets, infinite sets, equal, equivalent, inclusion, 1:1 correspondence, cardinality). However, there are other mathematical concepts that could influence a student’s reactions to the comparison of finite and infinite sets tasks, with which the student might (or might not) be familiar (e.g., function, series, sequences, limits). Decisions regarding the sequence of instruction should take account of the concepts that are embedded in the specific domain and of the concepts embedded in additional domains. The sequences that we developed took account of both these types of concepts. For example, the instructional unit for 10th graders was developed for students who were not familiar with the concept of limit. The instructional sequences designed for prospective teachers were sensitive to possible impacts of the participants’ familiarity with limits. The two sequences were similar with regard to the order of presentation of the set-theory embedded notions: from finite sets to infinite sets with attention to comparison methods that could be applied to both domains.

Summing Up and Looking Ahead In this chapter we have shown that the conceptual change-based recommendations for designing science learning environments could be valuable for mathematics education as well. The case that we discussed, the equivalency of infinite sets, could be regarded as a touchstone for examining the applicability of these recommendations. The related difficulties are well documented from both the phylogenetic and the ontogenetic viewpoints. In our case, Vosniadou et al.’s (2001) recommendations were used for reflection on instructional interventions that were formulated without reference to these principles. We have shown a way in which the given recommendations can serve as a tool for analyzing the environments that we developed. However, it seems that these recommendations could and should be used for designing learning environments for mathematics instruction that are apt to involve conceptual change. The eight recommendations, their discussion, illustrations, and labeling could assist in designing and formulating instruction in mathematics education. However, the impact of several inherent differences between mathematics and science education should be carefully considered when using Vosniadou et al.’s (2001) recommendations for designing learning environments for science instruction in mathematics instruction. Issues to be considered include the role of empirical facts in each of these domains, the consequences of confronting one specific misconception in each domain, and the relative nature of mathematics. Studies should be conducted to examine the impact of mathematics learning environments that are designed according to the recommendations of Vosniadou et al. (2001), regarding students and teachers’ mathematical conceptions and ways of thinking.

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References Borasi, R. (1985). Errors in the enumeration of infinite sets. Focus on Learning Problems in Mathematics, 7, 77–88. Carey, S. (1985). Conceptual change in childhood. Cambridge, MA: MIT Press. Duval, R. (1983). L’obstacle du dedoublement des objects mathematiques. Educational Studies in Mathematics, 14, 385–414. Falk, R., Gassner, D., Ben Zoor, F., & Ben Simon, K. (1986). How do children cope with the infinity of numbers? In: The Proceedings of the 10th conference of the international group for the psychology of mathematics education (pp. 7–12). London, UK: University of London Institute of Education. Fischbein, E. (1987). Intuition in science and mathematics: An educational approach. Dodrecht, The Netherlands: Reidel. Fischbein, E., Jehiam, R., & Cohen, D. (1995). The concept of irrational numbers in high-school students and prospective teachers. Educational Studies in Mathematics, 29, 29–44. Fischbein, E., Tirosh, D., & Hess, P. (1979). The intuition of infinity. Educational Studies in Mathematics, 10, 3–40. Galileo, G. (1954/1638). Dialogues concerning the new sciences. New York: Dover. Greer, B. (1992). Multiplication and division as models of situations. In: D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 276–295). New York: Macmillan. Hann, H. (1956). Infinity. In: J. R. Newman (Ed.), The world of mathematics (Vol. 3, pp. 1593–1611). New York: Simon & Schuster. Hart, K. (Ed.). (1981). Children’s understanding of mathematics. London: Murray. Hartnett, P., & Gelman, R. (1998). Early understanding of numbers: Paths or barriers to the construction of new understand? Learning and Instruction, 8, 341–374. Hefendehl, H. L. (1991). Negative numbers: Obstacles in their evolution from intuitive to intellectual constructs. For the Learning of Mathematics, 11, 26–32. Klein, R., & Tirosh, D. (1997). Teachers’ pedagogical content knowledge of multiplication and division of rational numbers. In: E. Pehkonen (Ed.), Proceedings of the 21st conference of the international group for the psychology of mathematics education (Vol. 3, pp. 144–151). Lahti, Finland. Lakoff, G., & Nunez, R. (2000). Where mathematics comes from. NY: Basic Books. Martin, W. G., & Wheeler, M. M. (1987). Infinity concepts among preservice elementary school teachers. In: J.G. Bergeron, N. Herscovics, & C. Kieran (Eds), Proceedings of the 11th conference of the international group for the psychology of mathematics education (pp. 362–368). Paris, France. Merenluoto, K., & Lehtinen, E. (2004). Number concept and conceptual change: Towards a systemic model of the processes of change. Learning and Instruction, 14, 519–534. Moloney, K., & Stacey, K. (1996). Understanding decimals. Australian Mathematics Teacher, 52, 4–8. Posner, G. J., Strike, K. A., Hewson, P. W., & Gertzog, W. A. (1982). Accomodation of a scientific conception: Toward a theory of conceptual change. Science Education, 66, 211–227. Putt, I. J. (1995). Preservice teachers ordering of decimal numbers: When more is smaller and less is larger! Focus on Learning Problems in Mathematics, 17, 1–15. Sierpinska, A. (1987). Humanities students and epistemological obstacles related to limits. Educational Studies in Mathematics, 18, 371–387. Stafylidou, S., & Vosniadou, S. (2004). The development of students’ understanding of the numerical value of fractions. Learning and Instruction, 14, 503–518. Streefland, L. (1996). Negative numbers: Reflection of a learning researcher. Journal of Mathematical Behavior, 15, 57–77.

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Tall, D. (1980). The notion of infinite measuring numbers and its relevance in the intuition of infinity. Educational Studies in Mathematics, 11, 271–284. Tall, D. (1992). The transition to advanced mathematical thinking: Functions, limits, infinity and proofs. In: D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 495–511). New York: Macmillan. Tall, D. (2001). Natural and formal infinities. Educational Studies in Mathematics, 48, 199–238. Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics, with special reference to limits and continuity. Educational Studies in Mathematics, 12, 151–169. Tirosh, D. (1991). The role of students’ intuitions of infinity in teaching the Cantorian theory. In: D. Tall (Ed.), Advanced mathematical thinking (pp. 199–214). Dordrecht, The Netherlands: Kluwer. Tirosh, D., & Tsamir, P. (1996). The role of representations in students’ intuitive thinking about infinity. International Journal of Mathematics Education in Science and Technology, 27, 33–40. Tirosh, D., Fischbein, E., & Dor, E. (1985). The teaching of infinity. In: L. Streefland (Ed.), Proceedings of the 9th conference of the international group for the psychology of mathematics education (pp. 501–506). Utrecht, The Netherlands. Tsamir, P. (1990). [Students’ inconsistent ideas about actual infinity]. Unpublished thesis for the Master’s degree. Tel Aviv University, Tel Aviv. [In Hebrew]. Tsamir, P. (1999). The transition from comparison of finite to the comparison of infinite sets: Teaching prospective teachers. Educational Studies in Mathematics, 38, 209–234. Tsamir, P. (2003a). From “easy” to “difficult” or vice versa: The case of infinite sets. Focus on Learning Problems in Mathematics, 25, 1–16. Tsamir, P. (2003b). Primary intuitions and instruction: The case of actual infinity. Research in Collegiate Mathematics Education, 12, 79–96. Tsamir, P., & Tirosh, D. (1999). Consistency and representations: The case of actual infinity. Journal for Research in Mathematics Education, 30, 213–219. Vamvakoussi, X., & Vosniadou, S. (2004). Understanding the structure of the set of rational numbers: A conceptual change approach. Learning and Instruction, 14, 453–46. Vlassis, (2004) Making sense of the minus sign or becoming flexible in negativity. Learning and Instruction, 14, 469–484. Vosniadou, S., & Brewer, W. F. (1992). Mental models of the earth: A study of conceptual change in childhood. Cognitive Psychology, 24, 535–585. Vosniadou, S., & Verschaffel, L. (Guest Eds.). (2004). The conceptual change approach to mathematics learning and teaching. Learning and Instruction, 14 (Special Issue). Vosniadou, S., Ioannides, C., Dimitrakopoulou, A., & Papademetriou, E. (2001). Designing learning environments to promote conceptual change in science. Learning and Instruction, 11, 381–419. Wiser, M., & Amin, T. (2001). “Is heat hot?” inducing conceptual change by integrating everyday and scientific perspectives on thermal phenomena. Learning and Instruction, 11, 331–356.

