Science Education Issues and Developments

SCIENCE EDUCATION ISSUES AND DEVELOPMENTS

SCIENCE EDUCATION ISSUES AND DEVELOPMENTS

CALVIN L. PETROSELLI EDITOR

Nova Science Publishers, Inc. New York

Copyright © 2008 by Nova Science Publishers, Inc.

All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Science education issues and developments / Calvin L. Petroselli, editor. p. cm. Includes index. ISBN-13: 978-1-60692-604-8 1. Science--Study and teaching. 2. Education. I. Petroselli, Calvin L. Q181.S37342 2008 507.1--dc22 2007031841

Published by Nova Science Publishers, Inc.

New York

CONTENTS Preface

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Expert Commentary Commentary

Why do We Lose Physics Students? Ronald Newburgh

1

Research and Review Articles Chapter 1

A Numerical Landscape Robert Adjiage and François Pluvinage

Chapter 2

Learning in and from Science Laboratories: Enhancing Students' Meta-Cognition and Argumentation Skills Avi Hofstein, Mira Kipnis and Per Kind

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The Crisis in Science Education and the Need to Enculturate All Learners in Science Stuart Rowlands

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Chapter 3

Chapter 4

Understanding Student Affect in Learning Mathematics Marja-Liisa Malmivuori

Chapter 5

The Challenge of Using the Multimodal Aspects of Informal Sources of Science Learning in the Context of Formal Education Krystallia Halkia and Menis Theodoridis

Chapter 6

Chapter 7

Understanding Scientific Evidence and the Data Collection Process: Explorations of Why, Who, When, What, and How Heisawn Jeong and Nancy B. Songer Constructivist-Informed Classroom Teaching: The Importance and Potential of Motivation Research David H. Palmer

5

125

151

179

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vi Chapter 8

Chapter 9

Chapter 10

Chapter 11

Index

Contents Oral Communication Competencies in the Science Classroom and the Scientific Workplace F. Elizabeth Gray, Lisa Emerson and Bruce MacKay

223

Supporting Future Teachers Learning to Teach Through an Integrated Model of Mentoring Pi-Jen Lin

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Strategies to Address Issues and Challenges Faced by Instructors in General Education Introductory Astronomy Courses for Non-Science Majors Michael C. LoPresto Getting it to Work: A Case of Success in Sustaining Science Professional Development Betty J. Young, Sally Beauman and Barbara Fitzsimmons

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PREFACE This book presents significant new analyses in the field of science education. This is hardly another field in education which is more important for a country's future than science education. Yet more and more students elect to concentrate on other fields to the exclusion of science for a variety of reasons: 1. The perception of degree of difficulty, 2. The actual degree of difficulty, 3. The lack of perceived prestige and earnings associated with the field. 4. The dearth of good and easy to use texts. 5. The lack of society in comprehending the significance of science and creating attractive incentives for those who enter the field. Chapter 1 - This study concerns the field of mathematics education. Today, for almost any technology, attaining the most advanced level relies on using digital systems. Therefore, the authors focus on number acquisition and use, emphasize major discussions about related topics, and introduce our personal contribution. The authors consider three areas: numbers in society, at school, and in the field of education. Numbers in society: Modern society needs two kinds of number users: the first has to deal with numbers, the second has to work with numbers. According to the framework for PISA assessment, dealing with numbers concerns every citizen facing situations in which the use of a quantitative… reasoning… would help clarify, formulate or solve a problem. Working with numbers is what a professional (marketing specialist, engineer, physician, artist…) does when he has to cope with numerical theoretical frameworks, or when he collaborates with a mathematician. This brings two types of questioning: what does competence in mathematics mean? What level of achievement is desirable in number-learning to meet either numerical need? Numbers at school: The authors examine what is proposed for teaching such a broad subject to 7-to-18-year-olds. The authors first observe and question the educational system, the designer of curricula, scenarios for teaching, training programmes, and national assessments. Secondly, the authors question the notions of problem solving and modelling as mere responses to mathematics-teaching issues. The authors then focus on what really happens in a standard classroom, particularly how teachers apply recommendations and directives, and how the generalization of assessments affects their practice. Numbers in the field of education: Three aspects are considered: epistemological, cognitive, and didactical. The authors distinguish four related levels which the authors have named: numeracy (competence linked to whole numbers), rationacy (competence linked to ratios and rational numbers), algebracy (competence linked to algebra), and functionacy (competence linked to calculus). The cognitive aspect evokes the essential issue of semiotic

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registers for representing and processing numerical objects, considers the discipline-ofexpression aspect of mathematics, and other issues taken into account by numerous researchers, such as process and object…. The didactical is devoted to exposing our conceptual framework for teaching numbers and understanding their learning. An experiment in ratio teaching is described and analysed. Chapter 2 - Laboratory activities have long had a distinctive and central role in the science curriculum and science educators have suggested that many benefits accrue from engaging students in science laboratory activities. More specifically, it has been suggested that, when properly developed, there is a potential to enhance students’ conceptual and procedural understanding, their practical and intellectual skills and their understanding of the nature of science. Research findings, however, have proven that “properly developed” laboratory work is less frequent than hoped for and that meaningful learning in laboratories is demanding and complex. The 21st century has offered new frames for dealing with the potential and challenges of laboratory based science teaching. This is an era of reform in which both the content and pedagogy of science learning and teaching are being scrutinized, and new standards intended to shape meaningful science education have emerged. The National Science Education Standards (National Research Council, 1996) and other science education literature (e.g. Lunetta, Hofstein and Clough, 2007) emphasize the importance of rethinking the role and practice of school laboratory work in light of these reforms. The new frames, however, also relates to the development in the understanding of human cognition and learning that has happened during the last 20 years. In the following chapter attention will be given to research on learning in and from the science laboratory. More specifically, the presentation will focus on the science laboratory as a unique learning environment for the following teaching and learning aspects: • •

Argumentation and the justification of assertions Development of metacognitive skill

It is suggested, that these are important aspects with a natural place in the science laboratory. They have, however, been neglected both regarding development of practical experiences provided to the student as well as in research on the effectiveness of practical work that is conducted in the context of science learning. A new approach is needed in which these two aspects are coordinated and seen in accordance with the general practice of teaching and learning in school science. Chapter 3 - There is a crisis in science education. Over the past two decades many organisations such as the American National Science Foundation, the Australian Audit of Science, Engineering and Technology and the UK’s Royal Society and the Confederation of British Industry, have reported a serious decline in students enrolling in science subjects and the failure of the science curriculum to inspire learners and to meet national needs. However, quite apart from instrumental reasons such as a national interest for having more scientists, science education is important for cultural reasons. Science permeates every aspect of modern life and arguably full citizenship in a technological society necessitates the understanding of science. Based on how the world is, science promotes critical thinking, a concern for evidence and an objectivity that is independent of personal opinion or the dictate of kings - yet few individuals have an elementary understanding of science. The failure of science education is

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reflected in science’s lack of popularity evident in the rise of mysticism, the rise in consensus of intelligent design, the postmodernist attack and the closing of many UK university science departments. There have been calls to remedy the situation, such as school visits by scientists and engineers, or overhauling the science curriculum by teaching the science deemed relevant to the everyday experience of children, consistent with the constructivist idea that there is a “children’s science”. However, there is no guarantee that exposure to the working lives of scientists will promote an interest in science and, moreover, science is not based on making sense of experience. To generate the interest and motivation of young learners requires an engagement with the nature of science (NOS) involving meta-discourse with the history and philosophy of the discipline. Contrary to the current wisdom of science educationalists, NOS has more to do with the rule-governed abstract possible world of the thought-experiment than hypothesis testing with a clipboard of data. Even the most concrete thinkers may be capable of thinking in the abstract and mechanics, because of its history and logical character (as opposed to the “soft sciences” such as ecology), provides the perfect opportunity to do this. This article consists of three parts: 1. Public perception of science and scientific literacy and understanding. 2. Why NOS is essential to science education. 3. Why “children’s science” and conceptual change, the largest domain in science education research, has failed to promote scientific understanding. Chapter 4 - Student affect has been one area of interest in mathematics education for decades. This applies in particular to rather large surveys of students in The United States since 1970’s. In general, education studies on affect have much focused on affective factors in the contexts related to mathematics achievements, learning of mathematics or solving mathematical problems. This is understandable since mathematics and mathematical problem solving carry many kinds of cognitive and sociocultural features that are not easily attached to the other school subjects. For example, the abstractness of mathematics and the differences in the symbol systems used in mathematical language set high demands on cognitive processes and also detach mathematics from the context and experience of everyday life. Furthermore, general views of mathematics as a difficult and demanding subject have caused it to be highly regarded and have been generally used to measure academic abilities. Mathematics tend to have a ritual value in societies that then cause powerful experiences with and important differences in mathematical performance. After showing passionate interest in human cognition and cognitive processes, education research paradigms have recently created new opportunities for and even laid emphasis on studies of student affect. Constructivism, together with applied socio-cognitive, cultural and contextual views of learning and education, has enriched our knowledge of affect in mathematics education research, as well. This theoretical chapter first discusses some conceptual features of affective factors traditionally applied in education research and especially in mathematics education studies. This short overview will then be followed by consideration of some of the most significant and often used affective variables in mathematics education research. More recently presented views of affect with cognition in learning will be considered as an introduction to the here suggested theoretical framework for understanding student affect in learning mathematics. Especially, perspectives on the coexistence of affect and cognition, on self-related cognitions and self-regulation are applied in constructing this suggested theoretical framework. It represents a dynamic, humanistic and socio-cognitive, viewpoint on the functioning and development of students´ powerful affect in their learning processes.

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Chapter 5 - In this work, an attempt has been made to study the plethora and the diversity of informal sources of science learning and the ways formal education may benefit by making use of these sources in its everyday school practice. Informal sources of science knowledge have many forms: they use several means of presenting scientific information, take place in several environments and use several ways to compose their “text”. Each one of them has its own communication codes and uses multiple ways (modes) to present its “meaning”. The material coming from them is chaotic, because it is diverse in terms of the means used, the purposes and the targets stated, the audience addressed, etc. To study them it is helpful to categorize them. Thus a three dimensional model has been developed. Each dimension describes one system of taxonomy: The first one refers to the environment and the conditions under which science learning takes place; the second refers to the way a science “text” is made up and the codes used; while the third one refers to the kind of mode used in the science “text”. Furthermore, the different learning environments in which informal science learning takes place have been studied. Three different learning environments have been distinguished: the organized out-of-schools visits to institutions and organizations (science museums, science centers, zoos, botanic gardens etc.), the students’/teachers’ personal navigation in several sources outside school and the use of informal sources of science learning by the teachers within their everyday classroom practice. The study reveals their particular characteristics, as well as their power and limitations. It also suggests ways of using them effectively in the context of science classroom. Chapter 6 - What is scientific evidence? How should scientific data be collected? These questions comprise essential components of scientific reasoning that are not well understood by students. This chapter explores conceptual challenges students face in inquiry-rich classrooms with respect to the notion of scientific evidence and the related data collection process. As students seek out evidence to support their inquiry, they are likely to ask and need to answer questions such as these: Why collect data? Who collects data? When should data be collected? What counts as scientific evidence? and How should scientific data be collected and analyzed? After examining conceptual issues involved in answering these questions, this chapter proposes that understanding what it means to collect scientific data and what scientific evidence is requires a complex understanding that involves conceptual, procedural, and epistemological knowledge. Chapter 7 - A constructivist paradigm has dominated science education research in recent years. According to this view, students use their existing preconceptions to interpret new experiences, and in doing so, these preconceptions may become modified or revised. In this way, science learning proceeds as children actively reconstruct their ideas as they become presented with new information. However, the implications of constructivism for classroom teaching are still open to question. This position paper refers to the science education literature to argue that strategies to arouse and maintain student motivation should be a crucial component of constructivist-informed classroom teaching. This is because constructivism is universally accepted to be an active process – students must make an effort to reconstruct their ideas, so it follows that if they are not motivated to make that effort then no learning will occur. However, extant models of constructivist classroom teaching make little if any mention of student motivation. In these models, the focus has typically been on strategies to elicit students’ prior conceptions and to guide and monitor their progress towards more scientific conceptions, but the motivational impetus for this process has received little attention. Perhaps one reason for this is that there are relatively few studies of student motivation in the

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science education literature. Another possible reason is the lack of a unified theory of motivation, which means that there is no clear consensus on how best to motivate students in the classroom. In view of this situation, there is a need for studies which can clarify motivational strategies in science classrooms. “Situational interest” is one motivation construct which appears to offer considerable potential, yet it has been largely ignored by science education researchers. Situational interest occurs when a particular situation generates interest in the majority of students in the class – a spectacular science demonstration might arouse transient situational interest even in students who are not normally interested in science. The potential of this construct lies in the fact that studies outside of science have shown that when situational interest is aroused on a number of occasions it can result in longterm personal interest and motivation in the topic. It is thus a potentially powerful construct for science education, and is one which should be further explored. Chapter 8 - This chapter investigates the importance of oral communication training in undergraduate scientific education. The authors examine the status of oral communication training in New Zealand universities and the debate concerning employer attitudes to this issue. The specific relevance of these issues to science education is explored through analysis of a case study and a qualitative and quantitative study of the attitudes of students and employers in science-related industries. Cronin, Grice and Palmerton (2000), Dannels (2001), and Morello (2000) argue that to significantly develop the rhetorical flexibility necessary to communicate competently, oral communication skills training needs to be discipline-specific and firmly contextualized in the genres, expectations, and conventions of the particular field. Responding to this call, a number of recent studies have examined the role of oral communication skill development in specific fields as diverse as design education, archaeology education, and engineering. This chapter moves the discussion of discipline-specific oral communication instruction to undergraduate science education. The recent inclusion of an oral communication component within a compulsory science communication class at Massey University, New Zealand remains a contentious issue. Possibly seeing oral communication training as a low priority in terms of student skills, knowledge, or preparedness for a future scientific career, both students and faculty have resisted the inclusion of oral communication into course curricula and assessment. The researchers designed a study to clarify whether oral communication skills were important to employers in science-related industries, what science employers meant by oral communication skills, and which skills they prioritized. At the same time, the team surveyed science students to better understand their attitudes to training in oral communication. Study findings strongly support the importance of oral communication skills in sciencebased employment in New Zealand. Science employers indicate that they require and value highly a wide variety of oral communication skills. The study also reveals that while science employers and university science students agree that oral communication skills will be important in scientific careers, the majority of employers find the desired level of these skills in new graduates only sometimes or occasionally. The retention of oral skills teaching and assessments, as currently exemplified by the Communication in the Sciences course at Massey University, is clearly indicated; study findings also make a strong case for an extended focus on oral competencies in undergraduate science education. Chapter 9 - The purpose of this article is to introduce an integrated model of mentoring for supporting future teachers learning to teach under the impact of teacher education reform

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of Taiwan, particularly, in the internship. This article begins with the introduction of teacher education reform and is followed by the description of the impact of teacher education on quality control. Then, it includes a brief description of six integrated reach projects investigated by teacher educators. One of the integrated research projects that was designed to improve mentors’ competence of mentoring for supporting future teachers learning to teach is reported in detailed and an integrated model of mentoring is developed. Finally, the views of mentors and the future teachers are described briefly and the issues of mentoring are addressed. Chapter 10 - The challenges currently faced by instructors of introductory general education college astronomy courses are numerous. Before effective instruction can even begin, student misconceptions must be addressed. This alone is a daunting task since astronomy is a field in which there are many misconceptions. If dispelling misconceptions is achieved, then effective methods of instruction must be identified and used. Since current research shows that most students learn very little from lectures, other approaches need to be employed. This then means that resources must be either located or created before implementation can occur. The recent movement to stress understanding of concepts rather than memorization and the regurgitation of facts requires that students be engaged and prompted to think critically, or scientifically. This is a challenge in itself, since, as useful as it may be, many non-science students are not used to thinking in this manner. In fact, many students come to class not even aware what science actually is, not a body of facts and figures, but rather a process of investigation. Mathematical illiteracy is not only rampant in our society, but in many cases condoned. Because of this, many non-science majors are math-phobic. They cringe at the site of an equation or graph, even if it is only used to explain a concept and they are not even required to actually use it. Many students are members of Carl Sagan’s “Demon Haunted World” mistaking not only astrology, but television shows, tabloid articles and internet sites about the “paranormal” for science. Many have learned all they know about science from movies and television. Also, some have deeply engrained religious beliefs that prevent them from approaching scientific ideas with an open mind. These challenges are not insurmountable. What follows are the details of various strategies that have been developed and employed to address these issues and challenges with the goal of improving instruction and the entire experience of introductory astronomy for both the students and the instructors. Chapter 11 - This article presents a case of a successful partnership between a university and nine school districts. Science educators, science and engineering faculty from the University joined forces with local school districts to attract funding and implement a high quality K-8 science curriculum supported by new materials and on-going professional development. There are five broad themes to the strategies that contributed to the success of the lasting the partnership: taking the load off central office administrators so that a high quality science curriculum with supportive PD “just happens” with another office managing the details, high quality communication among all partners, management/oversight/control, formative assessment of the quality of professional development implementation with redesign, and documenting results (e.g., parent interest, state-level school site visits, teachers’

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sense of preparedness to teach science, student achievement outcomes, and continued support by the University administration and faculty).

In: Science Education Issues and Developments Editor: C. L. Petroselli, pp. 1-4

ISBN: 978-1-60021-950-4 © 2008 Nova Science Publishers, Inc.

Expert Commentary

WHY DO WE LOSE PHYSICS STUDENTS? Ronald Newburgh1 Harvard Extension School, Cambridge, MA 02138, USA

ABSTRACT Why is it that we have such great difficulty in retaining students in physics? The answer lies, I believe, in the way most of us, as physics teachers, think. There is a wide spectrum in thought processes. Of the two major types one is linear (i.e. sequential) and the other lateral (i.e. seeking horizontal connections). Those who developed physics – from Galileo to Newton to Einstein to Heisenberg - were almost exclusively linear thinkers. The paradigm for linear thought is Euclidean thinking. Many physicists chose physics for their career as a result of their exposure to geometry. A consequence of this is that textbooks are usually written in a Euclidean format. Thus many beginning students look on physics as an exercise in Euclidean logic, with the attendant certainty that it implies. The sense of discovery is lost. Many students, male and female, do not recognize that the Euclidean format, though efficient, is not a valid description of how we do physics. Their way of approaching problems is different but just as valid. Too many physics teachers refuse to recognize the limitations of this approach, thereby causing would-be students who do not think in a Euclidean fashion to leave. Only when physics teachers are willing to make the effort to understand and even encourage other ways of thinking, will all students look on physics as a welcoming discipline.

The loss of physics students is an alarming trend. I believe that most physicists would agree that physics is an essential component for the development of an educated person. Certainly this belief was accepted in the eighteenth century when the study of natural philosophy was considered a sine qua non if a person were to be considered educated. Is this belief any less essential today? Moreover, the question of great importance to the nation. The renewal of our intellectual capital is a necessity for the maintenance of our standard of living. Yet the number of students who major in physics, as well as in other hard sciences and engineering, is continually 1

e-mail: [email protected].

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Ronald Newburgh

declining. This is especially true if we consider the number of native-born students. Until recently we have been able to cover the deficit with foreign-born students who remained in the country after graduation. However, for various reasons, many are now returning to their homelands. If we accept the need for more people trained in physics, we must try to answer the question posed in the title. The answer lies, I believe, in the way most physicists think, or perhaps more accurately, the way they appear to think. The term linear thinking is often used to describe scientific thinking. Frequently it is a synonym for logical thinking. This term is quite facile, and I feel that we should go beyond it. In this paper I wish to examine the influence of Euclid, not on the actual doing of physics but rather on the presentation of completed research and the writing of our textbooks. I suggest that Euclid may be the proximate cause of the flight from physics. Though I have no statistics, many physicists with whom I have spoken have said that their introduction to geometry was the reason for their going into physics. Einstein in his Autobiographical Notes [1] wrote the following. “At the age of 13 I experienced a second wonder of a totally different nature [The first was the gift of a compass when he was 5.] : in a little book dealing with Euclidean plane geometry, which came into my hands at the beginning of a school year. Here were assertions, as for example the intersection of the three altitudes of a triangle in one point, which – though by no means evident – could nevertheless be proved with such certainty that any doubt appeared to be out of the question. This lucidity and certainty made an indescribable impression on me. That the axiom had to be accepted unproved did not disturb me. In any case it was quite sufficient for me if I could peg proofs upon propositions the validity of which did not seem to me to be dubious. ... If thus it appeared that it was possible to get certain knowledge of the objects of experience by means of pure thinking, this “wonder” rested upon an error. Nevertheless, for anyone who experiences it for the first time, it is marvelous enough that man is capable at all to reach such a degree of certainty and purity in pure thinking as the Greeks showed us for the first time in geometry.”

Now I am no Einstein, but what he wrote is an exact description of my feelings on my first exposure to geometry, an experience that I can describe only as an epiphany. What has been the influence of Euclid on physicists? Think back on your own doctoral dissertation. In doing the research that led to the dissertation, the frustrations were enormous for nearly all of us – whether we were experimentalists or theorists. There were false starts, apparatus that did not work or broke, equations that did not describe the phenomena, and those that were insoluble. In conversations with advisors and fellow students we tossed out ideas, rejected some, argued about others and finally obtained definitive experimental results and developed a coherent description and resolution of the problem. One can hardly describe those years as an exercise in Euclidean logic. However, in writing the dissertation, we made the work conform to the Procrustean bed of Euclid. Very few dissertations describe or include the false steps. As a rule we present the research as starting from a carefully articulated thesis. It is designed to investigate and prove the effect of B on A and show how it relates to the principle C as enunciated by the eminent Dr. Pangloss. There is a sense of a monotonic progression – b follows a, c follows b, and therefore d is proven. The thesis is written with a strong sense of inevitability.

Why do We Lose Physics Students?

3

When the graduate student becomes a research scientist and teacher, both his scientific papers and textbooks have the same sense of inevitability. The format of our textbooks is Euclidean. Newton’s laws, Hamilton-Jacobi theory, and Maxwell’s equations are often presented as quasi-axioms in advanced texts. Elementary texts emulate them. After all, it is an efficient way of presenting complex material. It is just this approach that can confuse the beginning student. Seeing the material in this form and remembering Euclid, he will look on physics as a deductive discipline – exactly that which it isn’t. At the same time the instructor says that physics is an experimental science so that we discover our laws by induction. This contradiction, usually unrecognized, can create an intellectual malaise, especially for the questioning student. The logical certainty with which most textbooks are written is mirrored by the teacher in his presentation of the material. The problems accompanying the text do nothing to contradict this impression. Discussion of problem solving, whether in the text or by the teacher, usually becomes an application of sterile algorithms. Teachers talk of physics as an adventure in discovery, but most approach the subject in terms of meeting requirements such as the SAT’s. The laboratories become fixed exercises in which the student must confirm some principle already established. He knows the answer before he does the experiment. The result is that most students, in spite of all protestations to the contrary, look on physics as a matter of memorization. A few attempt to find the basic principles underlying the subject but then misuse them as the basis of deductive reasoning. It’s hardly surprising that many students decide that the subject is uninteresting and even illogical. No wonder they major in other fields. One teacher whose approach is a notable exception to this is Mazur [2] of Harvard. He lectures (or rather talks for 10 minutes), then poses a multiple choice question to the class. Using a computer interface, each students selects an answer. The results are recorded on a computer. Mazur then asks them to discuss their answers with the person beside them. This can take two or three minutes. They then vote again. Usually the second round leads to an overwhelming majority of correct answers. This method is one of real learning because the students are truly teaching themselves. Now I yield to no one in my admiration for Euclid. He has been an inspiration to many of us. We understand his genius but also see his limitations. Unfortunately there are many who do not follow his way of thinking. These, I submit, are the students we lose, both women and men,. By presenting alternate approaches to students, specifically uses of lateral thinking, false starts that must be corrected, and lessons that are discoveries not memorization, we can retain more students. We should remember that lateral thinking is essential to the formation of analogies, an activity that one cannot describe as Euclidean. Doing science without analogies seems to me an impossibility. At the same time I recognize that the introduction of alternate approaches must be a time consuming process. It is far less efficient than our current methods. It also requires teachers with greater flexibility, less rigidity, teachers who do not demand that the students parrot their analyses. Frost in his Mending Wall [3] writes of someone who “will not go beyond his father’s saying”. Mazur has gone beyond. If we are serious about the need for producing more scientists, so should we all. I feel that Euclid would agree.

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A POSTSCRIPT I realize that I have not conducted a scientific survey of the number of us who were influenced, indeed changed by Euclid in our formative years. However, many conversations going back to the 1940’s lead me to believe that the number is great. I also recognize that I have not proposed a specific program (or programs) for improving physics instruction. Too often, it seems, we believe that there is a single solution to a problem, if we could but find it. Just as there are many types of students, there are many ways to teach. I ask only that as teachers we are more open to the views of students and less rigid in our own thinking.

REFERENCES [1] [2] [3]

Albert Einstein, “Autobiographical Notes’ in Albert Einstein, Philosopher-Scientist, edited by Paul Schilpp (Library of Living Philosophers, Evanston, IL, 1949), pp. 9,11. Eric Mazur, Peer Instruction A User’s Manual, (Prentice Hall, Upper Saddle River, NJ, 1997). Robert Frost, “Mending Wall” in The New Modern American and British Poetry, edited by Louis Untermeyer (Harcourt Brace, New York, NY, 1941).

In: Science Education Issues and Developments Editor: C. L. Petroselli, pp. 5-57

ISBN: 978-1-60021-950-4 © 2008 Nova Science Publishers, Inc.

Chapter 1

A NUMERICAL LANDSCAPE Robert Adjiage1 and François Pluvinage ABSTRACT This study concerns the field of mathematics education. Today, for almost any technology, attaining the most advanced level relies on using digital systems. Therefore, we focus on number acquisition and use, emphasize major discussions about related topics, and introduce our personal contribution. We consider three areas: numbers in society, at school, and in the field of education. Numbers in society Modern society needs two kinds of number users: the first has to deal with numbers, the second has to work with numbers. According to the framework for PISA assessment, dealing with numbers concerns every citizen facing situations in which the use of a quantitative… reasoning… would help clarify, formulate or solve a problem. Working with numbers is what a professional (marketing specialist, engineer, physician, artist…) does when he has to cope with numerical theoretical frameworks, or when he collaborates with a mathematician. This brings two types of questioning: what does competence in mathematics mean? What level of achievement is desirable in number-learning to meet either numerical need? Numbers at school We examine what is proposed for teaching such a broad subject to 7-to-18-year-olds. We first observe and question the educational system, the designer of curricula, scenarios for teaching, training programmes, and national assessments. Secondly, we question the notions of problem solving and modelling as mere responses to mathematics-teaching issues. We then focus on what really happens in a standard classroom, particularly how teachers apply recommendations and directives, and how the generalization of assessments affects their practice. Numbers in the field of education Three aspects are considered: epistemological, cognitive, and didactical. We distinguish four related levels which we have named: numeracy (competence linked to whole numbers), rationacy (competence linked to ratios and rational numbers), algebracy 1 IUFM d'Alsace, 141, avenue de Colmar, 67089 STRASBOURG Cedex ,Personal address: 2, rue des Roses, F67170 MITTELHAUSEN,Tel : 00 33 (0)3 88 51 43 27, Mobile : 00 33 (0)6 16 35 32 01, [email protected], [email protected].

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Robert Adjiage and François Pluvinage (competence linked to algebra), and functionacy (competence linked to calculus). The cognitive aspect evokes the essential issue of semiotic registers for representing and processing numerical objects, considers the discipline-of-expression aspect of mathematics, and other issues taken into account by numerous researchers, such as process and object…. The didactical is devoted to exposing our conceptual framework for teaching numbers and understanding their learning. An experiment in ratio teaching is described and analysed.

INTRODUCTION Today, for almost any technology, attaining the most advanced level relies on using digital systems. Number acquisition has therefore become a central issue. As a consequence, we have decided to devote this chapter to numbers, learning them, and teaching them. We emphasize major discussions about related topics, and introduce our personal contribution. We will consider mathematics education from early childhood to grade 12. We mainly study the educational problems related to real numbers. Only a small subsection concerns complex numbers because they are mostly studied at university level. Mathematicians and advanced users of numerical domains face either the world of exact computation or the world of approximate computation. In the mathematical subject classification of the American Mathematical Society (2007, section 11), one finds a number theory section some of whose subsections (e.g. 11A Elementary number theory, 11D Diophantine equations, 11Y Computational number theory) are centred on exact computation. When performing exact computations, mathematicians consider certain subsets of real numbers (e.g. Ζ, Θ, Θ( 3 ), Ζ/2Ζ…) as separate entities. When performing approximate computations, educated people usually consider the field of real numbers as a whole. Nevertheless, although ancient Greek mathematicians already knew integers, rational numbers, and some irrational numbers like the square root of two, the construction of real numbers was only completed during the 19th century. This long period of maturation should suggest that mastering real numbers cannot be achieved quickly. Our main goal is to show that there are actually various stable levels of numerical acquisition. Mathematical problemsolving requires understanding the situation referred to, and then, adequate processing. Thus, for numerical problems, we can distinguish a level of access and a level of process. An individual facing a problem may: • • •

process the involved numerical objects correctly, when his own level allows him to both access and process the problem understand the statement but give an incorrect solution, when his own level allows him to access but not yet to process the problem not know how to proceed.

Modern society needs two kinds of number users. The first has to deal with numbers: “In real-world settings, citizens regularly face situations when shopping, travelling, cooking, dealing with their personal finances… in which the use of a quantitative… reasoning… would help clarify, formulate or solve a problem” (OECD, 2006, p.73). The second has to work with numbers. This is what a professional (marketing specialist, engineer, physician, artist…) does

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when he has to cope with numerical theoretical frameworks, or when he collaborates with a mathematician. What level of achievement is desirable in number-learning to meet either numerical need? Nowadays, this level is mainly defined in terms of competence, or competencies, by educational systems whereas it was previously mostly related to contents. The emphasis put on mastering real-world problems, in order to found well-advised judgments, by many institutions like OECD’s PISA2 project or the European parliament, reinforces this tendency. Piaget (1967, p.65), referring to Kant’s epistemological analysis of mathematics and physics, states: “Knowledge, [and thus acquiring knowledge] entails the relationship between subject and object”. Does not focusing on competencies, generally understood as a potential of action, neglect this relationship and therefore diminish the learning process? Does it not lead to mainly considering the product of teaching, when it has worked, instead of the development of this relationship? Briefly, does it not emphasize the end and thus put the means in the background? We first try to specify what society expects from number users, whether occasional or professional. We then observe and question the way educational systems design curricula, scenarios for teaching, training programmes, and national assessments, in order to enable people to take charge of the societal needs. We focus on modelling and applying mathematics, which has become a predominant topic, presented by many educational systems as a major response to mathematics-teaching issues. How are these official instructions and recommendations taken into account by teachers? What really happens in the classroom will be the theme of the fourth section. We thereafter introduce our framework of number acquisition. We emphasize cognitive aspects related to the specificity of mathematical objects: “Unlike material objects, however, advanced mathematical constructs are totally inaccessible to our senses… Indeed, even when we draw a function, or write down a number… the sign on the paper is but one among many possible representations of some abstract entity, which by itself can be neither seen or touched… for the mathematician…. It is important merely to know the rules or laws by which they may be combined.” (Sfard, 1991, p. 3). The semiotic nature of mathematical objects allows immediate access to them, whether they are already known or to be constructed. object “… unlike external mediation of instruments, semiotic mediation may be internalised, i.e. it may be transformed into infra conscious processing and thus become transparent from a phenomenological viewpoint” (Duval, 1998, p. 175). We then question the notion of competence, and its relation to knowledge. We wonder what should be the extent of this notion, for it to allow designing teaching objectives and assessments for number learning. We consider the epistemological aspect of number learning, focussing on numbers as objects inserted into a coherent network with its ruptures and continuities (Brousseau, 1997, pp. 79-99). Following our questioning about coherence and competence in number acquisition, we define four fields of competencies that allow to clearly specify the level of competence of an individual intended to deal or work with mathematics.

2

Programme for International Student Assessment, which assesses the abilities of 15-year-old students from 41 countries (including 30 of the most developed) to apply learning to problems with a real-world context.

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Robert Adjiage and François Pluvinage

NUMBERS IN SOCIETY Dealing with Numbers In order to define the social expectations for dealing with numbers, we can consult documents with rules or regulations that contain numbers. Understanding these rules is a social obligation, and legal pursuits may result from their lack of application, or from incorrect application. Rules requiring calculations are present in most fields of human activity, so that we shall consider a sample of official documents. Let us first assert that in all the documents that we have consulted, there are three levels in dealing with numbers. We shall describe these later in this paper. Therefore, the nature of the encountered phenomenon does not actually depend on the field of activity one chooses. At the moment we were writing this text, we read that World Water Day 2007 is celebrated each year on 22 March3 (by the way note the wrong use of the number 2007 in this sentence). As it appears anyway that water is something vital and possibly scarce, we suggest limiting our investigation to the domain of water: its consumption and its use. Texts of rules or regulations can refer to numbers in such a way that knowledge of decimal numbers and arithmetic operations, or numeracy, allows the reader to act as required. Such is the case in the following Mexican law, about consumers’ shares for water consumption. Each magnitude and arithmetic operation is completely described in a rhetorical way in this text (followed by its English translation).

Cámara de Diputados del H. Congreso de la Unión LEY DE CONTRIBUCIÓN DE MEJORAS POR OBRAS PÚBLICAS Y FEDERALES DE INFRAESTRUCTURA HIDRÁULICA Artículo 7 – II Nueva Ley D.O.F. 26/12/1990 Tratándose de acueductos o sistemas de suministro de agua en bloque realizados exclusivamente con inversión federal, el monto de la contribución obtenida en el artículo anterior se dividirá entre la capacidad de suministro del sistema, medida en metros cúbicos por segundo, y el cociente obtenido se multiplicará por el volumen asignado o concesionado por la Comisión Nacional del Agua a cada usuario del sistema, medido en metros cúbicos por segundo y el resultado será el monto de la contribución a cargo de cada contribuyente. Article 7-2 For aqueducts or water supply systems built with federal investment only, the amount of the contribution defined in the preceding article will be divided by the global capability of the system in cubic meters by second, and the obtained quotient will be multiplied by the volume in cubic meters per second attributed to each consumer, and the result will be the amount to be paid by each customer. 3

Extract from the UN-water webpage < http://www.unwater.org/wwd07/flashindex.html >: 'Coping with Water Scarcity' is the theme for World Water Day 2007, which is celebrated each year on 22 March.

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The following general part of a US federal text introduces tables (we reproduce only one table, but there are several). Compared to the preceding document, this one supposes from its readers practice in consulting mathematical presentations (some other documents also present diagrams). But it deals similarly with numbers: Every arithmetic operation is well defined (see below). Moreover, the section of the document devoted to definitions includes magnitudes and units not necessarily known by everyone, e.g. the so-called degree-day. We conclude that the expected competence is only numeracy. Hereafter we shall see that in a more specialised part of the same document, the reader is supposed to deal with formulas.

CODE OF FEDERAL REGULATIONS [2006] 10cfr434-- Part 434_Energy Code For New Federal Commercial And MultiFamily High Rise Residential Buildings Degree-day, heating: a unit, based upon temperature difference and time, used in estimating heating energy consumption. For any one day, when the mean temperature is less than a reference temperature, typically 65ºF, there are as many degree-days as degrees Fahrenheit temperature difference between the mean temperature for the day and the reference temperature. Annual heating degree days (HDD) are the sum of the degree-days over a calendar year. § 434.518 Service water heating. 518.1 The service water loads for Prototype and Reference Buildings are defined in terms of Btu/h per person4 in Table 518.1.1, Service Hot Water Quantities. The service water heating loads from Table 518.1.1 are prescribed assumptions for multi-family high-rise residential buildings and default assumptions for all other buildings. The same service water-heating load assumptions shall be made in calculating Design Energy Consumption as were used in calculating the Energy Cost Budget.

Table 518.1.1.—Service Hot Water Quantities Building type Assembly Office Retail Warehouse School Hotel/Motel Restaurant Health Multi-family High Rise Residential (2)

Btu/person-hour (1) 215 175 135 225 215 1110 390 135 1700

(1) This value is the number to be multiplied by the percentage multipliers of the Building Profile Schedules in Table 513.2.b. See Table 513.2.a for occupancy levels. (2) Total hot water use per dwelling unit for each hour shall be 3,400 Btu/h times the multi-family high rise residential building SWH system multiplier from Table 513.2.b. 4

The British thermal unit (BTU or Btu) is a unit of energy used in North America (1 Btu ≈ 1.055 Joule). See http://en.wikipedia.org/wiki/British_thermal_unit .

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We do not reproduce Table 513.2 in which we could find the multipliers. Observe a possible difficulty for interpreting certain units of magnitudes: Slash refers to division as usual in arithmetic, but dash indicates product in degree-days and Btu/person-hour When reading official texts, we are often surprised by the importance of the use of common language, even in cases in which the use of formulas would simplify heavy verbal formulations. Some legal texts however introduce mathematical formulas, which shorten expression. See for example a French text taken from J.O. (Official Journal of French Republic, 12-29-2002, n 303, p. 60 059 text 5) and followed by its English translation. Avis et communications Ministère de l'écologie et du développement durable Avis relatif à des délibérations des agences de l'eau AGENCE DE L'EAU ARTOIS-PICARDIE Délibération n° 2002-A-063 du 4 octobre 2002, NOR: DEVE0210424V Article 9 Mesure indirecte des volumes prélevés 1. Calcul du prélèvement en fonction de l'énergie électrique consommée Le volume prélevé est obtenu par application de la formule suivante : P = 250 × W/Z avec: P : volume prélevé en mètres cubes durant la période soumise à redevance ; W : énergie électrique consommée mesurée au compteur, exprimée en kWh ; Z : hauteur théorique minimale d'élévation en mètres.

MINISTRY OF ECOLOGY AND DURABLE DEVELOPMENT (FRANCE), ANNOUNCEMENT OF DECISIONS TAKEN BY THE WATER AGENCY OF ARTOIS-PICARDIE (10/4/2002) Article 9 Indirect measurements of volumes of pumped water 1. Water volume computed with measurement of consumed electrical energy The volume P in cubic meters is obtained using the following formula: P = 250 × W/Z, where W is the measured electrical energy in kWh and Z the theoretical minimal height of elevation in meters.

We find a formula in this text, but we can assert that using this formula does not require algebraic processing: substituting numbers for letters and then perform arithmetic calculation is sufficient to find the result. Therefore, the reader is only expected to be at numeracy level. But a question arises: Under what conditions does dealing with a formula entail the use of an effective algebraic language? Nevertheless, many texts, generally including at least ratios and proportions, require a more advanced knowledge from their readers. The following text belongs to this category. Drinking Water Standards

Priority Rulemakings

Arsenic The Safe Drinking Water Act requires EPA to revise the existing 50 parts per billion (ppb) standard for arsenic in drinking water. EPA is implementing a 10 ppb standard for arsenic. Ground Water Rule EPA proposed a rule which specified the appropriate use of disinfection in ground water and addressed other components of ground water systems to assure public health protection. Lead and Copper EPA estimates that approximately 20 percent of human exposure to lead is attributable to lead in drinking water.

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The quoted text is only part of a larger document from U.S. Environmental Protection Agency . The first pages of this document are devoted to a general presentation of public health protection. The considered text concerns everybody, so that it is supposed to be understood by any reader. Below, we present a second part of this document. It gives details of measures and procedures, so that we could think that it is only directed at specialists. But these specialists, in turn, are in charge of delivering the information to a large range of citizens who are supposed to understand the involved concepts. Electronic Code of Federal Regulations (e-CFR) e-CFR Data is current as of March 16, 2007 http://www.gpoaccess.gov/cfr/index.html (c)Lead and copper action levels. (1) The lead action level is exceeded if the concentration of lead in more than 10 percent of tap water samples collected during any monitoring period conducted in accordance with §141.86 is greater than 0.015 mg/L (i.e., if the “90th percentile” lead level is greater than 0.015 mg/L). (2) The copper action level is exceeded if the concentration of copper in more than 10 percent of tap water samples collected during any monitoring period conducted in accordance with §141.86 is greater than 1.3 mg/L (i.e., if the “90th percentile” copper level is greater than 1.3 mg/L). (3) The 90th percentile lead and copper levels shall be computed as follows: (i) The results of all lead or copper samples taken during a monitoring period shall be placed in ascending order from the sample with the lowest concentration to the sample with the highest concentration. Each sampling result shall be assigned a number, ascending by single integers beginning with the number 1 for the sample with the lowest contaminant level. The number assigned to the sample with the highest contaminant level shall be equal to the total number of samples taken. (ii) The number of samples taken during the monitoring period shall be multiplied by 0.9. (iii) The contaminant concentration in the numbered sample yielded by the calculation in paragraph (c)(3)(ii) is the 90th percentile contaminant level. (iv) For water systems serving fewer than 100 people that collect 5 samples per monitoring period, the 90th percentile is computed by taking the average of the highest and second highest concentrations. This text refers to ratios without mentioning how to process them. A precise idea of ratio seems to be necessary for understanding at least the last statement of each presented part (Lead and Copper in Priority Rulemaking, 3-iv in e-CFR). This would be a second level of numerical knowledge required. On the other hand, although the text refers to statistical notions such as percentiles, we will not go so far as stating that the intended readers are supposed to master statistics as the required processes are explicitly defined. In the Energy Code section mentioned above, the reader is expected to know how to process formulas, which implies familiarity with algebraic symbolic language (knowledge of

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symbol Σ and sub-indices, distributive rule). This brings us to the third level of mathematical social expectations, perceivable in the following quoted example (p. 466 of the Energy Code, loc. cit.). The overall thermal transmittance of the building envelope shall be calculated in accordance with Equation 402.1.2: Uo=ΣUiAi/Ao=(U1A1 +U2A2 + . . . +UnAn)/Ao (402.1.2) Where: Uo = the area-weighted average thermal transmittance of the gross area of the building envelope; i.e., the exterior wall assembly including fenestration and doors, the roof and ceiling assembly, and the floor assembly, Btu/(h·ft2·ºF) Ao = the gross area of the building envelope, ft2 Ui = the thermal transmittance of each individual path of the building envelope, i.e., the opaque portion or the fenestration, Btu/(h·ft2·ºF) Ui = 1/Ri (where Ri is the total resistance to heat flow of an individual path through the building envelope) Ai = the area of each individual element of the building envelope, ft2

Working with Numbers Professional mathematicians obviously work with numbers, and over the last three decades employment of mathematicians has been increasing in many sectors of human activity: astronomy, meteorology, aeronautic traffic, bank, marketing, health, quality control, industrial design, musical acoustics, and so on. If we add traditional mathematics users like physicists, and mathematics teachers, the result is a large number of people who dedicate much of their time to working with numbers. The major aims are to predict, to control, to optimise, and to decide (e.g. how to reduce cost or waste in industrial processing). Modelling (e.g. traffic regulation by programming traffic lights, designing a ship hull) is a usual tool at the present time. To be sure, computation and computers have largely contributed to such a development, particularly in designing models and checking their conformity with the reality they simulate. A classical use of number knowledge has been to discover relationships between realworld quantities, and thus find formulas. Renowned examples of results published during the 20th century are Pareto’s principle in economy (20% of the population earns 80% of the income) and Zipf’s law in linguistics. This law is a consequence of an economic principle (principle of least effort): A writer, or a speaker, has a more or less easy access to words, depending on the frequency in which he uses them (the more frequent, the easier). Originally, Zipf's law stated that, in a corpus of natural language utterances, the frequency of any word is roughly inversely proportional to its rank in the frequency table. (Wikipedia, 2007a). Let us for instance consider the following statistics related to a collection of 423 short TIME magazine articles (total number of term occurrences: N = 245,412).

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Table 1. Top 15 terms of short TIME magazine articles Rank r

Word

Frequency f

r*f

r*f/N

1

the

15861

15861

0.065

2

of

7239

14478

0.059

3

to

6331

18993

0.077

4

a

5878

23512

0.096

5

and

5614

28070

0.114

6

in

5294

31764

0.129

7

that

2507

17549

0.072

8

for

2228

17824

0.073

9

was

2149

19341

0.079

10

with

1839

18390

0.075

11

his

1815

19965

0.081

12

is

1810

21720

0.089

13

he

1700

22100

0.090

14

as

1581

22134

0.090

15

on

1551

23265

0.095

The product A = r*f/N in the last column tends to be equal to about 0.1. (http://linkage.rockefeller.edu/wli/zipf/cmpsci546_spring2002_notes.pdf, p. 3)

More recent kinds of work with numbers are: digitalisation, cryptography, and compression (zip). A typical example of digitalisation is that a coloured pixel on a screen is represented by a triple of numbers of the interval [0, 255]. In cryptography, a widespread method is NSA, related to public and private keys. It is based on the quasi-impossibility of finding the factors of a given product of two large prime numbers. Image compression produces JPEG files. The latter require the use of wavelets, a kind of mathematical function that allows decomposing a given function into different frequency components. All these domains are being developed on the base of constant contact between specialists and users. This supposes a sufficient general level of education. Thus, collaborative working becomes rather difficult between people whose competence differ by more than one level, for instance between somebody who is working at the level of functional analysis and somebody who does not yet master algebraic writing.

CURRICULA AND ASSESSMENT; THE CASE OF MODELLING After analysing the societal needs related to number learning, we investigate in this section curricular genesis and development, i.e. the process by which curricula are conceived and designed by institutions, as well as instructions for implementation. How does this

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process take account of both societal needs of dealing with and working with numbers? We will mainly consider an issue that has been in the foreground these last few years: mathematical modelling. Following the main line of this paper, we will focus on the numerical aspects of this topic.

PISA, Modelling, and Curricula We choose modelling because the way this domain of mathematics has emerged in just a few years in many curricula for teaching, teacher training, and the related research, as well as the emphasis put on it, seems to be characteristic of a modern process of reforming educational systems in the occidental world. What is intended by modelling? Let us refer to the Discussion Document (Blum, 2004, pp.152-153), prior to ICMI Study 2004 (International Commission on Mathematical Instruction), aimed at raising some important issues related to the theory and practice of teaching and learning mathematical modelling and applications. The authors of this Discussion Document (14 people from 12 countries), are the members of the International Programme Committee for this ICMI Study. “The starting point is normally a certain situation in the real world. Simplifying it, structuring it and making it more precise… leads to the formulation of a real model of the situation…. If appropriate, real data are collected in order to provide more information…This real model, still a part of the real world… is mathematised…” Mathematical results are then derived, using mathematical methods… and then “re-translated into the real world, that is, interpreted in relation to the original situation”. Checking whether the results are appropriate and reasonable now validates the model. “If need be… the whole process has to be repeated with a modified or a totally different model”. “…The obtained solution of the original real world problem is stated and communicated”.

Modelling is an important theme, taken into account these last few decades by numerous studies and international conferences in the field of mathematics education (see for instance: Niss, 1987; Blum et al.1989; Galbraith et al., 1990; Lesh et al., 2002). Nevertheless, “…genuine modelling activities are still rather rare in mathematics lessons” (Blum, 2004, p. 150). Now PISA has defined and tested mathematical literacy: “The emphasis in PISA is ‘on mathematical knowledge put into functional use in a multitude of different situations and contexts’. Therefore, real situations as well as interpreting, reflecting and validating mathematical results in “reality” are essential processes when solving literacy-oriented problems.” (Blum, 2004, p. p. 151-152). Major countries of the occidental world, including the USA, appeared to perform less well than expected at PISA 2000 and 2003. (For detailed results, see NCES, 2001, p. 43; OECD, 2004, p. 53 and pp. 89-95). As a consequence “an intense discussion has started, in several countries, about aims and design of mathematics instruction in schools, and especially about the role of mathematical modelling, applications of mathematics and relations to the real world.” (Blum, 2004, p. 151). One finds for instance, on the web site of the U.S. Department of Education (2004), a press release entitled: PISA results show need for high school reform, in which Secretary of

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Education Rod Paige said: “The PISA results are a blinking warning light”. In releasing the U.S. findings, Robert Lerner, commissioner of NCES, said, “PISA provides important information about education in the United States and in other industrialised nations, giving us an external perspective on U.S. performance. We need PISA in particular because it offers such a different measure of achievement, one that poses complex problems that students might realistically encounter in their lives.” (Ibid, 2004). National Center for Education Statistics (2002-2006) provides a huge amount of documents addressing PISA: access to PISA items and results, international comparisons, official and scientific reports, aids for educators…. The European Parliament on its part defined (2006, p.1) Key Competencies for lifelong learning. Concerning mathematics, it states that: “An individual should have the skills to apply basic mathematical principles and processes in everyday contexts at home and work, and to follow and assess chains of arguments” (2006, p.18 pdf version), putting the emphasis on applying mathematics in ‘everyday contexts’. What kind of echo to these statements and recommendations can we find in official learning curricula? We will give two examples drawn from a wealth of international material. In 1996, the New York State Board of Regents adopted learning standards for all content (subject) areas. Since then, the New York State Education Department (NYSED) has issued a series of core curricula, which provide an additional level of specificity to these learning standards. The Core Curriculum Standard 3, from Pre-kindergarten to grade 12 (New York State Education Department, 2005), appears as carefully thought-out and precise. It defines a mathematical proficiency relying on conceptual understanding, procedural fluency, and problem solving. In introduction to this Core Curriculum (ibid, p. 1), it is stated: “Most problems that students will encounter in the real world are multi-step or process problems. Solution of these problems involves the integration of conceptual understanding and procedural knowledge. …Many textbook problems are not typical of those that students will meet in real life. Therefore, students need to be able to have a general understanding of how to analyse a problem and how to choose the most useful strategy for solving the problem.” We first note this formulation that considers the complexity (‘multi-step’) of problems that students meet in the real world as a model for learning. The word ‘model’ and derived words like ‘modelling’ appear 88 times in the full document, throughout all grades, to which one can add 21 occurrences of expressions as: ‘everyday’ situations or ‘experiences or ‘real world’ problems or situations. In France, the Minister of National Education issued a document that defines a common base of knowledge and competencies5 to be acquired at the end of mandatory schooling. This document explicitly refers (MEN, 2006, p. 2) to two sources: the already quoted Recommendation of the European Parliament (2006, p.1) related to Key Competencies for lifelong learning, and “international assessments, particularly PISA. It specifies (ibid, pp. 5-6) that: “mathematics provide tools for acting, choosing, and deciding in everyday life”, and that “mastering the main elements of mathematics essentially depends on problem solving, particularly when the involved problems rely on situations stemming from the real world.” New learning curricula have therefore been published (or are in the process of being published). These mention as one of the main aims the capacity to use mathematics in different domains (everyday life, other disciplines). 5 Socle commun de connaissances et de compétences

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Thus, across many countries in the world: “Researchers and practitioners in maths education and policymakers have reached agreement that mathematics education should enable students to apply mathematics in their everyday life (PISA) and contribute to the development of active citizens (Council of the European Union following Lisbon Report, 2001, p.4, p. 13).” (LEMA, 2006, project-background page). Of course, this requires new interest and competencies in teachers who generally do not successfully integrate applications of mathematics into their course design and daily classroom practice. Appropriate pre- and inservice teacher training has therefore been, or is being, conceived. For instance, a European project, COMENIUS-LEMA (Learning and Education in and through Modelling and Applications) is being developed. Partners of the project are institutions from different countries: France, Germany, Hungary, Poland, Spain, and United-Kingdom. Mathematics education researchers from these countries, including one the authors of the present paper, participate or collaborate in this project, which “proposes to support teachers with development of their pedagogic practice in mathematical modelling and applications by developing a teacher training course”. Target groups are: “in-service and pre-service teachers at primary and lower secondary level and teacher trainers”. (LEMA, 2006, project page). We are now in measure to specify the leading strand, in the mathematical modelling case, of the current evolution in educational systems. An international assessment, PISA, has provided policy makers with benchmarks that allow to reconsider their systems' performances, and to identify potential strategies to improve, according to PISA’s Standards, student achievement. Their recommendations have determined important inflections of curricula intended to help students to better handle the real world. Researchers, already interested in this topic, have designed or are designing training programs for aiding teachers to take account of curricula and apply the related educational instructions in their lessons. Now, how are PISA-assessed competencies determined? PISA has been influenced by the Danish KOM Project, initiated by the Ministry of Education in order to profoundly reform Danish mathematics education from school to university, and its director, Morgen Niss: “It should be noted that the thinking behind and before the Danish KOM-project has influenced the mathematics domain of OECD’s PISA project, partly because the author is a member of the mathematics expert group for the project”. (Niss, 2003, p. 12). According to Winsløw (2005, p. 141-142) the empirical and theoretical bases of Niss’ model are of two orders: Scandinavian tradition and Project Pedagogy. Concerning the former, mathematics education has been considered since the seventies, in a society more and more influenced by mathematical models, as a mean of making sense of the world in which students live and will work, and, as a consequence, developing active citizenship. In particular, this tradition refuses to consider mathematics as politically neutral; it challenges a didactic that contents itself with passing on scientific knowledge, and thus is mostly oriented towards competence acquisition. Concerning the latter, students acquire knowledge when they feel a need for it, that is, according to real-world demands, and this explains the great interest of the KOM project in modelling. May one therefore consider that PISA results are absolute indicators (as often presented in press releases), although they in fact reflect only a particular measure of student performance? Moreover, must curricula be determined by assessments, risking the behaviourist shift feared by Brousseau (2007) and Chevallard (2002)? Reforming educational

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systems on only the PISA basis could only improve students’ performance at… PISA. Is this our unique goal? What sort of achievement do we expect for students and for what purpose?

Mathematical Coherence, Modelling, and Curricula We suggest another thought process. First of all, we must specify competencies for lifelong number learning, relying on a framework that takes account of personal development, and mathematical and didactical coherence. By personal development, we mean development that leads to either work or deal with numbers. By mathematical and didactical coherence we mean that our references for interpreting and structuring society and individual needs in number learning are mathematical objects, concepts, notions, domains (whole numbers, rational numbers, algebra, calculus, analysis) and the related studies in mathematics education. Assessments would be of course based on these competencies. As we can note, this process is opposite to the process described above: It stems from individuals and their needs in number learning and leads to assessments via curricula and competencies. May modelling remain a central reference in such a process of defining objectives for lifelong number learning? For debating this important question, we first go back to the ICMI Discussion Document (Blum, 2004), which is a very complete overall study on the subject. This document brings a real framework not only for appreciating and analysing the considered complexity and scope of modelling, but also for exhibiting any subject relevant to this theme. We especially focus on the chapter devoted to “Examples of important issues.” We particularly note that: •

• •

The interrelations between applications and modelling, and mathematics, from both a purely mathematical and a didactical viewpoint, are strongly taken into account. This goes in the direction of fostering mathematical coherence, which is one of the main lines we have indicated for lifelong learning. Shaping or restoring the image of mathematics is an important aim of applicationand-modelling promoters. Authenticity of the involved material is one the main considered issues.

And this leads us to examine the type of material, i.e. real-world situations, available and frequently used when implementing teaching sequences related to modelling. Let us refer to two authors that have conducted and reported many classroom-based studies on this subject: A. Peter-Koop, and K. Maass who participates in LEMA. In one of her papers (2004, p. 457), the former mentions that she has resorted to Fermi problems for the reported experience: “Enrico Fermi, who in 1938 won the Nobel Prize for physics, was known by his students for posing open problems that could only be solved by giving a reasonable estimate. Fermi problems such as ‘How many piano tuners are there in Chicago?’ share the characteristic that the initial response of the problem solver is that the problem could not possibly be solved without recourse to further reference material”.

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She gives a bit farther the following criteria as guidelines for the choice of problems posed in the research on work with 3rd and 4th-graders: • •

• •

The problems should present challenges and intrinsically motivate cooperation and interaction with peers. The wording of the problems should not contain numbers in order to avoid that the children immediately start calculating without first analysing the context of the given situation, and to challenge pupils to engage in estimation and rough calculation and/or the collection of relevant data. The problems should be based on a selection of real-world-related situations that include reference contexts for third and fourth graders. The problems should be open-beginning as well as open-ended real-world-related tasks that require decision-making with respect to the modelling process.

Four problems have therefore been posed to 3th- and 4th- graders: 1) How much paper does your school use in one month? (paper problem). 2) How many children are as heavy as a polar bear? (polar bear problem). 3) Your class is planning a trip to visit the Cologne Cathedral. Is it better to travel by bus or by train? (cathedral problem). 4) There is a 3 km tailback on the A1 motorway between Muenster and Bremen. How many vehicles are caught in this traffic jam? (traffic problem).

K. Maass (2005, p. 4), gives a list of problems she submitted to 7th- and 8th- graders. Among them, we retain the numerical items: 1) How many people can be found in a 25km-long traffic jam? 2) How can different charges of diverse mobile contracts be clearly arranged depending on customers’ habits? [….?…] 3) Is it possible to heat the water required in Stuttgart-Waldhausen with solar collectors on the roofs? 4) What is the connection between the height of fall and the subsequent height of rebound of a ball?

We lastly report a problem that has been experimented for LEMA: Giant’s foot. The following photo was taken in an English amusement park. How approximately tall is the entire figure, of which we can only see the foot? All these situations are much more than a “dressing up” of a “mere” mathematical problem, and the modelling process consists of much more than “undressing” the real-world problem. In this sense they bring great progress compared to numerous problems one finds in usual textbooks. The declared guidelines are respected. The quasi-absence of numbers in the statement is a main feature of these problems, and we have personally observed that this results in more considering relationships between the involved quantities rather than starting haphazard calculation. For instance, we have observed many 5th-graders, dealing with the “giant” problem, that have tried to find the number of men (or visible barriers, or boot-

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soles…) that one could put into the whole estimated length of the giant. This attitude brings them close to the proportionality model, whereas the presence of numbers frequently leads students to combining numbers, often by adding them, without reflecting the relationships linking the underlying quantities.

6

Figure 1. Giant’s foot .

One observes that many situations, almost all of them actually, rely on two pillars: data collection, and then applying a multiplicative model: proportionality, calculation of averages… So that pupils are supposed to resort to an already studied (and somehow standard) model. Very few existing situations lead to developing a mathematical model. K. Maass’s fourth problem quoted above is of that kind, although it mobilises linear functions. We have suggested, but not already experienced in classrooms, situations as car braking distance, which depends on the square of the speed. This could be a new issue to experiment. Great care should also be brought to situations that give real possibilities of validating the retained data and the findings. For instance, the traffic-jam problem (K. Maass, number one) depends on many choices made by students: the ratio cars/trucks, the average length of each, the average number of persons in a car… This problem is thus open-ended, the results depending on these choices, themselves having to be coherent with the retained time of the day or period of the year. Whatever ones choice, it seems difficult to directly validate the employed ratio cars/trucks or average length of cars and trucks, this information being not easily available, e.g. on Internet. On the other hand, it could be possible to check, e.g. on specialised radio channels, the number of persons involved in a given traffic jam, and this may validate the retained data. Can modelling effectively contribute towards promoting views of mathematics that extend beyond transmissive techniques to its role as a tool for structuring other areas of knowledge? (Blum, 2004, p. 161). Modelling is also intended to promote interaction, cooperation, and communication: “The real-world problems used in the study should intrinsically present challenges and thus motivate peer interaction during the solution process as opposed to problems that can be solved quite easily by an individual student.” 6

The authors are grateful to Richard Philipps from: http://www.problempictures.co.uk/, who has allowed them to use this picture.

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Robert Adjiage and François Pluvinage

(Peter-Koop, 2002, p.563). Succeeding in promoting these attitudes in the classroom does not only depend on the available didactical material. Teachers are often rushed by their schedule, whereas modelling activities demand giving students the time for acquiring the knowledge required by the real world situation, for collecting data, for testing hypotheses, for elaborating or identifying an appropriate mathematical model, for discussing all this matter between peers. So that teaching is also of great importance. We have observed many sequences, e.g. related to the giant situation, where productive peer interactions have been noted, leading to unexpected but correct procedures for calculating the giant’s height. But for lack of time, lack of training, the teacher did not take these ideas into account and favoured the procedures that he imagined. So that part of the expected benefit, in terms of the image of mathematics and interest of working groups, may be lost or invisible. In any case, the challenge of modelling has not been sufficient to change this teacher’s usual practise.For concluding this section, we would say that modelling still challenges research, training, and teaching. Important progress has already been made concerning the epistemology and the material related to modelling. Other issues, which we have mentioned above, remain to be explored. Modelling is coherent with the personal development of an individual and what society expects from him, whether he is intended to work or to deal with mathematics. However, let us beware of excessive enthusiasm. Let us consider other teaching entries, which take account of purely numerical obstacles and coherence, particularly those that help students to become aware of continuities and ruptures from numeracy to mastering calculus and analysis, going through ratios and algebra, because these themes structure all numerical development. Anyway, being capable of advanced modelling, particularly developing numerical models (and not only resorting to taught models), relies on mastering these fundamental areas and their interrelations.

STUDYING NUMBERS: HOW LONG AND UP TO WHICH DEGREE OF ACCURACY? In the previous section, we examined the way curricula are or might be designed. Here, we are interested in the general features and progression of curricula related to number acquisition, and the way they are actually implemented in the classroom. We take advantage of this presentation for sketching some continuities and ruptures through general numerical areas: whole numbers, ratio and rational numbers, algebra, calculus and analysis, which we consider essential for describing and understanding this learning. Numbers together with their operations constitute the first objects in learning mathematics, and pupils continually face the learning of numbers. At elementary school, teaching numbers is present each year, and has the most important place among the mathematical topics. After presentation of natural numbers and arithmetic operations, students learn about positive decimal numbers and fractions. Later, they study negative numbers, but from this stage, learning numbers does not follow a continuous path in curriculum development. That means that some properties, concerning for instance exponents, are examined without introduction of new concepts. And the theory of numbers, including theorems such as Fermat’s theorem (if p is a prime number, then for any integer a, the number a p – a is divisible by p), is actually taught only in advanced courses. Nowadays numeral systems, like binary system or hexadecimal, are only considered as a topic for

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prospective mathematicians or specialists in computing. At middle school the students are led to deal with some irrational numbers, canonical examples of them being 2 and π. But only a scarce minority of college students in scientific careers masters real numbers. Nevertheless, in many countries, core curricula in mathematics education formulate their objectives in such a way that a reader may think that at the end of grade 7 or 10, (it depends on the educational system) students are supposed to master real numbers. Moreover, complementary official documents, such as those of Federal Resources for Educational Excellence in the USA, strengthen this misleading impression. For example, many textbooks, or online documents, present a chapter or a list of exercises devoted to real numbers. If we carefully examine the content, it often appears that the presented properties (operations and order) do not characterise real numbers, but also apply to rational numbers. The most difficult one, and also the most important, the closure property (if a nonempty set of real numbers has an upper bound, then it has a least upper bound), is dismissed. Summarizing our experience as well as observations made by various researchers, we can assert that general tendencies while teaching mathematics are the following. • •

• • • • • •

At primary school, strong priority is given to teaching numbers, for instance in comparison with geometry or other topics. For this reason, magnitudes do not constitute an effective component of learning whole numbers, although researchers, such as Galperin quoted by Arievitch7 (2003, p. 280), have stated opposite educational principles. Teachers carefully introduce the mathematical tools (scripts and algorithms) Nevertheless, they do not systematically train students in mental calculation, nor present them arithmetical techniques such as casting out nines They do not consider computational tools as a positive resource of numerical problems They do not consider certain links between the various presented ways of expressing numbers, e.g. from repeating decimals to fractions At the secondary level, algebra progressively takes the place of numbers, but does not explicitly help to better understand numbers Teachers give little consideration to numerical proofs.

We are not surprised, in these conditions, that educational goals are difficult to achieve for many students. National and international assessments allow us to better understand what is currently expected from learning, and to observe the distance between the expectations (intended curriculum) and the obtained results (implemented curriculum). Learning numbers seems to be obvious for people who master them, for the apparent easiness that operative rules present. But if we want to build a mathematical world, rather than an arbitrary one, each general rule has to be justified and demonstrated, which leads us to consider both the universe of concrete applications and the management of representations. Let us report a case that we observed in a kindergarten classroom: An experienced teacher 7

The underlying assumption in Galperin’s approach to learning and activity is that there are two different types of actions: (1) an ideal action that is performed in the presence of objects and (2) an ideal action that is carried out separately, without any presence of objects.

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Robert Adjiage and François Pluvinage

organized a play, called “our boat is sinking”, with a group of 12 pupils. In a song mentioning a boat, suddenly come the words “our boat is sinking, for evacuation you have to form groups represents a number that the teacher chooses. When the chosen number was of ”, where 3 or 4, the task was easy for the pupils, but later, a choice was 8. Then the natural question arises: How many pupils are missing for forming two groups of eight? The teacher recommended using an abacus (representation) without allowing the pupils to consider the concrete situation that they could directly build: e.g. a row of 8 facing the remaining pupils, then constituting as many pairs as possible and counting how many pupils in the row of 8 remained alone.

Figure 2.

All students would easily understand this situation, whereas the link between the problem and its numerical representation and processing using the abacus (2×8 – 12) was not obvious for such young children. In this elementary situation we observe a possible link between learning numbers and exploiting a concrete situation. The observed lesson was very attractive for the pupils, so that we can only point out here some “loss of learning”, like one says “loss of earnings” in a commercial relationship. But we have also observed most important sources of difficulty in other situations. For instance, in a course about speed and velocity observed at grade 6, activities were proposed in the textbook, such as comparing the constant velocity of two swimmers. One of them takes 1 min 30 sec for 100 m, and a second swims 250 m in 4 min. The textbook also proposes comparing different students’ approaches. But instead of this, the teacher’s conception, i.e. applying the general formula v= d and comparing ratios, prevailed:

t

a female student built a table for the first swimmer’s times, under the hypothesis of constant velocity. Her correct result, 3 min 45 sec for 250m, was considered wrong and erased by the teacher. The latter reminded that the problem statement gave for the second swimmer 250 m in 4 min and that the question was to compare swimmers’ velocity by comparing ratios.

Figure 3. “A mistake”.

It is however well known that most 6th-graders need for this comparison such intermediary steps before being able to understand and apply the formula. These steps mark stages in the acquisition of processing involving ratios and proportions, and thus in acquiring competence related to these notions.

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What characterises the acquisition of competence is the length of learning and the need of a wide variety of activities. We have presented two examples, one referring to the acquisition of counting and calculating, the second to ratio and proportion. PISA2 assessment reveals that many 15-year-old people do not properly manage ratio problems, even if they have been taught for several years. At a little more advanced level, the teaching of algebra starts. But secondary school pupils encounter serious difficulties with algebraic processing. As an example, we give hereafter six typical multiple- choice questions extracted from a pre-test, and the table of answers given by a sample of Mexican students at the beginning of undergraduate engineering programs. Pocket calculators are not allowed. Q2. Calculate: 3 − 5

4 6

−1 ; -1; 12

38 ; Other 24 1 1 Q3. Calculate: − 2n − 2 2n 1 1 2n − 1 ; − ; ; Other 0; 2 n ( n − 1) 2n ( n − 1) 2 ; 24

Q4. Which of the following numbers is largest? 3.5; 2.46; 3.19; 0.546 -9.87 Q5. Calculate: 3 ( 5-8 ) + 3 ⎡⎣ 2 ( 4+9 ) -5 ( 2-6 ) ⎤⎦ 2

9; 30; -55; 229; Other Q8. The solution of −30 x + 4 ≤ 0 is:

2 2 ; ( −∞ , ∞ ) ; x ≤ ; Other 15 15 2 Q9. The solutions of 2 x − 3 x − 2 = 0 are: 2 2 −1 x1 = −2 and x2 = ; x1 = 2 and x2 = − ; x1 = 2 and x 2 = ; 3 3 2 2 x1 = −2 and x2 = − ; Other 3 x=

2 ; 15

x≥

Table 2. Empirical evidence - percentages of choice made by undergraduate students Choice Q2 % Q3 % Q4 % Q5 % Q8 % Q9 %

1 25 9 88 4 21 17

2 65 25 0 3 39 26

3 1 30 2 9 3 43

4 4 27 4 2 36 5

5 5 9 6 83 1 9

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Robert Adjiage and François Pluvinage

In the table above, the percentage of right answers lies in bold in gray cells. There is a strong contrast between numerical questions (Q2, Q4 and Q5) and questions including variables (Q3, Q8 and Q9, marked by the symbol). For Q3, Q8 and Q9 only a minority chose the correct answer. We particularly observe in Q8 that students were able to recognize the fraction

2 as representing 4 (choices 1, 2 and 4), but then had difficulty in 15 30

algebraically managing the inequality: The three corresponding choices are frequent. This underlines the difference between processing ratios and fractions and processing algebraic expressions. For this reason, it seems necessary to distinguish a level of mathematical competence that consists in managing algebra. If we consider the curricula up to grade 12, we can find other topics related to numbers in courses of Algebra. For instance, in the 2005 core curriculum of the State of New York, complex numbers are introduced in Algebra 2 and Trigonometry (grade 12). But in this case, the label “Algebra” may be questioned. Historically, complex numbers in the form of square roots of negative numbers were considered for solving polynomial equations: third grade by Niccolo Tartaglia and Gerolamo Cardano, and fourth grade by Lodovico Ferrari (all these results obtained during the 16th century). But later attempts at formalising encountered paradoxes,

such

as

the

extension

(−1)(−1) = 1 = 1 on one hand, and to

of

the

−1 −1 =

property

(

−1

)

2

ab = a b

leading

to

= −1 on the other hand, hence 1

= -1! For this reason, in Euler’s time, mathematicians introduced the symbol i (for imaginary) instead of the controversial −1 . Both numbers, i and its opposite –i, have -1 as square (then are acceptable square roots of -1), so that the paradox could be explained. Nevertheless, the full extent of complex numbers arose only two centuries later, with the proof of the fundamental theorem of algebra, also known as the d’Alembert/Gauss theorem. This theorem states that every non-constant polynomial with complex coefficients has at least one root. In other words, the field of complex numbers is said to be algebraically closed. As a consequence, a polynomial of degree n has n roots, counting multiplicities. The first book in which this result was stated (for polynomials with real coefficients) was published in 1629. Its author Albert Girard gave as an example that the equation x 4 = 4 x − 3 has the four roots: 1, 1, −1 + i 2, − 1 − i 2 (in modern writing). But the first attempt at proving the theorem was only made by d’Alembert in 1746. The proof given by d’Alembert was incomplete. Only in 1806, J. R. Argand (1768-1822), a non-professional mathematician who geometrically interpreted i as a rotation of 90º in the plane, published a nearly satisfactory proof (Argand, 1874, pp. 90-91), and Gauss later produced another proof. Although the theorem is about algebra, all proofs used analysis, e.g. Liouville’s theorem: “In complex analysis, every bounded entire function is constant”. Indeed, if a polynomial P(z) does not have any root, it 1 is entire and bounded, therefore P(z) reduces to a constant. follows that the function P( z ) From these epistemological considerations, it appears that an important jump was necessary: from discovering algorithms for solving some equations, to first formulating, and then proving general results about polynomials. Therefore, we cannot consider that manipulating algebraic expressions should constitute the highest level of mathematical competence that teaching up to grade 12 has to implement. Stating general results about

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functions, and solving problems, the solutions of which being functions, seem to be elements of a more advanced field of competence than simply solving numerical equations, or even parametrical equations.

COGNITIVE ASPECTS After examining institutional expectations as well as practices concerning the use and the acquisition of numbers, we will now introduce cognitive aspects of the general framework to which we refer. As already mentioned in the introduction, we focus on the semiotic features of mathematical objects, we regard numerical processes as if they were linguistic ones, and this leads us to propose a characterisation of algebra. We lastly consider the link between numbers and graphical representations.

Semiotic Registers Writing or representing numbers mobilizes various semiotic registers. Let us specify this assertion: “We do not have any perceptive or instrumental access to mathematical objects… as for any other object or phenomenon of the external world…”, “…the only way of gaining access to them is using signs, words, or symbols…” Duval (2000, p. 61). As seen above, realworld experiences are certainly a starting point for understanding the pertinence of numbers and then entering their universe. Duval’s assertion does not mean that we can do without referring to the real world for teaching numbers, and further large areas of mathematics. It only means that, at a certain point of learning, pure mathematical objects must be considered, if only because they can be dissociated from a particular context: for example fractions have to be detached from a measurement context, in which they are usually introduced, to apply to many contexts, e.g. mixture, enlargement… This is what an educated adult population can do, what 7th-graders, taken as a whole, are not yet able to do. This level of comprehension is certainly a condition for mathematical modelling. What Duval states is that symbols are the only way of accessing these pure mathematical entities. Now, there exist, and students are actually taught to face, many means of expression in mathematics. For instance, when expressing and processing rational numbers, we have at our disposal: fractions, of course, decimal numbers, two separate whole numbers (like in the expression “a player makes an average of 3 out of 4 basket attempts”), visual representations such as “pie charts” and number lines which are supposed to help students to better understand rational numbers… Duval (1995, pp. 15-85; 2000, p. 60-65) considers these diverse means of expression as separate and organised systems or “settings”, which he terms “semiotic registers”, necessary for mathematical activity and particularly teaching and learning, for at least four reasons (Duval, 1995, pp. 68-69; 2000, p. 62). • •

They are the only paths to mathematical objects, in the sense described above. They help students to distinguish a mathematical object from its representation: disposing of many representations allows considering what is invariable beyond these representations, thus outlining the underlying object.

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Robert Adjiage and François Pluvinage •

They highlight all aspects of a mathematical object: each representation is partial, and we need many representations for accessing all the complexity of a mathematical object. For instance, a fractional representation such as 37

5

•

highlights the

multiplicative relationship between 37 and 5, whereas the decimal representation 7.4 of the same number highlights its location (between 7 and 8, closer to 7 than 8…). They give alternatives when processing: a fraction can be useful for interpreting a ratio, but decimal numbers may make the comparison of ratios easier.

Duval (1995, pp. 39-44; 2000, p. 63) distinguishes two kinds of processing. “Treatment”: 3 6 = , and “conversion”: transformation of an object within a given register, e.g. 5 10 transformation of the representation of a mathematical object into a representation of the 3 same object in another register, e.g. = 0.6 . He states that the latter, unlike the former, 5 entails a rupture in the means of representing and processing, and thus in thinking. Conversion between semiotic registers is a cognitive operation essential for objectifying mathematical entities. Treatment and conversion are necessary for expressing mathematical rules and properties that legitimate processing. Constraints are determinant in the distinction between simple illustration and a semiotic register. For instance, it is usual to represent an addition like 5 + 4 on the number line with an arrow joining 5 to the point located 4 units ahead. But we note but we note in upper Figure 4 that the number 4 does not explicitly appear (we have to count 4 units). Thus, we consider this representation as a relevant illustration for 5 + 4, but not for a treatment in a semiotic register. It is truly different when the same operation is made with an “additive slide rule”: this kind of continuous abacus presents two rulers side by side, the upper one sliding at the user’s demand, and a cursor. Nowadays, students may use a virtual tool: the model presented in Figure 4 was built with CABRI. In this case, the upper ruler has its origin at point S facing 5, and the point A of the cursor faces 4. The cursor shows the result 9 on the lower ruler. Note that the same configuration may also be read: 9 – 4 = 5. Thus, the “additive slide rule” allows converting a given configuration into two distinct arithmetic equalities: 5 + 4 = 9 or 9 - 4 = 5. Moreover we observe that, if S remains constant, moving the cursor allows the expression of any sum of the kind 5 + a. In other words, the configuration: S constant, cursor variable, is the translation of the sentence “Add 5” expressed in the natural language or verbal register. We can also process a sum like 5 + 4 + 3 without determining an intermediary result. Starting from the initial position (the two 0’s facing one another), we first slide the upper ruler 5 units forward and we put the cursor on 4, then we slide the upper ruler a second time in order to put its origin under the cursor, and finally we move the cursor to 3. The result appears on the lower ruler. All these considerations show that the additive slide rule constitutes a semiotic register with its own treatments, distinct from both verbal and arithmetic registers. Software like Excel or the worksheet in Open Office give an apparent opportunity to see the same number expressed in various settings. For a given cell of a worksheet, you may select various number formats: decimal number, fraction, percentage, etc.

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Figure 4. representing 5 + 4 = 9 in two different ways.

In the fraction case, you must choose the denominator range (e.g. less than 10, 100…), making the involved representations and processing not exact, as is possible with software like Derive or Maple. If we calculate the sum: 11 + 23 with Derive or Maple in exact mode,

30 70 73 we obtain the correct result: . But suppose we enter each fraction in a cell of a worksheet, 105 e.g. cells A1 and A2, and then calculate the sum in cell A3. We choose, in the considered cells, fractions with denominator less than 100 (which is the largest option in Open Office) as number formats. We obtain 16/23 with Excel and 57/82 with Open Office. Such experiments or only their results could be interesting for 7-graders equipped with pocket calculators. Are the fractions 16/23 and 57/82 distant from or very close to one another? And in relation to the exact result 73/105…? Let us underline here that the distinctions we have introduced are important even when using software. Some software (with worksheets, as in our examples) has only one kind of “treatment” in Duval’s sense. Standard worksheets always operate with Binary Coded Decimal: even when

11 appears in a cell, the number taken into account by the computer is 30

0.32857142857…, and the software only converts when it displays the final presentation. Other software (as Derive or Maple) processes in distinct ways depending on the form of the given objects. We may say about the second that it uses distinct semiotic registers.

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Robert Adjiage and François Pluvinage

Components of Numerical Acquisitions Our general theory of number acquisition and numerical structures is based on a relationship between three components, which are to be studied separately and then linked with one another: physical experience, mathematical properties, and semiotic representations. We assert that a lack in presentation of one of the three components, or of their links, induces difficulties of understanding for many students and leads to unstable learning. In “The four competences” section below, we present typical physical situations, which may provide students with the needed experience in the conditions of an adequate learning milieu. In the following, we would like to emphasize the link between mathematical properties and semiotic representations. Example: In an observed active 7th-grade class, students tried to solve the following problem: obtain 1000 with 8 “8’s”. It is important to distinguish the heuristic phase from the written presentation of the answer. We consider the latter as one step toward algebraic treatment, although no variable is required. We explain below that the reason for this is that, when writing the solution, one is lead to process mathematical “sentences”. We assert that, in the considered case, the processing is a tool for solving the problem, not only a way to communicate or explain ideas. Two correct written answers are: 1000 = 888 + 88 + 8 + 8 + 8 1000 = (8888 – 888)÷8 Many students gave in this situation “equalities” like: 888 + 88 = 976 + 8 + 8 + 8 = 1000. Explaining why this kind of writing does not respect mathematical rules, and thus is incorrect, is a true challenge for teachers. Indeed, the correct use of the equality symbol is not something purely formal, as many students believe, but it constitutes an important element of the mathematical construction. Other students used seven “8’s” instead of eight, e.g. the following response that we summarise by the equality: 1000 = (8 + 8)×8×8 – 8 – 8 – 8

(*)

The students did not furnish this equality, they performed non-ordered calculations during the heuristic phase, and this may be the reason why nobody, including the teacher, noticed that there are only seven “8’s”. When calculating, the focus is on the result 1000 and this leads to forget the rest. If there had been a complete final writing of (*), this “concluding sentence” would have encouraged reconsidering what precedes, and thus noticing the errors. Moreover, from the above inadequate writing, it is easy (for someone familiar with algebraic process) to obtain a correct answer by dividing and multiplying by 8: 1000 = ((8 + 8)×8 – (8 + 8 + 8)÷8)×8.