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Appendix: A Stage 1: Counting Here are two sets A and B: A={1, t, }, B={7, w} Is the number of elements in sets A and B equal? Yes/No How did you reach this conclusion? _________________ Complete the following generalization: In order to compare the number of elements in any two finite sets one could ________________ Stage 2: Various Methods Problem 1 At a dance party all the students danced in couples, a boy and a girl in each couple. No pupils were left without a partner. Z = {The boys} W = {The girls} Is the number of elements in set Z equal to the number of elements in set W? Yes/No How did you reach this conclusion? _________ Problem 2 Given the sets: X = {1,2,3,4,5,6,7,8,9,10,11,12}, Y = {a,b,c,d,e} Is the number of elements in set X equal to the number of elements in set Y? Yes/No How did you reach this conclusion? _________ Problem 3 Given the sets: Y = {a,b,c,d,e,f}, V = {a,b,c,} Is the number of elements in set Y equal to the number of elements in set V? Yes/No How did you reach this conclusion? _________ Problem 4 Dan was ill. The doctor prescribed one green tablet every 3 hours for the first week. Then, in the second week he was ordered to take a red capsule every 3 hours. G = {The green tablets}, R = {The red capsules} Is the number of elements in set G equal to the number of elements in set R? Yes/No How did you reach this conclusion? _________ Problem 5 Along the new promenade, a lane of trees was planted; a tree every 200 m. Every 400 m a street light was placed adjacent to a tree. The first and the last trees had a street light next to them. L = {The street lights}, T = {The trees} Is the number of elements in set L equal to the number of elements in set s? Yes/ No How did you reach this conclusion? _________

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Stage 3: Reflection Let’s reflect — 1. Which methods did you use in order to compare the above sets? 2. Which method did you use in each problem? 3. Could the number of elements be compared even when counting was impossible? 4. In cases that counting was applicable, was it always the preferable method? 5. What is the meaning of ‘counting’?

Appendix: B Stage 1: The Four Criteria: The Finite Case Participants were asked to try to use all four methods to compare the finite sets in each of the problems that were given in Stage 2 of the activity, which is described in Section 2.2. One example follows: Sample Problem (Problem 1) At a dance party all the students danced in couples, a boy and a girl in each couple. No pupils were left without a partner. Z = {The boys} W = {The girls} Is the number of elements in set Z equal to the number in set W? Yes / No Try to apply all the four methods to each problem. In order to answer this question — Is ‘1:1’ correspondence applicable? Yes/No . If your answer is Yes — Use this method to solve the problemIs the number of elements in set Z equal to number in set W? Yes/No Is ‘inclusion’ applicable? Yes/No. If your answer is Yes — Use this method to solve the problemIs the number of elements in set Z equal to number in set W? Yes/No Is ‘intervals’ applicable? Yes/No. If your answer is Yes— Use this method to solve the problemIs the number of elements in set Z equal to number in set W? Yes/No Is ‘counting’ applicable? Yes/No. If your answer is Yes — Use this method to solve the problemIs the number of elements in set Z equal to number in set W? Yes/No Stage 2: The Four Criteria in the Finite Case At Stage 2 the participants were asked to reflect on their actions during Stage 1, by filling in a summary table, referring both to the problems and to the criteria.

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Let’s reflect — 1. List the methods applicable to each problem. Problem No.

1 Boys — Girls

a. b. c. d.

2 1 to 12 — a to e

3 a to f — a to e

4 tablets — capsules

5 lights — trees

Counting Inclusion Intervals Pairing — 1:1 correspondence

2. 3. 4. 5.

Is it OK to alternatively use these methods for the comparison of finite sets? Why? Is there, after all, a preferable method? How do the methods work? Does the “counting” method enable to compare the numbers of elements in two given sets when there is no way to count? 6. Does the “inclusion” method enable to compare the numbers of elements in two given sets when there is no inclusion relationship between the sets? 7. Does the “intervals” method enable to compare the numbers of elements in two given sets when there is no way to predict the intervals? 8. Does the “1:1 correspondence” method enable to compare the numbers of elements in two given sets when there is no way to pair? Stage 3: Summary — Finite Sets This was the final activity relating to the comparison of finite sets. Here students were expected to express awareness of the various methods that can be used for the comparison of finite sets, and of the advantage of 1:1 correspondence. Final conclusion for finite sets: Complete the following: In order to compare the number of elements in any two finite sets you can

COMMENTARIES

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Chapter 21

Nurturing Conceptual Change in Mathematics Education Brian Greer and Lieven Verschaffel Introduction Conceptual change lies at the heart of mathematics as a human creation. What is now considered the body of mathematical knowledge that a student may be expected to understand after schooling is the achievement of the collective intellectual activity of mankind over millennia. Consequently, as Sinclair (1990) pointed out, the challenge of mathematics education is that we expect children to master in a few years complexities that historically took millennia to evolve through conceptual change. Galileo (as noted by Tsamir & Tirosh, this volume), despite being a genius untrammeled by conventional thinking, could not comprehend how the number of points in a shorter line could be equal to the number in a longer line (though he did have the considerable insight that there was something puzzling that he did not understand). Yet a student of today is expected to understand the resolution of this apparent paradox. In order to achieve this miracle through education, it is essential to know how to nurture conceptual change in today’s students.

Conceptual Change Theory Applied to Mathematics Conceptual change theory (CCT), developed by Vosniadou and her colleagues and presented in this volume, originated within the domain of science education. Authors of chapters in Part 3 argue that its principles are equally applicable to mathematics education, despite certain differences such as the role of empirical facts in each of these domains, the consequences of confronting one specific misconception in each domain, and the relative nature of mathematics. (Tsamir & Tirosh, this volume)

Reframing the Conceptual Change Approach in Learning and Instruction Copyright © 2007 by Elsevier Ltd. All rights of reproduction in any form reserved. ISBN: 0-08-045355-4

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In particular, philosophers and historians of science and mathematics have tended to emphasize a distinction between the characters of the conceptual changes that occur in these two fields (see, especially, Gillies, 1992b). From examples such as the replacement of the Ptolemaic system by the Copernican, it is argued that the changes whereby old theories are replaced by new ones in science are revolutionary in the sense that the old one is displaced. By contrast, when conceptual change occurs in mathematics, the old structure is retained as a substructure of the new, the obvious example being that of number (e.g., Lakoff & Nunez, 2000). However, the answer to the question “Are there revolutions in mathematics?” (Gillies, 1992a) depends on the definition of “revolution”. Dunmore (1992) argued that there have been changes in mathematics at the meta-level whereby earlier views have indeed been displaced. For example, in the long struggle to establish the ontological status of negative integers, the doctrine that they are “impossible” was rejected. Likewise, the belief in the uniqueness of Euclidean geometry as the description of space has been irreversibly discarded. From this perspective, there are conceptual changes in mathematics that truly merit the title “revolutionary”. Dauben (1992) argued along the same lines, citing as four examples (a) the discovery of incommensurable magnitudes/irrational numbers, (b) Cantor’s transfinite set theory, (c) new standards of rigour for calculus introduced by Cauchy, (d) Robinson’s non-standard analysis, created in the 20th century. He concluded that when a true revolution has taken place, a significant part of the ‘older’ mathematics will come to be replaced or dramatically augmented by concepts and techniques that visibly change the vocabulary and grammar of mathematics. (Dauben, 1992, p. 80) There are already theories of conceptual change in mathematics education, of course — every theory of cognitive development is a theory of conceptual change, in some sense — and at some point a systematic review of how they relate to CCT would be valuable. Piagetian theory, in particular, has several core constructs that are highly compatible with CCT, notably those of equilibration and the complementary processes of assimilation/accommodation. However, Piaget was essentially an epistemologist and the weakness of his theory, from our perspective, is that he did not take sufficient account of the fact that cognitive development in mathematics takes place in historical/cultural circumstances, and, in particular, almost entirely under instruction. One of Piaget’s most trenchant critics, Freudenthal, was originally a mathematician who turned his attention to mathematics education, and was the leading founder of arguably the single most important theoretical/applied centre within mathematics education, whose work continues at the Freudenthal Institute. Freudenthal analysed in considerable detail the historical development of mathematics which, as he said, “grows by its self-organizing momentum” (1990, p. 15) through “the interplay between form and content” (1990, p. 12) from roots in practical experience and problem solving. One major thread of his philosophy of mathematics education, that is highly relevant to CCT, is the need for children to retrace the steps taken historically in the construction of mathematics: I asked the question of whether the learner should repeat the learning process of mankind. Of course not. Throughout the ages history has, as it