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And this would be true processing of (*), which can be compared to processing a linguistic sentence. Using letters in calculations is not specific to algebra. One can use letters without being competent in algebra, e.g. substituting values for time and velocity in the formula d = vt. Conversely, one can do algebra without using letters. We assert that a fundamental step in mastering algebra is being able to process sentences. A sentence is the unit of meaning. Mathematics form sentences, like common language does. In mathematical writing, we can recognize nouns that are: numbers, variables, or more generally all kinds of mathematical objects; verbs: =, ≠, <, ∈, etc.; conjunctions that are signs of operations: +, ×, →, etc.; punctuation marks: full-stop, comma, parenthesis, etc.; and, on a more advanced level, quantifiers as adjectives: ∀, ∃. We can perform linguistic operations such as the negation of mathematical sentences. These grammatical aspects of mathematics are often ignored in the teaching-learning process. Another element often neglected in teaching-learning is the necessity of considering the link between objects of two distinct semiotic registers in both directions: from register 1 to register 2 and reciprocally. A difficulty in some cases is that going in one direction is easy, whereas going in the opposite direction implies a more advanced level. Example: 21/37 = 0.567567567… = 0.567 . We obtain this result by ordinary decimal division. But for the inverse calculation, from a repeating decimal to a fraction, we do not have such an easy algorithm. A possible solution comes from the formula of geometric sequence, but this formula is only learned in an algebra course (grade 11 or 12). We give below a more elementary proof of the following theorem. Theorem: Let 0.ppp... = 0.p the periodic decimal development of a rational number, its period p having n decimal digits. Then 0.p =

p 10n − 1

=

p . 99...9 N n

Proof A first step is to observe that

(

)

1 10n − 1

= 0.0...01 N . It is obvious, because n

10n = 10n − 1 × 1 + 1 , so the Euclidian division of 10n by (10n – 1) leads to the remainder 1. We can now multiply each side of the relation by a whole number p such that 0 < p < 10n. So we obtain the conclusion of our theorem. In the example above, we will obtain 0.567 =

567 21× 27 = , and we can then simplify 999 37 × 27

by 27. If we have a repeating decimal, the period of which beginning at whatever place, e.g.

3.14567 , it is easy to convert it into a fraction putting apart the non periodic part and using the theorem for the periodic part:

3.14567 =

314 0.567 314 21 11639 . + = + = 100 100 100 3700 3700

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Robert Adjiage and François Pluvinage

As regards decimal treatment of repeating decimal, we suggest the following exercise. You will have an opportunity to observe yourself processing repeating decimals. The surprising result here is that our square of a repeating decimal with a period of only two digits has a period of 198 digits. Exercise: Prove that

( 0.10)

2

2

⎛ 10 ⎞ 1020304050607080910...979900 =⎜ ⎟ = 99...99 ⎝ 99 ⎠  

198

Note: Only when we know that the length of the period approaches 200, can we easily use software like Derive in approximate mode, in order to find the result introducing the number of digits (more than 198, for example 210) that we want.

Numbers and Coordinate Geometry In the previous subsection, we have emphasized pure numerical problems. Here, we go to dimension 2 in order to better apprehend numerical aspects of dimension 1. A frequent use of numbers occurs in graphical representations. Ordered pairs of numbers produce points in a plane with coordinate axes. We frequently have to represent relationships between heterogeneous magnitudes. A historical example is the study of motion. For instance, this study led Newton to publish his famous book De Motu Corporum (1684). Time and distance are heterogeneous magnitudes involved in the phenomenon. When we represent a point in movement, we frequently adopt the horizontal axis for time and the vertical axis for distance. So when a motion is uniform (constant velocity) its representation appears on a straight line, whatever the scales of time and distance. But, in mathematics, teaching such a result is often asserted as a truth, which does not need to be proved. Moreover in many texts and classes, we have noticed the ambiguous use of the term “slope”, in order to refer to velocity, or to other situations with heterogeneous magnitudes. Slope normally refers to the geometric plane with its Euclidian structure, i.e. representing situations in which the coordinate axes are homogeneous. In case of heterogeneous coordinates, we could say “directing coefficient” (coefficient directeur in French). In all situations we can consider ratios, or rates of change. We observe possible consequences later in calculus courses, between the derivative seen as the slope of a tangent and the derivative seen as a rate of change, as noticed by Habre and Abboud (2006, p. 65). A mathematical result is that alignment of points in coordinate geometry is preserved by change of scales of the axes. Teachers can choose between three possible attitudes: • • •

Only stating the results, for example “For every value of parameter a, the linear equation y = ax is represented in coordinate geometry by a straight line”, Giving the results and saying that they are admitted, Proving the results, by use of the invariance of ratios in parallel projection (Thales).

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In our didactical perspective, the last option is most mathematically accurate and propitious for learning. Indeed it links the three components we have mentioned above.

MATHEMATICAL COMPETENCE Reason’s whole pleasure, all the joys of sense, Lie in three words,—health, peace, and competence. Alexander Pope (1688–1744) Essay on Man. Epistle iv. Line 79.

When, in this document, we use the word competence, it must be taken as a whole. In its initial meaning, competence refers to a juridical concept. The competence of a law court determines the area within its jurisdiction, and the types of cases it is qualified to process. However, the word competence, or the derived adjective competent, has a wider meaning, which may introduce confusion. For instance the competent minister may be not competent…. It is also possible to confuse competence and competency. But here, competency must be considered as a unit taken among other units: competencies. Saying that Mr. So-and-so is a competent gardener does not have the same meaning as Mr. So-and-so has competencies in gardening, and the latter would probably not mean a competent gardener. The standard usage of competency refers to the ability to perform some task. In this sense, a competency relies on a set of knowledge, know-how, and behaviours stemming from experience or specific learning. In education, the notions of competence and competencies have become central these last four decades, since education specialists have emphasized the importance of specifying the learning goals for guiding curricula designers and course developers, for aiding students to better understand their learning process, and for better assessing acquisitions (Mager, 1962). Today, institutions as in Europe the European parliament, or in the USA the National Academies, officially recommend relying on competence and competencies for shaping training programs and defining the objectives of lifelong learning. In France, the programme entries of IUFM8 (teacher training college) are being based exclusively on general professional competencies. Concerning mathematics, Winsløw (2005, p. 131) states: “The description of goals of mathematics education has several potential ends, external (justification, declaration…) as well as internal (planning, evaluation…). Although these ends are not independent, the usual forms of description [i.e. classical forms relying on a description of contents only linked by considerations bearing on internal mathematical coherence9] have a tendency to serve but a part of these ends. …We examine the notion of “competency” as a possible solution to these problems.” A bit further, Winsløw (ibid, p. 134) 10 specifies: Whether one accepts it or not, [The use of the notion of competency] challenges the classic idea that training an individual mainly consists in acquiring knowledge…. What eventually counts – and what may be assessed – is

8 Institut Universitaire de Formation des Maîtres. 9 Notice of the authors. 10 Translated from French by the authors.

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the potential of action of the considered individual, linked to knowledge but also depending on contexts where the action is realised. Winsløw (ibid, p. 133) advances two arguments in favour of mathematical competencies. First, he notes that one cannot translate what society expects from mathematical education in terms of purely mathematical contents. What society expects is the precise description of potential of action, whether you deal or work with mathematics. Secondly, he observes that the different types of mathematical reasoning are articulated beyond their logical interdependence. For instance, reasoning in geometry is not independent from reasoning in analysis, and specialists in mathematics education should be able to explain this dependence. In terms of competence, or competencies? Didacticians do not always react favourably to this thinking. Two renowned French researchers in mathematics education, Guy Brousseau and Yves Chevallard are critical of the use, and also the relevance of, the notion of competence. Brousseau distinguishes ‘savoir’ et ‘connaissance’, two French words which both may be translated by ‘knowledge’. For Brousseau, ‘savoir’ refers to objective academic knowledge, whereas ‘connaissance’ refers to one subjective individual’s knowledge. While learning, the involved ‘savoir’ is internalised as ‘connaissance’. Brousseau (2007) states that a formal evaluation, which is supposed to determine to what extent competencies are acquired, mainly addresses ‘savoirs’ and ‘savoirfaire’ (know-how), systematically underestimates students’ ‘connaissances’ and, as a consequence, the results of the teaching they have received. Thus, a part of the students’ real acquisitions is subject to remaining unrecognised. Chevallard (2002, p. 55) fears that school is becoming a place where competencies are disseminated and validated, with no strict link to knowledge; the only aim is efficiency in professional life. Such a school would resemble a stock exchange where students would be lead to manage a portfolio of competencies in the same way one manages a portfolio of financial assets, updating it frequently as a rapid response to different market demands. The points to be discussed now are: what is the sense of the concept of mathematical competence? What is its extent? Is it possible to consider mathematical competencies as constituent of a mathematical competence? To what extent a competency might be specified to be relevant for designing curricula and assessments intended for individuals who will, all along their professional life, work or deal with numbers? According to Niss (2003, p. 7), who influenced PISA2 project as recalled above: “mathematical competence means the ability to understand, judge, do, and use mathematics in a variety of intra- and extra-mathematical contexts and situations in which mathematics plays or could play a role”; and a bit further: “A mathematical competency is a clearly, recognisable and distinct, major constituent of mathematical competence”. Having stated these definitions, Niss describes eight mathematical competencies (ibid, pp. 7-9): “Thinking mathematically (mastering mathematical modes of thought); Posing and solving mathematical problems; Modelling mathematically (i.e analysing and building models); Reasoning mathematically; representing mathematical entities (objects and situations); Handling mathematical symbols and formalism; Communicating in, with, and about mathematics; Making use of aids and tools (IT included)”. He specifies, in a note at the end of his article, that the influence of the Danish KOM project “is reflected in PISA’s notion of mathematical literacy and its constituents”. And, in the framework for PISA 2006, one actually finds this definition: “Mathematical literacy is an individual’s capacity to identify and understand the role that mathematics play in the world, to make well-founded judgements

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and to use and engage in mathematics in ways that meet the needs of that individual’s life as a constructive, concerned and reflexive citizen (OECD, 2006, p. 13). The notion of mathematical literacy is “the counterpart in mathematics of mastering a language” (Niss, 2003, p.6). And mastering a language leads to the linguistic competence as defined by Chomsky (1965). “Chomsky is concerned with establishing a science that would study … the language faculty, in analogy with other mental faculties like logic, which as a kind of intuitive reasoning power requires no accumulation of facts or skills in order to develop but rather seems to be present and fully functional in speakers fluent in a language. So competence in Chomsky’s sense implies neither an accumulated store of knowledge nor an ability or skill”. (Phillips and Tan, 2005). And a bit further: “An individual’s competence is defined by the grammar, or set of rules, that is represented mentally and manifested by their understanding of acceptable usage in a given linguistic idiom. Grammatical competence thus defines an innate knowledge of rules rather than knowledge of items and relations. It is said to be innate because one apparently does not have to be trained to acquire it and it can be applied to an unlimited number of previously unheard examples” (ibid). This leads us to think that such a notion of competence does not apply to large areas of mathematical activities, and, a fortiori, all elementary mathematics. In other words, the characteristics given by Niss in the KOM project do not fit the major constraints that characterize the linguistic competence: Learning mathematics requires an accumulation of facts or skills in order to develop, and one apparently has to be trained to acquire it. The only property of the linguistic competence valid in mathematics is that it can be applied to an unlimited number of previously unheard examples. Nevertheless, when describing the mathematical activity related to limited domains, we find similarities with the linguistic competence. We would term them competencies rather than competence. What could be the mathematical counterpart of a linguistic competency? Perhaps certain elements of logic as quoted above, but also subitizing, i.e. the capacity of visually discriminating small groups of one to four items in a very short time, without counting (Kaufman et al., 1949). Little children know by ear the first number-words and their meaning, i.e. they can associate one of these number-words to a set of a few items (two, three, maybe four) by subitizing. When they are able to count, i.e. knowing the counting-word list, and moving one number-word forward each time they point to one new item of a given set – the Successor Function (Sarnecka and al., 2005) – they may note that the last number-word uttered in the counting sequence is the same as that obtained by subitizing. Under this circumstance, they encounter what Gelman and Gallistel (1978) termed the cardinal principle, which allows expressing the cardinal number of larger sets by counting, insofar as it lies within the counting range capacity. An innate capacity (subitizing) has therefore encountered acquired knowledge (the number-word list) and skill (counting) to produce a competency (adequately expressing any cardinal number). For us, a fundamental aspect that is not apparent in Niss (2003), is the vision of mathematics as a discipline of expression. We have underlined above grammatical elements of mathematical writing. Thus, certain fields of mathematical activities allow a comparison with aspects of a written language (spelling among others), or acquisition of a foreign language. As a result of teaching-learning, individuals achieve a stable level in these fields of activities, and others do not. A very large majority achieves numeracy, but PISA has showed that many youngsters fail tasks related to the following level, that is, roughly mastering ratios. We encounter here the first major failure in mathematics education. And we know that a large

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percentage of adults are unable to understand, or even read, equations where unknowns are represented by letters, which is related to a third level. Let us now consider the eight competencies presented by Niss. They, of course, refer to real and essential aspects of mathematical expertise. Let us consider them using three criteria: their scope, their coherence, and their relationship to mathematical contents. The eight competencies have a very wide scope, to such an extent that their definition, except the eighth, requires the word mathematics or mathematical, risking a vicious circle. A first consequence is that they do not offer great assistance to designing activities for learning, teaching, and assessing. How can we assess that an individual is able to “think mathematically”? A second consequence is that they are not independent: we cannot think (competency 1) or reason (competency 4) mathematically, handling mathematical symbols and formalisms (competency 6) or communicate about mathematics (competency 7) without representing mathematical entities (competency 5). In addition, the former are not independent from one another and almost all of them are required in mathematical activity. Consequently, the lack of one of them risks masking acquisitions related to the others, and this is another real difficulty for assessing. If each competency has a wide scope, some of them are transversal, i.e. required by any mathematical activity, whether elementary or advanced, whereas others are related to a more restricted domain. “Modelling mathematically” is becoming a foremost preoccupation, particularly since PISA has put the emphasis on it. But this type of activity is much more specific than e.g. “thinking mathematically” or “representing mathematical entities”, as Winsløw (2005, p.144) remarks. Representing is essential in mathematics. It strongly distinguishes mathematics from other sciences like physics as already noted above. Furthermore, Duval (1995, p. 61; 2000, pp. 63-67) states that “Choosing and switching between representations”, which is an under-competency of the third competency (Niss, 2003, p. 8), is at the core of conceptualisation. Putting the “representing” competency on the same plane as the “modelling” competency displays elements of mathematical expertise instead of giving an organised vision of them (Winsløw, 2005, p.144). The eight competencies are not explicitly related to mathematical contents (objects, concepts, notions, techniques, theories…), which are actually relegated to a position of secondary importance. Consequently, the latter may appear as contexts necessary, but more or less arbitrary, for developing competencies, whereas a historical or an epistemological, and thus a didactical standpoint would lead to consider the inverse. Referring to Winsløw (2005, p.145): “historical developments of the [mathematical] discipline cannot be thought of in terms of collective progress bearing on general competencies”. A given mathematical content is inserted into a network of other mathematical contents that work both as supports and obstacles (Brousseau, 1997, pp. 79-114) and lend to the former, sense and necessity. This epistemological network determines progress much more than competence or competency development, both in the history of mathematics and in the individual’s own learning history. Niss’ eight competencies, which are set in a top-down movement leading from competence to contents, do not reflect this epistemological complexity, and in this sense are a-historic. Moreover, one takes a serious risk when using, as already mentioned above, the generic term “mathematics” to define them. Does e.g. “reasoning” have a sense, independent from a category of problems and methods related to a specified context? If reasoning in geometry is certainly related to reasoning in analysis beyond pure logical structures, as already mentioned, do these two kinds of reasoning require the same type of competence as Winsløw (2005,

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p.144) observes? As for us, we distinguish at least two kinds of competencies in reasoning: heuristic competencies (finding ideas for proving), and competencies related to expression (writing a demonstration). One may consider that both are dependent on the mathematical contents involved, if only because the involved objects, notions, and processes are quite different. For instance, the notion of “limit” in analysis (mostly visual – graphs – and involving formal objects and processing) and the notion of “figure” in geometry (mostly visual and involving verbal objects and processing). The overall character of mathematical competence and even competencies, and the topdown movement leading from competence to mathematical contents, via competencies, present disadvantages and risk confusing the mathematical complexity. But there is an opposite tendency that consists in reducing competency to know-how. Let us consider French mathematical curricula. In important sections, entitled “competencies”, one finds for instance: “Multiply a whole or decimal number by 10, 100, 1000…”, “Find the quotient and the remainder when dividing a whole number by a whole number” (grade 6); “Complete a proportionality table e.g. find a fourth proportional”, “Determine if a given complete table is a proportionality table or not” (grade 7). This detailed know-how is important for designing practical sections of a course, and for specifying success and failure, e.g. in problem solving, in order to bring well-advised aid. On the other hand, the risk is to generate a behaviourist effect if teachers and/or students consider know-how as prescriptions for developing action schemes regarded as ends in themselves. And this may lead to a school, feared by Chevallard (see above in this section), mostly aimed at quick efficiency. In French mathematical curricula, beside the description of the above mentioned knowhow, one finds also, under the same heading “competencies”, wide-scope competencies like: “solve problems using knowledge of whole and decimal numbers and of the studied operations” (grade 5). And this poses again the question that concerns us from the beginning of this section: to what extent must a competency be specified? We noticed in the first section that the readers of certain official texts are supposed to be competent in mastering numbers at a given level, which depends on the public to which the content of a text is directed. We therefore need a notion of competence which can be situated at different levels. We shall also take into account our analysis of the notions of mathematical competencies examined above. All this brings us to the following guidelines for determining numerical competences adapted to lifelong learning. • •

•

We suggest substituting a “bottom-up” movement for the “top-down” movement in the approach proposed by Niss. This bottom-up movement originates in numerical contents and their use for working or dealing with numbers. It leads to competencies supported by three pillars: numerical knowledge, know-how, and informed activation. The latter is very important, because it takes account of the capacity of coordinating knowledge and know-how, in order to solve a problem, whether stemming from real-, or pure mathematical-world. Defining a competence requires listing the inter-related elements of pertinent knowledge and know-how, and thus considering the mathematical complexity, especially its internal coherence and cohesion. A competence is complete and autonomous.

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The acquisition of a competence supposes mastering a new technique of expression. The aspects involved in such a technique must be both semantic and syntactic.

On this basis, four competences may be defined, each giving account of the capacity to master problems. In parallel with the term literacy used for the ability to read and write, we suggest the following term list. 1) Numeracy: mastering whole and decimal numbers, and processing the four elementary arithmetic operations; 2) Rationacy: mastering ratios and proportions, interpreting instructions leading to calculation of products or quotients of positive or negative rational numbers 3) Algebracy: using the mathematical sign system of algebra 4) Functionacy: using functional relationships and calculus. Let us underline that each listed competence introduces new syntactic rules. Comparing with the usual language, the first introduces a specific way of writing “words” and forming simple “phrases”: positional decimal notation, specific symbols for noting operations. The second breaks the writing line (one fractional number is expressed by two integers, the numerator and the denominator) and prohibits the denominator of a fraction to be zero. The third mixes known values with unknown: “Competent use of expressions with unknowns is achieved when it makes sense to perform operations between the unknowns and the data of the problem” (Puig, 2004, p. 3). It leads to processing whole sentences. Problems requiring equations containing the same variable on each side of the equality symbol, as 3x – 2 = x + 1, are the most significant. The last competence leads to introducing new rules for forming and processing words and sentences, as required by change of variables, composition of functions, and transformation of functions such as differentiation and integration, e.g. for composing in both senses the following f and g functions:

f( x) = x 2 ; g( x) = x + 1; f D g( x) = ( x + 1) ; gD f( x) = x 2 + 1 2

We will return in depth to each competence in the next section. We will content ourselves here with observing that acquiring one of these competences really certifies that an individual is competent in one indeed broad but cohesive domain of mathematics, necessary and sufficient to master a given degree of “dealing with numbers”. This definition of competence does not make obsolete more traditional forms of enumerating acquisitions. It enriches them and provides criteria for both teaching and assessment. It allows checking that success is systematic, and that a specific organisation underlies the processing performed.

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THE FOUR COMPETENCES This section is devoted to exploring the details or components of the four levels of competence defined in the previous section. Hereafter, we will also term them the four levels or the four competences, according to the aspect we wish to emphasise. Let us remark that at a given level, one is supposed to sufficiently master each component of the preceding competence. Nevertheless, working at level n may contribute to the achievement of level n-1. Thus, one competence may be considered as a component of the following competence and this will remain an implicit rule. Lastly, let us note that some components of a given competence are transversal to all competences, e.g. “Check whether the findings are plausible and compatible with the original situation”. After its first mention at one level, this kind of component will not be repeated further.

Numeracy Numeracy is the first level of competence that we distinguish. It is required for basic numerical modelling of real world, when the direct relationship with physical objects is replaced by numerical representations. Everybody is supposed to achieve this level and thus to master writing and using whole and decimal numbers, and resorting to integers in easy cases that will be specified below. The main features of this level are: • • • • •

Substitute numbers, and thus the digit symbols for quantities Calculate on numbers, even if the given data and relationships bear on quantities Interpret the obtained results in the numerical domain in terms of quantities and relationships between quantities Check whether the findings are plausible and compatible with the original situation Resort to a positive or a negative number, in problems that involve the location of this number e.g. on a number line, and the additive-subtractive rules.

Substitute numbers for quantities implies that one resorts to a unity, in both cases: discrete, e.g. counting pairs of socks, and continuous, e.g. counting how many cups a given quantity of water can fill up, this water being shared into cups and mugs (Galperin and Georgiev, 1969, pp. 189-196). Two main types of problems are related to numeracy: additive (including subtraction) problems and multiplicative (including division) problems. Vergnaud (1982, 1983, 1988) categorised both, resorting to mathematical concepts and notions like measure, transformation, linear function, and cognitive aspects referring to the nature of the involved physical objects and quantities and the relationships within and between them. The multiplicative problems may be considered as related to numeracy or rationacy. They fall within numeracy when a whole or decimal number may express the involved ratios and when:

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•

A ratio is given (“I have three times more money than John; gas costs 0.87 cents per litre”), and one has to apply this ratio (“John has $7, how much do I have?” “I buy 42.8 litre gas, how much do I pay?”) A measure, or a ratio between two constant quantities has to be worked out (“I paid $35 when I filled up. Gas costs 0.87 cents per litre. How much gas did I get?” Or “I paid $35 when I filled up. I got 40 litres of gas. How much does one litre of gas cost?”).

But, if they involve ratios only expressible by a fraction, or lead to consider ratios as objects (e.g. operators working on a variable quantity) and/or to perform calculations on them (comparing ratios, adding or multiplying fractions…), problems fall within rationacy. That is what we will develop in the next section.

Rationacy Rationacy has been central and constant in our research studies for ten years. We thus evoke these studies to deconstruct (unfold) the complexity of this level, especially in separating and articulating two main elements of this complexity, which are a physical and a semiotic component. Concerning the latter, we refer to Duval (1995) and his semiotic approach of object and concept genesis in mathematics. We take advantage of this section to point out what kind of experiment related to number acquisition one can conduct in the classroom. Achieving rationacy means that one is able to: •

• •

• •

•

Resort to a ratio rather than another mathematical object or notion for managing realworld data: for instance, when an individual’s leucocytes rate rises from 4 giga/l to 12 giga/l, the right indicator is not the difference 8 but the quotient 3; Interpret a ratio as a multiplicative relationship between two variable quantities; Identify, in a problem statement, the conditions that cause some ratios to be conserved although the involved quantities are variable (proportionality), e.g. in the case of constant speed, the ratio distance/time is conserved, and if one doubles, triples… the time, one doubles, triples the distance); Mobilise relevant expressions of ratios, as fractions, decimal numbers, visual representations… and process them; Interpret the numerical and non-numerical data and their relationships in terms of numbers and operations, in order to solve a problem involving ratios, especially in the case of proportionality. Understand the multiplicative rule for positive and negative rational numbers.

Rationacy refers to ratios that can be expressed by a rational number, i.e. a number r for which exist two integers (and, in a first approach, two whole numbers) a, and b ≠ 0, so that br = a. But a ratio may be related to any real number: “Two magnitudes of a continuous quantity stand in relation to one another as a ratio, which is a real number” (Wikipedia, 2007b). Rationacy is a crucial level in number acquisition, and this is historically and

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pedagogically attested. Historically, the notion of number is built upon the notion of quantity, and more precisely upon the notion of ratio of quantities (Dhombres referring to Euclide, 1997, pp. 5-8). The pedagogical value of ratio of quantities is considered as irreplaceable “because it practically and intuitively founds the number system.” (Rouche, 1997, pp. 43-44). Furthermore, proportionality, which is a main notion related to ratios, is a quasi-unavoidable step towards algebracy and functionacy, because it leads to considering variable quantities and links between or within them. Before the XVII century, proportionality was the unique way of expressing functions. Indeed, ratios of incommensurable quantities (like the ratio between the diagonal and the side of a square) were not considered as numbers until the XVI century. Thus, in order to express the functional link between two quantities, without resorting to an explicit numerical link that was not available, one used a rhetorical way involving proportion. For instance, to express the relationship between the area of a disk D and its radius R, one used a formulation as the following: “D is to D’ as R2 is to R’2.” We would say today that D is proportional to R squared. Let us report a real-life situation. In the desert, a group of tourists is observing camels watering at an oasis. One of them wonders loudly: “how many litres can a camel drink at once?” Somebody answers: “up to 100 litres! In comparison, a cow drinks 20 to 30 litres… per day”. Everybody remains impressed and silent for a while, when someone asks: “By the way, how much does a camel weigh?” In the beginning, everybody considers the quantity of water as an absolute indicator of the camel performance…. Until the last question introduces relative thought in relating the volume of water to the animal weight. Entering Rationacy implies at least two capacities: consider things as related; consider the multiplicative nature of the involved relationships, whenever the case requires. The camel situation leads to take into account a ratio of two heterogeneous quantities: volume and mass. Of course, there are many other types of contexts involving ratios. We proposed the following classification of the latter in Adjiage (2005, pp. 100-102), considering the number of quantities concerned, then it need be, the number of underlying objects, then as the case requires, the states in which these objects are examined: “ratio of two heterogeneous quantities” (e.g. speed, flow, production…); “measurement”; “mixture”; “frequency”; “enlargement” (or “dilatation”); “change of unit”. Hereafter, what we call “variable of context” will refer to these situations. Whereas the camel situation involves two quantities, a problem of mixture refers to one quantity (mass or volume), two ingredients or objects (e.g. black paint/white paint), examined in a state of fusion into one new entity (grey paint). A problem of enlargement involves one quantity (length), one object (e.g. a picture), in two different states or from two different aspects (before and after enlargement). In agreement with Tourniaire and Pulos (1985, p. 190), a survey conducted by Adjiage (2005, pp. 103-114) concluded the pertinence of this variable. A questionnaire was elaborated. It presented for solution six items, one representative of each of the six problemtypes of the classification. The survey was conducted with two types of population: 121 7thgraders, and 110 prospective primary school teachers. The latter, because they are not specialists in mathematics, can be regarded as a reference population made of mere educated adults. The findings show that the variations (success rate and procedures used) are important from one item to the other in the student group, and very slight in the teacher-trainee group. The differences observed are attributable to the real-world context, as the underlying mathematical problem was of the same nature (same type of numerical data, same problem

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structure and question: find out a fourth proportional). Two consequences can be drawn from this first study: •

•

there exists a component of rationacy complexity related to the diversity of the involved contexts, i.e. to the different types of real-world experiences referred to (what does tasting the flavour of a mixture have in common with enlarging?); generally, an “educated” adult overcomes this diversity. Most observed teachertrainees first interpreted any submitted problem, in a clear-sighted way, in terms of ratio and proportion, and then carried out the solution in using and automated algebraic procedure: the product of the means equals the product of the extremes (cross-product). This precise examination of their papers led us to state they are able to recognise a unique underlying mathematical model, effacing the diversity related to the variable of context.

Another component of Rationacy complexity stands in expressing ratios. When one has understood that a problem requires ratios to be solved, one has to choose a way of expressing these ratios in order to process them and find a solution. We have already described the nature of this semiotic complexity, which is so particular to mathematics, and given examples related to rational numbers. We will therefore content ourselves with recalling here that one disposes of many ways of expressing ratios: fractions, decimal numbers, two separate whole numbers, a number line, other visual representations… and that each of these semiotic registers in the sense of Duval (1995, pp. 15-85; 2000, p. 60-65) has its own way of making sense, especially through specific and non-congruent processing. Moving from one register to another is often a necessity or a facility, but it requires a real rupture in the thought process, as already mentioned. For completing the review of complexity of rationacy, we would like to consider more thoroughly the privileged representation of rational numbers that are fractions. Understanding and using fractions are important obstacles in learning, and all teachers and researchers in mathematics education can attest this. In syntactic terms, this representation entails a break in the writing line (see p. 36) into the numerator line and the denominator line, with different ways of processing in each line. Moreover, our personal research has led us to consider a fraction as a “double-triggered” object, namely: • •

An implicit numerical link between two explicit whole numbers, so that a given fraction represents one number expressed by two numbers. A numerical link between two series of whole numbers, so that a given fraction represents one number expressed by a set of so-called equivalent couples of whole numbers.

Misunderstanding arises when the teacher is considering the link while the pupils are taking into account two whole numbers separately. For example, in

3 , an expert considers 4

the link between 3 and 4, which may also express the link between 75 and 100 or 9 and 12… A teaching that gives greater place to “pie parts” (Streefland, 1993, p. 114; Adjiage and Pluvinage, 2000, p. 50) reinforces this misunderstanding. As we will see hereafter, our

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experimental teaching plan, which relied mainly on ratio problems, emphasised the link rather than the terms, and thus had a greater chance of reducing the misunderstanding. We have now described the complexity of rationacy, which mainly relies on two diversities and the articulations within and between them: the diversity of real-world ratio problems (physical or real-world domain) and the diversity of the means of expressing ratios (mathematical domain). These components are of course interwoven and a major challenge for teaching is to lead students to separate them in order to better articulate them. This was our main guideline for designing an experiment in teaching ratio and proportion, reported in (Adjiage and Pluvinage, 2007). We summarise below this experiment and its main findings. Two standard classes11 were followed at grades 6 and 7, involving 47 students. The objectives and the corpus of ratio problems were common to both classes. But in one class, called “Partial-experiment” (PEx), the teacher followed his usual method for designing and conducting the teaching sequences. In the other class, called “Full-experiment” (FEx), the teaching was based on precise guidelines relying on a systematic work of separation/articulation within and between the six types of ratio contexts described above, and three retained semiotic registers for expressing ratios in a computer environment: linear scale (a number line with resources such as subdividing, sliding along the line, zooming…), fractional writing, and decimal writing. The computer environment requires the software series ORATIO and NewOra (Adjiage and Heideier, 1998), which we designed and developed specifically. Our main hypotheses were (Adjiage and Pluvinage, 2007, p. 156): • •

Varying the ratio situation contexts according to a systematic classification allows pupils to perform relevant proportionality procedures and thus obtain better results. Working systematically within and between the physical and the mathematical domains involved in ratio problems helps pupils to better process rational numbers, identify common underlying features in ratio problems, build a model of proportionality, and perform advanced strategies.

We organised a pre- and a post-test in order to evaluate progress in both classes, and we compared the results at the post-test to the results of both samples (standard 7th-graders and prospective primary school teachers mentioned above). The findings show that both followed classes made clear progress, and their performance is much higher than the standard 7thgrader performance12. The learning appears to be more efficient in “FEx” than in “PEx”. Furthermore, the former approaches the teacher-trainee population, both in success rate and procedures, more than PEx does. Particularly, their procedure is much less dependent on the context, thus giving obvious signs of recognising a unique mathematical model underlying the diversity of real-world contexts. They perform advanced scaling strategies around procedures involving fractions: “The systematic work within the different rational registers helps them to secure their fractional processing and thus leads them to resort to it more willingly.” (Adjiage and Pluvinage, 2007, pp. 170). It is thus possible to train students to rationacy in an efficient way. Being able to resort to the ratio model, especially proportionality, whatever context it allows to interpret and process, demands acquiring more than mere know-how. It requires considering the whole complexity 11 12

According to their results at national mathematics achievement tests. Average success out of 6 items: 0.70 (pre-test, grade 6), 0.90 (standard 7th-graders), 2.36 (post-test, grade 7).

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of this topic, particularly its physical and mathematical coherence and cohesion, with a special mention to a clear-sighted use of semiotic registers that allows accessing the objects of the mathematical domain and processing. Of course, the learning of ratios is not finished at grade 7. The next step in the curriculum is algebra, which will complete the ability to use fractions and process ratio problems. In return, a sufficient level of acquisition in ratio and rational numbers is required to provide students with a solid base on which to build the new objects and concepts of algebra.

Algebracy Hereafter, we will term equation a mathematical writing stating that two expressions, containing one or more unknowns, are the same, as in: 3xy-1=x2+5y. We term identity a mathematical writing stating that two expressions, containing one or more variables, are the same regardless of the values of any variable that appears within them, as in: (a+b)2=a2+2ab+b2. We will term equality, or arithmetical equality, a mathematical writing stating that two expressions, only containing constants, are the same, as in: 2+5=7 or

1+ 2 = 5 . In order to lighten the reading, we will mainly consider equations, which are the 3 3 base of algebra. But most assertions are transferable to inequations. Let us consider the following problem, often quoted by Brousseau. A shopkeeper sells a bolt of fabric. He expects a 20% benefit. When he has sold the entire bolt, he realises that he has only done a 19% benefit. He incriminates the distance between the two marks, drawn on his counter, that he uses for measuring one metre. What is the exact distance that separates these two marks?

1. (Arithmetical) Solution at Rationacy Level This solution uses an arithmetic processing, i.e. roughly, without resorting to letters and combination of letters. As it requires the use of an advanced proportionality reasoning, we locate it at rationacy level. Let us consider the two-mark distance as variable. Let us call a unit the length of the material measured by a given couple of marks. • •

The selling price of the entire bolt is proportional to the number of units sold (if you double, triple… the number of units sold, you double, triple… the selling price). The number of units sold is inversely proportional to the two-mark distance (if you double, triple… the two-mark distance, you divide by two, three… the number of units sold).

Now, composing proportionality and inverse proportionality gives inverse proportionality. Thus, the selling price is inversely proportional to the two-mark distance. If the latter is 1 metre, the former is 1.20 (multiplied by the purchase price ); if the latter is the

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unknown two-mark distance used by the shopkeeper, the former is 1.19 (multiplied by the purchase price). Using the cross product related to inverse proportionality, it comes to: Unknown two-mark distance in metres: 1.2×1 ≈1.008

1.19

2. Solution at Algebracy Level Let x be the unknown two-mark distance, let P be the purchase price per metre, let n be the total number of units sold. We calculate the total selling price in two manners. 1. Every time the shopkeeper sells a unit, he thinks that he is selling 1 metre, and thus applies a 1.2 coefficient to P. Therefore, the total income is: 1.2 n P 2. The real length of the bolt is: n x, and the shopkeeper paid n x P at purchase. His actual benefit is 19%, so that the total income is: 1.19 n x P 3. Comparing 1 and 2, we obtain: 1.2 n P=1.19 n x P 4. Simplifying both members by n P, it comes: 1.2=1.19x, and thus: x= 1.2 , so that x≈1.008

1.19

Let us comment these two ways of finding the solution, and let us try to point out, on this particular example, the main features of continuities and ruptures from an arithmetical processing to an algebraic one. First of all, let us observe that, when mastering algebra, the second solution is more secure than the first one, because it frees the mind from sophisticated reasoning like resorting to a non-obvious proportionality model or making variable the involved constant quantities like the two-mark distance. But it also frees the mind from doubt: Are the hypotheses of direct and inverse proportionality relevant? Does one have the right to make variable the involved constant quantities? Lastly, it does not require the knowledge of rules such as: composing direct proportionality and inverse proportionality gives inverse proportionality. The second solution is “one” translation of the given data and their interrelations. No additional hypothesis is required, and one can check by a reverse translation whether the retained equation interprets the data properly. Of course, it also requires knowing and applying the rules of algebraic calculation. All the complexity of algebra lies in the “one”

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Robert Adjiage and François Pluvinage

related to translation, and in the specificity of algebraic calculation compared to arithmetical. This is what is to be discussed now. The most visible difference between Solutions 1 and 2 is that the former is expressed in a rhetorical way, except the final arithmetical calculation: 1.2×1 1≈1.008, whereas the latter

1.19

requires symbolic notations, i.e. roughly letters and combination of letters. This may refer to the historical development of processing numerical problems, which is usually described in three stages (Kieran, 1992): rhetorical, syncopated13, and symbolic. Rhetorical stage dates back to Antiquity and symbolic stage really commences at the end of the XVI century with Viete, and acquires its almost definitive features one century later with Leibniz. The length of this development leads us to think that substituting signs for objects and relationships, already defined with words, is a serious epistemological obstacle. However, whether one considers the arithmetical or algebraic solution of a numerical problem, one faces treatments and conversions in the sense of Duval (2000, pp. 62-63). Examples of treatments: within the natural language register, e.g. re-express the problem statement in such a way that it becomes easier to interpret; within the arithmetical register, e.g. calculate with numbers; within the algebraic register, e.g. calculate with symbols. Examples of conversions: between natural language and the arithmetical register, e.g. the interpretation of the last paragraph of Solution 1 (Thus, the selling price…) in terms of the (quasi) equality 1.2×1 ≈1.008; between

1.19

natural language and the algebraic register all through solution 2, e.g. when expressing the total income (2.1) or expressing the total selling price in two different manners (2.3). Therefore, comparing algebra and arithmetic mainly entails comparing treatments and conversions involved within and between both domains. According to Gascon (1994, p. 45) the prevailing implicit model at secondary school is to consider elementary algebra as generalised or extended arithmetic. In addition, this author specifies that numerous researchers or historians of algebra approve of this characterisation more or less explicitly. As a matter of fact, an equation mobilises the usual operations (+, -, × , ÷) and the equality sign (=). As an arithmetical equality, it may interpret part or all of a real-world and/or word problem. The visible difference is that an equation bears on letters and constants whereas an arithmetical equality only bears on numerals. The Piagetian tradition mainly distinguishes three uses of letters. • • •

An unknown to be specified after equation solving Generalised numbers as in identities: a(b+c)=ac+bc Variable of a function

But we can add at least three other uses: • •

13

Particular constants as π, e, i… Parameters or non-determined constants as a, b, c, in the general form of a quadratic equation (ax2+bx+c=0)

Syncopated algebra is an intermediate stage. Its main differences with modern algebra are that it lacks special symbols for operations, relations, and exponentials. Moreover, at this satge, processings do not really bear on symbols (mostly abreviations) but remain essentially rhetorical.

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Variable determined as an element of a set: let x be a real number; let 2n, n being an integer, be an even number.

This complexity entails major changes: algebraic activity resorts to signs (of operations, and equality) that, in the arithmetical context, have concrete references and thus a precise meaning (Gascon, 1994, p. 46). The same signs, in algebraic context, bear on objects of different kinds, as seen above, and mostly unspecified. Therefore, in arithmetic, an expression like: 2+3=5 is usually conceptualised operationally, i.e. as a process according to Sfard (1991, p.1), and the equality sign announces a result, whereas an algebraic expression may be considered either as a process or structurally as an object (ibid). For instance, in the fabric bolt problem, 1.2 n P refers both to a process (one way of calculating the income) and to an object (the income itself). And this has consequences for the meaning of the equality sign. If 1.2 n P is a process, how can it be that the following “=” in (2.3) does not produce a result, but another process or object: 1.19 n x P? And what about the permanence meant by “=” within an identity? Teaching algebra, seen as generalised arithmetic, requires that these differences are taken into account. This is actually done and reported in many studies that try to manage continuities from arithmetic to algebra, particularly in a computer environment. This type of approach is usually functional, thus leading students “to represent algebraic word problems in the form of computable algorithms, which serve as intermediate representations in the process of developing standard algebraic representations, and which also permit guess-andtest numerical strategies.” (Kieran et Al., 1990, p. 51). But other authors do not retain generalised arithmetic as the most relevant model for teaching algebra. Gascon (1994, pp. 49-55) considers that the analysis-and-synthesis pattern (shortened hereafter in A/S pattern), as described by Pappus14, can found an alternative model for algebra. Let us examine Gascon’s findings. The classical A/S pattern, which has mainly been applied in geometry, consists in two steps. First step, analysis or regression: starting from the unknowns, combining them under necessary conditions related to the constraints of the problem, and moving on to given data. Second step, synthesis: use the involved data as a basis for solving the problem, and in the algebra case, find the unknowns. Let us point out that algebraic reasoning, applied to problem solving, actually relies on the following analysis, e.g. in the case of one unknown x: “if x exists, it must verify…” and then continue in linking the necessary conditions until an equation is formed. And this is one of the major obstacles in mastering algebraic reasoning: accepting to calculate with quantities that may not exist, that one has not yet found, instead of calculating to find them! The synthesis consists in equation solving. Let us go back to the general A/S pattern. It assumes that, going up the chain of necessary conditions, you arrive in time at certain data. This is the case of most problems solvable by arithmetic, because they can be broken into a chain of easier problems. Let us consider a new example, which is a classic brainteaser: A newspaper and its supplement cost $1.10. The newspaper costs $1 more than the supplement. What is the cost of the newspaper?

14

Pappus of Alexandria is one of the most important Hellenistic mathematicians of antiquity. He flourished about the end of the third century A.D.

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Let p be the cost of the newspaper, s the cost of the supplement, t be the total cost. The following schema may avoid your giving the wrong answer, $1:

Using the normal direction of reading, left to right, leads to the equation: s+1+s=1.10, or 2s+1=1.10, and so: s=5cts, so that: p=1,05$. But, expressing the middle term p in two manners: p=s+1 and p=1.10-s might also lead to the equation: s+1=1.10-s, which is equivalent to the former, and this can be proved by algebraic transformations…. If one accepts to calculate with not-yet-known quantities. The first equation is obtained after applying the classical regression of the A/S pattern, because it leads to the numerical given, 1.10. This entails that an arithmetical solution is possible. On the other hand, the second equation is obtained in considering a quantity that is expressed in two different ways, both depending on the unknown. That means that the classical A/S pattern does not work, because we cannot rely on a given to go back to the unknown and find it out. In this case, the regression follows two different ways that lead to two symbolic expressions, and this is a true discontinuity between arithmetic and algebra. Let us observe that Solution 2 of the fabric bolt problem is the same nature when expressing the total income in two different ways. Gascon suggests modifying the classical A/S pattern, in order to take into account this important rupture: symbolic expressions, along with numerical data, are allowed at the end of the analysis process, so that the synthesis has to go through algebraic calculations. After a second modification, mainly taking account of parameters, Gascon concludes that this revised A/S pattern may model algebra teaching. Until now, we have pointed out ruptures from arithmetic to algebra, mainly in terms of processing within either expression system (treatments, in Duval’s sense). We still have to analyse ruptures in term of conversion. Therefore, we have to consider the two main semiotic registers involved in algebra, i.e. the natural language and the symbolic expression system, and the way an algebra user has to articulate both of them. This entry is mostly a cognitive one. It specifically consists in analysing differences between the ways we designate objects, and differences between the ways they are interrelated, in either register. We mostly refer to a conference by Duval held in “IREM15 de Strasbourg” on the 7th of December 2005. What are the main stages in interpreting the newspaper problem in terms of equation? 1. Re-designate with letters what has already been designated with words: p for “the cost of the newspaper”; s for the “cost of the supplement”; t for the “total cost”. Let us observe that p is an individual designation whereas “the cost of the newspaper” is descriptive (noun and complement), and this is a first difficulty for students that may either confuse the quantity (cost) for the object (newspaper), or focus on “cost” and use the same unknown for designating any cost of the problem (Kourkoulos, 1990, p.119). 15

Institut de Recherche sur l’Enseignement des Mathématiques.

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2. Reduce the lexicon in re-re-designating (functional designation). Many unknowns are not independent, and obtaining a relevant equation requires expressing some unknown in function of others, e.g. p=s+1, so that the designation of the supplement and the newspaper in the final equation rely on the only letter s: introducing one letter is not only intended to designate one object, but many. 3. Double designation of the same quantity. As noted above, posing an equation comes down to identifying two different expressions of the same quantity in one formula that contains part or all of the re-designations. In linguistic terms, it comes down to making explicit a referential equivalence. All these actions, and the consecutive processing, differ from the usual practise of natural language. A mere letter, used as an unknown, may refer to a whole syntagm. Functional designation allows the expression of different syntagms in using one letter, and this way of designating is specific to algebraic expressions. Not to mention the numerous uses of letters recalled above… Furthermore, there is a rupture between discursive practises of a natural language and processing in algebra. Chaining syntagms in a linguistic phrase is oriented; it is not for both members of an equation, and this may lead to two well-known obstacles: •

•

An oriented reading of an equation, left to right, that may entail the expectation of a result, instead of considering both members simultaneously in order to find the best way of processing. Moving any quantity from one member to the other, with respect to usual rules that differ from additive terms to multiplicative factors, left to right and right to left.

More generally, reading any algebraic, or even arithmetical, expression must not always be sequential, as when reading a phrase, according to the rules of priority. For instance, in an expression as easy as: a +b×c , the focus should be first put on × , then on b and c, and then on +, and lastly on a. In conclusion, should we retain more ruptures than continuities in going from arithmetic to algebra? The answer is a general one in mathematics, because all previous knowledge is both an obstacle and a support for new knowledge. Of course we must rely on arithmetic when teaching algebra. But we must keep in mind what deeply differs, and conceive activities that lead students to become aware of the necessity of changes, if only because these changes permit avoiding a loss of time. Let us take a last example. A headmaster buys 25 copies of a book. Another headmaster buys the same book 1.4 dollar less, so that he can get 5 more copies than his colleague for the same amount. How much does the first headmaster pay for one book? Let x be the unknown price. When one masters algebra, one is able to pose the following equation: 25x=30(x-1.4) This reduces to one expression two open lists, based on all imaginable hypothetic values for x. This is what Duval terms the “function of condensation”.

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Robert Adjiage and François Pluvinage Table 3. opening lists for solving a numerical problem involving unknowns

Unknown price 15 10 …

Total paid 1st headmaster 375 250 …

Total paid 2nd headmaster 408 258 …

Duval suggests asking students to establish these two lists and continue until they are aware they are always writing the same thing, and they can make explicit this “thing”. He recommends then leading students to identify the known quantities, the indirectly known quantities, and the unknown quantities. When the teacher introduces x, he should specify that he intends to use this letter to designate not only one quantity, but two. This conducts students to be aware that they are searching for two different expressions of the same quantity and the teacher must underline this. Arithmetic is present in this development. But algebra does not appear as an extension of arithmetic. The teacher waits until students are ready to consider and express the encountered obstacle. The rupture is taken into account and emphasised. It is now possible to go further.

Functionacy As Algebracy, the competence that we term Functionacy has a much wider extent than pure numerical knowledge and processing. However, it includes elements specifically linked to numbers, as distinguishing open and closed intervals of real numbers. One encounters this kind of problems when one studies questions related to domain and range of real functions. Solving problems of global or local maxima and minima of real functions also implies a good knowledge of the world of real numbers. Let us remind the reader here that in the historic development of mathematics, the study of functions preceded and induced the discovery of the complete field of real numbers. The actual basis for teaching functions in a Calculus course are the Euler’s and Lagrange’s notations and standpoints. Euler systematically used the notation f(x), and Lagrange introduced the notation f’(x) for its derivative. The set of studied functions at that time was constituted by algebraic functions and some transcendent functions (trigonometric, logarithmic and exponential). Thus, it seems to be an unnecessary obstacle for students to immediately face the modern general concept of function. Moreover, for us, functionacy does not necessarily comprise the knowledge of this general concept. The notion of function related to mapping refers to the language of logic and sets, so that its usual definition requires the knowledge of this language. A correct formal definition of a function is the following.

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Definition A A correspondence f between a set A and a set B is a sub-set of the Cartesian product of the sets A and B, A× B =

{( x, y ) , x ∈ A, y ∈ B} . A correspondence f between a set A and a

set B is called a function, and will be written as f: A → B if and only if:

∀x ∈ A, ∀y ∈ B, ∀z ∈ B, ⎡⎣( x, y ) ∈ f ⎤⎦ ∧ ⎡⎣( x, z ) ∈ f ⎤⎦ ⇒ y = z . Does the trouble lie in the presentation? As the use of the symbolic language is not necessary, one could think that a statement in common language is sufficient and facilitates the understanding of this definition. But the difficulty remains, as it appears for example in the definition given in the following French version (translated by the authors) of Multilingual Online Encyclopedia Wikipedia (2007c).

Definition B A function f of a set E in a set F is a correspondence or functional relation; thus it is an ordered triple (E, F, G) where G is a sub-set of ExF in which each element of E does not appear more than once. E is said to be the domain of f; F is said to be the co-domain of f; G is the graph of f. G is sometime denoted “Gf” or “G( f )” for specifying the function referred to. Moreover, other authors give slightly distinct definitions of function. In the English version (Wikipedia, 2007d) of the preceding article, a condition of existence is added, so that two conditions are stated. In formal writing, this gives:

Definition C 1 (Existence) - ∀x ∈ A, ∃y ∈ B, ( x, y ) ∈ f 2 (Unicity) - ∀x ∈ A, ∀y ∈ B, ∀z ∈ B, ⎡⎣( x, y ) ∈ f ⎤⎦ ∧ ⎡⎣( x, z ) ∈ f ⎤⎦ ⇒ y = z The difficulty in both, the French and the English versions, are comparable and does not disappear when using the common language instead of the symbolic. The verbal definition noted from MathWorld illustrates this fact:

Definition D A function is a relation that uniquely associates members of one set with members of another set. More formally, a function from A to B is an object f such that every a ∈A is uniquely associated with an object f(a) ∈ B. A function is therefore a many-to-one (or sometimes one-to-one) relation. The set A of values at which a function is defined is called its domain, while the set B of values that the function can produce is called its range. The term "map" is synonymous with function.

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The level of the linguistic complexity of this text is very high: problems with the meaning of “uniquely” in the first sentence, repetition of “object” in association with f in the second sentence, parenthesis in the third sentence, problems with the meaning of the verb “can” in the fourth sentence. Moreover, something about f is missing: what kind of object is f? The response requires the Cartesian product A×B, which is absent here and explicitly or implicitly present in the other mentioned definitions. Lastly, specialists will observe that the author introduces terms like “range” or “map” in an uncertain meaning. We conclude that we lack a definition totally shared by all mathematicians, so that it is preferable to reserve formal introductions at the undergraduate level. Fortunately, the functions used by most people who have to deal or even to work with mathematics can be introduced from the 18th century standpoint, i.e. without general definition. For us, applying the principles stated in the “Components of numerical acquisitions” subsection facilitates this introduction. Nevertheless, all difficulties related to this teaching topic do not magically disappear. Perhaps, functions are, among elementary mathematical objects, those that have generated the most specific cognitive terms in the recent past. In the eighties and nineties, it is mainly about functions that, in France, both concepts of tool-object dialectic (Régine Douady, 1984) and semiotic registers (Raymond Duval, 1995) were established. During the same period, in the United Kingdom, the construct procept, contraction of process and concept, was created by Eddie Gray and David Tall (1994). For example, the notation f(x) = x3 refers both to the mapping x → x3 as well as the outcome of this process, i.e. the function f. “An elementary procept is the amalgam of three components: a process which produces a mathematical object, and a symbol which is used to represent either process or object. A procept consists of a collection of elementary procepts which have the same object.” (Gray and Tall, 1994, p. 119).

One could argue that some examples given by Gray and Tall are not related to functions, e.g. the procept 3 + 2. But a preceding text shows that the main goal was the concept of function and its learning. “For example, functions can be regarded structurally as aggregates of ordered pair, or operationally as certain computational procedures. These two approaches, ostensibly incompatible (how can anything be a process and an object at the same time?) are in fact complementary. (…) Similarly, the ability of seeing a function or a number both as a process and as an object seems to be indispensable for solving advanced mathematical problems.” (Sfard, 1989, p. 151)

Another obstacle to learning functions is that it is necessary to be competent at the preceding levels in order to directly understand what kind of objects functions are. And we have already noted that an important proportion of high school students are not at the algebracy level, and that some of them do not even master rationacy completely. Recall that in spite of the intensive training given in an experimental context, the observed percentage of failures at the end of the course of calculus for undergraduates fell between 30% and 50% (Tall, 1996). In some attempts, remedial courses were designed, but in any case they remained unsuccessful.

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Instead of repeating a teaching that produces failure, a possibility is to use modelling. When the situation is easily understood by students, and allows them to control their results, the possible teaching deficiencies do not have the same consequences as in merely theoretical teaching. y

P

100 N

100 N

N

100 N 100 N

1 1

x M

Figure 5. Fixed pulley (Wikipedia, 2007e) and graph of a corresponding function.

Figure 5 shows such an example. The model of a fixed pulley with a pulling point M moving on the ground leads to consider independent variable (x-coordinate of M), dependent variable (y-coordinate of N), and parameters (length of cable and height of P). Before studying this relatively complex situation, students may begin with the model where P is at the ground level (situation of a donkey pulling water from a well). In this simple case, the trajectory of the point (x, y) is a segment. Computations made in such situations introduce the algebraic concept of real function with all its essential properties, as domain and range. Functionacy obviously is not reduced to understanding properties of functions. Searching for solutions of optimisation problems, and solving problems having a function of a given type for the unknown, are reliable indicators of mastering functionacy. Simple linear regression summarizes most activities related to this competence. From data constituted by a set of ordered pairs ( xi , yi ) , we look for (a, b) that

minimizes the sum of the squares of the error terms

ε i = yi − axi − b (method of least

squares). Here the unknown function is f( x) = ax + b . Understanding the nature of this problem and its solution is highly significant, although it does not require a general definition of function. A more formal concept of function is only needed later, when studying transcendental functions, but functionacy will be reached at that moment.

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Robert Adjiage and François Pluvinage Table 4. Examples of activities, and numerical domains, at each level and age range

Numeracy

Theoretical age of learning 1 < y < 10

Rationacy

8 < y < 14

Algebracy

11 < y < 16

Functionacy

13 < y < 18

Examples of activities

Mathematical topics

Classifying Counting Choosing unit Comparing Playing exchange games Sharing a set of objects Calculating Solving arithmetical problems Writing Using a pocket calculator Using a computer Locating rational values on a number line Sharing a whole Evaluating durations Solving mixture problems Enlarging Scale drawing Using mathematics software Introducing variables in problem solving Drawing straight lines in a coordinate plane Proving the irrationality of the square root of 2 Approximating π Using computer worksheets Using formal calculation Describing a model in terms of variables and parameters Designing models and working with models Approximate computing with real numbers Drawing graphs

Finite sets Whole numbers Arithmetic operations Positive decimal numbers Simple fractions

Ratios Common multiples and divisors (lcm and gcd) Fractions Field of rational numbers Priorities of operations

Inverse of an operation Polynomials Square roots and exponents Coordinate plane

Countable sets Field of real numbers; it is complete and not countable Real-value functions Trigonometry Exponential and logarithm Complex numbers: definition and plane representation

CONCLUSION Whether we need to understand basic documents or meet more advanced requirements of modern society, we are led to distinguish essential competences in number learning. They are

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in fact unavoidable, because these competences are related to mathematical epistemology. The growing complexity of problems that humanity is facing, has entailed necessary successive extensions of the numerical domain from whole numbers to real numbers. These successive extensions are interrelated with new concepts, notions, methods, and processing: whole numbers with arithmetic operations, rational numbers with proportionality and algebra, real number with algebra and analysis. Each extension relies on previous stages, but also implies reconsidering the whole construction, so that knowledge is always an epistemological obstacle to knowledge (Bachelard, 1983, pp.17-22). The global didactical coherence cannot remove ruptures and continuities in the successive steps necessary for constructing mathematics. Deconstructing the numerical complexity is as important for teaching as taking account of differences between writing words, processing words, constructing phrases and sentences, and processing them. Leading students to be aware of the involved processes provides them with the insight for possibly highlighting large areas of mathematics and resorting to chosen tools instead of applying blind algorithms. In order to make apparent the coherence and the cohesion between and within the units of numerical knowledge, we have suggested referring to the natural16 levels of competence we have pointed out. Teaching objectives and assessment items should be based on these levels. In this manner, teaching could aim at enabling students to implement and articulate knowledge and know-how in a coherent numerical area in order to solve problems. Assessment could aim at validating these students’ capacities. In any case, elaborating and testing questionnaires or other forms of significant assessment, including those resorting to computational tools, remain the task for researchers and educators. Modelling, currently considered so important, is one activity among others that may be linked to any level of competence. It can also favour the development of these competences, or be useful for detecting misconceptions or gaps in learning. But for us, it cannot be the unique base for mathematics learning without risks of misunderstanding the global design of mathematics. Empirical and systematic experiments are to be conducted in order to determine the relevance of modelling and to merge it into an educational environment. Following our own reflection and class observations, and others’ studies in this matter, we have pointed out six issues to be taken imperatively into account. •

• • •

16

Extend the concept of modelling to any re-presentation: replace processes and objects, not necessarily stemming from the real world, by mathematical processes and objects intended to explain the former and make predictions. The domain to be modelled may be itself a mathematical one: Three paradigms for modelling geometry have for instance been proposed by Houdement and Kuzniak (2000). Deepen the relationship between the four levels of competence and modeling. Elaborate patterns for designing and developing modelling tasks. Develop modelling tasks. Deepen what really means “authentic” for a modelling task; specify also the impact of its relevance, both to real society needs and to students’ concern, on engagement and success of the latter.

“Natural” has the sense that the construction of the whole numerical domain is not the product of a human design. Its diverse constituents hang together by necessity.

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Specify recommendations for regulating teacher interventions during modelling sessions. Promote orders of magnitude and approximate calculation for helping students to mobilise an adequate procedure (round numbers, being easier than exact numbers, often facilitate a well-considered choice of procedure), and to anticipate and control the results.

These six items will be guidelines for our research and teacher training courses in the next years.

REFERENCES Adjiage R. and Heideier A. (1998), Didacticiels des séries Oratio et NovOra, http://www.alsace.iufm.fr/web/ressourc/serveur_cd_et_video/tout_oratio.htm Adjiage R. et Pluvinage F. (2000), Un registre géométrique unidimensionnel pour l'expression des rationnels, RDM, La Pensée Sauvage, Grenoble, Vol.20.1, pp.41-88. Adjiage R. (2005). Diversité et invariants des problèmes mettant en jeu des rapports. Annales de Didactique et de Sciences Cognitives de l’IREM de Strasbourg, Vol. 10, pp. 95-129. Adjiage R. and Pluvinage F. (2007), An experiment in teaching ratio and proportion, Educational Studies in Mathematics, Vol. 65, pp. 149-175. American Mathematical Society, (2007), Mathematical Subject Classification, number theory section. http://www.ams.org/msc/. Argand, R. (1874, 2nd ed.) Essai sur une manière de représenter les quantités imaginaires dans les constructions géométriques, Paris: Gauthier-Villars Arievitch, I. M. (2003). A potential for an integrated view of development and learning: Galperin’s contribution to sociocultural psychology. Mind, Culture and Activity, 10(4), pp. 278–288 Bachelard G. (1983, 12th ed.). La formation de l’esprit scientifique, Paris: J . Vrin. Blum, W. et al. (1989). Applications and Modelling in Learning and Teaching Mathematics, Chichester: Ellis Horwood. Blum W. (2004). ICMI Study 14: Applications and modelling in mathematics education – Discussion document. Educational Studies in Mathematics, Volume 51, N° 1-2, pp. 149171 http://www.springerlink.com/content/p1l244802942w921 Brousseau G. (1997). Theory of didactical situations in mathematics – 1970-1990 –, Berlin, Germany: Springer. Brousseau, G. (2007). La didactique « spontanée » et le tracassin des réformes. http://www.ardm.asso.fr/didactique/brousseau_seminaire/didactique_spontanee.html Chevallard, Y. (2002). Organiser l’étude. 3. Ecologie et régulation. In Dorier J.L. et al. (Eds), Actes de la 11e école d’été de didactique des mathématiques (pp. 41-56). Grenoble, France: La Pensée Sauvage. Chomsky, N. (1965). Aspects of the Theory of Syntax. Cambridge: The MIT Press. Council of the European Union, (2001). Report from Education Council to the European Council on the Concrete Future Objectives of Education and Training Systems (following Lisbon Report). http://ec.europa.eu/education/policies/2010/doc/rep_fut_obj_en.pdf

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Dhombres J. (1997). Les grandeurs, évolution d’un concept flexible. Groupe de Contact : Enseignement des Mathématiques, Grandeurs physiques et grandeurs mathématiques, n° 1, pp. 1-57. Nivelles Belgique : CREM A.S.B.L. Douady R. (1984), Jeux de cadres et dialectique outil-objet dans l'enseignement des mathématiques, une réalisation de tout le cursus primaire, Doctorat d'état, Paris : Université de Paris VII. Duval, R. (1995). Sémiosis et pensée humaine, Bern : Peter Lang,. Duval, R. (1998). Signe et objet (II) : Analyse de la connaissance. Annales de Didactique et de Sciences Cognitives de l’IREM de Strasbourg, Vol. 6, pp. 165-196. Duval, R. (2000). Basic Issues for Research in Mathematics Education. Proceedings of the 24th Conference of the International Group for the Psychology of Mathematics Education (PME). (July 23-27 2000, Vol. 1, pp.55-69). Hiroshima, Japan. http://eric.ed.gov/ERICDocs/data/ericdocs2sql/content_storage_01/0000019b/80/1a/39/ 1a.pdf European Parliament, (2006). European Parliament legislative resolution on the proposal for a recommendation of the European Parliament and of the Council on key competences for lifelong learning. http://www.europarl.europa.eu/sides/getDoc.do?pubRef=-//EP//TEXT+TA+P6-TA-20060365+0+DOC+XML+V0//EN Galbraith, P., Clathworthy, N. (1990). Beyond Standard Models – Meeting the Challenge of Modelling. Educational Studies in Mathematics 21 (2), pp. 137-163. Galperin P.Y.and Georgiev L.S. (1969). The Formation of Elementary Mathematical Notions. In J. Kilpatrick and I. Wirszup (Eds.), Soviet Studies in the Psychology of Learning and Teaching Mathematics. Vol. 1: The Learning of Mathematical Concepts (pp. 189-216). Chicago, USA: University of Chicago. Gascon J. (1994). Un nouveau modèle de l’algèbre élémentaire comme alternative à « l’arithmétique généralisée ». Petit x, Vol. 37. pp. 43-63. Gelman, R. and Gallistel, C. R. (1978). The child's understanding of number. Cambridge, Mass: Harvard University Press. Second printing, 1985. Gray, E. and Tall, D. (1994). Duality, Ambiguity, and Flexibility: A “Proceptual” View of Simple Arithmetic. Journal for Research in Mathematics Education 25(2), pp.116-140. Habre, S. and Abboud, M. (2006). Students’ Conceptual Understanding of a Function and its Derivative in an Experimental Calculus Course. Journal of Mathematical Behavior 25, pp. 57-72. Houdement C. and Kuzniak A. (2000), Formations des maîtres et paradigmes géométriques. Recherches en Didactique des Mathématiques. Vol 20/1. La Pensée sauvage. Grenoble Kaufman, E. L., Lord, M. W., Reese, T. W., and Volkmann, J (1949). The Discrimination of Visual Number. American Journal of Psychology, Vol. 62, pp. 498-525. Kieran C. (1992). The learning and teaching of school algebra. In D. Grouws (Ed.), Handbook of Research in Mathematics Teaching and Learning, pp. 390-419. New York: Macmillan. Kieran C., Garançon M., Boileau A. (1990). Introducing algebra: A functional approach in a computer environment, proceedings of the fourteenth PME Conference, pp. 51-58. Kourkoulos, M. (1990). Modélisation mathématique de situations aboutissant à des équations du premier degré (thèse). Strasbourg : IREM de Strasbourg, France

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LEMA (2006), Learning and Education in and through Modelling and Application http://www.lema-project.org Lesh, R. A. and Doerr, H. (2002). Beyond Constructivism: A Models and Modelling Perspective on Teaching, Learning, and Problem Solving in Mathematics Education, Mahwah: Lawrence Erlaum. Mager, R. (1962). Preparing instructional objectives. Atlanta: Center for Effective Performance, third edition – May 1997. Maass K. (2005). Barriers and Opportunities for the Integration of modelling in Mathematics Classes- Results of an Empirical Study. Teaching Mathematics and its Applications, Vol 24, pp. 61-74. MEN (Ministère de l’Education Nationale), (2006). Socle commun des connaissances. Journal officiel de la République Française, n° 160. http://www.legifrance.gouv.fr/WAspad/UnTexteDeJorf?numjo=MENE0601554D# NCES (National Center for Education Statistics), (2002-2006) http://nces.ed.gov/surveys/pisa/publications.asp NCES (National Center for Education Statistics), (2001). Results from the 2000 PISA in Reading, Mathematics, and Science Literacy. http://nces.ed.gov/pubs2002/2002115.pdf NCES (National Center for Education Statistics), (2000-2006). International Comparisons in Education. http://nces.ed.gov/surveys/international/intlindicators/ NYSED (New York State Education Department), (1996). Learning Standard 3-Mathematics. http://www.emsc.nysed.gov/ciai/mst/pub/mststa3.pdf NYSED (New York State Education Department), (2005). NYS Mathematics Core Curriculum. http://www.emsc.nysed.gov/3-8/MathCore.pdf Niss, M. (1987). Applications and Modelling in the Mathematics Curriculum - State and Trends. Science and Technology 18, pp. 487-505. Niss, M. (2003). Mathematical competencies and the learning of mathematics: The Danish KOM project. http://www7.nationalacademies.org/mseb/Mathematical_Competencies_and_the_Learning_o f_Mathematics.pdf OECD, (2006). Assessing Scientific, Reading, and Mathematical Literacy, A Framework for PISA 2006, http://213.253.134.43/oecd/pdfs/browseit/9806031E.PDF. OECD, (2004). A profile of student performance in mathematics, http:// www.oecd. org/dataoecd/58/41/33917867.pdf Peter-Koop, A. (2002). Real-world Problem Solving in Small Groups: Interaction Patterns of Third and Fourth Graders. In B. Barton, K. C. Irwin, M. Pfannkuch and M. O. J. Thomas (Eds.). Proceedings of the 25th Annual Conference of the Mathematics Education Research Group of Australia, (pp. 559-566). Auckland, Sydney: MERGA. Peter-Koop, A. (2004). Fermi problems in primary mathematics classrooms: Pupils’ Interactive Modelling processes. In I. Putt, R. Faragher and M. McLean (eds.), Mathematics education for the third millennium: Towards 2010. Proceedings of the 27th annual conference of the Mathematics Education Research Group of Australasia, Townsville, (pp. 454-461). Sydney: MERGA. http://www.merga.net.au/documents/RP542004.pdf Phillips, J. and Tan, C. (2005). Competence, Linguistic. The Literary Encyclopedia. http://www.litencyc.com/php/stopics.php?rec=trueandUID=208#

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Piaget, J. (1967). Biologie et connaissance. Paris : Gallimard. Puig L. (2004). History of Algebraic Ideas and Research on Educational Algebra. ICME-10 Regular Lecture, http://www.uv.es/puigl/icme-10.pdf Rouche, N. (1997). Faut-il enseigner les grandeurs ? Groupe de Contact : Enseignement des Mathématiques, Grandeurs physiques et grandeurs mathématiques, n° 1, pp. 1-57. Nivelles Belgique : CREM A.S.B.L. Sarnecka, B., Cerutti, A., and Carey, S. (2005). Unpacking the Cardinal Principle of Counting: A Last-Word Rule plus the Successor Function www.cogsci.uci.edu/cogdev/sarnecka/Sarnecka%20and%20Cerutti%202005%20FT.pdf Sfard A. (1989). Transition from Operational to Structural Conception: The notion of function revisited. Proceedings of PME XIII. Vol. 3, pp. 151-158, Paris. 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, Vol. 22, pp. 1-36. Streefland L. (1993). The Design of a Mathematics Course, a Theoretical Reflection, Educational Studies in Mathematics, Vol. 25, pp. 109-135. Tall D. (1996). Functions and Calculus. International Handbook of Mathematics Education, chapter 8. The Netherlands: Kluwer Academic Publishers. Tourniaire, F. and Pulos, S. (1985). Proportional reasoning: A review of the literature, Educational Studies in Mathematics, Vol. 16, pp. 181-204. U.S. Department of Education (2004). PISA Results Show Need for High School Reform.http://www.ed.gov/news/pressreleases/2004/12/12062004a.html Vergnaud G. (1982). A classification of cognitive tasks and operations of thought involved in addition and subtraction problems In T.P. Carpenter, J.M. Moser andT.A. Romberg (Eds.), Addition and subtraction: A cognitive perspective, pp. 39-57. Hillsdale: Lawrence Erlbaum Associates. Vergnaud G. (1983). Multiplicative Structures, in R. Lesh and M. Landau (eds.), Acquisition of mathematics concepts and processes, Academic Press, New York, pp. 127-174. Vergnaud, G. (1988). Multiplicative Structures In J. Hiebert and M. Behr (Eds.), Number Concepts and Operations in the Middle Grades, NJ: Lawrence Erlbaum Association, pp. 141-161. Winslow, C. (2005). Définir les objectifs de l’enseignement mathématique : la dialectique matières – compétences (Defining goals of mathematics education: the contents – competencies dialectic). Annales de didactique et de sciences cognitives, volume 10, IREM de Strasbourg France, p. 131-155. Wikipedia, (2007a). Zip’s law. http://en.wikipedia.org/wiki/Zipf's_law Wikipedia, (2007b). Quantity. http://en.wikipedia.org/wiki/Quantity Wikipedia, (2007c). Correspondance et relations. http://fr.wikipedia.org/wiki/Correspondances_et_Relations Wikipedia, (2007d). Relation (mathematics). http://en.wikipedia.org/wiki/Relation_%28mathematics%29 Wikipedia, (2007e). Pulley. http://en.wikipedia.org/wiki/Pulley

In: Science Education Issues and Developments Editor: C. L. Petroselli, pp. 59-94

ISBN: 978-1-60021-950-4 © 2008 Nova Science Publishers, Inc.

Chapter 2

'LEARNING IN AND FROM SCIENCE LABORATORIES: ENHANCING STUDENTS' META-COGNITION AND ARGUMENTATION SKILLS Avi Hofstein1, Mira Kipnis and Per Kind2 1

Department of Science Teaching The Weizmann Institute of Science, Rehovot, 76100, Israel, 2 Department of Education, the University of Durham, UK

ABSTRACT Laboratory activities have long had a distinctive and central role in the science curriculum and science educators have suggested that many benefits accrue from engaging students in science laboratory activities. More specifically, it has been suggested that, when properly developed, there is a potential to enhance students’ conceptual and procedural understanding, their practical and intellectual skills and their understanding of the nature of science. Research findings, however, have proven that “properly developed” laboratory work is less frequent than hoped for and that meaningful learning in laboratories is demanding and complex. The 21st century has offered new frames for dealing with the potential and challenges of laboratory based science teaching. This is an era of reform in which both the content and pedagogy of science learning and teaching are being scrutinized, and new standards intended to shape meaningful science education have emerged. The National Science Education Standards (National Research Council, 1996) and other science education literature (e.g. Lunetta, Hofstein and Clough, 2007) emphasize the importance of rethinking the role and practice of school laboratory work in light of these reforms. The new frames, however, also relates to the development in the understanding of human cognition and learning that has happened during the last 20 years. In the following chapter attention will be given to research on learning in and from the science laboratory. More specifically, the presentation will focus on the science

1 2

[email protected]. [email protected].

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Avi Hofstein, Mira Kipnis and Per Kind laboratory as a unique learning environment for the following teaching and learning aspects: • •

Argumentation and the justification of assertions Development of metacognitive skill

It is suggested, that these are important aspects with a natural place in the science laboratory. They have, however, been neglected both regarding development of practical experiences provided to the student as well as in research on the effectiveness of practical work that is conducted in the context of science learning. A new approach is needed in which these two aspects are coordinated and seen in accordance with the general practice of teaching and learning in school science.

THE LABORATORY IN SCIENCE EDUCATION At the beginning of the twenty-first century, we are entering a new era of reform in science education. Both the content and pedagogy of science learning and teaching are being scrutinized, and new standards intended to shape and rejuvenate science education are emerging (National Research Council 1996; 2005; AAAS, 1990; Millar and Osborne, 1998; Bybee 2000). In general, one of the characteristics of the current reform is the change in the goals articulated for science teaching and learning namely, that science education should be targeted to all students (attaining scientific literacy for all students) and should be extended beyond the preparation of science oriented students for academic careers in the sciences (van den Akker, 1998). This is in fact a call for also rethinking the goals for the learning in and from laboratory work. There are several buzz words that characterize current reform. Among these are student's centered learning, learning by the inquiry method, and development of high learning skills such as argumentation and metacognition. Inquiry in the context of science learning in general and inquiry in the science laboratory in particular are amongst the important components of this reform (Bybee 2000; Lunetta, 1998; Hofstein and Lunetta, 2004; Sere, 2002; Tibergien, Veillard, LeMarechal, Buty, and Millar, 2001). Bybee (2000) suggested that inquiry in terms of skills and abilities includes the following components identifying and posing scientifically oriented questions, forming hypotheses, designing and conducting scientific investigations, formulating and revising scientific explanations, and communicating and defending scientific arguments. It is suggested that many of these abilities and skills are in alignment with those that characterize inquiry-type laboratory work (practical work to include project-based learning), an activity that puts the student in the center of the learning process (see also Hofstein, Shore, and Kipnis, 2004; Hofstein, MamlokNaaman, Navon, and Kipnis, 2005; Dori and Sasson 2007). Laboratory activities have long had a distinctive and central role in the science curriculum, and science educators have suggested that many benefits accrue from engaging students in science laboratory activities (Dori, Sasson, Kaberman, and Herscovitz, 2004, Hofstein and Lunetta, 1982; Woolnogh, 1999; Hegarthy-Hazel, ; Tobin, 1990; Hodson, 1990; Lazarowitz and Tamir, 1994; Garnett, Garnett and Hacking, 1995; Lunetta, 1998; Hofstein and Lunetta, 2004; Lunetta, Hofstein, and Clough, 2007). More specifically, they suggest that, when properly developed, inquiry-centered laboratories have the potential to enhance students’ meaningful learning, conceptual understanding, and their understanding of the

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nature of science. Hofstein and Walberg (1995) suggested that inquiry-type laboratories are central to learning science, since students are involved in the process of conceiving problems and scientific questions, formulating hypotheses, designing experiments, gathering and analyzing data, and drawing conclusions about scientific problems or phenomena. Tobin (1990) based his review on constructivist ideas of providing students with experiences that would enable meaningful learning in the science laboratory. He wrote that: “Laboratory activities appeal as a way of allowing students to learn with understanding and, at the same time, engage in the process of constructing knowledge by doing science” (p.405).

To attain this goal he suggested that students should be provided in the laboratory, with opportunities to reflect on findings, clarify understandings and misunderstandings with peers, and consult a range of resources that include teachers, books, websites and other learning materials. His review reported that such opportunities rarely exist since teachers, in the laboratory, are so often preoccupied with technical and managerial activities. Similarly, Hodson (1993) suggested that although teachers generally professed belief in the value of student-driven, open, practical investigation, in general, their teaching practices in the laboratory failed to support that claim. He also argued that the research literature failed to provide evidence that standard school laboratory activities encouraged knowledge construction. He was critical of the research literature: “Despite the very obvious differences among, for example, practical exercises designed to develop manipulative skills or to measure ‘physical constraints’, demonstration-type experiments to illustrate certain key-concepts, and inquiries that enable children to conduct their own investigations, there is a tendency for researchers to lump them all together under the same umbrella title of practical work” (p. 97).

Tobin wrote that teachers’ interpretation of practical activity should be elaborated and made a part of the research design since a laboratory session could be open-ended inquiry in one classroom and more didactic and confirmatory in another teacher’s classroom. Based on their review of the literature regarding the laboratory Lazarowitz and Tamir (1994) joined the long list of writers claiming that the potential of the laboratory as a medium for teaching and learning science is enormous. They wrote that the laboratory is the only place in school where certain kinds of skills and understanding can be developed. They are among those who have suggested that one of the complicating factors associated with research on the effectiveness of the school laboratory is that often the goals articulated for learning in the laboratory have been almost synonymous with those articulated for learning science more generally. Subsequently, Hodson (2001) wrote that unique outcomes for laboratory/practical work had been articulated in the more recent past, but the nature of students’ experiences in the laboratory and related assessment practices have remained relatively unchanged. In addition, Hart et al (2000) claim that much practical work is purposeless and often the explicit objectives of the practical work does not coincide with the purpose of practical experiences. They also claim that many practical tasks have too many different teaching/learning objectives to focus on during instruction. Similarly, Sere (2002) in France, reporting on a long-term project (Lab-Work in Science Education) conducted in seven European nations

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wrote that: “The intention [of the study] was to address the problem of the effectiveness of lab-work, which in most countries is recognized as being essential to experimental sciences, but which turns out to be expensive and less effective than wished” (p. 624). The project focused mainly on the effectiveness of lab-work conducted in the context of science learning in the upper secondary schools. Information on practice was gathered through 23 case studies, surveys, and a tool that helps to map and describe the domain of laboratory work. Sere reported that the objectives typically articulated for laboratory work (i.e., understanding theories, concepts, and laws; conducting various experiments; learning processes and approaches; and applying knowledge to new situations) were too numerous and comprehensive for teachers to address successfully in individual laboratory sessions. In response, she suggested that the scope of the objectives for specific laboratory activities should be limited. Science curriculum developers and science teachers should make conscious choices among specific learning objectives for specific lab activities and clearly articulate the specific objectives for their students. Sere’s “targeted lab-work” project produced a series of recommendations including the need for each laboratory activity to be supported by a particular strategy organized within a coherent long-term program plan with varied-kinds of laboratory work. The main goal of this chapter is to argue and demonstrate that the laboratory in science education is a unique learning environment (see also Kelly and Lister, 1965; Yeany, Larossa and Yale, 1989; Hofstein, 2004; Lunetta, Hofstein, and Clogh, 2007) that if designed in an articulated and purposeful manner with clear goals in mind has the potential to enhance some of the more important learning skills such as metacognition and argumentation.