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were, corrected itself, by avoiding blind alleys, by cutting short numerous circuitous paths, by rearranging the road-system itself. We know nearly nothing about how thinking develops in individuals, but we can learn a great deal from the development of mankind. Children should repeat the learning process of mankind, not as it factually took place but rather as it would have done if people in the past had known a bit more of what we know now. (Freudenthal, 1990, p. 48) A second major theoretical perspective from within mathematics education is that of Fischbein, originally a psychologist, and his followers (including Tsamir and Tirosh). In particular, Fischbein (1987) analysed the difficulties in overcoming intuitions in the face of situations requiring conceptual change, and abundantly illustrated them with historical examples. He pointed out that: From the educational point of view there is an important problem to be considered by curricula writers and by teachers. A certain interpretation of a concept or an operation may be initially very useful in the teaching process as a result of its intuitive qualities (concreteness, behavioral meaning etc.). But as a result of the primacy effect that first model may become so rigidly attached to the respective concept that it may become impossible to get rid of it later on. The initial model may become an obstacle which can hinder the passage to a higher-order interpretation — more general and more abstract — of the same concept. (Fischbein, 1987, p. 198) This analysis clearly bears on several examples in the chapters of Part 3. While CCT shares characteristics with these other theories, its particular value, in our opinion, is that it proposes a number of specific and testable principles that can guide instruction and research on instruction, as illustrated, in particular, by Tsamir and Tirosh (this volume). In the rest of this chapter, we consider a number of key themes prompted by the chapters in this part of the volume.

Nobody Learns Alone There is no universal, “natural” course of development of understanding in mathematics. Otherwise, presumably, Galileo would have understood Cantor’s theory of countable infinity! How, and what, a student learns (apart from learning that takes place in environments other than school) is dependent on his/her instruction. As a simple and clear example, Tsamir and Tirosh (this volume) found that students who were learning about limits, addressing infinity as “a single sized entity”, were liable to say that two infinite sets had the same cardinality on the grounds that all infinite sets have the same number of elements, whereas students in another class mainly relied on considerations of inclusion of one set within the other. For ascertaining research (i.e., research that probes students’ existing understanding) in CCT, accordingly, attention should be given, as far as possible, to the students’ instructional

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history. Details of the curriculum provide some information, but there is often a large gap between the intended curriculum and the implemented curriculum. Moreover, as pointed out by Merenluoto and Palonen (this volume) teachers’ own understanding of the relevant mathematics may be fragile. In order to develop CCT fully, therefore, it is necessary to conduct studies within classes to document the nature of the teaching and attempt to relate it to the development of students’ conceptions. The most direct way of doing this, of course, is through intervention studies in which the researchers, either personally or in collaboration with teachers, design and carry out the teaching. An example is provided by Tsamir and Tirosh (this volume), who retrospectively evaluated the principles of CCT in relation to their investigations. They recommend that: Studies should be conducted to examine the impact of mathematics learning environments that are designed according to the recommendations of Vosniadou et al. (2001), regarding students’ and teachers’ mathematical conceptions and ways of thinking. (this volume)

Learning from History While avoiding a simplistic interpretation (for example, through concentrating on vignettes and individuals rather than conceptual shifts in communities of practice), there are many powerful ways in which the historical record can inform attempts to improve instruction through understanding of cognitive obstacles and the conceptual changes needed to overcome them (e.g., Fischbein, 1987; Kaput, 1994; and see Greer, 2004). Most obviously, the tenacity of resistance to conceptual change is easy to illustrate, the struggle to accept negative integers being perhaps the best example (Fischbein, 1987, Chapter 8). As late as 1831 an eminent mathematician could write that “3–8 is an impossibility, it requires you to take from 3 more than there is in 3, which is absurd” (De Morgan, 1910 [originally 1831], pp. 103–104). The initial use of the negative sign was extremely restricted in arithmetic, to the subtraction of a smaller natural number from a larger, with well-known results. In recent years — exemplifying a case where research has impacted practice — considerable attention has been given to the variety of situations modelled by subtraction, thus addressing the effects of the limited intuitive “take away” model (Fischbein, Deri, Nello, & Marino, 1985). As Christou et al. (this volume) and Vlassis (2004) show, the extension to manipulation of algebraic expressions brings multiple new challenges. (For a fascinating historical example, see De Morgan [1910, originally 1831, Ch. 9].)

The Role of Representational Tools Throughout the historical record can be traced the central role of physical devices and cognitive tools, including language, representations, and notations, in extending the domain of

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mathematical entities (think of Cartesian geometry, notational advances in arithmetic and algebra [e.g., for exponentiation, see Greer, 2004], Argand diagrams for the representation of complex numbers and operations on them, the whole range of new representational means provided by computers [Kaput, 1992], etc.). A problem with representations, in general, is that they are transparent to those who understand them and opaque to those who do not. When the former group includes the teacher and the latter the students, the dangers for the teaching/learning process are obvious, as constructivists pointed out. A further problem is that students may focus on superficial aspects (sometimes unwittingly abetted by teachers seeking “short-cuts”). To the mathematician’s eye, the number line is a way par excellence to represent numbers — starting with the natural numbers, adding rationals, negative numbers, and finally the continuum of the real numbers (see, for example, the case of Eric, described by Merenluoto and Paronen [this volume], describing his imagery of the real numbers in terms of neighbourhoods). However, Vamvakoussi and Vosniadou (this volume) found that the presence of a number line in a test had a limited effect on students’ performance on tasks related to density, and that this effect did not last when the number line was removed.

Language The great mathematician, Poincaré, defined mathematics as “the art of giving the same name to different things”. Often an interesting question to ask is: “Why are X and Y both called A?” — for example, why are 3, −3.14, , and −7i ⫹ 6 all called numbers? In the development of mathematics, the bestowing of a name reifies the integration of previously disparate entities, thus creating mathematical objects that can be conceptually manipulated. An excellent historical example is the definition of group (through axioms) by Cayley in 1854 (see Freudenthal, 1990, p. 31). Likewise, we may ask, “Why are various mathematical objects all called tangents?” (and what have they to do with the trigonometrical usage of the same word — and what, in turn, does that have to do with tangent as a function of a real or complex variable?). This particular means of growth in mathematics reflects some notion of the essence of a concept, with inessential elements (such as that a tangent only cuts a curve once and it divides the plane in two parts, one of which contains the whole curve) being discarded. As pointed out throughout the chapters, and as central to CCT, problems arise when students attempt to maintain such elements when the concept is extended to a wider domain in which they no longer apply. The examples above illustrate that, despite frequent assertions to the contrary, mathematical language, notation, and representations are inherently ambiguous. This ambiguity is an advantage to those who are familiar with it, can effortlessly disambiguate through contextual cues, and understand the conceptual linkages underlying the ambiguity, but for the neophyte it can be a major source of confusion. A clear case in point is the extension of use of the minus sign (Christou et al., this volume; Vlassis, 2004). Another is the polysemous usage of literals in algebra (Christou et al., this volume).

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The Weakness of Formality Freudenthal commented, “Traditionally, mathematics is taught as a ready made subject” and added, “Only a small minority learn mathematics in this way” (1990, pp. 47–48). Unfortunate results can occur when mathematicians exert undue influence over the curriculum, as in the time of the “New Math”. As Freudenthal commented: “New Math’s wrong perspective was to replace the learner’s insight with the adult mathematician’s” (1990, p. 112). From the perspective of CCT, mathematicians are prone to see mathematics as a complete structure that can be systematically and formally transmitted to the learner, forgetting their own conceptual struggles (which may have been minor, in any case — see Merenluoto & Paroten, this volume). For these reasons, it is appropriate to be suspicious of the rather fashionable characterization of mathematics education as the enculturation of students into the mathematical community. As Merenluoto and Paronen point out, ‘doing’ mathematics in the school context is very different from ‘doing’ mathematics within the mathematics community. (this volume) Their final recommendation is to facilitate conceptual change in the number concept, we need to give the students some access to the shared practices in the community of mathematicians. (this volume) While we agree with that, up to a point, it needs to be clarified how much, and of what kind, “some” is, and, further, if reference to “the community of mathematicians” implies research mathematicians then that is problematic, given how esoteric the practices of that group have become (Davis & Hersh, 1981, p. 39 et seq.). A particular feature of (many) formal mathematicians that has considerable bearing on CCT is their belief in the efficacy of definitions (and other formal apparatus such as axioms). Yet there are abundant examples in mathematics education where a learner correctly repeats a definition but then responds to a question in a way that contradicts his/her own definition just stated. For example, a student may give a correct definition of a quadrilateral but immediately say that a concave quadrilateral (or a square) is not a quadrilateral. A useful theoretical construct here is the contrast between “concept definition” and “concept image” (Vinner, 1982, 1991). As Thorndike (1922) emphasized long ago, children construct concept images and properties inductively from the sample of instances that they encounter. For example, long experience with whole-number multiplication and division create a strong and persistent intuition that multiplication always makes bigger and division always makes smaller, which is no longer true when the operations are extended to apply to rational numbers, with well-documented results (Greer, 1994). Another term that emphasizes that people do not generally act in accordance with definitions is “prototype”. Research in psychology (e.g., Rosch, 1978) has shown that people think of some birds — for example, a robin — as more “bird-like” than others, such as a penguin. In mathematics, the circle is the “paradigmatic intuitive model” (Fischbein, 1987, Chapter 13) of a