LEARNING IN AND FROM SCIENCE LABORATORIES Many research studies have been conducted to investigate the educational effectiveness of laboratory work in science education in facilitating the attainment of the cognitive, affective, and practical goals. These studies have been critically and extensively reviewed in the literature (Blosser, 1983; Bryce and Robertson 1985; Dori and Sasson, 2007, Hodson, 1990; Hofstein and Lunetta 1982; 2004; Lunetta, Hofstein, and Clogh, 2007; Lazarowitz and Tamir 1994). From these reviews it is clear that in general, although the science laboratory has been given a distinctive role in science education, research has failed to show simplistic relationships between experiences in the laboratory and student learning. Hodson (1990; 1993) has criticized laboratory work and claimed that it is unproductive, and confusing, since it is very often used unthinkingly without any clearly thought-out purpose, and he called for more emphasis on what students are actually doing in the laboratory. Constructivist learning theory suggests that learners use ideas and constructs already in their minds to make sense of their experiences. Learning is an active, interpretive, iterative process (Bransford, et al., 2000). Gunstone (1991), however, wrote that helping students’ develop scientific ideas from practical experiences is a very complex process and that students generally do not have sufficient time or encouragement to express their interpretations and beliefs and to reflect on central ideas in the laboratory. Research on learning in the school laboratory makes clear that to understand their laboratory experiences, students must manipulate ideas as well as materials in the school laboratory (White and Gunstone, 1992), and they must be helped to

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contrast their findings and ideas with the concepts of the contemporary scientific community. Manipulating materials in the laboratory is not sufficient for learning contemporary scientific concepts and this accounts for the failure of “cookbook” laboratory activities and relatively “unguided” discovery activities to promote desired scientific understanding. Expecting students to develop scientific understanding solely though their laboratory experiences, reflects misconceptions of the nature of science (Wolpert, 1992; Matthews, 1994) and how people learn science. Several studies suggested that while laboratory investigations offer excellent settings in which students can make sense of phenomena and in which teachers can better understand their students’ thinking, laboratory inquiry alone is not sufficient to enable students to construct the complex conceptual understandings of the contemporary scientific community (Lunetta, 1998). In the laboratory, students should be encouraged to articulate and share their ideas, to help them perceive discrepancies among their ideas, those of their classmates, and those of the scientific community. At the end of the twentieth century there was increasing understanding from cognitive sciences that learning is contextualized and that learners construct knowledge by solving genuine, meaningful problems (Brown et al., 1989; Roth, 1995; Williams and Hmelo, 1998; Wenger, 1998; Polman, 1999). The school science laboratory can offer students opportunities to have some control of their activities, enhance their perception of ownership and motivation (Johnstone and Al-Shuaili , 2001). It can be an environment particularly well suited for providing a meaningful context for learning, determining and challenging students’ deeply held ideas about natural phenomena, and constructing and reconstructing their ideas. Though a complex process, meaningful learning in the laboratory can occur if students are given sufficient time and opportunities to interact, reflect, explain, and modify their ideas (Barron, et al, 1998). In addition, it is suggested that engaging students in metacognitive behaviors enables them to elaborate and to apply their ideas; the process can promote conceptual understanding as well as the development of problem-solving skills. The challenge is to help learners take control of their own learning in the search for understanding while providing opportunities that encourage them to ask questions, suggest hypotheses, and design investigations, “minds-on as well as hands-on” (Gunstone, 1991). Gunstone (1991) suggested that using the laboratory to have students construct and restructure their knowledge is straightforward; however, he also claimed that this view is naïve. This is true, since the picture relating to practical work, as derived from constructivism, is more complicated. In addition, Gunstone and Champagne (1990) suggested that learning in the laboratory would occur if students were given ample time and opportunities for interaction and reflection in order to initiate a discussion. This approach, according to them was under-used, since students in the science laboratory are usually involved in technical activities with only few opportunities for the development of metacognitive activities. We claim that an inquiry-type laboratory that is properly planned and performed have the potential to give students the opportunity to practice metacognitive and argumentative skills (Kipnis and Hofstein, 2007; Dori and Sasson 2007), which are regarded as important goals for the development of scientifically literate citizens who in the future will be in situations that they will need to make decisions, and be able to think critically. Metacognitive skills, it is suggested, will be developed if the conditions in the laboratory will be provided with the goal in mind to promote collaborations, reflection and a community discourse and negotiations (Brown et al, 1989).

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The main goal of this chapter is to explore the potential of the science laboratory for the development of such skills. In recent years we have observed that the science laboratory has lost its unique status in both research and practice (e.g. Osborne,1993; Hofstein and Lunetta, 2004). Thus, the authors of this chapter believe that there are many good reasons to rethink the potential of science laboratories, not at least for the development of high-order learning skills such as argumentation and metacognitive skills.

METACOGNITION: THEORETICAL BACKGROUND Metacognition refers to higher order thinking skills that involve active control over the thinking processes involved in learning. Activities such as planning how to approach a given learning task, monitoring comprehension, and evaluating progress toward the completion of a task are metacognitive in nature (Livingstone, 1997). Because metacognition plays a critical role in successful learning, it is important for both students and teachers. Metacognition has been linked with intelligence and it has been shown that those with greater metacognitive abilities tend to be more successful thinkers. Metacognition is the main issue underlying the theory, research, and practice of teaching for thinking, but there is no single definition that describes it and its diverse meanings are represented in the literature that deals with thinking skills. According to Flavell (1976), who is one of the pioneers of metacognition research, metacognition refers to one's knowledge concerning one's own cognitive processes and products or anything related to them. He recognized the importance of developing metacognition for improving learning and he claimed that "good schools should be hotbeds of metacognitive development" (Flavell, as cited in Georghiades, 2004. p. 366). The definition of metacognition as 'thinking about cognition' enhances the possibility of relating to the components of cognition and classifying the metacognition according to the cognition component that is related to it. According to Schraw (1998), "knowledge of cognition includes at least three different kinds of metacognitive awareness: declarative, procedural and conditional knowledge" (p. 114) (see figure 1). Kuhn (1999a) too, claimed that knowing about declarative knowledge (as a product) is addressed in the section, Metacognitive Knowing. Knowing about procedural knowing (as a process) is addressed in the section, Metastrategic Knowing. Other researchers made their own divisions (Baird and White, 1996; Flavell, Miller and Miller, 2002; Schraw, 1998, Yore and Treagust, 2006). Schraw (1998) referred to knowing of cognition and regulation of cognition. The function of Schraw's regulation of cognition is similar to that of Kuhn's (1999a) metastrategic knowing, which includes the knower's awareness, monitoring, and management of procedural knowledge. Similarly, Yore and Treagust (2006) suggested that metacognition is composed of metacognitive knowledge and regulation of cognition. In this regard, Baird and White (1996) referred to three components of metacognition: metacognitive knowledge, metacognitive awareness, and metacognitive control, and the combination between the two last component functions as Kuhn's metastrategic knowing.

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Figure 1. The metacognition's structure. Based on the model of Shraw (1998).

METACOGNITION IN SCIENCE LEARNING Metacognition, is regarded as an important component of learning and has been investigated lately with regard to science teaching because of several reasons. For the purpose of this chapter we choose to elaborate on the following three reasons: a.

In many research studies in the area of science education, it was found that metacognitive processes promote meaningful learning, or learning with understanding (Baird, 1986; Baird and White, 1996; Gourgey, 1998; White and Mitchell, 1994; Conner, 2000; Rickey and Stacy, 2000; Thomas and McRobbie, 2001; Davidovitz and Rollnick, 2003; Kuhn, 1999a). Meaningful learning, which, as a result of it, students improve their ability to apply what they have learned in a new context, is one of the goals of teaching (Kuhn, 1999a). b. In view of a constantly changing technological world when, not only is it impossible for individuals to acquire all existing knowledge, but it is also difficult to envisage what knowledge will be essential for the future (Georghiades, 2004), the developing of metacognitive abilities that will enable the student to study any desirable knowledge in the future is particularly important. c. One of the goals of scientific education is the development of an independent learner (NRC, 1996; 2005). Efficient independent learning requires the learner to be aware of his knowledge and of the options to enlarge it, meaning that the student must utilize metacognitive skills. In several studies dealing with meaningful learning of science, it was found that one of the characteristic factors in this kind of learning is the involvement of metacognitive processes (Baird, 1998; Baird and White, 1996; Conner, 2000; Davidovitz and Rolnick, 2003;

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Georghiades, 2004; Gourgey, 1998; Rickey and Stacy, 2000; White and Mitchell, 1994). Most of these researchers agree that one of the main characteristics of meaningful learning is that the student must control the problem-solving processes and the performance of other learning assignments. These researchers link this control to the student's awareness of his physical and cognitive actions during the performance of the tasks (Baird, 1998; Gourgey, 1998; White, 1998). Another element that is related to metacognition is the student's monitoring of knowledge (Baird, 1986; Gourgey, 1998; Mayer, 1998; Rickey and Stacy, 2000; Tobias and Everson, 2002). A learner who properly monitors his knowledge can distinguish between the concepts that he knows and the concepts that he does not know and can plan his learning efficiently. When discussing metacognitive processes and scientific operations, we have to take into account that the boundary between cognition and metacognition is fuzzy and not absolutely clear (Zohar, 1999, 2004). For example, Gunstone and Mitchell (1998) argue that some conceptual change processes are metacognitive because "the learner who must recognize his/her conceptions, evaluate these conceptions, decide whether to reconstruct the conceptions, and, if they decide to reconstruct, to review and restructure other relevant aspects of their understanding in ways that lead to consistency" (p. 134). Therefore, an outside person has great difficulty in distinguishing between the cognitive change that represents a cognitive process, and the metacognitive process that represents the intellectual process that caused the learner to decide that there is a need for a conceptual change, and the learner's awareness of the change and of the reasons that caused it. Flavell (1976) suggested that cognitive strategies facilitate learning and task completion, whereas metacognitive strategies monitor the process. Accordingly, "asking oneself question about an article that he is reading might function either to improve one's knowledge (a cognitive function) or to monitor it (a metacognitive function), hence demonstrating co-existence and interchangeability of cognitive and metacognitive functions" (Georghiades, 2004, p.371). Another difficulty in metacognition research, besides the difficulty in distinguishing it from cognition, is to identify and to measure it (Georghiades, 2004; White, 1998). These difficulties originate from the fact that metacognition is cognitive and represents an inner awareness or process rather than an overt behavior (White, 1986). Some strategies for identifying metacognition are suggested in the literature, such as asking the students being investigated about their cognition, asking them to 'think aloud' while performing a task and asking them to teach a younger child a good solution to a problem. Some tools were developed for measuring pupils' metacognitive abilities, mainly in mathematics (Georghiades, 2004), but any strategy and any tool has its limitation.

DEVELOPMENT OF METACOGNITIVE SKILLS IN AN INQUIRY-TYPE SCIENCE LABORATORY White and Mitchell (1994) specify students' behaviors that, in their opinion, are characterized as "good learning behaviors" for students who developed certain metacognitive skills. A large part of these behaviors (and skills) are actions that constitute an integral part of the inquiry laboratory activity, such as: asking questions, checking work against instructions, correcting errors and omissions, justifying opinions, seeking reasons for aspects of current

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work, suggesting new activities and alternative procedures, and planning a general strategy before starting. Since the students who participate in the inquiry laboratory activities are obliged to act according to the activities that are typical of students with developed metacognition, it is logical to assume that during the study of the inquiry laboratory unit the students can practice and develop their metacognitive skills. Kuhn, Black, Keselman, and Kaplan (2000) argued that students that experience inquiry activity "come to understand that they are able to acquire knowledge they desire, in virtually any content domain, in ways that they can initiate, manage, and execute on their own, and that such knowledge is empowering" (p. 496). Baird and White (1996) claimed that: "If carried out thoughtfully, this process of purposeful inquiry will generate a desirable level of metacognition; the person will know about effective learning strategies and requirements, and will be aware of, and be capable of exerting control over, the nature and progress of the current learning task" (p. 191).

They also claimed that four conditions are necessary in order to induce the personal development entailed in directing purposeful inquiry: time, opportunity, guidance, and support. In the inquiry laboratory activity the students get the time and the opportunity to practice metacognitive skills, and the teacher gives them the guidance and the support that they need. Thus, one can conclude that the inquiry laboratory activity is specially suitable for enhancing metacognition and meaningful learning, because during the activity the students perform open inquiry, which integrates strategies that are known in the literature as metacognition's promoters: working in small groups, supplying time for group discussion, observing phenomena that should be explained at the particle level, and exploring a question that was asked by the students. Emerging attention to a social constructivist theoretical framework has special potential for guiding teaching in the laboratory (e.g. Tobin, 1990; Lunetta, 1998). Social learning theory emphasizes that learning is situated in interactions with those around us and conceptual development is associated with the medium of language. Thus, learning depends, in part, on interactions with adults and peers. Social learning theory makes clear the importance of promoting group work in the laboratory, so that meaningful, conceptually focused dialogue takes place between students, as well as between the teacher and students. Moreover, laboratory experiences in which students discuss ideas and make decisions can present many opportunities for teachers to observe students’ thinking as they negotiate meaning with their peers. Carefully observing students’ actions and listening to their dialogue creates opportunities for teachers to focus questions and make comments within learners’ zones of proximal development (Vygotsky, 1978, 1986; Duschl and Osborne, 2002) that can help the students construct understandings more compatible with the concepts of expert scientific communities. It should be noted that developing metacognitive skills is time consuming process. In order to allow for the development of these skills, students should be provided with practical (laboratory) activities that have potential for such processes. The science teacher should provide his/her students with experiences, opportunities, and the time to discuss their idea about the problems that they have to solve during the learning activity. The role of the teacher is to provide continuous guidance and support to ensure that the students develop control and awareness over their learning. This, it is suggested, can be accomplished by providing the

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students with more freedom to select the subject of their project, to manage their time and their action in the problem solving process. It is suggested that cooperative learning in groups has the potential to enhance students' development of metacognitive skills too, since while conducting the group activity, the student can clarify their ideas and the way they had developed them, in order to explain those ideas to their class mates. All these are critical characteristics of practical work that have potential to develop learning with understanding in which the student is in the center of learning.

RESEARCHING THE SCIENCE LABORATORY FOR THE DEVELOPMENT OF METCOGNITIVE SKILL Since metacognition has many definitions and meanings, we chose to present here a model that was used in this study for the purpose of analysis of the data. This model, we suggest, is highly aligned with the nature of metacognition that has the potential to be developed in inquiry-type science laboratories. The model is based on Schraw's (1998) definition of metacognition and is similar to Yore and Treagust's (2006) conception of metacognition. According to this model, there are two main components in the metacognition: 1. Knowledge of cognition refers to what individuals know about their own cognition or about cognition in general. It includes three different kinds of metacognitive knowledge: declarative, procedural, and conditional knowledge. • Declarative knowledge includes knowledge about oneself as a learner and about factors that influence one's performance (knowing 'about' things). • Procedural knowledge refers to knowledge about doing things. Much of this knowledge is represented as heuristics and strategies (knowing 'how' to do things). • Conditional knowledge refers to knowing when and why to use declarative and procedural knowledge (knowing the 'why' and 'when' aspects of cognition). 2. Regulation of cognition refers to a set of activities that help students control their learning. Although a number of regulatory skills have been described in the literature, three essential skills are included in all accounts: planning, monitoring, and evaluation. • Planning involves the selection of appropriate strategies and the allocation of resources that affect performance. • Monitoring refers to one's on-line awareness of comprehension and task performance. • Evaluating refers to appraising the products and efficiency of one's learning. Regarding the inquiry laboratory activity, knowledge of cognition should be reflected during the discussion about the observations that students conduct, by asking appropriate questions, and operating a suitable inquiry stage. Regulation of cognition should be expressed

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during the planning of the experiment, while performing it, and evaluating the results regarding the assumption.

FROM THEORY TO PRACTICE: DEVELOPING METACOGNITIVE SKILLS IN AN INQUIRY LABORATORY-A CASE-STUDY The Inquiry Laboratory Program A laboratory program titled: "Learning in the chemistry laboratory by the inquiry approach" was developed in Israel at the department of Science Teaching at the Weizmann Institute of Science. For this program, about 100 inquiry-type experiments were developed and implemented in 11th and 12th grade chemistry classes in Israel, amongst students who opted to specialize in high school chemistry (Dori, 2003). Almost all the experiments were integrated into the framework of the key concepts taught in high-school chemistry, namely acids-bases, stoichiometry, oxidation-reduction, bonding, energy, chemical-equilibrium, and the rate of reaction. These experiments were implemented in the school chemistry laboratory in Israel in the last eight years. This implementation took place in a situation in which control was provided over such variables as the professional development of teachers, the continuous assessment of students’ achievement in the laboratory and the allocation of time and facilities (materials and equipment) for conducting inquiry-type experiments. Over the years, this program was researched intensively regarding the development and implementation, assessment of students’ achievement and progress, professional development of the chemistry teachers, and the class room laboratory learning environment (Hofstein Levy Nahum and Shore, 2001; Hofstein, Shore and Kipnis, 2004; Hofstein, Navon, Kipnis, and MamlokNaaman, 2005; Kipnis and Hofstein, 2007). Typically in this program, the students perform the experiments in small groups (3-4), following the instructions given to them by the laboratory manual. The basic assumption within Vigotskian frame of reference is that reasoning in students is mainly manifested in the externalized form of discussing and arguing with others. Table 1 illustrates the various stages that each of the groups undergoes in order to accomplish the inquiry task. In the first phase (the pre-inquiry phase), the students are asked to conduct the experiment based on specific instructions. This phase is largely ‘close-ended’, in which the students are asked to conduct the experiment based on specific instructions given in the laboratory manual. Thus, this phase provides the students with very limited inquirytype experiences. The ‘inquiry phase’ (the second phase) is where the students are involved in a more ‘open-ended’-type experiences such as asking relevant questions, hypothesizing, choosing a question for further investigation, planning an experiment, conducting the experiment (including observations), and finally analyzing the findings and arriving at conclusions. It is suggested that this phase allows the students to learn and experience science with understanding and consequently to enhance their metacognitive abilities.

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Abilities and Skills

Phase 1: Pre-inquiry • •

•

Describe in detail the apparatus in front of you. Add drops of water to the small test tube, until the powder is wet. Seal the test tubes immediately. Observe the test tube carefully, and record all your observations in your notebook.

• Conducting an experiment

• Observing and recording observations.

Phase 2: The inquiry phase of the experiment 1. Hypothesizing • Ask relevant questions. Choose one question for further investigation. • Formulate a hypothesis that is aligned with your chosen question 2. Planning an experiment • Plan an experiment to investigate the question • Ask the teacher to provide you with the equipment and material needed to conduct the experiment • Conduct the experiment that you proposed. • Observe and note clearly your observations. • Discuss with your group whether your hypothesis was accepted or you need to reject it.

• Asking questions and hypothesizing

• Planning an experiment

• Conducting the planned experiment • Analyzing results, asking further questions, and presenting the results

CONCLUSION In this research we realized that the inquiry laboratory supplies the students with an opportunity to practice metacognitive skills. We found that while performing the inquiry activity, the students, whose activity was observed above, practiced their metacognition in various stages of the inquiry process. This was expressed particularly in the following stages: (a) while asking questions and choosing an inquiry question, the students revealed their thoughts about the questions that were suggested by their partners and about their own questions. In this stage, the metacognitive declarative knowledge is expressed. (b) While choosing the inquiry question, the students expressed their metacognitive procedural knowledge by choosing the question that leads to conclusions. (c) While performing their own experiment and planning changes and improvements, the students demonstrate the planning

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component of regulation of cognition. (d) At the final stage of the inquiry activity, when the students write their report and have to draw conclusions, they utilize metacognitive conditional knowledge. (e) During the whole activity, the students made use of the monitoring and the evaluating components concerned with regulation of cognition. In this way, they examined the results of their observations in order to decide whether the results are logical. The fitting between the inquiry stage and the metacognitive component is shown in Table 2. Table 2. Matching the metacognitive components that were observed with the various inquiry stages. (N=3) The inquiry stage

The students' metacognitive activity

Asking questions and choosing an inquiry question

The students revealed their thoughts about the questions that were suggested by their partners and about their own questions.

Choosing the inquiry question Performing their own experiment

Drawing conclusions and writing the final report

The students choose the question that leads to conclusions. The students plan changes and improvements for the experiment. At the final stage of the inquiry activity, the students use knowledge that was acquired in a different context to write their report and to draw conclusions.

The metacognitive component that is expressed Metacognitive declarative knowledge

Metacognitive procedural knowledge Planning component of regulation of cognition.

Metacognitive conditional knowledge

As was previously mentioned, metacognition is an inner awareness or process (White, 1986) and therefore it is not constantly shown. To discern it during the activity, the students under observation should be verbal and willing to reveal their thoughts. In the analysed activity there was a unique combination of talkative students and a situation where a new student was trying to understand what had been going on in the group in the past, and therefore their thoughts were revealed to us. The fact that other observations lacked prominent expressions of metacognition does not mean that the students did not use their metacognition. We concluded that there are opportunities for metacognitive activity in the inquiry laboratory, but it is revealed in observations that demand special conditions, as mentioned above. A further investigation should be conducted in order to determine the factors that influence metacognitive activity in the inquiry laboratory: is this activity depended on external conditions such as the teacher's behavior, the experiment type, the class, etc., or is it depended only on the student's mentality. In the interviews it is easier to find metacognition aspects, because when a person talks about his/her knowledge and learning, he/she reflects his knowledge about his own cognition and this reflection is actually metacognition (Georghiades, 2004). All students that were

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interviewed demonstrated metacognitive knowledge. All the components of metacognitive knowledge, namely knowledge about people's knowledge, knowledge about strategies, and knowledge about tasks, exist in the students' conversations. The students mentioned metacognitive activity that took place during the performance of the inquiry and was the focus of the activity, according to them, in checking, searching for logical explanations, and correcting errors. Those actions involve monitoring and evaluation, which are important components of regulation of cognition (Flavell et al., 2002). Also Yore and Treagust (2006) suggested that strategic planning, monitoring progress and regulation actions are the 'realtime' executive control of cognitive operations central to science literacy involved in doing science and learning science. The students' metacognition is also demonstrated in their reflection essays in which they refer to their learning processes and to the role of the inquiry activity regarding those processes. Most of the students indicated that the inquiry activities help them to understand the theoretical concepts: "I understand what exists under the formula. I think that the inquiry program helped me very much regarding this point, because it gave me the opportunity to think by myself about things, and it helped me to understand them better." "We see how the theoretical subject becomes practical by experiments that help us to understand it better."

According to the reflection essays, the help in understanding is expressed in several ways, such as realization: "The inquiry program caused us to see the subject matter that was learnt in the class in a different way, because it made it concrete."

The inquiry activities also aid in remembering things: "When I perform something by myself, than the subject matter is transferred better and there is a better chance that I shall remember it."

They also provide an opportunity for students to make mistakes and to learn from them: "I learnt to accept the fact that sometimes our results differ from our expectations and that we can learn a lot from doing mistakes."

In addition to the literature sources that claim that the inquiry laboratory activity has the potential to enhance students' meaningful learning, their conceptual understanding, their inquiry skills, and their understanding of the nature of science (Hodson, 1990; Hofstein and Lunetta, 1982; 2004), we can add that the inquiry laboratory provides the students with the opportunity for metacognitive activities. The utilization of this opportunity depends on many factors, such as the teacher's behavior, the inquiry activity, and the laboratory environment. It should be noted that the most important variable for the development of metacognitive abilities is the students' motivation to utilize the time and the activity for meaningful learning.

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ARGUMENTATION: THEORETICAL BACKGROUND Our next topic, argumentation, is fundamental to many situations in professional and everyday life: it is found in academic work, in political and civil debates, in juror work and many more. Due to the variation in where it is found, it is not surprising that is approached in a wide variety of ways also as a research topic. We will be looking at argumentation in science, and more particularly the science laboratory, and will restrict our approach to elements relevant for this. We will start by following Toulmin (1958) and separate argumentation from the much older field of studying logic. In logic focus is on rules for relating premises to conclusions; in other words, deciding what could in principle be concluded from a set of premises. Toulmin claimed this is different from the practice of argumentation, i.e. how people actually go about with their discursive activities. Following his view, our approach is to look at how argumentation is practiced in science. We should, however, keep in mind that science may have different meanings in different contexts. Professional scientists doing research, school students with their school-based experiments, and citizens trying to analyse and act upon socio-scientific issues may all practice scientific argumentation, but within very different frames. A next step is to be aware of different types of arguementations. Kuhn (1992) separates between rhetorical and dialogic arguments. The former is the persuasive argumentation done to convince somebody that something is right or true. The latter normally happens in dialogues when two or more people with opposing views try to reach agreement. Keeping the science classroom in mind we easily can imagine both types of arguments happening. For example, the teacher trying to convince the class about the correctness of a scientific claim or theory is demonstrating a rhetorical argumentation, while the group of students who are sharing views about best explanation for an observation or a problem are demonstrating the dialogic argumentation. Our main focus will be on dialogic argumentation, i.e. argumentation trying to establish consensus among many arguments. We will, however, not see this only as a group activity. An individual person may in a similar way as a group weigh up arguments to try to reach a best conclusion. For example, when a scientist is faced with a range of evidences and tries to find out how these relate to alternative theoretical models. This may still be called a dialogic argument, because it is a dialog between arguments, or we may use the term monological argument (Newton, Driver and Osborne,1999) to point out that it happens within one person. An essence of dialogical and monological arguments is interplay between the arguments and the evidence that back them up. Mason and Santi (1994) describe arguments as the “empirical fact of communication” (p.4). It is a critical engagement with ideas; such as when we look for reasons, advance justifications, try to explain something, oppose rebuttals, suppose solutions, evaluate evidence or consider alternative positions. A definition of argumentation capturing this is presented by Krummheuer (1995, 231): “the intentional explication of the reasoning of a solution during its development or after it”. In other words, argumentation is to be explicit about the reasoning behind a statement, or a solution. Some, like Mason and Scitica (2006), also describe argument as informal reasoning: “reasoning applied outside the formal context of symbolic logic and mathematics” (p.493). One of the most commonly used theories to explain argumentation is presented by Toulmin (1958). He identified a set of constitutive elements involved in arguments and

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described the functional relationship between them in a model. These may be illustrated in an example from science: We put water in a bowl and later observe that it has disappeared. Based on this “data” we make a claim that water has evaporated. We justify the claim with the warrant of a rational scientific explanation (water can go from liquid to gas) and the backing that the water has been untouched (nobody tipped or drank the water). We also could strengthen the argumentation by using qualifiers, such as “the temperature in the room was high” or “the air was very dry”, and rebuttals, such as “the evaporation would not have happened if the bowl was covered up”.

Toulmin’s model shows how the elements data, warrants, qualifiers, and rebuttals, are used in argumentation to support a claim. Although the model has proved very useful for analysing argumentation in educational research, and many other fields, it has weaknesses. One of these is to ignore the context of the argumentation (Driver, Newton and Osborne, 2000). The argumentation above, for example, would appear very different in a science classroom today and in a science laboratory at the time when the theory of phases of matter was still in development; although the structure of the argument could have been exactly the same. We also find that Toulmin’s model draws the attention towards a “philosophical” perspective on argumentation and ignores psychological abilities: It analyses the actual argument, but does not say anything about the skills or abilities lying underneath and that make a person good at arguing. Searching for such underlying skills and abilities, we again find obvious links to reasoning. However, as Mason and Scitica (2006) explain, reasoning abilities are just one factor affecting a person’s ability to argue. There are also social factors, such as “assertiveness”, i.e. being able to express yourself and your rights without violating the rights of others, and the desire to maintain a warm and friendly relationship. Beside, for most argumentation it is obviously important to be knowledgeable in the topic that is discussed. Mason and Scitica also add “epistemological belief” to the list. We will return to several of these skills when we look at how argumentation may be fostered in practical work, but first, to understand the full extend to which argumentation has established as a topic in science education, we will look at the more specific issue of scientific argumentation.

ARGUMENTATION IN SCIENCE We are often led to think of data and observations as the bedrock on which science is built, but there are good reasons to offer this position to scientific argumentation. At both a personal and a social level we find that scientists continually work in an argumentation process of weighing empirical and theoretical evidence in light of warrants, backings and rebuttals to reach an understanding of how natural phenomena may be explained. Characteristics of this process may be illustrated within three different theoretical frames. The first frame is an epistemological view, which deals with the logical and philosophical grounds for scientific knowledge. In early versions of science philosophies, scientists were seen to collect observations without any predisposed understandings and to reach true knowledge by use of “inductive logic”. Popper (1972) and other science philosophers at the

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beginning of the 20th century challenged this view and made argumentation a fundamental principle in any scientific inquiry. They suggested that scientists rather use their imagination to conjecture patterns in the nature and thereafter test these in experiments. The science approach therefore involves a rational argument among scientists to find out which conjectured theories are best. The “tool” for doing this is, of course, scientific experiments. In experiments scientists test specific elements of a theoretical model to see how this match expected observations. The main challenge in this process, that makes argumentation such a fundamental principle in science, is that experimental data never speaks for themselves; in the meaning of giving a straightforward picture of which theory is right. Experimental data rather must be interpreted and given meaning in light of what scientists already know. Scientists are therefore left with a set of possible theoretical claims and models describing what they think the nature is like and a set of empirical evidence that may be interpreted in different ways. Holding all this information up against each other is the core of a scientific approach and fundamentally based on argumentation. A second frame for argumentation in science is based on a sociological view. Focus is then on scientists as a group and how they work together; i.e. on science as a collective activity and a culture. We then find the same principle of argumentation as presented above but in a different scale and setting: The argument is not carried out just in the laboratories by scientist reflecting on claims and evidence but at conferences and in journals when scientific findings are made public and criticised by peers. Reviewers evaluating scientific papers, for example, are fundamentally in a process of argumentation when they assess the quality of conclusions and in light of their backing and warrants. Newton, Driver and Osborne (1999) on this basis describe arguments as an institutionalised practice. Science philosophers who have studied this practice, such as Thomas Kuhn (1962), have demonstrated clearly that it is not a purely rational or “technical” activity: Scientists are humans influenced by social commitments and personal values and preferences. This makes the scientific argumentation more complicated, including both logical and sociological factors. A third frame for describing scientific argumentation is the psychological view of science, describing characteristics of scientific thinking. Obviously logical reasoning is very important to scientists (Simonton, 2004; Feist, 1998) and Kuhn, Amsel and O’Loughlin (1988) identify the most essential and general skills in this reasoning to be the coordination of theory and evidence. This is seen as general cognitive skills, but with domain specific features. Scientists and other groups who depend strongly on identifying and using evidence, Kuhn et al. claim, benefit strongly from conscious control of the skills; in other words, on metacognition as discussed earlier in this chapter. Their research (ibid.) make them conclude that people who master coordination of theory and evidence have the ability to think explicitly about theories they hold and about evidence that might support the theories. Central themes in our further discussion are, therefore, what types of teaching that may help student develop knowledge about theory and evidence and how this understanding may help them becoming better at conducting scientific argumentation.

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ARGUMENTATION IN SCIENCE EDUCATION Searches on argumentation in science education on the databases, such as ERIC (www.eric.ed.gov), or in specific science education journals, reveal that the concept was almost none-existing up to the mid 1990s, but has later flourished in popularity. One reason for this is the development of scientific literacy as a main target for science education all over the world (Mayer, 2002), but we also find other influences. We will look into some of these influences and how they have manifested in science education. One reason that has brought argumentation to its important position is the social constructivist view of learning, influenced amongst others by Vygotsky (1978). In this view, knowledge is seen as “socially constructed” and therefore strongly related to discursive activities: the individual learn through the processes of dialog and discourse with others. Learning science then becomes very much like learning a foreign language (Driver, Newton and Osborne, 2000) – the learner has to formulate meaningful sentence and learn from the feedback from others. This view of learning is matched by a social constructivist view of science, emphasising science knowledge as shared and agreed knowledge among scientists. Emphasising discursive teaching and learning in science, therefore, has a double role in, first, being an efficient and “natural” way of learning science and, next, by illustrating a core principle of the nature of science (Duschl and Osborne, 2002). A pedagogical consequence of this development has been to give students a range of opportunities and possibilities to practice their use of scientific knowledge through language-based activities (Wellington and Osborne, 2001). Relating argumentation more directly to scientific literacy, we find emphasise on two main targets. First, students should understand the role of argumentation in science as part of a more general emphasis on “nature of science” and how science works (Millar and Osborne, 1998). The emphasise on argumentation in this context follows naturally from role it plays in both the epistemology of science and the social practice of science, as outlined above. The other target is to make students themselves better at doing scientific argumentation. The plausibility of this also follows from the science literacy perspective: Citizens in a society meet situations were they are expected to express a voice about public science-based issues. In many of these situations they have to use scientific information, identify claims, weigh up evidence and draw their own conclusions. For example, when engaging with socio-scientific issues such as global warming or the prospect of BSE, the individual needs to: separate conjectured claims and predictions made by scientists from well established knowledge and theories; they need to ask critical questions about quality and origin of the evidence; and they need to weigh up all this information to draw their own conclusions about risk and what political actions to support. Following from these perspectives it is natural that science educators now search for ways of implementing and teaching argumentation. Laboratory teaching is one of the obvious ways of doing this.

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ARGUMENTATION IN PRACTICAL WORK TEACHING One of the early papers to trigger a wider interest for scientific argumentation among science educators was presented by Driver and Newton (1997) at the ESERA conference in Rome in 1997. In the paper they introduce scientific argumentation as a following up of the critique of “process science” presented ten years earlier (in Millar and Driver, 1987). The main argument in the1987-paper was that “process science” was based on a false picture of how science works, and the 1997-paper claims that “argumentation” represents a better and more authentic platform for science teaching. This critic against “process science” united with a more general critic against the role of practical work in science education (Hodson, 1990; 1993; 1996). “Process science”, it was claimed, had been part of a culture that too strongly saw practical work as the right way of teaching science and Hodson, and other critics, argued for a less dominant role. Several researchers, like Osborne (1993), on such bases started to seek alternatives to practical work; for example, language-based activities in which students articulate and discuss science problems in oral and written form. In this way we get a sense of contrast and tension between argumentation and practical work. However, we have also illustrated that argumentation is at the core of a scientific approach, and as such would think the development should strengthen the position of practical work. Driver and Newton (1997), in their analysis, do give credit to role of practical investigations in the teaching, but emphasise a strong need for changed perspectives. The teaching, they claim, should give less attention to the tradition of mechanic training of procedures and instead make students understand the overall purpose of investigations. More specifically they claim that rather than training students’ skills on controlling variables and repeating measurements students’ attention should be given to planning investigations and to interpretation of data; “both which are processes that require argument” (ibid. p. 15). To understand more fully how “argumentation” influences upon practical work teaching we need to separate between the different aims that have been indicated earlier. We will look into argumentation as a tool for learning science, as an activity among scientists that students should understand and identify as a core element of the “nature of science”, and last, as a skill we would like students to learn and transfer to their lives outside school.

Teaching Science in the Laboratory Through Argumentation As we have seen, argumentation is regarded as a way of teaching science with background in the social constructivist perspective on learning. In this perspective, critical engagement in using and practicing the science language in a range of structured activities is a prerequisite for learning science (Duschl and Osborne, 2002: 41): “[I]f the structures that enable and support dialogical argumentation are absent from the classroom, it is hardly surprising that student learning is hindered or curtailed”. Students, of course, through history have learned science in most science classroom without there being much emphasise on discursive activities, but the point made by Duschl and Osborne, and other educators supporting argumentation, is that the discursive element fits more naturally into students’ natural way of learning and that such activities therefore may prove to be more efficient teaching and leading to “better” understanding of scientific knowledge. One reason for

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claiming this is that engagement in argumentation when learning science gives students a better feeling for “how we know” and why we believe something to be right. These epistemological perspectives are thought to deepen the understanding of the scientific theories and concepts (Lemke, 1990; Sutton, 1992). The best way of organising such teaching is small group discussions, which provides an opportunity for the individual student to socially construct, and reconstruct, their own personal knowledge through a process of dialogic argument (Driver, Newton and Osborne, 2000). Research, however, has shown that discursive argumentation is rare in science teaching (Newton et al., 1999; Mortimer and Scott, 2003). The study in Newton et al., for example, based on observation of 39 lessons, revealed that less than 5 per cent of the time was devoted to group discussions and less than 2 per cent of the teacher-pupil interactions were genuine discussions with an exchange of differing views. Obviously, ordinary classroom teaching orchestrated by the teacher may also bring forward discursive activities, for example if the students are presented with alternative theories and invited to evaluate these and move towards an agreed outcome, but most teaching in the lecture-based classrooms rather follow a dialog pattern referred to as the “IRE” or “IRF” structures (Lemke, 1990); controlled entirely by the teacher: The teacher starts with an initiation, the student(s) respond and the teacher evaluates or follow-up the response. This gives little room for discussion as students are more engaged with satisfying the teacher’s expectation for a right answer than coming up with their own points of views. Wellington and Osborne (2001), therefore, suggest that discussion activities should be carried out in groups with no more than 3-4 students. There are of course other factors than group size that may stimulate or hinder students’ learning outcome in group-based argumentation, such as the learning environment within the group (Alexopolou and Driver, 1996; Hogan, 1999). Groups with best learning outcome are characterised by more willingness to consider one another’s opinions, while groups with a lack of cohesiveness are less successful (Alexopolou and Driver, 1996). In a group, the same “teacher-led” dynamic as in the class may happen if one student take a too “persuasive” leader style and persuade other students to follow his/her way of thinking (Richmond and Striley, 1996). This may be a good learning experience for that particular student, but not for the group as a whole. Hogan (1999) has shown that all individuals in a group may have different socio-cultural “roles”, some which may stimulate learning and others that may hinder it. Practical work, we would think, should offer many of the sought after conditions for engaging students in scientific argumentation and therefore be an efficient ground for social constructivist learning of science. All arguments against conceptual science learning in the laboratory, however, warn us that this is not a straightforward matter (see for example Hodson, 1993). We are bound to admit that much teaching in the science laboratory has been no more successful with creating argumentation than teaching in the ordinary classrooms. Two reasons seem obvious for this. First, the “recipe” nature of practical work activities that has made them into “hands-on” but “minds-off” activities. Second, that teaching in the laboratory invites for “doing the lesson” rather than “doing science” (Jimenez-Aleixandre et al., 2000): It invites talks about what the task is, what to write, what is required as an answer rather than ways in which observations and data justify or contradict theoretical claims. Watson (2004) concluded from a study of students’ discussion in practical scientific inquiries that the main factor that seemed to militate against the development of argumentation was “the routinzed nature of the practical activity” (p. 40). Both students and the teacher seemed

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for most parts to view scientific inquiry as learning to carry out a set of fixed procedures. Other studies, however, are more optimistic and prove situations in which students engage uniquely in discussion about empirical observations and findings (Kelly, Drucker and Chen, 1998; Kelley and Chen, 1999). Kelly et al. (1998) focused particular on student use of warrants in laboratory tasks and found a range of situations in which warrants were used, but not always as expected in given tasks. The reasons for coming up with warrants were more “social” than “scientific”. For example, students felt no need to give warrants if they thought their laboratory partner shared the same view as themselves. The analysis above give no reason to claim that laboratory tasks automatically stimulate argumentation among students more than classroom based activities, but focus on argumentation has helped understand some of the problems regarding conceptual learning in the laboratory. One obvious outcome is that the potential for learning would be much greater if students engaged more uniquely in discussing theoretical claims in light of observations and data. A way forward is therefore to identify ways in which this may happen. As we have seen, one way would be to give more attention to the planning and the analysis phases of investigations (Driver, Newton and Osborne, 2000). Research by Chin (2006) and Oh (2005) also indicates that much may be achieved by having teachers asking the right questions and giving appropriate feedback, while Watson’s (2004) research suggests that the teacher’s and students’ conception of practical tasks should be more coordinated. Another suggestion is to look at argumentation activities developed for group work in classrooms (Wellington and Osborne, 2001). In many science educators’ mindset practical work activities are supposed to follow the structure of a scientific experiment: with problem, data gathering, analyses and conclusions. Wellington and Osborne have presented a range approaches to structure argumentation activities which follow a very different approach. Many of these may work as well in a laboratory setting as in ordinary group work in the classroom. For example, observations in the laboratory may be used as a starting point for making concept maps, experiments may be presented as “discussions of instances” or students may be given written “reasoning tasks” about evidence for theoretical claims that they discuss first and then bring into the laboratory for further investigation. By thinking more creatively around how laboratory tasks are organized this natural group-based learning environment may open up a range of argumentation-based learning task in line with the social-constructivist view of learning.

Teaching About Scientific Argumentation in the Laboratory With the increasing interest for scientific literacy discussed in the opening of this chapter, there has been established a much stronger awareness among science educators about the needs of ordinary citizen and how these needs differ from those of students who will carry on with a career in science or technology. One issue that has been discussed is the need for explicit rather than implicit knowledge about science. A scientist needs to be able to use scientific methods, and for him or her, the first experiments at school are steps towards doing the same types of activities at a more advanced levels later in the career. Through engaging with increasingly advanced and independent research tasks a science researcher develops tacit knowledge about how to do science experiments. For non-specialist “citizens” the situation is very different. They do not need the tacit knowledge that makes them able to carry out

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experiments, as these will never be repeated after they have left school. Rather, they need to be able to understand and engage with science in public media and everyday events. This requires an explicit, declared understanding about science rather than the implicit and tacit competence related to science investigations and experiments (Driver et al., 1996). Citizens need to be able to tell the difference between good and bad science and why some science conclusions are more trustworthy than others. For these reasons teaching about science has become a major topic in the scientific literacy movement (AAAS, 1993, Mayer, 2002, Millar and Osborne, 1998; NRC, 1996). A range of terms are used to describe this topic; for example, “nature of science” (NOS), “ideas-about-science” and “how science works”. The concept of argumentation fits comfortably into all these areas as a major “idea” about science. The question we will be asking is, therefore, what opportunities the school science laboratory offers for teaching about argumentation. Before discussing this question, however, we need a somewhat more precise account for what we want students to understand about argumentation. Several different approaches are found to identify what epistemological knowledge students should learn about science, and subsequently about scientific argumentation. One approach use information from situations outside formal education and is based on analysing problems and situations occurring in society and how people reflect on and handle these (Ryder 2001. Tytler et al., 2001; Aikenhead, 2005). A second approach is based on studying students who work with investigative task in the school laboratory (e.g. Millar et al., 1995), who do project work including the laboratory (e.g. Ryder, Leach and driver, 1999; Ryder and Leach, 1999) or who more generally discuss socio-scientific issues (e.g. Sadler, Chambers and Zeidler, 2004). A third approach is based more on theoretical analysis of features of scientific methods (Gott and Duggan, 1995). A fourth approach is doing “Delphi” studies to find agreed elements among experts (Osborne et al. 2003). Still there are more, and this variation naturally draws attention towards a range of different aspects. There is, however, a clear general agreement across all these studies about the value of understanding how scientists approach the task of establishing trustful and valid information, and about the problems and challenges this involves. This includes being aware of reliability and validity issues; for example, the issue of uncertainty when making measurements and how this affects the interpretations and conclusions that may be made from the data. A more divided question is whether training towards this understanding should happen in a laboratory or nonlaboratory environment. The laboratory environment draws focus towards methodical issues in experiments and investigations (how reliable and valid is empirical data), while the nonlaboratory environment put more general focus on epistemological backing of science knowledge (how do we know that the Earth is spinning?) and on how to draw valid conclusions in socio-scientific problems (do mobile telephones cause a health risk?). We also find a contrast between teaching about science in school contexts and out of school (real life) contexts. “School issues” normally operate within established knowledge and clearly framed questions, such as, “do temperature have an effect on how sugar dissolves?”. While out of school issues are more complex and involve knowledge in the frontier of science (what is the evidence for human caused climate change and what should we do about it). Although the same scientific principles or rationale operate underneath all these issues and problems, there is a huge question mark to whether students see it this way. Some information which shed light upon this problem is the attempt to teach “ideas and evidence” in UK.

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Few other countries have had such a systematic effort on teaching the principles of science as England and Wales since the introduction of the national curriculum in the late 1980s. This was done mainly through practical work investigations. Surveys, however, revealed that the conceptions students develop are far from the ones intended in the guidelines (Donnely, 1996; Solomon, 1994; , Driver et al. 1996). Although students develop the skills to do certain types of experiments and investigations, few students acquire the intended understanding that experiments in science are used to test theoretical ideas. Some (mostly younger) students see experiments simply as activities in which you make something happen (like baking a cake). At end of primary and at early secondary level most students establish the idea that science investigations and experiments aim to find out something, but restrict this idea to identifying relationship between variables and not to establishing theoretical claims. Student are also empiricist in their view and see the process of drawing a conclusion as a simple matter of stating what happened in terms of data from an experiment. At an older age (upper secondary and first year at university) students are found sometimes to go to the opposite extreme and take “radical relativist views” (Ryder, Leach and Driver, 1999), stating that nothing certain may be stated from experiments. The overall picture from UK (i.e. England and Wales) is that students become good at doing specific types of experiments, but that these are conducted “mechanically” rather than as a result of the aimed-for understanding of science (Kind, 2003). Mostly it is the exam system that is blamed for this situation (Donnelly et al., 1996), students are coached to do specific activities for the purpose of having a high mark, but the problem has a more general relevance. It shows very clearly that it is not just enough to do science, i.e. do investigations, in order to establish and understanding about science. Something more is needed, and if teachers have not got an understanding of this “something more” practical work teaching for the purpose of understanding about science is bound to fail. To be fair towards the development in UK, it should be mentioned that curriculum guidelines have now changed and made it much more explicit to teachers that it is students’ knowledge about science that is the aim for the teaching rather than their ability to carry out a specific type of investigative activity. New educational programs, like the 21st Century Science (University of York/Nuffield Foundation, 2004), also have followed up the development and treat teaching about science in a systematic and coherent way all through the curriculum. From these developments and the accompanying research we get a clearer picture of what understanding is needed and how it may be implemented in science teaching. Evaluation from the changes, however, is still to come. Contrasting the negative experience overall in UK we find many studies that report positively on learning about science in the laboratory teaching. For example this is documented during the last 20 years in studies of students’ development of procedural knowledge in laboratory work (Gott and Duggan, 1995; Lubben and Millar, 1996; Millar et al., 1995). When properly taught laboratory tasks may make students aware of quality issues of experimental data and of experimental design, and through this get a realistic feel for the experiment as a specific type of argumentation. The repeatedly heard phrase, however, is that just doing investigations is not enough; this type of teaching requires explicit focus on the procedural concepts for the intended learning to happen (Gott and Duggan, 2003). The teacher’s understanding of the aim for the teaching is, for example, a crucial factor (Kang, and Wallace, 2005).

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A further question is to what degree this possible successful learning in the science laboratory may transfer to other contexts? Will it apply, for example, to students’ understanding of science in of science in socio-scientific issues? In interview situations when students are asked about their understanding of what real scientists do compared to what they do in the school laboratory, they tend to draw many similarities (Kind, 1999). For example, students with an “empiricist” view about school investigations revealed similar views when asked about science methods used by scientists. However, these interviews trigger a comparison and do not prove that students use this understanding on their own initiative. As pointed out by Ryder, Leach and Driver (1999), understanding about science may be developed from a range of sources; including school science, science documentaries on television and scientific issues reported in the news. As the understanding about science tends to be very “episodic”; i.e. related to particularly memorable events and anecdotes, experiences from school experiments may not necessarily be the one recalled by students themselves in other situations. Some studies, however, have indicated, with various amount of evidence, links between students’ understanding of the nature of science and how they approach and handle argumentation in out-of-school affairs (e.g. Sadler, Chambers and Zeidler, 2004).

Training Argumentation Skills in the Laboratory The final aim for teaching argumentation in the science laboratory is to develop students’ argumentation skills as a tool for drawing their own conclusions and handling decisions in “socio-scientific” issues. The underlying rationale is that students through learning about science and through systematic training in scientific argumentation at school will develop a scientific mind-set that prepares them for meeting such issues in everyday life. The last question we will be asking in this section on argumentation is, therefore, what role practical work in the science laboratory play in this training. However, before answering this we will give a more extensively outline what we mean by a skill of argumentation. In doing so, we will revisit the three frames for scientific argument: the psychological, the sociological and the epistemological. Explaining argumentation skills in a psychological frame draws close correspondence to reasoning, i.e. the process of drawing conclusions (Leighton, 2004), and critical thinking, i.e. thinking that is goal directed, logical and purposeful (Halpern, 2007). These are both areas of the cognitive domain that underlies and inform problem-solving and decision making. Leighton (ibid.) describes reasoning as a “mediator” and gives an analogy of a middleman in a business: someone who works behind the scenes, coordinate ideas, premises or beliefs in the pursuit of conclusions. These conclusions may sometimes find their way to the surface in terms of explicitly expressed ideas, arguments and claims, but will often stay beneath the surface and feed into some chain of thinking. Others, like Gilhooly (2004), make the claim that calling something reasoning it should be explicit, and that it is an application of some sort of formalized rule (deductive logic, mathematics, statistics, probability, decision theory or inductive/deontic logic). The latter does not match our definition of argumentation, which is closer related to informal logic, but the former gives a point of interest in teaching argumentation skills. In the psychological approach to scientific argumentation that we presented earlier (based on Kuhn, 1999b and Kuhn, Amsel, and O’Loughlin, 1988) it is

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emphasized that scientists need conscious control of their ability to coordinate theory and evidence: In the absence of this ability, one’s belief are utilized as a basis for organizing and interpreting experience, but only by means of this second-order, reflective thinking ability can one think about, evaluate, and hence be in a position to justify these beliefs. Kuhn, 1999b, p. 14

Such control is established though epistemological understanding (which is outlined below). Contrarily to a cognitive psychological approach, argumentation may be seen more as a skill in social discourse. Reasoning, even when it is made explicit, is primarily an individual activity, whereas argumentation is more typically a social activity, i.e. claims put forward, criticised and defended in a group of people. The skill therefore obviously requires more than systematic reasoning. One way of seeing this, which is suggested also by Kuhn (1999b), is that the reasoning ability presented above is a cognitive prerequisite of argumentation, but that a fuller understanding of the skill needs more extensive perspectives. It is not obvious what specific skills make a person good at doing social discourse, but Brown (1992), who presents a training program for science discourse, gives some examples: group members have to ask questions, clarify comprehension problems, summarise the gist what have been said and make predictions about future content. The discourse abilities are also described as showing critical attitude and abilities to justify and evaluate one’s own and other people’s reasoning (Duschl and Osborne, 2002). Obviously, this social element of argumentation is vaguer and harder to operationalize in measurable constructs. In science education research studies argumentation skills have often been categorised and “measured” by counting the number of claims, warrants, data, and backings in a discussion; i.e., the elements from Toulmin’s (1958) model for argument pattern (Erduran, Simon and Osborne, 2004; Osborne, Erduran, Simon and Monk, 2001). Many researchers, however, have felt a need to go beyond this counting to grasp the fuller complexity of social discourse (Walker and Zeidler, 2007). Although there is some tension between a cognitive psychological and a social discourse approach to understand scientific argumentation, there also exist an extensive body of research that support the integration of these two dimensions. Duschl and Osborne (2002) describe this integration as a crucial element of today’s understanding of learning; based on the theories of Vygotsky (1978), Rogoff (1990), Saxe (1991) and many others who see thinking and reasoning as socially driven. According to Vygotsky reasoning patterns that are practiced at an intra-psychological level, are copied and used at an inter-psychological level by the individual in his or her reasoning. Allowing students to engage in high-quality argumentation in groups therefore, in theory, should develop the cognitive argumentation skills at an individual level. Several places in this chapter we have hinted that the skill of argumentation is linked to knowledge and understanding. This claim has two dimensions as the knowledge that is taken into consideration is both epistemological and conceptual. The latter of this is most discussed, and we find, on one hand, researches, like Kuhn et al. (1988), who start with the view that certain reasoning skills in argumentation are domain general (rather tied to specific content knowledge), and on the other hand, researchers who claim argumentative discourse operate primarily on knowledge and understanding rather than on procedures (De Vries et al., 2002).

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The analysis conducted by Erduran (2007), however, shows that either of these two claims is far too simple as a statement on their own. There are different types of domain-specific knowledge that interact in reasoning processes in many different ways, depending on what type of issues or problems that are discussed. The epistemological knowledge, contrarily, has gained a much greater agreement over its role in argumentation skills, even if this knowledge is more complicated to comprehend. For example, epistemological understanding is crucial in the metacognitive control described above. Kuhn et al. (1988) identify three types of epistemological understanding involved in scientific argumentation. Firstly, a person needs to be able to think about a theory, rather than only think with it. “A theory” may mean both a claim and a more comprehensive model, but in either case it is important to know that this is a tentative statement which is put forward as a possible explanation. Secondly, the person must be able to encode and think about the evidence in a similar way. Theory and evidence must be differentiated, such that there is a concept of evidence standing apart from the theory and bearing on it. Third, a person needs to put aside his or her own interpretation of what is “right”; i.e. his or her opinion if the theory should be accepted or rejected. The theory rather should be “weighted” in light of the available evidence. From Kuhn’s perspectives we see the obvious role of epistemological understanding taught through “the nature of science” (NOS). Through teaching about NOS students will understand the tentative nature of science and the role of evidence. Many researchers have therefore asked if and how such teaching influence upon students’ cognition (Bell and Lederman, 2003; Sadler, Chambers and Zeidler, 2002; Walker and Zeidler, 2007; Zeidler et al., 2002). This research to some degree gives support to Kuhn’s claims that epistemological understanding gives better reasoning, but when it comes to actual decision making NOS-understanding is reported to play “the role of the back seat” (Walker and Zeidler, 2007). Bell and Lederman (2003), for example, managed to show that a student group divided into two subgroups following epistemological beliefs about science, but found not difference in how these groups approached decision making. Rather, there are student’ attitudes and not their knowledge that has a final saying about what students think is the best options. Walker and Zeidler (2007) therefore conclude that it is not enough to teach epistemological perspectives in NOS topics, but that students should also be explicitly guided in applying that knowledge as he or she evaluates competing scientific claims. For example, students should be guided on what it means that knowledge is tentative when they weigh data and evidence when deciding upon an alternative in a particular socio-scientific issue. We sense a great potential for laboratory teaching to offer a frame for teaching in which meaningful understanding about science may be linked to arguementation in authentic scientific problems. When assessing this potential, however, we need to take into consideration that nature of inquiry tasks, which vary greatly in their level of being authentic (Chinn and Malhotra, 2002). Simple inquiry tasks, such as when students are introduced to an independent and a dependent variable and told to find the relationship between them, require very simple type of reasoning and represent an incomplete environment for students to practice authentic argumentation. Text book analysis by Chinn and Malhotra of nearly 500 inquiry tasks showed that few if any captured the fuller cognitive processes of authentic science. The tasks rather assume an epistemology that sometimes is entirely at odds with the epistemology of real science. We find a strong tradition in science laboratory teaching of making tasks that algorithmically leads students to the intended conclusion. More authentic tasks do exists, but often these are more comprehensive project work tasks and not practical tasks done in a single or double lesson. This problem may be an important reason that many

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teachers rather turn to computer-based simulations when they want to illustrate and teach issues of argumentation in inquiry in an authentic contexts (Walker and Zeidler, 2007). There is, however, some evidence that “simple” inquiry tasks also may be used for teaching argumentation skills. Gott and Duggan (1995, 2003), for example, argue for building up an understanding of scientific evidence through a systematic and stepwise process using guided tasks. They present an approach in which students are introduced to the underlying structure of an investigation and then taught elements of design, data gathering and data analysis that affects validity and reliability. They make a strong case about the value of data in science and claim exercises illustrating reliability and validity in simple inquiry tasks may lead to better argumentation skills in authentic socio-scentific issues. Research show that students improve their understanding of evidence in science and improve their school investigations (Gott, Roberts and Glaesser, 2007), with some transfer to socio-scientific issues (Roberts and Gott, 2007). The conclusions to be made from research which attempts to understand the use of laboratory teaching for developing students’ skills of argumentation are similar to the previous sections. Firstly, there is an obvious problem that practical work traditionally is carried out in a stereotyped way with recipe-like activities that require little discussion about validity and reliability of data. This makes students “do science” with little understanding about science. This problem may be omitted if validity and reliability issues are made more explicitly the aim for the activity and if teachers help stimulate discussion and reflection among students (Simon, Erduan and Osborne, 2006). Secondly, properly designed practical activities with supportive teaching can give students valuable epistemological knowledge and an understanding of how science works. The advantage of teaching about science in the laboratory is to underline the importance of data as evidence in science (Gott and Duggan, 1995). Thirdly, the school science laboratory can not offer a fully authentic scientific environment (Chinn and Malhotra, 2002). Compared to the complexity of science as it is carried out in real research, and similarly the socio-scientific issues and problems students meet in society, the laboratory teaching will always be a simplified version. This, however, does not mean it is without value, but rather that teachers and other educators need to be aware of the limitations in what the laboratory can offer. Fourthly, which appear as an overall conclusion, at the moment there are many promising and interesting ideas about what the role of the laboratory might be, but a lack of research to prove the full line of transfer of learning from laboratory teaching to the argumentation students are required to carry out in real life contexts.

CONCLUSION In this chapter, we define laboratory with special attention to scholarship associated with learning argumentation, and justification of assertions and the development of metacognitive skills. We claim that by working in science laboratory students can establish understanding about science, about scientific knowledge namely how science works in principle. Developing assertions about the natural world in school science and then justifying those assertions with data collected in investigations within or beyond the science classroom walls is considered increasingly to be an important element of school science learning (Newton ,

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Driver, Osborne,1999; Zeidler, 1997). The National Science Education Standards (NRC, 1996) also indicates the importance of engaging learners in describing and in using observational evidence and current scientific knowledge to construct and evaluate alternative explanations "based on evidence and logical argument" (p.145). All these means that one should have the opportunities to develop epistemological understanding of science, in which we see the laboratory teaching to have a foremost role. . Laboratory activities that engage the mind as well as the hands have students “thinking out loud, developing alternative explanations, interpreting data, participating in cognitive conflict (constructive argumentation about phenomena under study), develop[ing] alternative hypotheses, [designing] further experiments to test alternative hypotheses, and [selecting] plausible hypotheses from among competing explanations” (Saunders, 1992, p 140). Effective laboratory activities require significant student engagement, thinking, and decision-making. However, in effective laboratory activities, teachers play a crucial role in helping students have productive experiences. The teaching models and strategies teachers employ and the ways they interact with students determine the extent to which well designed laboratory activities promote the desired learning.

Metacognition and Argumentation: Are they Related? Zohar (2004) in her book regarding the development of high-order thinking skills in the science classroom, focused on the issue of declarative metacognitive skill. She suggested that this skill comprise of an explicit knowledge about both the cognitive procedures and the conceptual structures that are manipulated. Among the cognitive procedures she included analyze causal relationships, ability to construct a good argument, and the ability to test hypotheses. She wrote: The pertinent metacognitive knowledge includes explicit awareness of the type of cognitive procedures being used in specific instances and the knowledge of when, why and how they should be applied (p 180).

Regarding knowledge of conceptual structures, she mentioned knowledge of taxonomies, knowledge about the logic of hypothesis testing, and finally relevant to this chapter, the knowledge about what constitutes a good argument. Mason and Santi (1994) found a strong link between argumentation and metacognition. They found four levels of reflection which allow to think "about" a conception as a thinking target, and to think about the thinking process. Those levels include: (1) awareness of what one knows, (2) awareness of why one knows something, (3) awareness of knowledge construction procedure, and (4) awareness of changes in one's own conceptual structure. They explain that the argumentative structure of the collective reasoning induces reflection on what, why, when, and how one knows. In their research they found that the deepest steps of argument were characterized by the higher level of metacognitive thinking. Let us now return to the main issue of this chapter, namely the interrelationship between metacognitive skills and argumentation, in the context of the science laboratory. It is suggested, that the two high order skills that are in the focus of this chapter, could be developed through practical work under certain laboratory conditions that provide the

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students with time and opportunities to discuss and reflect on their findings. Our key assumption is that in order to construct an articulated-type and good argument, the student should be able to operate several metacognitive skills. It is suggested, that those metacognitive skills should be operated in addition to cognitive skills that are necessary in order to include knowledge and underlining theories. Based on Schraw's (1998) model (presented in figure 1), it is suggested, that this could be accomplished if the student will regulate the cognition, namely will monitor, plan, and evaluate the task. In other words, in the laboratory, during the construction of an argument, the students has to plan his/her activity, namely how he/she could construct an argument that will be a persuasive. In order to accomplish that, he must monitor his knowledge in order to be able to compose the argument, to bring about evidence to support his/her argument, and to find the proper scientific principle that supports the argument. In addition, he/she needs to evaluate whether the scientific principle is valid one in an attempt to connect the argument with the evidence. Falvell et al (2002) model includes also a metacognitive knowledge component that relates to knowledge about persons. It is suggested that in presenting an argument, one has to know about the person's knowledge to which the argument is presented. This type of knowledge is important in order to convince those to whom the argument is presented, if the goal is to convince them. For example, if the students are arguing about results obtained from a certain experiment, it is important that those in the group who are presenting an argument, will have an insight regarding the way of thinking of the other members of the group. The claim we have made in this chapter is that, based on Kuhn, Cheney, and Weinstock (2000), epistemological understanding is vital for the development of metacognitive skills, and that this further will lead to the development of more advanced and mature argumentation skills. We have to remember that in order to develop such high order skills students should be provided with opportunities in the science laboratory that will allow these to happen

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In: Science Education Issues and Developments Editor: C. L. Petroselli, pp. 95-123

ISBN: 978-1-60021-950-4 © 2008 Nova Science Publishers, Inc.

Chapter 3

THE CRISIS IN SCIENCE EDUCATION AND THE NEED TO ENCULTURATE ALL LEARNERS IN SCIENCE Stuart Rowlands Centre for Teaching Mathematics, School of Mathematics and Statistics University of Plymouth, United Kingdom

ABSTRACT There is a crisis in science education. Over the past two decades many organisations such as the American National Science Foundation, the Australian Audit of Science, Engineering and Technology and the UK’s Royal Society and the Confederation of British Industry, have reported a serious decline in students enrolling in science subjects and the failure of the science curriculum to inspire learners and to meet national needs. However, quite apart from instrumental reasons such as a national interest for having more scientists, science education is important for cultural reasons. Science permeates every aspect of modern life and arguably full citizenship in a technological society necessitates the understanding of science. Based on how the world is, science promotes critical thinking, a concern for evidence and an objectivity that is independent of personal opinion or the dictate of kings - yet few individuals have an elementary understanding of science. The failure of science education is reflected in science’s lack of popularity evident in the rise of mysticism, the rise in consensus of intelligent design, the postmodernist attack and the closing of many UK university science departments. There have been calls to remedy the situation, such as school visits by scientists and engineers, or overhauling the science curriculum by teaching the science deemed relevant to the everyday experience of children, consistent with the constructivist idea that there is a “children’s science”. However, there is no guarantee that exposure to the working lives of scientists will promote an interest in science and, moreover, science is not based on making sense of experience. To generate the interest and motivation of young learners requires an engagement with the nature of science (NOS) involving meta-discourse with the history and philosophy of the discipline. Contrary to the current wisdom of science educationalists, NOS has more to do with the rule-governed abstract possible world of the thought-experiment than hypothesis testing with a clipboard of data. Even the most concrete thinkers may be capable of thinking in the abstract and mechanics, because of its history and logical character (as opposed to the “soft sciences” such as ecology), provides the perfect opportunity to do this. This article consists of three parts: 1. Public perception

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INTRODUCTION The crisis in science education has been generally perceived in terms of national interests, but the crisis is also more than this. Arguably an understanding of science is necessary for the full enculturation into technological society, yet the majority of people do not have a basic understanding of basic scientific concepts. The fundamental argument of this article is that science education should not only engage all learners with the content of science but also with the nature of science itself (NOS). Perhaps the biggest mistake regarding the nature of science is that it is fundamentally an empirical affair, but science doesn’t begin with experience, it begins with intellectually constructed objects – it is the product of the thinking head (Matthews, 1980). This is not to undermine the role of observation, but underestimating the ideal and abstract nature of science not only undermines its very nature but also the difficulties that learners have in understanding it. Section 1 highlights the nature of the crisis in science education, but it also delves into fundamental philosophical issues regarding science and the world. It was felt necessary to do this because of the attack on science by the prevailing relativism that exists in education. Defending science from relativism requires a critique of what is being claimed, a critique that cannot avoid the nature of science in terms of its logic, something that relativism claims is irrelevant. Because of this, Section 1 is very long, but it is hoped that the discussion of NOS in this section will prepare for the discussion in Section 2 of the importance of NOS in science education. However, despite the length of Section 1, its discussion of fundamental issues is in no way adequate in representing the volume and length of the debate in the philosophy of science, but it is hoped that the author will be forgiven and that the discussion will prompt further interest in the debate. Section 3 criticises the science education literature’s emphasis on the learner at the expense of promoting the disciplines and argues that cognition must be viewed through the subject matter as a lens. Promoting the disciplines has the danger of evoking the ‘science’ or ‘math wars’, and there simply is no space to take on the whole debate. But it is good to provoke, if only to knock us out of our complacency and to question what could become new orthodoxies. This is a discussion of science education, but many points are made with reference to mathematics education. There is a similarity between science and mathematics in that both consist in highly abstract and ideal theoretical objects. The ideal and abstract nature of science is discussed in sections 1 and 2 and it was not felt out of place to discuss mathematics education in relation to science education, especially since both as disciplines have been under attack by relativists in education. However, there is a common misconception that the difference between science and mathematics consists in the former being based on measurement and observation and the latter on the highly theoretical and ideal. This misconception has existed within mechanics textbooks since Jeans (1907) up until the present. Consider the following from a popular theoretical mechanics textbook of the 1970’s:

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Mechanics, which deals with the effects that forces have on bodies, is a science. So the laws of Mechanics are scientific laws. They come from observation and experiments and so can never be considered as universally true. The most that can be said of several of these laws is that they agree with observed results to the extent that they are accurate enough for most purposes. Pure mathematics, on the other hand, is an art and its theorems are universally true. (Bostock and Chandler, 1975, p. 1). How the laws of motion come from observation and experiment has never been made clear, but educationally the problem with this assumption is that it fails to engage the learner with the nature of the subject matter, which in turn might affect the understanding of the content. This article attempts to explain how NOS is inextricably bound in resolving the crisis in science education

1. PUBLIC PERCEPTION OF SCIENCE AND SCIENTIFIC LITERACY AND UNDERSTANDING The Crisis in Science Education as a Socio-Cultural Crisis Since the launching of Sputnik in 1957 there has been a persistent sense of crisis in science education and one that is perceived in terms of national interests, that sense is international. For example, the Australian Government Audit of Science, Engineering and Technology recently reported a serious decline in students enrolling in science subjects. According to the UK’s Confederation of British Industry, too many young people are ‘turning their backs’ on science and technology and that 16 -18 year old students studying physics and chemistry have declined by 56 percent and 37 percent respectively over the past two decades. Advanced technological nations are deemed ‘at risk’ if education fails to develop the next generation of scientists, mathematicians engineers and especially teachers who can teach science and mathematics effectively. This sense of crisis is a sense of science education failing as ‘cultural capital’, a kind of investment that is necessary not only for the survival and economic well-being of any technological society, but also for the very thing that makes the investment possible – science itself. This sense of crisis is acute and goes beyond national boundaries: if science as a discipline vanished in the school and university curricular internationally then in one or so generation’s time science as a discipline would cease to exist. This is a crisis in education but arguably its roots extend beyond education to the way society is organised. Currently in the UK and perhaps internationally, the survival of university subject departments depend on enrolment; consequently and despite the protest, many chemistry departments and more recently physics departments have closed. Despite the time, effort and money gone into highlighting the crisis in science education and popularising science, those very institutions that are necessary for the survival of science as a discipline, namely the universities, are closing science departments due to economic survival. Why study science and indeed why is science so important if universities are closing their science departments? It becomes a vicious circle: the popularity of science affects enrolment which in turn affects the popularity of science. The problem is that in terms of national well-being, the return on science as a form of ‘cultural capital’ is very long-term. Commercial technology

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utilises the outcome of science long after publication and the outcomes established, quantum theory and its effect on the electronics industry being a case in point, but in practice such (‘academic’) arguments are redundant when it comes down to surviving what could be termed ‘market-forces’. Valuing science is one thing, the survival of institutions in a period of economic instability quite another. Although in the UK many physics and chemistry departments have either closed or are under the threat of closure, there are many science courses that are ‘soft’ in the sense of servicing popular degrees such as environmental science, (called Surf Science at one university), perfumery and golf-course management. One reason why physics and chemistry are dubbed ‘hard sciences’ is because they are relatively autonomous theoretical practices (in the sense given by Chalmers, 1982), which means to say that their development is intrinsic to the very subject matter itself. They are relatively autonomous because a space-race or a thermonuclear project does have its influence (see Gillot and Kumar, 1995). However, the products of that practice, namely laws, theories, data and the associated theoretical objects, are taught for their own sake as well as for any instrumental value. The reason to teach science for its own sake has a disciplinary corollary and this is explored later, the point here is that this contrasts with science becoming more and more of a service module. Although the science modules in these courses may be deemed to have the same rigours that are associated with what is regarded as ‘hard science’, science is less and less becoming taught for its own intrinsic and disciplinary value. In addition to the anxiety over falling admission rates in university science courses such as physics and chemistry, there are concerns over the ‘dumbing-down’ of undergraduate programmes of study, a dumbing-down that is presumably aimed to counter this falling admissions rate. Two decades ago the American National Science Foundation reported the scope and quality of undergraduate programmes in science, mathematics and technology no longer met national needs (Matthews, 1994). That trend appears to continue, demonstrating that falling admission rates aren’t due to any increase in the rigours of the subject-matter. Admission rates fall despite the lowering of standards. There seems to have been the assumption that the lowering of standards will increase the motivation to learn, the UK’s school mathematics curriculum being a case in point: the curriculum has been ‘dumbeddown’ (Rowlands and Carson, 2002a), yet there presently exists a national crisis whereby many learners are underachieving in mathematics at all levels (Smith Report, 2004). There exists a perceived failure of education to inspire the learning of science. Recently the UK’s Royal Society reported the failure of school science to inspire and interest learners as a major contributory factor in the decline of school science. The report may be a little harsh given that studying for a science degree in the past gave you more of an assurance of a science based job at the end of it – nowadays there is less extrinsic motivation to learn science – but the report raises a challenge as to how an intrinsic interest in science can be developed amongst our learners. This is a tall challenge and assumes that it is indeed possible to develop an intrinsic interest. This article take the more optimistic view that it is not only possible, but also necessary for cultural reasons, such as citizenship in a technological society and the Enlightenment value of furthering science for the benefit of humanity. However, it must be stated at the outset that there are limits to what education can do with respect to changing lives for the better. Social ills such as exploitation and oppression will not be eliminated by education, as if education can solve the world’s most basic and ‘fundamental’ problems.

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Solving the world’s problems may begin with education, but these problems have a social and material basis and that has more to do with the way society is organised than education. This is reflected in the unanimous agreement that education must be a priority, yet ‘nobody knows what to do with it’ (Gillott and Kumar, 1995, p.187). Solving the world’s problems is more complex than educational, and Gillott and Kumar locate that complexity in the very structure of society itself: The view that a broad and scientific education can rescue society from its deep-seated economic and social problems speaks more of a culture of despair than of a rolling-up of intellectual and practical sleeves. It ought to be obvious that today’s crisis in education is a symptom, not a cause, of those problems. By themselves, schools and universities are not magically going to mobilise science for good, whatever the plans now drawn up for them might suggest. Science education will only fully help humanity control its fate once other, more basic wellsprings of wealth and power – labour, capital the state – are fundamentally transformed for the twenty fist century. (Gillott and Kumar, 1995, p.187, emphasis given).

The crisis in education is a symptom, not a cause. Unfortunately, the argument that science education will only fully help humanity if society has been fundamentally transformed was not elaborated by Gillott and Kumar (1995). Perhaps understandably so if, as I suspect with Gillott and Kumar, the elaboration would have to make explicit the proletariat as the agent for that transformation. It’s not easy to write about science from the Marxist perspective of labour and capital, the Leninist perspective of the state and the need to build the revolutionary party, and in a way that not only shows the relevance of all this to science education but doesn’t put off the reader. From the Leninist perspective, such writing can only be aimed at the potential vanguard (Lenin, 1973), which today doesn’t exist, hence the difficulty. This does suggest that the ambiguity in Gillott and Kumar is purposeful. However, the crisis in education as a symptom notwithstanding, if science education is to benefit humanity (and hopefully one day in controlling our fate) then we cannot wait for that transformation (if at all). By and large this crisis in science education has its roots in the way society is organised but is reflected in instrumental concerns such as the need to have more scientists or the wellbeing of the nation-state. For liberal educators, however, there is an altogether different sense in which science education is in crisis. That sense is cultural and has to with the nature of science juxtaposed with a general lack of understanding what science is and the social consequences. On the one hand we have science which promotes rationality, objectivity, critical thinking, a concern for evidence and judging ideas on how the world is (Matthews, 2000). On the other hand studies in scientific literacy reveal widespread antiscientific views and illogical thought amongst large proportions of the population, who do not know the meaning of basic scientific concepts (Matthews, 1994). Matthews states the case rather poignantly: Studies of scientific illiteracy reveal a situation that is culturally alarming, not just because they indicate that large percentages of the population do not know the meaning of basic scientific concepts, and thus have little if any idea of how nature functions and how technology works, but because they suggest widespread antiscientific views, and illogical thought. Newspaper astrology columns are read by far more people than science columns; the tabloid press, with their Elvis sightings and Martian visits, adorn checkout counters and are

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Perhaps ‘self interest’ refers to what is deemed immediate rather than learning science because it is in your interests to do so, ‘self interest’ seems to suggest ‘false consciousness’, but Mathews is making the point that if science is to serve the whole of humanity materially and spiritually then it follows that the whole of humanity ought to understand science. As Matthews (2000) states, to live in a ‘scientific age’ requires understanding science. The corollary is that the ignorance of science in terms of both content and enterprise becomes an ignorance of the way society operates, contrary to any sense of what full democratic participation in any technological society might mean: Science has an ever present but often subtle impact on virtually every aspect of modern life – both from the technology that flows from it and the profound philosophical implications arising from its ideas. However, despite this enormous effect, few individuals even have an elementary understanding of how the scientific enterprise operates. (McComas and Almazroa, 1998, p.115).

It is more than knowing the content of science or how the scientific enterprise works; it also has to do with the profound philosophical implications of science. Science has impacted deeply on all the fundamental questions that humans ever posed and has changed our perception of the world; the problem is that none of this comes out in the ordinary course of science education (Carson, 1997). For Carson, the great tragedy of modern education is its lack of coherent cultural vision; so-much-so that mathematics and science are taught for their instrumental value and not for their disciplinary value. This is explored further. The logic of teaching for instrumental value actually implies very little science and mathematics precisely because it puts into question the value of teaching the abstract concepts of science and mathematics. Why learn the quadratic formula if you are going to become a machine operator or salesperson? Why learn mechanics if its concepts are counterintuitive compared with your everyday experience (the ‘misconceptions’ literature is perhaps the largest domain in science education). A disciplinary argument for teaching science is that science education can engage all learners in the abstract nature of the discipline and how this abstraction relates to the world, thus providing the opportunity for intellectual development. The disciplinary argument for teaching science becomes the developmental argument and in Section Two it will be argued that a cultural-historical approach can make that development a conscious one irrespective of some ranges of ‘ability’. The point here is that if science is not taught for its disciplinary value then there is danger that school science either becomes something that makes sense of experience (a radical constructivist position, discussed in Section 3) or something required for employment or living in the everyday. If so, then we might have a two-tier system whereby the children of the poor learn what’s deemed good for them and the children of the rich learn the disciplines for their developmental value.

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This article takes the initial premise that all children ought to be afforded the opportunity to receive a liberal education in the sense given by Hirst (1973); that is, an education that requires some proficiency in most areas of intellectual culture. Hirst (originally) argued that a liberal education is necessary if the learner is to enjoy a broad range of experiences that would be otherwise inaccessible and this requires teaching/learning the concepts of the disciplines (he was slated for this argument and has since changed his position). He was not suggesting that learning science in-itself constitutes a liberally educated mind; rather that science education is a necessity for a liberal education. Unfortunately, most school learners are not going to become scientists, mathematicians or chartered engineers, and if much of the science and mathematics taught in schools have no relevance to the subsequent employment of learners then the instrumental reasons for teaching these disciplines becomes redundant. Unfortunately there is no ‘great debate’ as to why we should teach the disciplines for their disciplinary value. In the UK the great Education Acts of the past were at least based on philosophic considerations (Fox, 2004). Philosophy provides a meta-analysis of the public debates over education, examining issues such as such as what, how and why ought to be taught (Carson, 1997). Presently in the UK no such consideration is given, while heaps of reform serves to plague the teacher with mind-numbing bureaucracy (Fox, 2004). In the UK at least, we are left with bureaucracy, an uninspiring science and mathematics curriculum and batteries of tests with pupils who feel they don’t have a stake in the system. There appears to be many intelligent pupils who ‘play-dumb’; intellectually ‘dropping-out’ as a form of strategic resistance to the legal obligation to learn something that neither interests them intrinsically (what is taught) or extrinsically (qualification). With such bureaucracy and accountability it would be reasonable to assume that the UK can present to the world an exemplary model of education. Instead, there is a crisis in science and mathematics education with the UK failing in international comparisons. It is arguable that the crisis in science and mathematics education is not in spite of but due to the bureaucracy. It must be stated at the outset that the crisis is not due to any fault of the teachers, they have already been accounted by endless inspections. Despite many laypeople having a fascination for science (discussed below), science has an unpopularity the roots of which go back to the scientific revolution of the Seventeenth century. This unpopularity augments the crisis in science education not only in the sense of science as ‘cultural capital’ but also in any sense of science as an integral part of a liberal education. This unpopularity extends to some science educationalists downplaying science as a discipline. This is explored in the next few sections.

The Unpopularity of Science The unpopularity of science extends beyond the classroom. In the minds of many people, science has contributed to so many ills of the world that it has a dark side. Indeed, pollution, weapons of mass destruction and the domination of the third world would not have been possible without science; but is science the cause of these ills? The unpopularity of science may not have a rational basis. Arguably, these ills have a social and cultural origin that is independent of the logic of science. Science is an explanation of the world in terms of laws and the logic of that

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explanation is independent of nationality, culture, class, gender or individual. Science is a socio-cultural and historical product, but it is also more than that. Through its theoretical objects, science actually explains the world and that explanation is independent of origin. It’s horrors notwithstanding, the thermonuclear bomb shows that science does indeed explain the world – that science is able to understand the laws of nature and in a way that makes the manufacture of the thermonuclear bomb possible – but it is the adherence of populations to the nation state and the conflict between nations that brings about the manufacture of thermonuclear bombs. To put this point another way, the thermonuclear bomb would not have been possible without science, but the decision to manufacture and use the bomb is a political one. Science may be utilised to make weapons of mass destruction and to dominate, but that is because it can explain - it says something about the world that is true – and by explaining the world science can serve the interests of those who invest in science. Advanced economies have utilised science, and may have even developed science, in their quest for political and military domination (the Manhattan Project and the space-race are cases in point), but science is not the cause of the hegemony of advanced economies. If anything, the hegemony of advanced economies illustrates the argument for the democratisation of science, that science should not be the preserve of oppressor nations in their quest to dominate. Indeed, science has the potential to serve the whole of humanity, but then that is a political issue, not an issue regarding the nature of science in terms of its logic. Both, however, can become educational issues. Although science makes such things as pollution possible, arguably it will be science that will enable pollution to be a thing of the past. Science may have contributed to the ills of the world and the ills of the world may not have been possible without science, but ultimately it will be science that will contribute to saving humankind. This presupposes science as the mastery of nature for the benefit of humankind as a whole and this presupposition is antithetical to the notion that science should retreat to a limitation of being, in a sense, ‘at one’ with nature [1]. This is a highly controversial position to take and the criticism of Gillott and Kumar on this have been highly charged. Perhaps the essential point to be made from all this is that in the sense of a ‘hard’ science such as physics and chemistry, the logic of science transcends the social and ideological. The Enlightenment valued science because the logic of science is such that it can benefit humankind; but that logic is independent of ideology. There can be no proletarian science, national socialist science, culturally diverse science, Eurocentric science or feminine science; that is, what is meant here as a ‘hard’ science. Despite any inequality in the history of the profession or any cultural inequality, science is not masculine or Eurocentric, just as rationality itself is not a male or western preserve. Science is independent of origin, but the opposition to science sees science as nothing over and above its origin. Science has served the oppressor, so the very nature of science is seen as oppressive. What is overlooked, however, is the distinction between science itself (its logic of explanation) and the multifarious ways in which it can be utilised (e.g. warfare). Once created, the theoretical objects of science exist independently of the community or the individuals that created them. Science is a practice without a knowing subject and the products of that practice, namely laws, theories and related data, have an autonomous existence (Chalmers, 1982). Prior to the First World War the opponents of scientific and social progress were in a minority. After the bloodshed, books that associated the decline of western civilisation with scientific values, such Oswald Spengler’s The Decline of the West, became very popular

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(Gillott and Kumar, 1995). The Twentieth Century has been a barbaric one, out of which has come a contemporary rejection of science based on a collapse of confidence in progress; that rejection of progress, however, becomes the umbrella for any anti-scientific trend: Contemporary anti-science trends are bad news. But they are just a small component of a broader rejection pf progress. Only a concerted drive to challenge the anti-progress consensus can offer a challenge to the suspicion with which science is greeted today. A failure to do this – and most if not all recent campaigns to popularise science fail to do this – will leave anti-science trends untouched. (Gillott and Kumar, 1995, p.199) Gillott and Kumar challenge the Green movement’s emphasis on the natural limits to human activity; the problem is that the authors do not make explicit how progress is possible. If it is to do with changing the way society is organised, or, to be a little more specific, changing the ownership of the means of production, then perhaps there is cause for pessimism. The anti-science trend has its origins well before the horrors of the First World War in the humanist tradition as a twin pole of the Enlightenment. The Scientific revolution of the 1600’s ushered a new era of an emerging bourgeoisie that freed itself of the constraints of the Feudal period, constructing in the process a world-view of a rational universe governed by impersonal laws contrary to the will of kings and church. This world-view saw science as a mastery over nature for the benefit of humanity; but opposing this world-view was a romantic humanism that regarded the new science as contrary to the nature of ‘man’ as portrayed in Rousseau’s Emile, or of nature itself. Since then there has been a romantic humanist tradition in education in which the twin-pole still exists. With Rousseau, Pestalozzi, Fröbel, and Piaget we have a child orientation, such as the potential of the child to blossom in the absence of intervention or knowledge as the construction of the epistemic subject, and there is a sense in which this tradition exists today in science education. There is a tendency to focus on the learner with a secondary concern for science as a discipline. For example, the tendency to record student thinking and the formation of student concepts divorced from the consideration of how the teaching of the concepts of science as a discipline can arouse the mind to life. It seems more to do with how the child thinks and sees the world than it has with the concepts of science and how the child can think and can see the world. Much of the literature in science education and indeed mathematics education downplay science and mathematics as disciplines. Not surprisingly, this tradition in both science and mathematics education today sits very comfortably with the advocates of postmodernism. This isn’t to say that the crisis in science education has its roots in the educational establishment or the community of educationalists, the crisis is a reflection of something far wider, but the promotion of scientific values can begin with education. This requires a critique of the relativism that seems to pervade the educationalist community.

The Postmodernist Attack on Science and Mathematics: The ‘New-Left’ Versus the Old On reading the literature you may be forgiven for forming a distinct view that many science and mathematics educationalists do not like science and mathematics as disciplines. For many constructivists, an ‘objectivist’ view of science, that science is a discipline

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comprising of body of tried and tested knowledge, or an ‘absolutist’ view of mathematics, that mathematics comprises of infallible truths, implies a transmission model of teaching, that children are ‘empty vessels’ to be filled with knowledge (Rowlands and Carson, 2001; Rowlands et al. 2001). This view can be traced to Piaget (1977/1972) and there doesn’t appear to have been any attempt to make explicit how an objectivist or absolutist view necessarily implies a transmission model of teaching. It is just assumed. Nor is there much criticism of the liberal education premise that the disciplines ought to be taught for their developmental value. Instead, there is a ‘critique’ of science and mathematics as if these disciplines are somehow ‘myths’. Apparently, for example, mathematics has lost its certainty and any seeming objectivity is conferred by consensus (e.g. Ernest, 1991, 1998; Jaworski, 1994. See Rowlands et al. 2001). Accordingly, the mathematics in mathematics education ought to be something to be constructed by the child or what is deemed relevant by the class, with the teacher acting as ‘guide on the side’. In exactly the same vein, science is no longer about the world but about the child making sense of experience, or a social form of activity to which children can play. Mathematics becomes open-ended investigations or strategies developed by children to subtract 21 from 42. Science becomes measurement, hypothesising why the school pond has turned green and researching under laboratory conditions the behaviour of insects (with lab coat, clipboard, hypothesis and a data base to record discussion and state conclusions). The questions raised and the issues explored are those deemed relevant by the child rather than those deemed relevant in learning and understanding content. Science and mathematics becomes debased; and education an expression of the logic of the child or the way the child sees the world. The debasement of science and mathematics is the denial of the objectivity of science and the certainty of mathematics. What’s left is the science and mathematics of the child. Constructivism and postmodernism seem hand in glove. There are many articles in mathematics education with a postmodernist slant. Indeed the Falmer series in mathematics education edited by Paul Ernest contains articles that explicitly express the postmodernist perspective sitting comfortably with the whole series. Number one in the series is Ernest’s (1991) social constructivism as a philosophy of mathematics which argues that there is no certainty in mathematics; rather, it is mutable, fallible and is always open to revision. The ‘philosophy’ is in fact a sociology, for even the logic of mathematics is something that cannot be separated from its social creation, with ‘if-thenism’ a matter of convention. At the end of the day it is all by consensus (for a critique see Rowlands et al. 2001). It has been said that post-modernism is the umbrella of constructivism (Phillips, 1998), but it seems that social constructivism has given post-modernism its tour-de-force: if mathematical truth is, in the final analysis, by convention, then so is everything else and perhaps even ‘more’ so. If mathematics is no longer a grand-narrative then lesser grandnarratives no longer exist. Take away mathematics as a certain body of truths then what is left, apart from a sociological reductionism. Postmodernism does raise relevant questions such as what does it mean to be convinced by a mathematical proof, and it does raise a healthy scepticism concerning issues of certainty, but all this seems to be a reforming zeal to liberate the child from both the oppression of the disciplines and the oppression of having to learn them. A sociology of postmodernism would be complex, but it would be fair to say that in education, postmodernism is very ‘new-left’. By contrast, many liberal educators are of the ‘old-left’. The physicist Alan Sokal (of ‘Sokal’s hoax’), regarded himself as a ‘stodgy old leftie’.

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In the UK there has been a working class tradition of learning the disciplines. In the UK the Workers Association and various miners’ libraries and colleges were institutions established so that workers could receive a liberal education in science and the humanities. That tradition is not unique to the UK but exists internationally. This is a tradition that has more to do with science as a discipline than science as a creative process of hunches and guesses. This is not denying the creative processes involved in the development of science or indeed the importance of promoting creativity in the classroom, the point is that hunches and guesses are vacuous if they are unrelated to content. To investigate scientifically some aspect of what is given in everyday or school life requires the acquisition of the relevant concepts necessary to not only to carry out the investigation but to raise the kind of questions that would allow for any scientific investigation in the first place. The old-left tradition sees learning content as intrinsic to everything else, which contrasts with the new-left tradition of regarding content as one aspect of science education. For example, in the book Challenging New Zealand Science Education, the Australian Michael Matthews (1995) acknowledges himself as part of the ‘old-left’ tradition, and criticises the constructivist based New Zealand science curriculum. In response, Beverly Bell’s (1997) review of the book tends to be adhominem without reference to the logic of the book’s argument. There is a war on, central to which is the importance of science and mathematics as disciplines. Of course, the attack on science extends beyond education and leading the attack seems to be cultural studies. Science has been attacked for not only being a tool of the oppressor but also being constitutive of the oppression itself. Newton’s Principia as a rape-manual and fluid-dynamics typically male has been well documented by Gross and Levitt (1998), and such extreme views continue. These views are extreme because they are explicit in denying the logic of science as having any significance. Without being so explicit, we have a postmodernist influence that seems to have permeated education. For example, the popularity of radical constructivism, social constructivism, feminism and ethnomathematics are illustrations of relativism that serves to downplay mathematics as a discipline (see Rowlands et al. 2001; Rowlands and Carson, 2001, and Rowlands and Carson, 2002b). Science is a socio-cultural-historical product but it is also much more than this, it is a system that actually explains the world. So-much-so that for Marx, stodgy old lefties and liberal educators who aren’t Marxists, science has the potential to serve the material and spiritual interests of humanity because it can explain the world by revealing the laws that nature obeys. Learning knowledge for its own sake but, paradoxically, because it has developmental value. To put the ‘new left’ and ‘old left’ in perspective, it might be instructive to put forward the Marxist scheme of things regarding science, bearing in mind that many liberal educators would not like to be included under the category of Marxist. According to this scheme, science explains and in so doing has the potential to raise the productivity of labour, reduce nature imposed necessity and increase the time for cultural pursuits; only the trouble is, society has never been organised in a way that an increase in the productivity of labour directly results in a decrease in the amount of work needed to have your needs met. Labour-saving devices increases the productivity of labour; they don’t tend to be used to reduce the working day. Marxism was meant to be post-Enlightenment project by which society becomes organised and in a way that science and technology liberates us from nature imposed necessity. Implicit in all this is the notion of truth, that science does indeed speak of the world and in a way that can be said to be true. This does need qualifying but the point

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here is that we have diametric opposites. We have the old-left and the notion of truth (for example, Trotsky is meant to have said ‘truth is revolutionary’), which is exemplified in Lenin (1976), and we have the new-left questioning the meaning of ‘truth’, as if it doesn’t exist or somehow doesn’t make sense. In many a conversation, ‘What is truth?’ is said in the abstract by ‘new-left’ educationalists and with a confidence that there can be no answer. The three angles of a plane triangle equals two right-angles is a good answer to give, although it is a limited one because it is not a formal definition of what truth is. Despite its unpopularity with the new-left, however, we all nevertheless have a sense of truth, and I contend it is that which is the case. From the very basic example of proclaiming whether the cat is on the mat and the very real situation of whether it is or not, to the hydrogen model and the way it can explain phenomena, there is a sense in which we do speak about the world that is true. This is a theory of truth: we say that the world is as we say it is and if the world isn’t then what we have said is false. The theory is compelling but becomes problematic when it comes to science. Consider the statement ‘the hydrogen atom has one electron’. For Searle (1995) this is a brute fact, but it isn’t. The statement is embedded in the model of the hydrogen atom. The world is made up of hydrogen atoms, but any statement about the hydrogen atom refers to the model of the atom, not the world. The world is explained by the model, but there is a difference between the world and a model of the world. Despite any political persuasion of its advocates, I would contend that liberal education implies a realist philosophy regarding the relation of science to the world, and this is explored next. Take away what science refers to, and you have a relativism that inevitably downplays the importance of science.

Realism and Relativism Acceleration is not only an abstract mathematical concept, it is also what objects do (Chalmers, 1982), and it took Galileo years to define acceleration as the rate of change of velocity with respect to time. During these years he did consider acceleration as the rate of change of velocity with respect to displacement, but through analysis and experiment Galileo eventually realised that this was the wrong definition (see Matthews, 2000). The point is the right definition was hard-won, but it was nature (what objects do) together with reason (the concept of what they do) that revealed the ‘truth’ in the sense of that which is the case. Revealing what is actually the case requires more than direct observation or establishing a consensus, it requires reason and the necessary construction of theoretical objects to explain the way nature reveals itself. That there is an inverse-square law of gravity (and not an inverse-cube law) has more to do with what is the case and the necessary theoretical objects to describe it than simply by measurement or the consensus that it is the case. Revealing what is actually the case implies a form of realism, that what is the case, nature, is as spoken by science. This is problematic, especially as this may be seen to imply a correspondence theory of truth and the theory is certainly problematic when it comes to the philosophy of science. The correspondence theory is discussed below, but to give an indication of how problematic the notion of truth, compare the inverse square law of gravity with the inverse square law of acceleration. According to (Russell, 2000), the former is superfluous since the latter can explain configuration and acceleration, which does raise the issue as to how ‘truth’ can be ascribed to the inverse square law of gravity. However, if your

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unit of analysis is acceleration and not force then ‘truth’ can apply to the inverse-square law of acceleration as compared with an inverse-cube law of acceleration. What is force? For Hertz (1956) the question is illegitimate since it implies force having some kind of ontological status – and it seemingly it would if science is regarded as saying something about the world. If force is a (fundamental) concept in science then, supposedly for the realist, force must exist; but this presupposes realism as synonymous with the correspondence theory of truth. However, an answer to the question would depend on what kind of answer is expected. For example, ‘Force is that which is defined by laws of motion’ would be a formal answer. ‘Force is the agent which changes the state of uniform motion’ would be a formal but qualitative answer. ‘Force is a concept that represents the interaction between two bodies’ is an epistemological answer that implies the distinction between Popper’s World One of physical objects and their processes with World Three of the objective content of thought, but this distinction need not imply a correspondence between the two. Two bodies interact and we represent that interaction with force, which is defined in a way that we can predict the future of a system of two bodies. We can model the world even though force is a concept. So in what sense does force exist, especially since Mach’s (1960) concept of mass could replace force as the central unit of analysis, with the same results? Indeed, all we observe are bodies interacting, so force is a somewhat of a ‘metaphysical’ concept and if metaphysical, does force speak about the objects of the world? The irony here is that a realist, or even stronger, a materialist, would hold the concept of force as a valid concept for science, whereas Mach was an idealist in the sense that science was an economy of describing what is given in sense perception to which metaphysical notions have no place. The difficulty with realism is that in the history of science there have been incommensurable paradigms that can account for the same data, compounded by the fact that nature doesn’t make explicit which paradigm is correct. It’s more to do with the logic of the explanation than it does nature laid bare. The difficulty with relativism is that it’s all down to the behaviour of scientists, the practice of science or the relation of science to society, as opposed to the logic of science. For realism, truth plays a part in establishing consensus whereas for the relativist, consensus establishes what is to count as truth. – according to which we accept that the US landed people on the moon and reject the conspiracy argument that it was all a fraud because of the prestige of the scientific community. Evidence for and against supposedly cannot be the final arbiter because it can always be countered. We supposedly accept the orthodox version because of the prestige of the community. But do we? Again, take a look at the history of science. If a sociologist were to go back in time to observe Galileo, she might regard his behaviour against such overwhelming opposition as odd or eccentric to say the least. With history we can with hindsight proclaim Galileo right, but right about what? The only possible answer is his explanation of the world despite the overwhelming consensus at the time. His explanation didn’t become right due to the eventual change in consensus, just as the world didn’t become round due to the change in consensus that the earth was flat. The point is the world and the way science explains it are influential in our choice of theory and we know this because science changes the world – we can fly, go to the moon, destroy the world and cure disease - which would not have been possible if there was no understanding of the laws of nature. Can a philosophy of science be prescriptive or should it only be descriptive? Can two scientists, each belonging to different incommensurate paradigms, go to a philosopher with the possibility of having a reasoned judgement or advice on the logic of each paradigm, or is

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judgement only restricted to the credentials of the scientists and the communities they represent? For the relativist, any ‘truth’ of what is proclaimed by the protagonists doesn’t come into it; that is, if the notion of truth involves the very objects of discourse which arise in dispute. The very content of science is dismissed; for the relativist what counts is authority, socio-cultural context, vested interest, even equity issues, but not the very thing that is under dispute. What confounds the issue is that relativists have been known to U-turn, starting with a strong claim or ‘interesting position’, such as the truth content of what is being asserted is irrelevant, and then resorting to a weaker claim or ‘boring position’ such as the truth content is relevant (Phillips, 1998). This is what Phillips (1998) refers to as ‘shillyshallying’ and is well documented (e.g. Phillips, 1998; Rowlands et al. 2001).

The Relation of Science to the World I have seen bright young education academics who, being into activity theory or situated cognition, would gleefully announce the question ‘truth, what is truth?’ whenever possible in scholarly exchange. In a mathematics education conference I have experienced a room full of laughter when it was suggested that nature obeys the laws of science. I have seen an audience murmur in concern and censure when, referring to Vygotsky, someone uttered ‘he is a Platonist’. This really is the ‘post-modern’ age when science and mathematics educationalists deny science and mathematics as ‘grand narratives’ and view them as merely socio-cultural products and nothing more. The fact that science does explain the world and reveals how the world is has something to do with truth. It has to do with how things are rather than someone’s opinion of what she or he thinks they are. This is one of the reasons why science should be taught. Having said that, it is understandable when, viewing incommensurate domains with their different ways in explaining the same phenomena in the history of science, some philosophers express an instrumentalist view with regard to the theoretical objects of science. Consider Hanson’s (1958) ‘picture’ of Kepler and Tycho Brahe watching a sun-rise with Tycho Brahe ‘seeing’ the sun rising and Kepler ‘seeing’ the horizon roll below a stationary sun: both are watching the same event but there is also a sense in which they don’t see the same thing. Can we say that one is the ‘true’ way of perceiving the event while the other isn’t? One can think of many other examples, such as the earth orbiting the Sun due to their masses creating a force of gravity, or the earth orbiting the Sun due to their masses creating a curvature in space. Is the trajectory of a projectile due to the laws of motion, or is it due to what may be described as the synthesis between Fermat’s principle of least time and Maupertuis’ principle of least action? If the law of gravity or the laws of motion are laws of nature, then how is it possible to describe events with laws or principles that are alternative to what we might ascribe as laws of nature? The answer really depends on what we mean by a law of nature and the domain in which the law holds. It can be argued that Newton’s temporal expression of the first law of motion is a law of nature (a transfactual tendency of macroscopic objects at low velocities, Chalmers, 1982), while the conservation laws are not. The conservation laws are derived from the laws of motion and thus become ‘manners of speaking’. We say that ‘momentum is conserved’ because of the third law of motion, and we say that energy is ‘converted’ or ‘transferred’ because the work than a body can do is equal to the work done on the body (Carson and

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Rowlands, 2005), although it is arguable that nothing really is converted or transferred. A law of nature is also domain specific. Newtonian mechanics has not been falsified by relativity and quantum mechanics, the latter have merely established the domain in which the former can be said to hold (this is true of the angle property of the triangle; non-Euclidian geometry hasn’t falsified it, it merely sets the domain for its truth, namely plane triangles). For macroscopic objects at small speeds predicated on the inertial framework the laws of motion hold and the proof is in the practice. That people have walked on the moon demonstrates the truth of those laws, albeit the qualification these truths require, such as the domain in which these laws are said to hold. Appearances are ‘deceptive’. The sun does appear to move relative to a ‘fixed’ horizon. What science does is explain, not in terms of appearances, but in terms of laws that can account for the appearances. But the simplicity of accounting for so much in terms of so little can itself be deceptive, because it seems to suggest that simplicity is the essential criteria that would make us choose one theory over another. Mach’s Science of Mechanics was a struggle to make mechanics ‘simpler’. But simpler in form does not necessarily make it ‘simpler to understand’. Nor is simplicity an essential characteristic of science, although Augustus Comte, the father of positivism, may be right in thinking that science develops towards simplicity in terms of formalised systems. The fact is, Copernicus’ heliocentric system took hundreds of years of scientific labour for the system to mach in any way the geocentric model held by astronomers (Chalmers, 1982). The former wasn’t established because it was ‘simpler’ in that it contained fewer assumptions. In fact, initially the former still had to use epicycles to match data; thinking that the sun was at the centre of circular motion required it. Science and technology have moved on and to the extent that the geocentric model would have been contrary to that development, but people didn’t change paradigms because one is simpler than the other. The sociologist is quite correct when pointing out that many factors are always involved such as vested interests when it comes to paradigms and their attacks/defences. The sociologist is wrong if he or she thinks that it is only these other factors which determines behaviour, one very big contributing factor is the truth in what the paradigm of the scientist proclaims. The situation is far more complex than the dichotomy between the coherence of a theory and any correspondence a theory might have with reality (e.g. as presented by, for example, Staver, 1998. See Rowlands and Carson, 2001). On the one hand simplicity and parsimony are not essential criteria for acceptance, the history of the heliocentric model a case in point. On the other hand it is difficult to maintain a correspondence theory of truth when it comes to scientific explanation. What state of affairs does the temporal expression of the first law of motion, the forceless case, refer to? We can show the law to be truth though, either by approximating to the forceless case with an air-track (experiment) or building on the forceless case by considering friction (modelling). A theory isn’t true because it is beautiful. Indeed, a theory can be beautiful in the way it is true, but there is a danger in allowing ascetic judgement to be the arbiter in truth [2]. Experimentation must play a leading role. As stated by Marx, science is the ‘ascent from the abstract to the concrete’ (see Matthews, 1980) in which experiment has an inseparable role, not only as judge of theory but in the very consideration of theory itself. There is an inseparable relation between the concept of light-ray, for example, and the construction of a light-box to produce a parallel beam of light. This relation between theory and experiment has educational implications.

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Modelling the world is a complicated affair; we begin with how the world can be modelled by creating an ideal representation. Many learners, however, often complain that the ideal representation given is somehow unrelated to real, practical examples. Of course, an attempt should always be made to relate theory to practical examples, particular observations or concrete exemplars, but it may be difficult and in many cases impossible to begin theory from practical examples. There is a sense that theory comes first, but the intrinsic beauty of the theory may not be enough to instil an intrinsic motivation to learn science. It may require making explicit the way science can model the world, to discuss possible-worlds such as the thought experiment and to see how possible worlds can explain the real one. An interest in science may be promoted in making conscious the way science explains the world, which is more than learning the content of science. The philosophy of science can also have a role to play in developing an understanding of the very content of science as well as an understanding of science. This is explored in Section 2 but before then the possibility of engaging all learners in the abstract nature of science is explored. It is argued that, despite its unpopularity, science holds a fascination for many people and much of that fascination has to do with their own ability to reason and to think in the abstract, despite any categorisation that suggests they might be concrete thinkers.

The Popularity of Science Science may not be popular, especially in post-modern circles, yet many books that attempt to popularise science indeed are. Sobel’s Longitude and Galileo’s Daughter are best sellers and the plethora of books on topics in modern physics such as quantum mechanics, relativity, thermodynamics and cosmology written by physicists for laypeople seems to suggest that despite negative experiences of schooldays many people find fascination in the way science speaks of the world. It is an irony that some people reel at the memory of their experience of learning school mathematics, yet love Simon Singh’s Fermat’s Last Theorem (my evidence is anecdotal). It would appear that people get on better with what science and mathematics are about than learning the content of science and mathematics. Of course, it could be argued that the former has to do with the mature mind and the latter the experience of schooling, but there is evidence to suggest (e.g. the work of Peter Davson-Galle) that children are not only capable but also inclined to engage with what science and mathematics are about. Indeed, in my past experience as a mathematics teacher, children have tended to regard philosophic and historic discussion of the theoretical objects concerned as an enjoyment and a distraction to the ‘real’ work of doing pencil-and-paper exercises. I found their attempts at encouraging further discussion a positive thing. If used wisely, the nature of science can be employed to generate the incentive to learn the content as well as to understand it. I have met laypeople who have explained to me (usually over a drink) the twin-paradox, the doors opening by a beam of light in a train carriage, the diffraction/interference paradox of the single electron or the nature of black-holes in their fascination of the special theory of relativity, quantum mechanics or the general theory of relativity. Indeed, I have met a treecutter with no formal qualifications in education yet the depth of his qualitative understanding of science and the philosophical issues surrounding them incredible (we are now good friends). A disenchanted student at school, he became self-taught by reading popular books

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on modern science purely out of fascination. For him and for many others like him this fascination is of the world coupled with their ability to understand the way science speaks of that world. Quite often that understanding is expressed in the form of the thought-experiment. Thought-experiments seem something that many people are quite prepared and willing to engage with, voluntarily so, which does seem rather contrary to the notion that your ‘average’ layperson would rather think in the concrete! This suggests an educational corollary: why not utilise thought-experiments to enhance an understudying of NOS, and at the same time develop an understanding of the content of science? In section 2 an attempt is made to situate the thought-experiment as central to NOS and the educational implication in learning the content of science. Meanwhile we shall examine the attempt to popularise science amongst schoolchildren.

Promoting a Positive Image of Science (and Mathematics) There have been several attempts to promote a positive image of science and mathematics amongst schoolchildren (and indeed the general public) with the primary aim to generate an interest in science and the motivation to learn (e.g. Monash University). The problem is that positive images of science, scientists, mathematics and mathematicians will not necessarily captivate the imagination and create the incentive to study science and mathematics. Making the subject ‘sexy’, such as the UK’s television presenter Carole Vorderman’s introduction to modelling in mechanics with reference to the rides at Alton Towers (Mechanics in Action video, produced in 1987), is limited to showing that interesting, intelligent and sexy role models know and like mechanics. It’s limited because their interest in the subject may not be enough to convince why anyone else should take an interest, chairo-plane rides and big-dippers notwithstanding. I fancy Carole and I love gut-churning rides, therefore I love mechanics! It would depend more on the way it was presented than who presented it and Madison style presentations and celebrations of science and mathematics can be counterproductive. Take, for example, the UK’s ‘Math Year 2000’. Expensive high quality paper pamphlets accompanied the events of ‘Math Year 2000’, proclaiming we need maths to space out the candles on a cake, or to help us draw. On their website there were hundreds of celebrities stating that they needed mathematics in what they do as celebrities. The top house/techno/trance/break-beat DY Pete Tong needing maths to mix or famous football (‘soccer’) players and athletes needing maths to play or compete. All these examples are in fact reasons why you don’t need to learn maths, all you need is aesthetic judgement, spatial awareness and the ability to count four-to-the-four beats. This is not to undermine such activities or aesthetic tastes, the point is these are examples as to why you don’t need to learn mathematics – any ‘mathematics’ is already inherent in the activity. Apparently, many children have negative images of scientists (Matthews and Davies, 1999) and mathematicians (Picker and Berry, 2000) in the sense of scientists and mathematicians as ugly beasts, social outcasts, ‘not-normal’ and ‘uncool’. The methodology involves asking pupils to draw their impressions of what a scientist or mathematician looks like and to write comments next to each drawing. One problem with the methodology is that scientists and mathematicians have been singled out so as to ascertain images of mathematicians and scientists. In so doing the very exercise of ascertaining these negative

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images becomes ‘self-fulfilling’ – ask someone to draw a scientist and chances are you are going to get a ‘mad’ one, independently of whether, in fact, any of them are, indeed, ‘mad’. The recommendation is an exposure to scientists and mathematicians and to their multifarious activities. The hope is to spark interest amongst children, but it also justifies the negative image in the first place. Included in these negative images was the ‘Einstein’ appearance and no matter how eccentric it might appear, it could be seen a form of flattery. To be fair though, many of the images weren’t flattering. Picker and Berry suggest that pupil exposure to mathematicians and their activities might promote a positive image and hence enhance the motivation to learn, reinforced by the knowledge that many mathematicians work in many different professions. There are philosophical and educational problems here as well as the methodological one. On the one hand many mathematicians and indeed scientists may actually be ugly beasts and uncool. This is not to say that they are, we are speaking philosophically here and have to entertain all possibilities. On the other hand, if it could be shown that scientists and mathematicians are not as popularly imagined, that in fact they are human just like the rest of us (and perhaps even prettier, more hansom and sexier to boot), then why would that encourage the motivation to learn science and mathematics? An assurance that mathematicians are normal people who lead normal lives and therefore it’s OK to learn mathematics is not a reason for why you should learn mathematics in the first place. There is no short cut to infuse a passion to learn science and mathematics. That passion has to come from within the teacher or textbook and the way the subject can be engaged by the learner. It also has to do with the content. There is a paradox in mathematics education: The virtues of mathematics can be easily extolled in terms of its applicability and utility (mathematics as the ‘servant of science’), but the virtues of mathematics are not so easily extolled in terms of its intrinsic beauty (mathematics as the ‘queen of science’). Applied mathematics would not be possible without pure mathematics, but pure doesn’t require applied as a justification for learning it for its own sake. The appreciation of the intrinsic beauty of pure mathematics comes with learning and understanding it in some way. It is an acquired taste, but you try telling kids that! It’s the same with mechanics and science in general. Like mathematics, the theoretical objects of mechanics are exact but this seems to go against a popular conception that the theoretical objects of mechanics are empirical and therefore not exact. The theoretical objects of science have to be acquired before the beauty of science can be revealed in the way it can explain the world. This doesn’t mean that a teacher should not begin with real objects. Real objects come first; the point is that the theoretical objects of science have to be learnt. The next section, the nature of science, shows how an historical narrative treatment can become a very useful device in teaching these theoretical objects, because such a treatment contextualises these objects by showing the intellectual struggles involved in formulating them and the difficulties that may be experienced in understanding them. An historical narrative can also capture the imagination, take the learner to a different time and place, and pose the epistemological problem that was considered at the time. Such considerations can enrich the very concept to be learnt and consequently enrich the understanding of that concept. Once the pupil becomes engaged, with the realisation that she is able to become engaged, you have the basis for the pupil to develop an intrinsic motivation to learn science – a form of learning that needs no extrinsic motivation.

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2. NOS AND SCIENCE EDUCATION Why NOS is Necessary in Science Education One only has to read the highly acclaimed, informative and very readable What is this thing called Science? by Alan Chalmers (1978, 1982, 1999) to realise the diverse theories of science in the history and philosophy of science. The literature shows that NOS will always be debated and as to be expected there are revisions or rather refinements in each edition of Chalmers’ book. For example, in the first two editions he argues very eloquently that in science all observations are theory laden, yet his third edition gives the counterexample of Faraday’s electric motor. Given the diversity of theory in the history and philosophy of science, how can NOS be introduced in the classroom? There is no definitive answer and not surprising given the unlikelihood of a definitive answer to NOS. That’s philosophy, needless to say that NOS in the school classroom does not necessitate introducing the philosophy of science as if it were an undergraduate course in philosophy. Nor is science education a course in philosophy. Science education ought to include the philosophy and history of the subject because of the science’s abstract (‘ideal’) nature. The learner having to accommodate this abstract nature of science may be eased by making this nature explicit. Consider for example Newton’s temporal expression of the first law of motion: A body will continue in a state of rest or uniform motion until an external force acts.

Easy enough to understand as a statement, but what is it actually referring to? It refers to the forceless case to which no one on earth has ever seen. A pupil may be able to state the law, but will the student be able to apply it in explaining the physical world? What does it mean to ‘apply’ the law? There appears to be a regard of the law as if it is somehow either irrelevant or circular (e.g. MA 1965, Arons, 1990. See Carson and Rowlands, 2005). The law doesn’t appear explicit in the quantitative treatment of mechanics and once quoted seems forgotten. The law is fundamental, however, when it comes to modelling the world. Knowing the law is one thing, explaining the world with the law is quite another, somuch-so that many learners world-wide, independent of culture, age, gender, ability and expertise, have ‘misconceptions’ of force and motion, and these misconceptions are of the first law. According to the law, what would be required to move an object on a frictionless surface is to give it a quick push. There is no such thing as a frictionless surface, but if one existed a quick push would be needed. We can use an air-track to show that we can approximate to this ideal of a frictionless surface; and for real surfaces we can apply the concept of friction and Coulomb’s law of friction built onto the forceless case. The point is that mechanics is an ‘ideal’; a structure of theoretical objects to which the first law of motion is an axiom. The difficulty that learners have with mechanics is that it is axiomatic; there are no state of affairs that the laws of motion depict, although a ticker-tape experiment can demonstrate F = ma (which assumes the understanding of the first law. See Carson and Rowlands, 2005). The laws are not empirical, they are ideal, and the difficulty for learners is not so much the abstract nature of the subject but how the ideal models the world.

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The irony with mechanics is that it explains the physical world but from the ideal. The question is, how can students become engaged with the ideal in a meaningful way that can model the world? The answer is the thought-experiment. Ask a class of fifteen year olds what forces act on a thrown ball and you can almost guarantee that some will give a force pushing up. Say to them “this is not the right answer, but it is a very good answer because this was the answer given by Aristotle and accepted by ‘scientists’ for two thousand years until challenged relatively recently by Galileo” and the chances are you will have a captive audience. If in the discussion Galileo’s thoughtexperiment is introduced and a series of concept-questions asked, such as “gravity is switched off, describe the motion of a thrown ball”, “what forces are acting on the ball?” and “what forces act on the ball when gravity is switched back on?”, then there is the possibility of engaging even the most concrete thinkers in a rule-governed possible world that can explain the real one. That rule-governed possible world is what Hestenes (1992) refers to as the ‘Newtonian world’ and the whole point is that the history and philosophy of the subject can provide mediation into that world. History and philosophy can make the abstract nature of mechanics explicit for the learner: the learner not only learns content and develops an understanding of its nature but also becomes aware of her own understanding of the subject. This can be considered as conceptual change and this is discussed in section 3. The point is, the history and philosophy of science can encourage meta-cognition - thinking about concepts and thinking about thinking - necessary for the understanding of the content of science. The offshoot from this is that the learner can have a better understanding of NOS necessary for constructing a view of science and its relation to society. Introducing NOS through the history of the subject can serve as illustrations of the abstract nature of science and can serve as a more human introduction to an otherwise purely esoteric, formalised and symbolic one. It can serve to present a problem-space and a metadiscourse as to the nature of that space and can encourage the meta-thinking necessary to engage learning the subject. Introducing NOS in the classroom can aid the understanding of the content of science. NOS can encourage a ‘conceptual understanding’ of the concepts of science; that is, a qualitative understanding of the way the concepts of science can explain the world as well as the ability to apply the appropriate concepts to derive a quantitative result. Perhaps the thought-experiment is the most important ingredient in NOS precisely because it is a requirement in understanding the abstract nature of scientific law. All too often learners are expected to work with ideal situations presented in science such as frictionless surfaces or ideal gases without any consideration as to the terms of the discourse. To model the world using abstract and ideal laws requires thinking in the abstract and is best achieved by inviting the learner to think of the law as a thought-experiment: it makes explicit that you are working with the ideal and within a rule-governed possible world. Apparently there are some people who can only think in the abstract when presented with something concrete. Apparently there are peoples whose thinking is entirely context-bound such that any thought beyond the immediate considered almost as having no virtue. But nearly all people are capable of thinking in imaginative ways that are rule-governed. Crossculturally and almost independent of age we all use imagination in our interactions with the world and how we see the world. We nearly all use reason to guide our behaviour towards desired outcome in one form or another. Science requires and can develop imagination structured by reason, and mechanics in particular can aid that intellectual development.

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A Justification of NOS so Far Presented There is no such thing as an inertial framework, or frictionless surface or any position in space where there is no gravity. These are ‘metaphysical’ entities, but they are not chimeras. They are employed to tell us how the world would behave under the conditions presented and as such, they tell us the logic of the world. That logic is representation and mathematics. It doesn’t say that God is a great mathematician and everything that happens physically has been calculated, what it does say is the physical world is rational in the sense that it can be explained by the concepts of a rational mind. Actually it says much more. Rationality itself cannot explain the world and send people to the moon. It takes the changing of nature and to explain nature as a result of that change. How can we demonstrate the forceless case? An answer is with an air-track. By using an air pump, power source, tubing and a particularly designed perforated tube, that is, by changing nature, we can come up with a frictionless surface (well, almost, but enough to fulfil a demonstration). Science begins with theoretical objects, not real ones, but science comprising of theoretical objects seem ‘metaphysical’, which can be problematic for many people. Mach (1960) abhorred the notion that there could be metaphysical entities in science. For example, he argued that there is no such thing as an inertial framework and that the motions of objects are predicated on the fixed stars. Many science textbooks take this for granted without explicit reference to Mach. The stars aren’t fixed and a century before, Kant argued that the laws must be predicated at the centre of the universe (Friedman, 1992). Mach might agree, but he was making a point: science is about what is given in sense perception and not about ‘metaphysical’ entities such as inertial frameworks. The irony is that a realist might agree, the study of nature is about what is real and anything metaphysical is not physical and hence not real. But Mach was an idealist; science is not about what is ‘real’ but an economy of thought on what is given in sense perception. The problem with Mach’s empiricism is that it underrates the role of ‘metaphysical’ entities in understanding the physical world and worse, it excludes the possibility for the learner to engage with the abstract necessary in understanding NOS and science itself. Frictionless surfaces may be a limiting case of real ones; they can also present possible world scenarios for the learner to understand how science explains the real world. The Newtonian concept of force and the temporal expression of the first law of motion may be ‘metaphysical’, there is also a sense in which they are real: the former an interaction variable, the latter a transfactual tendency. Mach didn’t banish force altogether, he replaced force as the central unit of analysis with mass and relegated force as that which is the ‘ma’ product of a moving body, as if they were somehow synonymous. It is compelling! Two bodies collide, whack! And it is their respective masses that change each other’s motion. Force is a concept that represents the interaction and may seem superfluous compared with what actually changes their motions, namely their masses, but it is also a ‘shorthand’ way to represent the influence of the other body on the body modelled. The influence of the earth on an object can be modelled as gravity acting at the centre of the object, even though gravity acts on the whole body. When arrows representing forces are drawn on a point or extended body, we can ignore drawing other bodies because their influence is represented by the arrows. One of the difficulties in learning mechanics must be the understanding of what the arrows drawn on a point or extended body represent. It is one thing understanding what they do in terms of their effect on the point or body of interest, but it is quite another to know

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where they came from. Text-book illustrations don’t help when an arrow is drawn coming out of the front of a car; or a normal reaction of the ground drawn at the top of the car, or between the wheels, or both. How conscious are students of each arrow representing the interaction of an object?

The Role of Theory in Understanding NOS In science, all (or rather, nearly all) observations are theory-laden. At the Fifth History and Philosophy of Science Teaching Group conference in conjunction with the History of Science, Alan Chalmers explained how he, as an experimental physicist, had to spend most of the time eliminating unwanted variables, which took much theoretical consideration. In contemporary physics, much observation is of a dial of some sort, but the calibration of the dial is one of the very fruits of theory that can be expressed in mathematical form, and to actually make one requires greater theoretical considerations. The angle a bob at the end of a string makes to the vertical can be calibrated to give the acceleration of a car. The calibration cannot be formed by mapping each angle to each magnitude of acceleration, unless there was another device that gave the acceleration, but then you would be calibrating one device from another. Calibrating acceleration using a bob is the product of the application of the laws of motion. The behaviour of one object can tell us the behaviour of a connected object if we understand the nature of that connection. In the example of the bob, the tension in the string and the fact that the bob has the same acceleration as the car can give us the angle of inclination of the string and the opportunity for calibration. It is theory rather than measurement that gives the calibration and theory enables us to be creative and come up with scenarios that can either demonstrate in some way that the theory is true or can be utilised, such as calibration. These scenarios are usually experiments. This is by no means to undermine observation and experiment, but they are governed by theory. The difficulty facing the learner is seeing the real world reflected in the Newtonian world, and Newtonian mechanics is not ‘commonsensical’. Aristotelian mechanics was ‘commonsensical’. If you want a cart in uniform motion then you have to push or pull it and if you don’t believe it can be shown! Newtonian mechanics is contra-commonsensical because it explains with reference to idealised possible worlds that not only don’t exist but couldn’t. ‘Frictionless surfaces indeed, whatever next?’ But approximating to a frictionless surface or modelling with friction not only explains the motion of an object in terms of nature obeying laws but will predict the future of the system (a gust of wind may spoil the prediction unless by system we mean the possible-world’s correlation with the real one). The degree of accuracy depends on the correlation. It is not so much the laws of motion as approximations. The laws are ideal; it is more how much nature approximates to the laws. The world is too complicated to be known directly, so we simplify it. It would seem that, for many, this simplification is a stumbling-block and probably because it seems so unreal and hence not so ‘tangible’. People can relate to ‘real’ examples, until it comes to realising the modelling involved and the assumptions that have to be made in order to simplify the model. The irony is that the simplification of the model, the idealisation, seems to be the difficult part. Here the history of science plays a part because it contextualises for the learner the difficulties in forming these idealisations.

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The essential point is that, yes, education ought to be about learner cognition, but cognition as to the abstract and ideal nature of science. That nature of science comes first in the sense that all other considerations follow; top-down from which the bottom moves up. Unfortunately, science education seems to be losing its centre: the learning of science as a discipline.

3. WHY “CHILDREN’S SCIENCE” AND CONCEPTUAL CHANGE, THE LARGEST DOMAIN IN SCIENCE EDUCATION RESEARCH, HAS FAILED TO PROMOTE SCIENTIFIC UNDERSTANDING As stated above, it has been assumed that an objectivist view of science or an ‘absolutist’ view of mathematics necessarily means the transmission model of teaching science and mathematics. There exists a false dichotomy between the ‘bottom up’ and ‘top down’ in terms of teaching and learning. Indeed, for some constructivists the emphasis ought to be on the learning and not on the teaching. The former a creation by the child, the latter an imposition on the child. The dichotomy is false because an objectivist or absolutist stance does not necessarily imply a transmission model of learning. For example, Plato argued for the ontological existence of the Forms, which is a very objectivist and absolutist position to take, but he also put forward the dialectic as mediation in understanding these Forms. There is no contradiction between an objectivist position on NOS and the consideration of learner cognition in understanding science. The problems begin when there is an emphasis on the latter, on how the learner (mentally) constructs, at the expense of the concepts embodied in the discipline.

Top Down Versus Bottom Up Broadly speaking, the science education literature is in two almost opposing camps. One camp is child orientated and the other content orientated. ‘Almost’ because there are elements of commonality between both, such as the consideration of learner understanding of scientific concepts. What separates both is their orientation to the learner. With the former the focus is the learner to the extent that science becomes, in a sense, a secondary consideration. It becomes making sense of experience (a radical constructivist position), or something that should relate to the everyday experience of the learner (a socioculturist position). That there is such a thing as a ‘children’s science’ (e.g. Gunstone, 1990), an alternative framework of preconceptions that have their counterpart in the concepts of science as a discipline, is an example of this orientation. The focus of the latter, however, is the content of the discipline which includes the consideration of how it is possible for learners to understand/grasp/comprehend in a qualitative way as well as quantitatively this content. The focus is more on the subject matter in conjunction with learning it than it is how learners ‘see’ the world or what concepts the learner should acquire in order to live in it. The majority of educators might consider themselves constructivists in the sense that learners are not passive recipients of knowledge but that some kind of construction of sense and meaning is involved in learning that knowledge. This is what von Glasersfeld (1995)

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rather sneeringly refers to as ‘trivial constructivism’ [3] and although it doesn’t deny science as an objective body of knowledge, it is not explicit about the relation between science and cognition. Although Hestenes (1992), a constructivist and a scientific realist at the same time, draws the link between Popper’s Three Worlds, there has been a tendency in the science education literature to overemphasise the cognition at the expense of science as an objective body of knowledge. For example, the conceptual change literature is perhaps the largest field of research in science education and has documented misconceptions of force and motion that are independent of age, ability, expertise or culture. One common assumption in the literature is that these misconceptions are constructed prior to instruction, that pre-instructed children carry with them into the classroom Aristotelian conceptions of force and motion and other concepts held by our predecessors. How this is possible and other related issues such as what happens in conceptual change when these ideas are challenged has yet to be resolved (Rowlands et al. 2007). The difficulty is that the literature has focussed on learner ideas divorced from the very thing that may have prompted those ideas in the first place – the consideration of scientific concepts within a scientific context. Ask a pre-instructed learner what forces act on a thrown ball and the response will be determined by how the learner has made sense of the question. The response that a force must be pushing the ball for the ball to go upwards may be spontaneous and not pre-formed (see Rowlands et al. 2007). The point is, rather than focus on how misconceptions are formed in the context of mediating scientific concepts, the focus is on how the learner ‘saw’ the world prior to the engagement. Again, the emphasis is on the learner rather than the (mediation of the) subject-matter and has more to do with how the learner ‘sees’ the world (making sense of experience) than the cognitive response to scientific concepts (and the enculturation into scientific thinking). An emphasis on the learner in the sense of cognition viewed independently of the subject-matter as lens loses the opportunity to promote a scientific understanding and a means by which nearly all children can develop abstract thought. If thought experiments are the key to NOS, then they have a key role in the understanding of cognition in learning science. The primary emphasis ought to be NOS rather than cognition, and this is the conceptual change literature’s shortcoming. However, this is not to imply a formalist treatment of science without the consideration of the cognition involved in the learning of science. Much of Warren’s (1979) Understanding Force is a catalogue of misconceptions written in text books with a critique on each one showing why they are misconceptions. It would seem as though misconceptions arise because they already exist, at least in text-books, and so are in a sense ‘passed on’ to the learner. It’s as if misconceptions have to do with instruction, not cognition. One of the classic misconceptions presented in the book is the bottle on the table as an example of action being equal and opposite to reaction as an illustration of the third law of motion [4]. Warren’s book must have had an effect apart from a number of references in the science education literature. Not so many contemporary mechanics as applied mathematics textbooks in the UK have the same misconceptions, which isn’t surprising given that these textbooks have to deal with presenting mechanics, not as a highly theoretical subject it once was, but the one that can model the world. Warren’s book was valuable not only because it showed what misconceptions there were and where they existed, it also explained why they were misconceptions relative the Newtonian mechanics. His formalist approach to the subject treated mechanics as a rational subject and criticised those who have created confusion by stepping into rotating frames of reference without being

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explicit that they have overstepped the domain of Newtonian mechanics. There seems to be an implication by Warren in his formalist approach that misconceptions arise because people (such as some text book writers) do not know what they are talking about. Indeed, he made disparaging references to psychology’s interest in the learner and mechanics because the psychologist is not an expert and therefore cannot know the nature of misconceptions. It seems that, for Warren, misconceptions arise because some teachers and textbooks do not understand the concept of force as defined by the laws of motion AND how it is applied, either in possible worlds or with the real one. The corollary is that if textbooks (and possibly teachers) no longer presented misconceptions then the misconceptions would no longer exist. If a misconception is a lack of understanding the Newtonian concept of force, then it falls to the teacher and textbook to present a complete and correct understanding as clearly as possible. Unfortunately, the situation is not so simple, as it doesn’t allow for idiosyncratic interpretations in the sense making process. Clarity of explication will not guarantee a comprehension of mechanics; there no guarantee that students will not form misconceptions given 100% correct and good instruction. Correct and good instruction, meaning getting it right about what you teach and teaching it effectively, is a necessary condition, but it is not sufficient. There is still a tendency to form misconceptions despite the instruction. For example, in a sort of ‘experiment’ (Berry and Rowlands, unpublished), 17 year old students of high ability who already demonstrated their understanding of force (due to previous science lessons) as the agent which causes changes in motion and who could derive F = Ma from force as proportional to the rate of change of momentum, were placed in a learning environment in which, through ‘brain storming' activities in creating posters, group discussion and teacher dialogue, they collectively constructed a qualitative understanding of force. Yet subsequently all agreed that the resultant force on a thrown ball going vertically up is up and zero what the ball is at instantaneous rest. This is a misconception that was spontaneous and not pre-formed, yet arose despite the instruction. Attention has to be paid to how learners can construct idiosyncratic interpretations despite the clarity and the accessibility of the instruction. The problem is, can a student’s understanding of the concept as presented in an example and explained by the teacher or textbook be transferred to a different example requiring the same explanation? The chances are, no. There has to be many different examples to show the Newtonian concept of force to be invariant to all these examples. These many examples not only include the many concerning basic force and motion, such as the incline plane and pulleys (the Attwood machine) but also projectiles, circular motion, periodic motion, etc. All these topics in mechanics are different examples of the laws of motion. Know how to identify and treat the interactions of various bodies on the one of interest, know how to represent these interactions as force and know how to apply the laws of motion then with some mathematics you can predict the future of the system (concerning the object modelled). Essentially what differs between each topic is how we represent mathematically the motion in terms of acceleration. Mechanics can appear fragmented and a lot to learn; but in fact, there is very little to learn. Mechanics consists of the three laws of motion (plus, of course, the kinematics), a few auxiliary ones (Hooke’s law, Coulomb’s law of friction, the law of restitution) and conservation laws which are derivative of the laws of motion. But in applied mathematics the subject is often treated as consisting of many different topics with not much similarity between them, taking the discerning eye to realise the underlying theme is the application of F = Ma. Indeed, with projectiles F = Ma was downplayed for the constant acceleration

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formulae, treating projectiles as a topic in kinematics rather than dynamics (see Rowlands, 2003). By alleviating the headache for detail and heading straight for the kernel of the issue, much of everything else can be discerned. There is evidence to show that mathematics undergraduates tend to assimilate the axioms of a vector space as if they were an extension of vector algebra (Tall, 1991). You can imagine this if the axioms were presented as merely something else you have to learn. If the axioms are made to become the discerning focus then much subsequently can be seen to be subsumed by the axioms. The same with the laws of motion, they are not some kind of appendage, but form the basis of how we can model and hence explain the world according to Newtonian mechanics. As stated by Wittgenstein: Mechanics determines one form of description of the world by saying that all propositions used in the description of the world must be obtained in a given way from a given set of propositions – the axioms of mechanics. It thus supplies the bricks for building the edifice of science, and it says. ‘Any building that you want to erect, whatever it may be, must somehow be constructed with these bricks, and with these alone’. (Wittgenstein, 1974. proposition 6.341, p.68).

Understand qualitatively the axioms in how they speak of the objects of the world and you have understood the nature of science: ‘‘Mechanics is an attempt to construct to a single plan all the true propositions that we need for the description of the world’’ (Wittgenstein, 1974, proposition 6.343).

NOTES 1. Gillott and Kumar (1995) have taken the very radical stance that ecological science, contrary to the Enlightenment view that science can benefit humanity in its mastery over nature, is committed to limiting science to nature’s mastery. They also argue that influential interpretations of quantum, chaos and complexity theory suggest conceding to nature (Gillott and Kumar, 1995). 2. There is a problem when what is taken to be science is a departure from what has been assumed to be the ‘scientific method’, an assumption that involves the role of observation and experiment. Science may begin with theoretical objects, the products of a thinking mind, but at the end of the day theory is inextricably bound with observation and experiment. Super-string theory has not been experimentally verified, and many scientists regard some unverified theories as science because of their inherent beauty and not because nature has shown itself to be explained in that way (Gillott and Kumar, 1996). It seems that speculation with metaphysics but without observation has become taken to be science itself, rather than some kind of intuition as to how science can proceed. 3. He adds to this notion to formulate the not so trivial ‘radical constructivism’ and in so doing everything is reduced to making sense of experience, including science. But science is not about making sense of experience, as indeed many concepts and laws are counter-intuitive and appear falsifiable by direct observation.

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4. It’s a misconception because the normal reaction and the weight are two forces acting on the bottle, the fact they are equal and opposite is a consequence of the first law of motion. The third law concerns the one force of interaction between the bottle and the earth.

REFERENCES Arons, A.: 1990, A Guide to Introductory Physics Teaching, Wiley and Sons, New York. Bell, B. (1997), Review of Challenging NZ Science Education. International Journal of Science Education, 19(3), 360-362. Berry, J. and Rowlands, S. (unpublished), Are misconceptions of force and motion spontaneous or formed prior to instruction? Bostock, L. and Chandler, S. (1975), Applied Mathematics, Thornes, Cheltenham. Carson, R. (1997), Science and the Ideals of Liberal Education, Science and Education, 6(3), 225-238. Carson, R. and Rowlands, S. (2005), ‘Mechanics as the logical point of entry for the enculturation into scientific thinking’, Science and Education, 14, 473-492. Chalmers, A. (1978), What is This Thing Called Science? First edition. The Open University Press, Milton Keynes. Chalmers, A. (1982), What is This Thing Called Science? Second edition. The Open University Press, Milton Keynes. Chalmers, A. (1999), What is This Thing Called Science? Third edition. The Open University Press, Milton Keynes. Ernest, P. (1991), The philosophy of mathematics education. London, The Falmer Press. Ernest, P. (1998). Social Constructivism as a philosophy of mathematics. New York, SUNY Press. Fox, C. (2004), The Philosophy Gap, in (D. Hayes, ed.) The RoutledgeFalmer guide to key debates in education. London and New York, Taylor and Frances Group. Friedman, M (1992), Kant and the Exact Sciences. London, Harvard University Press. Gillott, J. and Kumar, M. (1995), Science and the retreat from reason. London, Merlin. Gross, P. and Levitt, N, (1998), Higher Superstition: The Academic Left and its Quarrels with Science. Baltimore, John Hopkins. Gunstone, R. F. (1990). "Childrens' science": A decade of developments in constructivist views of science teaching and learning. Australian Science Teachers Journal, 36(4), 919. Hanson, N.R. (1958), Patterns of Discovery. Cambridge, Cambridge University Press. Hertz, H. (1956), The Principles of Mechanics (first published in 1894), Dover, New York. Hestenes, D. (1992), ‘Modeling Games in the Newtonian World’. American Journal of Physics. 60(8), 732-748. Jaworski, B. (1994), Investigating Mathematics Teaching: A Constructivist Enquiry, Falmer, London Jeans, J, H. (1907), Theoretical Mechanics, Ginn and Company, Boston. Lenin, V. I. (1973), What is to be done? Peking, Foreign Languages Press.

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Lenin, V. I. (1976), Philosophical Notebooks, Collected Works, volume 38. Moscow, Progress Publishers. MA. (1965), A Second Report on the Teaching of Mechanics in Schools, G. Bell and Sons Ltd., London. Mach, E. (1960), The Science of Mechanics, (first English edition, 1906). Open Court, La Salle. Matthews, M. R. (1980), The Marxist Theory of Schooling: A Study of Epistemology and Education, Harvester Press, Sussex. Matthews, M. R. (1994), Science Teaching: The Role of History and Philosophy of Science, Routledge, New York. Matthews, M. (1995), Challenging NZ Science Education. New Zealand, Dunmore Press. Matthews, M. R. (2000), Time for Science Education, Kluwer, New York. McComas, W. F., Clough, M. P. and Almazroa, H. (1998), ‘The role and character of the nature of science in science education. Science and Education, 7(6), 511-532. Matthews, B. and Davies, D. (1999), “’Changing children’s images of scientists: can teachers make a difference?’, School Science Review, 80(293), 79-85. Nuffield. (1994), Mechanics 1. Longman, Essex. Phillips, D. C. (1998), ‘Coming to Terms with Radical Social Constructivisms’, in (M. Matthews, ed.) Constructivism in Science Education, Kluwer, Dordrecht Piaget, J. (1977/1972, ‘Comments on Mathematics Education’ in (H. E. Gruber and J. J. Vonèche, eds), The Essential Piaget. New York, Basic Books. Picker, S. and Berry, J. (2000), ‘Investigating Pupils’ Images of Mathematicians’, Educational Studies in Mathematics, 43, 65-94. Rowlands, S., Graham, E. and Berry, J. (2001), ‘An Objectivist Critique of Relativism in Mathematics Education, Science and Education, 10(3), 215-241 Rowlands, S. and Carson, R. (2001). “The contradictions in the constructivist discourse” in Philosophy of Mathematics Education Journal. No. 14 (May 2001). http:// www. people.ex.ac.uk/PErnest/pome14/rowlands.pdf Rowlands, S. and Carson, R. (2002a), ‘The state’s attack on mathematics education research in Britain: A response’, Themes in Education, 3(1), 79-101. Rowlands, S. and Carson, R. (2002b), ‘Where would formal, academic mathematics stand in a curriculum informed by ethnomathematics? A critical review of ethnomathematics’. Educational Studies in Mathematics,50, 79-102. Russell, B. (2000), History of Western Philosophy, (first published, 1946), Routledge, London. Searle, J., R. (1995), The Construction of Social Reality. Penguin, London. Staver, J. R. (1998), ‘Constructivism: Sound Theory for Explicating the Practice of Science Education’, Journal of Research in Science Teaching 35(5),p.501-520 Smith Report (2004), Making mathematics count: The report of Professor Adrian Smith’s inquiry into Post-14 mathematics education. The Stationary Office, London. Tall, D. (1991), ‘The Psychology of Advanced Mathematical Thinking’, (in D. Tall, Ed) Advanced Mathematical Thinking, Kluwer, The Netherlands. Von Glasersfeld, E. (1995), Radical Constructivism: A Way of Knowing and Learning.Falmer, London Warren, J. (1979), Understanding Force, John Murray, London

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Wittgenstein, L. (1974), Tractatus Logico-Philosophicus, (first published, 1922), Routledge and Kegan. Paul, London.

In: Science Education Issues and Developments Editor: C. L. Petroselli, pp. 125-149

ISBN: 978-1-60021-950-4 © 2008 Nova Science Publishers, Inc.

Chapter 4

UNDERSTANDING STUDENT AFFECT IN LEARNING MATHEMATICS Marja-Liisa Malmivuori1 University of Helsinki, Finland

ABSTRACT Student affect has been one area of interest in mathematics education for decades. This applies in particular to rather large surveys of students in The United States since 1970’s. In general, education studies on affect have much focused on affective factors in the contexts related to mathematics achievements, learning of mathematics or solving mathematical problems. This is understandable since mathematics and mathematical problem solving carry many kinds of cognitive and sociocultural features that are not easily attached to the other school subjects. For example, the abstractness of mathematics and the differences in the symbol systems used in mathematical language set high demands on cognitive processes and also detach mathematics from the context and experience of everyday life. Furthermore, general views of mathematics as a difficult and demanding subject have caused it to be highly regarded and have been generally used to measure academic abilities. Mathematics tend to have a ritual value in societies that then cause powerful experiences with and important differences in mathematical performance. After showing passionate interest in human cognition and cognitive processes, education research paradigms have recently created new opportunities for and even laid emphasis on studies of student affect. Constructivism, together with applied sociocognitive, cultural and contextual views of learning and education, has enriched our knowledge of affect in mathematics education research, as well. This theoretical chapter first discusses some conceptual features of affective factors traditionally applied in education research and especially in mathematics education studies. This short overview will then be followed by consideration of some of the most significant and often used affective variables in mathematics education research. More recently presented views of affect with cognition in learning will be considered as an introduction to the here suggested theoretical framework for understanding student affect in learning mathematics. Especially, perspectives on the coexistence of affect and cognition, on self1

Department of Education, P.O. Box 9, 00014, University of Helsinki, Finland.

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INTRODUCTION Traditionally, the abstractness and the difficulty of mathematics have been suggested as reasons for the appearance of negative views, attitudes, or affect regarding mathematics. The effects of these reactions are most commonly considered negative in nature and related to differences in students´ achievement and participation in mathematics. Viewing mathematics as a difficult and demanding subject has caused it to be highly regarded, and ability in it has frequently been used to measure academic ability in general. Today, the increased complexity of everyday life has increased the significance of mathematics for all citizens, at least in welldeveloped countries. The increased significance of mathematics in society and the recent fundamental changes taking place in mathematics instruction (e.g., non-routine problems, open-ended problems) create new challenges to all students´ performances and mathematics learning contexts, and it is here where affect is playing a significant role. In mathematics education studies, affective factors have been traditionally considered as minor components in explaining the differences in mathematics achievements between males and females, between low- and high-ability students, or between students with a low or high socio-economic status or with dissimilar ethnic backgrounds (Aiken, 1970; Fennema and Hart, 1994; Frost, Hyde and Fennema, 1994; Ma and Kishor, 1997; Reyes, 1984). Today, affect instead of acting as a less significant personal mediator of educational outcomes, has become a focus of attention for its own sake. This need for change of perpective was emphasized especially by McLeod (1992) and is reflected in recent mathematics education studies of affective factors (e.g., Evans, 2000; Goldin, 2000; Malmivuori, 2001). But, increased interest can be perceived in affective components also in general educational psychology and in cognitive science (e.g., Boekaerts, 1995; Collis and Messick, 2001; Dalgleish and Power, 1999; Schutz and DeCuir, 2002; Snow, Corno and Jackson, 1996). The new emphasis on affect has also produced more unambiguous conceptualizations with affective variables, as well as efforts to link affective qualities more closely to cognitive, social, and contextual aspects of learning and performance. In this chapter, we continue the theoretical efforts to understand the role and functioning of affect in personal learning processes. We are focusing especially on mathematics learning processes in school learning situations.

THE AFFECTIVE DOMAIN OF PERSONALITY Lack of detailed specifications for the usually-applied affective terms or concepts, of explicit expressions, and of consistency or asymmetry in the definitions has hampered a systematic consideration of the affective domain. The reasons for this inconsistency can be found in at least three different areas (Malmivuori, 2001). One relates to the low traditional interest in a detailed and serious study of affective variables in the scientific field of education

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(e.g., Gable, 1986; Oatley and Nundy, 1996; Schutz and DeCuir, 2002). This leads to another important factor: the complicated and diversified nature of phenomena that are intertwined with human affect (e.g., Frijda, 2000; McLeod, 1992). The third reason can be traced to the variability in the theoretical starting-points or paradigms behind the studies of human affect, ranging from behavioristic to psychoanalytic views, and from biological to cognitive or sociocognitive views of human nature and behavior, together with the variation of the methods used in studying affect.2 The traditional definitions given for affect or the affective domain of personality often vary according to the extent they deal with biological and physiological elements of human actions or socio-psychological aspects related to socialization processes. Affective responses are viewed either as based mainly on the functioning of autonomic nervous system (physiological changes in the body) and on evolution or as learned personal arousals constructed and reflected in social interactions (Ciompi and Wimmer, 1996; Corradi Fiumara, 2001; Davidson, 1999; LeDoux, 1998; Johnson-Laird and Oatley, 2000). In addition to the term “affective domain,” such concepts or terms as “affective development of a personality,” “affective learning,” “affective educational objectives,” or “educational affective outcomes” have also been applied in educational scientific fields. They all are used to explain the perceived individual differences in students´ academic achievements, and education researchers often join them with the generic psychological term ‘affect’ (Malmivuori, 2001; Oatley and Nundy, 1996). Even though variation appears in the definitions, most education and mathematics education researchers are of the opinion that the affective domain refers to factors that “differ from pure cognition,” or that “cannot be classified as cognitive or psychomotoric objectives of learning,” or that “in some way or another are connected with feelings and emotions” (e.g., DeBellis and Goldin, 1997; Martin and Briggs, 1986; McLeod, 1989; Reyes, 1984). A very traditional way to define affective domain or affective educational objectives is merely to list certain significant affective terms or to develop specific affective taxonomies for educational objectives. The use of operationalized concepts or generalized expressions for affect has produced taxonomies operating on varying scopes of human affective aspects or behavior (Corradi Fiumara, 2001; Hart, 1989a; Malmivuori, 2001; Martin and Briggs, 1986; Ringness, 1975). For example, Eraut (1989) viewed the objectives in the affective domain to be linked to the norms and values of the educators, including socializing with general school norms and values, moral and social education, and feelings and sensitivities. Taxonomies of affective human characteristics are designed also to express hierarchical structures. Krathwohl, Bloom and Masia´s (1964) early taxonomy of affective educational objectives distinguishes between different stages involved in the internalization of an affective construct. In turn, taxonomies of affective mathematics learning objectives have often been consistent with Wilson´s (1971) division between attitudes, interests, motivation, anxiety, and selfconcept and, on the other hand, external, internal, and operational values. Today, affective taxonomies often differentiate between mathematical beliefs, attitudes, and emotions (DeBellis and Goldin, 1997; 2006; Kloosterman, 1996; Malmivuori, 2001; McLeod, 1992). One option in recent education research is to deal with affect as the equivalent of or manifestation of different kinds of emotions (cf., Oatley and Nundy, 1996; Pekrun, 1992; Schutz and DeCuir, 2002; Weiner, 1986). For example, Anderson´s (1981) perspective on 2 E.g., quantitative methods in measuring social attitudes vs. projective or associative techniques in dealing with transitory affective states in psychoanalytic approaches.

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affective characteristics includes “qualities which present people´s typical ways of feeling or expressing their emotions” (p. 3). These qualities are said to have three properties: intensity, direction, and target. Emotions are considered especially in cognitive approaches to affect (see, e.g., Lyons, 1999), as in G. Mandler´s (1989) attempts to describe emotions in learning or problem solving as varying in intensity, direction, and magnitude. An additional approach to powerful affect is represented by connected notions of the self or identity. The significant relationship between the self and affect is acknowledged in the classical psychological theories of self (e.g., by James, 1890/1963; Kelly, 1955; Rogers, 1983) and it is also essentially included in the descriptions of ego functioning in the psychoanalytic tradition (Corradi Fiumara, 2001; Sigel, 1986; Zimiles, 1986). These perspectives of the role of the self are reflected recently especially in social psychological literature of affective issues (e.g., Emmons and Kaiser, 1996; Halberstadt et al., 1996; Johnson-Laird and Oatley, 2000) and in notions of self-determination (Deci and Ryan, 2002) or of self-system structures (e.g., Borkowski et al., 1990; Connell, 1990; Markus and Wurf, 1987) and more thoroughly in humanistic psychological or phenomenological perspectives (McCombs, 2001; Polkinghorne, 2001). But, the self has gained attention for long also in education psychological studies. For example, Beatty (1976) connected affect in education simply with experiences of positive or negative feelings and the awareness of pleasantness or unpleasantness, which is caused by self-perceptions and closely related to motivation. And, for Sonnier, Fontecchio and Dow (1989) affective learning outcomes are associated with the question of whether or not students enjoy their learning. In this, three possibilities are distinguished: students´ positive feeling of themselves and their experience, neutral, or negative feelings about themselves and their experience. The interaction between affect and the self appears in education research often as results of the close relationship between students´ self-concept and their affective responses, such as anxiety (Bandura, 1993; Epstein, 1986; Higgins, 1987; Reyes, 1984; Weiner, 1986). More generally, we may relate these definitions to the so-called self-concept theories of learning or of motivation that deal with behavior or learning outcomes as the consequences of individuals´ self-concept, self-esteem, or self-confidence (e.g., Burns, 1979; Covington and Roberts, 1994; Deci and Ryan, 1991; Dweck, 2000; Harter, 1985; Wylie, 1974). These approaches have significant relations to our theoretical model of affect and of the interplay between students´ affect and cognition in their personal learning processes.

SIGNIFICANT AFFFECTIVE VARIABLES IN MATHEMATICS EDUCATION RESEARCH Even though attention has been given to such specific affective reactions as mathematics anxiety since the 1970s (Buxton, 1981; Fennema and Sherman, 1976; Hembree, 1990; Tobias, 1978), attitudes have dominated the research on affect in mathematics education (Aiken, 1970; McLeod, 1994). Traditional measurements of students´ general attitude toward mathematics and its learning, called attitude toward mathematics, represented the basic view for dealing with affective aspects in mathematics education (see Aiken, 1970; Leder, 1993; Ma and Kishor, 1997). Understanding of this attitude as students´ general emotionally toned disposition toward the school subject of mathematics (Haladyna, Shaughnessy, and Shaughnessy, 1983), or as a learned predisposition or tendency to respond positively or

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negatively to mathematics (Aiken, 1970), reflected the vagueness both in the theoretical backgrounds of these studies and in the scales designed to measure affect in learning mathematics (cf., Fennema, 1989; Hart, 1989a; McLeod, 1994).3 Currently, mathematics education researchers tend to agree upon the view of mathematics attitudes as rather stable or consistent affective responses of long duration, accompanied by slight affective arousals (cf., DeBellis and Goldin, 2006; Hart, 1989a; Leder, 1993; McLeod, 1992), and possibly to be developed through experiences of frequent and highly intense positive or negative affective responses in mathematics learning situations (Hembree, 1990; Marshall, 1989). The later developed multidimensional attitude scales produced the idea of different domain or object specific mathematics attitudes, hypothesized to act in a mathematics learning context in particular. Along with the more accurate conceptualizations and measurements being developed in the field, it became also more appropriate to speak of various mathematical attitudes, rather than a unidimensional affective variable. Specific concepts were applied that also seemed to have separate impacts on mathematics learning or performance outcomes (Leder, 1993). Attitude scales were designed to measure separately such constructs as self-concept or self-confidence in mathematics, motivation in mathematics, enjoyment of mathematics, value of mathematics, view of mathematics usefulness, anxiety toward mathematics, and views of mathematics as a male domain, or measures of parents´ or teacher´s perceived views (Eccles et al., 1983; Fennema and Sherman, 1976; Kulm, 1980; Leder, 1987; McLeod, 1994; Reyes, 1984; Sandman, 1980). Rather than speaking about mathematics attitudes, later studies made also an effort to bring their forms of approach and affective concepts in line with those in use within mathematical problem-solving research, cognitive science, socio-cognitive perspectives, or socio-cultural studies (Bishop, 1993; Goldin, 1992; 2000; Ma and Kishor, 1997; Mandler, 1989; McLeod, 1990; Op’t Eynde, DeCorte and Verschaffel, 2001; Silver, 1985). With these efforts especially the concepts of beliefs and emotions entered into mathematics education studies (DeBellis and Goldin, 1997; Evans, 2000; Malmivuori, 2001; McLeod, 1992; McLeod and Adams, 1989; Schoenfeld, 1992; Törner and Pehkonen, 1996). Also, current research on affect may deal with such constructs as self-efficacy, causal attributions of success or failure in mathematics, expectations of success, values, esthetic experiences, learned helplessness, help-seeking, metacognition, and autonomy or independence (DeBellis and Goldin, 1997; Lester, Garofalo and Kroll, 1989; McLeod, 1992; 1994; Meece, Wigfield and Eccles, 1990; Newman, 1990; Stipek and Gralinski, 1991; Yackel and Cobb, 1996). All these constructs have significant connections to student cognition and behavior, both theoretically and in related empirical studies. In addition to apparent connections to mathematics achievements and performances, these constructs tend to reveal consistent and significant differences in the patterns of students´ mathematics learning experiences and processes. This applies especially to mathematics education research results on students´ self-referenced cognitions and genderrelated differences (Evans, 2000; Fennema and Hart, 1994; Frost, Hyde and Fennema, 1994; Malmivuori, 1996; 2001; Marsh and Yeung, 1997; Reyes, 1984; Seegers and Boekaerts, 1996; Skaalvik, 1997).

3 These views can be viewed as supported by such as Allports’ (1935, p. 810) early definition of attitude in social psychology as “a mental and neural state of readiness, organized through experience, exerting a directive or dynamic influence upon the individual’s response to all objects and situations with which it is related” (cf., Kulm, 1980; Leder, 1993).

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The use of more specific and several different theoretical constructs have produced new approaches to affect and more accuracy in the field, but there are still uncertainties in the use of affective concepts. For example, as in general education research, various kinds of affective variables (e.g., feelings, preferences, values) or attitudes are referred to through the psychological concept of affect. On the other hand, the term affect has been used to denote an intense “hot” short-term affective state or an emotion, such as anxiety, shame and pride, panic reactions, fear, frustration, joy, and the so-called “aha” experience (Buxton, 1981; Evans, 2000; Hart, 1989a; Kulm, 1980; Linnebrink and Pintrich, 2002; Mandler, 1989; Meyer and Turner, 2002). Furthermore, variations that concerned the attitude construct seem to overshadow both the theoretical descriptions and use of the term mathematical beliefs. More agreement relates to the increased use and views of the term emotion, considered basically as an intense (whole-level) but short-cut affective response perceived in mathematics learning situations (DeBellis and Goldin, 1997; 2006; Evans, 2000; Hart, 1989a; Malmivuori, 2001; McDonald, 1989). D.B. McLeod´s conceptualization of affect in mathematics education (1988; 1992) represents an effort to adjust the significant affective variables in regard to the affective and cognitive involvement in them, or to the intensity and stability of affective responses. Accordingly, beliefs, attitudes, and emotions represent constructs with increasing affective involvement, decreasing cognitive involvement, increasing intensity, and decreasing stability, in this order. Goldin (1988, 2000) proposes an affective system representation that is joined with several cognitive representational systems. He also makes a difference between global affect (i.e., attitudes or traits) and local affect by which he refers to changing states of feeling during problem solving (see also Gomez-Chacon, 2000). In turn, DeBellis and Goldin (1997) suggest categories of affect through local emotions, stable attitudes, stable and highly cognitive beliefs, and deeply held preferences of values, ethics, and morals. Today, there appear suggestions for more structural or complex presentations of affect that take several perspectives or influential factors into account in explaining the role of affect in mathematics learning or problem solving. In efforts to understand affect in mathematical problem solving, Fennema and Peterson (1985) presented their Autonomous Learning Behavior model (ALB). It presented various affective variables such as confidence in one´s ability, perceived usefulness of mathematics, and causal attributions in mathematics as student internal belief systems that would mediate the effects of social context on such autonomous learning behaviors as independent thinking about the problem, persistence in solving the problem, and success in solving it that then produce variation in students´ mathematical outcomes (Fennema, 1989, p. 215). The expectancy-value model of Eccles et al. (1983) again connected the perspectives involved in the causal attribution model of achievement motivation (Weiner, 1986) and the social cognitive learning model of selfefficacy perceptions (Bandura, 1986). Students´ current and future expectancies in a given mathematical situation, together with their values attached to the task, were viewed to determine the qualities of such achievement behaviors as choice of activity, intensity of effort, and actual performance (p. 81). The former were interpreted as influenced especially by their task-specific beliefs or self-concept of ability and perceptions of task difficulty, and the latter more directly by goals and general self-schemata. Another achievement motivational model relevant to affective variables and mathematics learning is proposed by Dweck and Elliot (1983). Accordingly, such beliefs as students´ beliefs about the nature of mathematics and mathematics learning, or about the nature of intelligence and mathematics learning goals, are seen to influence the quality of their affective experiences and

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motivational patterns (i.e., performance vs. learning orientation) through their learning goals, values, and expectancies of mathematical performances. This perspective is consistent with often applied Nicholls´s (1984) distinction between ego-orientation and task-orientation behaviors, as well as with the difference between performance and mastery goals made by Ames and Archer (1988). More recent perspectives consider affect in mathematical problem solving as representational structures or pathways that interact with cognitive configurations by providing useful information, facilitating monitoring, and suggesting heuristic problemsolving strategies (DeBellis and Goldin, 2006; Goldin, 2000). Structural analysis presented by Evans, Morgan and Tsatsaroni (2006) again connects emotions to school practices and sociocultural interactions with discursive positioning involving different degrees of power and values. By applying Scherer’s (2000) component systems approach of emotions, Op’t Eynde, De Corte and Verschaffel (2006) again refer to five different systems (cognitive, autonomic nervous, monitor, motor, and motivational) that interact in social context in emotional process. We will continue to develop more structural and complex viewpoint of affect. In this, we will construct a dynamical theoretical framework for understanding student affect in learning and doing mathematics, in which several recent perspectives on affect or emotions presented in psychology, socio-cognitive theories, and education psychology will also be taken account (Malmivuori, 2004). Our model of mathematics learning attaches the important affective variables and mathematics education research results on affect to students’ significant selfreferent constructs (e.g., self-confidence, self-efficacy, autonomy, personal agency) and to their achievement behaviors (e.g., persistence, choices, goal construction) that relate further especially to the aspects of their metacognition and self-regulation. Specifically, we consider students´ personal and unique situational mathematics learning processes under the idea of self-systems and self-system processes (Malmivuori, 2004; 2006). This humanistic-cognitive idea links affect more strongly, naturally, and in a dynamical way to cognition and, further, to social environment in mathematics learning situations. For presenting this viewpoint, we first consider few important and related perspectives on the interplay of affect and cognition.

PERSPECTIVES ON THE INTERPLAY OF AFFECT AND COGNITION Deficiencies, restrictions, or variation in the theoretical models, definitions, or levels of abstraction applied, together with the complexity or inaccessible nature of the mental processes that underlie affective responses, can all be viewed to be related to the traditionally accepted gap between feeling and thinking or affective and cognitive domains of personality (Leventhal, 1982; Lewis, 1999; Scherer, 1999; Wozniak, 1986; Zajonc, Pietromonaco, and Bargh, 1982). Basically, the dichotomy between physical and mental or phenomenological aspects of human characteristics has much flavoured the general tendency to treat affect in isolation from cognition (cf., Corradi Fiumara, 2001; Damasio, 1996; Sigel, 1986; Sun, 2002). The related long debate on the separateness or linkages between cognition and emotion (see Lazarus, 1999; Solomon, 2000) characterizes this situation, in which the opinions and theories for understanding the relations between these two human aspects vary from neurophysiological approaches to philosophical illustrations. Both philosophers´ and psychologists´ cognitive views of affect tend to stress the primacy of cognition over emotion,

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but they also differ in perspectives they take (cf., Lyons, 1999). For example, the well-known cognitive psychologist, Piaget (1981) believed that the affective domain plays an essential part in human mental processes and that it forms an “energizing” context for personal knowledge construction. And, many contemporary cognitive theorists consider that affect interacts with cognitive actions in important ways, and that it may dominate and direct one´s cognitive resources and processes or the development of aptitudes in learning (cf., Bearison and Zimiles, 1986; Parrott and Spackman, 2000; Power and Dalgleish, 1997; Snow and Farr, 1987). In addition to various socio-cognitive models (e.g., Kemper, 2000; Mischel and Shoda, 2000), more recent perspectives lay emphasis, for instance, on psychodynamic or psychoanalytic viewpoints to understand the interplay between affect and cognition (see, e.g., Corradi Fiumara, 2001; Evans, 2000; Westen, 1999). Also, such notions as minding the body through feelings, embodiment of cognition, affective knowledge or systems, emotional networks, and the duality of implicit and explicit cognitive processes have been suggested for linking affect and cognition (Brown and Reid, 2006; Damasio, 1996; Johnson-Laird and Oatley, 2000; LeDoux and Phelps, 2000; Sun, 2002; Westen, 1999). Today, the causal-evaluative theories of emotion represent the most favoured cognitive alternatives to the cognition-affect debate applied in education research. Emotions are viewed to be aroused by a person´s evaluations or appraisals of a future or an ongoing event as relevant to a concern or a goal that is important (Carver and Scheier, 2000; Ellsworth and Scherer, 2003; Emmons and Kaiser, 1996; Hunsley, 1987; Lazarus, 1991; Lewis, 2000; Mandler, 1989; Oatley and Jenkins, 1996; Schutz and DeCuir, 2002; Stein, Trabasso and Liwag, 2000). In addition to cognitive resources, related education research results and perspectives thence connect academic emotions significantly to such aspects as students´ motivation, learning strategies, and self-regulation processes in classroom interactions.4 The underlying meaning-making concepts and processes behind emotions have been connected to various social-historical, contextual and personal aspects of the learning events (cf., Kemper, 2000; Schutz and DeCuir, 2002). But, especially in the case of personally significant, complex, and individually constructed emotions (e.g., shame, guilt, anxiety), the most apparent and significant framework for these processes and evaluations can be viewed to be associated with self-referenced cognitions (Borkowski et al., 1990; Damon, 1986; Epstein, 1986; Harter, 1985; Higgins, 1987; Johnson-Laird and Oatley, 2000; Kemper, 2000; Lewis, 1999; Malmivuori, 2001; 2006; Singer and Salovey, 1996; Turner, Husman, and Schallert, 2002). We follow these perspectives and view students´ powerful affective experiences and responses to mathematics to be closely linked to their personal mental constructions, structures, and processes, but more particularly, to their personal and situational selfperceptions, efforts, goals, and self-regulation in the social and contextual mathematics learning environment. 5 This kind of approach is reflected, for example, in the perspective given by Hatfield (1991) who has linked high quality mathematical experiences to emotional states (feeling) and inquiry states (thinking) in which the emotional elements involve a sense 4

See, e.g., Bandura, 1993; Bessant, 1995; Boekaerts, 1995; Borkowski et al., 1990; DeBellis & Goldin, 1997; Dweck & Legget, 2000; Halberstadt, Niedenthal & Setterlund, 1996; Hannula, 2006; Markus & Wurf, 1987; McLeod, 1988; Malmivuori, 2001; Meyer & Turner, 2002; Op’t Eynde et al., 2001; Sansone & Harackiewicz, 1996. 5 Cf., Bandura, 1993; Boekaerts, 1995; Carver & Scheier, 2000; Deci & Ryan, 1991; Eccles et al., 1983; Goldin, 2000; McCombs & Marzano, 1990; Pekrun et al., 2002; Schunk, 2001; Skaalvik, 1997; Zimmerman, 1994.