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curve. Fischbein cites data from Vinner (1982) who found that being able to state the definition for a tangent correctly was far from a guarantee that the student could escape the influence of the prototype or concept image. A particularly useful tool to test the solidity or fragility of definition-based understanding is to present a singular case. For example, Vinner (1982) found that most students could not comprehend the notion of a tangent to a point on a straight line segment. One suggested remedy for this general pattern of results is to “let the student into the secret” about how definitions work in the community of mathematicians, which is different in systematic ways from their use in everyday parlance.

The Pedagogical Dilemma Fischbein and colleagues identified “a fundamental didactical dilemma”, pointing out that if one tries to avoid building the ideas related to arithmetical operations on a foundation that is behaviorally and intuitively meaningful, one certainly will violate the most elementary principles of psychology and didactics. (Fischbein et al., 1985, p. 15) As a consequence: The initial didactical models seem to become so deeply rooted in the learner’s mind that they continue to exert an unconscious control over mental behavior even after the learner has acquired formal mathematical notions that are solid and correct. (Fischbein et al., 1985, p. 15) The dilemma may be unavoidable, but prevalent patterns of curriculum design exacerbate it. A common guiding principle, consistent with the perspective of the person who already knows the system, is to build up mathematics systematically — in the case of arithmetic, for example, first come operations with the natural numbers, then rationals, then integers, then real numbers. As a consequence the intuitive models can become very firmly established. Bearing in mind that the student will see many examples of numbers other than natural numbers in use both in and out of school, we propose that it would make sense to deliberately and pre-emptively introduce a wider variety of numbers earlier than currently is typical. A related point is the predominant selection of “easy” numbers that one can see in almost any textbook, based on the well-intentioned but misguided (in our view) assumption that this will make it easier for the student to understand — which may well be true in the short term, but not the in the long term (see next section). Have you ever, for example, seen a quadratic expression like this: 2.67x2 – 3.86x ⫹ 12.23? Yet, with a calculator there is no reason why a student should not be able to solve it. By planfully varying the examples, it should be possible to avoid the establishment of expectations such as that when a general formula for a function, such as f(x) ⫽ ax2, is presented, the parameter “a” ranges only over the natural numbers.

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The Long-Term Perspective CCT clearly implies that curricular design must be long term. For example, attention must be given to the series of conceptual restructurings that take place as numbers, and operations on them, are progressively extended. Yet, as Merenluoto and Paronen point out, in … teaching practices the problems inherent in the extensions of the number concept are often implicitly passed over. (this volume) Recent treatments of algebra (e.g., Kaput, 1999; and see Verschaffel, Greer, & Torbeyns, 2006) have emphasized that, rather than conceiving of a transition from arithmetic to algebra, the two strands should be intertwined from an early age (see Verschaffel, Greer, & De Corte, in press). A particularly clear justification for this stance comes from recent work on young children’s restricted interpretation of the equals sign whereby they think of it solely as an instruction to carry out computation on the expression on the left, resulting in a single number on the right. As a result, researchers recommend early exposure to arithmetic equations such as 7 ⫽ 3 ⫹ 4 to promote the broader interpretation of the equals sign as indicating equivalence, an interpretation essential for algebra (e.g., Carpenter, Franke, & Levi, 2003; Seo & Ginsburg, 2003). An extremely strong example of the building of a case for long-term curricular design — with painstaking analysis of the historical record — was provided by Kaput who explained: I look closely at the origins of the major underlying ideas of calculus for clues regarding how calculus might be regarded as a web of ideas that should be approached gradually, from elementary school onward in a longitudinally coherent school mathematics curriculum. (Kaput, 1994, p. 78)

Revolutions in Mathematics Education? “Are there revolutions in mathematics?” asked Gillies (1992a). We may similarly ask: “Are there revolutions in mathematics education?” Lerman (2000) referred to “the social turn in mathematics” (and a turn is a revolution!). As regards the research community, the work in this part of the book suggests the onset of a conceptual shift away from the view of mathematics learning as additive to the view that it is characterized by conceptual restructurings. Since affective and metacognitive factors are also given prominence (Merenluoto & Lehtinen, 2004) and the historical record is mined, this shift is in alignment with the “second wave” (De Corte, Greer, & Verschaffel, 1996) of what has been called the Cognitive Revolution (Gardner, 1985), during which the importance of historical, cultural, social, and effective factors became recognized (Greer & Verschaffel, 1990). However, there are only limited signs that this shift is impacting school mathematics, and there are powerful forces that resist change and militate against long-term strategic planning — most noticeably, but by no means exclusively, in the United States. These forces include the pressure to maximize scores on the next test, which is a powerful motive

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for not thinking about the longer-term consequences of short-term pedagogical expediency. Moreover, the positivistic style of research now being demanded in the United States favours short-term local interventions, the results of which can be quantified, with little hope that the findings will accumulate into any comprehensive curricular blueprint (see Verschaffel, Greer, & De Corte, 2007). So, much remains to be done, in terms of exposing the problems caused by insufficient attention to conceptual change, exploring ways to facilitate it, and trying to ensure that they enter the classroom.

References Carpenter, T. P., Franke, M., & Levi, L. (2003). Thinking mathematically: Integrating arithmetic and algebra in elementary school. Portsmouth, NH: Heinemann. Dauben, J. (1992). Conceptual revolutions and the history of mathematics. In: D. Gillies (Ed.), Revolutions in mathematics (pp. 15–20). Oxford: Oxford University Press. Davis, P. J., & Hersh, R. (1981). The mathematical experience. Brighton,UK: Harvester. De Corte, E., Greer, B., & Verschaffel, L. (1996). Mathematics teaching and learning. In: D. Berliner & R. Calfee (Eds.), Handbook of Education Psychology (pp. 491-549). New York: Macmillan De Morgan, A. (1910). Study and difficulties of mathematics. Chicago, IL: University of Chicago Press. Dunmore, C. (1992). Meta-level revolutions in mathematics. In: D. Gillies (Ed.), Revolutions in mathematics (pp. 209–225). Oxford: Oxford University Press. Fischbein, E. (1987). Intuition in science and mathematics. Dordrecht, The Netherlands: Reidel. Fischbein, E., Deri, M., Nello, M. S., & Marino, M. S. (1985). The role of implicit models in solving verbal problems in multiplication and division. Journal for Research in Mathematics Education, 16, 3–17. Freudenthal, H. (1990). Revisiting mathematics education: China lectures. Dordrecht, The Netherlands: Kluwer. Gardner, H. (1985). The mind’s new science. New York: Basic Books. Gillies, D. (1992a). Introduction. In: D. Gillies (Ed.), Revolutions in mathematics (pp. 1–14). Oxford: Oxford University Press. Gillies, D. (Ed.) (1992b). Revolutions in mathematics. Oxford: Oxford University Press. Greer, B. (1994). Extending the meaning of multiplication and division. In: G. Harel, & J. Confrey (Eds), The development of multiplicative reasoning in the learning of mathematics (pp. 61–85). Albany, NY: Suny Press. Greer, B. (2004). The growth of mathematics through conceptual restructuring. Learning and Instruction, 14, 541–548. Greer, B., & Verschaffel, L. (Eds). (1990). Mathematics education as a proving ground for information-processing theories. Special issue of International Journal of Educational Research, 14(1). Kaput, J. (1994). Democratizing access to calculus: New routes to old roots. In: A. H. Schoenfeld (Ed.), Mathematical thinking and problem solving (pp. 77–156). Hillsdale, NJ: Lawrence Erlbaum Associates. Kaput, J. (1999). Teaching and learning a new algebra. In: E. Fennema, & T. Romberg (Eds), Mathematics classrooms that promote understanding (pp. 133–155). Mahwah, NJ: Lawrence Erlbaum Associates. Kaput, J. J. (1992). Technology and mathematics education. In: D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 515–556). New York: Macmillan.

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Lakoff, G., & Nunez, R. E. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. New York: Basic Books. Lerman, S. (2000). The social turn in mathematics education research. In: J. Boaler (Ed.), Multiple perspectives on mathematics teaching and learning (pp. 19–44). Westport, CT: Ablex. Merenluoto, K., & Lehtinen, E. (2004). Number concept and conceptual change: Towards a systemic model of the processes of change. Learning and Instruction, 14, 519–534. Rosch, E. (1978). Principles of categorization. In: E. Rosch, & B. Lloyd (Eds), Cognition and categorization (pp. 27–48). Hillsdale, NJ: Lawrence Erlbaum Associates. Seo, K.-H., & Ginsburg, H. P. (2003). “You’ve got to carefully read the math sentence …”: Classroom context and children’s interpretations of the equals sign. In: A. J. Baroody, & A. Dowker (Eds), The development of arithmetic concepts and skills (pp. 161–188). Mahwah, NJ: Lawrence Erlbaum Associates. Sinclair, H. (1990). Learning: The interactive recreation of knowledge. In: L. P. Steffe, & T. Wood (Eds), Transforming children’s mathematics education: international perspectives (pp. 19–29). Hillsdale, NJ: Lawrence Erlbaum Associates. Thorndike, E. L. (1922). The psychology of arithmetic. New York: Macmillan. Verschaffel, L., Greer, B., & De Corte, E. (2007). Whole number concepts and operations. In: F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 557–628). Greenwich, CT: Information Age Publishing. Verschaffel, L., Greer, B., & Torbeyns, J. (2006). Numerical thinking. In: A. Gutierrez, & P. Boero (Eds), Handbook of research on the psychology of mathematics education — Past, present, and future (pp. 51–82). Rotterdam/Taipei: Sense Publishers. Vinner, S. (1982). Conflicts between definitions and intuitions — the case of the tangent. In: A. Vermandel (Ed.), Proceedings of the sixth international conference for the psychology of mathematics education (pp. 24–28). Antwerp, Belgium: Universitaire Instelling. Vinner, S. (1991). The role of definitions in the teaching and learning of mathematics. In: D. Tall (Ed.), Advanced mathematical thinking (pp. 65–81). Dordrecht, The Netherlands: Kluwer. Vlassis, J. (2004). Making sense of the minus sign or becoming flexible in “negativity”. Learning and Instruction, 14, 469–484. Vosniadou, S., Ionannides, C., Dimitrakopoulou, A., & Papademetriou, E. (2001). Designing learning environments to promote conceptual change in science. Learning and Instruction, 11, 381–419.

Chapter 22

Reconceptualizing Conceptual Change Anna Sfard This commentary is an exercise in reinterpretation: I will be looking at the four studies reported in this part of the volume and asking myself what would have changed — for better or for worse — had the researchers used somewhat different theoretical lenses. An attempt at theoretical modification is a useful thing to do. Recurring acts of reinterpretation keep us aware of the fact that different theories may have different relative advantages, whereas no one theory can “do it all.” While undertaking the task of theory modification, I will be recapitulating my own past experience. A quarter of a century ago, when I was beginning my career as an explorer of mathematical thinking, my research worldview was closer to the one adopted by the authors of this book than it is today. If I am now in a somewhat different place, it is exactly because of my habit to question my own work in the never-ending pursuit of answers to puzzles which the current approach left open. All four chapters discussed in this commentary focus on the same type of phenomena, ask the same questions, and, in their attempt to answer, use the same types of analysis. The focus is on the regularly observed, systematically non-standard (“misconceived”) ways in which mathematics learners solve certain types of problems. Thus, three of the four teams — Vamvakoussi and Vosniadou, Merenluoto and Palonen, and Tsamir & Tirosh — show that different kinds of numbers — rational, real, infinite — when first introduced, are treated as if they possessed all the properties of the previously learned numbers. The fourth team, Christou, Vosniadou, and Vamvakoussi, analyzes a similar phenomenon occurring in the transition from numbers to algebraic variables. All the authors agree that the student’s non-standard responses are indicative of the fact that she has reached a point in her mathematical development where learning cannot occur by a straightforward accrual. To be effective, the learning requires a change in the former knowledge. The existence of such critical developmental junctures has always been stirring much interest among students of human cognition. While in agreement on many related issues, such as the location of the “singular points” on the developmental continuum, the difficulties with which these transitions face the learner, and the behavioral indices of these difficulties, researchers have been divided on the question of how to conceptualize what they were able to observe. Consequently, they differed on two basic issues: What is it that changes? and What is the mechanism of the change? For Piaget (1950), the object of Reframing the Conceptual Change Approach in Learning and Instruction Copyright © 2007 by Elsevier Ltd. All rights of reproduction in any form reserved. ISBN: 0-08-045355-4

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change was the cognitive scheme sitting in the head of the learner, and the critical change was one that required the accommodation of a scheme rather than a straightforward assimilation of a new item into the existing mental structure. Vygotsky (1987) preferred to speak about the change of concepts, and viewed the critical transitions as the developmental junctures where everyday “complexes” reincarnate into scientific concepts. Although the adherents of the conceptual change (CC) approach adopt the Vygotskian unit of analysis, their tendency to talk in terms of models (e.g., synthetic) and of assimilating aspects of the new, incompatible information in their existing knowledge (e.g., Vamvakoussi & Vosniadou, this volume) makes their approach closer to that of Piaget than that of Vygotsky. The theoretical consistency of the four chapters is remarkable and not any less impressive is the authors’ explicitly stated belief in the power of CC to unify the existing research in mathematics education under a common theoretical framework. The contributors are also in the full agreement that this theoretization can then be used to predict students’ difficulties in mathematics learning and inform mathematics teaching (e.g., Vamvakoussi & Vosniadou, this volume). The authors’ enthusiasm is contagious. Indeed, there is much appeal in the CC framework: one cannot but admire its elegance, its unifying power, and its applicability as a tool for informing educational practice.1 This said, CC, like any other theory, has aspects that may benefit from additional refinement, and the operationality of its vocabulary is one of them. The authors of the chapters contained in this volume speak about concepts, conceptions, misconceptions, knowledge structures, information, models (e.g., synthetic), explanatory frameworks, conceptual change, etc., but none of these terms is explicitly defined. While reading the chapters I felt the need for some guidance about criteria used in identifying concepts or conceptions and in telling the conceptual change from any other. I also wished I had been given some operational explanation about the differences between misconceptions, synthetic models, and the models that should not be called synthetic. Finally, I wished to be told where the concepts, conceptions, or synthetic models are sited and how they can be retrieved from the data. What follows is an attempt to provide what is missing. While trying to operationalize the term concept it is natural to turn to the work of Vygotsky (1987), whose life project was devoted to the investigation of conceptual development, and who defined his key term, concepts, as referring to words together with their meaning. This description, to count as fully operational, requires that the word meaning be provided with an operational definition of its own. An inspiration for such definition may be found in Wittgenstein’s (1953) famous statement that meaning is the use of a word in language. When combined together, these two descriptions, Vygotskian and Wittgenteinian, define the term concept as referring to words together with their discursive uses. At this point, I could stop my attempts to clarify CC to myself. My aim seems to have been attained: the Vygotsky/Wittgenstein definition introduced above presents the key

1

CC’s long history both in science education and mathematics education research (its roots go back at least to Fischbein’s theory of intuitive-tacit models; see Fischbein, 1987, 2001) is an apt, if somewhat indirect, evidence of its attractiveness.

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notion concept in terms of features that are accessible to public inspection. As a result, we now have a pretty good idea about what we should look at while trying to pinpoint people’s concepts. And yet, since theories are well-organized, tightly interconnected systems of concepts, operationalization of one of these concepts is likely to have an effect on all the others. In the rest of this commentary I argue that this, indeed, is the case. More specifically, I show that the introduction of Vygotsky/Wittgenstein’s definition makes a difference in the unit and methods of analysis, in the resulting interpretations of data, and in practical implications.