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of purpose, self-perception of potential for success, and a willingness and capacity to monitor and control the effects of one´s feelings. We consider the arousal and development of students´ highly influential affective responses to mathematics to be importantly intertwined with their situational or learned habitual beliefs, perceptions, and appraisals of the self in social mathematics learning contexts (Malmivuori, 2001; 2004). These are aspects of the key personal processes that commonly differentiate between students´ powerful mathematics learning experiences. Moreover, we interpret that the self, together with the varying levels of consciousness or awareness in the appraisals, affective responses and self-regulation, offer an unconstrained path from cognition to affect and vice versa (Malmivuori, 2001; 2006; cf., Clark, 1982; LeDoux, 1998; Lewis, 1999; Öhman, 1999; Piaget, 1976). Consistently with the current structural or process-based views of learning, we emphasize complex mutual interrelations between affect and cognition in student behavior. Moreover, examination of unique situational aspects and constantly ongoing processes of personal learning and affect will offer opportunities for going beyond the traditional static and/or inaccurate concepts and personality domains. In this, we refer to the constantly operating mind and flow of mental processes and affective states, in which different appraisals and processing activities can coexist at different levels of consciousness or self-awareness, and cause several (continuously flowing or changing and perhaps conflicting) affective experiences or self-states that are influential in students´ mathematics learning processes (see also Corradi Fiumara, 2001; Evans, 2000; Lazarus, 1991; LeDoux, 1998; Lewis, 1999; Stein and Levine, 1999; Sun, 2002). Then, personal and unique situational appraisals, as well as regulation, with affective responses also represent different degrees of abstraction in students´ mental processes or cognition.6 We may call the scene of these mental operations a student´s contextual consciousness, in which his or her mental co-constructions, referred to by Wozniak (1986, p. 41) as experience, symbolic discourse, and action (i.e., affect, cognition, and action), are intertwined in complex ways, and also conditioned by personal aspects and various external features of a particular socio-cultural mathematics learning context. Within this scene, affective responses have important organizing, motivating, and adaptive functions in students´ learning processes that are related to or independent of their specific learning goals and behaviors (cf., Davidson, 1999; Evans, 2000; Goldin, 2000; Lazarus, 1991; Leventhal, 1982; Piaget, 1981; Scherer, 1999; Taylor et al., 1997). They also generally serve students as a significant source of information about their own mental content and ongoing processing activities, of their action conditions, and of their self-states with respect to mathematics and mathematical activities in a learning situation (DeBellis and Goldin, 2006; Malmivuori, 2001; 2004). Assessment and reactions to this information are then involved in students´ self-regulatory processes at different levels of their selfawareness.7 Thus, affective responses and experiences constitute the significant personal and situational attributions of students´ learning mathematics as well as the essential, personal and unique situational, regulators of their mathematics learning processes (see also DeBellis and Goldin, 1997; Hunsley, 1987; Marshall, 1989; McLeod, 1988). With these viewpoints we want to emphasize especially the dynamic aspects of affect and cognition in student learning processes. These aspects make it more possible to understand the multiplicity and concerted 6 cf., Carver & Scheier, 2000; Dodge, 1991; Kuhl, 1987; Lazarus, 1991; Leventhal, 1982; Scherer, 1999; Stein & Levine, 1999, Sun, 2002; Taylor, Bagby, & Parker, 1997.

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functioning of affect in learning processes but also students´ personal agency and selfregulatory features with their affect.

SELF-SYSTEMS AND SELF-SYSTEM PROCESSES Students as historical and social individuals or selves constantly constitute, evaluate, develop, and regulate themselves and their own affective experiences and learning processes in relation to mathematics. We consider these activities as the essential dynamical features of students´ personal learning processes and significant affect. In our learning model, the qualities of students´ self-reflections and experiences of self with affective experiences and self-regulation represent the primary individual scene for the integration of affect and cognition in their personal mathematics learning processes. We call this schene students’ selfsystems and self-system processes. They create the links among affective, cognitive and behavioural processes in learning. By self-systems we refer to stable internal structures that derive from students’ past history and experiences with mathematics in social environment. They consist of content-based mathematical knowledge; learned socio-cultural beliefs about mathematics, its learning and problem solving, beliefs about the self in mathematics; affective schemata; and habitual behavioural patterns in mathematical situations (Malmivuori, 2001; 2006). Even more focus is here attached to dynamical features of the self and self-system processes. The stable self-systems, once activated in a learning context, are the basis for the functioning of students’ self-system processes in mathematical situations. But, their metacognitive, cognitive and affective capacity are always conditioned by situation-specific factors and, thence, result in unique form of functioning of personal processes in a mathematics learning situation. Our emphasis on the dynamical features of the self relates to this functioning of self-system processes. We connect these dynamical features of the self and self-system processes in particular to metalevel, i.e. self-reflective and self-regulatory, aspects of students´ personal learning processes and affective experiences. Metalevel constructions and processes constitute the basis for individual and situation-specific variation and development of students’ mathematical knowledge and skills, affective responses, and use of personal resources in mathematics learning (Malmivuori, 2001). They also ultimately determine the power and role of affect in each of students’ mathematics learning situations. Unique situational self-appraisals and self-judgments constitute central occasions for the dynamic interplay of students’ cognition and affect in learning mathematics. They have to do with students´ self-consciousness, self-beliefs and self-knowledge, as well as with learned cultural and individual standards, rules and goals with respect to school, achievement, mathematics and mathematics learning. The related aroused powerful affective responses are connected with students´ experiences of self-esteem, self-worth, and personal efficacy and control (or responsibility) with respect to mathematics learning and performance, which can be described as the aspect of “how one feels about one´s worth and power” (Malmivuori, 2001; cf., Bandura, 1993; Covington and Roberts, 1994; Harter, 1985; Lewis, 1999). Consequently, they relate to students’ appraisals of their personal capability and agency in mathematical situations and, thence, represent the primary factors in the arousal and 7 cf., Boekaerts, 1995; Carver & Scheier, 2000; Corno, 1989; Folkman et al., 2000; Kuhl, 1987; Malmivuori, 2001; McCombs, 2001; Schunk, 2001; Taylor et al., 1997; Zimmerman, 1994.

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development of students´ powerful emotions with respect to mathematics. We call these powerful affective responses “self-emotions” (Malmivuori, 2006; cf., Blasi, 2004; Harter, 1985) that involve such powerful emotions like shame, guilt, anxiety and enjoyment. Mathematics education research has found these kind of highly influential affective responses characteristic for mathematics achievement situations (e.g., McLeod, 1992). They often appear to be negative and inhibiting in nature, resulting in disturbance of students´ personal mathematics learning processes, performances, and achievements. Moreover, these negative effects tend to apply more to females´ experiences and mathematics learning processes than that of males´ (Aiken, 1970; Fennema and Hart, 1994; Hembree, 1990; Malmivuori, 1996; McGraw et al., 2006; Seegers and Boekaerts, 1996). Self-regulation processes represent the central combining feature of self-system processes with affect. In addition to self-appraisals and self-judgments, these metalevel mental processes involve students’ more or less conscious self-directive constructions, self-control and self-regulatory actions (Malmivuori, 2001; 2006). They represent the significant aspect of the dynamic affective-cognitive interplay that are then accompanied by and/or directed towards affective responses and states. Related to our notion of self-system processes, we emphasize the dynamic nature of the self or the self as process (cf., Polkinghorne, 2001). More specifically, we relate the qualities of self-experiences and self-regulation processes with affect to the degree of students´ self-awareness and emergence or their lack of personal agency. Thus, students´ unique situational affective experiences as well as the development of their more stable and highly influential affective responses toward mathematics are importantly connected to their personal beliefs about or possibilities and development of selfknowledge, self-determination and self-control with respect to their own mental contents, responses or arousals, and actions in mathematical situations. These significant features of personal agency and efficiently self-regulated learning then have a strong influence on the sources, variation, and sustenance of students´ important affective responses toward mathematics, as well as on the role that their affective responses play in determining the qualities of their personal mathematics learning processes and performances. They are the key qualities of students´ self-system processes and their contextual consciousness in any mathematics learning situation. With respect to these features and affect in self-system processes, we may contrast active regulation of affective responses with automatic affective regulation (Malmivuori, 2004; 2006). Automatic affective regulation refers to an affective feedback system dominating the evaluation system and behaviour at a relatively low level of control (cf., Carver and Scheier, 2000; Dodge, 1991; Taylor et al., 1997). Mental blockages, simpification of cognitive processes or information processing strategies, and hindering of higher order mental processes due to strong negative affective responses are example of this kind of affectivecognitive dynamics (e.g., Mandler,1989; McLeod, 1988; Mischel and Shoda, 2000). In turn, increased flexibility or efficiency of decision making in problem solving processes and change of thought contents or instant positive memory retrieval caused by promotive affective responses (e.g., positive moods) provide positive examples of automatic affective regulation (see, e.g., Hirt, McDonald and Melton, 1996; Isen, 2000). Affective responses also give rise to, accelerate or sustain additional interpretations, personal meanings, and beliefs with several evaluation processes going on at the same time at different levels of consciousness (e.g., Wegener and Petty, 1996). They further establish a set of additional behavioural goals related to or independent of students’ specific goals or objectives with ongoing original learning

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intentions or behaviours, and cause differing and possibly conflicting action tendencies (e.g., Brown and Reid, 2006; Evans, 2000; Frijda, 1986; Izard and Ackerman, 2000; Lazarus 1991; Leventhal, 1982; Stein et al., 2000). In this way affective responses have important organizing, motivating and adaptive functions, and they also induce or regulate other affective responses (e.g., interest attenuating fear and sadness or shame attenuating joy; cf., DeBellis and Goldin, 1997; Taylor et al., 1997). Integration of this kind of whole-level affective cognitive dynamics or self-systems processes has a major impact on the organization of personality with important individual differences in affective development as well as in the development of self-systems. We consider this integration to be importantly related to the activation and development of students’ self-conscious monitoring, assessment and judgments of their own affective arousals, responses and self-states, to their self-conscious decisions and choices directed towards affective responses or states and the causes or effects of these, and to their conscious control over their own affective responses (Malmivuori, 2001; 2004). Students’ affective arousals and the sates of the self then become objects of their conscious evaluation and regulation. With these metalevel processes, students have significant power in affecting the arousals, experiences and effects of their affective responses in learning and doing mathematics. We refer to these self-system processes as active regulation of affective responses. The essential difference between these two forms of interplay of affect and cognition is linked here with the varying degrees of students’ conciousness or states of selfawareness and reflectively directed activity in the functioning of their self-system processes. Affective regulation represents automatic or habitual self-regulation with lower states of selfawareness and weak personal agency, while active regulation of affective responses is involved in high personal agency, high levels of self-awareness and efficient self-regulatory processes. Moreover, we connect the former activity to students’ habitual structures in their stable self-systems with mathematics, characterized by arousal of similar, often interfering affective responses leading directly to habitual behaviours in mathematics learning situations (e.g., particular defensive action or intentions; Malmivuori, 2001). Actions and affective responses in mathematical situations are then determined mainly by students’ stable selfsystems. In contrast, we connect active regulation of affective responses to students’ high personal agency with effective, situation-specific and creative self-system processes. Here students’ actions and affective experiences may be relatively independent of their stable selfsystems as well as of the social features of the mathematics learning context. The quality of students’ self-systems and the functioning of their self-system processes are influenced not only by their personal aspects but also by various situation-specific, contextual and socio-cultural, features of mathematics and mathematics learning. Thus, important self-perceptions and self-appraisals mediate not only the effects of students´ past personal mathematical history (e.g., beliefs about their own mathematical abilities), but also those of the fundamental socio-cultural and contextual features of mathematics learning on their affective responses to mathematics (cf., Bishop, 2001; Boekaerts, 1995; Deci and Ryan, 2000; Dweck,and Leggett, 2000; Eccles et al, 1983; Hart, 1989b; Hembree, 1990; Lave, 1988; Marshall and Weinstein, 1987; Mischel and Shoda, 2000; Risness, Hannula and Malmivuori, 1999; Tesser, Millar and Moore, 2000; Weiner, 1986; Yackel and Cobb, 1996). In this mental and behavioural individual-environmental interaction, we view the characteristics of an actual mathematics learning and task context, or unexpected, new, or rapidly changing occurrences in this context, to represent more direct environmental

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influences on students´ self-appraisals and on such self-emotions as anxiety or fear. In turn, less direct or more clearly mediated environmental influences on students´ self-appraisals and emotions (e.g., shame, pride, test anxiety) are again linked to particular kinds of socio-cultural forms of interpretation or beliefs about mathematics and mathematics learning or about performance situations that are reflected by students as well as by the larger social environment (e.g., perceptions of the difficulty or importance of a mathematics learning situation, beliefs about mathematics learning goals, attributions for mathematical successes or failures 8). For example, significant and constant gender-related differences are measured in students´ perceptions of their mathematical abilities and performances as well as in their selfemotions, such as anxiety or pride and shame (Eccles et al., 1983; Evans, 2000; Fennema and Hart, 1994; Hembree, 1988; McLeod, 1992; Reyes, 1984; Seegers and Boekaerts, 1996; Stipek and Gralinski, 1991). Behind these differences may be found socio-cultural beliefs about mathematics as a male domain. With respect to individual-environmental interaction we further characterize active regulation of affective responses as individually and situationally directed personal processes with affect. The interaction between environmental features and students’ mathematical affective responses can then be considered less direct and more flexible or independent of the instant environmental conditions and specific social features of school mathematics learning, but also of their own stable or habitual self-systems (i.e., self-beliefs, mathematical belief systems, affective schemata, behavioural patterns). Instead, automatic affective regulation can be considered as basically retaining personal functioning, in which the interaction between environmental features and students’ affective responses is rather direct. Arousal, repetition and effects of similar strong and often hindrance affective responses (cf. global affect; DeBellis and Goldin, 1997) depend then mainly on the qualities of students’ stable selfsystems and/or particular contextual and socio-cultural features of school mathematics learning. This distinction between the described opposite forms of individual-environmental interaction relates again to the qualitative variation considered here due to the activation and development of students’ higher order metalevel self-system processes with high states of self-awareness and high personal agency as essentially included in the former.

CONCLUSION Newly rediscovered and less restricted terms or theoretical constructs, such as metacognition, consciousness, and self-regulation, afford opportunities to consider cognition as more closely linked to affect and behavior in learning and education. Moreover, application and more thorough understanding of various aspects of and viewpoints on affect, together with extensive empirical research on human information-processing, brain structures or neuroscientific systems, will increase our holistic understanding of the nature, functioning, and development of student affect. In this chapter, we emphasize the role of personal constructive and self-regulatory aspects of affective responses in social learning environment. The findings of mathematics education research on affect are complemented here by recent general cognitive, socio-cognitive, constructivist, and phenomenological or humanistic8 cf., Cobb et al., 1989; Dweck, 2000; Eccles et al., 1983; Malmivuori, 2001; McLeod, 1992; Op’t Eynde et al., 2006; Skaalvik, 1997.

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cognitive perspectives on learning. More specifically, we connect these aspects and perspectives to our notions of the functioning, qualities and development of students´ selfsystems and self-system processes in respect to learning mathematics. Special importance is here attached to the co-constructive and dynamic nature of affect and cognition. Students´ personal stable systems (i.e., self-systems) and their previous mathematics learning experiences play a significant role in the interplay of their affect and cognition in new mathematics learning situations. But, always their specific interpretations, appraisals and the forms of behaviours and self-regulation (i.e., self-system processes) aroused and constructed against a unique mathematics learning context and situation will finally determine the (affective) qualities of their personal learning processes and experiences in that situation. In this, self-regulation processes represent the central combining feature of our dynamic considerations and self-system processes intertwined with affect and personal learning processes. Then, the qualities and functioning of self-system processes depend on the emergence and development of students´ self-awareness and personal agency that then ultimately determine the nature of the interplay of their affect and cognition. These conceptualizations and perspectives introduce a possibility to bridge the underlying gap between the affective, cognitive and behavioural personality domains, to overcome the restrictions of the traditional static concepts of affect applied in education research, and also to deal with the complexity of affect-cognitive interplay in learning situations. Moreover, the viewpoints are designed to combine the significant variables and mathematics education research results of affect. For illustrating our dynamic perspective, we may distinguish between two essentially different lines in the qualities and development of students´ self-systems and self-system processes in respect to affect and mathematics learning (Malmivuori, 2001; 2004). In the case of inefficient learning of mathematics, students´ learning processes, experiences and states are more engaged in the activation, operation, and influence of their closed and inflexible selfsystems or defectively functioning self-system processes with mathematics. This kind of basically retaining or self-defending personal functioning is often dominated by students´ negative self-appraisals with highly intense negative self-emotions, further intertwined with their habitual or preventive mathematics learning behaviours and inefficient self-regulatory processes with sense of low self-esteem, weak personal agency and low control. In this case, the qualities of students´ personal and habitual (stable) self-systems (i.e., beliefs, affective schemata, behavioral patterns) as well as the external contextual and socio-cultural influences have a decisive impact on their mathematics learning and affective experiences that most commonly cause disengagement from mathematics and its learning. On the other hand, efficient and personally powerful learning of mathematics is attached here to students´ open and flexible self-systems or to their fully activated and functioning selfsystem processes with respect to mathematics learning situations (see also Rogers, 1983). The more efficient functioning of students´ higher order self-regulatory processes and self-system processes means that they can actively influence, take control over, give rise to, construct, direct, and/or revise their ongoing mental processes and affective states in learning mathematics. Students are more capable of consciously acting on their habitual mathematical self-appraisals, affective responses, and behavioural patterns (i.e., stable self-systems) irrespective of the instant environmental conditions and socio-cultural factors. They will also understand the relationships between their beliefs, affective responses and actions and, thence, they may choose the level of influence that their specific beliefs and affective

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responses have on their mathematics learning behaviours and intentions. These significant features and experiences of reflectively directed affective experiences and learning behaviours will significantly empower students’ mathematics learning (Malmivuori, 2001; 2006).

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In: Science Education Issues and Developments Editor: C. L. Petroselli, pp. 151-177

ISBN: 978-1-60021-950-4 © 2008 Nova Science Publishers, Inc.

Chapter 5

THE CHALLENGE OF USING THE MULTIMODAL ASPECTS OF INFORMAL SOURCES OF SCIENCE LEARNING IN THE CONTEXT OF FORMAL EDUCATION Krystallia Halkia1 and Menis Theodoridis2 1

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University of Athens Greek National Television Broadcasting

ABSTRACT In this work, an attempt has been made to study the plethora and the diversity of informal sources of science learning and the ways formal education may benefit by making use of these sources in its everyday school practice. Informal sources of science knowledge have many forms: they use several means of presenting scientific information, take place in several environments and use several ways to compose their “text”. Each one of them has its own communication codes and uses multiple ways (modes) to present its “meaning”. The material coming from them is chaotic, because it is diverse in terms of the means used, the purposes and the targets stated, the audience addressed, etc. To study them it is helpful to categorize them. Thus a three dimensional model has been developed. Each dimension describes one system of taxonomy: The first one refers to the environment and the conditions under which science learning takes place; the second refers to the way a science “text” is made up and the codes used; while the third one refers to the kind of mode used in the science “text”. Furthermore, the different learning environments in which informal science learning takes place have been studied. Three different learning environments have been distinguished: the organized out-of-schools visits to institutions and organizations (science museums, science centers, zoos, botanic gardens etc.), the students’/teachers’ personal navigation in several sources outside school and the use of informal sources of science learning by the teachers within their everyday classroom practice. The study reveals their particular characteristics, as well as their power and limitations. It also suggests ways of using them effectively in the context of science classroom.

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INTRODUCTION In most countries, formal education is determined and guided mainly by the educational authorities of the official state and takes place in institutions (schools). Until recently, society considered schools as the prime educational institutions which -among others- lead towards citizens’ scientific literacy. But, during the last few decades, a rapid development of science in several fields (information and communication technology, space, biology, health, etc.) has taken place. Many of these developments have a remarkable influence on everyday life (Millar 2006). Thus, every citizen should be appropriately equipped (with the necessary background knowledge and skills), to enable quick access to reliable information and make meaning out of it. In that way he/she would be able to take the proper decisions for his/her own life and participate in the social discourse on scientific matters (e.g. about bioethics issues). It might be for that reason that citizens -among them students and teachers- show a great interest in such scientific issues (Halkia 2003). However, formal education usually seems to resist the transformation of new knowledge into school knowledge. As a result, new knowledge very rarely is a part of a science curriculum. Thus, mass media and other forms of informal sources of science learning seem to play a major role in filling the gap between school science knowledge and some areas of contemporary science knowledge (Millar 2006). Contrary to the school system, informal sources have developed appropriate mechanisms (e.g. the use of known scientists or accredited journalists) to popularize scientific knowledge (i.e. to transform scientific knowledge into public knowledge). Thus, informal sources of science learning gradually command a large portion of citizens’ scientific literacy (Martin 2004, DeBoer 2000). According to Miller (1998), a citizen could be considered as scientifically literate if he/she can read, understand and handle the scientific knowledge presented in the articles of high status newspapers. But, while so much attention is paid to the way formal education addresses students, little attention is paid to the impact that informal sources of science knowledge may have on students and teachers. Many people still believe that learning takes place mainly through a well organized and specially designed system, like formal education. However, fragments of science knowledge coming from sources beyond formal education usually intervene in school science and affect students’ learning of science. Formal science education is often confronted with bits of information coming from a plethora of informal sources of science learning. Many of these sources are of high status and are considered to be vehicles of culture and knowledge (like museums, planetariums, science centers and, to a lesser extent, mass media, the Internet etc.). Their characteristics and content may vary considerably (e.g. exhibits in science museums, several animal species in zoos, sea environments in aquariums, science articles in newspapers, science documentaries on television or DVD etc.), and the public meets with them in a number of different ways. Sometimes, this (informal) knowledge may contribute to the objectives of the school science curriculum, because it can serve as a rich background (preexisting knowledge) to the acceptance of new science concepts. In some other cases however, it may act as a “noise” which distracts students from the negotiation of science concepts in science classrooms (i.e. it

The Challenge of Using the Multimodal Aspects of Informal Sources of Science… 153 can be the cause of the creation of students’ alternative ideas or it may strengthen their already existing ones). Today, the informal sources of science learning (cultural institutions and the Media) seem to be actively involved in the development of the citizens’ scientific literacy. They largely affect students background knowledge in science, while they also contribute to teachers self education in scientific matters (Halkia 2003). Their role is more and more decisive in shaping the public understanding of science and the attitude of citizens towards science. Science educators have realized the contribution of informal sources of science learning to formal science education, considering them as important complements to the science curriculum (Millar 2006).

CORE ISSUES RELATED TO INFORMAL SOURCES OF SCIENCE LEARNING There is an ongoing discussion as to what constitutes informal, out-of-school learning and the ways it occurs. The theories of learning, used to study learning in out-of-school settings, are based on two different perspectives: the constructivist ideas (Anderson et al. 2003, Hein 1995) and the socio-cultural theory (Schauble et al. 1997, Rennie 2007). Relevant research shows that effective processes of learning demand a socio-cultural framework of analysis and a theory based on a constructivist model of knowledge (Hein 1995). To study the degree to which informal sources of science learning can contribute to students’ learning, one has to investigate: questions about the meaning of learning in science (is it a body of knowledge, or a process or both), cognitive psychology questions about the way mental models are developed and questions about how different types of discourse may influence conceptual learning and change (Martin 2004). These questions have to do with learning in science generally and formal education is also concerned in answering them. They seem to constitute a good start when investigating whether learning occurs out of any source of knowledge (formal or informal). In this paper, we will primarily focus on the particular characteristics of the different informal sources of science learning and the ways in which teachers may benefit by using such sources in relation to their formal classroom activity. This would require: a) identifying and considering different kinds of informal sources of science learning. b) studying each particular learning experience offered, in terms of learning environment and context, types of activities suggested, kind of scientific knowledge promoted, teaching design adopted, etc. c) suggesting ways in which teachers will develop their own teaching strategies for using creatively all kinds of learning experiences offered by the different informal sources of science learning, according to the science curriculum. In the present work a mapping of informal sources of science learning is attempted and a system of taxonomy for the many different forms of these sources is suggested. This system will help the understanding of the peculiarities of these forms as well as their degree of interaction with formal science education.

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INFORMAL AND NON FORMAL SOURCES OF LEARNING:CLARIFICATION OF TERMS The sources of learning which do not belong to formal education usually are divided into two types: a) informal sources of science learning, where learning develops on a personal basis (voluntarily, relaxed and instinctively), outside of an organized educational system (e.g. popularized science books, science documentaries on television or DVD, science articles in magazines and newspapers etc.) and b) non formal sources of science learning, where learning develops within organized educational systems (e.g. museums, science centers, planetariums etc.) offering educational programs built on an implicit curriculum. The distinction seems to be functional. However from formal education’s point of view both kinds of sources, regardless of their status, require a critical consideration of the educational experiences which they offer in order to become science curriculum relevant. For this reason -and for brevity’s sake- we are going to use the term informal sources of science learning for both types of sources. Also we will use the term “museums” as a general term referring to all kinds of cultural organizations which offer educational experiences, such as science museums, science centers, planetariums, zoos, aquariums, botanic gardens etc. Informal sources of science learning have many forms and use multiple ways (modes) of presenting scientific information. Most of them have not yet been studied thoroughly to investigate the kind and the level of knowledge targeted and achieved. The review of the relevant literature shows that science museums and science centers are the most studied sources of informal science education. The other forms are not equally well studied. A special edition of Science Education (2004) was dedicated to the research carried out on science museums. A considerable portion of the bibliography used in this present work comes from this special issue.

INFORMAL SOURCES OF SCIENCE KNOWLEDGE AND LEARNING: BASIC PARAMETERS As far as informal sources of science knowledge are concerned, according to Rennie et al (2004), research may focus on three basic characteristics of this kind of learning: a) learning is personal, i.e. it is approached according to each student’s personal will and needs (psychological, cognitive, etc.); b) learning is contextualized, i.e. it takes place in a context that determines -more or less- the kind of “knowledge” constructed; and c) learning takes time, i.e. its outcomes need time to be detected. It seems that the context in which learning takes place is the most determining factor of the whole procedure of learning. Falk and Dierking (2000) studying the outcomes of a visitor’s experience in museums, have suggested three contexts (the personal, the sociocultural and the physical) interacting to shape the kind of learning taking place. Falk and Dierking in their work, consider the educational design as part of the physical context. Still,

The Challenge of Using the Multimodal Aspects of Informal Sources of Science… 155 from the point of view of formal education, a fourth context (the educational) can be identified. The teacher responsible for organizing a school visit in a museum faces the challenge of developing a specific teaching plan which will link the museum’s educational experience with the needs and priorities of the science curriculum. To him/her, distinguishing a fourth context, the educational context, signifies the necessary extra care required to relate the informal sources of science learning to his/her everyday classroom activity. These four contexts interact with each other and produce “knowledge”. For the needs of the present work, we think that these contexts can be applied to any kind of informal source: a) The personal context: this refers to the background of the “student-visitor” (his/her personal experiences; preexisting knowledge; interests; social skills; mental skills; values, etc.) and his/her personal agenda of learning implicit when visiting any informal source of science learning. According to Contextual Model of Learning, the main factors of the personal context are: motivation and expectations; prior knowledge, interests and beliefs; choice and control (Falk and Dierking 2000). b) The socio-cultural context: this refers firstly to students’ interaction with persons associated with the particular visit experience (e.g. parents, other relatives, teachers, friends, museums’ staff etc.) and secondly to the social and cultural characteristics of the “exhibited items” (like the exhibits in a science museum or the science articles in the newspapers and magazines or the science documentaries on the television and the DVDs or the site of an institution/organism on the Internet) presented by the informal sources of science learning. According to Contextual Model of Learning, the main factors of the socio-cultural context are: within group socio-cultural mediation; facilitated mediation by others, cultural background and upbringing (Falk and Dierking 2000). c) The physical context: this refers to the physical aspects of the surrounding space, which may host some kinds of exhibits or the screening of a science film etc. It includes the architectural characteristics of the environment, the arrangement of the exhibits in a science museum (Semper 1996), the lay-out of a science article (or of a website), the caption and the music which accompany an exhibit, etc. According to Contextual Model of Learning, the main factors of the physical context are: advanced organizers and orientation; design; architecture and large-scale environments; reinforcing events and experiences outside the museum (Falk and Dierking 2000). d) The educational context (which in Falk and Dierking analysis falls under physical context): this refers to the educational design, whether explicitly or latently signified, and is expressed in the way the learning environment is constructed (the navigation routes, the work sheets, the extent of interactivity of the “exhibits” or of the software, the layout of a science article or of a webpage etc.). It is reflected in the curriculum of learning opportunities created by the informal sources for the visitors to their space (Rennie et al. 2004). It represents the intentional or unintentional educational conceptions of the people behind each informal source of science learning; their values, their purposes, etc. (e.g. which model of teaching they adopt and why). It interacts strongly with the rest of the learning contexts and, depending on the quality of the design, it can prove to be the dominant context.

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Thus, the context of learning seems to be a decisive parameter for the way students will perceive new information provided by the informal sources of science learning. Since this information seems to make sense to them, they add it as “knowledge” (among others) to their existing conceptual frameworks. Students, when exposed to the items of informal sources in a pleasant way, usually transform subconsciously any information offered into “knowledge”. The experiences (usually indirect), through which students construct the new “knowledge”, can be very intense and for that reason very powerful. This happens because: •

•

the information presented in these sources refers to phenomena and events of the physical world (like the phenomena which happen in the micro cosmos or mega cosmos) to which students -in most cases- can not have a direct access (thus lacking the corresponding sensory experiences); and many of these sources use sophisticated technological means, like virtual reality presentations, which help students to create powerful (virtual) experiences, thus imposing their “message” whatever it may be.

However, such “knowledge” which comes from informal sources may differ significantly from the scientific knowledge it refers to. It may differ: • •

•

either because students have not constructed the appropriate conceptual frameworks which would anticipate such new concepts; or because these sources are not reliable enough in presenting information about scientific concepts and phenomena. They may present them in a confused way, misleading students into arbitrary, alternative interpretations; or because these sources -although their target population is mainly students- do not base the construction of their “products” on an educational design, but instead the didactic model implied in the design of their “products” is confined by an empirical approach.

In that way, the conceptions students construct about the phenomena of the physical world could be characterized as “indirect” alternative conceptions and, sometimes, are equally as powerful as the alternative conceptions constructed through direct experiences. The teacher’s role is decisive because he/she has to elicit his/her students’ alternative conceptions of phenomena which they could not approach through their senses. Also the teacher should check the scientific content of these sources and the way they present their “exhibits” or their “products” (e.g. are they appropriate to the mental skills of his/her students? are the concepts presented in an organized way? etc.) and not uncritically accept their validity. It is interesting to notice that there is a wider variety of informal learning sources than one would expect.

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TAXONOMY OF INFORMAL SOURCES OF SCIENCE KNOWLEDGE: A THREE DIMENSIONAL MODEL Informal sources of science knowledge have many forms: they use several means of presenting scientific information, take place in several environments, and use several ways to compose their “text”. Each one of them has its own communication codes and uses multiple ways (modes) to present its “meaning”. To study them it is necessary to categorize them somehow. The organization of such chaotic material, diverse in terms of the means used, the purposes and the targets stated, the audience addressed etc., revealed three basic fields where similarities and differences could be traced. Thus, a three dimensional system of taxonomy has been developed.

Dimension A: System of Taxonomy According to The Environment and the Conditions under which Science Learning Takes Place This system focuses on the fact that the ability of students to elaborate the “message” sent by the informal sources and to construct a meaning out of it, depends decisively on the dynamics of the environment and the conditions under which learning takes place. Thus, it makes a great difference if the students watch a specific science documentary film: • • •

in an organized way with their teacher and classmates in a planetarium or; personally, in conditions of free attendance, at home on the TV or; on a TV or video screen in their classroom as part of a presentation by their teacher.

According to this system of taxonomy, and having formal education as a point of reference, three major categories of informal sources of science learning can be distinguished: 1. Organized school visits (outside school): Formal education taking advantage of educational programs designed by the informal sources of science learning (like museums and science centers, planetariums, zoos and botanic gardens, etc.), arranges organized school visits to enrich the school science curriculum; 2. Private activity (outside school): Students’ or teachers’ personal navigation into informal sources of science learning (like newspapers and magazines, TV, the Internet, DVD, CD, etc.), in any place outside school (e.g. in their homes) or even private (or family) visits to a public institution; 3. As educational material inside school: Teachers may design educational material (aiming to support the objectives of the school science curricula) with the inclusion of science material coming from all sorts of informal sources of science learning (like those mentioned above as well as newspapers and the magazines, TV, the Internet, DVD, CD, DVD-ROM, CD-ROM etc.), to be used in the science classrooms (inside school). (Further down the above three categories are analytically discussed)

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Dimension B: System of Taxonomy According to the Way a Science “Text” is Made up and the Codes Used (Genre) In this paper we make use of to the term “text” to refer to an overall presentation of a thematic entity, which may combine several modes: written passages, sound (commentary voices or music), static or moving pictures, models, etc., in order to convey the intended meanings effectively. This kind of “text” seems suited to science, because the communication code of science is multimodal (Lemke 1998). It facilitates the learning of science, because if the representation of a concept is multimodal, its meaning is “multiplied” (Lemke 1998). The genre of each “text” refers to the different approaches conveying the scientific content. Each approach imposes the kind of “code” used: 1) Fiction: where the scientific information or a scientific event is presented as part of a fictional narration (which is usually characterized by plot, development of the characters, conflicts, etc.) 2) Documentary: where the scientific information or the scientific events are presented as documents, which could have the form of a simple lining up of relevant scientific information (e.g. presenting the case of a factory polluting the environment: interviews with opposing sides, scientific data, historical data etc.) or the form of a recording of a specific scientific event (e.g. recording of a moon eclipse or of a laboratory experiment), etc. 3) Educational instruction on a given subject (e.g. educational television): where according to a chosen educational design, attention is being focused on some concepts, questions are being put forward, data presented are discussed and explained in a particular frame of reference, explanatory graphics are being used, etc. Often, these three approaches are mutually exclusive. But, in multimodal “texts”, all three approaches could be components of the same complex “text”.

Dimension C: System of Taxonomy According to the Kind of Mode Used in the Science “Text” Depending on the particular characteristics of the medium used, we may identify five (5) basic modes of presenting a “text”: 1. The printed “text”, which combines written text and static pictures. It could have several functions: 1.1. the work sheets (like the ones either distributed by the museums or created by teachers, so that the students are able to navigate the museum’s exhibits and follow specific routes); 1.2. the popularized science articles (like the ones which refer to several contemporary science issues and appear every day in the press);

The Challenge of Using the Multimodal Aspects of Informal Sources of Science… 159 1.3. the books of popularized science, which either use narration (like the ones that narrate science stories usually written by known scientists) or just present scientific information (like science encyclopedias or books of general knowledge); 1.4. the science comics (like the ones referring to “crucial episodes” of science, or the biographies of scientists appearing in the press or in popularized science books). 2. The cinematic “text”, which is linear, is based on moving pictures and sounds and can also have several functions: 2.1. Science films screened in a large movie theater (public or in a museum, planetarium etc.); 2.2. Science films in a compact medium (DVD, VCD etc.) for private viewing; 2.3. Science films broadcast by T.V. channels. As in all above cases we may refer to one specific film, we should mainly distinguish between examples of dimension A (environment and conditions of viewing). Also, depending on their genre (or combination of genres) such science films may: • • •

narrate a science story (like a movie about Einstein’s life including dramatized recreation of moments from his life), where the code of fiction mostly prevails; present a scientific event in the form of a documentary including actual recordings, virtual or animated representations, interviews etc. (like a wild life documentary); elaborate in teaching specific concepts, possibly using all the above codes to support a chosen teaching design (like an educational TV program)

3. The theatrical “text”, which can be based on: 3.1. a dramatized recreation of an event taken from the history of science, presented by actors – heroes of the story (e.g. Bertold Brecht’s Galileo, Michael Frein’s Kopenhagen, or Piter Parnel’s QED or What Mr Feynman has proved), 3.2. a theatrical game (drama playing) designed to teach a science issue. 4. The stage-designed “text” (including settings, scenery, posters, models, installations, etc.), which is based on the representation of “snapshots” (tableau vivant) taken from the natural world (e.g. representations of several ecosystems, or representations of several scenes from the history of life on earth or the representation of a specific “century” of the planet earth, etc.). It consists of special settings, which can combine models of objects or species of the natural world, special lightings, music, authentic sounds (e.g. an animal’s cry), oral narration (or written captions). All these aim to provide the proper information in order to orient and organize the visitor’s perception of the exhibits. The contemporary science museums very often use this kind of “text”. 5. The digital “text”, which combines the written text, the pictures (static or moving) and sound (e.g. oral speech, imitation of sounds from the physical world, accompanied music etc.). It demands from its viewers-visitors skills of a non linear way of “reading”, since it offers them the ability to intervene and follow any route they want. The Internet allows greater interactivity, while the DVD-ROM and the CD-ROM allow free navigation through specific and usually predetermined routes.

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Regarding these three dimensions (systems of taxonomy), when discussing any particular example of an informal source of science learning, we may choose to analyze the particular environment determining the educational activity taking place and move on to the genre characterizing the specific “text” and then define the kind of “text” mode used or, vice versa, we may discuss a particular “text” and further analyze its genre and its impact according to the particular environment that it is implemented, as indicated in the hypothetical examples below.

Hypothetical example (1) A DVD of a science documentary about volcanoes, shown in the classroom: • • •

Mode of text: Cinematic text (linear navigation). Genre: Documentary presentation of data and educational instruction. Environment: Video projection in the classroom and use of a work sheet. Students work in small groups and have access to facilities of non-linear re-navigation of the DVD.

Hypothetical Example (2) A school visit in a natural sciences museum • • •

Environment: Organized school visit to a collection of exhibits. Students are allowed to move around freely. No preparation. Genre: Combination of documentary presentation of data, fictional representations and educational instruction. Mode of text: Stage-designed text including installations and explanatory, written captions.

Hypothetical example (3) A DVD-ROM used in the classroom • • •

Mode of text: Multimodal, digital text (non-linear navigation). Genre: Documentary presentation of science information, educational instruction. Environment: Students in groups of 5-6, are allowed to navigate freely for an hour.

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FURTHER DISCUSSION OF THE CHARACTERISTICS RELATED TO DIFFERENT ENVIRONMENTS (DIMENSION A) As mentioned earlier, the way different educational environments may characterize informal sources of science learning, is of particular interest, as it suggests the corresponding ways through which formal education may benefit from these sources, as well as suggesting the anticipated teacher-role, in order to allow teachers to create their own educational context according to their curriculum objectives and methodological priorities.

1. Organized out-of-School Visits (Informal Sources of Learning which Present Several Kinds of “Exhibits”) Searching in the relevant bibliography, we realized that a lot of research has been done about museums and science centers. Several researchers have studied the way learning happens in these places and have suggested ways of using it in formal education (Griffin 2004, Rennie 2007). Furthermore, several other places of informal science learning are included in this category, which we will attempt to present.

The Kinds of Sources • • •

science museums, science and technology centers, museums of natural history, museums of ecology etc. ; aquariums, zoos, botanic gardens, sea parks, etc; planetariums, cinemas (where science documentaries are screened) and theaters (where performances about the nature of science or the history of science are enacted).

Their Basic Characteristics These sources: a) are of high status (as far as the validity of knowledge is concerned); b) have their own premises which schools usually visit to attend special educational programs. The whole environment is mounted in such a way (the stage design, the technological means, the conditions of a film projection or a navigation through a museum’s exhibits, etc.) that a special dynamic is developed, which imposes and determines the “message” (the kind of knowledge) transmitted by that source. Thus, the aesthetics and the architecture of the building, the presence of gift shops and recreation areas often convey an overall message and attitude; c) constitute the meeting point of culture – science – society (variety of visitors), and for that reason usually offer rich experiences of knowledge in several regions; d) demand a permanent – or in some cases occasional – connection with formal education.

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Their Exhibits Today, science museums of all kinds avoid the simple juxtaposition of exhibits. Their “exhibits” usually are designed either as a synthesis of several images of the natural world (e.g. natural history museums), or as a synthesis of the representations of the history of the natural world as deduced by science (e.g. planetariums), or finally as activities for the exploration of science laws and technological applications (e.g. science centers). Thus, they create virtual environments (e.g. planetariums) or adopt physical environments (e.g. sea parks) to represent snapshots of the physical world, or to represent the dynamic evolution of the physical world in billions, millions or thousands of years, or to allow the free development of an ecosystem in a region. In all cases, the experiences, direct or indirect, gained by the students could be catalytic for their understanding of the physical world’s function. Allen (2004) points out the fact that in museums and in science centers, the cognitive load carried out by very many of the interesting, but -at the same time- mentally demanding “exhibits”, make it difficult for their “visitors/students” to handle this load. Therefore, one of the main challenges these sources have to face is how to design interactive exhibits which will stimulate their visitors’ every step in a meaningful way while developing their overall understanding of the relevant scientific concepts. Their Target Population Their main target population is students of primary and secondary education, which make prescheduled visits to these places mainly organized by their schools. The population of adults (parents, grandparents, other family members), who usually accompany students during the weekends or during the school vacations could be considered as a secondary target. Their Goal Their main goal is the development of educational programs for several science issues. Thus, their main function is educational (the public understanding of science) in an imaginative or enjoyable way, with a further aim of their possible intervention in formal education for issues concerned with their particular science interest and specialties. The Syntax of their “Text” Every one of these sources has its own syntax code and modes for shaping its “text”. Thus, museums and science centers are based on the printed and stage designed “text”, planetariums and movies use the cinematic “text” and the theaters use the theatrical “text”. Their Strong Points Their contribution to formal education could be proved substantial for several reasons, some of which are: a) The use of alternative ways of educating people in science. Students (and teachers) are educated in a way complementary to formal education in regions which either are not included in the thematic entities of the school science curriculum or are only partially covered by it (Michie 1998). The environment of these informal sources makes easy the enrichment of the school experience, by using several and alternative ways of

The Challenge of Using the Multimodal Aspects of Informal Sources of Science… 163 representing, studying and approaching the physical world. Their “exhibits” use models, images and representations of the physical world extending students’ perceptions (virtual experience) far away from the limits of science textbooks or other educational materials. b) Learning in an interesting and pleasant environment Students, who visit these special informal sources, usually participate in their activities with great interest and pleasure. This happens -among other things- because their participation is not obligatory, there is not a certain amount of knowledge which has to be “covered”, and not even any particular thing they should necessarily learn. Also, their knowledge, if acquired there, is not going to be evaluated. They can freely choose which “exhibit” to focus on and to what degree to pay attention to. Thus, knowledge (if any) evolves freely. Students become familiar with the physical world and its phenomena mainly through their experiences gained when interacting with “objects” of the physical world or through images representing the development of physical phenomena and less through printed text (which demands mental labor to be comprehended). The interactive activities developed in these sources are usually novel, amusing, interesting and full of heuristics, mobilizing all senses (Allen 2004). Also, the spaces where these exhibits and performances take place are usually well designed, interesting from an aesthetic point of view and attractive to children.

Their Weak Points Despite their potential, these sources have some organic weaknesses, like: a) Students’ participation In some of these sources, students are supposed to be actively involved responding to the interactive “exhibits”, while in some others they may only passively attend the scientific “happenings”. Indeed, science centers (like museums of technology), invite the students’ active participation through interaction with their exhibits. Their whole structure is determined by their purpose to represent crucial “snapshots” (crucial experiments or crucial technological constructions) from the history of science. They have been designed in such way as to stimulate the interactivity of the visitor/students with their exhibits. They consider that if this purpose could be achieved, students may realize the creativity of science and the way it develops. Among the various science centers there are some where this goal is achieved, while in some others this objective is difficult, because it would demand a much more systematic effort (Allen 2004). An alternative suggestion would be to engage students-visitors not in the full number of a museum’s exhibits but in a specific educational program involving a limited number of exhibits, focusing the effort and interest of students on them (Rennie 2007). Such programs demand the design and arrangement of the exhibits in a separate set-up for presenting a thematic entity. Thus, students could construct meanings for the phenomena referring to this thematic entity. On the other hand, a phenomenon which often characterizes the practices of the museums is that the design and construction of impressive exhibits aims primarily at the amusement and astonishment of their visitors. In that case, the students’ interactivity with the exhibits is confined just to the limits of a game, which is not accompanied by the necessary mental elaboration and profound cognitive functions about the concepts and phenomena of science.

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As relevant researches show, our attention should be focused on the fact that the interactivity by itself does not guarantee science learning (Allen et al 2004). Another problem is associated with the design of interactive “exhibits” which should take into account that each “exhibit” is addressed to both students and adults, who however follow different paths of learning and are at different stages of knowledge (Borum et al 1997). In some other informal sources, students are passive receivers-spectators: •

•

either of exhibits (alive or model) functioning as a gallery of the physical world which satisfy the curiosity of the public, as happens in the natural history museums or in the zoos and botanic gardens; or of multimedia presentations of scientific events (e.g. films representing the history of the universe or a future catastrophe of the earth and the solar system), as happens in the planetariums where students’ attention is riveted and confined in an aweinspiring virtual reality environment. In this case, students can not challenge or further elaborate the information given. Virtual experiences are imposed on them in a catalytic way and very rarely are checked or reexamined.

b) The educational design. As has been already mentioned, the main aim of all the above sources -to a certain degree- is students’ education in their specialized field. They imply their educational model when designing educational programs which are based on: • •

•

the choice/construction of particular exhibits, in the way they have staged the surrounding space (e.g. Natural History Museums) or they have filmed, animated and presented the representations shown (e.g. performances in the planetariums), and in the way they have organized the educational activities intended to be performed mainly by the students.

But, often, the study and contextual analysis of the surrounding space staged, of the educational programs designed, of the representations shown, of the activities offered and of the educational materials, reveal an empirical approach. Rarely, if ever, are opportunities for the negotiation of scientific knowledge anticipated as high priorities of the offered activities. As Rennie (2007) points out a major weakness characterizing the science center exhibits is that they present science as a final product and not as the result of an ongoing process. Thus, it seems that despite their intentions and statements, the effort to impress often precedes and determines the educational design and the pursuit of fascinating experiences imposes on and downgrades any experiences related to the scientific investigation. Most of these sources are not structuring their educational interventions according to the constructivist theories of knowledge. As Hein (1995) states, there are four types of museums according to the way they approach knowledge: the traditional, the behaviorist, the discovery and the constructivist. Research shows that most of the museums belong to the first two types, some others make an effort to adopt the discovery way of learning (through the interactive exhibits), while very few of them follow the constructivist model. Griffin (2004) suggests that

The Challenge of Using the Multimodal Aspects of Informal Sources of Science… 165 informal educational institutions have to update their educational methodology to match that adopted by the school science curriculum. c) Teachers’ responses Rennie (2007) points out that although many science centers, museums etc. take special care to develop educational programs related to schools science curricula, teachers appear to be uninformed about them. A number of them actually view out-of-school visits as occasions for amusement and relaxation from the demanding everyday school program. Research also shows that teachers are often reluctant to prepare their students “visit” to most of the informal sources of science learning (Griffin 2004). The most important reasons given are: lack of time, difficulties met in organizing a school visit (economical demands, etc.), lack of knowledge about the scientific matters these sources deal with (Tofield et al. 2003). As a final assessment, and despite the fact that it is not always feasible to attempt generalizations for such a broad category of informal sources of science learning, we could point out that the contribution of these sources is important for formal education because they help students: • • •

to develop a positive attitude towards science, to achieve sensory motor and emotional skills, and to socialize (Kariotoglou and Papasotiriou 1999).

On the other hand, it seems that they are lacking in elaboration of their cognitive targets (which concepts are chosen and why, how the selected concepts are organized and presented, how students will be involved, which is the adopted model of learning). Such questions are related to the way scientific knowledge is negotiated and is transformed into school or public knowledge, as well as to the epistemological framework that is suggested (what is the nature of science).

Formal Education and the Utilization of These Sources Formal education could use the opportunity offered to utilize the positive aspects of the above mentioned sources (students’ interest, alternative learning environment, etc.) and maximize the educational benefit coming from them, if the teachers prepare their school “visit” to them very carefully. Before the visit of their students, teachers should “visit” the premises/space of the informal sources of science learning and be informed about the content and the kind of the educational programs or other educational material offered. Then, they have to study them and if they have been persuaded that these programs and materials address the mental age of their students and could function complementarily to the concepts and objectives of the curriculum, they can proceed to the preparation of the visit. A few days before the visit, they could then introduce their students to the content of the informal source they are going to visit. They could explore their students’ preexisting knowledge in respect to the specific content and elicit their possible alternative conceptions about it. They may also need to develop appropriate background knowledge, in order for their students to be able to join constructively in the visit planned. It should be noted that in all preparatory schoolwork regarding the design of routes and the selection of key-exhibits

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around which the student activities will be directed, students may be invited to participate in planning their own visit, thus having an opportunity to actively express their personal priorities and interests (Griffin 2004). Teachers may also design relevant worksheets, which will act in three ways: • • •

as a students’ navigator to the routes within the new space visited, as a guide for them, focusing on certain elements (concepts, information) of the program or material offered and as a tool encouraging small group activities within the museum venues. Worksheets may ask students to work in small groups and experience the exhibits by discussing and negotiating their group’s final answers to a number of open questions. Small groups may also be encouraged (depending on the museum’s regulations and restrictions) to record their observations by taking digital photos or by brief short interviews with other visitors, asking pre-designed questions.

Thus, students will not be “lost” in the chaos of stimuli and information met with and their “visit” will be fruitful. Back at school, after the “fireworks” are over, is the time for what can be characterized as the most creative educational part of the whole “visit experience”. Post visit activities may involve discussions of the different small groups’ answers on their worksheets, drawing of the most interesting “exhibits”, further processing and elaboration on any photos and interviews recorded by the groups of students, evaluation of the visit experience etc. In this context, the teacher can reveal an altogether new frame of reference of the visit experience relating impressions, bits of information and additional explanatory comments to the relevant aspects of the school curriculum. The above suggestions may indeed describe an ideal situation of an organized school visit. But, as the research shows only a small percentage of teachers adopt a methodology for out-of-school visits integrating it in their work back at school (Anderson et al. 2000, Griffin and Symington 1997).

2. Students’/Teachers’ Personal Navigation in Several Science Sources Outside School These sources are not yet being comprehensively studied, because the literature is not rich in researches on this subject (Rennie et al. 2003, Rennie 2007). This happens because they are reached in a personal way and in several places, making difficult to create a proper research design. Thus, our remarks are not conclusive since they are based on preliminary studies only. Among these sources the more studied ones are those referring to press science (Elliott 2006, Jarman and McCluney 2002, Halkia and Mantzouridis 2005, Christidou et al 2004, Halkia 2003, Philips et al 1999) which is considered decisive for citizens’ scientific literacy (Parkinson et al 2004, Korpan et al 1997).

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The Kinds of Sources • •

•

• •

Magazines and newspapers (supplements) of popularized science addressed to the public (press science); Books of popularized science: science encyclopedias or books with rich visual material presenting information about a specific science matter (e.g. The human body) or biographies of scientists or books which use fiction to popularize and present theories of modern science; The TV (when presenting science documentaries or other science programs), the DVD or the DVD-ROM and the VCD or the CD-ROM, which are distributed by the newspapers and magazines or can be rented from video clubs (mainly science documentary films); The Internet (WebPages referring to science matters). Private visits to a museum.

Their Basic Characteristics As Halkia (2006) points out, these sources: a) are not necessarily considered as high status sources, as far as the validity of knowledge offered is concerned; b) are mostly used in a private context (in each student’s home environment, in the internet cafés, or even in the school library); c) usually choose science issues which are related to current publicity (this happens especially to press science articles and to the TV programs); d) are met with the students occasionally or -in most cases- accidentally (students usually by accident watch a science documentary when zapping from channel to channel or riffle through a science magazine or watch a science DVD which was distributed with their parent’s newspaper); e) do not require systematic or disciplined navigation or attention (e.g. the student could run through them quickly, or just concentrate on only one picture, or read only one passage, or watch only one part of a TV program or a documentary on the TV, or navigate to a part of a path in a DVD-ROM, etc.); f) usually are addressed to the general public and not especially to students (thus they are referring to levels of knowledge which are not necessarily compatible with children’s cognitive skills and level of knowledge); g) contribute to citizens’ scientific literacy (Parkinson and Adendorff 2004, Korpan et al 1997); h) can be connected with formal education through students’ homework and projects.

Their Target Population Their main target population is every citizen (adults mostly), without any special knowledge on the science issues these sources deal with. Thus, all teachers belong in this target population, often declaring that these sources contribute to their self education (Halkia 2003). Also all students may be part of their target population, usually approaching the content of these sources with interest (Halkia and Mantzouridis 2005).

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Their Goal Their main goal is to inform citizens about important science issues and critical “episodes” of science (e.g. DVD’s about the solar system, about the human body etc. or TV documentaries about Einstein’s life), also about current science issues (e.g. popularized press science articles and TV documentaries about bird flu, human cloning, the exploration of space, etc.). Their primary goal seems to be the public understanding of science and the development of the scientific literacy of the citizens. The Syntax of their “Text” Every one of these sources has its own mode and syntax code for shaping its “text”. Thus, press science articles and popularized science books are based on the printed “text”, science documentary films (on TV, DVD) are based on the cinematic “text”, while the Internet, the DVD-ROMs and the CD-ROMs are based on the digital “text”. Their Strong Points Formal education seems helpless in comparison with the “charm” and fascination characterizing informal sources, which seem to be very powerful sources of mediated experiences for concepts and phenomena of the physical world. Every day, students and teachers bring to school -through these sources- a plethora of information and knowledge on a lot of science issues. The power of these sources is based on the fact that: a) they provide students with a broad range of audiovisual information about the phenomena of the physical world; b) students approach them on their own initiative (free choice) often in a private context (Wellington 1991); c) students can navigate them in a non linear way (even when these sources anticipate a linear way of presenting the science information), following their personal routes (e.g. by focusing their attention on some minor elements, or starting from the end towards the beginning of the “text”, etc.); d) students feel that the content of these sources “makes sense” to them (Halkia 2003); e) students can be informed about issues of current scientific achievements, beyond the ones studied in their science classrooms (Halkia 2003); f) students are motivated to participate in social discourse about current science issues (Hand 1999); g) students consider that the kind of “language” used (with a lot of metaphors and analogies, without difficult terms, avoiding mathematical formalism, making extensive use of narration techniques, etc.) is much more interesting and friendly than the one used in their science textbooks (Halkia and Mantzouridis 2005); h) students, in some cases (e.g. film documentaries on Biology), are informed in a thorough manner about issues that the school curriculum chooses not to deal with. Such issues may often convey a positive attitude towards science. Research also shows that although students often bring into the classroom science knowledge gained by the mediation of mass media or the Internet, teachers rarely build on this knowledge (Dhingra 2003, Mayoh and Knutton 1997).

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Their Weak Points The most powerful and -at the same time- the weakest point of this kind of source comes from the fact that students reach them freely, with the minimum of supervised control. Thus, some of their weak points are the following: a) knowledge acquired can often be fragmented and the science issues covered accidental (due to the demands of current publicity and the flexibility of the medium) (Halkia 2006); b) conditions of personal leisure (possibility of frequent interruptions, noises, etc.) reinforce an attitude of “browsing” (superficial navigation) rather than thorough study of their content (Halkia 2006); c) these sources often promote information and statements, the validity of which is not guaranteed or supported by research (Laugksh 2000). As a result, students may construct alternative conceptions about the science issues they present. Often, science fiction productions tend to blur the limits between facts adopted and assumptions made for the sake of their story (Rennie 2007). Also, in expensive productions recreating science history events, “scientific uncertainty” characterizing most scientific breakthroughs is usually overlooked for the sake of dramatic emphasis and emotional excitement. In all such cases, the role of the teacher who can bring these issues to classroom discussions can be crucial. Students may be asked to point out specific examples in the film presented, differentiating between scientific and fictional elements (Allday 2003); d) such sources often adopt an anthropocentric point of view of physical phenomena, attributing social values to them (as happens in some documentaries e.g. The Emperor’s Journey, or in some TV programs, or in some popularized science books). But, this anthropocentric point of view confronts the scientific view and raises severe obstacles when formal education tries to develop students’ scientific attitude towards the study of the physical world (Halkia 2006); e) In many popularized presentations (science documentary films, comics, or books) addressed to young students, anthropomorphic analogies are adopted (e.g. presenting the particles of the micro cosmos or the blood cells as small human beings) often leading to alternative conceptions. The above mentioned weaknesses are often responsible on one hand for the construction of students’ alternative conceptions of the phenomena of the physical world, especially of those for which they can not have direct experience and on the other for the development of students’ illusion of having a solid knowledge of physical phenomena, which actually consists of loosely connected fragments of information.

Formal Education and the Utilization of These Sources The above mentioned sources of knowledge can be used creatively in the framework of formal education if the teacher suggests students’ navigation within them for a specific purpose (e.g. for projects, homework), which often fulfills the needs of the science curriculum (Wellington 1991). In this case, informal sources of science knowledge can constitute a bank of multimodal educational material to which teachers may refer students for their projects and school research (Jarman and McCluney 2005). Thus the existing context of science curricula

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provides the necessary “point of view” to allow students relate fragments of information from informal sources to a systematic understanding of physical phenomena. For this to happen, the specifications of the proposed school project should be clearly stated, and the inquiry questions should be novel and attractive. Actually these suggestions imply an active role by the teacher who is asked to fit such sources into the context of formal education, thus leading to the next category of learning environment.

3. Use of Informal Sources of Science Learning by the Teachers, Within Their Everyday Classroom Practice In this thematic unit we are going to present the way informal sources of science learning can be the raw material for teachers to design and develop their own educational material in accomplishment or support of the objectives of the science curriculum. Apart from the use of the Internet in science classrooms, the use of the rest of informal sources for the development of educational material -though often suggested by the teachers- has not yet been comprehensively studied.

The Kinds of Sources • • • • •

press science articles (passages and pictures); popularized science books (passages and pictures); science documentary films (as a whole or parts of them) on TV, or on DVDs and VCDs (often distributed by the press); educational kits, distributed to schools by science museums, which have been designed according to the exhibits and the aims of the museums; Websites and downloaded software, DVD-ROMs and CD- ROMs which present science issues.

Their Basic Characteristics The sources of this third category, while seeming to have the same characteristics as those of the second category, are very much differentiated in the way they are used by formal education. In other words we now refer not to the items themselves (books DVDs, exhibits etc.) but to a teaching plan which embraces these items into an educational process. Thus: a) they constitute a prime source of teachers’ self education in science matters (Halkia 2006); b) their use in the classroom may be the results of teachers’ own evaluation, who choose them according to their teaching priorities, imposing specific criteria of selection; c) as a result of their evaluation by the teachers, they gain educational status and function in a context of formal education; d) they provide teachers with an opportunity to transform public knowledge into school knowledge (Mantzouridis et al. 2005, Jarman and McCluney 2002) according to their students’ mental abilities, cognitive and emotional needs;

The Challenge of Using the Multimodal Aspects of Informal Sources of Science… 171 e) they provide a supplementary support to the science curriculum (Dimopoulos Koulaidis 2003); f) they update the content and examples of the science curriculum, because very often they present current issues of science in a simple and imaginative way; g) they offer opportunities to focus students’ attention on science issues, which are currently predominant subjects of social discourse, thus allowing them to participate in this dialogue (Millar and Osborn 1998); h) they motivate students, because in the presentation of their content they may combine several sources with different kinds of “texts”, which enhance students’ ability to understand science concepts.

Their Target Population In as far as these informal sources of science learning have been chosen by the teachers and their content has been transformed by them according to the needs of their students, their main target population is students of a specific school grade. Their Goal Their main goal is to enrich and update the science curriculum with science issues which are on the cutting edge of social discourse about science (e.g. popularized press science articles and TV documentaries about bird flu, human cloning, the exploration of space, etc.). The Syntax of Their “Text” Since teachers produce their educational material with the combination of several kinds of informal sources, it is reasonable to assume that this material makes use of several types of “texts”. These “texts” have been commented on, above (Dimension C). Their Strong Points The diversity of forms which characterizes these sources, as well as their multimodality (in the way they present science concepts) make them a valuable and rich source from which teachers can draw ideas for designing educational material for their everyday science classrooms (Jarman and McCluney 2005, Freudenrich 2000). Thus, their strong points are that: a) They offer multimodal ways of presenting a physical phenomenon, thus enhancing scientific understanding (Halkia 2003, Lemke 1998); b) They use a communication code which seems to be very appealing to students (Halkia and Mantzouridis 2005, Freudenrich 2000); c) They use an emotional - “poetic” language, with a lot of metaphors and analogies, to introduce difficult scientific concepts. This language is highly appreciated by students (Halkia et al. 2005, Christidou et al 2004). d) Teachers realize the fact that the narrative style of science “stories” and the teaching tools used (metaphors, analogies, models, etc.) catch students’ interest and facilitate the understanding of scientific concepts and accordingly affect their everyday teaching practices (Halkia 2003, Carson 2002, Lemke 1990).

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Their Weak Points The limited ability of many programs of study to accept and embody the material of these sources in everyday school practice, the lack of teachers’ training in the transformation of public knowledge into school knowledge and the lack of teachers’ time for designing their own educational material seem to be the main weaknesses of the sources belonging in this category. Thus, the main weak points of these sources are (Halkia 2006): a) The rigidity of the educational system in embracing the informal sources of learning on a regular basis. This rigidity could make teachers passive and unwilling to design and produce innovative material. b) The overloaded science curriculum sometimes combined with the use of a particular textbook prevents the use of educational material from sources which are not under the educational authorities’ control. c) They are often used as they are found, without any educational transformation into school knowledge, which results in students finding it difficult to handle their content. d) The lack of adequate teacher training in the educationally creative use of such sources (e.g. fitting impressively presented information into a context of discourse and constructivist teaching). e) The lack of formal education provision permitting the regular use of such sources (e.g. limited funding for multiple photocopying or transferring into transparencies, limited facilities for scanning and video projection available in every classroom, limited funding for the purchase of popularized science books, lack of free computer access for navigation on the Internet in the context of everyday lesson, etc.). f) The vast quantity of available information discourages teachers from developing proper initiatives, for fear of unreliable data.

Formal Education and the Utilization of These Sources The informal sources of science learning could be creatively used in the context of everyday school practice only if teachers are free to design their own educational material according to the educational needs of their students and the demands of society (which is not always the case for the programs of study in many countries). This would help towards the emancipation of teachers, so that they could intervene innovatively in educational practice. For example, lesson plans could be designed to utilize informal sources in a number of different ways by developing different teaching strategies:

The Challenge of Using the Multimodal Aspects of Informal Sources of Science… 173 • • •

Provide a work sheet which invites students to focus on particular questions and suggest their own ideas. Ask students to organize the information provided in a summary or design a conceptual map or develop short fictional snapshots in the form of a comic strip, etc. Consider some of the concepts presented as starting points for further collection of data and research (investigation into reference books and the Internet or distribution of relevant questionnaires to their social environment) etc.

Such teaching strategies may focus only on some parts of the informal source, may suggest a non-linear navigation of a linear text, may direct students to individual or group activity, may invite students to further develop or critically oppose the informal source content etc. But for such initiatives to be developed, teachers need a proper training in the ways of evaluating the informal sources, of selecting material from them and of designing their own material. On the other hand, students, as future citizens, could gain a lot from their interactivity with the multimodal way these sources use when presenting science issues, as well as with a non-linear way of “reading” them. Thus, students could develop skills in navigating contemporary forms of “texts”. Those skills, today, are essential for a citizen to be considered scientifically literate. Perhaps the most crucial latent aspect of all experiences mediated by informal sources of science learning is the fact that realism in their presentation usually hinders us from realizing that such experiences are indirect and differ considerably from the corresponding direct ones. Experiencing a volcano through the vivid presentation of a National Geographic documentary should never be confused with the relevant field experience. Even experiencing a wild animal behind the zoo bars can be a very useful educational stimulus, but should not be confused with the kind of impressions gained in a field observation. We do not maintain that mediated and reconstructed experiences are to be considered as deceiving or disorienting. However, we feel that a wise strategy elaborating on such experiences would require from the teachers, during post visit discussions, to raise among their students questions like: “What would be some significant differences if we were to observe the actual volcano activity from a nearby position?”, “How would this animal be different if we were observing it in its own natural environment? (try to consider differences related to as many senses as possible)”. And vice-versa: “What possible aspects of our understanding of these exhibits might have been lost if we were experiencing them directly in their own original context?” Such elaboration would contribute to an overall critical perception of science by the students.

CONCLUSION The above attempt to map the informal sources of science learning had the intention of showing the wealth of sources of science knowledge, much of which formal education seems to underestimate (or even ignore) making a very limited use of. It is very important to stress once more that the contribution of these sources to citizens’ scientific literacy is considered

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decisive today. Formal education seems to lose its prime and undisputable role for citizens’ education in science. This is especially true, when realizing the low levels of scientific literacy many countries achieve (e.g. PISA, TIMMS), which reveal the dead ends of their formal education. It seems that formal education should reexamine its role, abandon linear and one-way didactic practices and instead look for alternative (non-linear and multimodal) practices. This attempt could be supported effectively by the use of informal sources of science learning. The variety of their forms, the multimodality of their addressing code, their science issues which are of the current publicity and the possibility of allowing the non-linear “reading” of their content are elements which are considered attractive to students. Maybe this is why students consider that science learning, through these sources, “makes sense” to them (Halkia and Mantzouridis 2005). In the present work an attempt has been made to study in part and to reveal the very many informal sources of science learning. It is interesting to notice that the use of the term “informal source” almost exclusively for the kind of learning occurring through the museums, has downgraded the importance of other informal sources in science learning. But, these informal sources have a great influence on everyday school practice. Students are exposed in many more ways to informal sources, such as science documentary films on TV, DVDs and VCDs, popularized science articles appearing in the press, popularized science books, or the science Websites on the Internet, than they are to those in science museums. This suggests that in the coming years a lot of research should be carried out to investigate the way these sources contribute to science learning and interact with formal education. In any case, the constructive and intriguing use of every kind of informal sources in the context of formal education depends upon the teachers’ own creativity and deeper understanding of the science curriculum objectives. Teaching qualities such as actively combining multimodal educational stimuli with science curriculum requirements, may suggest the need for a systematic teacher training, the content and specific character of which might constitute the object of further research.

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The Challenge of Using the Multimodal Aspects of Informal Sources of Science… 177 Wellington J. (1991). Newspaper science, school science: friends or enemies? International Journal of Science Education, 13(4), 363-372.

In: Science Education Issues and Developments Editor: C. L. Petroselli, pp. 179-199

ISBN: 978-1-60021-950-4 © 2008 Nova Science Publishers, Inc.

Chapter 6

UNDERSTANDING SCIENTIFIC EVIDENCE AND THE DATA COLLECTION PROCESS: EXPLORATIONS OF WHY, WHO, WHEN, WHAT, AND HOW Heisawn Jeong1 and Nancy B. Songer Hallym University, South Korea University of Michigan, USA

ABSTRACT What is scientific evidence? How should scientific data be collected? These questions comprise essential components of scientific reasoning that are not well understood by students. This chapter explores conceptual challenges students face in inquiry-rich classrooms with respect to the notion of scientific evidence and the related data collection process. As students seek out evidence to support their inquiry, they are likely to ask and need to answer questions such as these: Why collect data? Who collects data? When should data be collected? What counts as scientific evidence? and How should scientific data be collected and analyzed? After examining conceptual issues involved in answering these questions, this chapter proposes that understanding what it means to collect scientific data and what scientific evidence is requires a complex understanding that involves conceptual, procedural, and epistemological knowledge.

As science education emphasizes inquiry over the acquisition of factual knowledge (Duschl and Osborne, 2002; Duschl, Schweingruber, and Shouse, 2007; Minstrell and van Zee, 2000; National Research Council, 2000), students are increasingly engaged in classroom activities that mirror some of the essential dimensions of the practices of scientists (e.g., Reiser, Tabak, Sandoval, Smith, Steinmuller, and Leone, 2001; Songer 1996, 2006). In these 1

Please send all correspondence to Heisawn Jeong at [email protected] or Department of Psychology, Hallym University, 39 Hallymdaehak-gil, Chuncheon, Gangwon-do, 200-702, Korea (South), +82-33-248-1725 (phone) and +82-33-256-3424 (fax). This chapter was written when the first author was on sabbatical leave at Rutgers University. The authors thank Cindy Hmelo-Silver and Clark Chinn for their suggestions on earlier versions of the manuscript.

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inquiry-rich environments, students often participate in the full inquiry cycle including posing their own questions, gathering and analyzing data, and evaluating evidence against hypotheses and theories. Scientific evidence refers to empirical data collected in accordance with scientific methods accepted in a given discipline. Data refers to numbers, characters, images, or outputs from devices and instruments that convert physical characteristics of the phenomena into these formats. Not all data is evidence, but data becomes evidence when it is processed and used in the context of determining the truth of a theory or hypothesis.2 The collection and the use of data is a critical aspect of scientific inquiry that needs to be appreciated by and cultivated in students. Although many studies on scientific reasoning have examined students’ understanding of evidence, these were mostly in the context of how they evaluate evidence with respect to certain beliefs and theories (Chinn and Brewer, 2001; Chinn and Malhotra, 2002b; Driver, Leach, Millar, and Scott, 1996; Kuhn, Amsel, and O’Laughlin, 1988; Sodian, Zaitchik, and Carey, 1991). Evidence evaluation is clearly one of the most central and general scientific reasoning skills, but competencies that students need to develop with respect to evidence are not limited to theory-evidence coordination. In actual scientific investigations, even before they evaluate any evidence, scientists typically wrestle with questions such as whether they need new evidence to evaluate a claim, and if they do, what kinds of evidence they need (e.g., randomized experiments or qualitative observations?), and how they would go about collecting the data (e.g., how many conditions they would have, how to operationalize variables, what kinds of instruments or devices they would use, etc.). The notion of evidentiary competence was proposed in Jeong, Songer, and Lee (2007) to refer to the concepts and reasoning skills involved in the collection, organization, and interpretation of evidence up to a point where the data can be readily used for evaluating theories and explanations. 3 In this chapter, we explore further what it means for students to understand data collection and evidence in the context of scientific inquiry by posing a set of questions that students might need to ask and answer as they seek out evidence for their hypotheses and questions in inquiry-rich classrooms. (See Table 1 for the list of questions addressed in this chapter.)

Why Collect Data? In order to become competent with the notion of scientific evidence and the data collection process, students need to understand how knowledge is justified and why evidence is needed in science in the first place. This does not necessarily mean that students should possess a refined constructivist or hypothetico-deductive epistemology that views scientific observation and experimentations as purposeful and theory-driven activities (e.g., Carey, Evans, Honda, Jay, and Unger, 1989). However, students should at least possess an ‘appreciation for evidence’ so that they could use evidence to address questions and to

2 3

Since scientific evidence is in the form of empirical data collected and used for the purpose of testing or justifying knowledge claims, the two terms are sometimes used interchangeably in this chapter. The term interpretation is used in a narrower sense of interpreting the patterns in the data itself instead of a wider sense of evaluating the data with respect to specific hypotheses. The distinction between interpretation and evaluation is only conceptual since the workings of the two stages not only overlap but also interact with each other.

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support arguments and recognize the difference between simple assertions and arguments that are backed by evidence. Table 1. Questions about Scientific Evidence and the Data Collection Process Main Questions

Sub-questions

Why collect data?

Why is data collection important in science? How is knowledge justified in science?

Who collects data?

Whose job is data collection? How is scientists’ data collection related to the development of my own thinking? How should the potential biases in data collection be reduced?

When should data be collected?

When should new data be collected? If data is collected, how is my evidence related to the existing evidence in the domain?

What counts as scientific evidence?

What are the disciplinary norms and standards for acceptable evidence in a given discipline? How is evidence different from theory and explanations? Which piece of data is relevant to answer the question?

How should scientific data be collected and analyzed? a

How should data collection be designed and planned? What kinds of strategies can be used to manipulate and control variables? How can the data collection process be better informed by the use of the right methodology and domain knowledge? How should the data be represented and recorded? How should data be organized, summarized, and represented? How should the variability in data be interpreted? How could meaningful features and patterns be identified in the data?

a

The How question consists of two parts, data collection and data analysis, and was later discussed in separate sections.

Appreciation for evidence has often been investigated in the context of students’ epistemology (Abd-El-Khalick, 2004; Abd-El-Khalick and Lederman, 2000; Carey et al., 1989; Carey and Smith, 1993; Lederman, Abd-El-Khalick, Bell, and Schwartz, 2002; Sandoval and Morrison, 2003; Smith, Maclin, Houghton, and Hennessey, 2000; Smith and Wenk, 2006). The general conclusion from this line of research, as far as appreciation for evidence is concerned, has been that students often do not demonstrate the basic appreciation for evidence in science. Using the Nature of Science Interview, Carey and her colleagues (Carey et al., 1989; Smith et al., 2000; Smith and Wenk, 2006) categorized children’s epistemology into three general levels. The key feature that differentiates these levels is the distinction between ideas and experimentation. In Level 1, students make no clear distinction

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between ideas and experiments, equating science as doing things or simple gathering of information. In Level 2, students start making a distinction between ideas and experiments and understand that the motivation for experiments is to test the idea in order to determine if it is right. In Level 3, students make a clear distinction between ideas and experiments and understand that the motivation for experiments is verification or exploration of theories. Students need to possess at least Level 2 epistemology in order to appreciate the role of evidence in justifying knowledge claims. Results indicated that many seventh graders were solidly Level 1 in their epistemological thinking (Carey et al., 1989; Carey and Smith, 1993) and even college students remained mostly at Level 2 (Smith and Wenk, 2006). Using the Views of Nature of Science Questionnaires (VNOS), Lederman and his colleagues investigated overlapping but different aspects of the nature of science concepts (Abd-El-Khalick, 2004; Abd-El-Khalick and Lederman, 2000; Lederman, 2004; Lederman et al., 2002). They examined seven interrelated facets about the nature of science, one of which was the empirical nature of science. They assessed mostly undergraduate and graduate students including pre-service teachers and coded students’ answers as naïve versus informed. Abd-El-Khalick (2004) reported that a majority of the participants (57%) indicated that science is empirical or has empirical components, but 70% of them held a naive view. AbdEl-Khalick and Lederman (2000) examined undergraduate (and some graduate) students and pre-service teachers who were enrolled in a history of science class and reported that 82% of the college participants and 40% of preservice teachers had naïve conceptions about the empirical nature of science. These investigations of students’ epistemology have revealed a great deal about the general state of students’ epistemological thinking. However, these questionnaires and interviews assessed diverse aspects of students’ epistemology in an integrated fashion. As a result, it is often difficult to evaluate where students stand with respect to specific aspects of epistemological understanding such as appreciation for evidence. Given the possibility that students’ epistemology can be quite fragmented (Sandoval and Morrison, 2003, but see also Smith and Wenk, 2006), it would be wise to examine students’ appreciation for evidence separately from the rest of the epistemological understanding. In addition, the majority of the epistemological questionnaires consist of mostly abstract questions (e.g., ‘What do you think science is all about?’, ‘What is an experiment?’) and do not attend to contexts (Elby and Hammer, 2001). It needs to be examined how students’ epistemological ideas would be used in the context of specific inquiries (Sandoval, 2003). It is still unclear how concrete contexts of inquiry influence students’ appreciation for evidence, but the general level of appreciation for empirical evidence is not at a desirable level even when questions are posed in the context of specific investigations or knowledge claims. Jeong et al. (2007) posed a set of questions and knowledge claims about atmosphere science to sixth grade students and assessed their understanding about scientific evidence and the data collection process. In one question, students were presented with a claim (i.e., it would snow the next day) along with a set of justifications. Most (92.5%) of the 40 students who participated in their study selected the correct choice (i.e., weather forecast) over justifications with no empirical grounds (e.g., dream, father said so, intuition), but only 37.8% of them demonstrated even an rudimentary understanding about the role of evidence in justifying claims or the data collection process behind the weather forecasts. Fortunately, appreciation for empirical evidence seems to develop with age. Driver et al. (1996) examined what types of warrants (if any) students use to justify their acceptance or

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rejection of theories. Students were presented with a familiar theoretical proposition (e.g., ‘the Earth is round like a ball’ and ‘an electric current flows round a simple circuit when it is switched on’) and asked to give their reasons for accepting the ideas as true or correct. Students’ warrants then were coded into three categories. The first category of answers, observation and explanation not distinguished, referred to naïve answers that did not differentiate observations and evidence. The second category of answers, warrants based on acceptance of authority, referred to answers where students accepted the beliefs based on authority either blindly (e.g., it all comes from the man who discovered electricity) or with some reasoning for why they placed trust in authority sources (e.g., they learn about it all the time). The last category of answers, warrants involving evidence, referred to answers where students accepted the beliefs on the basis of direct perceptual observation (e.g., photograph from space) or more complex reasoning about the relationship between evidence and idea (e.g., the Moon seems round as well; working electrical gadgets such as TV exist that are built on the same idea). Note that the second and the third categories of answers consist of a naïve and a sophisticated version. The results indicated that children’s warrants for beliefs became more sophisticated with age, although even at age 16, only about one third of the warrants relied on more sophisticated reasoning about evidence or authority. For the Earth statement, only 10% of 9 year olds gave responses based on the more sophisticated forms of reasoning using evidence or authority, which had risen to only 30% in 16 year olds. For the electricity statement, about 50% of 9 and 12 year olds appeared not to be able to distinguish clearly between the phenomenon of a bulb lighting and its explanation. The number of warrants using more sophisticated reasoning about evidence or authority increased with age, but even at the age of 16, only 40% of warrants fell into this category. Krajcik, Blumenfeld, Marx, Bass, Fredricks, and Soloway (1998) described how seventh grade students handled their first encounter with inquiry. Students participated in several inquiry projects with driving questions such as ‘Where does all our garbage go?’ that lasted from two to four months. In each project, students explored scientific ideas related to the driving questions (e.g., decomposition, recycling), generated sub-questions related to the driving questions (e.g., How much garbage can a worm eat in two and a half months?), planned and carried out data collection to address their questions, and presented their findings to their classmates. Data from eight target students and their interaction with other students in their groups showed that although students were capable of conducting inquiry in a ‘realworld’ classroom, they still experienced difficulty with using data in their conclusions. Even after several months of conducting their own empirical investigations, students did not use their data in their final presentation and instead relied on background knowledge to justify conclusions. These reports suggest that participation in data collection, even an extended one driven by students’ own questions, is not enough to prompt students to appreciate and use data in making conclusions of their investigations. Sandoval (2003) examined high school students’ usage of evidence as they construct explanations about evolution in a inquiry-rich curriculum. In this study, students’ explanation construction was heavily scaffolded by the curriculum and also by a software program that integrated domain-specific guidance about what to explain in a particular problem with domain-general guidance about what a good scientific explanation looked like. Students were also explicitly instructed by their teacher to use the data in their explanations before they constructed the explanations. Although these efforts appeared effective in enabling students to focus on important aspects of the explanation and encouraged an orientation toward data as

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something to be explained, results indicated that student groups often failed to cite data to support key claims of their explanations. Data citation was common for certain claims (e.g., environmental pressure and individual effects), but not for others (e.g., differential traits and selective advantage). In spite of the scaffolding and guidance these high-school students had, they still experienced difficulty with making explicit citation of evidence as a crucial component of their explanation, suggesting again that appreciation for evidence is difficult to cultivate. Although appreciation for evidence seems to develop over time and with inquiry experience and scaffolding to some extent, the role of evidence is still under- appreciated in college students. Brickhouse, Dagher, Letts, and Shipman (2000) examined the appreciation for empirical evidence in an investigative setting in college students who were not science majors. They asked students who enrolled in an astronomy course to imagine being a reporter for a local newspaper. As a reporter, they were given a press release about a new scientific finding (e.g., new evidence for life on Mars). Their task was to anticipate the reaction of scientists to the new finding. Students also had to state their own views regarding how much confidence they had that this new finding would turn out to be valid. They coded students’ answers in four categories: (1) understands evidence, (2) appreciates evidence, (3) naïve, and (4) other. Understands evidence referred to a case where students showed awareness of the need for evidence and were able to document the need with specific examples or questions. Appreciates evidence referred to a case where students expressed the need for evidence but only in a general ways. Naïve referred to cases where students did not express the need for evidence and just naively accepted the claim (e.g., ‘Based on the fact that the news release is from NASA, I think that the information is pretty solid’). Other referred to cases where students’ answers were not responsive to questions. Examination of students’ answers revealed that at the beginning of the course, 56% of them manifested naïve views on the need for the evidence, not recognizing the need for evidence, and accepting the claim as is. Fortunately, at the end of the course, which emphasized the use of evidence for justifying claims, the proportion of students with naïve views dropped to 3%, while the proportion of students who understand evidence increased to 24% from 6% at the beginning of the course. Research so far indicates that students often do not appreciate evidence in justifying knowledge claims, and even when they do, their appreciation for evidence is mostly shallow. They may understand that empirical evidence is important, but they do not understand why or how. Appreciation for evidence seems to develop with age, but even college students do not appreciate the role of evidence fully. This was true regardless of whether their understanding was examined in the context of broad and abstract epistemological probes or in the context of carrying out specific investigations. In addition, it seems that participating in data collection or being scaffolded to use evidence by teachers and computer programs was not enough to make them appreciate evidence. Just going through the activities of data collection and evaluation does not mean much if students do not understand why they do such activities or actually use evidence in evaluating claims and explanations. Inquiry activities need to be connected and aligned with the development of students’ understanding so that students can understand clearly what the purpose of the data collection is in scientific inquiry and use evidence consistently in their justifications and arguments across diverse investigations.