Unit of Analysis Vygotsky/Wittgenstein’s definition suggests, among others, that in order to get a sense of, say, a person’s concept of number, one needs to consider the totality of this person’s discursive activities in which the term number may appear. The combined Vygotskian/Wittgensteinian rendition of the term concept thus makes it clear that in research on development of numerical thinking, nothing less than the entire discourse on numbers must constitute the unit of analysis. Since discourse is the activity of communicating, this latter fact justifies describing the resulting research framework as communicational.2 The new unit of analysis is, of course, much broader than the former one, the concept. In fact, the term numerical discourse (or discourse on numbers) is the communicational counterpart of the term knowledge of numbers.3 Almost any tenet of conceptual change theory may now be translated into a claim on discourse. Thus, for example, the CC statement [S]tudents are assimilating new information into prior knowledge instead of making radical changes. (Merenluoto & Palonen, this volume) corresponds to the communicational claim “Students are incorporating new numerical words and symbols into their current numerical discourse without introducing any changes in this discourse.” Similarly, the CC assertion [M]isconceptions … are caused when learners assimilate aspects of the new, incompatible information in their existing knowledge. (Vamvakoussi & Vosniadou, this volume) may be translated into the communicational statement “Many non-standard narratives produced by students result from their attempts to apply former discursive routines to new, incompatibly defined words and symbols.”

2

For detailed presentation of the communicational framework see, for example, Sfard and Lavie (2005). This, replacement is, indeed, necessitated by several theoretical developments, and the one due to Michael Foucault (1972) is probably most prominent among them. 3

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Methods of Analysis One may wonder whether these latter reformulations are any more than an idle semantic game. I thus now wish to claim that they do have far-reaching consequences. Among others, they are bound to inform methods of study. To begin with, the term discourse, unlike knowledge, is fully operational. There are public criteria for distinguishing discourses and thus also for detecting changes induced by processes of learning. Numerical discourse, like any other, is recognizable by its four characteristics: (1) its keywords (e.g., number words, words that refer to quantitative comparisons) and their routine uses, (2) its visual mediators (e.g., numerals, symbols for operations and for numerical relations) and their routine uses, (3) its routines (e.g., computational algorithms, heuristics for numerical estimations), and (4) its endorsed narratives (known also as facts about numbers, such as “2 ⫹ 2 ⫽ 4” or “rational numbers have the property of density”). All these characteristics are publicly accessible and straightforwardly investigable. The public nature of the numerous aspects of discourse which need to be attended to in research impose a number of methodological principles, some of which differ from those that guided the authors of the studies collected in this volume. True, human talk, either vocal or written, is the basic type of data for both CC and communicational explorer. The distinctive features of communicational approach, however, are, first, its uncompromising attention to the verbatim version of the interlocutors’ utterances and second, the fact that the interactions conducted for the sake of data collection are always documented and analyzed in their entirety (as opposed to their being attended only partially, as is the case when one chooses to analyze just the interviewees’ parts of oral or written exchanges). Thus, the communicational researcher begins her report with what was said, rather than with her own story about it. She is also aware that being a product of interaction rather than of individual doing, students’ answers may be sensitive even to minute changes in the wording of interviewer’s questions, not to speak about the interviewer’s facial expressions, her institutional membership, the seemingly unrelated things she says before and during the interview, the order in which she asks the questions, etc. All this means, among others, that the communicationalist would be reluctant to satisfy herself with written questionnaires, which do not give much access to the ways in which all these factors and, in particular, the formulations of the questions might have framed the answers. Thus, for example, one needs to consider the possibility that discursive routines evoked by Christou et al.’s (this volume) written request to “assign numbers to the following expressions” would have been quite different if the instruction were formulated in terms of substituting rather than assigning and if it spelled out explicitly what it was that the numbers should be substituted for (in other words, I am speculating that replacing the question “Are there any numbers … that you think ‘cannot’ be assigned to 4g?” with “Are there any numbers … that you think ‘cannot’ be substituted for g in the expression 4g?” might have brought a change in responses). Similarly, my experience taught me that asking “Which of the two sets is larger?” may elicit responses quite unlike those induced by the seeming equivalent question “Which of the sets has a greater number of elements?” (This fact is relevant to Tsamir and Tirosh’s (this volume) study; the explicit use of the word number may be found confusing by the student to whom it never occurred to associate infinite sets with numbers.) If not followed with an interview, the written answers will always remain subject to multiple interpretations and non-testable guesses.

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Results of Analysis (Interpretation) To substantiate the claim about the impact of the units of analysis on our interpretation of observed phenomena, let us consider one of the central claims of Christou et al.’ s (this volume) study, which was tested and reportedly corroborated also by Vamvakoussi and Vosniadou’s (this volume) and Merenluoto and Palonen’s (this volume) findings, namely that “many students even in secondary education tend to think that all numbers share all the properties of natural numbers.” This sentence can be translated into the communicational claim that “the students continue to use the word number and all the related terms and symbols the way they did so far.” From our own studies, in which we scrutinized children’s numerical discourse in its diverse manifestations, we know that the learner’s use of numerical keywords may be quite different from that of expert interlocutors, and that this may be true even in the case of the discourse on natural numbers. In particular, children tend to use the word “number” the way experts use the word “numeral,” that is, as referring to strings of decimal digits rather than to intangible entities for which the numeral is supposed to be but a representation.4 This interpretation of students’ non-standard answers, while fully consistent with the findings of Christou et al., Vamvakoussi and Vosniadou, and Merenluoto and Palonen studies,5 is somewhat different from the explanations given by these researchers. This difference, I wish to claim now, is not without practical consequences.

Teaching The interpretations provided by Christou et al. (this volume), Merenluoto and Palonen (this volume), and Vamvakoussi and Vosniadou (this volume) for their findings rest on the assumption that in all the cases, the interviewees and questionnaire respondents could be regarded as fully competent in the domain of natural numbers. Indeed, according to the authors, it was the familiarity with this basic type of number, accompanied by the conviction that what was true now would also be true in the future, that constituted the main obstacle to further development. Communicational researcher is less upbeat about the students’ prior learning. According to the alternative interpretation presented above, this former discourse may suffer from exactly the same weakness as its various extensions are known to have, except that in the discourse on natural numbers, this weakness — the lack of objectification6 — does not lead to visibly “misconceived” responses to questions such as those asked in the present studies. This interpretation of what the students can or cannot do has clear implications for what the teachers should aim for and how they should act to attain their goal.

4

Unfortunately, there is no space here to explain the operational criteria with which communicational researcher evaluates similarities and dissimilarities between discursive uses of different words; let me just say that one of the indicators of the use of number words as signifying abstract objects rather than numerals is that the user views such different symbolical and verbal forms as 6, 1 + 5, 3·2, 24/3, etc. as fully exchangeable — the feature that is missing as long as number words are interpreted as referring to specific numerals. 5 For the lack of space, I leave verification of this claim to the reader. 6 To objectify numerical discourse means to be able to use number words as if they signified intangible entities rather than decimal numerals. (see Footnote 4 ).

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Our ability to make sense of what we see depends on our uses of words. As illustrated above, the interpretation of the notion of concept that gave rise to the communicational framework is likely to make a difference in both research and practice. This said, I wish to stress again that the communicational approach was presented here not as a replacement for CC but rather as a theoretical lens of choice, to be applied whenever most appropriate. As I am never tired of reminding to myself, no one theory would ever satisfy all our needs. Indeed, as suitable and helpful as a theory may be in answering some questions, it will leave some other queries unanswered. To decide which theory to use in a given context, the researcher needs to consider, among others, its explanatory power and its potential to lead to consensus and accumulation of knowledge. The ultimate test of any interpretation, however, would be in practice. Each of the two approaches discussed in this commentary yields its own practical advice, and it is the empirically testable effectiveness of these didactic implications that may constitute the major consideration in our choice of theoretical lens. In the present two cases, such empirical examination is yet to come.

References Fischbein, E. (1987). Intuition in science and mathematics: An educational approach. Dodrecht, The Netherlands: Reidel. Fischbein, E. (2001) Tacit models and infinity. Educational Studies in Mathematics, 48(2,3), 309–329. Foucault, M. (1972). The archaelogy of knowledge. New York: Pantheon Books. Piaget, J. (1950). The psychology of intelligence. New York: Harcourt, Brace. Sfard, A., & Lavie, I. (2005). Why cannot children see as the same what grownups cannot see as different? Early numerical thinking revisited. Cognition and Instruction, 23(2), 237–309. Vygotsky, L. S. (1987). Thinking and speech. In: R. W. Rieber, & A. C. Carton (Eds), The collected works of L. S. Vygotsky (Vol. 1, pp. 39–285). New York: Plenum Press. Wittgenstein, L. (1953). In: G. E. M. Anscombe (Trans.), Philosophical investigations. Oxford, UK: Blackwell.