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Who Collects Scientific Data? Driver et al. (1996) pointed out that students could form two separate conceptions of science, one involving the real science practiced in scientific community by professional scientists and another involving school science practiced in science classrooms by themselves. Students may well construe that what they do in science classroom is unrelated to what scientists do in the laboratories, especially if what they do in their classroom is perceived as remote from the actual practices of the scientists. However, even if what students learn and do in their classrooms closely resembles the activities of scientists, students might still think that these classroom activities apply to scientists but not to themselves once they leave the schools. For example, a student in Krajcik et al. (1998) responded to interview probes that “I do not think I am going to use science much later in life unless to tell my kids about their science homework.” Such failure to see the connection between science and their own life can be partly due to the division of labor in the wider society. Verifying knowledge claims by collecting and using empirical evidence is not a common practice outside of the scientific community. Individuals neither necessarily possess the relevant knowledge and skills nor have time for it. People instead rely on scientists and researchers whose job is to generate and verify knowledge. Thus, it is not surprising that students associate evidence and data collection strongly with professional scientists but not with themselves. As a result, in spite of the emphasis put on evidence in the classrooms, students might still resort to other forms of justification and arguments in contexts outside of schools. This seems to be the case, at least with some students. As mentioned earlier in Driver et al. (1996), students’ warrants for claims relied heavily on authority. Jeong et al. (2007) also reported that when presented with a claim (e.g., arthritis gets worse before it rains), sixth grade students often opted for consulting authorities such as doctors or teachers. It was as if the students did not consider data collection as their job. Of course, consulting an authority is not necessarily a bad thing; we do not want our students to think that they should all become scientists or discredit scientists’ findings by verifying all of their assertions. A blanket distrust of authority is no more sophisticated than a blanket trust (Elby and Hammer, 2001; Jeong et al., 2007). However, although scientists may specialize in collecting data and evaluating evidence, this does not mean that students can or should be ignorant of the related concepts and processes. They need to understand how scientists carry out these tasks. There might be times when students need to question or be suspicious of scientists’ claims or when students need to undertake their own data collection because no one has addressed their question in the past, or their question is too specific to a personal situation. In addition, even when students rely on the authorities of the scientists, they need to do so knowing why scientists’ ‘telling’ is more valued than others and how scientists reached their conclusions. Students also need to perceive that data collection, whether it is carried out by scientists or by themselves, as informing the development of their own ideas and conceptions. Research from social psychology demonstrated that people reason differently about social and personal matters (e.g. Why did I fail the exam?) depending on whether things are related to others versus themselves (i.e., fundamental attribution error). It seems that science is not immune from this kind of bias. It is reported that students can engage in perfectly rational reasoning when it is about somebody else’s conceptions but cannot do so when it is about their own conceptions. Greenhoot, Semb, Colombo, and Schreiber (2004) presented college students results from a study (e.g., How does the angle of the ramp’s incline and the weight of the ball

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affect the speed of a ball rolling down a ramp?) and asked them to identify the correct conclusion. Students were asked to indicate conclusions in two conditions: experimenter conclusion and personal conclusion conditions. In the experimenter conclusion condition, students identified what the fictional experimenter should conclude about the effect of a variable, whereas in the personal conclusion condition, they identified their own conclusion about the effect of the variable. Although many participants were able to draw accurate scientific conclusions in both contexts, about one-fourth of the participants made different conclusions depending on whom they made the conclusions for. They were more likely to reason accurately when asked what a hypothetical experimenter should conclude than what they themselves would conclude. It seems that these students understood, in the abstract, how the data should be related to the validity of scientific conclusions about the phenomena. However, when the data was applied to their own conclusions, their prior beliefs about the phenomena often took precedence over the data. These results suggest that students need to understand clearly that what they learn in science classrooms, be it experimental findings or reasoning strategies, is applied to their own thinking and beliefs as well. Science instruction should also work with and from students’ own conceptions and reasoning strategies, instead of merely emphasizing the correct conceptions of science and scientific ways of thinking that might be correct but too abstract and disembodied to make a claim on students’ conceptions and ways of thinking. With respect to the question of who collects data, students also need to understand who can make a difference; that is, data collection can be influenced by who plans and collects the data. The theoretical inclination, experience, race, or gender of the data collector can unintentionally influence the planning, collection, and analysis of data, biasing them to choose a certain methodology, to observe and interpret the finding in specific ways, to look for only confirming evidence, or to ignore alternative hypotheses. Researchers like Norris (1985) emphasized that one of the characteristics of good observations is ‘not allow[ing] his or her emotions to interfere with his or her making sound judgments’ or ‘be[ing] skilled in observing the sort of thing observed and in the technique being used’. The underlying assumption has been that the data in principle can be unbiased and objective if only we try harder and become more skillful at collecting them. However, even when a simple act of observation is theory-laden and constructive in nature (Finley and Pocovi, 2000; Goldenberg, 2006), scientists cannot avoid subjectivity altogether but can only minimize its influences. This, however, does not mean that students can or should treat data collection as an arbitrary activity. Although students need to understand the inherent subjectivity in data collection, they should also understand how to deal with the issues of subjectivity so that they could transcend individual perspectives and make their observations more precise and objective.

When Should Data be Collected? The accumulation of scientific knowledge is an on-going process. In some cases, scientists and researchers have worked on problems for centuries. Even when the exact question has never been studied, there is always a wealth of information related to the problem. Science is a collaborative endeavor. Scientists rely on each other in carrying out their investigations. They read about discoveries and results made by other scientists and learn from their investigations. When a question is raised (e.g., why do things decompose?),

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scientists do not jump into data collection right away. Before they initiate their own data collection, they try to find out whether and what kinds of evidence have been provided by other scientists and how their own questions and future data collection and analysis would fit in with the existing evidence. Scientists engage in their own data collection when their question is not sufficiently answered by existing evidence, when there is a reason to believe that the existing findings might be flawed in some way, or when existing evidence provides conflicting stories. What does this practice in science mean for students’ learning? First, it suggests that students should be encouraged to research the literature as part of their inquiry. Researching the literature does not mean merely learning about past discoveries and theories in the field. It means aligning one’s question and data collection with the existing evidence and theories in the field. Performing a literature search as part of scientific inquiry has not been the norm in science classrooms yet. Even in inquiry classrooms where students pose their own questions and carry our their own data collection, students seldom research the literature to answer questions such as what kinds of data have been collected in the past and what kinds of evidence their data would provide about the question. However, in order to develop an ability to initiate and carry out scientific data collection in an authentic setting, students need the experience of engaging in the disciplinary practice of incorporating existing evidence into the development and evaluation of their own data collection. By contextualizing their data collection within existing inquiries, students not only learn about specific findings or theories, but they also learn how existing evidence relates to their own evidence and data collection efforts. Second, students should be given an opportunity to decide themselves whether and at which point they need to collect new data. This decision requires a level of understanding about the domain where students need to integrate different pieces of evidence across different studies. Students need to consider issues such as whether sufficient evidence exists for a given claim and/or whether there exists any conflicts in existing evidence before they decide whether a new data collection is warranted. Abilities to make these kinds of decisions are not easy to develop. It requires years of training even in professional scientists. However, in an age when people are bombarded daily with new scientific findings, some of which are often mutually conflicting, it is essential to develop abilities to integrate different kinds of evidence within a domain and to recognize a need for more or stronger evidence. Even though students might not be able to sift through existing evidence carefully and decide how their own evidence would fit into the existing evidence, they should be encouraged to make the decision themselves instead of being told by a teacher or some other person (e.g., advisors) that they need to collect new data. Such an experience of being in the ‘driver’s seat’ would be important in promoting students’ productive engagements in inquiry as well (Engle and Conant, 2002). There has not been much research on how students incorporate existing evidence with their own investigation. Engle and Conant (2002), however, demonstrated that under the right conditions, children can be quite adept at incorporating textual evidence (e.g., documentary evidence, anatomical evidence, lexical evidence, and evidence about the credibility of sources) within scientific arguments. They reported an episode of students’ controversy about whether a killer whale is in fact a whale or a dolphin. Although there were cases where students relied on ‘position-driven’ arguments in which the primary goal was to win and intimidate other students, many of them engaged in arguments that included rich references to

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the textual evidence and were able to make sense of often contradicting claims. Although such impressive usage of textual evidence by fifth graders would be the exception rather than the norm in most classrooms, efforts need to be made so that students would be more adept at integrating evidence from different investigations and making decisions about data collection themselves.

What Counts as Scientific Evidence? Understanding what counts as legitimate evidence is not easy. In spite of the ubiquity of scientific evidence and the high emphasis put on evidence, the notion of evidence is quite contentious even among professionals (e.g., Goldengerg, 2006; Thompson, 2007). Some of the questions raised include these: Is there a hierarchy of evidence in terms of its acceptableness, as the recent ‘evidence-based’ practices in medicine and science education suggest? Does the evidence need to be certain type (e.g., experimental data or epidemiological data) to be considered as acceptable evidence? Are the standards for evidence different between natural and social sciences? In the case of natural sciences, does the evidence need to be experimental, or can be it observational? If experiments are carried out, do they need to be a completely randomized experiments? What about quasi-experiments or design experiments? If other methods of data collection are acceptable, what are the considerations for using qualitative observations and case studies? What if the data collection process was flawed in some way so that a complete control was not achieved during experimentation (e.g., procedures were carried out by different experimenters and/or at different times of the day)? What kinds and how many violations are fatal? What if the evidence is statistical in nature? How are we to compare results that are significant but inconsistent with previous findings with results that are consistent but only marginally significant? Does the data need to be published in order to be accepted as evidence? In its most basic sense, scientific evidence refers to some kind of empirical data collected in relation to a scientific question or a hypothesis, but the form and the nature of the evidence vary widely depending on the specific questions, disciplines, and times. In natural science, controlled experiments are generally considered to be the methodology of choice. Experimentation manipulates variables to test the potential causal relations underlying them. Even within natural sciences, however, there are considerable methodological variations. There are many natural science disciplines in which experimentation is not the norm or even possible. Astronomy, geology, and evolutionary biology are all areas of study that investigate objects and processes that cannot be brought into and/or manipulated in the laboratory. In these areas of study, planned and structured data collection often replaces experiments. In some cases, disciplines rely on historical records such as fossils since the objects of their study are no longer alive. In these disciplines, detailed, often qualitative descriptions of the events and available records are taken as evidence instead of systematic recording from randomized experiments (Driver et al., 1996; Lehrer, Schauble, and Petrosino, 2001). Understanding what counts as evidence means understanding what counts as legitimate evidence in the context of specific disciplines and questions (Driver et al., 1996; Sandoval, 2003). In other words, the conceptions about what is the legitimate evidence need to be grounded in specific disciplines and questions. Students need to understand clearly what are the specific disciplinary norms and standards in a given context of inquiry (Duschl and

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Osborne, 2002). They also need to understand that the criteria for acceptable evidence can vary considerably depending on the disciplines and the questions. If students fail to understand the discipline- and question-specific nature of evidence, they may become confused about what constitutes sufficient scientific evidence and subsequently treat arguments as arbitrary or capricious (Zeidler, 1997). In order to understand what counts as evidence, one should be able to make a distinction between evidence and theory. Researchers like Kuhn et al. (1988) claim that children are unable to distinguish instances or non-instances of theory from theory itself. They presented third, sixth, and ninth grade children and adults with evidence that either confirmed or disconfirmed their existing causal theory (e.g., certain type of cereal can make you sick, the size of a tennis ball affects the quality of one’s serve) and examined how they reconciled their prior beliefs with evidence. A typical finding was that children, even adults, confuse evidence and theory. When asked to provide evidence for their proposed theories, they often provided a further statement of their theory rather than specific evidence. They also ignored or distorted evidence and adjusted theories to fit the evidence, but then were unaware of the fact that they modified their theory to fit the evidence. Kuhn et al. concluded that children were unable to distinguish between the theory itself and instances or non-instances of that theory. As mentioned previously, after assessing seventh graders’ epistemology of science, Carey et al. (1989) and Carey and Smith (1993) also reported that many seventh graders were at Level 1 in their epistemological thinking in which students make no differentiation between ideas, activities, or evidence. Level 2 thinking where they distinguish ideas and experiments started emerging in high school but still remained in an incoherent state (Sandoval and Morrison, 2003). An alternative view has been that children can distinguish causal claims from evidence in various settings (Chinn and Malhotra, 2002b; Koslowski, 1996; Sandoval, 2003; Sodian et al. 1991). Sodian et al. (1991) investigated first and second grade students and found that the majority of them were able to choose conclusive tests from inconclusive tests, elaborate the logic of such tests, and spontaneously generate empirical procedures for gathering evidence to decide between alternative hypotheses. Several factors seem to have contributed to this differing conclusion. As is the case with other abilities that show developmental changes, the age of the students seems to be a contributing factor. Sandoval (2003) examined high school students’ explanations about natural selection and reported that even when students’ claims were not warranted from data, students made causal claims about the data that were distinct from the observed data. In addition to age, it seems that factors such as the difficulty of the task and the personal commitment to theories also played a role. Students were able to distinguish evidence from theory when the task was less complex (e.g., evaluation of theory versus selection of data patterns), when the data is more perceptible (e.g., directly observable data versus second-hand reports of data), or when they were less committed to the theory personally (e.g., whether they were asked to evaluate their own theories versus theories proposed by others that they were not committed to) (Chinn and Malhotra, 2002b; Sodian et al., 1991; Zimmerman, 2000). Another important aspect of understanding what counts as evidence is to understand what kinds of data are relevant for the given questions or hypotheses. Collecting relevant data is an issue in all types of data collection including experiments, but it becomes even more difficult when data collection is more open-ended and observational. In their work with elementary students, Lehrer and Schauble (2002) observed that in the course of measuring the length of

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tobacco hornworms, many students insisted on including the name of the data collector for each piece of data and whether that individual was a girl or a boy. Similar problems persisted in the middle school students studied by Krajcik et al. (1998). The students did not always specify what they were looking for in data collection (e.g., weight and height of the decomposition columns) or what it would indicate about their question. They often included measures with which they were familiar but that were not appropriate for their purpose. Jeong et al. (2007) reported that roughly half of the sixth grade students they studied experienced difficulty with determining the relevance of data. In one question, students were provided with a set of data gathered for a question (e.g., how the distance from the equator affects the temperature during the winter) and were asked to choose the data that was unnecessary to answer the question. 50% of the students correctly identified the irrelevant data (i.e., data collected in the summer), but only 30% of them were able to give an adequate explanation about why it was irrelevant to answer the question. In another question, students were asked to provide two pieces of weather data needed to answer a question (e.g., whether they could ski). Only a little over half of their suggestions were relevant to the question. Students’ ability to determine the relevancy of data has not been the subject of much research, but it should be one of the main goals of inquiry when teaching in this information age. While collecting too little data as well as too much data is problematic, with the growth of the Internet and digital technology, the latter problem has become more prevalent. The temptation to collect all available data is understandable since more data is often equated with stronger methodology and more evidence. More experienced researchers, however, recognize that more data does not necessarily lead to stronger evidence or more understanding. In order to navigate the ever-increasing amount of data, it is important for students to learn to differentiate relevant and important data from irrelevant and tangential data. Students should be able to sift through different kinds of data and select the data that are pertinent to answer the question, weighing them in terms of their importance, credibility, and soundness of methodology used to collect them.

How Should Scientific Data be Collected? One of the major approaches in scientific reasoning literature has been to identify scientific reasoning with the acquisition of methodological concepts and processes skills such as experimentation strategies (e.g., control of variables strategy) or data collection skills (e.g., instrument usage, measurement accuracy, or record keeping). When the potential causal mechanisms of a set of variables are investigated, the use of the right experimental strategy is critical in producing unambiguous experimental results. The general finding from the literature is that children did not always identify informative experiments when provided with a set of hypothesis and potential experimental outcomes (Kuhn et al., 1988), although they could select appropriate experiments from a range of possible experiments (Samarapungavan, 1992; Sodian et al., 1991). Children were less likely to employ experimental strategies such as the control of variables strategy (CVS) or plan structures across multiple experimental comparisons. They were also less comprehensive in their search of the experimental space (Schauble, 1996; see Zimmerman, 2000, 2007 for reviews). With practice and age, children’s ability to design experiments improved. Chen and Klahr (1999) asked seven to ten year old children to make a series of paired comparisons in order to

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test the effect of particular variables (e.g., How does the length of a spring affect how far it will stretch?) in three different domains (i.e., springs, slopes, and sinking). Children explored the effect of variables in three training conditions that differed in terms of whether they received explicit instruction on CVS and whether they received systematic probe questions about why they designed the tests as they did and what they learned from the tests. The results indicated that children learned best when they were provided with explicit training on the strategy within domains along with probe questions. Children were also able to transfer the strategy to another, remote situation that they never encountered, and the ability to transfer improved with age. By middle school, many students seem to become quite sophisticated with planning and designing data collection, although they might not be always systemtic and consistent about it. Krajcik et al. (1998) reported that middle school students, even though many of them were designing their own investigations for the first time, were capable of designing quite complex multivariate designs (e.g., 3 X 4 design) that enabled them to obtain data to answer their questions. Many student groups understood the need for experimental control and sources of potential confounds. For example, one group, in setting up decomposition columns, wore gloves to put items into decomposition columns and used a standard for cutting each type of item, so that samples to be placed in the different columns would be exactly the same. Students also manifested understanding for other aspects of data collection such as types of samples to use, ways to create or obtain samples, and the construction of tables and charts for recording data. Understanding methodological concepts and skills involved in data collection is useful. It aids the development of scientific knowledge and scientific reasoning skills. Chen and Klahr (1999) reported that learning CVS facilitated conceptual change. Children who received the instruction on CVS and probe significantly improved their domain knowledge, whereas other children did not. Also, children who were ‘good experimenters’ (i.e., who designed at least 12 unconfounded tests out of a total of 16 trials) gained a more accurate understanding about the causal variables in the domain. Similarly, Schauble (1996) reported a relationship between the kinds of plan structure adopted and the percentage of valid inferences made. She studied fifth and sixth graders and noncollege adults and found that participants who used more structured plans tended to make higher percentage of valid inferences. Greenhoot et al. (2004) examined the relationship between college students’ understanding of methodological concepts such as objectivity and experimental control and their ability to change beliefs in the face of contradicting evidence. Students with a higher understanding of the concept of objectivity were more likely to draw an accurate conclusion than students with a lower understanding of the concept. In addition, students with higher objectivity score were more likely to overcome their own personal biases; that is, they were more likely to draw a personal conclusion that was consistent with what they had recommended for the hypothetical experimenter. Although these concepts and skills play a valuable role in guiding students’ thinking, the problem has been that these methodological concepts and process skills are often treated as general heuristics and strategies that can stand on their own and were taught in an abstract fashion (Chinn and Malhotra, 2002a; Windschitl, 2004). As Lehrer et al. (2001) argued, concepts like experimentation require a more complex understanding than typically assumed. Experimentation is a complex form of argument deeply embedded within the domain-specific practice of modeling, representing, and materially manipulating the world. Selecting variables for experimental manipulation presupposes an understanding for the phenomena, however

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rudimentary or biased it might be. Generating an experiment involves identifying what is theoretically central to a phenomenon, devising a situation that models it, and justifying it as a legitimate model of the phenomenon (Schauble, 2003). When these aspects of experimentation are stripped away from instruction, students can construct a distorted image of scientific reasoning and experimentation (Lehrer et al. 2001). In order to be effective, instruction of experimentation strategies and other methodological concepts and skills needs to be firmly embedded in the development of students’ domain knowledge. Domain knowledge and process skills interact in achieving richer theories and explanations in science (Koslowski, 1996; Schauble, 1996; Zimmerman, 2000). Process skills or methodological concepts alone cannot prescribe which variables need to be manipulated or controlled, how strong the treatment should be, or whether variance in outcomes is due to a change in an underlying causal variable or due to some other random fluctuations that can be ignored. Knowledge plays an essential role in informing these decisions. At the same time, however, knowledge cannot be the sole criteria or guide. If so, explanations that are inconsistent with the current belief system cannot be entertained or accepted (Chinn and Malhotra, 2002b; Schauble, 1996). Sound understanding of methodological concepts and data collection strategies are indispensible, especially when there is little domain knowledge to rely on or when their existing conceptions run counter to the underlying phenomena. Another competency students need to develop with respect to data collection is the ability to represent and record the data properly. When scientists observe the world, they do not necessarily work with raw observations. The observation is often described and transformed in representations such as detailed records, drawings, diagrams, mathematical formulae, or various kinds of outputs from instruments. These ‘inscriptions’ abstract important characteristics of the phenomena and later serve as a useful record when inferences are made about the target variables and underlying mechanisms (Lehrer, Carpenter, Schauble, and Putz, 2000; Lehrer et al., 2001). Lehrer et al. (2000) described in detail how first grade students, with aid from a very skillful teacher, handled this task. In this study, students engaged in a year-long investigation where they examined the effects of numerous variables (e.g., heat, light) on tomato decomposition. Faced with the problem of how to record the decomposition process, they initially decided to draw the tomato. As their investigation progressed, however, they struggled with the issue of how to represent and record the discoloration, discharge, and changes in turgidity that accompanied the decomposition process. Decisions on how to record and represent data (e.g., pictorial representation, numerical representation, or use of specific devices) is important since different representational formats allow different information to be preserved or discarded. Students need to understand the characteristics of different data representations and recording formats and use them appropriately in a given investigation. Yet another kind of competency is involved in successful data collection, which is an ability to carry out collection procedures faithfully and consistently. Krajcik et al. (1998) reported that even though middle school students were capable of coming up with quite impressive designs and plans for collecting data, they were not good at actually carrying them out and varied considerably in how systematic they were in following through their own plans for collecting and recording data, especially when the investigation lasted over an extended period of time. For example, students did not take accurate or precise measurements, contaminated their measurements (e.g., they did not rinse out the instrument before taking a new measurement), or copied data with an assumption that it would be the same. Students

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were also not always eager to help their partners or share the responsibilities of data collection with them, which exacerbated the problems when the collection procedure was complex and/or when the group was under the pressure of time. The researchers reported that these problems arose because students were not good at tracking their progress and maintaining a focus on the investigation. It was also because students were not good at sharing and coordinating the tasks’ requirements with their partners. Metacognitive and collaboration abilities were as essential as scientific reasoning skills per se for a successful completion of inquiries.

How Should Scientific Data be Analyzed? When data is collected, it needs to be processed and analyzed. Analysis can take various forms depending on the nature of the data, but it typically involves organizing and summarizing the data, looking for patterns in the data, representing them in an appropriate format (e.g., charts, graphs and tables), and testing for statistical significance if needed. One important concept involved in this process is the notion of variability and error.4 Error is an inherent part of any measurement, and yet with increasing variability it is harder to distinguish errors from true effects in the data. Understanding the nature of measurement errors and knowing how to interpret the variability in data is important in data analysis. Studies report that students typically do not understand that measurements can differ from one another. Most of the time, students do not plan replications and collect data with no apparent awareness of the uncertainty associated with the measurement process or the reliability of their measurements. As a result, they look confused and disturbed when they inadvertently generate replications (Schauble, 1996). Even when they repeat measurements, they see it as a matter of correcting a flawed initial measurement without understanding the inherent uncertainty related to measurement and do not know what to do with the different values obtained through replication (Lubben and Millar, 1996; Varelas, 1997). At the same time, studies also report that students possess rudimentary understanding for measurement errors and its causes. As they guide children’s exploration on potential variables affecting how far a ball rolls down a ramp, Masnick and Klahr (2003) asked second and fourth grade students questions such as ‘What would happen if the identical experiment were to be repeated?’ and ‘Can you think of some reasons why the results came out differently even though we rolled the same ball down the same ramp five times?’ Students were able to name sources of errors (e.g., the particular way the ball was released from the gate) when replication failed. The researchers concluded that elementary students understood quite a bit about errors, especially measurement and execution errors, although their understanding was not yet integrated into a coherent concept. In an effort to replicate and extend Masnick and Klahr’s findings in a more open-ended and unguided investigation, Kanari and Millar (2004) studied 10, 12, and 14 years olds using a different task (e.g., exploration of the relation between the length and the weight of a pendulum). Like Masnick and Klahr, they found that many children had some awareness of the measurement errors (e.g., they were not surprised when repetition yielded different values), but estimated students’ level of understanding at a 4

Although we discuss this notion in the context of analysis, the notion is an important consideration in data collection as well, influencing decisions about the sample size and the number of repetitions needed.

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lower level and concluded that only a minority of them could be said to have developed their awareness of the variability of measurements into even an elementary notion. A potential reason for these different estimations of the level of students’ understanding seems to be the domain knowledge that students have. Masnick, Klahr, and Morris (2007) reported that domain knowledge helped children to figure out which variability was due to error and which was not. For example, children could reason about the sources of variation in data to some degree in a well-understood context (i.e., a ball rolling down a ramp), but not in a poorly understood context (i.e., pendulum swing). Lehrer and his colleagues’ recent works describe in detail other kinds of challenges students face when they deal with complex forms of data (Lehrer et al., 2000; Lehrer and Schauble, 2000). Lehrer and Schauble (2000) examined how elementary students dealt with the task of abstracting features from qualitative data. Children were provided with drawings of self-portraits and two houses, one near and one far, and asked to classify them into appropriate grade levels. In doing so, they were asked to come up with a description or rules that would guide their classification of the drawings into different grade levels. The rules or classification systems had to be clear and specific so that even an uninformed participant could use them to classify the drawings. Students, especially younger children (first and second grade students) preferred holistic descriptions and ‘eyeballing.’ Their descriptions were replete with vague expressions and qualifiers. For example, they would discuss that older children made portraits that included ‘more detail’ than younger children, but failed to specify what ‘detail’ meant. With discussion and scaffolding by the teachers and others (e.g., researchers and older students), they began to question whether a proposed feature was accurate and precise. For example, some students questioned whether it was really true that the kindergarten pictures had ‘no feet’ and noted that in order to decide that one would have to determine what counts as feet. The older children experienced similar challenges encountered by younger children, but they were able to advance to the next level more readily and use their ‘model’ to classify drawings into different grade levels. Data needs to be processed after it is collected before it can be used for evaluating hypotheses and theories. The kinds and amounts of processing vary a great deal depending on the nature of the data, but students need to be aware of the necessary considerations and steps involved in organizing and summarizing the data.

CONCLUSION In this chapter, we have examined the kinds of conceptual challenges that students face as they set out to collect scientific data in inquiry-rich environments. Even when concepts and skills immediately related to initiating and carrying out data collection are considered, the challenges that students face are substantial. As students begin to collect data in the context of scientific investigations, they need to deal with various conceptual, procedural, and epistemological issues. More specifically, students need to understand the why’s, who’s, when’s, what’s, and how’s of data collection as we have discussed in this chapter. The first question of why is about the purpose of data collection in science. Students can answer this question by understanding how knowledge is justified in science. The second question of who concerns the agency of data collection. It is also about understanding how the activities of

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scientists are related to the development of one’s own thinking, how individual perspectives can influence data collection, and how such influences can be reduced. The third question of when relates to the timing of data collection. It is about judging whether and when data collection is needed and justified based on their understanding of how their data collection would fit within the wider disciplinary context. The fourth question of what is about understanding the discipline-specific nature of evidence. It deals with understanding that the norms and standards for legitimate evidence can differ from discipline to discipline and what the norms are in specific investigative contexts. This question is also about understanding how specific pieces of data are relevant to answering the questions posed in their inquiries. The last question of how is about methodological and procedural aspects of data collection and analysis that need to be developed and practiced in the context of building disciplinary knowledge. As we have seen in this chapter, there is a complex set of concepts and skills that students need to develop in order to carry out the collection and analysis of data successfully. In order to scaffold students more effectively in developing these competencies, we need to understand better what kinds of conceptual challenges they face as they deal with the task of collecting and using data in scientific investigations. It should be noted that only a subset of the concepts and skills needed for the development of evidentiary competence has been discussed in this chapter. A more exhaustive list of concepts and skills involved in the successful collection and analysis of data can be found elsewhere (e.g., Gott and Dugan, 1996; Jeong et al., 2007), which include concepts such as variable selection, sample size, and reliability, to name a few. It should be noted, however, that these lists are not only incomplete, but also quite flexible. The set of relevant concepts and skills involved in the collection and analysis of data varies depending on the contexts and types of the investigations. Since concepts or skills that are necessary in some investigative settings might not be applicable in other settings, the discussion of a more comprehensive set of relevant concepts and skills would be better undertaken in the context of specific investigations. As we wrap up our conclusion, we would like to pose yet another question: the question of how to cultivate this understanding in students. Researchers have been searching for ways to foster students’ scientific reasoning abilities and inquiry skills. Unfortunately, research has shown that scientific reasoning abilities such as distinguishing between theory and evidence do not simply emerge as a result of experiencing inquiry (e.g., Sandoval and Morrison, 2003). Many aspects of scientific reasoning require experience and instruction to develop (Duschl et al., 2007), and we suspect that the same is true with respect to the development of the concepts and skills involved in the collection and use of evidence. What kinds of experiences and instructions are critical to their development? When it comes to teaching scientific reasoning, past curricular programs often focused on teaching a single skill (e.g., the control of variables strategy). More recently with the appreciation of scientific inquiry as the integration of content and thinking skills (Koslowski, 1996; Schauble, 1996; Songer, 2006), curricular programs are moving away from isolated skills to the integration of several different aspects of inquiry skills such as hypothesis generation and evaluation. Integration has also occurred at the dimension of content knowledge versus thinking skills. Instruction used to focus either on content knowledge or process skills, but in recent years the development of content knowledge and inquiry skills are increasingly addressed simultaneously (e.g., Songer, 2006). Yet another kind of integration is happening in science education, this time with epistemological understanding. Although it is unclear yet whether

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epistemological thinking constitutes a separate dimension of scientific thinking or just another form of scientific reasoning (C. Chinn, personal communications, May 31, 2007), it is increasingly viewed that the epistemological understanding can and should be fostered along with the development of scientific reasoning skills and conceptual understanding of the domain in the same context (Driver et al., 1996; Kenyon and Reiser, 2005; Sandoval, 2003; Sandoval and Reiser, 2004; Smith and Wenk, 2006). Although these attempts have met with only partial success so far (e.g., Kenyon and Reiser, 2005; Sandoval, 2003), given the complex nature of understanding involved in scientific inquiry, integrated instructions that carefully orchestrate different aspects of inquiry across multiple inquiry cycles over an extended period of time seem to be the right direction to take. As we continue our efforts to foster students’ inquiry competence, we need to give sufficient attention to the development of students’ competence with respect to collecting and making sense of the scientific evidence and data collection process. Fuller competence with these concepts would not only help students to become better at collecting evidence but also to become better at using evidence and scientific inquiry in general. At the same time, although the goal is to make every student to think like a scientist, we should be careful not to overwhelm students. The kinds of understanding that we expect from our students are quite complex and often underdeveloped in college-educated adults, graduate students, and perhaps even in scientists. How much sophistication we want to cultivate in our children’s thinking and how much scientific literacy would be required to become a productive and criticallyminded citizen in the future are additional questions to ponder in achieving these goals.

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conceptions of nature of science. Journal of Research in Science Teaching, 39(6), 497521. Lehrer, R., Carpenter, S., Schauble, L., and Putz, A. (2000). Designing classrooms that support inquiry. In J. Minstrell and E. van Zee, E. (Eds.). Inquiring into inquiry learning and teaching in science. Washington, D.C.: American Association for Advancement of Science. Lehrer, R., and Schauble, L. (2000). Inventing data structures for representational purposes: Elementary grade students’ classification models. Mathematical Thinking and Learning, 2(1and2), 51-74. Lehrer, R., and Schauble, L. (Eds.). (2002). Investigating real data in the classroom. New York: Teachers College Press. Lehrer, R., Schauble, L., and Petrosino, A. J. (2001). Reconsidering the role of experiment in science education. In K. Crowley, C. D. Schunn, and T. Okada (Eds.). Designing for science: Implications from everyday, classroom, and professional settings. Mahwah, NJ: Erlbaum. Lubben, F., and Millar, R. (1996). Children's ideas about the reliability of experimental data. International Journal of Science Education, 18, 955-968. Masnick, A. M., and Klahr, D. (2003). Error matters: An initial exploration of elementary school children's understanding of experimental error. Journal of Cognition and Development, 4(1), 67-98. Masnick, A. M., Klahr, D., and Morris, B. J. (2007). Separating signal from noise: Children’s understanding of error and variability in experimental outcomes. In M. Lovett and P. Shah (Eds.). Thinking with Data: The 33rd Carnegie Symposium on Cognition. Mahwah, NJ: Erlbaum. Minstrell, J., and van Zee, E. (Eds.). (2000). Inquiring into inquiry learning and teaching in science. Washington, D.C.: American Association for Advancement of Science. National Research Council. (2000). Inquiry and the National Science Education Standards: A guide for teaching and learning. Washington, DC: National Academy Press. Norris, S. P. (1985). The philosophical basis of observation in science and science education. Journal of Research in Science Teaching, 22(9), 817-833. Reiser, B.J., Tabak, I., Sandoval, W.A., Smith, B. K., Steinmuller, F., and Leone, A.J. (2001). BGuILE: Strategic and conceptual scaffolds for scientific inquiry in biology classrooms. In S. M. Carver and D. Klahr (Eds.), Cognition and Instruction: Twenty-five Years of Progress. Mahwah, NJ: Erlbaum. Samarapungavan, A. (1992). Children’s judgments in theory choice tasks: Scientific rationality in childhood. Cognition, 45, 1-32. Sandoval, W. A. (2003). Conceptual and epistemic aspects of students’ scientific explanations. Journal of the Learning Sciences, 12(1), 5-51. Sandoval, W. A. and Morrison, K. (2003). High school students’ ideas about theories and theory change after a biological inquiry unit. Journal of Research in Science Teaching, 40(4), 36-392. Sandoval, W. A. and Reiser, B. J. (2004). Explanation-driven inquiry: Integrating conceptual and epistemic scaffolds for scientific inquiry. Science Education, 88, 345-372. Schauble, L. (1996). The development of scientific reasoning in knowledge-rich contexts. Developmental Psychology, 32, 102-119.

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In: Science Education Issues and Developments Editor: C. L. Petroselli, pp. 201-222

ISBN: 978-1-60021-950-4 © 2008 Nova Science Publishers, Inc.

Chapter 7

CONSTRUCTIVIST-INFORMED CLASSROOM TEACHING: THE IMPORTANCE AND POTENTIAL OF MOTIVATION RESEARCH David H. Palmer1 School of Education, University of Newcastle, NSW, 2308, Australia

ABSTRACT A constructivist paradigm has dominated science education research in recent years. According to this view, students use their existing preconceptions to interpret new experiences, and in doing so, these preconceptions may become modified or revised. In this way, science learning proceeds as children actively reconstruct their ideas as they become presented with new information. However, the implications of constructivism for classroom teaching are still open to question. This position paper refers to the science education literature to argue that strategies to arouse and maintain student motivation should be a crucial component of constructivist-informed classroom teaching. This is because constructivism is universally accepted to be an active process – students must make an effort to reconstruct their ideas, so it follows that if they are not motivated to make that effort then no learning will occur. However, extant models of constructivist classroom teaching make little if any mention of student motivation. In these models, the focus has typically been on strategies to elicit students’ prior conceptions and to guide and monitor their progress towards more scientific conceptions, but the motivational impetus for this process has received little attention. Perhaps one reason for this is that there are relatively few studies of student motivation in the science education literature. Another possible reason is the lack of a unified theory of motivation, which means that there is no clear consensus on how best to motivate students in the classroom. In view of this situation, there is a need for studies which can clarify motivational strategies in science classrooms. “Situational interest” is one motivation construct which appears to offer considerable potential, yet it has been largely ignored by science education researchers. Situational interest occurs when a particular situation generates interest in the majority of students in the class – a spectacular science demonstration might arouse transient situational interest even in students who are not normally interested in science. 1

email: [email protected].

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David H. Palmer The potential of this construct lies in the fact that studies outside of science have shown that when situational interest is aroused on a number of occasions it can result in longterm personal interest and motivation in the topic. It is thus a potentially powerful construct for science education, and is one which should be further explored.

INTRODUCTION Over the last three decades our understanding of how students learn has been dominated by constructivist theory. The main purpose of this chapter is not to extensively analyse constructivism itself, but rather to focus on the instructional models which have developed from it. In the first part of the chapter it will be argued that the instructional implications of constructivist theory are: (1) the teacher needs to ensure that students’ conceptions are changed towards more scientific ideas; and (2) the teacher needs to ensure that students are motivated. The extent to which these two factors are represented in models of constructivist classroom teaching will then be examined, and it will be argued that motivation has been under-represented. Some possible reasons for the lack of emphasis on motivation will then be proposed, in order to identify some potential issues and challenges for future research in science education. The type of motivation that is addressed in this chapter is “motivation to learn” and for present purposes it will be defined as, any process which initiates and maintains learning activity. This is to be contrasted with other types of motivation. For example, motivation to enrol in further studies in science is a type of motivation which has been linked to studies of students’ interest in various science topics and their attitudes towards science (see Osborne, Simon, and Collins, 2003; Tamir and Gardner, 1989). However, the focus of this chapter will only be on motivation to learn. This is essentially a position paper because it is intended to argue a point of view. One of the main factors influencing one’s point of view is the research that one has carried out in the past, and its relationship to the research of others. Consequently this chapter will, when appropriate, describe some of the author’s own previous research as examples of relevant findings or issues. In particular, some parts of this chapter draw strongly from a previous publication (Palmer, 2005) which analysed constructivist-informed teaching models from a motivational perspective, but the material has been re-worked to emphasise the issues and challenges for science education research.

THE CONSTRUCTIVIST PERSPECTIVE According to constructivist theory, learning is not simply a matter of filling students’ minds with knowledge. Instead, individual students develop their own ideas, or conceptions, about the world around them as a result of their daily interactions with people and events. Then, when they are learning about science, they actively interpret any new information from the point of view of their existing conceptions. This may result in their conceptions becoming revised or modified. In this way, science learning proceeds as each individual’s existing conceptions are progressively “reconstructed”.

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However, there is disagreement amongst constructivists regarding the exact manner in which this learning process occurs. Many varieties of constructivism have been described, but two have been especially influential. Firstly, “cognitive constructivists” have emphasised the importance of the cognitive processes that occur within individuals (Osborne and Wittrock, 1983; Piaget, 1978). Supporters of this view argue that learning mainly occurs as individuals try to make sense of their everyday experiences (von Glasersfeld, 1987). Children for example, investigate the world around them and attempt to make sense of what they find by creating mental models. Thus, the metaphor of the “child as scientist” has often been used to describe how children develop their intuitive ideas as a result of their experiences in everyday life (Driver and Erikson, 1983). These experiences can include: (1) physical interaction with objects in the environment; (2) mental experiences in which children think about things they have observed; and (3) social experiences such as interactions with adults and peers. Cognitive constructivists therefore emphasise the very personal construction of knowledge which occurs as individuals strive to make sense of their own unique experiences. On the other hand, “social constructivists” have emphasised the sharing of sociallyconstructed knowledge, through the medium of language (Lemke, 2001; Vygotsky, 1978). According to this perspective, structured social interaction provides children with ways of interpreting the physical and social world, so learning primarily occurs when competent individuals such as adults and teachers can educate students into ways of thinking that are commonly used in that specific community. Through a process of scaffolding (i.e., using techniques that will assist learning) a teacher can make connections with students’ existing conceptions then gradually guide them towards more scientific understandings. Through language, the teacher is able to share ideas with the students, and the students are able to seek clarification until they understand. Although cognitive constructivism and social constructivism differ in many details, there are two characteristics which they have in common. The first is that students do create personal conceptions, and the second is that motivation is essential for the construction of knowledge. These are explained in the following two sections.

THE IMPORTANCE OF STUDENTS’ CONCEPTIONS Marín, Benarroch, and Jiménez Gómez (2000) analysed cognitive and social constructivism and found that the nature of students’ conceptions was essentially the same in both – students’ conceptions represent their ways of perceiving reality. In science education, a large amount of research has been carried out to describe students’ conceptions in various science topics (see Pfundt and Duit, 1991). It has been found that in many cases the conceptions are not compatible with scientific views, and these have been referred to as “misconceptions” or “alternative conceptions”. Many students can hold the same or similar misconception about a particular phenomenon. In one previous study for example (Palmer, 1999) individual interviews were carried out with 63 11-12-year-old students and 44 15-16year-old students to describe their conceptions about ecological role (i.e., every living thing has a role to play in nature). It was found that, of the students who did have misconceptions, 59% of the younger students and 75% of the older group had the same misconception that,

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“Some organisms do not have a role in nature because they don’t actually do anything significant – they are just there” (p. 645). In addition, a number of studies have found that any individual student can have multiple conceptions in relation to a particular scientific concept. For example, Engel Clough and Driver (1986) reported that students had a mixture of conceptions in relation to concepts such as pressure, heat and inheritance. In another study (Palmer, 2001) 112 students from grades 6 and 10 were individually interviewed to determine their conceptions about gravity. They were presented with a series of diagrams showing everyday situations in which objects were either motionless or moving vertically upwards or downwards. In each case students were asked whether gravity was acting. It was found that many students had a scientifically acceptable idea that gravity acted downwards on objects which were falling, but they also had a misconception that gravity did not act upon objects which were moving vertically upwards. The finding that students can have some scientifically acceptable ideas alongside their misconceptions has important implications which will be explained later in this chapter. Finally, a number of authors have commented that misconceptions can be quite tenacious and resistant to change, as they can be unaffected by traditional science teaching (see Wandersee, Mintzes, and Novak, 1994). Misconceptions therefore pose a considerable challenge for science educators, who have the task of changing them towards more scientific viewpoints. For this reason, the process of learning in science has often been referred to as “conceptual change”. Probably the most widely-accepted model of conceptual change is that proposed by Posner et al. (1982). They argued that conceptual change could take two forms: “assimilation” occurs when individuals add information to existing knowledge structures; but “accommodation” is a more radical change that occurs when a central concept is replaced or reorganised. The advantage of this model was that it provided an explanation of the conditions necessary for conceptual change. Accommodation will begin when there is dissatisfaction with an existing conception, then it will proceed as the student considers a new conception to be more intelligible (able to be understood), plausible (makes sense) and fruitful (having the potential to explain more situations).

THE IMPORTANCE OF MOTIVATION The second basic feature of constructivist theory is that learning is an active rather than a passive process – in any learning situation individuals are required to access their existing conceptions, link them to what is currently being experienced, and modify them as necessary (Driver and Oldham, 1986; Phillips, 1995; Roth, 1994; von Glasersfeld, 1987). Thus, according to all forms of constructivist theory, the reconstruction of meaning requires effort on the part of the learner. If effort is required for learning then it follows that motivation is also required, because students will not make that effort unless they are motivated to do so. Motivation would therefore be required to initially arouse students to want to participate in learning, and it would also be needed throughout the whole process until knowledge construction has been completed. Constructivist theory thus implicates motivation as a necessary prerequisite and co-requisite for learning. The role of motivation has also been recognised by other authors. Pintrich, Marx and Boyle (1993) argued that conceptual change would largely be influenced

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by three factors – the choice to engage in the task, the level of engagement in the task, and the willingness to persist at the task – all of which are behavioural indicators of motivation. More recently, Sinatra and Pintrich (2003) argued that motivation is an important controlling factor in the conceptual change process. The importance of motivation can be illustrated empirically as follows. The Aristotelian idea that a continuous action of a force is needed to keep an object in motion has been widely studied, and is often referred to as the “motion-implies-force” misconception. Most authors have agreed that students appear to hold onto this idea tenaciously, so that even courses specifically designed to change this conception may have only limited success. For example, Gunstone, Champagne and Klopfer (1981) found that, after an 8-week course designed to change their students’ misconceptions about force and motion, most students still retained an Aristotelian view. In another study (Palmer, 1997) it was decided to further investigate conceptual change in relation to this misconception. The methodology employed a stratified sampling technique with a pretest – intervention – posttest – delayed posttest design, which was delivered in individual interviews rather than to complete classes. Forty grade 6 students and 47 grade 10 students were diagnosed to have the “motion-implies-force” misconception. During the intervention, these students were simply asked to read a short “refutational text”, which consisted of a sheet of written information which refuted the misconception. Most of this information consisted of specific examples designed to persuade students that on a freely moving object, there is no force in the direction of the motion. The students typically took only one or two minutes to read this information, yet it was successful in causing roughly 40% of the grade 6 and grade 10 students to revise their understandings (conceptual change was demonstrated by understanding, application and metacognition). This finding was rather startling considering the brevity of the intervention (1-2 minutes) and the form of the intervention, which consisted of simply reading a sheet of information, with no accompanying discussion, demonstration or follow up of any sort. In addition, Hynd et al. (1994) reported that many students do not like refutational texts, so the effectiveness of the one used in this study was intriguing. After reflection, the most plausible explanation was that student motivation was artificially enhanced by the interview technique and that this impacted on the results. The interviews were held in quiet rooms where there was little to distract the students, and they typically displayed behavioural indicators of motivation, by concentrating and persisting at the task at hand. One likely cause of this motivation would have been the novelty of participating in an individual interview. The opportunity to sit down in a one-to-one situation with an adult professional who was obviously interested in their ideas, and non-judgemental about those ideas, would be an unusual experience which many adolescents would find challenging, enjoyable and a welcome break from the normal routine of their science lessons. The term “novelty effect” has been used to describe this type of situation in which educational interventions may artificially enhance student motivation and performance by introducing a source of novelty and variety into the classroom (Gay, 1987). Thus, although the motivation was an artefact created by the study design, the results provided a powerful practical demonstration of the importance of motivation in conceptual change.

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CONSTRUCTIVIST-INFORMED TEACHING Implications of Constructivism for Teaching Practice Constructivism is a learning theory, but if it is to be useful to practitioners it should have clear implications for classroom practice. In the previous section it was argued that all forms of constructivism have two features in common: students create personal conceptions; and motivation is essential for the construction of knowledge. Consequently, it can be argued that, regardless of to which form of constructivism one adheres, there are two basic implications for classroom practice:

1. The teacher needs to ensure that students’ conceptions are changed towards more scientific ideas. 2. The teacher needs to ensure that students are motivated. These two points should therefore represent the essential characteristics of all constructivist-informed teaching. A number of authors have proposed instructional models based on constructivism, and these will be analysed below in order to determine whether the two essential characteristics have been included.

Extant Models of Constructivist Classroom Teaching The following is a chronologically-based selection of models of constructivist-informed teaching. Nussbaum and Novick (1982) proposed that conceptual change could be achieved by the following three steps: (1) using an “exposing event” that would stimulate students to state their existing conceptions and debate them with each other; (2) creating conceptual conflict through the use of a “discrepant event” demonstration (i.e., a demonstration with a surprising or unexpected result) that the students cannot explain using their existing conceptions; and (3) students propose various solutions, one of which will be the desired conception, which is then elaborated upon. The authors stated that the discrepant event demonstration would arouse motivation as well as create the need for conceptual change. In the “generative learning model” proposed by Cosgrove and Osborne (1985) the teaching sequence consisted of four phases: (1) the teacher ascertains the pupils’ views through surveys or other activities; (2) the pupils’ attention is focussed on a phenomenon and their ideas about that phenomenon; (3) pupils present their views to the group, the teacher presents the scientific view and they are discussed and compared in order to facilitate accommodation; and (4) the students use the accepted scientific viewpoint to solve a range of problems. The authors stated that motivating experiences would occur in the second phase but the exact nature of these experiences was not described. However, they did emphasise that in general, content should be related to real, everyday situations as this may help motivation. Driver and Oldham (1986) proposed five phases of instruction: (1) pupils are oriented to the topic and develop a sense of purpose and motivation; (2) pupils make their ideas explicit

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through discussion or writing; (3) students are exposed to the conflicting views of other students or to conflicting evidence in a “surprise” demonstration, and in which the scientific view is presented by the teacher; (4) students apply their new conceptions to a variety of situations, both familiar and novel; and (5) students reflect upon their change of ideas. Although the authors stated that pupils would develop motivation during the orientation phase, the source of the motivation was not identified. Neale, Smith and Johnson (1990) proposed that conceptual change teaching should include: eliciting children’s conceptions; activities which allow children to discover contradictory evidence and contrasting explanations; children presenting their ideas to each other and debating them; and the children summarising and contrasting alternative views. The authors emphasised that instruction should be directed towards contradicting children’s mental schemes, and that links should be made to prior lessons and to children’s informal experiences. Motivation was not specifically mentioned in this model. The Language Oriented Learning Cycle (Glasson and Lalik, 1993) consisted of three phases: students experience cognitive disequilibrium through experiences in the exploration phase; students construct new knowledge in the clarification phase; and students engage in divergent problem solving in the elaboration phase. It was intended that during the exploration phase, students would participate in stimulating activities designed to engage their curiosity, but otherwise motivation was not an explicit component of this model. Banet and Núñez (1997) designed a teaching sequence centred on attractive or surprising activities that would capture the attention of the students and promote cognitive conflict. In the first phase, students become motivated and their ideas are elicited. In the second phase, cognitive conflict is used to reveal the inadequacy of pupils’ ideas and encourage the formation of new knowledge. In the third phase, the new ideas are shown to be valid and students review the change in their thinking. The authors stated that the students would become motivated early in the sequence, but exactly how this would be done was not explained. The “conceptual replacement” approach (Dekkers and Thijs, 1998) was used to teach the concept of “force”. It had three main phases: (1) find a shared meaning of force in limited contexts; (2) refine the students’ partial understandings of force and expand it into other contexts; (3) resolve dissonance by finding shared meanings. The authors proposed that motivation was aroused through the use of teaching activities that created cognitive dissonance. Hewson et al. (1999) proposed that teaching for conceptual change should contain the following components: the ideas of both students and teachers needs to be discussed; students display metacognition about their ideas; the status of their conceptions will change as some ideas become more acceptable and other ideas less acceptable; and students need to explain the justification for their ideas. The authors did not specifically mention motivation as a factor in teaching for conceptual change. The Metacognitive Learning Cycle (Blank, 2000) began with the concept assessment phase, in which students reflected on their science ideas and the status of those ideas (i.e., the extent to which they were intelligible, plausible and fruitful). In the concept exploration phase the students explored phenomena related to the concept. This was followed by the concept introduction phase in which the teacher introduced the main concept in the lesson and the students reflected on any changes in their ideas. Finally, in the concept application phase, the

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students were presented with other examples of the concept and again considered the status of their ideas. Motivation was not mentioned as a factor in this model. She (2004) proposed a Dual-Situated Learning Model (DSLM) which contained a dualsituated learning event, which was an experience designed to create dissonance with students’ existing beliefs, and to provide a new mental set for them to achieve a more scientific view. The model consisted of six stages: (1) examining attributes of the science concept; (2) probing students’ misconceptions of the science concept; (3) analysing which mental sets students lack; (4) designing dual-situated learning events; (5) instructing with dual-situated learning events; and (6) instructing with a challenging situated learning event. She argued that one of the purposes of dissonance was to create motivation, as it would arouse students’ curiosity and interest. Liang and Gabel (2005) described the Powerful Ideas in Physical Science (PIPS) model which had been developed by the American Association of Physics Teachers. This model consisted of five major phases: (1) eliciting and elaborating the students’ ideas; (2) testing and comparing the ideas with nature; (3) resolving the discrepancies between ideas; (4) applying the ideas; and (5) reviewing and summarising of ideas. Motivation was not mentioned as a component of this strategy. It is apparent that these models have many features in common. For example, the teacher is typically required to elicit the students’ prior conceptions, present students with experiences which challenge their prior conceptions (cognitive conflict for example, can be introduced through the use of discrepant events or through argumentation), present the students with the scientific viewpoint, allow the students to practise using the scientific view, and ask the students to reflect upon and evaluate their process of learning. Macbeth (2000) similarly concluded that most models of constructivist teaching have these types of features in common. These models have undoubtedly made an important contribution to our understanding of classroom practice, as they have emphasised the importance of considering students’ conceptions. However, with respect to motivation, there are two features of these models which deserve mention: 1) The extent to which motivation has been included in the models is rather limited. Two sources of motivation were suggested in some of the models. Firstly, activities such as discrepant events demonstrations which cause surprise or cognitive dissonance (e.g., Driver and Oldham, 1986; Nussbaum and Novick, 1982; She, 2004) and secondly, relating content to everyday life (e.g., Cosgrove and Osborne, 1985) were recommended as motivational strategies. Some of the models (Cosgrove and Osborne, 1985; Driver and Oldham, 1986) included a particular phase that was intended to arouse motivation, but the actual motivating strategies were not described. In several models (Blank, 2000; Glasson and Lalik, 1993; Hewson et al., 1999; Liang and Gabel, 2005; Neale, Smith, and Johnson, 1990) motivation was not explicitly mentioned at all as a component of the model. 2) One central strategy advocated by the models may have the potential to reduce motivation. Most of the models imply that the students’ conceptions will typically be incorrect, so there will be a need to directly challenge those misconceptions (by using argumentation or cognitive conflict) in order to achieve accommodation and the development of more scientific understandings. The problem with such a

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confrontational approach is that it may have a negative effect upon motivation, as students will be regularly faced with evidence of their own scientific inability. Some authors such as Bencze (2000) and Gil-Pérez and Carrascosa-Alis (1994) have suggested that, if students have their ideas systematically critiqued with evidence of their scientific inadequacy they may lose confidence in their ability to construct knowledge. Similarly, Dreyfus, Jungwirth and Eliovitch (1990) found that bright, successful students reacted enthusiastically to confrontation, whereas unsuccessful students were “characteristically apologetical when confronted with a conflict which, to them, seemed to represent just another failure” (p. 566). It is therefore possible that repeated experiences of this type of failure in science classes could negatively impact on students’ motivation. In summary, the extant models of constructivist-informed teaching have a strong focus on student conceptions, but relatively little attention has been given to motivation. Whether this is a problem or not, will be discussed in the next section.

Does Motivation Need to be Explicitly Considered in Instructional Models? As argued above, motivation is recognised as being essential for learning, yet there is very little mention of it in models of constructivist-informed teaching. In addition, one of the central strategies advocated by the models may actually be detrimental to motivation. Are these really problems though? Some people might argue that: (1) we can disregard motivation in instructional models because there is little that teachers can do to influence student motivation anyway; or (2) it is acceptable for a model to focus on one aspect of instruction and ignore other aspects; or (3) there is no need to explicitly consider motivation because it might be inherent in the constructivist teaching techniques. Each of these arguments is addressed as follows.

Argument 1 There is little that teachers can do about motivation. It is generally agreed that there is a strong downturn in student motivation during the middle school and high school years (Anderman and Maehr, 1994) particularly in science (Butler, 1999; George, 2006; Tobin, Tippins, and Gallard, 1994). In fact it has been argued that lack of student motivation is one of the greatest challenges facing science teachers in our schools – according to many secondary students, science is a difficult and boring subject which fails to be engaging (Rennie, Goodrum, and Hackling, 2001). Teachers are thus faced with a sizable problem which to some may seem insurmountable. On the other hand, motivation theorists would suggest that there is much that teachers can do. In contrast to older views in which motivation was assumed to be a relatively stable personality trait, the more recent research has adopted a social cognitive perspective which emphasises that motivation can be significantly influenced by aspects of the classroom context (Pintrich and Schunk, 1996). This is a particularly important point because it indicates that classroom experiences should have a strong effect on student motivation. To suggest that there is little that teachers can do is therefore not convincing.

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Argument 2 It is acceptable for a model to focus on one domain only. Some people might argue that, in a learning model, it can be quite acceptable to focus on cognition, and exclude motivation, for the purpose of simplicity. For example, the conceptual change model proposed by Posner et al. (1982) did not contain a component for the motivational forces that would be necessary to drive the conceptual change process. Instead, the authors acknowledged their focus to be on the rational or cognitive component of learning: Our central commitment in this study is that learning is a rational activity. . . . It does not, of course, follow that motivational or affective variables are unimportant to the learning process. The claim that learning is a rational activity is meant to focus attention on what learning is, not what learning depends on (p. 212; italics added).

This conceptual change model continues (deservedly) to be very highly regarded, and in fact many of the extant models of constructivist teaching have used it as a referent (e.g., Banet and Núñez, 1997; Driver and Oldham, 1986; Hewson et al., 1999; She, 2004). There is an important distinction to be made however, between models of learning (i.e., those, such as the conceptual change model, which describe how learning occurs) and models of instruction (i.e., those which describe classroom teaching strategies). While it can be acceptable for a learning model to focus on only one aspect of learning, the same cannot be said for instructional models. Any instructional model, that is intended to guide practical classroom teaching, cannot afford to ignore any factor which the theory suggests is essential to learning. An instructional model which does so should simply not work. Consequently, models of constructivist-informed teaching should comprehensively address both of the factors which constructivism implies to be essential – student conceptions as well as motivation.

Argument 3 There is no need to explicitly consider motivation because it might be inherent in the student-centred techniques that are typical of constructivist teaching. Some people might argue that the use of specific techniques such as discrepant events demonstrations, hands-on activities and the use of real life examples, which are typical of many constructivist teaching models, will be enough to automatically motivate students. It is worth analysing some of these techniques to determine the extent to which this might be the case. •

Discrepant events demonstrations (i.e., demonstrations in which there is an unexpected or surprising result) do have the ability to arouse interest and motivation through cognitive dissonance (e.g., Nussbaum and Novick, 1982). Recent research however, suggests that the motivational effects of such experiences can be relatively transient (Hidi and Harackiewicz, 2000). The extent to which a discrepant event demonstration will be able to maintain student motivation is therefore debatable. Additional use of more discrepant event demonstrations in order to maintain student motivation is not the answer because the repetitious use of any teaching technique will soon lead to boredom and loss of effectiveness – Loughran, Northfield and Jones (1994) for example, found that even discrepant events demonstrations can become monotonous to students if they are overused. Use of other motivational strategies in addition to discrepant events demonstrations would seem to be advisable.

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Hands-on activities have been advocated for most constructivist teaching models, and can have a number of purposes: to generate students’ ideas at the beginning of a unit (Nussbaum and Novick, 1982); to present students with real world problems that can provide a basis for discussion (Lee and Brophy, 1996; Nussbaum and Novick, 1982); to collect evidence in order to evaluate alternative ideas (Driver and Oldham, 1986); to explicitly test students’ assumptions, discover contradictory evidence and promote cognitive conflict (Banet and Núñez, 1997; Neale, Smith, and Johnson, 1990); and to provide examples of a range of contexts to which students might apply their ideas (Beeth and Hewson, 1999). There is no doubt that hands-on activities can be motivating to students (Weaver, 1998) and most science teachers consider them to be the main source of classroom motivation (Zahorik, 1996), but they have a similar problem to that described above for discrepant events demonstrations. In other words, we have little idea of whether the motivation aroused by a hands-on activity will be maintained after the activity has been completed. In fact, the relationship between motivation and hands-on activities has been remarkably under-researched. We have very little understanding of why hands-on activities are motivating. Is it purely due to physical activity for some students? Or is it only due to a combination of physical and mental activity? Is it due to the opportunity for collaboration? What is the motivational construct that adequately explains this type of hands-on/minds-on motivation? Given the central importance attached to hands-on activities in science instruction, it is surprising that more attention has not been given to these questions. Use of real life examples is often advocated in constructivist teaching models (e.g., Banet and Núñez, 1997; Liang and Gable, 2005). Relating concepts to real life examples should have the potential to enhance motivation as it makes the work appear relevant and valuable to the students by helping them to understand the real world (Zusho, Pintrich, and Coppola, 2003). One issue though, is that there are some aspects of real life with which students would not have had personal experience. For example, the motor car engine is often referred to as a real life example of the effect of lubrication in reducing heat from friction. Yet many students would never have seen the engine pistons or cylinders which require lubrication, so although this example is from real life, it is not from the students’ lives. Its relevance to students is therefore debatable. It is important that educators make the distinction between real life examples which can be directly experienced by students and those which cannot, and they should take the motivational implications of these into consideration.

Thus, it is clear that techniques such as discrepant events demonstrations, hands-on activities and real life examples do have the potential to motivate students on occasion, and if used carefully. However, this does not mean that use of these techniques will automatically motivate students to desirable levels. Banet and Núñez (1997) observed that their constructivist teaching program was not successful in maintaining the interest of about one third of their 13- and 14-year-old students. Similarly, Lee and Brophy (1996) investigated teaching that was guided by a conceptual change approach and found that it failed to motivate half of the sixth-grade students in their study. These reports suggest that motivation can still be problematical in constructivist classrooms, so one cannot simply assume that it is inherent in the teaching techniques. Thus, although some constructivist teaching techniques do have the potential to motivate students, they need to be supported by other motivational strategies.

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In summary, the points above suggest that motivation does need to be included as an explicit component of constructivist teaching models. The fact that previous instructional models have not done this is therefore a problem. Any future models of constructivist teaching should be expected to explicitly consider motivation strategies as well as students’ conceptions. However, this might not be a simple thing to do, as explained in the next section.

Difficulties in Identifying Motivation Strategies Two main motivation-related problems have been identified in the extant models of constructivist teaching. The first was that motivation has been generally under-represented, and the second was that one of the central strategies in constructivist teaching might actually inhibit motivation. In order to address these problems it would be necessary for any future models of constructivist teaching to include a comprehensive description of exactly how student motivation should be enhanced to optimum levels in the classroom, and an elimination of motivationally-negative teaching strategies in favour of motivationally-positive ones. At this time however, it is questionable whether the research data would be able to supply the necessary information. In the sections below it will be argued that: (1) educational research has not been able to provide a clear direction on exactly how to motivate students to optimum levels, particularly in science; and (2) a bias in the science education research has meant we do not have enough information about student conceptions to enable us to plan for motivationally-positive teaching. 1. There is a lack of a clear direction on how to motivate students in the classroom. Unfortunately, the research has been unable to provide a clear consensus on exactly how motivation can be enhanced to optimum levels, especially in the critical middle school and high school years. There are two main reasons for this: •

The first reason is that motivation theory is very diffuse. Currently, there is no unified or cohesive theory of educational motivation that adequately describes all its features. Instead, motivational researchers have identified a range of affective and/or cognitive constructs, each of which can influence motivation to some degree. These include “intrinsic motivation” (i.e., doing something because it is inherently enjoyable; Ryan and Deci, 2000), “self-efficacy” (a belief in one’s ability to do a task successfully; Bandura, 1997), “achievement goals” (a desire to understand the content or perform better than one’s peers; Ames, 1992), “interest” (directed attention towards a particular topic area or a particular event; Ainley, Hidi, and Berndorff, 2002), and “task value beliefs” (beliefs about the value of the material to be learnt; Wigfield and Eccles, 2000). Unfortunately, the relative importance of these constructs and their relationship to each other is yet to be established, but each construct does have classroom strategies which are advocated for it. Some examples of these include: intrinsic motivation can be enhanced by providing challenge and allowing students to make choices (Lepper and Hodell, 1989); self-efficacy can be enhanced by authentic mastery experiences (Bandura, 1997); positive achievement goals can be enhanced by reducing classroom competition (Kaplan and Maehr, 1999); and task value

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beliefs can be enhanced by emphasising the relationship of the content to everyday life (Zusho, Pintrich, and Coppola, 2003). A number of classroom strategies are therefore available, but as the relationship and relative importance of the theoretical constructs have not yet been established, so the relative importance of the classroom strategies cannot be exactly determined either. As a result, the general educational research offers no clear consensus on how best to motivate students in the classroom. It is even possible that the complexity of motivational theory has, to some extent, hindered its usefulness as a guide for classroom instruction. The second reason is that in the science education research, motivation has been understudied in comparison to some other research areas. Tsai and Wen (2005) analysed research in science education from 1998 to 2002 and found that the largest quantum of papers was empirical studies of students’ conceptions. By contrast, student motivation studies were not common enough to warrant their own category, and were counted with studies of other classroom and learner characteristics such as learning environment, reasoning, peer interactions, cooperative learning, writing, discourse and economic factors. An additional problem is that in some cases, the science motivation research that does exist has focussed on tertiary level, so its implications for the critical elementary, middle school and high school levels are not well understood. The research on the science teaching self-efficacy of elementary teacher education students is an important body of work which exemplifies this issue. It has been found that many elementary teachers and elementary teacher education students have a negative confidence in their ability to teach science (Skamp, 1991) and this is of concern because teachers who lack self-efficacy for the subject will teach science poorly or not at all (Andersen et al, 2004; Enochs, Scharmann, and Riggs, 1995). It is therefore important to change negative self-efficacy to positive during their teacher education studies, and a significant body of research has investigated how to do this (e.g., Appleton, 1995; Mulholland and Wallace, 2001; Rice and Roychoudhury, 2003). In one study for example (Palmer, 2006) surveys were used to measure changes in self-efficacy before and after a science methods course for elementary teacher education students. A series of three open-ended questionnaires throughout the course showed that positive changes in selfefficacy were mainly due to the students’ increased understanding of science teaching techniques (called “cognitive pedagogical mastery”) and imagining themselves teaching using those techniques (“cognitive self-modelling”). However, neither of these sources of self-efficacy would be relevant to high school level, so this study had very limited applicability for the crucial secondary levels of schooling. Unfortunately, few studies have directly focussed on selfefficacy in school science classes, and the study by Britner and Pajares (2006) claimed to be the first study of this type at middle school level. They found that ensuring successful experiences in authentic, inquiry-oriented science was the most effective technique. Self-efficacy is therefore one example in which a lack of balance in the research has severely limited the information available to inform motivation in school settings. Thus, the research has not been able to provide a clear direction for how to best motivate students in the classroom.

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David H. Palmer 2. A bias in the science education research has meant we do not have enough information about student conceptions to enable us to plan for motivationallypositive teaching. Earlier in this chapter, a study was described in which students were shown to have scientifically acceptable conceptions alongside their misconceptions (Palmer, 2001). Some other authors have found similar patterns. Stavy (1990) investigated children’s ideas about changes of states of matter and found that, “children who recognised weight conservation in one of the tasks did not necessarily recognise the same in the second task” (p. 247). Finegold and Gorsky (1991) found that some university and high school students had an acceptable understanding of the forces on objects in linear motion, but invoked the idea of a “motion force” when the object was in projectile motion. However, relatively little has been made of these types of findings. A huge amount of research has described students’ conceptions in various science topics, but in general, the focus has been on their misconceptions (or alternative conceptions) rather than their scientifically acceptable conceptions. A graphic example of this emphasis is that the relevant chapter in the Handbook of Science Teaching and Learning (Wandersee, Mintzes, and Novak, 1994) was titled “Research on Alternative Conceptions in Science” (italics added). This chapter provided an extensive review of the research on students’ conceptions, but did not contain any descriptions of those which might be scientifically correct or partially correct. Thus, we now know quite a lot about student misconceptions, but not very much at all about their scientifically acceptable conceptions. In other words, the research literature has focussed on one side of the coin only. This is an issue because it restricts the strategies which can be potentially used to teach students. For example, it might be useful to know the particular contexts or situations in which students invoke scientifically acceptable ideas in each science topic, as teachers might then be able to directly refer to these contexts to begin instruction. There is evidence that this approach can be successful. Brown (1992) referred to students’ correct conceptions as “anchoring intuitions”, and showed that they could provide a firm basis for reducing the misconceptions which these students also held. He used a “bridging analogies” technique to extend the students’ correct understanding of action-reaction forces to a wider range of situations, thereby reducing the misconception. A more recent study has confirmed the effectiveness of the bridging analogy technique (Savinainen, Scott, and Viiri, 2005). It should also be pointed out that one of the extant models of constructivist teaching which was described in an earlier section has, unlike the other models, adopted an approach utilising students’ scientifically acceptable understandings. Dekkers and Thijs (1998) argued that many students’ misconceptions arose from mistaking the meaning attached to scientific terms such as “force”, whereas their actual ideas were in many cases not incorrect but were limited to a small range of contexts. By using concept refinement and context expansion it should therefore be possible to move students towards a more scientific view. They tested the idea in preuniversity courses in Botswana and South Africa, and found increases in correct answers and answer patterns. Thus, students’ scientifically acceptable conceptions could be used as a basis for instruction, and this approach would not involve potentially damaging confrontation. At the present time however, we do not have enough information about students’ scientifically acceptable conceptions

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(particularly the contexts or situations in which they would be most likely to be invoked) to enable this to occur on a wide scale. In summary, the arguments above have suggested that: (1) the research on educational motivation has not provided enough clear direction on how to best motivate students in the classroom; (2) there has been relatively little research on motivation in school science classes; and (3) there has been a lack of balance in the science education research on student conceptions, which has restricted the teaching approaches which can be used in constructivist classrooms. Considering the factors above, it is perhaps unsurprising that motivation has been under-represented in models of constructivist-informed teaching. In fact, it is possible that one of the main reasons that motivation has not been included in the extant models is our lack of research data on exactly how to include it. In view of this situation, there is a need for studies which can identify motivational strategies in science classrooms. There is one motivation construct which has the potential to be particularly informative to instruction because it focuses directly on classroom events and their motivational impact on all students in the class. This construct, “situational interest” is described below.

SITUATIONAL INTEREST “Interest” is characterised by “focussed attention, increased cognitive and affective functioning, and persistent effort” (Ainley, Hillman, and Hidi, 2002, p. 545). It is a psychological state that is thought to be very closely related to intrinsic motivation (Deci, 1992; Tobias, 1994). Interest is a very powerful form of motivation – Pintrich and Schunk (1996) reviewed a number of previous studies and concluded that interest is related to increased memory and attention, greater comprehension, deeper cognitive engagement and thinking. There is also a positive relationship between interest and achievement (Wang, Wu, and Huang, 2007). Hidi (1990) proposed two distinct types of interest: “personal interest” is a long-term preference for a particular topic or domain (a student might have a personal interest in biology, for example) whereas “situational interest” is interest that is generated by aspects of a specific situation, and is usually transient (a spectacular science demonstration could arouse short-term interest even in students who are not normally interested in science). Personal interest is a relatively enduring phenomenon, but its disadvantage is that it is very difficult for teachers to take all students’ personal interests into consideration when planning lessons (Hidi and Anderson, 1992). Consequently, some researchers have given particular attention to situational interest because it concentrates on classroom events and their direct impact on student engagement. Mitchell (1997) suggested that by providing sources of situational interest in their lessons teachers could substantially increase student motivation: If learning environments are to be motivationally effective, they need to be perceived as high in situational interest for a substantial percentage of the students in the classroom (p. 5).

Situational interest can have a range of positive effects on students. Schraw, Flowerday, and Lehman (2001) reviewed a number of earlier studies of situational interest in written text.

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They concluded that, “Virtually all studies reveal that higher levels of situational interest are related to better learning, including propositional recall and the construction of a holistic text interpretation (i.e., an integrated understanding of the text’s main themes) . . . and often deeper, text processing” (p. 42-3). These authors also noted the possibility that situational interest could, if aroused on a number of occasions, develop into long standing personal interest. They argued that, if so, this would have tremendous educational implications for teachers who are striving to promote student interest in their domains. Mitchell and Gilson (1997) for example, found that although situational interest is a transient occurrence, there was evidence that multiple experiences of situational interest in mathematics classes could develop into long-term, personal interest in mathematics. In another study (Palmer, 2004) involving teacher education students, it was found that repeated arousal of situational interest in a university elementary science methods course had positive effects on other attitudinal dimensions such as self-concept in science, science anxiety, perception of the science teacher, enjoyment of science and motivation in science. These findings suggest that repeated experiences of situational interest can have powerful and wide-ranging effects on students. Most of the previous studies of situational interest have concerned text-based interest, or features of written text that arouse interest in the majority of readers (e.g., Ainley, Hillman, and Hidi, 2002; Hidi, 2001). In studies such as these, situational interest is usually recorded when subjects report their excitement, attention or interest during or immediately following the reading of a passage containing carefully selected content (Alexander, Kulikowich, and Jetton, 1994). Using this approach, a number of sources of situational interest have been identified. These include novelty (i.e., a new or unusual experience), surprise (i.e., unexpected or discrepant stimuli), autonomy (i.e., giving students meaningful choices), suspense, social involvement, ease of comprehension of text and emotional appeal in topics such as death, sex and violence (see Anderman et al., 2004; Deci, 1992; Hidi, 2001; Hidi and Anderson, 1992). In addition, Mitchell (1993, 1997) investigated situational interest in high school mathematics classes. His findings suggested two main sources of interest: (1) “meaningfulness” occurs when students perceive activities as being as being relevant to their present lives; and (2) “involvement” refers to the degree to which students are active participants in the learning process. Chen and Darst (2001) studied middle school students in physical education classes, and found that cognitive demand was the critical factor in generating situational interest in physical learning tasks. However, our understanding of situational interest in school science is still rudimentary. Zahorik (1996) asked elementary and secondary teachers to write essays on how they created situational interest in science lessons, and found that teachers believed that hands-on activities were the primary source. Laukenmann et al. (2003) found that in eighth grade physics classrooms, successful learning experiences were linked to interest. Other studies however, are notable by their absence, and it is clear that we still have much to learn about the sources of situational interest in school science classrooms. It might be expected that further research in this area would be profitable, as science lends itself to providing factors such as novelty and surprise (through discrepant events demonstrations), autonomy (through student-centred investigations), social involvement (through group work in hands-on activities) and meaningfulness (through relevance to real life phenomena). Of course, the motivational impact of these factors will be moderated by students’ motivational beliefs, such as their selfefficacy and achievement goals, but these also can be significantly changed as a result of

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other strategies that the teacher uses in the classroom. A direct focus on classroom strategies and their affects on motivation might therefore be a profitable direction for future research.

CONCLUSION This chapter has attempted to identify a number of issues and challenges relevant to research in science education. The main issue was the under-representation of motivation in models of constructivist-informed classroom teaching. It was argued that this is a problem which needs to be addressed in the future. However, it was noted that this may not be a straightforward process, as there are some other issues which have impacted on the situation: -

the lack of a unified or cohesive theory of motivation to learn, which means there is no clear consensus on how best to motivate students in the classroom; the lack of research on motivation in school science, which means there is limited empirical data available to inform instructional practices; a bias in the research on student conceptions, which means we have a lot of information about students’ misconceptions but very little information about their scientifically acceptable conceptions as starting points for instruction.

In view of these problems, it is possible that one of the main reasons that motivation has not been included in the models of constructivist teaching is our lack of research data on exactly how to include it. Finally, situational interest is one motivation construct which is potentially of value because it focuses directly on classroom events and their immediate effects on student motivation. There is a need to further explore the sources of situational interest in school science classes, and any possible relationship between these and students’ other motivational beliefs.

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In: Science Education Issues and Developments Editor: C. L. Petroselli, pp. 223-238

ISBN: 978-1-60021-950-4 © 2008 Nova Science Publishers, Inc.

Chapter 8

ORAL COMMUNICATION COMPETENCIES IN THE SCIENCE CLASSROOM AND THE SCIENTIFIC WORKPLACE F. Elizabeth Gray1, Lisa Emerson and Bruce MacKay Department of Communication and Journalism Massey University,New Zealand School of English and Media Studies Massey University,New Zealand Institute of Natural Resources Massey University,New Zealand

ABSTRACT This chapter investigates the importance of oral communication training in undergraduate scientific education. The authors examine the status of oral communication training in New Zealand universities and the debate concerning employer attitudes to this issue. The specific relevance of these issues to science education is explored through analysis of a case study and a qualitative and quantitative study of the attitudes of students and employers in science-related industries. Cronin, Grice and Palmerton (2000), Dannels (2001), and Morello (2000) argue that to significantly develop the rhetorical flexibility necessary to communicate competently, oral communication skills training needs to be discipline-specific and firmly contextualized in the genres, expectations, and conventions of the particular field. Responding to this call, a number of recent studies have examined the role of oral communication skill development in specific fields as diverse as design education (Morton and O’Brien, 2005), archaeology education (Chanock 2005), and engineering (Darling, 2005; Dannels, 2002). This chapter moves the discussion of discipline-specific oral communication instruction to undergraduate science education. The recent inclusion of an oral communication component within a compulsory science communication class at Massey University, New Zealand remains a contentious 1

[email protected].