Subject Index affective, 3, 149–151, 160, 165, 202, 207–209, 211, 232, 326 algebra, 239, 242, 283–287, 289–291, 293–295, 310, 323, 326 algebraic, 243, 252, 284–295, 322, 329 expressions, 243, 284, 286, 288–295, 322 notation, 287 anti-realism, 22, 54–55 asymmetry, 9, 71–72, 77, 83 approach to learning and studying, 102, 151–154, 157–159 approaches to studying, 101, 123, 135, 140 background assumptions, 9–11, 86–87, 92, 241 beliefs, 2–8, 10, 20, 27, 36, 41, 49–50, 53–55, 59, 72, 90–91, 99–102, 105–120, 125, 145–155, 157–160, 165–171, 173–193, 197, 208, 213–214, 222–233, 240, 243, 295, 299, 301, 303, 305–306 cantorian set theory, 300–301, 304, 308, 310 certainty, 64–65, 86, 100, 117–118, 129, 147, 170, 225–226, 252–253, 256–257, 259, 290 classification, 51, 53, 58, 91, 124 coherentism, 107–111, 114 cognitive, 1–7, 12, 20–22, 27, 39, 41, 43, 90, 93, 100, 106–107, 109–112, 116–117, 120, 125, 131, 140, 149–150, 159–160, 165–166, 191–193, 202, 204, 207, 209–213, 230, 244, 250, 255, 306, 320, 322, 326, 330 conflict, 2–3, 159–160, 204, 250, 306 development, 1, 4–6, 27, 320 processes, 2, 7, 109–110, 112, 116–117 science, 93

communication breakdown, 70–71, 78 communicational framework, 244, 331, 334 comprehension, 100, 147, 167–172, 175, 179–180, 183, 190–192, 305 community of mathematicians, 239, 251, 258, 324–325 concept, 6–7, 9, 11, 37, 40–42, 48, 50–55, 57–61, 63–64, 66, 68–69, 73, 78, 84–87, 89–90, 111, 124–128, 136, 139, 149–150, 168, 171, 202, 204, 211–213, 224, 231, 239, 241–242, 247–257, 259–261, 265–269, 279–280, 283–285, 299, 302, 310–311, 321, 323–326, 330–331, 334 conceptions, 2–3, 11, 57, 101, 111, 116–117, 123–133, 135, 137, 139–140, 159, 165–170, 175, 179, 183, 190–193, 199–201, 203–204, 207–209, 213, 222, 224, 242, 247, 301–302, 311, 322, 330 conceptions of learning, 123, 125, 127, 129–133, 135, 137, 139, 159 conceptions of knowledge, 123, 128–130, 140, 159 conceptual, 1–12, 17, 19, 21, 25–27, 31–32, 35, 37, 39, 42–43, 47–55, 57, 59, 63–78, 83–87, 89–94, 97, 99–102, 105–120, 123–129, 133, 135–137, 139–140, 145–155, 157–160, 165–169, 171, 183, 189–193, 197, 199, 201, 203–205, 207, 209–215, 221–233, 237, 239–245, 247–253, 255, 257, 259–261, 265–269, 276, 278–280, 283, 285–287, 294–295, 299–301, 303–307, 309, 311, 319–327, 329–331, 333 asymmetry, 83

336

Subject Index

change, 1–12, 17, 19, 21, 25–26, 31–32, 35, 47–55, 57, 59, 63–69, 71, 73, 75, 77, 83–87, 89–93, 97, 99–102, 105–107, 109–120, 123–126, 145–151, 153, 155, 157–160, 165–169, 171, 183, 189–193, 197, 199, 201, 203–205, 207, 209, 211–215, 221–233, 237, 239–245, 247–249, 251–253, 255, 257, 259–260, 265–267, 269, 276, 278–280, 283, 285–287, 294–295, 299–301, 303, 305–307, 309, 311, 319–325, 327, 329–331, 333 development, 101, 123–124, 126–128, 133, 137, 139–140, 228–229, 261, 330 revolution, 93–94 system, 3, 65–66, 70–76, 90, 94 understanding, 101–102, 106, 123, 135–136, 148–150, 152, 154, 159–160, 193 constructivist, 1, 10–11, 100–102, 146–149, 151, 153, 158–160, 166, 197–201, 203–204, 206, 208–209, 212–213, 224, 226–228, 240, 269 contextual, 11, 47, 100–101, 108, 123, 127, 130, 146–147, 165, 229, 323 contextualism, 90 counter-intuitive, 7, 126, 147 counterintuitive, 63, 71, 241, 304 culture of mathematics, 251, 253, 256–259 development, 1, 4–6, 10–11, 19–20, 25, 27, 29–31, 48, 53, 60, 64, 66, 91, 99–101, 119, 123–129, 131–133, 135–137, 139–140, 146–147, 157, 159–160, 197–198, 202, 204, 207–211, 214–215, 221–222, 226–233, 241, 247–250, 252, 261, 265–267, 269, 283, 300, 302–303, 320–323, 329–331, 333 density, 249, 251, 254, 256, 259–260, 266, 269–270, 275–278, 284, 286, 323, 332 discourse, 21, 107, 117, 119, 125–126, 128–129, 131, 137, 139, 204, 206, 210–211, 213, 244, 331–333

discreteness, 11, 242, 266–268, 270, 273, 275–278, 284 domain learning, 228 domain specific, 4–5 education, 1–2, 11, 20–21, 26–27, 64, 78, 113, 117, 126, 130–131, 135, 148, 150, 167, 192, 239–240, 243–244, 248, 261, 267, 283–284, 299, 302, 311, 319–321, 323–326, 330, 333 enumeration, 66–67, 242 epistemic, 10, 55, 99, 101–102, 105–106, 108–110, 113–114, 116–120, 145, 183, 222–233 beliefs, 10, 99, 101–102, 105–106, 108, 113–114, 117–118, 145, 222–233 cognition, 99, 113, 116, 118, 183 epistemological, 10, 12, 20, 23, 99–102, 105–111, 113, 115, 117–120, 123, 128–129, 145–155, 157–160, 165–193, 197, 208, 214, 222–227, 229, 231, 233, 240 development, 101, 123, 129, 147 beliefs, 10, 99–102, 105–106, 118, 120, 145–155, 157–160, 165–171, 173–193, 197, 208, 214, 222–227, 229 epistemology, 38, 57, 97, 99–102, 105–108, 110, 112–113, 116–120, 130, 145–147, 149, 151, 158, 166–167, 221–223, 225–228, 230, 232, 258 expository text, 167, 170, 193 exemplar, 48, 91, 93 experts, 112, 120, 250, 259, 333 expertise, 5, 125, 222, 228, 233, 250, 252, 259 foundationalism, 105, 107–111, 114–115, 119 grammar, 22, 63, 65, 67, 69, 71, 73, 75, 77, 86, 92, 241, 320 grammatical conditions, 22, 64–65, 77

Subject Index historicist turn, 20 holism, 84, 90 idealism, 22, 64, 70, 92 incommensurable, 2, 6, 8–9, 26, 40–41, 43, 52, 279, 320 incommensurability, 3, 8–9, 11, 20–22, 30–31, 36, 40–43, 47, 52, 70, 72, 77, 83–84 local, 8, 22, 41, 83, 84 global, 8, 14, 42, 43, 83 infinity, 242, 275, 279, 300–302, 305, 307, 310, 321 initial explanatory frameworks, 243, 266–268, 270, 279 intuitions, 9, 41, 53–54, 60, 90, 240, 249, 300–301, 321, 324 intuitive, 4, 7, 38, 114–115, 126, 147, 209, 240, 243, 248, 251, 299–301, 307, 321–322, 324–325, 330 instruction, 1–2, 4, 7–8, 19, 25, 35, 47, 63, 83, 89, 99, 102, 105, 113, 115, 117, 123, 126, 145, 149–150, 160, 165, 197–198, 201–202, 206–210, 212–214, 221, 239–244, 247, 265–267, 270, 280, 283, 295, 299–302, 304–307, 311, 319–322, 326, 329, 332 justification, 66, 89, 105–110, 112, 117, 119–120, 145, 152, 166, 224–226, 326 knowledge, 1–8, 10–12, 25–27, 36, 38–40, 48–50, 53, 57, 63, 89–90, 99–102, 105–110, 114–120, 123–124, 126, 128–131, 133–135, 137–140, 145–152, 159–160, 165–167, 169–172, 175, 179, 183, 190–193, 198–204, 206, 208, 212–214, 222–233, 239–241, 243, 247–252, 255, 258–259, 261, 267, 269–270, 278, 280, 283–288, 294, 299–302, 304, 311, 319, 329–332, 334