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F. Elizabeth Gray, Lisa Emerson and Bruce MacKay issue. Possibly seeing oral communication training as a low priority in terms of student skills, knowledge, or preparedness for a future scientific career, both students and faculty have resisted the inclusion of oral communication into course curricula and assessment. The researchers designed a study to clarify whether oral communication skills were important to employers in science-related industries, what science employers meant by oral communication skills, and which skills they prioritized. At the same time, the team surveyed science students to better understand their attitudes to training in oral communication. Study findings strongly support the importance of oral communication skills in science-based employment in New Zealand. Science employers indicate that they require and value highly a wide variety of oral communication skills. The study also reveals that while science employers and university science students agree that oral communication skills will be important in scientific careers, the majority of employers find the desired level of these skills in new graduates only sometimes or occasionally. The retention of oral skills teaching and assessments, as currently exemplified by the Communication in the Sciences course at Massey University, is clearly indicated; study findings also make a strong case for an extended focus on oral competencies in undergraduate science education.

INTRODUCTION Effective oral communication skills help a person to function productively in education, in employment, and in everyday social interactions. Communication education research suggests that universities that fail to provide adequate oral communication training may be putting their students at a personal and professional disadvantage. Internationally, this research has informed and been in turn informed by the growth of Communication Across the Curriculum Programs (CXCPs), the rationale for which is based in social constructivist theory. As expounded by Driscoll (1994), social constructivist theories of knowledge acquisition see meaning essentially as a social product, created from and out of social interactions, and naturally fluid and open-ended. Knowledge acquisition thus takes place through social interaction with others, and may be best achieved when students must engage with complex, real world problems in which there are multiple perspectives and no clear ‘right’ answer. Oral communication, in which students actively gather, synthesise, and critique information and ideas, thus provides a critical venue for knowledge acquisition, and represents a fundamental mode of learning (Cronin and Grice, 1993; Davis, 1992; Garside, 1998; Modaff and Hopper, 1984; Palmerton, 1992). Cronin and Glenn (1991), Cronin, Grice, and Palmerton (2000), Morreale (1990), and Steinfatt (1986), among others, specifically argue the importance of oral communication training in university education. CXCPs emphasize the fundamental importance of developing students’ oral communication competencies across subjects and majors, and experts agree that skill in communicating orally is not solely relevant to a ‘communication studies silo,’ but is a foundational skill in all fields (Cronin and Grice, 1993; Modaff and Hopper, 1984; Rubin and Morreale, 1996). A number of recent studies have narrowed their focus, examining the vital role of oral communication skill development in specific fields, as diverse as design education (Morton and O’Brien, 2005), archaeology education (Chanock 2005), and engineering (Darling, 2005; Dannels, 2002).

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This chapter considers the status of oral communication training in New Zealand universities, the factors contributing to a lack of focus on oral communication at university level, and the debate concerning employer attitudes to this issue. Through analysis of a case study and a qualitative and quantitative study of the attitudes of students and employers in science-related industries, the chapter argues the case for establishing a stronger base for teaching oral communication skills in university science programs.

BACKGROUND Employer Perspectives While academic research has supported the assertion that graduates need strong communication skills, a lack of specificity in the research has limited its impact on curriculum development. Over the course of the last fifteen years, several studies of the New Zealand and Australian job markets have confirmed the findings of a number of large international studies, delineating a consistently expressed desire on the part of employers for “strong communication skills” (Andrews, 1995; Australian Association of Graduate Employers, 1993; Higher Education Council, 1992; Reid, 1997; Tapper, 2000; Victoria University, 1996). In New Zealand, the job market’s need for employees with strong communication skills has also been widely reported in the popular press (see for example Bland, 2005; Hart, 2004). However, these studies have maintained only a general approach to the issue of communication skills; they have not inquired closely into what specific attributes are indicated by the umbrella term “communication skills.” When responding to survey questions, the surveyed employers might have had in mind solely written communication skill, or interpersonal communication skill, or oral communication skill, or a combination. For researchers specifically interested in oral communication, these results are too general to be enlightening: employers may value oral communication skills quite differently from skill in written communication or in interpersonal communication and teamwork. For universities developing curricula or policy relating to communication, such findings are so general that they fail to give a clear indication of what skill instruction needs to be integrated into university programs. As a further limitation, the studies have tended to report on employers en masse, not differentiating between employers from different industries and their potentially different communication requirements.2 It is doubtful that all employers ascribe precisely the same value to oral communication skills in their employees. It is even more doubtful, given the wide range of skills which are included in the umbrella term “oral communication skills,” that all employers agree on the value for their business of the same kinds of oral communication skills. 2

Burchell et al (2001) and Rainsbury et al (2002) have reported on the competencies required of business graduates in the New Zealand context, and Coll and Zegwaard (2006) draw on the same survey instrument used by both earlier studies to examine the competencies required of New Zealand science and technology graduates. All of these studies omit oral communication as a competency. In their study of business students’ attitudes towards communication skills, Waller and Hingorani (2006) included in their survey only the single, undifferentiated category of “oral communication skills.”

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Media Interest Employer concern has been noted by a wider audience than just academics in the field. For some time now, the need for stronger broad-based communication skills amongst recent graduates has been reported in New Zealand’s mainstream media. Recognition of the particular importance of oral communication skill in the workplace has also begun filtering through to the broader population, with a recent article in New Zealand’s largest daily newspaper specifically focusing on the need for oral communication skills for employment (Cain, 2006). Once again, however, these issues are generally discussed only at the most generic level, and media reports have had little direct impact on universities.

University Attitudes The precise nature of the university’s responsibility to prepare students for the workforce is a subject of much debate, in New Zealand as internationally (see for example Coll, 1996; Coll and Zegwaard, 2006; Jones 1994; Rubin and Morreale, 2000; Zorn, 1998). While Zorn warns against the pitfalls of the increased vocationalisation of university education, he also admits that the need to promote study programs as valuable to career preparation is a reality New Zealand universities can ignore only at their financial peril. In New Zealand, the question of communication training for the workplace is complicated by the vexed issue of university education funding. The percentage of university operating revenue coming from government funding (as a percentage of total operating revenue) dropped from 75.8% in 1992 to 42.2% in 2002 (NZUSA, 2006). The government’s total funding to the tertiary sector dropped again between 2003/2004 and 2004/2005, and until a shift in policy mid-2006 this funding was tied to enrolment numbers. In response, broad budget cuts have been rolled out, and university departments across New Zealand have eliminated many courses that had relatively low enrolments or that were perceived as “inessential.” Many undergraduate degree programs are now compressed into three tightlystructured years in which students have very little opportunity to take elective courses. Because compulsory courses have become already strained for time, many faculty members are highly resistant to the idea of expanding course content and assessments. As universities have sought to trim “luxuries,” communication competencies, and particularly oral competencies, have in many cases been jettisoned from university courses and from assessments. New Zealand universities’ diminishing emphasis on teaching oral communication competencies seems counter-intuitive in the face of the employer demand cited above. This lack of emphasis also disregards the recommendations of contemporary communication education literature concerning the increasing need for discipline-based oral communication instruction. Cronin, Grice and Palmerton (2000), Dannels (2001), and Morello (2000), among others, argue forcefully that for students to significantly develop the rhetorical flexibility necessary to communicate competently, communication skills training needs to be disciplinespecific and firmly contextualized in the genres, expectations, and conventions of the particular field of study. This principle, argues Garside (2002), has been all too often overlooked or poorly implemented in university education. New Zealand faces a need more urgent than ever before for research into the specific need for oral competencies in particular

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sectors of the workforce, research that will provide university departments with data and arguments that either support or recommend against incorporating training in oral communication into their programs.

Case Study Massey University, New Zealand, is a public university with an enrolment of approximately 18,000 students, many of those enrolled extramurally in the Distance Learning program. While the university recently approved an overarching language, literacy, and communication skills policy, no implementation of that policy has occurred, and no compulsory Oral Communication requirement exists within undergraduate courses at Massey. Most Bachelor’s degrees are taught in three intensive years, without a general education program, with students beginning specialized programs from their first year of entry. Because many degree programs are extremely tightly structured with compulsory courses, students have very constrained opportunities to take elective communication classes. There is, in any case, a very limited supply of oral communication elective classes for students to enroll in: Massey has no Schools of Rhetoric or Composition, which are relatively unknown in New Zealand, and even the newly-created Bachelor of Communication degree (in which students may major in Expressive Arts, Media Studies, Journalism, or Communication Management) contains no compulsory speech class component. While the School of English and Media Studies does offer one lower-division course each semester in Speaking: Theory and Practice, which students across the university may take as an elective, the course only enrolls approximately 35 students per semester. In the College of Business, one course offers an introduction to business presentations, and enrolls approximately 70 students per year. In addition to the dearth of dedicated oral communication courses, existing university structures pose further difficulties for instructors wishing to incorporate communication across the curriculum. Many faculty and administrators strongly resist the idea of making room for communication instruction by shoehorning more content (and potentially more assessment) into set classes that are already strained for time. Furthermore, anecdotal evidence reveals that many instructors encounter implacable student resistance to having to undertake any oral communication tasks in the classroom. Massey’s lack of communication requirements may be grounded in student avoidance at least as much as resource availability; in a budget-conscious institution in which government funding is tied to enrolment numbers, the two problems are intimately connected. At the end of their undergraduate education, most Massey students will graduate with a bachelor’s degree having never received instruction in oral communication, and having never given a speech for assessment or feedback. Given the consistent viewpoint of New Zealand and international employers, students graduating without training in oral communication might reasonably be expected to be at a disadvantage on the job market. In an effort to address this imperfect preparation for the workplace, and to acknowledge the need for communication training that reflects discipline-specific conventions, a new course was introduced at Massey: Communication in the Sciences. Championed by the ProVice Chancellor of the College of Science, this was the first communication in the disciplines course to be established in a New Zealand university. The course was grounded in Writing Across the Curriculum (WAC) theory, and in 2000 made compulsory for almost all students

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enrolled in a science major. Designed through several years of close consultation between faculty of the School of English and Media Studies and the College of Sciences, the course included several written assessments and one oral communication assessment, an individual seminar presentation. The seminar presentation was designed to be a seven minute extemporaneous speech, based on a research essay the student had previously submitted in the same course, which was to be followed by a three minute question and answer period. Because of the logistical difficulties of administering and assessing the seminar for students studying at a distance, the Distance Learning version of the course omitted the seminar presentation (compensating by weighting the written assessments differently). An unanticipated consequence of this was that a number of students who were enrolled in their degree programs internally chose to enroll in the course externally, voluntarily giving up the learning benefits provided by face-to-face lectures and interactive workshops, in order to avoid the oral communication assessment. From the very beginning of the establishment of the course, both students and science faculty members repeatedly suggested that the seminar presentation be dropped from the curriculum. While a number of faculty members, particularly from the applied sciences, were strong supporters of the seminar presentation, a very vocal minority of science faculty (primarily from the pure sciences) questioned its usefulness and argued strongly for it to be discarded. At one stage, a proposal was made to develop an alternative course for students majoring in the pure sciences, which would not include any oral communication component. At the same time, many students expressed the view that the seminar presentation caused unwarranted and extreme anxiety; some students, as described above, took drastic steps to avoid the assessment. One preliminary study suggests that student resistance is caused by a belief that the skills developed by this assessment are not important to a career in science (Emerson, 1999). However, the teaching staff of Communication in the Sciences, supported by faculty from the Applied Sciences, felt very strongly that the skills taught and practiced by the seminar presentation were too important, both to the development of student learning and to the development of necessary career skills, to be jettisoned. Clearly, the inclusion of an oral communications component within a compulsory science communication class remains a contentious issue. Students resist the acquisition of oral communication skills for a number of reasons including communication apprehension, and faculty resist the inclusion of oral communication into course curricula and assessment, either because they see it as a low priority in terms of student skills and knowledge, and/or because they see it as irrelevant to a future career. Student and Employer Surveys In the light of the debates outlined above, and given the lack of specific information from the existing literature, the teaching team for the Communication in the Sciences course decided that hard data was needed to clarify whether oral communications were important to employers in science-related industries, what science employers meant by oral communication skills, and which skills they prioritized. At the same time, the team surveyed students within the Communication in the Sciences class to confirm the hypothesis that science students did not see the value of oral communication skills for their future careers within science-related industry. The research team generated a number of research questions in order to investigate these issues:

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R1 What do New Zealand science employers mean by ‘strong communication skills’? R2 What specific oral communication attributes, separate from written and interpersonal communication skills, may be identified as desirable to New Zealand science employers? R3 Do science students have an accurate understanding of employers’ requirements of oral communication skills? R4 Are science employers finding the oral communication skills they desire in graduate employees? Answers to these questions would, the team hoped, provide quality data on the importance of specific oral communication skills, which could then be used both to strengthen the case for communication courses within tertiary science programmes and to help in curriculum design and assessment decisions.

Method Questionnaires were supplied to approximately 300 science students and sent to 50 New Zealand employers. These questionnaires divided written communication, oral communication, and interpersonal communication into separate categories and identified specific attributes of each. The first, general question asked if the respondent agreed or did not agree with the statement: “Good communication skills are among the five most important attributes for employment.” The survey then asked respondents to value the importance of each attribute on a seven-point Likert scale, with 1 being “not at all important” and 7 being “extremely important,” and also to rank the relative importance of each attribute. The seven oral communication attributes listed in the survey were: • • • • • • •

The ability to speak professionally with clients The ability to give an effective seminar or speech (to a non-academic, non-scientific audience) The ability to listen carefully to others The ability to persuade other people to do something The ability to present a paper (to an academic/scientific audience) The ability to orally instruct other people on how to do something The ability to interact socially with different kinds of people.

Space was also made available on the questionnaire for respondents to add comments or clarify their views. Of the 50 questionnaires mailed to employers, 23 were returned, the response rate of 46% robust in the absence of reminder messages. The questionnaire invited interested respondents to volunteer for a follow-up telephone interview, and six did so, a participation rate of 26.1% of the total number of respondents. A range of businesses were represented in those who responded to the survey: 52.38% worked for a national organization; 33.33% for an international organization; 9.52% for a small business employing less than 25 people; and 4.76% from regional organizations. The kinds of industries represented also varied widely,

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including general business, agribusiness, research institutes, and a variety of others including financial and consulting firms. 15 of the 23 respondents reported hiring employees with an undergraduate degree in Science or in Applied Science within the last three years; seven of the remaining eight reported hiring students with postgraduate qualifications in science. The student questionnaire was distributed in a first year course compulsory to all science students and achieved a response rate of over 92% (N=280, although due to incompletions the number of answers to individual questions was in several cases <280). Of these respondents, 58.9% were enrolled in a Bachelor of Science and 20.7% in a Bachelor of Applied Science. Almost two-thirds (65.2%) were female and over 80% were aged less than 21. We did not request that students engage in follow-up interviews, as we considered employers to be our primary research focus for this study; student perspectives were elicited in order to investigate significant difference in responses between students and employers.

Findings At the most basic level, the survey showed that New Zealand science employers agree with the undifferentiated employers in previous surveys: 100% of responding science employers agreed that good communication skills were in the top five qualities they sought in new hires. This feedback was not unexpected, although the unanimous agreement was something of a surprise. The questionnaire further yielded more nuanced results. To find specific answers to the first research question, What do New Zealand science employers mean by ‘strong communication skills’? we divided up written, oral, and interpersonal communication skills into separate categories, and identified 15 written communication attributes, seven oral communication attributes, and 13 interpersonal communication attributes in these categories. The questionnaire revealed that the attributes employers scored most highly in the three categories were, respectively, the ability to write clearly, the ability to listen carefully to others, and the ability to work within teams. A complete list of the thirty-five communication attributes, and the scores employers and students gave to each category, may be found in Table 1. The seven specific oral communication skills listed in the questionnaire were designed to provide answers to the second research question, What specific oral communication attributes, separate from written and interpersonal communication skills, may be identified as desirable to New Zealand science employers? Employer responses revealed clear distinctions in the values accorded different attributes, and one clear winner emerged. Employers ranked listening carefully to others as the most important area of oral communication, giving this skill the value of 6.75 out of a possible 7 (SD=0.6). Comments from telephone interviews underlined employers’ concern with the critical importance of listening skills. Employers variously identified managers, clients, and colleagues as possessing information critical to a new employee’s successful completion of any given task, and stressed that the employee had to recognize the importance of being receptive to and processing that information. One employer stated: “From a feedback perspective, when an employee has a brief from a project manager, it’s incredibly important to get that as right as possible. Listening’s important in order to pick things up, understand what the issues are, sell products and services to clients… A whole range of reasons.” Echoing this emphatic evaluation, another employer stated:

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“Listening is fundamental. Firstly, you have to have the ability to understand technical instructions, that’s really important – and lots of our work is planned, developed, reviewed, and implemented at team meetings, so listening skills are fundamental.” Table 1. Science Employer and Science Student Ratings of Communication Competencies

Written communication competencies The ability to spell correctly The ability to use correct punctuation The ability to use correct grammar The ability to express ideas clearly The ability to write in business format (for a non-academic audience) The ability to write an academic report (for an academic audience) The ability to write persuasively The ability to convey information accurately The ability to write in styles appropriate to different readers (clients, employees, govt agencies) The ability to write logically The ability to collect relevant information from a variety of sources The ability to condense material from a variety of sources and convey it clearly The ability to use a professional writing style The ability to write clear instructions The ability to write an academic paper (for publication)

Employer Mean

SD

Student Mean

SD

5.4 5.4 5.9 6.9

1.0 1.1 0.9 0.4

5.3 5.1 5.5 6.2

1.3 1.4 1.3 0.9

5.5

1.1

5.3

1.3

5.8 5.7 6.7

1.1 1.0 0.5

5.7 5.4 6.4

1.2 1.2 0.7

5.9 6.4

0.9 0.7

5.8 6.1

1.1 0.9

6.5

0.9

6.2

0.9

6.0 5.8 6.3

0.9 1.0 0.7

6.0 5.6 6.1

1.0 1.2 0.9

5.1

1.5

5.0

1.5

Oral communication competencies The ability to speak professionally with clients The ability to give an effective seminar or speech (to a nonacademic audience) The ability to listen carefully to others

6.3

0.7

6.1

1.2

5.4 6.8

1.2 0.6

5.2 6.3

1.4 0.9

The ability to persuade other people to do something

5.3

0.8

5.4

1.3

The ability to present a paper (to an academic audience) The ability to orally instruct other people on how to do something

4.6

1.3

5.0

1.5

5.7

1.1

6.2

1

The ability to interact socially with different kinds of people

5.9

0.8

6.2

1.1

Interpersonal communication competencies The ability to lead a team

4.5

1.6

5.7

1.1

The ability to work within a team (not a leadership role) The ability to encourage and support co-workers

6.5 5.8

0.7 1.0

6.3 6.1

0.9 1.0

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Patience with others The ability to interact professionally with other clients or other professionals The ability to interact socially with clients The ability to accept criticism Tolerance of different kinds of people The ability to listen to other people’s needs The ability to draw people together A sense of humour Kindness and generosity The ability to see other people’s weaknesses

Employer Mean 5.5

SD 1.2

Student Mean 6.3

SD 0.9

6.1 5.2 6.1 6.2 6.4 5.0 5.1 4.8 4.5

1.4 1.2 0.7 0.8 0.8 1.3 1.5 1.4 1.2

6.3 5.8 6.1 6.3 6.2 5.4 5.6 5.6 5.3

0.9 1.2 1.0 1.0 1.0 1.2 1.3 1.3 1.5

One anonymous respondent wrote into the questionnaire that he/she considered another attribute of oral communication to be essential in a graduate employee: “Questioning skills – interviewing clients.” This comment helps highlight a difference in the kinds of oral communication skills most valued by science employers. Overall, the values science employers accorded specific oral communication skills indicate they value more highly an employee’s ability to receive information as opposed to an employee’s ability to deliver information (for the relative rankings of oral communication attributes given by science employers and science students, see Table 2). Science employers gave the highest rankings to the three oral communication attributes that foregrounded the contribution of others to the communication process: listening, speaking professionally with clients, and interacting with others socially, respectively. The highest ranked one-way communication attribute was “The ability to orally instruct other people on how to do something,” which employers ranked fourth in importance out of the seven attributes, and which received the score of 5.65 (SD=1.1). The implicit position that science graduates must be receptive to information and instruction, and must recognise that they still have a great deal to learn, was expressed succinctly by one employer: “They [new graduates] have a perception of their abilities much higher than they actually are.” Results also show that science employers differentiate between oral communication skills as applied to professional and social contexts. Thus “the ability to speak professionally with clients” was ranked by employers as the second most valuable oral communication skill, with a value of 6.3 (SD=0.7), while “the ability to interact socially with others” received the lower score of 5.9 (SD=0.8). Despite the burgeoning sub-genre of texts and manuals on the subject of presentations in science, science employers ranked the ability to deliver presentations lowest in importance amongst the specified oral communication skills, although again they drew a distinction between the ability to present to a general audience and the ability to a present to a specialist, scientific audience. With a score of 5.36 (SD=1.2), “the ability to give a seminar/speech to a non-specialist audience” was valued by employers as more important than “the ability to present a paper to an academic or scientific audience” (valued at 4.6, SD=1.3).

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Table 2. Science Employer and Science Student Rankings of Communication Competencies

Employer

Student

1 2

2= 1

The ability to collect relevant information from a variety of sources The ability to write logically The ability to write clear instructions The ability to condense material from a variety of sources and convey it clearly The ability to use correct grammar The ability to write in styles appropriate to different readers (clients, employees, govt agencies)

3 4 5

2= 4= 4=

6 7=

6 10

7=

7

The ability to write an academic report (for an academic audience) The ability to use a professional writing style The ability to write persuasively

9= 9= 11

8 9 11

The ability to write in business format (for a non-academic audience) The ability to spell correctly The ability to use correct punctuation The ability to write an academic paper (for publication)

12 13= 13= 15

12= 12= 14 15

Oral communication competencies The ability to listen carefully to others The ability to speak professionally with clients The ability to interact socially with different kinds of people

1 2 3

1 4 2=

4

2=

5 6 7

6 5 7

The ability to work within a team (not a leadership role) The ability to listen to other people’s needs Tolerance of different kinds of people

1 2 3

1= 5 1=

The ability to interact professionally with other clients or other professionals The ability to accept criticism The ability to encourage and support co-workers

4= 4= 6

1= 6= 6=

Written communication competencies The ability to express ideas clearly The ability to convey information accurately

The ability to orally instruct other people on how to do something The ability to give an effective seminar or speech (to a non-academic audience) The ability to persuade other people to do something The ability to present a paper (to an academic audience) Interpersonal communication competencies

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Patience with others The ability to interact socially with clients A sense of humour The ability to draw people together Kindness and generosity The ability to lead a team The ability to see other people’s weaknesses

Employer 7 8 9 10 11 12= 12=

Student 1= 8 10 12 11 9 13

However, both kinds of presentations were ranked lower than the less formal forms of oral communication, “persuasion” and “instruction,” the more common kinds of spoken interaction undertaken in the course of the working day. Science employers valued “the ability to persuade other people to do something” at 5.3 (SD=0.8), but, as mentioned above, employers rank the ability of an employee to instruct as well as to persuade well below an employee’s ability to carefully listen to the instructions (and persuasions) of others. Despite our intentions of comprehensiveness, our list of seven categories of oral communication was not exhaustive. In the phone interviews, science employers were asked if there were other specific oral communication skills which they considered particularly important in a graduate employee. Several respondents also took the opportunity to write into their paper questionnaire additional oral communication skills they considered important. Additional skills mentioned included the aforementioned “questioning skills, interviewing clients,” as well as the ability to give critical feedback “in a constructive, non-confrontational way,” and “participation in group meetings and workshops.” Our third research question, Do science students have an accurate understanding of employers’ requirements of oral communication skills? elicited a number of disparities as well as unexpected agreement. While we were somewhat surprised by science employers’ 100% unanimity on the importance of communication skills in employees, we were even more surprised to find the degree to which science students concurred: 97.9% agreed that communication skills were among the top five important attributes for employment. Students also agreed with employers that the ability to listen carefully was the most important of the listed oral communication skills, although they gave it a value of 6.3 (SD=0.9), as compared with the employers’ value of 6.75. When science students were asked to consider the importance of other specific oral communication skills, the results become further differentiated and more revealing. While science employers valued an employee’s ability to receive information over his or her ability to impart it, science students, on the other hand, perceived a far greater importance in telling others things. Students ranked “the ability to give others oral instruction” as equal second in importance of the listed oral communication skills, and valued it at 6.2 (SD=1.0). Interestingly, students also valued “the ability to persuade others to do something” slightly more highly than employers (valuing this attribute at 5.4 as compared to employers’ 5.3). With these disparities reinforcing employers’ qualitative comments, it seems students may be overconfident about the level of knowledge they will have when they graduate,

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overestimating how much information they will have to impart and underestimating how much they will still have to learn when they commence their careers. Science employers valued an employee’s ability to speak professionally with clients over the ability to speak socially in a general context. Science students saw the respective values of these two skillsets differently, giving social interaction skills a 6.2 (SD=1.1) and client interaction skills a slightly lower 6.1 (SD=1.2). However, the small difference between the scores attached to these skillsets may indicate that students perceive relatively little difference between speaking in a professional context with clients and conversing socially, more generally. As the difference between the two oral communication situations is perceived as more significant by employers, these findings reinforce the case for more training of students in analyzing particular audiences and tailoring communication content, style, and form, with an emphasis on practical and situational application. While several students wrote into their questionnaires the importance in the workplace of presentation skills, including the use of visual aids, and coalface experience indicates considerable student anxiety about this area, science students like science employers ranked the ability to deliver presentations as least important amongst the oral communication skills specified in the questionnaire. Both stakeholder groups agreed that the ability to present to a generalist audience was more important than the ability to address a specialist scientific audience, and both kinds of presentations were ranked by both groups lower than the less formal forms of oral communication, “persuasion” and “instruction.” Interestingly, while presentation skills seem to be a focus for anxiety and resistance amongst science students, students seem to have an understanding of the significance of this particular oral communication attribute in their future careers that is reasonably close to the opinion of employers. Similarly significantly, while specialist scientific presentations were ranked least important by both groups, the values given were nonetheless relatively high (5.0 of a possible 7 on the part of students, and 4.6 of a possible 7 on the part of employers.) The ability to deliver an oral presentation clearly remains a recognised and valued oral communication skill in the New Zealand scientific workplace. While the questionnaire revealed striking agreement amongst both employers and students concerning the importance placed on oral communication generally, the survey data related to the fourth research question: Are science employers finding the oral communication skills they desire in graduate employees? found disappointingly negative answers. While no science employer reported “never” finding these skills in the science graduates they had hired, 55% stated their new hires demonstrated the required oral communication skills only “sometimes” or “occasionally.” One respondent wrote trenchantly: “Engineers and scientists can rarely write and only occasionally present clearly. It is definitely a key area for improvement within the curriculum.”

Discussion The findings of this study clearly support the case for oral communication skills to be included in tertiary science programmes and in the design of communication courses. Perhaps the most surprising result of the survey was that students clearly saw the value of oral communication skills for their future careers. This finding made us reassess our approach to handling students who resisted this aspect of assessment. Where we had previously

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attempted to address student resistance by trying to convince them of the usefulness of the assessment, we now approached the issue in terms of how to handle fear and anxiety. The findings have broader implications for tertiary curricula. The fact that students and employers in science-related industries all confirm the importance of oral communication skills in science graduates renders more pressing the question of why science faculty, and administrators who design overall program curricula, are not equally convinced of the importance of such skills. Further work is needed to clarify the root of resistance, and to suggest ways to address it. Applying a more specific focus, the findings give clear indications of what skills need to be incorporated into science communication courses. While seminars are the easiest form of oral communication skills to incorporate into course content and assessment, our findings suggest the need for further investigation into ways to incorporate some of the more preferred skills, such as listening skills, into assessment structures.

CONCLUSION Findings from our study strongly support the importance of oral communication skills in science-based employment in New Zealand. Science employers indicate that they require and value highly a variety of oral communication skills, from listening, to speaking with clients, to making presentations for general audiences. The study also demonstrates that while science employers and university science students agree that oral communication skills will be important in scientific careers, the majority of employers find the desired level of these skills in new graduates only sometimes or occasionally. The retention of oral skills teaching and assessments, as currently exemplified by the Communication in the Sciences course at Massey University, is clearly indicated. There is also a strong case to be made for an extension of the focus on oral competencies, despite the difficulties of limited resources of time and funding. Further work is clearly needed to locate the exact causes of faculty resistance to including oral communications into science programmes. Furthermore, it is important that these findings are disseminated to science faculty to challenge their resistance. Finally, program administrators in tertiary institutions need to be made aware of the breadth of research which establishes the importance of integrating communication, and oral communication in particular, into the science curriculum.

REFERENCES Andrews, R. J. (1995). A survey of employer perceptions of graduates of the University of Otago. Dunedin, New Zealand: University of Otago. Australian Association of Graduate Employers. (1993). National survey of graduate employers. Sydney, Australia: Direct Connection Marketing Consultants. Bland, V. (January 12, 2005). Levelling the playing field. New Zealand Herald Online. Retrieved 18/2/2005 from http://www.nzherald.co.nz

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Burchell, N., Hodges, D. and Rainsbury, E. (2001). What competencies do business graduates require? Perspectives of New Zealand stakeholders. Journal of Cooperative Education, 35, 11-20. Cain, R. (March 6, 2006). Speech as vital as writing. New Zealand Herald Online. Retrieved 27/6/2006 from http://www.nzherald.co.nz Chanock, K. (2005). Investigating patterns and possibilities in an academic oral genre. Communication Education, 54:1, 92-99. Coll, R. K. (1996). The BSc (Technology) degree: Responding to the challenges of the education marketplace. Journal of Cooperative Education 32, 39-45. Coll, R. K. and Zegwaard, K. E. (2006). Perceptions of desirable graduate competencies for science and technology new graduates. Research in Science and Technological Education, 24:1, 29-54. Cronin, M. and Glenn, P. (1991). Oral communication across the curriculum in higher education: The state of the art. Communication Education, 40, 356-367. Cronin, M. and Grice, G. (1993). A comparative analysis of training models versus consulting/training models for implementing oral communication across the curriculum. Communication Education, 42, 1-9. Cronin, M. W., Grice, G. L., and Palmerton, P. R. (2000). Oral communication across the curriculum: The state of the art after twenty-five years of experience. Journal of the Association of Communication Administration, 29, 66-87. Dannels, D. P. (2001). Time to speak up: A theoretical framework of situated pedagogy and practice for communication across the curriculum. Communication Education, 50, 14458. Dannels, D. P. (2002). Communication across the curriculum and in the disciplines: Speaking in engineering. Communication Education, 51, 254-268. Darling, A. L. (2005). Public presentations in mechanical engineering and the discourse of technology. Communication Education, 54:1, 20-33. Davis, B. (1992). Critical thinking and cooperative learning: Are they compatible? In W. Oxman (ed.), Critical thinking: Implications for teaching and teachers. Conference proceedings of the New Jersey Institute for Critical Thinking Conference. ERIC Document Reproduction Service ED352358. Driscoll, M. P. (1994). Psychology of learning for instruction. Boston: Allyn and Bacon. Emerson, L. (1999). A collaborative approach to integrating the teaching of writing into the sciences in a New Zealand tertiary context. Unpublished doctoral thesis: Massey University, Palmerston North, NZ. Garside, C. (1998). Can we talk? The role of oral communication in student learning. In J. Tapper and P. Gruba, (Eds.), Proceedings of the Australian Communication Skills Conference (68-77). Melbourne, Australia; University of Melbourne. Garside, C. (Jan 2002). Seeing the forest through the trees: A challenge facing communication across the curriculum programs. Communication Education 51:1, 51-64. Hart, S. (October 18, 2004). Txting’s no wy 2 gt a jb. New Zealand Herald Online. Retrieved 18/2/2005 from http://www.nzherald.co.nz Higher Education Council (1992). Higher education: Achieving quality. Canberra, Australia: NBEET/AGPS. Jones, E. A. (1994). Defining essential writing skills for college graduates. Innovative Higher Education 19:1, 67-78.

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Modaff, J. and Hopper, R. (1984). Why speech is “basic.” Communication Education, 29, 332-347. Morreale, S. (1990). The competent speaker: Development of a communication competency based speech evaluation form and manual. (ERIC Document Reproduction Service ED325 901). Morello, J. T. (2000). Comparing speaking across the curriculum and writing across the curriculum programs. Communication Education, 49, 99-113. Morton, J. and O’Brien, D. (2005). Selling your design: Oral communication pedagogy in design education. Communication Education, 54:1, 6-19. New Zealand Union of Students’ Associations. (2006). Campaign to increase government funding for public tertiary education. Retrieved 7/7/2006 from http: //www. students.org. nz/index.php?page=publicfunding Palmerton, P. R. (1992). Teaching skills or teaching thinking? Journal of Applied Communication Research, 20, 335-341. Rainsbury, E., Hodges, D., Burchell, N. and Lay, M. (2002). Ranking workplace competencies: Student and graduate perceptions. Asia-Pacific Journal of Cooperative Education, 3, 8-18. Reid, I. (1997). Disciplinary and cultural perspectives on student literacy. In Z. Golebiowski and H. Borland (Eds.), Selected proceedings of the first national conference on tertiary literacy: Research and practice, vol. 2 (1-11). Melbourne, Australia: Victoria University of Technology. Rubin, R. and Morreale, S. (1996). Setting expectations for speech communication and listening. In E. Jones (Ed.), New directions for higher education: Vol. 96, Preparing competent college graduates: Setting new and higher expectations for student learning (19-29). San Francisco: Jossey-Bass. Rubin, R. and Morreale, S. (2000). What college students should know and be able to do. Journal of the Association for Communication Administration, 29, 53-65. Steinfatt, T. (1986). Communication across the curriculum. Communication Quarterly, 34, 460-470. Tapper, J. (2000). Preparing university students for the communicative attributes and skills required by employers. Australian Journal of Communication 27, 2, 111-130. Tapper, J. and P. Gruba, eds. (1998). The Proceedings of the Australian Communication Skills Conference. Melbourne, Australia: The University of Melbourne. Victoria University of Wellington, Careers Advisory Service (1996). Skills requested by employers in 1996 employer visit programme. Student information sheet, Wellington, New Zealand: Author. Waller, D. S. and Hingorani, A. Perceptions of business students towards skills and attributes for industry: How important is communication? Proceedings of 2006 ANZCA Conference. Retrieved 13/2/2007 from http:// www. adelaide. edu.au /anzca2006 /conf_ proceedings/ Zorn, T. E. (1998). Educating professional communicators: Limiting options in the new academic ‘marketplace.’ Australian Journal of Communication, 25:2, 31-44

In: Science Education Issues and Developments Editor: C. L. Petroselli, pp. 239-255

ISBN: 978-1-60021-950-4 © 2008 Nova Science Publishers, Inc.

Chapter 9

SUPPORTING FUTURE TEACHERS LEARNING TO TEACH THROUGH AN INTEGRATED MODEL OF MENTORING Pi-Jen Lin1 National Hsin-Chu University of Education, Taiwan

ABSTRACT The purpose of this article is to introduce an integrated model of mentoring for supporting future teachers learning to teach under the impact of teacher education reform of Taiwan, particularly, in the internship. This article begins with the introduction of teacher education reform and is followed by the description of the impact of teacher education on quality control. Then, it includes a brief description of six integrated reach projects investigated by teacher educators. One of the integrated research projects that was designed to improve mentors’ competence of mentoring for supporting future teachers learning to teach is reported in detailed and an integrated model of mentoring is developed. Finally, the views of mentors and the future teachers are described briefly and the issues of mentoring are addressed.

Keywords: Future teachers, integrated model, internship, metnors.

INTRODUCTION Teacher preparation programs across countries have made considerable efforts to the content and the process of the practicum. The practicum stipulated allows the future teacher (FT) to have field experience in school settings throughout the entire school year with the support of university faculty and school teachers. Due to much of the responsibility for 1

All correspondence to: Pi-Jen Lin, Email [email protected], National Hsin-Chu University of Education, Taiwan,521, Nan-Dah Road, Hsin-Chu City 300. Taiwan, R. O. C.

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mentoring FTs in Taiwan lie with the mentor in the schools who are not subject specialists rather than with university faculty. As a result, FTs have little professional learning with school teachers during practicum. With drastic changes of economy, policy, and society, quality control is an emerging issue since the teacher preparation reform was issued by the Ministry of Education (MOE ) of Taiwan in 1994 (MOE, 1994). A great deal of researchers on teacher education have paid a lot of attention to the studies of teacher preparation, but these studies are limited on the learning opportunities for FT provided by the teacher preparation program in teacher education institutes, the criteria of recruitment and selection, university-based course and practicum requirements, and accreditation systems for teacher education (Fwu and Wang, 2002). Teacher education reformers of Taiwan did not focus on the supports of FTs with creating opportunities for improving quality of teaching, in particular in the internship until the privilege of teacher colleges or Normal university for teacher preparation is deprived. The main focus of the article is on the introduction of a new program that was designed to improve mentors’ competence of mentoring for supporting FTs’ quality of teaching with the support of school-university partnership in the internship.

TEACHER EDUCATION REFORM OF TAIWAN The teacher education reform under the impact of economic, political and social contexts has demonstrated a drastic change since the Teacher Education Act (TEA) of Taiwan was issued in 1994. The major changes are: 1) School-based practicum is reduced to a half year from a whole year and attached in the fourth year of a four-year teacher preparation program; 2) Teachers are certified by the processes consisting of qualifying a teacher while completing four-year courses, half-year practicum, and then certifying a teacher after passing a certified teacher examination. 3) The teacher preparation is opened to any institution which has a teacher education program (MOE, 1994).

The Impact of Teacher Education Reform The TEA declared that all four-year public and private universities and colleges are allowed to run teacher education programs for training teachers as long as they meet the requirement of the MOE. As a result, deprived of the privileges in teacher training, teacherscolleges suffered from lower popularity among high school graduates and a decrease in students’ academic level. Under situations, some teacher colleges upgraded to a university of education or a comprehensive university. The declining government budget for higher education and the limited quantity and quality in faculty and facilities made the transformation of universities of education or to seek opportunities to integrate them into nearby universities. In addition, with the decreasing population of babies born, the supply of teachers from teacher education programs is higher than the demand. The number of teachers to be prepared from each university of education is required to be reduced by 50% as many as before (MOE, 2005).

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The establishment of teacher education programs by any university or college needs to be approved by the MOE according to a set of official criteria for quality in the faculty, curriculum, and facilities of the programs. However, the process of training, curriculum, instruction, and practicum vary among the different teacher education programs. Some programs in universities have an inadequate number of faculties and a lack of practical experience in internship. To control teacher quality, a national examination of teacher inspection was ignited in 2004. However, only FTs’ knowledge of general pedagogy instead of subject matter pedagogy is assessed in the inspection. The inspection is not able to assess how well the future teachers (FTs) performed in teaching learned from teacher education program. With the exception of the high percentage of FTs passing the national examination of teacher inspection to be achieved, the teacher educators of Taiwan were aware of the importance of FTs’ ability in performing in the classroom. They recognized the practicum as an important component in teacher education. Teaching is a form of highly complex and skilled practice depending on teachers’ knowledge and skill. A knowledge base including a theoretical and a professional component underpins teaching. The theoretical component is taught in the years of teacher preparation program, while the professional component needs to be developed in the professional practice. However, the practicum provides FTs with an opportunity to develop the professional knowledge but it often results in FTs developing the technical skills of classroom management, rather than the wisdom of professional practice. Within ten years, a method of assisting FTs to develop professional knowledge in partnership formed between schools and universities where FTs have opportunities to be involved with the day-to-day activities of professional practice. The studies on teacher preparation show that FTs complained that they are required to devote a great deal of time to administrative affairs of schools. They were mentored by the mentors who do not have enough professional knowledge in mentoring (Lo, Hung, and Liu, 2002). Thus, their professional knowledge was not developed during the internship although the school-university partnership was implemented. The failure of the partnership could have resulted from the unsuccessful mechanism of collaboration between school and university. When comparing the successful experience of the USA implementing the partnership of school-university in practicum, teacher educators of Taiwan attempted to reconstruct a new concept of the school-university partnership that was designed to enhance the mentors’ knowledge and skill such that improving the quality of practicum by providing FTs with greater involvement with mentors in teaching. Since teacher education reformers of the USA indicate that the structure of the relationship creates the opportunities for the FTs to relate the theoretical knowledge to the practical realities of schools and classrooms. They regard through the school-university partnership FTs as one of the important strategies to support FTs’ learning to teach, and thus, to improve the quality of teaching (Odell, Huling, and Sweeny, 1999). In these experiences, the FTs are not focused on the technical skills of classroom management. Instead, the FTs are engaged in meaningful professional-related tasks.

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The Issues of Internship Emerged Under the Reform Since the enactment of the TEA in 1994, the issues of the ambiguity of FT’s role, government’s over-loaded on allowance of internship, and diversity in outcome quality control have emerged. The ambiguity of FTs’ role during practicum is an issue under the TEA impact. FTs in school placement were neither a student (because of their completion of courses of TE program) nor a teacher (because of no salary). FTs were dominated by mentors by school administrators to devote a great deal of time to doing school administrative affairs. The FTs were afraid of rebelling school teachers’ authorities because the part of their grade of internship was graded by school teachers or mentors. Therefore, it leads to lack of professional learning during the internship. Each FT gets NT$8000 monthly allowance for internship during school placement. It is overloaded for government in finance. On the other hand, for FTs, monthly allowance with NT$8000 is not sufficient for affording FTs’ living. It is imbalance between the hours FTs worked and the pay they gained. Diversity is another essential feature of the teacher education reform. Due to the huge variance in FTs’ quality and the training process at individual universities, the MOE is worried that the mushrooming of TE programs in the past few years might result in a decline in teacher quality. Thus, it is necessary to establish a uniform standard for assessing FTs’ quality through a nationwide licensing examination to assure teacher quality.

MUSHROOMING OF INTEGRATED RESEARCH PROJECTS ON MENTORING Due to the enactment of the TEA, the number of TE programs set by regular universities has accelerated, from the initial 9 programs in 1994 to 88 programs approved by the MOE in the year 2006 (MOE, 2005). Mushrooming of these programs has indicated the variance of the teacher quality. Due to the variance in training process at individual universities, the MOE is worried that the increment of the TE programs in the past few years might result in a decline in teacher quality. To overcome this problem, recently, the National Science Council associated with MOE funded a goal-oriented research grant to call for research proposals from teacher educators. The goal-oriented program is to improve the quality of teacher education in mathematics and science. Within three years, there were six integrated research projects approved by NSC were investigated by mathematics and science teacher educators from University of Education and Normal universities (National Hsinchu University of Education, 2006). Four of them were at the primary level and two were at the secondary level. The six integrated research projects are displayed in Table 1 by level, subject-matter area, and goals.

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Table 1. Introduction of Six Integrated Research Projects projects

MP1

Mentors

○

○

○

○

○

○

FTs

○

○

○

○

○

○

○ ○

○ ○

○

○ ○

○

○

○

○

○

○

○

○ ○ ○

○ ○ ○

○ ○

MP2

SP1

SP2

MC1

MC 2

characteristics

Establishing professional Standards

Mentoring program School-based Mathema tics Subject Chinese Science Primary Level Secondar y

○ ○

○

○

○

M: mathematics S: Science P: Primary level C: Secondary level.

Table 1 shows that only one project (SP2) investigated mentoring in science and only one project (MC1) involving in mathematics. There were two integrated research projects (MP1 and MP2) involving in three subject-matter areas, mathematics, Chinese, and science. Each integrated research project is described briefly as follows.

MP1. A Project of Professional Development of Mentors at Primary Level The integrated research project MP1 includes four sub-projects which explored the literacy of subject-based teaching including mathematics, Chinese, science and non-subject based technology at primary level. The sub-project involving in technology was intended to be integrated into the three subject-matters. MP1 is a three-year research project. It was begun by establishing a set of professional standards for mentors and for FTs, respectively. The second year was to develop and design a mentor training program. The third year was to develop an interactive model of mentoring by integrated three subjects. The mentoring model of MP1 was to meet the need of teaching a range of subjects for a teacher at the primary level. It was called as a model of one-subject mentors with multiple-subject future teachers (OSMMSFT). It means that each mentor was only trained to be specialized in one subject by subject-matter teacher educator of the university.

MP2. A Project of Developing Mentors’ Professional Development The integrated research project MP2 including four sub-projects was designed to develop a set of professional standards for mentors in teaching mathematics, Chinese, science at primary level. It was followed by setting up a mentoring program and developing a

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collaborative model between mentors and FTs. Then, it was ended by assisting professional growth in a school-based context.

SP1. A Project of Mathematics and Science Teachers’ Professional Standards and Professional Development The integrated research project SP1 was designed to develop a set of professional standards and develop a training program for mentors at primary level for improving the effectiveness of mentoring. SP1 was a three-year research project. The first two years were working on the establishment of professional standards and the design of mentoring program, respectively. The third year was to evaluate the effectiveness of the training program and revised. The mentors and FTs involving in SP1 are in mathematics and science area.

SP2. A Project of Science Mentors’ Professional Development and Training The integrated research project SP2 was designed to construct a theory and a model of training a science teacher to be a mentor at primary level. This was a three-year research project. The first year was to establish a set of professional standards for mentors. It was followed by developing a training program for mentors. The second year was to develop a model of mentoring with a university-based approach. The third year was designed to develop a model of mentoring with a school-based approach.

MC1. A Project of Professional Development for Mathematics Mentors at Secondary Level The integrated research project MC1 was designed to establish a set of professional standards for mathematics mentors at secondary level. The standards consisted of two dimensions: mathematics and teaching mathematics. MCI consisted of six sub-projects. There were three sub-projects set up professional standards with respect to mathematics dimension including geometry, algebra, and statistics. The other three sub-projects were related to the establishment of professional standards of teaching mathematics including mathematics teaching, meta-cognition, and culture of mathematics. MC1 was a three-year research project. The first year was to set up a set of ideal professional standards for mentors and for FTs, respectively. The second year was to develop a set of professional standards for FTs to be a qualified teacher. The third year was to develop an instrument of inspecting a FT to be an initial teacher.

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MC2. A Project of Establishing Professional Standards and Professional Development for Mentors and FTs The integrated research project MC1 was designed to establish a set of professional standards for mathematics and science mentors at secondary level and a set of professional standards for mathematics FTs at secondary level. There were five sub-projects involving in the integrated project MC1. The first year was to set up a set of professional standards for mentors and FTs in mathematics and science. The second year was to design a training program for mentors and FTs. The third year was to evaluate the effect of the mentoring program. In sum, the establishment of professional standards for mentors and for FTs and followed by setting up a mentoring program is common goal among the six integrated research projects. Although setting up a set of professional standards for mentors and for FTs respectively were the purpose of the integrated projects, the distinction among them was varied by subject area, level, and methodology. The design of a mentoring program, the development of a model of mentoring, and the evaluation of the mentoring program were the main focus among the six integrated projects. The introduction of each integrated project was not the purpose of the article. Instead, only one integrated research project MP1 is reported in this article.

AN INTEGRATED MODEL OF MENTORING PROGRAM FOR IMPROVING QUALITY OF INTERNSHIP A Mentoring Program: MP1 The goal of the half-year mentoring program for mathematics mentors group as part of the integrated research project MP1 was to enhance mentors’ knowledge and skill of mentoring. The mentoring program was based on the professional standards of mentors that were conducted by the author in the first year of MP1 project (Lin and Tsai, 2007). The professional standards describe the indicators of preliminary knowledge and skill of a teacher to be a mentor. The course of mentoring program includes two parts, professional knowledge and skills of mathematical teaching and mentoring. Each part includes five topics: curriculum, pedagogy, assessment, social mathematics norm, topics about individual students. Curriculum topics refers to the objectives for instruction, the scope and sequence of the content to be learned, the sequence of activities, textbook, resources of teaching, and the plans and schedules for teaching. Pedagogical topics cover the discussions on subject mater knowledge, instructional strategies, clarity of explanation, questioning, problem-posing, and analyzing students’ various solutions. Assessment topic related to assessing students’ learning and performance as well as their progress. Social mathematics norm topic is the issues about social interaction in mathematics classroom, the norms of groups of students in a class. Topic about individual students included discussions about the background, learners’ needs, behavior, and progress of an individual student (Lin, 2007).

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The courses of mentoring program provided by the researcher, who is the teacher educator of university, including theory and practice of mathematics teaching and mentoring were implemented in a six-day summer workshop with 36 hours and half school-year with 42 hours. The summer workshop was to conceptualize mentors’ and FTs’ knowledge of teaching mathematics toward learner-oriented, while the course of the school year was to enhance mentors’ knowledge and skills in mentoring and FTs’ knowledge of teaching.

An Integrated Model of Mentoring An integrated model of mentoring was explored in the integrated research project MP1. The purpose of the MP1 was to improve the knowledge and skill in mentoring for mentors via the partnership of school-university. In developing the school-university partnership on FT preparation, there were five main considerations. First, the school to be recruited was dependent on the willing of the mentors and the FTs. Second, the school to be recruited at least consists of the mentors from mathematics, Chinese, and science. Third, the school has a commitment to maximize the FTs’ involvement in the community of mentors while at the same time minimizing the possible disruption this participation might cause the mentors and schools. Fourth, some kind of ancillary benefits and feedbacks for giving back to the school from the partnership were also a consideration for the university when designing the mentoring program, for instance, minor finical support and certified hours of institutes. Fifth, the school needs to offer mentors and FTs supports on professional practices. The final consideration was that the supervisors (or teacher educators) of the FTs during practicum are the researchers involving in the integrated research project. This consideration is intended to reduce FTs’ burden from the researchers and supervisor of practicum as possible. It is not possible for developing FTs’ professional knowledge if the mentors’ mentoring knowledge and skills have not been developed well. The mentoring program associated with school-university partnership was designed to assist mentors in developing mentoring knowledge and skills, and then to enhance FTs’ professional practice during practicum. The mentoring program included three subject mentoring programs, mentoring in mathematics, mentoring in Chinese, and mentoring in science, as part of an integrated research project. Due to FTs to be a primary school teacher who teach several subjects, mathematics and Chinese are required to be taught for a home-room teacher. To meet this need, an integrated model of mentoring was developed. The integrated model of mentoring was that each participant FT was mentored by a mentor in mathematics and mentored by another mentor in Chinese. Mathematics future teachers group consisting of FTm1, FTm2, FTm3, and FTm4 were mainly mentored to be a professional teaching in mathematics. They were mentored by mathematics mentors group consisting of A, B, C, and D, and were also mentored to be a professional teaching in Chinese assisted by Chinese mentors group consisting of P, Q, R, and S. Each FT in mathematics group was mentored by a mentor from mathematics mentor group and a mentor from Chinese mentor group. Likewise, Chinese future teachers group consisting of FTc1, FTc2, FTc3, and FTc4 were mainly mentored to be a professional teaching in Chinese. They were mentored by Chinese mentors group and also were mentored by mathematics mentors group. Each FT in Chinese group was mentored by a mentor from Chinese mentor group and by a mentor from mathematics mentor group. Science future teachers group consisting of FTs1 and FTs2 were mentored to be a professional teaching in

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science and were mentored by science mentors group consisting of I and J. Each FT in science group was mentored by a mentor in science mentor group.

Figure 1. The integrated model of one-subject mentors with multiple-subject future teachers (OSMMSFT).

Figure 2. The Partnership of School-University.

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However, to reduce mentors’ tension and burden from their participation in the mentoring program, each mentor was only trained to be specialized in one subject by subject teacher educator of the university. The subject teacher educators were the researchers, who participated in the integrated research project. For instance, the mentors A, B, C, and D were trained to be an expert in mathematics teaching assisted by the researcher from mathematics department, while mentors P, Q, R, and S were trained to be an expert in Chinese teaching assisted by the teacher educator from Chinese department. The integrated model displayed in Figure 1 is called as a model of one-subject mentors with multiple-subject future teachers (OSM-MSFT). The connection among future teachers, teacher educators of university, and mentors in a school is described in Figure 2. The partnership of school-university was designed to form three different professional mentoring groups, mathematics, science, and Chinese in the school. Mentors I, and J are trained to be an expert in science teaching assisted by a science teacher educator. This creates the maximum opportunity for FTs to learn the professional knowledge. The integrated model of one-subject mentors with multiple-subject future teachers (OSM-MSFT) forms several teams of mentors with FTs. For instance, two mentors A and P working with two FTs, Fm1 and FTc1. FTm1 and FTc1 learned from Mentor A about how to teach mathematics, while they learned from Mentor P about how teach Chinese, as displayed in Figure 3.

Figure 3. The Collaborative Team of Mentors Working with FTs under the OSM-MSFT Model.

To let FTm1 and FTc1 presented in mentor A classroom simultaneously to watch Mantor A’s lesson, they were required to appear in Mentor P classroom simultaneously to watch Mentor P’s Chinese lesson at other time. Mentor A and P needed to sit together to arrange

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their class schedules for FTm1 (displayed by ) and FTc1 (displayed by ☺), as depicted in Table 2. Table 2 is an example of a weekly class schedule of mentor A (or mentor P) during school day. The mathematics class of the two mentors was arranged at the same time. Same as the Chinese class. Both Mentors A and P are fifth grade teachers. FTm1 and FTc1 were always appeared in the same classroom at the same time. Table 2： A Weekly Class Schedule of Mentor A (or P) During School Day

Class 1 8:40-9:20 Class 2 9:30-10:10

Monday Chinese ☺

Tuesday Chinese

Wednesday Mathematics

Thursday Mathematics

Friday Mathematics

☺

☺

☺

☺

Mathematics

Chinese

Mathematics

Chinese

☺

☺

☺

☺

Class 3 10:2511:05 Class 4 11:1511:55 Class 5 13:2014:00 Class 6 14:1014:50 Class 7 15:0015:40 ☺: FTm1

(or ☺)

(or ☺)

(or ☺)

(or ☺)

(or ☺)

(or ☺)

(or ☺)

(or ☺)

(or ☺)

(☺)

(or ☺)

(or ☺)

(or ☺)

(☺)

(or ☺)

(or ☺)

(or ☺)

(or ☺)

Chinese

☺

(or ☺)

(or ☺) Chinese

☺

:FTc1

THE PROCESS OF HELPING IN IMPROVING MENTORS’ PROFESSIONAL TEACHING AND MENTORING The four mentors participating in the project had no experience of mentoring. To help them putting their visions for mentoring into practice, the mentors were supported via four phases. There was a one-hour classroom observation on every Thursday morning and a follow-up three-hour mentoring group meeting in the afternoon throughout each phase of the mentoring program. There are two groups. One is mathematics mentors group consisting of the researcher and four mentors. The other is mathematics FTs group consisting of four pairs of mentors-interns. Afterwards, each mentor required immediately share FTs with main ideas discussed in the mathematics mentor group meeting. The integrated model took the critical constructivist perspective on mentoring that knowledge is actively built by learners through

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the process of active thinking (Wang, and Odell, 2002). The researcher and the mentors were viewed as learners and generators of new knowledge and practices of mentoring. Likewise, the mentors and the FTs were also viewed as learners and generators of new knowledge, and they had to count on each other. The integrated model stressed mentors’ active construction of mentoring knowledge through what they have leaned in practice and constant dialogue with teacher educators. The collaborative inquiry model of mentoring in school-university partnership is depicted in Figure 4 as follows.

Figure 4. Collaborative Inquiry Model of Mentoring in School-University.

Phase 1: In the first two weeks of the mentoring program as the first phase, the mentors were supported in gaining the idea of induction through mutually sharing among them. The mentors were encouraged to offer emotional support for interns to overcome reality shock and reduce psychological stresses caused by the conflicts between their personal lives and professional requirement. Each mentor took turns to report in public how she introduced her intern to students and parents in the first few days of the school year. Each FT was asked to report their feelings about the induction treated by mentor. Phase 2: In the second phase, from week 3 to 6, we supported the mentors in gaining a general picture of the kind of teaching and in understanding the basic procedures in their teaching through observation and reflection about other mentors’ lessons. Each mentor was asked to teach several lessons for FTs in their own classroom. In this way, each FT could see how their mentor taught a lesson on the content that was going to teach. It was followed by a short conversation with the mentor concerning the relationship between the syllabus, students’ performance in classroom, and the lesson actually taught. Before the FTs’ observation, each mentor must elaborate the purposes of teaching she had for that lesson. Then, each mentor required FT to observe her lesson with these purposes in mind and to understand the reasons underlying the teaching. This phase provided the mentors an opportunity to support FTs on learning how to observe a lesson focusing on learners and supported mentors’ learned the teaching with a learned-oriented approach. Phase 3: The third phase, from week 7 to 10, we supported the mentors working with the FTs together in preparing a lesson and a peer observation (called as LPPO). The first opportunity was observing a mentor preparing a lesson with her FT sitting together and then

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observed the mentor to teach the lesson. It was followed by peers’ observation on how the mentor carried out the lesson, and then observing the mentor asking her intern a series of questions, such as explaining how well the lesson plan was carried out, how well the objectives she have achieved in the lesson, identifying the changes she made in the lesson compared to the lesson plan. During the third phase, other mentors not only learned from the pair of mentor- intern about mentoring on lesson plan and teaching, but also gave the mentor comments or suggestion on mentoring. Each pair of mentor-intern took turns engaging in the activities of LPPO. The FT in each pair was asked to report what she learned in the activity of LPPO. Phase 4: Each FT’s teaching was arranged in the fourth phase, from week 11 to 14. The final phase was allowed the assigned mentor to passively work with FT altogether on lesson plan. The phase was to examine the effect of mentoring on interns’ performance on mathematics teaching. The result accounts for an aspect of the effect of the mentoring program. During this phase, each FT was evaluated by other FTs, mentors and a researcher. The evaluation of mathematics teaching consists of two aspects: teaching preparation and teaching behavior.

MENTORS’ AND FTS’ VIEWS OF THE INTEGRATED MODEL The four pairs of mentors and FTs participating in mathematics group as part of the integrated research project were interviewed individually about their views of integrated model. The consensus they made was on the class schedule. The two mentors working with same two FTs arranging at the same time for both mathematics and Chinese class respectively are preliminary requirement in the model of one-subject mentors with multiple-subject future teachers. Otherwise, the FTs were not allowed by their mentors to watch lesson without their mentors’ permission. All mathematics mentors committed the function of the integrated model because this model created the opportunity for them to learn the new pedagogy for teaching Chinese from their FTs who participated in the Chinese mentoring group. Conversely, the Chinese mentors have the same agreement. They also mentioned that two FTs working with each mentor had more opportunities to stimulate multiple perspectives than only one FT working with each mentor. The suggestion of the model the mentors made was that the two FTs worked with two same grade mentors since their concerns had readily on the same focus. For FTs, the integrated model afforded them rich professional learning. For instance, on the phase of lesson plan, their mentors guided them the use of teachers’ guide or resources. They said that they learned how to work on lesson plan for an effective teaching, since mentors asked them to predict students’ various anticipated strategies or solutions and to ask students follow-up key questions in align with students’ responses. These concerns should be written preciously on the lesson plan. Besides, the FTs learned to pay more attentions to the sequence of the activities to be taught. They also learned that the sequence of the activities were relied on the objectivities of the lesson, the context of the problems to be posed, the numbers involving in the problems, and students’ prior knowledge.

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THE EFFECT OF THE INTEGRATED MODEL OF MENTORING ON FTS PERFORMING ON TEACHING The effect of the integrated model of supporting FTs on the preparation of mathematics teaching and classroom teaching are depicted in Table 3 and Table 4. The 5-scale average scores shown in Table 3 were evaluated by the participant mentors from the mathematics mentors group who observed the FTs’ teaching. The data of Table 3 indicates the four FTs well-equipped on preparing a lesson before teaching, the average score of the most of the items is more than 4.0, excepting the consideration of assessment. With the help of mentors, they had good understanding on objectives and logic structure to the content to be taught. Their good preparation of the lesson and good organization of teaching activities were revealed in the performance on classroom teaching.

Table 3. Average Score of FTs on Readiness of Preparing a Mathematics Lesson FTm1

FTm2

FTm3

FTm4

Mean

1. Understanding of instructional objectives.

4.2

4.0

4.4

4.8

4.4

2. Understanding of the structure of materials.

4.6

4.2

4.3

4.6

4.4

3. Understanding of the mathematics content.

3.9

4.6

4.4

4.5

4.4

4. Preparation of lesson.

4.3

4.6

4.9

4.8

4.7

5. Activities building on students’ preexperience 6. Adaptation of teaching activities.

4.6

5.0

4.6

4.6

4.7

4.3

4.0

3.9

4.1

4.1

7. Lesson plan including assessment.

3.3

3.1

3.7

4.1

3.6

Readiness of Preparing a Lesson

Table 4 shows that of the 15 items of mathematics teaching, FTs performing on the 13 items had average scores more than 4.0, other than two items with respect to dealing with students’ thinking and solutions. FTs still had the difficulty with realizing the distinction among students’ various solutions. They also needed to learn how to polish the strategy of stimulating students’ thinking.

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Table 4. Average Score of FTs Performing on Mathematics Teaching Teaching Behaviors

FTm1

FTm2

FTm3

FTm4

Mean

1. Draw students’ attention by various strategies. 2. Using resources (e.g. manipulatives, ICT etc.). 3. Asking questions for evoking kids’ thinking.

3.7

4.8

4.5

4.6

4.4

3.6

4.9

4.9

4.6

4.5

3.8

4.4

4.1

4.5

4.2

4. Posing daily life problems.

4.3

4.4

3.5

4.5

4.2

5. Posing problems by solving a specific strategy. 6.Good interaction between students and teacher

3.7

4.0

4.3

4.4

4.1

3.6

4.9

4.5

4.0

4.3

7. Questioning students’ thinking.

3.9

4.6

4.1

4.5

4.3

8. Stimulating students’ various strategies.

4.1

3.6

4.1

3.9

3.9

9. Comparing various solution given by students.

3.7

3.9

4.0

3.8

3.9

10. Default students’ misconception.

3.6

4.3

4.3

3.8

4.0

11. Feedback to students’ responses.

3.9

4.7

4.4

4.1

4.3

12. Affording equal opportunity for students.

3.8

4.0

4.3

4.4

4.1

13. Create secure environment of learning.

3.7

4.7

4.6

4.6

4.4

14. The activities are interesting.

3.3

4.9

4.8

4.8

4.5

15. Achieving instructional objectivies.

3.4

4.1

4.3

4.4

4.1

DISCUSSION With reconceptualizing the meaning of a school-university partnership, the integrated model of mentoring provides some evidence for the crucial importance of the mentor in the development of the FTs’ professional learning. It gives the view that simply placing FTs in school without adequate mentoring support would give FTs little chance to develop their classroom teaching skills and understanding. The teacher educators of the university offered the support of an integrated model of mentoring for mentors in school. However, there were several tensions and difficulties which emerged under the integrated model of mentoring. Most of the mentors and FTs had the agreement of joining the partnership of the university, but they felt that neither the school nor the university had provided a detailed enough brief about what was involved. There is little doubt that mentors were surprised by the unexpected tensions their role generated in the school. Initially the mentors showed hostility due to a belief that they had gained additional work. They struggled with the additional work

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and the improvement of professional knowledge. However, the factors of additional work did not appear to play a significant part in influencing mentors choosing to take on the role. Gaining their professional confidence and professional knowledge became an internal incentive. The difficulties mentors encountered in the integrated model included additional work, tight schedule, and FTs’ will to participate. Likewise, additional work and tight schedule were the difficulties for the FTs need to face during practicum. The willingness of FTs participating in the integrated model of mentoring is drastically decreasing, science they have little opportunity to become an initial teacher needed in school. Some of the FTs who planned to change their profession to other occupation lacked professional engagement during practicum. FTs had unequal professional knowledge before getting into school placement. It is suggested that some items of professional standards required for FTs should be achieved in the coursework of the teacher education program prior to practicum. Preliminary literacy of elementary school teachers teaching several subjects and practice-oriented methods of teaching in the subject area should be covered in the coursework of the teacher education program. This indicates that it is necessary to construct an operational system for qualifying a teacher. Nation-wide professional standards either across subject matter or subject-bounded, various professional standards for future teachers, internship, initial teachers, and expert teachers should be established in the contemporary teacher education in Taiwan. In addition, the policy-makers of teacher education are encouraged to associate with the researcher of teacher professional development such as the model of mentoring and the model of evaluation of teacher professional development to set up decisive policies of teacher education.

REFERENCES Field, J. C., and Latta, M. M. (2001). What constitutes becoming experienced in teaching and learning? Teaching and Teacher Education, 17, 885-895. Fwu, B. J. and Wang, H. H. (2002). From uniformity to diversification: Transformation of teacher education in pursuit of teacher quality in Taiwan from 1949 to 2000. International Journal of Educational Development, 22 155-167.Inventing a new role for cooperating teachers. Paper presented at the annual Lin, P. J. (2007). The effect of a mentoring development program on mentors’ conceptualizing in mathematics teaching and mentoring. Paper will be presented in the 31th Annul Meeting of International Group of Psychology of Mathematics Education. Korea: Soul Lin, P. J., and Tsai, W. H. (2007). The establishment and development of processional standards of mentors. Journal of Hsinchu University of Education. Lo, J. J., Hung, C. C., Liu, S. T. (2002). An analysis of teacher education reform in Taiwan since 1994 and its potential impact on the preparation of mathematics teachers at the elementary school level. International Journal of Educational research, 37, 145-159. Ministry of Education (1994). Teacher Education Act, Ministry of Education, Taipei (in Chinese).

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Ministry of Education (2002). The statistics of the teacher education programs permitted in the first semester of 1995 school year) Retrieved 2 March, 2002, Taiwan, from http://www2.edu.tw/high-school/ii1320/bbs/63.doc. Ministry of Education (2005). Yearbook of teacher education statistics, Ministry of Education, Taipei (in Chinese). National Hischu University of Education (2006). Proceedings of the conference on the teacher development of the mathematics and science mentors and interns. National Hischu University of Education (in Chinese). Odell, S. J., Huling, L., and Sweeny, B. (1999). Conceptualizing quality mentoring: Background information. In S. J. Odell, and L. Huling (Eds.), Quality mentoring for novice teachers (pp. 8-17). Indianaplolis, IN: Kappa Delta Pi. Wang, J., and Odell, S. J. (2002). Mentored learning to teach according to standards-based reform: A critical review. Review of Educational Research, 72(3), 481-546.

In: Science Education Issues and Developments Editor: C. L. Petroselli, pp. 257-269

ISBN: 978-1-60021-950-4 © 2008 Nova Science Publishers, Inc.

Chapter 10

STRATEGIES TO ADDRESS ISSUES AND CHALLENGES FACED BY INSTRUCTORS IN GENERAL EDUCATION INTRODUCTORY ASTRONOMY COURSES FOR NONSCIENCE MAJORS Michael C. LoPresto1 Henry Ford Community College, USA

ABSTRACT The challenges currently faced by instructors of introductory general education college astronomy courses are numerous. Before effective instruction can even begin, student misconceptions must be addressed. This alone is a daunting task since astronomy is a field in which there are many misconceptions. If dispelling misconceptions is achieved, then effective methods of instruction must be identified and used. Since current research shows that most students learn very little from lectures, other approaches need to be employed. This then means that resources must be either located or created before implementation can occur. The recent movement to stress understanding of concepts rather than memorization and the regurgitation of facts requires that students be engaged and prompted to think critically, or scientifically. This is a challenge in itself, since, as useful as it may be, many non-science students are not used to thinking in this manner. In fact, many students come to class not even aware what science actually is, not a body of facts and figures, but rather a process of investigation. Mathematical illiteracy is not only rampant in our society, but in many cases condoned. Because of this, many non-science majors are math-phobic. They cringe at the site of an equation or graph, even if it is only used to explain a concept and they are not even required to actually use it. Many students are members of Carl Sagan’s “Demon Haunted World” mistaking not only astrology, but television shows, tabloid articles and internet sites about the “paranormal” for science. Many have learned all they know about science from movies

1

[email protected].

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Michael C. LoPresto and television. Also, some have deeply engrained religious beliefs that prevent them from approaching scientific ideas with an open mind. These challenges are not insurmountable. What follows are the details of various strategies that have been developed and employed to address these issues and challenges with the goal of improving instruction and the entire experience of introductory astronomy for both the students and the instructors.

INTRODUCTION Instructors of introductory astronomy, or “astronomy 101” [1] as it is often called, at colleges and universities face numerous challenges when attempting to teach a course that is both interesting and engaging to the students while also providing an appropriate and meaningful experience in astronomy and with science in general.

Research Much research has been done on teaching and learning in many fields, largely with the goal of determining how students learn and how instruction can be improved and tailored to best serve different learning styles. Instructors of astronomy’s sister science, physics, have been actively attempting to improve instruction with educational research for about 20 years now. Physics departments at major research universities, first among them the Universities of Washington and Maryland and the Ohio State University began forming education research groups. Instructors of Astronomy, which at many institutions, especially community colleges and smaller 4-year teaching oriented schools, including this author, are the same people, have been following suit for about 10 years. The University of Arizona, housing departments in both astronomy and planetary science, became the first major-research university to establish an education research group within its astronomy department. This group, which has come to be known as the CAPER (Conceptual Astronomy and Physics Education Research) Team, [2] and its affiliates, including the University of Montana, where CAPER had its origins, have discovered and disseminated, through books, journal articles, meeting and conference presentations and workshops throughout the country, much of what is currently known about teaching and learning in introductory astronomy.

Expectations Many students come to introductory astronomy with no idea what to expect from the course or what will be expected of them, in fact many do not even know what astronomy or even science actually are. Surveys conducted [3] have shown that although students do expect to learn about stars, planets, galaxies, black holes, comets. the moon and the sun, they are often disappointed that the names and locations of bright stars and constellations is not major focus of a course. This is also true of UFO’s and space-travel. Some students even expect to learn about the atmosphere and weather in introductory astronomy. A simple way to gather some information

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about the students and theirs expectations is to ask them to write on a piece of notebook paper their name, major, hobbies or interests, their reason for taking astronomy; what they think astronomy is and what they hope to learn in astronomy. This may reveal what false expectations are present.

Misconceptions Even once what will and will not covered in introductory astronomy is established, misconceptions about the concepts are rampant. This was first shown convincingly to many in the video “A Private Universe” [4] where Harvard graduates were polled after commencement about the cause of Earth’s changing seasons. Most, greater than 90% gave answers that showed they that they did not know. If this true of Harvard graduates, it does not speak well for the general population of students. Students of an introductory astronomy course are a good cross-section of the general population. Most of them are not science majors, they have very little background in mathematics and their introductory astronomy course is likely one of the few if not only science class they will ever take. [5] Some of the more common misconceptions are that the varying distance between Earth and the sun is the cause of the seasons, that since there is no gravity in space, the moon, being in space, has no gravity, that meteors really are “shooting-stars” falling out of the sky, the north-star is the brightest or closest star and that he asteroid –belt is extremely crowded like shown in movies like “Star-Wars” or in the once-popular video game “asteroids”. [6] Students do not easily let go of misconceptions, even when taught to the contrary, their minds are not “empty and open vessels” in which the instructor can pour information, but rather that misconceptions must be addressed directly and dealt with before new information can be processed. [7] One possibility would be with a “misconception-survey”, a pre-test given early in instruction. Common misconceptions can be used as the distracters (incorrect responses) in multiple-choice questions. Once the misconceptions are identified, instruction can be designed to dispel them. The same test could be given post-instruction to see which misconceptions were corrected and how effectively. Another method could be to provide a list of common misconceptions to students with an announced course goal being to dispel as many of them as possible.

Astrology Probably the largest misconception and source of false expectations in introductory astronomy comes from confusion between astronomy and astrology. There are students who come expecting to learn about their zodiacal sign and casting horoscopes and are surprised and disappointed when this is not the case. This misconception is not surprising since horoscopes are found in many daily papers and believers in their predictions, who will even pay for astrological readings are part of the general public. It is important to point out to a class the difference between astronomy and astrology. Astronomy is a science and astrology a belief. Science requires proof, it is based on testability, but beliefs do not and are not. Astrology is scientifically falsifiable, and the

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confusion about astronomy that it causes is very frustrating to astronomers and teachers. This makes it very tempting to spend class-time berating astrology and its believers. Evidence nonwithstanding, it is very hard to talk someone out of something they believe. Once an instructor has made it clear that astrology is not a science and not going to be studied in the course, they have fulfilled their responsibilities in the matter. Further attempts to diminish astrology or its believers will accomplish little more than making the instructor appear to be an authoritarian that some students may simply turn off, which is obviously detrimental to student learning. [8]

Other Beliefs Teaching the difference between science and belief again brings up the above mentioned issue that many students of introductory astronomy arrive not really knowing what science actually is and the general issue of dealing with beliefs (other than in astrology) when teaching science. If, as suggested, belief is astrology is not condoned, but tolerated, the door will be open to questions about things students have seen on television, the internet or in tabloid journalism such as the claims of psychics, sightings of ghosts, UFO encounters and alien abduction, government cover-ups about both and conspiracies to fake the moon landing and other space missions and likely other things as well. Although those who want to believe what they read or hear may still be unlikely to change their opinions, it is necessary and appropriate to dismiss claims of these types based on a lack of scientific evidence, especially since so many have been exposed as untrue. [8]

Religion The religious beliefs of some students can cause them to, or at least think they have to reject some scientific theories, evolution and the big-bang theory being chief among them. The former is a much bigger issue in society and education in general, but the latter is of more immediate concern to instructors of astronomy. An approach of tolerance can be useful here as well. A science teacher’s job is not to convert student beliefs, but rather to teach the scientific explanations of a phenomenon. If a student’s religious beliefs prevent them from accepting the science, it is their choice. However, whether they “believe” (which is a bit ironic since science is not based on belief) the scientific explanations or not, they are still responsible for understanding them and the evidence for them as part of the course material. What can also be useful in these situations, similar to the above distinction between science and belief, is to explain the difference between science and faith. Science requires proof, faith does not and if you need proof, you do not have faith. Also, that many great scientists have been and are people of faith. The fact that science and faith are two different things and not mutually exclusive can, for some students, put to rest perceived conflicts and threats to their beliefs. [9]

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Science Everything discussed thus far has been pointing to a realization that teaching students about science and its methods is just as important as teaching astronomical concepts in an introductory course. Most students come to class thinking that science is just another subject, albeit one of the more difficult ones, in school and largely is a body or facts and figures about nature, and often involving something even worse; mathematics. Although it is true that science is the study of natural phenomena, what is not well understood by students and the general public is that science is the process or method by which we study, not the subjects themselves. Simply listing the steps of the “scientific method” as was probably done for the students in middle school, may not be the best way to do this. A better way is to try to demonstrate the process through examples of its use. Some useful examples from every-day life that students should be able to relate to are; a mechanic trying to figure out what is wrong with a car that will not run, or doctor making a diagnosis at an office visit. Both involve observation, followed by explanations, and then testing of explanation by further observation (after repairs or treatment in these examples) and then acceptance of the explanation if it agrees with further observation (Does the car now run? Is the medical condition cured?) and rejection and another attempt if it does not. A good exercise after coverage of the scientific method is to ask students to write down an example from everyday life of use of the process, possibly in the form of a concept-map or flow-chart. Many introductory astronomy textbooks begin with the basic observed motions in the sky, those of the stars, sun and moon and then the attempts at explanations for them throughout the history of astronomy. This is because this is an excellent way to use astronomy to teach the process of science. Arguably, ancient astronomers while attempting to explain the sky’s motions with a geocentric or earth-centered, system invented the scientific method. The method was later perfected into its current state during the Copernican revolution to the heliocentric or sun-centered system. Covering the observed motions of the sky and the stories of the attempts at explanations is class time well spent. It can provide students a basic understanding of the scientific process that will last them throughout the course, which will be very useful when they are confronted with explanations of many other astronomical occurrences. A better appreciation of science and its methods may even produce a life-long interest in science for a non-science student or at least provide them with tools with which to better evaluate what they will hear in the future about science from the media. The correct use of terminology is very important when attempting to teach about the process of science. The scientific use of the terms theory and law are widely misunderstood. In criminal justice and other fields, like, as mentioned previously, auto-repair, a theory is really just an educated guess or hypothesis that has to be proven before a conviction or other course of action can occur. In science, a theory is the best explanation based on available data. It is really the highest status a scientific idea can achieve. Many also mistakenly believe that theories eventually become laws when there is actually no “final-test” for a theory that makes it a “law.” Theories must continually pass any test put to them and will be discarded or amended if they do not. Some theories, such as those of geocentric astronomy, can last a long time before they are discarded, often when new technology, such as the telescope, is developed that can test them in ways they have not been before.

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A scientific law is an observation that is known to occur or be true, even if the reason for it is not know. Laws are actually the empirical data or evidence that is used to verify or refute theories. Kepler’s Laws of Planetary motion are an excellent example of scientific laws for introductory astronomy. They were determined by Kepler from data on Tycho’s years of observations of planets’ motion through the sky. The reason the planets orbit the sun the way they do, the “why”, which eventually turned out to be Newton’s theory of Universal Gravitation was not yet know when Kepler’s laws were discovered. Often referred to as a law, the observation that “what goes up, must come down” is the “law” for gravity, but the explanation of why it happens, the theory, is Universal Gravitation. An important point to make to students is that, based on the above, an attempt to diminish a scientific idea, such as evolution or the big bang, by saying it is “just” a theory, betrays a lack of understanding of scientific terminology and likely the scientific process as well. [10]

Mathematics Very closely related to science is its language, or its greatest tool, mathematics. Mathematical illiteracy is not only rampant, but condoned in our society. It is often a matter of pride for some to not know or not be good at math and to have avoided it as much as possible in school. The non-science majors that make up the majority of the introductory astronomy audience generally consider themselves not good with or not very well prepared in math. [5] One of the reasons they likely took introductory astronomy is that it is often advertised as non-mathematical or descriptive. This causes instructors to have to be very cautious how mathematics, which is sometimes unavoidable in astronomy, at any level is used. Introductory courses and the textbooks for them vary in mathematical level. The twosemester surveys, often taught out of longer texts usually at larger universities, will often make use of physics equations by numerically evaluating formulas for different values of their parameters. Examples include using Wein’s Displacement Law to determine the temperature of a star, based on the peak wavelength of its emissions. This even involves simple algebra, as the expression has to be solved for the temperature. One –semester courses, often taught from shorter versions of the same textbooks, are often described as non-mathematical and the texts are in fact shortened partly by making very limited use of equations. Sometimes use of mathematics often only appears in cordoned-off sections that can be omitted without loss of continuity. A good general rule for using mathematics is for the instructor to ask himself or herself what is to be gained by taking the “risk” of scaring off some students and having others cry foul. If the answer to this question is that the use of an equation helps to better explain a concept then it is likely worth the risk, otherwise perhaps it should be just left out. Examples of useful equations include Kepler’s third law of planetary motion and the relationship between the speed, wavelength and frequencies of waves. The expression for escape velocity is useful when explaining how collapsing stellar-cores become black holes and even making use of rudimentary algebra to invert the expression to the Schwarzschild radius of the eventhorizon can be informative.