337

knowledge objects, 123, 137–140 Kuhn, 1–3, 5–6, 8–9, 11, 17, 19–23, 25–32, 35–43, 47–49, 52, 60, 63–65, 69–70, 72, 75–76, 78, 83–84, 89–94, 100, 146, 193, 247 Kuhnian, 2, 22, 26, 28–30, 36–37, 40, 43, 65, 74, 78, 91 language, 4, 41, 47, 64, 67, 71, 73, 78, 86, 90, 93, 224, 248, 250, 261, 322–323, 330 learning, 1–11, 19–20, 25, 35–36, 39–41, 47, 63, 83, 89, 91–92, 99–102, 105, 108, 113, 117–118, 120, 123, 125–137, 139–140, 145–147, 149–155, 157–160, 165–171, 173, 175, 177, 179, 181, 183, 185, 187, 189, 191–193, 197–206, 208–215, 221–224, 226–233, 237, 239–245, 247–248, 251–255, 258–261, 265, 267, 280, 283, 299–303, 309–311, 319–323, 326, 329–330, 332–333 from text, 167 goals, 10, 101, 149, 151, 230 literal symbols, 239, 242–243, 283–291, 293–295 logical positivists, 1, 37 meaning, 8, 22, 25, 37, 40, 42–43, 47–51, 54, 64–66, 69–73, 78, 89–90, 93, 125, 127–132, 136–137, 139, 151–152, 154–155, 159, 172, 209, 241, 244, 265–266, 269, 278–279, 286, 294–295, 307, 321, 330 meaning change, 51, 70 mediators, 209, 332 metacognitive, 131–132, 135, 140, 149, 159, 190–191, 193, 228, 230, 326 metaconceptual awareness, 4–5, 7–8, 10, 12, 125, 131, 150–152, 154, 159, 168–169, 171–172, 190, 192, 228, 244, 303

338

Subject Index

mathematics, 1, 5, 8–11, 36, 38, 40, 66–67, 72, 168, 198, 214, 237, 239–245, 247–261, 265, 267–269, 280, 283–285, 289, 291, 299–302, 304, 310–311, 319–326, 329–330 mathematics learning, 11, 237, 239–245, 247, 260, 300, 310–311, 322, 326, 330 methodological principles, 332 motivation, 3, 134, 149, 202, 222, 305–306 motivational, 3, 149–151, 159–160, 165, 230 naive 13–14, 162, 163, 195 negative sign, 243, 288, 293–294, 322 normal science, 2, 9, 26, 28, 36–37, 47, 65, 91 number, 4, 7–8, 11, 22, 27, 30, 67, 73, 89, 101–102, 111, 140, 148, 160, 167, 199, 203, 206, 208, 212–214, 232–233, 239–243, 247–261, 265–280, 283–287, 291, 294, 299–300, 302–310, 319–324, 326, 331–333 concept, 11, 239, 241–242, 247–249, 251, 253, 255, 257, 259–261, 265–269, 280, 283–284, 324, 326 line 249, 251, 254, 256, 257, 259, 265, 269, 272, 274, 275, 323 numbers, 8, 67, 242–243, 247–261, 265–273, 275–280, 283–295, 299–300, 302–305, 307, 309–310, 320, 323–326, 329, 331–333 natural, 67, 242–243, 248–250, 265, 284, 333 negative, 266, 268, 280, 287–290, 300, 323 rational, 8, 242, 243, 249, 261, 284, 300, 324, 332 real, 247, 260, 267, 269, 284, 288, 299, 323 novices, 250, 259–260, 267–268, 285 Objectification, 333

paradigm, 2, 8–11, 21–22, 26, 28, 30–31, 35–38, 40, 43, 47, 63–78, 83, 86–87, 90, 92 paradigm shift, 26, 31, 35 personal epistemology, 97, 99–102, 118, 145–147, 151, 166–167, 223, 227–228 phenomenal sign, 243, 286–289, 291–294 philosophy, 2, 19–22, 27, 35, 37–38, 43, 47–48, 90, 99, 105–107, 120, 225, 241, 320 physics, 1, 5–6, 9, 11, 23, 25–26, 28–32, 51, 64, 69, 72, 75, 78, 84, 93–94, 101–102, 134, 145–155, 157–160, 167, 197–198, 201–202, 206, 209–214, 226–228, 242, 295 principles, 4, 11, 31, 36–37, 56, 90–91, 112, 117, 147, 151, 155, 159, 198–199, 239–240, 242–243, 256, 259, 299–301, 311, 319, 321–322, 325, 332 prior knowledge, 1, 107, 114, 118, 126, 133, 135, 150, 167, 169–171, 175, 179, 183, 199–201, 203–204, 206, 208, 227, 239–241, 243, 247–248, 251, 267, 269, 280, 283–284, 286–287, 294, 299, 301–302, 304, 311, 331 puzzle solving, 21, 36–37, 91 realism, 22, 47–49, 51–55, 57, 59, 61, 70, 107–108 reference, 41, 43, 53–55, 58–61, 75, 85, 90–91, 93, 113, 127, 168, 201, 214, 249, 268, 287, 305, 311, 324 refutational text, 102, 118, 147, 165, 167–171, 175, 179, 183, 190–193, 226–227, 230 relativism, 22, 64, 76–77, 92, 128–130, 137 reliabilism, 105, 107–108, 112–113, 117 representations, 11, 39, 94, 114, 125–126, 130, 169, 191, 193, 243–244, 256, 259, 261, 267–270, 275–280, 301, 303, 306, 308–310, 322–323

Subject Index revolutionary science, 26 revolutions in mathematics, 320, 326 science, 1–3, 7–9, 11, 19–23, 25–32, 35–41, 43, 47–49, 51–52, 58, 60, 65, 70, 73, 76, 78, 83, 89–93, 116–118, 124, 139, 146–148, 150–151, 165, 171–172, 191–192, 197–200, 203–204, 206, 208–210, 212–214, 226–227, 240–243, 247, 258, 299–302, 304, 306, 311, 319–320, 330 scientific, 1–5, 7–9, 11, 19–23, 25–31, 35–38, 40–41, 43, 47–61, 63–66, 70, 75–77, 89–93, 108, 110, 112, 114–117, 126–127, 146–147, 150, 166–169, 172, 183, 191–193, 198–200, 206, 211, 226, 247–248, 299, 306, 308–309, 330 community, 3, 21, 36 practice, 36, 56, 92 progress, 26, 77, 92 realism, 22, 47, 49 revolution, 1–2, 8, 11, 20–21, 25–27, 29, 31, 35, 40, 47 school context, 198, 248, 250–251, 259–261, 268–269, 324 shared practices, 248–249, 251, 260, 324 similarity, 6–7, 84, 86, 91–93, 138, 251 skepticism, 55, 106–108 socio-cultural, 2–4, 7, 10, 12, 150 sociocultural, 101, 111, 150, 232 social epistemology, 107–108, 112, 117 sociologist turn, 21 structure change, 89, 91, 93

339

study strategies, 10, 101, 135, 151, 158–159 symbolic representations, 243, 267–268, 270, 275–280, 309 synthetic models, 7–9, 11, 115, 150, 242–243, 247, 265, 269–270, 276–278, 283, 299, 311, 330 teaching, 3, 11–12, 20, 22, 25–26, 68, 71, 78, 101, 120, 123, 126–128, 133–135, 140, 192–193, 197–199, 201–202, 204–206, 208–215, 224, 226, 239–240, 243, 245, 247, 251–252, 256, 260, 280, 283–284, 299–301, 303, 305, 307, 309–311, 321–323, 326, 330, 333 teaching and learning processes, 197, 215 theory-laden, 36, 40, 75 theory-ladness, 36 understanding, 1, 3–4, 9–10, 19, 21, 26, 43, 56–57, 64, 68, 72, 74, 76, 78, 89, 94, 99–102, 105–107, 111, 113, 115, 117–118, 120, 123–139, 147–152, 154–155, 158–160, 165, 167, 190–193, 209–210, 227–229, 239, 242, 244, 248, 252–256, 261, 266–270, 275–276, 279, 283–284, 294, 310, 321–322, 325 unit of analysis, 12, 330–331 unobservable entities, 48, 51, 53, 55–56, 59 variable, 42, 112, 133, 154, 168, 175, 179, 183, 190, 192, 233, 284–287, 294, 323