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The use of graphs is unavoidable in the case of Hertzsprung-Russel diagrams, which are graphs. It is also often useful to have students plot the diagrams themselves or for students to plot data for the Hubble-law and determine their own estimates for the age of the universe. Whether formulas and/or calculations are used on tests probably should be determined by the prerequisites of the specific course. But if challenged for the use of equations or graphs in explanations, the instructor could fairly reply that the students are not being asked to deal with anything that they were not expected to know in order to graduate from high school. This is another instance where frustration can make it very tempting to berate students for their shortcomings, but this again should be resisted.

Active and Collaborative Learning Recently there have been major paradigm shifts in both physics and astronomy education from teacher-centered education to learner-centered education with the mantra “It is not what the instructor does that matters; it is what the students do.” [1, p.1] This means that rather than have students passively sit-back and listen to lectures about the subjects, they should be actively engaged in activities meant to teach them the concepts. Often done in groups this has come to be known as “active and collaborative learning.” [1 p. 53-58] There are many resources available for those who wish to employ these methods, chief among them are what are known as lecture-tutorials. [11, 12] These are activities that walk the students through concepts one step at time, building on what came previously, asking questions and prompting them to think each step of the way. Short lectures are still permissible and even desirable, prior to an activity to set up what the students will be dong and afterward to sum-up the most important points. Two important points on lecture tutorials are that since they are designed to deal with only a few concepts at a time a large amount of time should not be spent on any one activity, maybe 15-20 minutes each. Also, instructors should not just sit back and let the students work; they need to be active too by circulating around the room, offering encouragement and answering questions. In small classes like at community colleges and smaller education-oriented 4-year schools one instructor can do this. In the large-group lecture-hall classes often found at universities the help of teaching assistants is useful if they are available. A useful bonus with this approach is that the instructor (s) will get to know many more students as individuals and become of aware of their strengths and weaknesses, thus being better able to aid them in overcoming difficulties. This often makes the entire experience more satisfying for both the student and the instructor. Some students prefer not to work in groups. If they are initially resistant they can be encouraged by being flattered if they are told that what they know could help others, or coaxed that they may received needed help if they work with others. However, if a student is adamant about working alone, he or she should not be pushed further. Those who choose to waste the time given for in-class activities should be encouraged to behave differently, but if they do not, as long as they are not disruptive to those who want to work, they can be ignored and will likely pay the price for their lack of attention come exam time. A list of introductory astronomy topics that can be taught with currently available lecturetutorials: [11]

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Motions-Stars, Sun, Moon, Planets Solar System-Size and Scale, Formation, Extra Solar-Planets Stars-Radiation, Spectra, Magnitudes, Distances, HR-Diagram, Lifetimes, Stellar Evolution Galaxies-Size and Scale, Galaxy Types Cosmology-Look-back time, Expansion of the Universe

Another type of group activity that can be used in lieu of lectures are known as rankingtask exercises. A list of items is presented with the task of placing them in order by a given criterion. Perhaps objects: Earth; the moon; Jupiter; the Sun and Pluto, or distances: Earthmoon; Earth-Sun; Earth-Mars; Sun-Mercury from smallest to largest or vice-versa. Items could all be associated with the one subject, those given are all solar-system objects and distances or items could be over several subjects and thus more comprehensive, such as putting solar system and stellar and galactic objects and distances in the same exercise. It is likely that activities that different instructors find useful for various topics will be a combination of various types. A course can be taught with a combination of lecture-tutorials, ranking-task-exercises, topical discussions and/or writing-assignments, graphing activities and even still a few traditional lectures on subjects known to be of high interest to students, such as impacts, black holes, the big bang or life in the universe. A combination of methods is likely the best approach, as students may tire of repeatedly being asked to participate in any one type of activity all the time. [13]

An Active Class Session A brief example of an activity based class session for the solar system overview, commonly the beginning of coverage of a solar-system unit. First require that they read the text-chapter on the material before class and turn in questions about their reading. Most textbooks have qualitative review questions at the ends of the chapters and even accompanying websites where questions can be answered with immediate feedback and electronically submitted scores. This is certainly a useful option for larger classes. Reading about a topic prior to classroom coverage should almost always be required, the common reasoning being that in classes such as English, students are always asked to read a book before discussing it, so why should astronomy or other subjects be any different? [1 p. 21] Now, instead of hearing a lecture, have students do several activities. First, have them break into groups of 3-5 students and without looking up the “right” answers in the text, discuss and the write down group agreed-upon definitions of the terms: planet; moon; asteroid; comet and meteor. Allow groups to deliberate for 10-15 minutes with the instructor actively circulating and encouraging students by assuring them that you really want to know their definitions and that they will learn more if they try to figure something out rather than just looking it up or being told. The latter comment can be repeated often if students ask why they cannot just be told answers. After discussions wind down, this can usually be determined by a circulating instructor, conversations begin to be about things other than the discussion topic, ask different groups for their definition of each term and invite members of other groups to add to or debate it. The result will often be a lively discussion and the comment that the exercise turned out much more difficult, and therefore useful, then first anticipated.

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Next, refer the students to the planetary data appendices in their books, or provide data if necessary, and ask different groups to create bar-graphs comparing a different data item for each of the planets including, diameter, mass, density, surface gravity, distance from the sun, orbital and rotational period and number of moons. After 10-15 minutes have one example of each graph transferred by a group representative to the chalk or white-boards for everyone to see. Then focusing on one graph at a time, ask if there are any trends or similarities they see between different planets. Invariably, the conclusion will be that Earth, Venus, Mars and Mercury have enough in common and opposite Jupiter, Saturn, Uranus and Neptune that each group can be considered a planet-type with recently demoted Pluto not fitting into either category. The graphs can be collected and copies of graphs for each data item can be made and distributed to all class members. In regards to Pluto, if time permits, a survey on student opinions about Pluto’s recent declassification as a planet will yield interesting results. [14] Group or class discussions are a viable option too. Finally, bring different size objects to class, a basketball, softball, baseball, golf-ball, marble and even smaller ball-bearings, or a small starter-earring and at the front of the room or even better, outside if possible, have students select different objects to represent relative sizes and pace out relative distances on the same scale, i.e., ask one student, that if the basketball represents Earth, to choose which ball would be closest in size to the moon. Solicit other opinions before verifying the answer. Have another student then show the class how far away the chosen moon object should be from the basketball and then one does that not agree with the first to show what they think the distance should be. Then switch to the basketball being the sun and begin to model the objects and distances in the solar system. Students usually begin underestimating distance and express surprise at how vast the distances between the objects are compared to their sizes. All of the above can be done comfortably in a typical class period and is more enjoyable for everyone involved than a lecture on the same subjects.

Feedback Prompt and frequent feedback is encouraged when using active and collaborative learning. [1 p. 16] This can be achieved with the use of various classroom response systems. [15] Although electronic response systems are becoming more common in classrooms, they are not absolutely necessary, similar results can be achieved with flash cards. After coverage of a topic, perhaps towards the end of the period in the previous example, project and pose a multiple-choice question that represents a basic idea of the topic covered. For instance; Which is NOT a characteristic of terrestrial planets? -

Low Mass Close to Sun Low density* A, B and C are ALL characteristics of terrestrial planets. None of the above, neither A, B or C are characteristics of terrestrial planets. *Correct Response

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Give the students a moment to consider their answer then ask them all at the same time to vote. Voting all at the same time prevents some students from waiting and looking to see how others voted. If 75% or more students respond correctly, it is time to move on, if less than that, down to about 50%, the question should be discussed and if less than 50% are correct, more time needs to be spent on the topic. [16] These questions can also be discussed and answered in groups, when done in this fashion; the format is referred to as “Think- pairshare.” [1 p. 44, 17] There exist many sources for questions that can be used in this fashion [1 pp. 107-129, 18,19] including test-banks that are usually provided with textbook adoptions.

Attendance and Grading Obviously, active and collaborative learning is very attendance driven, those who are present in class are more likely to reap its benefits. Attendance can be encourage by collecting and giving points for assigned homework and in-class activities or simply counting attendance as a “participation” grade. The viability of this, as with some other issues, is dependent on class size. There is not a consensus on whether or not in-class activities should be graded and/or attendance should be taken. Individual instructors should determine this for their own classes. If large amounts of homework and activity papers are going to be graded a simple check-mark system can be used giving credit for completion only, or a simple 3-2-1 system differentiating only slightly for the quality of the work. The author has noted that awarding points for homework, activities and attendance does positively affect both class participation and attendance; simple rewards seeming to go a long way.

Assessment The above-mentioned prompt-feedback questions are initial examples of assessment. The effectiveness of any instructional method needs to be assessed, the results of assessment being useful in improving instruction and possibly in comparing different methods of instruction. Assessment can occur for an entire course, a single portion or unit of a course or even a single topic. The previously mentioned misconception-survey is an example of assessment over an entire course, the test being given first early in the course and again late in the course. Comparison of average scores before and after instruction can give a general idea of overall gains. Numbers of correct responses for each item can show which topics were taught more and less effectively. The Astronomy Diagnostic Test, (ADT) [20, 1 p. 131, 18 p. 127 and p. 159] has been in use at many colleges and universities, including Henry Ford Community College (HFCC), [21] throughout the country since 1999 and provides a good general assessment of the success of instruction in an introductory course as a whole. [22, 23] Single “units” in a course can also be assessed through use of instruments such as the Star Properties Concept Inventory (SPCI) [23] or Light and Spectroscopy Concept Inventory, (LSCI) [23, 24]. This more specific type of assessment could be used not only to judge the effectiveness of instruction in one area (in this case light and stars) of introductory astronomy, but perhaps also to compare instructional methods. For example, the SPCI could be given

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prior to and after instruction on stars to several sections of introductory astronomy in which the different methods of instruction are going to be employed. One section could be taught entirely with lecture tutorials while another could be taught by lecture only. Depending on the size of the sections, it may take several semesters to collect meaningful statistics, but insight to which type of instruction is more effective may be gained. Recent use of the SPCI to measure gains in a section of introductory astronomy at HFCC showed an increase in average score of 26% pre-instruction to 51% post-instruction with 21 of 23 individual test items showing an increase in correct responses of anywhere from an 11% to 71%. Although the percentage scores may seem low, the gains are significant. ADT results over the entire course both at HFCC and throughout the country are similar in both percentage scores and gains. [21, 22] Initial implementation of the first version of a solar system concept inventory currently under development at HFCC also showed similar results. Assessment instruments for one topic also exist, such as the Lunar Phases Concept Inventory (LPCI). [23, 25] Again, results of pre and post testing can be used simply to measure gains or compare different methods of instruction. This could be done with nearly any course topic such as the above example of the solar system overview. One section could spend the period doing the described activities, another could receive a lecture on the topics and scores on a short, perhaps 10-item, multiple-choice pre and post-test from the two sections could then be compared. Again, several semesters of data may be required, but could provide viable statistics for comparing the instructional methods. Good assessment questions are those that stress the understanding of the basic concepts related to the goals of instruction. With that in mind, instructors certainly can create their own assessment instruments by writing their own questions, selecting appropriate questions from test-banks designed to accompany the course textbooks or from other existing references. [1 pp. 107-129, 18,19]

CONCLUSION The process of identifying issues and challenges and developing strategies to overcome them in introductory astronomy and education in general is, as it should be, ongoing. Many of the issues discussed here are not new or exclusive to astronomy and the list is certainly not all-inclusive, nor will they be last issues that will need to be addressed. Similarly, the strategies and suggested instructional methods listed are not intended to be considered the only ones possible or even necessarily the best. They are simply among those that have been tried, ones that seem to show promise. As time and teaching go on there are sure to be new issues that will arise and new ways to cope with them, but in the relatively short amount of time that learning and teaching in astronomy have been being carefully researched, much initial progress seems to have been made toward the goal of improving instruction and thus the experience for both the students and instructors. Due to this encouraging start, there is little doubt that with further research and development and implementation of methods, the improvements will continue.

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REFERENCES [1] [2] [3]

[4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]

[24]

Slater, T.F; Adams, J.P. Strategies for Astro 101. Upper Saddle River, NJ: Prentice Hall; 2003, p. 1. http://caperteam.as.arizona.edu/ Lacey, T.; Slater, T. First Contact: Expectations of Beginning Astronomy Students. Bulletin of the American Astronomical Society, 1999 31(2). A Private Universe. 1987, Video, produced by the Harvard-Smithsonian Center for Astrophysics. Deming, G.; Hufnagel, B. Who’s Taking ASTRO 101? The Physics Teacher, 2001 39 (2) pp. 368-369. Comins, N.; Heavenly Errors. New York. Columbia University Press; 2001. National Research Council. How People Learn. Washington DC: National Academy Press; 2000, p. 19. LoPresto, M.; Tolerance for Astrology? Astronomy Education Review, 2002 1(2). LoPresto, M.; Dealing with Potential Conflicts Between Religion and Science in Introductory Astronomy. Mercury, 1999 28 (6). LoPresto, M.; Teaching the Scientific Method in Introductory Astronomy.Astronomy Education Review, 2003 2(2). Adams, J., Prather, E., and Slater, T. Lecture Tutorials for Introductory Astronomy, Englewood Cliffs, NJ: Prentice Hall; 2005 http://astronomy101.jpl.nasa.gov/tips/index.cfm?TeachingID=24 LoPresto, M.; Some Useful Resources for a Student –Centered Introductory Astronomy Class. Astronomy Education Review, 2005 2 (2) LoPresto, M.; A First Glimpse of Student Attitudes about Pluto’s Demotion. Astronomy Education Review, 2006, 5 (2). http://astronomy101.jpl.nasa.gov/tips/index.cfm?TeachingID=18 NASA/JPL-Center for Astronomy Education, Teaching Excellence Workshop; http://astronomy101.jpl.nasa.gov/workshops/index.cfm http://astronomy101.jpl.nasa.gov/tips/index.cfm?TeachingID=20 Green, P.J.; Peer Instruction for Astronomy. Upper Saddle River, NJ: Prentice Hall; 2003, pp. 41-123 Duncan, D.; Clickers in the Astronomy Classroom. San Francisco, CA: Pearson, Addison Wesley; 2006, pp. 66-93. Hufnagel, B.; Development of the Astronomy Diagnostic Test. Astronomy Education Review, 2002 1(1) LoPresto, M.; Astronomy Diagnostic Test Results Reflect Course Goals and Show Room for Improvement. Astronomy Education Review, 2006 5 (2) Deming, G.; Results from the Astronomy Diagnostic Test National Project. Astronomy Education Review, 2002 1(1). Information about the SPCI and the other assessment tests referenced, including the ADT, can be found at; http://astronomy101.jpl.nasa.gov/tips/index. cfm?teaching ID=32 Bardar, E.M.; Prather, E.E.; Brecher; Slater T.F. Development and Validation of the Light and Spectroscopy Inventory. Astronomy Education Review, 2007, 5 (2)

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ISBN: 978-1-60021-950-4 © 2008 Nova Science Publishers, Inc.

Chapter 11

GETTING IT TO WORK: A CASE OF SUCCESS IN SUSTAINING SCIENCE PROFESSIONAL DEVELOPMENT Betty J. Young, Sally Beauman and Barbara Fitzsimmons University of Rhode Island, USA North Kingstown School District, USA

ABSTRACT This article presents a case of a successful partnership between a university and nine school districts. Science educators, science and engineering faculty from the University joined forces with local school districts to attract funding and implement a high quality K-8 science curriculum supported by new materials and on-going professional development. There are five broad themes to the strategies that contributed to the success of the lasting the partnership: taking the load off central office administrators so that a high quality science curriculum with supportive PD “just happens” with another office managing the details, high quality communication among all partners, management/oversight/control, formative assessment of the quality of professional development implementation with redesign, and documenting results (e.g., parent interest, state-level school site visits, teachers’ sense of preparedness to teach science, student achievement outcomes, and continued support by the University administration and faculty).

INTRODUCTION Everyone in the science education community is aware of the call for improved science achievement that requires reform in the way science is taught (NRC, 1995; AAAS, 1993). Obviously, there is a great need for on-going professional development for elementary and middle school teachers if these reforms are to reach the classroom and benefit students (AERA, 2007). The National Science Foundation has funded a number of projects nationally to try out models of programs that provide for a comprehensive curriculum, high quality

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materials, and re-tooling of teacher skills and science content knowledge. The federallymandated statewide science assessments are focusing all districts on the adequacy of the science teaching and learning in their districts. More than ever, there is a need for models to learn from the successes, missteps and outright failures of science reform initiatives. Because funded projects often need to show how wonderful they are in order to keep the funding agency happy and the funds flowing, there is generally a lack of candor related to the challenges and obstacles and how they were handled, managed, or side-stepped. The purpose of this paper is to present the lessons learned from a particular project that has continued to provide professional development support for the last 10 years, and has district commitment to continue for the next three years and hopefully beyond. Our project has been dynamic in addressing challenges and building on successes. Our team has developed a way to “trim the sails, but stay the course” toward our goal of providing high quality science instruction that is informed by national science standards and supported by high quality materials and continuing professional development support. Here is our story and our lists of suggestions on ways to improve the likelihood that your project will take hold and be sustained with on-going, job-embedded professional development.

CONTEXT The Guiding Education in Math and Science Network (GEMSNET) provides professional development and organizational support for a K-8 science curriculum that uses kit-based materials from Full-Option Science Systems (FOSS), Science and Technology for Children (STC) and STC Middle. This National Science Foundatation Local Systemic Change (LSC) group consisted of six suburban districts and one urban district in Rhode Island. The initiative was started in 1996 by a group of teachers in one of the suburban districts who asked for the assistance of a science educator from the University of Rhode Island in establishing a K-6 science curriculum. With funding from an Eisenhower Higher Education grant1, this group, joined by a research scientist, began by constructing science units that in the end were disappointingly traditional and far too text-oriented. Based on a curricular reform institute put on by the American Physical Society (APS) in the summer of 1996, the superintendents of 5 suburban districts decided to collaborate on a common curriculum across districts that could be supported by professional development. At this point the first of the scientists and engineers from the University connected to the project. The interest in this project coincided with the rollout of the Rhode Island Science Frameworks based on the AAAS Benchmarks. In the summer of 1996 a group of 35 teachers from the five districts worked with the science educator and staff from the KIts in Teaching Elementary Science (KITES) LSC to formulate a K-6 curriculum working from the Framework and examining the NSF-approved materials to develop the science curriculum strands of life, earth/space, physical science and technology. The curriculum for grades 7 and 8 would be added later after the successful implementation of the K-6 kits.

1

The somewhat confusing name of the GEMSNET LSC was derived from this earlier 3-year Eisenhower grant that was well-known to districts and provided PD in mathematics as well as science.

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Starting with Eisenhower funding, a group of lead teachers, one per grade level from each district, were trained and began using the first two science kits in their classrooms. The parents and children from the kit classes raved about the experience. Almost immediately parents (the most vocal from science-related fields) in classrooms that did not use the kits contacted principals and superintendents to find out why their children did not have this science learning opportunity. The science educator worked out a cost scenario and implementation plan for kit purchase and PD for the superintendents who then pooled funds to purchase the first kit at each level. GEMSNET utilized the Rhode Island Materials Resource Center to assure the maintenance of the kit materials by continual refurbishment. Two more districts joined the effort and worked with the science educator and scientists to put together an NSF LSC proposal in the Spring/Summer of 1997. The LSC was awarded in Spring of 1998. At this point we were able to build the “GEMSNET Central” team including Teachers-in-Residence (TIRs). A number of us took the extraordinary workshops offered by Exploratorium’s Institute for Inquiry (IFI) and used the ideas in our PD and pre-service elementary science methods courses. All of the TIRs attend IFI sessions to give them the opportunity to develop further their notions of inquiry pedagogy and high quality PD techniques. We developed kit and training schedules with the districts and handled the notification and record-keeping for who was trained on what kit. By the third year, GEMSNET had completed the basic training and implementation of the K-6 curriculum and began organizing advanced workshops that would further consider science content and push ways to increase the level of openness and inquiry in using the kits. Issues of assessment and ties to other elementary subjects, particularly literacy, were also addressed. During this time we began to work more intensively with the 7th and 8th grade teachers, many of whom were very resistant to the idea of a common curriculum and changing their individual practices. The more comprehensive instructional materials (e.g., STC Middle and FOSS Middle modules) were becoming available to support the direction of common topics, professional development on inquiry teaching approaches, and depth over breadth of content. The middle school teacher leadership only began to develop in the last years of the LSC. As the NSF grant was ending, the GEMSNET staff brought the districts together to create a continuing Memorandum of Understanding, a contract that extended the GEMSNET project for another 3-year period, Fall of 2003 until Spring of 2006. To our delight, the science mentors have continued to work with each training session and even classroom visits though the small participation stipends are no longer in our skeletal budget. Additionally, the four Co-Driectors have continued to work with the project in frequent consultations without monetary support. In the Summer of 2006, the 7 districts, now joined by an two more districts, re-contracted for another three years, extending the partnership through the Spring of 2009. In addition, we have two other districts who have joined us on a pay-per-view basis and pay, per teacher, to join our existing professional development sessions. These funds are then used to enhance the GEMSNET project.

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PROCESS OF DEVELOPING THE GEMSNET PARTNERSHIP There are four main aspects to the development of a multi-district partnership with a college or university: serving as an important resource, recruiting the right people, training the people, and monitoring results.

Serving as an Important Resource • •

• •

•

Responding promptly to a need perceived by teachers and superintendents Connecting districts to national and local resource networks (e.g., University science/engineering faculty, materials resource centers, Exploratorium, American Physical Society, National Science Resources Center) Providing realistic cost scenarios and implementation plans for superintendents so they could assess the feasibility and visualize the process. Connecting the University’s School of Education by having TIRs co-teach preservice elementary and middle level science methods course with the education faculty. Having GEMSNET TIRs and Director participate in the development and review of the Rhode Island grade-span expectations and assessment targets that will be connected to the Tri-state assessment being developed for Rhode Island, Vermont, and New Hampshire. This connection allows the GEMSNET to be responsive to the demands that districts face in terms of the NCLB-required science assessment that is soon to be required.

Recruiting the Right People •

Finding just the right project leadership team to provide expert and somewhat autonomous leadership in the key areas of the grant, be creative collaborators, and see the goals of the grant as an important mission.

•

•

•

- PI team (The ‘gang of five’ including a science education faculty member, 3 University research scientists- with life, earth, and physical areas of expertise, and a district curriculum administrator) - Project Manager Full-time organizer to coordinate trainings, arrange meetings, handle budget, and act as the communication hub, the central point of contact for all partners Teachers-in Residence (TIRs) rotating teams of two outstanding teacher leaders from the 7 districts who contribute and get to know the project from the ‘inside’ out Recruiting scientists and engineers from the University and science-related industry to serve as science mentors or science buddies to teachers and classrooms. These science mentors co-present each training session with the classroom teacher leaders or TIRs. Creating a cadre of professional development teacher-leaders from among the enthusiastic teachers in PD sessions

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Creating a teacher-leader steering committee to meet monthly to acknowledge successes, air concerns, and problem solve solutions.

Training the Team • • • •

Training for GEMSNET Central staff on cutting-edge inquiry science pedagogical approaches at the Institute for Inquiry at the Exploratorium Enhancing leadership talents of district teachers and letting support for the initiative be led by a large group of well-respected classroom teachers. Training for the science mentors to suggest how to be helpful and to encourage only those who respect teachers for the pedagogical skills they have. Providing yearly leadership retreats to recommit to the vision and explore new ideas and approaches.

Monitoring and Reporting Results • • •

•

Informing superintendents and curriculum directors through updates in biyearly meetings, periodic newsletters, and telephone contacts. Utilizing a frugal and transparent budget process and helping districts figure ways to cut costs of materials purchase and PD. Following up with districts on teachers who miss trainings or are excessively late to PD sessions. (This task indicates how seriously we take our work as PD providers during the school day as well as the need to have teachers receive formal training BEFORE using the kit materials.). Conducting, presenting and publishing research on the achievement and attitude toward science outcomes associated with kit-based curriculum and job-embedded professional development in science and inquiry pedagogy. This activity provides valuable information to justify district expenditures and adds to the research university’s academic currency.

STRATEGIES There are five broad themes to the strategies that contributed to the success of the lasting GEMSNET partnership: taking the load off central office administrators so that a high quality science curriculum with supportive PD “just happens,” high quality communication among all partners, management and control of activities, formative assessment of the PD quality and kit implementation, and measuring and reporting results in terms of parent interest, state-level school site visits, teachers’ sense of preparedness to teach science, student achievement outcomes, and continued support by the University administration and faculty.

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Taking the Load off Central Offices: •

•

• • •

•

•

Consider the districts’ budgetary and instructional situations in terms of the larger environment in which they operate. The project acknowledges that science education is the center of our universe, but the districts need to respond to many competing demands for money as well as teacher time and attention. We always arrange our design and demands by showing the relative merits of the GEMSNET activities to the other initiatives that the districts consider. Keep the district central offices informed, but manage all the logistics involved in implementation of a high-quality science curriculum so that the district curriculum directors or superintendents can use their time on other projects. Utilize a broad conception of partnership by bringing many sources of expertise to the districts and helping them staying tuned to the emerging research literature. Help teachers establish their “highly qualified” status using GEMSNET records of coursework and PD sessions. Reach out to new districts to interest them in the process. The Director has consulted on developing kit-based curricula supported by PD resulting in a number of nonpartner districts who pay, per teacher, for PD sessions. The extra revenue reduces the costs to the partnership districts. In one case we have acquired a new partner whose share will reduce the costs of other districts while allowing us to expand our offerings. Keep the training mandatory and work as a unified force with the central offices to enforce training attendance. This process works well for everyone involved and provides a clear message to the faculty on the importance of the training. Reinforce the attendance message by reporting and tracking no shows. If a teacher does not attend training, it is reported the same day and it is also logged into the files at the project office. In most instances, there is a good reason for the lack of attendance but this process can also demonstrate where the level of resistance is. Standardize the document flow so it becomes part of the central office routine. Analyze the business cycle of the school administrations and establish your procedures to mesh well with them. For example, we routinely draft our training calendar for the next academic year when the districts are seeking school board approval on their district calendar. Once they have approval, we send our training calendar to them for their final approval. This minimizes conflicts with other district objectives, allows the administrator to assess the demand for substitute hiring and keeps the training prominent. Other routine items include sending financial information during their budget preparation time, distributing a master training roster to them several weeks prior to the start of the school year, and mailing all training letters directly to the teachers 30 days prior to their assigned sessions.

Communication: •

Pay constant attention to communication with expert project manager serving full time administrative support at GEMSNET Central.

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Hold an annual retreat to refocus on purpose with GEMSNET staff, teacher leaders, science mentors, and district administrators (both superintendents and curriculum directors). Provide annual teacher recognition presentations at various school committee meetings along with mini-inquiries for board members to demonstrate the ‘thinking curriculum’ in science. Rotate TIR positions (15 so far) with TIRs returning to districts as confident supporters and, so far, four becoming principals in the partner districts. Publish a periodic newsletter and maintain a user-friendly website to distribute information. Assign teacher PD leaders to work in cross-district pairs for trainings along with science mentors to keep the collective sense of ownership alive.

Management, Oversight, and Control •

•

•

•

Recognize the challenges the specialized knowledge and viewpoints of all parties bring to the communication process. Getting teachers, scientists and engineers to communicate, understand and appreciate each others’ specialty proved to be more challenging than we had anticipated. It was evident early on that communication difficulties went beyond understanding the acronyms involved. Many teachers thought the scientists and engineers had an expertise in other fields and some mentors quickly became uncomfortable when pressed for answers. Mentors also had little understanding of the realities of teaching in an elementary or middle school classroom. We were fortunate to have three scientist PI’s who were very interested in bridging the communication gap to reach the right level of dialogue. This in turn allowed the mentors to provide the optimal level of science content necessary for the professional development. It was also very important for mentors and teachers to understand it is okay to say you don’t have the answers. Establish a management structure. We have a relatively flat organizational hierarchy and consider the project staff, administrators, mentors and education faculty to be working in unison toward the same goal. Teachers-in-Residence establish their own objectives as well as work toward project objectives. This arrangement allows individual creativity in addressing the reform while still moving the project forward with long term objectives. Assure that the project manager is clear on the long range plan for the project. Higher education faculty, teachers, central office administrators and volunteer science mentors have little time to devote to strategic planning. Try to stay informed of current trends and anticipate what changes they will bring to the project. Manage the flow of information so everyone is on the same page. Standardize the basic training sessions. Decide on key lessons to communicate content and show tricky set-ups. Get to modeling hands-on right away. Inform teachers the system: doing accurate kit inventories, communicating with the Materials Resource Center, and GEMSNET classroom-based resources.

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•

•

•

• • •

•

• •

• • •

Stress the importance of follow-up sessions. These sessions help teachers refocus on the content and push toward opening up lessons to include more critical thinking and inquiry pedagogy. Assessing student knowledge and connecting lessons to state science target objectives. Limit your affiliations early on. Funding and positive feedback causes big ripples in many other outreach efforts. Some programs want to use the success of your project to launch theirs without really contributing to the project. While many of them are worthy of collaboration, they can stress your resources and weaken your program if not controlled. Monitor and control the level of outreach and training efforts. Promoting the project is great, but with a limited staff you can quickly become overburdened if you stray too far from your primary objectives. The instinct from a business marketing perspective is to meet the “customer” expectation to strengthen and expand “the business,” but the first objective always has to be on training quality. A rapid and uncontrolled expansion can bring training to a shallow surface-level rather than the inquiry and content rich sessions you developed. Make sure time is allowed for the project office to prepare materials and plan for tightly-run group meetings. Last minute preparation and planning can lead to unfocused meetings that partners can see as a waste of their time. Communication has to be on-going to stay on track of the long-term objectives. Support your trainers with leadership development. Monitor and control the budget. Anticipate the level of resistance. If you monitor this aspect as a regular activity, you can judge the classroom level impact and adjust your activities to increase the level of implementation. You can also see how internal decisions impact the classroom teachers. Keep it realistic. Teachers, scientists and engineers are highly creative individuals. Brainstorming sessions often yield fantastic ideas that are unrealistic for a small business with a limited budget and staff to maintain. Maintain training quality above all else. The word will spread very quickly if the training was the best session ever, just okay, or total waste of time. Ensure data integrity. Accurate tracking of training attendance protects the integrity of the project. One error can quickly become a public relations nightmare and deter your training efforts for an entire grade or building. Be visible! Be responsive! Return phone calls and e-mails in a timely manner. Focus on classroom support-not project spying. Being the “kit police” is a bad idea.

Make sacrifices to sustain. When our project first began, it was very common to hear from teachers that the University repeatedly formed partnerships with the local districts for funding purposes, but left them high and dry when the funds ran out. In most instances, this occurance is the reality of soft money and the biggest obstacle to any lasting reform. We were determined to break this cycle and have succeeded by making the decision to sustain to the local level. This decision has not been without sacrifice. The expenditure is significant for the districts, the University does not collect overhead on the project offices, and the mentors and

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PIs work on a volunteer basis including the Director. Everyone is doing their part, which is the true essence of a successful partnership.

Formative Assessment/Project Evaluation: • • • •

Utilize multiple means of receiving constant feedback about teachers’ challenges and concerns as well as success stories. Hold bi-monthly teacher steering committee meetings. Evaluate PD sessions continuously and periodically examine for trends…strengths and weak spots. Collect and track anonymous “Sun and Clouds” feedback in which participating teachers in second PD session write to us about successes and concerns. The GEMSNET Central team uses these to generate solutions and suggestions.

Document and Disseminate Results: •

•

•

•

Conduct and report research on science achievement and attitudes toward science, comparing growth in project districts relative to non-kit districts with similar demographics. Present informative evening sessions for school committees and parents by taking them through mini-inquiries and then showing them work samples from elementary and middle school students in their districts. Show off the type of active, critical thinking instruction that is valued by the state and others. Rhode Island has a process of school site visits in which classroom teachers and administrators analyze their test and social/instructional data as well as invite classroom observations to state-level teams of evaluators. Universally, GEMSNET schools have been commended on the cohesiveness and instructional practices in K-6 science. Transfer high quality instructional techniques to the University classrooms. While this was an unanticipated outcome, the scientists and engineers who work on the project have gained a great deal of pedagogical and assessment knowledge from the GEMSNET classroom teachers and project staff. We have capitalized on this outcome by assisting with a number of instructional reform grants aimed at undergraduate and graduate science and engineering education. These activities have strengthened the connection with the University and helped us gain recognition as one of the most extensive and effective outreach programs in the University. The common interest in instruction has also connected the teacher education programs to the content specialists in the science-related departments, thus, providing more ideas and cooperation in enhancing teacher content knowledge. These efforts have facilitated our requests for University office space, telephones, and use of office equipment without assessing an overhead charge from the GEMSNET districts during the continuation contracts.

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Betty J. Young, Sally Beauman and Barbara Fitzsimmons

CHALLENGES Over the years of the GEMSNET partnership we have encountered a number of challenges that are similar to those faced in other projects across the country. First, we have endured a large turnover in superintendents and curriculum directors/assistant superintendents in the past 10 years. With the first big change we put together an “intervention” team consisting of the PIs (science educator, scientist, and curriculum director) and the project manager. We scheduled a meeting with the new superintendent or curriculum administrator, provided a project overview, and answered any questions they had. We also alerted some of our teacher leaders in the district to “talk up” the project and keep us abreast of any proposed changes related to science. In some instances, our curriculum director Co-PI alone arranged a ‘mini-orientation’ with a new curriculum administrator in any of our districts. We found this to be a highly effective way to socialize our new district central office personnel. GEMSNET has run its main PD sessions during the regular school day in order to make attendance mandatory and emphasize the importance of preparing all teachers, new or grade changers, to work with the content depth of the kits and the inquiry-based instructional approach. The availability of substitutes as well as the district funds for substitute pay have been issues at times. We have spread out the way sessions have been scheduled. The funding shortages are a more recent development that has also affected the appointment of TIRs. In the words of one of our superintendents, “Just so you can feel my pain ---I was supposed to eliminate 15.2 positions, I have eliminated 20.3 just to try and impact the $1,140,000 deficit we are already projecting for next year. We have a potential shortfall for this year and I cannot even hire subs. Times are tough.”

Most of our districts are facing large budget shortfalls due to a change in the pension contributions at the state level in combination with property tax revolts at the local level. Districts are closing schools and laying off teachers. We held an Oversight Committee meeting in early spring to discuss a strategy for renewing our partnership agreement that was to expire in June, 2006. Each of the districts indicated that they would continue as the professional development and other services offered by GEMSNET were well worth the funds. Uncertainty of refurbishment costs charged to the districts by the Materials Resource Center is another concern. Our partners have not enjoyed the economies of scale that were promised to them as more districts in the two-state area utilized the service. The cost of refurbishment is currently $117 per kit/per rotation and has gone as high as $125, no matter how many materials needed to be replaced. A second concern in this area has been the great increase in the cost of the STC and FOSS kit materials as some have increased $200-$300 per kit. This huge increase discourages districts from using these materials that can be so well supported by PD. In the face of the coming mandatory science testing under NCLB, these increases have the appearance of price gouging. Another challenge has been the teachers’ focus on mathematics and, especially, reading and language literacy in the face of the NCLB testing and performance demands in those areas. Science has been pushed aside in some classrooms as anxious teachers prepare their students to do well on the math and reading tests. We have addressed this issue in two ways:

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by demonstrating the growth in writing and expository reading that comes from the study of science and by alerting teachers to the coming state-level assessment in science. We have been slow to counteract resistance of 7th and 8th grade science teachers to a common science curriculum supported by extended modules such as the STC Middle materials. We encountered this early on when we tried to extend the curriculum into the middle grades. We backed off, reasoning that as the students who had the kit-based elementary experience entered middle school, the teachers would be more convinced of the power of this approach. Also the superintendents were more reluctant to insist on change at this level. In elementary we were filling a near void, but in the middle we were making changes that affected the whole of the teachers’ practice. We have been able to recruit lead teachers from these grades who are helping us to move faster in the desired direction. PD training in this area has increased dramatically.

RE-DESIGN IDEAS In addition to the re-design ideas that came as we encountered challenges over the last 10 years, there are a number of areas that we would change based on our experiences. First, we would seek a greater commitment from superintendents to push for changes at the 7th and 8th grade. Second, we would involve the principals more directly as they need to assist with monitoring the implementation of the kit-based curriculum. They needed to buy into the project to a greater degree. Third, we would need to set up a kit-use monitoring system that would be requested by superintendents, enforced by principals, and eventually, accepted by teachers as a normal routine. We have evidence of kits returned virtually unused to the MRC. Though this practice is not widespread, it presents a risk to the success of the science curriculum. Finally, we would increase the classroom level coaching from the beginning. As we moved into the maintenance stages with districts, our TIRs have spent more and more time in classrooms co-teaching to demonstrate more open and inquiry-based lessons. This aspect of GEMSNET has been highly effective and helps us ground our suggested approaches in the realities of classrooms. This exercise in reflecting on the successes and challenges of partnerships in GEMSNET project has been a valuable effort to inform our strategic planning. In the end the success of the GEMSNET partnership, or any other for that matter, depends on continuous attention to the cost/benefit analysis that continuously assessed by all partners, either formally or, more often, informally. The currency is not only monetary, but also the factors of usefulness and TIME. Is the budget reasonable and well spent? Are there significant positive outcomes? Is the project a worthwhile use of the time devoted to it by GEMNET staff, science mentors, classroom teachers, children, school principals, superintendents, and curriculum administrators? Our success indicates that we have done well in balancing the costs and benefits to our partners.

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Betty J. Young, Sally Beauman and Barbara Fitzsimmons

REFERENCES American Association for the Advancement of Science. (1993). Benchmarks for science literacy. New York: Oxford University Press. American Educational Research Association. (2007). Science education that makes sense. Research Points: Essential Information for Education Policy, 5(1), 1-4. National Research Council. (1995). National science education standards. Washington DC: National Academy Press.

INDEX

A academic tasks, 217 academics, 108, 226 access, 6, 7, 12, 15, 25, 152, 156, 160, 172, 204 accommodation, 204, 206, 208, 220 accountability, 101 accounting, 109 accreditation, 240 accuracy, 116, 130, 190 achievement, vii, 5, 7, 15, 17, 37, 41, 69, 126, 130, 131, 134, 140, 141, 144, 145, 146, 147, 148, 212, 215, 216, 218, 219, 222, 271, 275, 279 achievement test, 41 acquisitions, 31, 32, 34, 36, 50 activation, 35, 136, 137, 138 activity theory, 108 administration, xiii, 271, 275 administrators, xii, 227, 236, 242, 271, 275, 276, 277, 279, 281 adolescents, 146, 205 adult population, 25 adults, 34, 39, 67, 162, 164, 167, 189, 191, 196, 203 aesthetics, 161 affective experience, 130, 132, 133, 134, 135, 136, 138, 139 affective reactions, 128 affective states, 127, 133, 138 African American, 219 age, 2, 52, 81, 100, 108, 113, 114, 118, 182, 184, 187, 189, 190, 263 agent, 99, 107, 119, 145 aggregates, 50 algorithm, 29 alternative(s), 26, 45, 55, 67, 73, 77, 78, 84, 86, 88, 108, 117, 132, 140, 153, 156, 162, 163, 165, 169,

174, 186, 189, 203, 207, 211, 214, 220, 221, 222, 228 amalgam, 50 ambiguity, 99, 242 American Educational Research Association, 220, 281 anxiety, 98, 127, 128, 129, 130, 132, 135, 137, 139, 141, 143, 146, 147, 148, 216, 220, 228, 235, 236 applied mathematics, 118, 119 appraisals, 132, 133, 134, 135, 136, 138, 143 argument, 60, 62, 64, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 96, 99, 100, 101, 102, 105, 107, 149, 191, 197, 208 Aristotle, 114 arithmetic, 8, 9, 10, 20, 26, 36, 42, 44, 45, 46, 47, 48, 53 Arizona, 258 arousal, 133, 134, 136, 140, 141, 216 arsenic, 10 arthritis, 185 articulation, 41 Asia, 238 assertiveness, 74 assessment, vii, xi, xii, 5, 16, 23, 36, 53, 61, 69, 88, 90, 91, 136, 165, 197, 207, 224, 227, 228, 229, 235, 236, 245, 252, 266, 267, 268, 271, 273, 274, 275, 279, 280 assets, 32 assimilation, 204 assumptions, 9, 109, 116, 169, 211 asymmetry, 126 Athens, 151 attention, viii, x, 59, 67, 74, 77, 79, 80, 85, 126, 128, 152, 158, 163, 164, 167, 168, 171, 196, 201, 206, 207, 209, 210, 211, 212, 215, 216, 240, 253, 263, 275, 276, 281 attitudes, xi, 20, 30, 84, 126, 127, 128, 129, 130, 142, 146, 175, 202, 219, 220, 223, 224, 225, 279

284

Index

attribution, 130 Australasia, 56 Australia, 56, 201, 236, 237, 238 authority, 108, 183, 185 autonomic nervous system, 127 autonomy, 129, 131, 149, 216 awareness, 64, 66, 67, 68, 71, 79, 86, 93, 111, 128, 133, 136, 184, 193

B banks, 266, 267 barriers, 18 behavior, 66, 71, 72, 127, 128, 129, 137, 245, 251 belief systems, 130, 137 beliefs, 62, 82, 83, 84, 127, 129, 130, 133, 134, 135, 136, 137, 138, 144, 147, 148, 155, 180, 183, 186, 189, 191, 197, 199, 208, 212, 216, 217, 218, 219, 221, 259, 260 benchmarks, 16 benefits, viii, 59, 60, 228, 246, 266, 281 bias, 185, 212, 214, 217 bioethics, 152 bird flu, 168, 171 black hole, 258, 262, 264 blood, 169 bloodshed, 102 board members, 276 bonding, 69 boredom, 210 Botswana, 214 brain, 119, 137, 144 brain structure, 137 Britain, 122 browsing, 169 budget cuts, 226 bureaucracy, 101 business cycle, 276

C calculus, vii, 6, 17, 20, 30, 36, 50 calibration, 116 campaigns, 103 case study, xi, 88, 90, 94, 219, 223, 225 casting, 21, 259 causal attribution, 129, 130 causal relationship, 86 cell, 26, 27 Chalmers, 98, 102, 106, 108, 109, 113, 116, 121 channels, 19, 159 chaos, 120, 166

Chicago, 17, 55, 90, 91, 140, 220 child development, 146 childhood, 6, 140, 198 children, ix, x, 18, 22, 33, 61, 89, 95, 100, 101, 104, 110, 111, 117, 118, 122, 140, 148, 163, 167, 181, 183, 187, 189, 190, 191, 193, 194, 196, 198, 199, 201, 203, 207, 214, 218, 273, 281 Chinese, 243, 246, 248, 249, 251, 254, 255 citizenship, viii, 16, 95, 98 classes, 30, 41, 69, 205, 209, 213, 215, 216, 217, 227, 263, 264, 266, 273 classification, 6, 39, 41, 57, 143, 194, 198 classroom, vii, x, 5, 7, 16, 17, 20, 21, 38, 61, 73, 74, 77, 78, 79, 85, 86, 87, 88, 91, 92, 93, 101, 105, 113, 114, 118, 132, 139, 142, 146, 151, 153, 155, 157, 160, 168, 169, 170, 172, 175, 179, 183, 185, 197, 198, 201, 202, 205, 206, 208, 209, 210, 211, 212, 213, 215, 217, 220, 221, 227, 241, 245, 248, 249, 250, 252, 253, 264, 265, 271, 273, 274, 275, 277, 278, 279, 281 classroom activity, 153, 155 classroom events, 215, 217 classroom management, 241 classroom practice, x, 16, 151, 206, 208 classroom settings, 88 classroom teacher(s), 274, 275, 278, 279, 281 classrooms, x, xi, 19, 56, 78, 79, 88, 89, 92, 94, 152, 157, 168, 170, 171, 179, 180, 185, 186, 187, 188, 197, 198, 199, 201, 211, 215, 216, 219, 241, 265, 273, 274, 279, 280, 281 clients, 229, 230, 231, 232, 233, 234, 235, 236 climate change, 80 cloning, 168, 171 closure, 21, 98 codes, x, 151, 157, 159 cognition, ix, 64, 66, 68, 70, 71, 84, 87, 88, 96, 108, 114, 117, 118, 125, 127, 128, 129, 131, 132, 133, 134, 136, 137, 138, 140, 141, 144, 146, 147, 148, 149, 210, 222, 244 cognitive development, 93, 139 cognitive dissonance, 207, 208, 210 cognitive domains, 131, 145 cognitive function, 66, 163 cognitive involvement, 130 cognitive load, 162 cognitive models, 132 cognitive perspective, 57, 129, 138, 209 cognitive process, ix, 64, 66, 84, 125, 132, 135, 203 cognitive psychology, 153 cognitive science, 63, 126, 129 cognitive tasks, 57 coherence, 7, 17, 20, 31, 34, 35, 42, 53, 109 cohesion, 35, 42, 53

Index cohesiveness, 78, 279 collaboration, 193, 211, 241, 277 college students, 21, 139, 182, 184, 185, 191, 199, 238 colleges, 105, 240, 258, 263, 266 communication, x, xi, xii, 19, 73, 151, 152, 157, 158, 171, 175, 223, 224, 225, 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 238, 271, 274, 275, 276, 277 communication skills, xi, 223, 224, 225, 226, 227, 228, 229, 230, 232, 234, 235, 236 community, 63, 92, 102, 103, 107, 185, 197, 203, 246, 258, 263, 271 competence, vii, xii, 5, 7, 9, 13, 16, 22, 23, 24, 31, 32, 33, 34, 35, 36, 37, 48, 51, 53, 80, 180, 195, 196, 197, 239, 240 competency, 31, 32, 33, 34, 35, 192, 225, 238 competition, 212 complement, 46 complex numbers, 6, 24 complexity, 15, 17, 26, 34, 35, 38, 40, 41, 43, 45, 50, 53, 83, 85, 99, 120, 126, 131, 138, 146, 149, 213 components, x, 10, 13, 28, 31, 37, 41, 50, 60, 64, 68, 71, 126, 158, 179, 182, 207 composition, 36 comprehension, 25, 64, 68, 83, 119, 215, 216 computation, 6, 12 computers, 12 computing, 21, 52, 143 concentrates, 215 concentration, 11 concept map, 79 conception, 22, 68, 79, 86, 112, 204, 205, 206, 220, 221, 276 conceptualization, 130 concrete, ix, 21, 22, 45, 72, 95, 109, 110, 111, 114, 182 condensation, 47 Confederation of British Industry, viii, 95, 97 confidence, 103, 106, 130, 143, 144, 184, 209, 213, 218, 254 configuration, 26, 106 conflict, 86, 102, 206, 207, 208, 211, 218, 220 conformity, 12 confrontation, 209, 214 confusion, 31, 118, 259, 260 consciousness, 100, 133, 135, 137, 146 consensus, ix, xi, 73, 95, 103, 104, 106, 107, 201, 212, 213, 217, 251, 266 conservation, 108, 119, 214 conspiracy, 107 constraints, 33, 45, 61, 100, 103

285

construction, 6, 28, 53, 87, 103, 106, 109, 117, 131, 156, 163, 164, 169, 183, 191, 196, 203, 206, 216, 219, 250 consulting, 9, 185, 230, 237 consumers, 8 consumption, 8 contaminant, 11 content analysis, 222 continuity, 262 control, xii, 12, 51, 54, 63, 64, 66, 67, 68, 69, 72, 75, 83, 84, 99, 133, 134, 135, 136, 138, 140, 155, 169, 172, 181, 188, 190, 191, 195, 196, 219, 241, 271, 275, 278 conversion, 26, 46 cooperative learning, 68, 213, 237 copper, 11 correlation, 116 costs, 38, 45, 275, 276, 280, 281 Council of the European Union, 16, 54 course content, 226, 236 course design, 16 coverage, 261, 264, 265 creative process, 105 creativity, 105, 163, 174, 277 credentials, 108 credibility, 187, 190 credit, 77, 266 critical thinking, viii, 82, 89, 93, 95, 99, 277, 279 criticism, 102, 104, 232, 233 cryptography, 13 cultural beliefs, 134, 137 cultural influence, 138 cultural perspective, 238 culture, 75, 77, 88, 99, 101, 102, 113, 118, 144, 152, 161, 244 curiosity, 164, 207, 208 currency, 275, 281 curriculum, viii, xii, 20, 21, 24, 42, 59, 60, 62, 81, 87, 88, 89, 90, 92, 93, 95, 98, 101, 105, 122, 152, 153, 154, 155, 157, 161, 162, 165, 166, 168, 169, 170, 171, 172, 174, 183, 218, 225, 227, 228, 229, 235, 236, 237, 238, 241, 245, 271, 272, 273, 274, 275, 276, 279, 280, 281 curriculum development, 20, 218, 225 Cyprus, 147

D danger, 96, 100, 109 data analysis, 85, 181, 193 data base, 104

286

Index

data collection, x, 19, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 196 data gathering, 79, 85 data structure, 198 decision making, 82, 84, 88, 135, 142, 143 decisions, 63, 67, 82, 136, 152, 187, 188, 192, 193, 229, 278 declarative knowledge, 64, 70, 71 decomposition, 183, 190, 191, 192 deductive reasoning, 3 deficit, 2, 280 definition, 32, 34, 36, 48, 49, 50, 51, 52, 64, 68, 73, 82, 106, 129, 264 demand, 3, 20, 26, 71, 153, 161, 163, 216, 226, 240, 276 democratisation, 102 demographics, 279 denial, 104 density, 265 dependent variable, 51, 84 designers, 31 desire, 67, 74, 212, 225, 229, 235 developed countries, 126 developmental change, 189 developmental psychology, 140 differentiation, 36, 189, 199 diffraction, 110 direct observation, 106, 120 directives, vii, 5 discipline, viii, ix, xi, 1, 3, 6, 33, 34, 89, 95, 97, 100, 101, 103, 105, 117, 180, 181, 189, 195, 223, 226, 227 discontinuity, 46 discourse, ix, 63, 76, 83, 88, 89, 90, 92, 94, 95, 108, 114, 122, 133, 152, 153, 168, 171, 172, 197, 199, 213, 237 discrete emotions, 143 disequilibrium, 207 disinfection, 10 disorder, 146 displacement, 106 disposition, 128 dissatisfaction, 204 distracters, 259 distribution, 173 diversification, 254 diversity, x, 40, 41, 113, 151, 171, 242 division, 10, 29, 37, 127, 185, 227 division of labor, 185 domain-specificity, 89 DVD, 152, 154, 157, 159, 160, 167, 168, 170

E earnings, vii, 22 earth, 107, 108, 113, 115, 121, 159, 164, 261, 272, 274 ecology, ix, 95, 161 economic status, 126 economies of scale, 280 ecosystem, 162 education, i, iii, v, vi, vii, viii, ix, x, xi, xii, 5, 13, 14, 15, 16, 31, 32, 54, 55, 56, 57, 59, 60, 61, 62, 65, 74, 76, 77, 83, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 103, 104, 105, 108, 110, 113, 117, 118, 121, 122, 125, 126, 127, 128, 129, 131, 132, 135, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 151, 152, 153, 154, 157, 164, 165, 167, 168, 169, 170, 172, 174, 175, 176, 177, 179, 188, 195, 196, 197, 198, 199, 201, 202, 203, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 224, 225, 226, 237, 238, 239, 240, 241, 242, 254, 255, 257, 258, 260, 263, 267, 268, 271, 272, 274, 275, 277, 279, 281 education reform, xii, 88, 239, 240, 241, 242, 254 educational institutions, 152, 165 educational objective, 127, 143 educational process, 170 educational programs, 81, 154, 157, 162, 164, 165 educational psychology, 148 educational research, 74, 212, 258 educational system, vii, 5, 7, 14, 16, 17, 21, 154, 172 educators, xii, 15, 53, 77, 85, 99, 104, 105, 117, 127, 153, 211, 239, 241, 242, 246, 248, 250, 253, 271 ego, 128, 131, 147 Einstein, 1, 2, 4, 112, 159, 168 elaboration, 99, 163, 165, 166, 173, 207 electric current, 183 electricity, 90, 174, 183 electron, 106, 110 elementary school, 20, 198, 254 elementary science teachers, 221 elementary students, 189, 193, 194 elementary teachers, 213, 218, 220 email, 201 emancipation, 172 emotion(s), 127, 129, 130, 131, 132, 135, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 186 emotion regulation, 141 emotional experience, 148 emotional processes, 146 emotional responses, 148 emotional state, 132 emotional well-being, 141

Index employees, 225, 229, 230, 231, 233, 234, 235 employment, xi, 12, 100, 101, 224, 226, 229, 234, 236 encouragement, 62, 263 enculturation, 96, 118, 121 energy, 9, 10, 69, 108 energy consumption, 9 engagement, ix, 53, 73, 77, 86, 95, 118, 175, 197, 205, 215, 254 England, 81, 91, 142, 197 enlargement, 25, 39 enlightenment, 98, 102, 103, 105, 120 enrollment, 146 enthusiasm, 20 environment, x, 41, 45, 53, 55, 63, 72, 80, 84, 85, 88, 137, 151, 155, 157, 158, 159, 160, 161, 162, 163, 164, 167, 203, 218, 253, 275 environmental conditions, 137, 138 environmental influences, 137 Environmental Protection Agency, 11 EPA, 10 epistemology, 20, 53, 76, 84, 93, 180, 181, 182, 189, 197 equality, 28, 36, 42, 44, 45 equating, 182 equilibrium, 69 equipment, 69, 70, 279 equity, 108 estimating, 9 ethics, 130 ethnic background, 126 ethnicity, 145 Euclidian geometry, 109 Europe, 31 European Parliament, 15, 55 evacuation, 22 evaporation, 74 evidence, viii, x, 23, 61, 73, 74, 75, 76, 79, 80, 82, 83, 84, 85, 86, 87, 89, 93, 95, 99, 110, 120, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 195, 196, 197, 199, 207, 209, 211, 214, 216, 227, 253, 260, 262, 281 evolution, 16, 127, 162, 183, 260, 262 exclusion, vii execution, 193 exercise, 1, 2, 30, 111, 261, 264, 281 expenditures, 275 experimental design, 81 experimental space, 190 expertise, 34, 113, 118, 274, 276, 277 explicit knowledge, 86 exploitation, 98 exposure, ix, 1, 2, 95, 112

287

extrinsic motivation, 98, 112, 221

F fabric, 42, 45, 46 factual knowledge, 179 failure, viii, 33, 35, 51, 63, 95, 98, 103, 129, 144, 148, 185, 209, 241 faith, 260 family, 9, 157, 162, 174 fanaticism, 100 fear, 32, 130, 136, 137, 172, 236 feedback, 76, 79, 88, 135, 227, 230, 234, 264, 265, 266, 278, 279 feelings, 2, 127, 128, 130, 132, 133, 250 females, 126, 135 feminism, 105 fifth grade teachers, 249 film, 155, 157, 159, 161, 168, 169 films, 159, 164, 167, 168, 169, 170, 174 finance, 242 Finland, 125, 141, 144 First World, 102, 103 flexibility, xi, 3, 135, 169, 223, 226 fluctuations, 192 fluid, 105, 224 focusing, 7, 126, 163, 166, 168, 226, 250, 265, 272 foreign language, 33, 76 formal education, x, 80, 151, 152, 153, 154, 155, 157, 161, 162, 165, 167, 169, 170, 172, 173, 174, 175 France, 10, 15, 16, 31, 50, 54, 55, 57, 61 fraud, 107 free choice, 168, 176 freedom, 68 friction, 109, 113, 116, 119, 211 fruits, 116 frustration, 130, 263 functional analysis, 13 functional approach, 55 fundamental attribution error, 185 funding, xii, 172, 226, 227, 236, 238, 271, 272, 273, 278, 280 funds, 272, 273, 278, 280 fusion, 39

G galaxy, 264 gases, 114 gender, 94, 102, 113, 129, 137, 142, 145, 186 gene, 165

288

Index

general education, xii, 126, 130, 213, 227, 257 general knowledge, 159 generalization, vii, 5 generation, 97, 195 genetics, 90 genre, 158, 159, 160, 232, 237 geology, 188 Germany, 16, 54, 91, 221 goals, 21, 31, 57, 60, 61, 62, 63, 65, 130, 132, 133, 134, 135, 137, 139, 141, 142, 143, 147, 148, 190, 196, 212, 216, 218, 242, 267, 274 God, 115 government, 226, 227, 238, 240, 242, 260 government budget, 240 grades, 15, 41, 197, 204, 217, 272, 280 graduate students, 182, 196 grants, 279 graph, xii, 49, 51, 257, 265 gravity, 106, 108, 114, 115, 204, 220, 259, 262, 265 Greeks, 2 ground water, 10 group activities, 166 group size, 78 group work, 67, 79, 216 groups, 16, 22, 33, 67, 68, 69, 75, 78, 83, 84, 160, 166, 183, 235, 245, 248, 249, 258, 263, 264, 265, 266 growth, 92, 190, 224, 279, 280 guidance, 67, 183 guidelines, 18, 35, 41, 54, 81 guilt, 132, 135, 148

H hands, 2, 63, 78, 86, 210, 211, 216, 277 Harvard, 1, 3, 55, 91, 93, 94, 121, 146, 222, 259, 268 health, 12, 31, 80, 152 heat, 12, 18, 192, 204, 211 hegemony, 102 height, 10, 18, 20, 51, 190 Henry Ford, 257, 266 high quality instruction, 279 high school, 14, 50, 69, 88, 90, 183, 189, 209, 212, 213, 214, 216, 219, 221, 240, 263 higher education, 237, 238, 240 hiring, 230, 276 homework, 167, 169, 185, 266 Honda, 180, 196 host, 155 hostility, 253 hub, 274 Hubble, 263 human actions, 127

human activity, 8, 12, 103 human brain, 140 human cognition, viii, ix, 59, 125 human development, 146 human exposure, 10 human nature, 127 humanism, 103 humanistic psychology, 146 humanity, 53, 98, 99, 100, 102, 103, 105, 120 Hungary, 16 hydrogen, 106 hydrogen atoms, 106 hypothesis, ix, 22, 43, 70, 86, 95, 104, 180, 188, 190, 195, 228, 261 hypothesis test, ix, 86, 95 hypothetico-deductive, 180

I identity, 42, 45, 94, 128 ideology, 102 idiosyncratic, 119 illiteracy, xii, 99, 257, 262 illusion, 169 images, 88, 93, 111, 122, 162, 163, 180, 197 imagination, 75, 111, 112, 114 imitation, 159 implementation, xii, 13, 69, 90, 227, 257, 267, 271, 272, 273, 274, 275, 276, 278, 281 implicit knowledge, 79 in situ, 63, 215 in transition, 140 incentives, vii inclusion, xi, 157, 223, 228 income, 12, 43, 44, 45, 46 independence, 129 independent variable, 51 indication, 106, 225 indicators, 16, 51, 205, 245 indices, 12 individual differences, 127, 136 individual students, 202, 245 induction, 3, 220, 250 industrial processing, 12 industry, 98, 228, 238, 274 inequality, 24, 102 inferences, 191, 192 information processing, 135 inheritance, 204 initiation, 78 innate capacity, 33 insight, 53, 87, 267 inspections, 101

Index inspiration, 3 instability, 98 instinct, 278 institutions, x, 7, 13, 16, 31, 97, 105, 151, 152, 153, 236, 258 instruction, xi, xii, 4, 14, 61, 89, 91, 94, 118, 119, 121, 126, 140, 145, 148, 158, 160, 175, 186, 191, 192, 195, 197, 206, 207, 209, 210, 211, 213, 214, 215, 217, 219, 220, 222, 223, 225, 226, 227, 232, 234, 235, 237, 241, 245, 257, 258, 259, 266, 267, 272, 279 instructional materials, 273 instructional methods, 266, 267 instructional practice, 147, 217, 279 instructors, xii, 227, 257, 258, 260, 262, 263, 264, 266, 267 instrumental concerns, 99 instruments, 7, 149, 180, 192, 266, 267 integration, 15, 36, 83, 89, 134, 136, 144, 195 integrity, 278 intellectual capital, 1 intellectual development, 100, 114 intellectual skills, viii, 59 intelligence, 64, 89, 130 intensity, 128, 130 intentions, 136, 139, 146, 164, 218, 221, 234 interaction(s), 18, 19, 20, 56, 63, 67, 78, 87, 88, 107, 114, 115, 116, 119, 121, 127,128, 131, 132, 136, 137, 140, 142, 145, 149, 153, 155, 163, 175, 183, 202, 203, 213, 221, 224, 234, 235, 245, 253 interactivity, 155, 159, 163, 173 interdependence, 32 interface, 3, 196 interference, 110 internalised, 7, 32 internalization, 127 Internet, xii, 19, 152, 155, 157, 159, 167, 168, 170, 172, 173, 174, 190, 257, 260 internship, xii, 239, 240, 241, 242, 254 interpersonal communication, 225, 229, 230 interpretation, 44, 61, 77, 84, 137, 180, 216 interrelations, 17, 20, 43, 133 interval, 13 intervention, 103, 162, 205, 279 interview, 82, 185, 205, 229 intrinsic motivation, 110, 112, 212, 215 intuition, 120, 182 invariants, 54 investment, 8, 97 isolation, 131 Israel, 59, 69, 88 Italy, 175

289

J Japan, 55, 139, 144 journalism, 260 journalists, 152 junior high school, 222 jurisdiction, 31 juror, 73 justice, 261 justification, viii, 31, 60, 85, 112, 185, 207

K kernel, 120 Keynes, 89, 94, 121 kindergarten, 15, 21, 194 knowledge acquisition, 224 knowledge construction, 61, 86, 132, 204 Korea, 179, 254

L labour, 99, 105, 109, 163 language, ix, 10, 11, 12, 26, 29, 33, 36, 44, 46, 47, 48, 49, 67, 76, 77, 125, 168, 171, 203, 227, 262, 280 laughter, 108 laws, 3, 7, 62, 97, 98, 101, 102, 103, 105, 107, 108, 109, 113, 114, 115, 116, 119, 120, 162, 261, 262 leadership, 231, 233, 273, 274, 275, 278 learned helplessness, 129 learners, viii, 62, 63, 67, 86, 92, 95, 96, 98, 100, 101, 110, 113, 114, 117, 119, 197, 245, 249, 250 learning, vii, viii, ix, x, xii, 3, 5, 6, 7, 13, 14, 15, 17, 20, 21, 22, 23, 25, 28, 29, 31, 32, 33, 34, 40, 41, 42, 50, 52, 53, 54, 55, 56, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 72, 76, 77, 78, 79, 81, 82, 83, 85, 86, 87, 88, 90, 91, 92, 93, 94, 98, 100, 101, 104, 105, 110, 111, 112, 114, 115, 117, 118, 119, 121, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 142, 143, 144, 145, 146, 147, 148, 149, 151, 152, 153, 154, 155, 156, 157, 158, 160, 161, 164, 165, 170, 171, 172, 173, 174, 175, 176, 187, 191, 196, 197, 198, 199, 201, 202, 203, 204, 206, 208, 209, 210, 213, 215, 216, 217, 218, 219, 221, 222, 224, 228, 237, 238, 239, 240, 241, 242, 245, 250, 251, 253, 254, 255, 258, 260, 263, 265, 266, 267, 272, 273 learning activity, 67, 202 learning behavior, 66, 130, 142 learning environment, viii, x, 60, 62, 69, 78, 79, 90, 119, 132, 151, 153, 155, 165, 170, 213, 215

290

Index

learning outcomes, 128 learning process, ix, 7, 29, 31, 60, 62, 72, 126, 131, 133, 134, 135, 138, 203, 210, 216 learning skills, 60, 62, 64, 90 learning styles, 258 learning task, 64, 67, 79, 216, 218 lens, 96, 118 lesson plan, 172, 251 liberal education, 101, 104, 105, 106 lifelong learning, 15, 17, 31, 35, 55 limitation, 66, 102, 225 linear function, 19, 37 linguistics, 12 links, 21, 28, 31, 39, 74, 82, 131, 134, 207 listening, 67, 230, 232, 236, 238 literacy, ix, 14, 32, 36, 60, 76, 79, 89, 92, 93, 94, 96, 99, 152, 153, 166, 167, 168, 173, 175, 176, 196, 218, 227, 238, 243, 254, 273, 280 literature, viii, x, 57, 59, 61, 62, 64, 66, 67, 68, 72, 90, 96, 100, 103, 113, 117, 118, 128, 154, 166, 187, 190, 201, 214, 220, 226, 228, 276 location, 26, 37 logical reasoning, 75 logistics, 276 love, 110, 111

M magazines, 154, 155, 157, 167, 175 magnetism, 174 malaise, 3 males, 126, 135 management, xii, 21, 64, 98, 271, 275, 277 manipulation, 191 manners, 43, 44, 46, 108 mapping, 48, 50, 116, 153 market(s), 32, 98, 225, 227 marketing, vii, 5, 6, 12, 278 Mars, 184, 264, 265 Marx, 105, 109, 183, 197, 204, 221 Marxism, 105 Maryland, 258 masking, 34 mass media, 152, 168 Massachusetts, 197 mastery, 102, 103, 120, 131, 212, 213 mathematical knowledge, 14, 134 mathematical methods, 14 mathematics, vii, viii, ix, 5, 6, 7, 12, 14, 15, 16, 17, 19, 20, 21, 25, 29, 30, 31, 32, 33, 34, 36, 38, 39, 40, 41, 47, 48, 50, 52, 53, 54, 56, 57, 66, 73, 82, 87, 96, 97, 98, 100, 101, 103, 104, 105, 108, 110, 111, 112, 115, 117, 119, 120, 121, 122, 125, 126,

127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 141, 142, 143, 144, 145, 146, 147, 148, 149, 216, 220, 222, 242, 243, 244, 245, 246, 248, 249, 251, 252, 254, 255, 259, 261, 262, 272, 280 mathematics education, vii, ix, 5, 6, 14, 16, 17, 21, 31, 32, 33, 40, 54, 57, 96, 101, 103, 104, 108, 112, 121, 122, 125, 126, 127, 128, 129, 130, 131, 137, 138, 144, 145, 147 maturation, 6 meanings, 64, 68, 73, 135, 158, 163, 207 measurement, 10, 25, 39, 96, 104, 106, 116, 139, 140, 190, 192, 193 measures, 11, 94, 129, 190 media, 80, 226, 261 mediation, 7, 114, 117, 118, 144, 155, 168 medicine, 188, 197 memory, 89, 110, 135, 146, 147, 215 memory retrieval, 135 mental activity, 211 mental age, 165 mental model, 153, 203 mental processes, 131, 133, 135, 138 mentor, 240, 243, 244, 245, 246, 248, 249, 250, 251, 253 mentoring, xii, 239, 240, 241, 243, 244, 245, 246, 248, 249, 250, 251, 253, 254, 255 mentoring program, 243, 244, 245, 246, 248, 249, 250, 251 Mercury, 264, 265, 268 meta-analysis, 101, 144 metacognition, 60, 62, 64, 65, 66, 67, 68, 70, 71, 72, 75, 86, 87, 88, 89, 92, 93, 94, 129, 131, 137, 140, 144, 147, 205, 207 metacognitive knowledge, 64, 68, 71, 86, 87, 93, 94 metacognitive skills, 64, 65, 66, 67, 68, 70, 85, 86, 87, 91 metaphor, 203 meteor, 264 methodological implications, 146 Ministry of Education, 16, 240, 254, 255 minority, 21, 24, 102, 194, 228 misconceptions, xii, 53, 63, 100, 113, 118, 121, 203, 204, 205, 208, 214, 217, 218, 257, 259 misunderstanding, 40, 53 modeling, 53, 191, 277 models, x, 12, 16, 20, 32, 52, 73, 75, 86, 111, 113, 131, 145, 146, 158, 159, 163, 171, 192, 198, 201, 202, 206, 208, 209, 210, 211, 212, 214, 215, 217, 237, 271, 272 modern society, 52 modernism, 104 modules, 98, 273, 280

Index momentum, 108, 119 money, 38, 97, 275 Montana, 258 mood, 143, 148 Moscow, 122 motion, 30, 97, 107, 108, 109, 113, 114, 115, 116, 118, 119, 120, 121, 205, 214, 220, 262 motivation, ix, x, 63, 72, 95, 98, 111, 112, 127, 128, 129, 130, 132, 139, 140, 141, 144, 146, 148, 155, 182, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 222 movement, xii, 30, 34, 35, 80, 103, 257 multidimensional, 129 multimedia, 164 multiple-choice questions, 259 multiplicity, 133 multiplier, 9 multivariate, 191 murmur, 108 music, 90, 155, 158, 159

N narratives, 104, 108 nation, 1, 99, 102 National Center for Education Statistics, 15, 56 national interests, 96, 97 National Research Council, viii, 59, 60, 92, 179, 198, 268, 281 National Science Foundation, viii, 95, 98, 271 natural environment, 173 natural science, 160, 188 natural sciences, 160, 188 natural selection, 189 NCES, 14, 15, 56 Nebraska, 141, 149 negative experiences, 110 negotiation, 152, 164 Netherlands, 57, 91, 92, 93, 94, 122, 175 network, 7, 34, 199 New Jersey, 237 New Orleans, 88, 91, 92, 93 New York, iii, iv, 4, 15, 24, 55, 56, 57, 87, 91, 92, 94, 121, 122, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 198, 199, 218, 268, 281 New Zealand, xi, 105, 122, 218, 223, 224, 225, 226, 227, 229, 230, 235, 236, 237, 238 newspapers, 152, 154, 155, 157, 167, 175, 176 Nobel Prize, 17 North America, 9, 142 Northern Ireland, 175 Norway, 144

291

novelty, 205, 216 number-words, 33 nutrition, 218

O objectivity, viii, 95, 99, 104, 191 obligation, 8, 101 observations, 21, 53, 68, 69, 70, 71, 74, 78, 79, 110, 113, 116, 166, 180, 183, 186, 188, 192, 262, 279 OECD, 6, 7, 14, 16, 33, 56 oppression, 98, 104, 105 orbit, 262 organism, 155 organization, 136, 142, 157, 180, 229, 252 organizations, x, 151, 154, 229 orientation, 103, 117, 131, 141, 147, 155, 183, 207, 280 original learning, 135 outreach programs, 279 oversight, xii, 271 ownership, 63, 103, 277 oxidation, 69

P Pacific, 238 pain, 148, 280 paradigm shift, 263 parameter, 30, 156 parents, 129, 155, 162, 250, 273, 279 Pareto, 12 Paris, 54, 55, 57 Parliament, 55 partnership(s), xii, 240, 241, 246, 248, 250, 253, 271, 273, 274, 275, 276, 278, 279, 280, 281 passive, 117, 164, 172, 204 pathways, 131, 142 pedagogy, viii, 59, 60, 92, 237, 238, 241, 245, 251, 273, 275, 277 peers, 18, 20, 61, 67, 75, 203, 212, 251 percentile, 11 perception, vii, ix, 63, 95, 100, 107, 115, 133, 159, 173, 216, 232 perceptions, 130, 133, 137, 163, 236, 238 performance, ix, 15, 16, 39, 41, 56, 66, 68, 72, 88, 90, 92, 94, 125, 126, 129, 130, 134, 137, 142, 143, 144, 146, 149, 205, 245, 250, 251, 252, 280 permit, 45, 47 personal, vii, viii, x, xi, 5, 6, 17, 20, 40, 67, 74, 75, 78, 95, 126, 127, 128, 131, 132, 133, 134, 135, 136, 137, 138, 143, 144, 151, 154, 155, 157, 166,

292

Index

168, 169, 185, 189, 191, 196, 202, 203, 206, 211, 215, 216, 224, 250 personal communication, 196 personal efficacy, 134 personal learning, 126, 128, 133, 134, 138, 144 personal values, 75 personality, 127, 131, 133, 136, 138, 139, 140, 141, 142, 143, 146, 148, 209 persuasion, 106, 148, 234, 235 pessimism, 103 philosophers, 74, 75, 108, 131 physical activity, 211 physical education, 216, 218 physical environment, 162 physical interaction, 203 physics, 1, 2, 3, 4, 7, 17, 34, 87, 97, 98, 102, 110, 116, 216, 218, 219, 221, 258, 262, 263 PISA, vii, 5, 7, 14, 15, 16, 32, 33, 34, 56, 57, 174 planets, 258, 262, 265 planning, 18, 31, 64, 67, 68, 69, 70, 77, 79, 166, 186, 191, 215, 278 Plato, 108, 117 plausibility, 76 pleasure, 31, 148, 163 Poland, 16 policy makers, 16 pollution, 101, 102 polynomials, 24 population, 12, 39, 41, 99, 162, 167, 226, 240, 259 portfolio, 32 Portugal, 90, 175 positive feedback, 277 positive mood, 135 positive relation, 215 positive relationship, 215 positivism, 109 postmodernism, 103, 104 power, x, 33, 99, 115, 131, 134, 136, 151, 168, 280 practical activity, 61, 78 prejudice, 100 preparedness, xi, xiii, 224, 271, 275 presentation skills, 235 preservice teachers, 182, 199 pressure, 184, 193, 204 prestige, vii, 107 price gouging, 280 primacy, 131 primary school, 21, 39, 41, 246 prior knowledge, 155, 221, 251 probability, 82 probe, 191

problem solving, vii, ix, 3, 5, 15, 35, 45, 52, 68, 89, 92, 125, 128, 130, 134, 135, 140, 141, 142, 143, 144, 145, 147, 207 problem-solving, 63, 66, 82, 129, 131, 144 problem-solving behavior, 144 problem-solving skills, 63 problem-solving strategies, 131 procedural knowledge, 15, 64, 68, 70, 71, 81 production, 39, 103 productivity, 105 profession(s), 102, 112, 254 professional development, xii, 69, 94, 254, 271, 272, 273, 274, 275, 277, 280 professional growth, 244 program, 4, 62, 69, 72, 83, 146, 159, 163, 165, 166, 167, 175, 183, 211, 227, 236, 240, 241, 242, 243, 244, 245, 246, 254, 277 programming, 12 promote, ix, 19, 63, 65, 86, 95, 111, 112, 118, 169, 207, 211, 216, 222, 226 proportionality, 19, 35, 38, 39, 41, 42, 43, 53 proposition, 120, 183 psychiatric illness, 148 psychological processes, 93, 217, 222 psychological stress, 250 psychologist, 119, 132 psychology, 89, 91, 119, 126, 131, 139, 141, 142, 143, 146 public health, 10, 11 pupil, 78, 112, 113, 219

Q QED, 159 qualifications, 110, 230 quality control, xii, 12, 239, 240, 242 quantitative research, 146 quantum mechanics, 109, 110 quantum theory, 98 questioning, vii, 3, 5, 7, 88, 106, 234, 245 questionnaires, 53, 173, 182, 213, 229, 235

R race, 98, 102, 145, 186 racism, 100 radiation, 264 radio, 19 radius, 39, 262 range, 11, 27, 33, 48, 49, 50, 51, 52, 61, 73, 76, 77, 79, 80, 82, 101, 168, 190, 206, 211, 212, 214, 215, 225, 229, 230, 243, 277

Index rape, 105 rationality, 99, 102, 198 reading, 10, 42, 46, 47, 66, 94, 103, 110, 159, 173, 174, 175, 205, 216, 219, 264, 280 real numbers, 6, 21, 48, 52, 53 realism, 106, 107, 173 reality, 12, 14, 109, 144, 156, 164, 203, 226, 250, 278 reasoning, vii, x, 5, 6, 32, 33, 34, 42, 43, 45, 57, 69, 73, 74, 75, 79, 82, 83, 84, 86, 89, 90, 91, 94, 179, 180, 183, 185, 190, 191, 192, 193, 195, 197, 198, 199, 213, 264, 280 reasoning skills, 83, 94, 180, 191, 193, 196, 199 recall, 216 recalling, 40 recognition, 276, 279 reconstruction, 204 recreation, 159, 161 recruiting, 274 recycling, 183 reductionism, 104 reflection, 53, 63, 71, 72, 85, 86, 103, 148, 205, 250 reforms, viii, 59, 271 regional, 229 regression, 45, 46, 51 regulation(s), 8, 12, 64, 70, 71, 72, 133, 135, 136, 137, 138, 144, 147, 148, 149, 166 regulators, 133 rejection, 103, 183, 261 relationship(s), 7, 18, 22, 26, 28, 30, 34, 36, 37, 38, 39, 44, 53, 62, 74, 81, 84, 128, 138, 144, 146, 149, 183, 191, 202, 211, 212, 217, 218, 219, 220, 241, 250, 262 relative size, 265 relatives, 155 relativity, 109, 110 relaxation, 165 relevance, xi, 32, 53, 81, 99, 101, 190, 211, 216, 223 reliability, 80, 85, 91, 193, 195, 198 religious beliefs, xii, 258, 260 repetitions, 193 replication, 193 research design, 61, 166 residential buildings, 9 resistance, 12, 101, 227, 228, 235, 236, 276, 278, 280 resolution, 2, 55 resource availability, 227 resources, xii, 41, 61, 68, 132, 134, 236, 245, 251, 253, 257, 263, 277 restitution, 119 retention, xi, 224, 236 revenue, 226, 276

293

rhetoric, 175, 176 Rhode Island, 271, 272, 273, 274, 279 rigidity, 3, 172 risk, 34, 35, 76, 80, 97, 262, 281 rolling, 99, 186, 194 Royal Society, viii, 95, 98

S SA, 271 sacrifice, 278 sample, 8, 11, 23, 193, 195 scaling, 41 scepticism, 104 schema, 46, 145 scholarship, 85 school, vii, viii, ix, x, xii, 2, 5, 14, 16, 18, 21, 23, 32, 35, 44, 55, 59, 60, 61, 62, 63, 64, 69, 73, 77, 79, 80, 82, 85, 88, 90, 91, 92, 93, 95, 97, 98, 99, 100, 101, 104, 105, 110, 113, 125, 126, 127, 128, 131, 134, 137, 139, 141, 143, 149, 151, 152, 153, 155, 157, 160, 161, 162, 165, 166, 167, 168, 169, 170, 171, 172, 174, 175, 176, 177, 184, 185, 190, 191, 192, 197, 198, 199, 209, 212, 213, 215, 216, 217, 218, 220, 221, 239, 240, 241, 242, 244, 246, 248, 249, 250, 253, 254, 255, 258, 261, 262, 263, 271, 273, 275, 276, 277, 279, 280, 281 school learning, 126, 153 schooling, 15, 110, 146, 213, 217 science, vii, viii, x, xi, xii, 3, 33, 59, 60, 61, 62, 63, 64, 65, 67, 68, 69, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 120, 121, 122, 139, 140, 141, 142, 146, 148, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 179, 180, 181, 182, 184, 185, 187, 188, 189, 192, 194, 195, 196, 197, 198, 199, 201, 202, 203, 204, 205, 207, 208, 209, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 224, 225, 228, 229, 230, 232, 234, 235, 236, 237, 242, 243, 244, 245, 246, 248, 254, 255, 257, 258, 259, 260, 261, 262, 271, 272, 273, 274, 275, 276, 277, 279, 280, 281 science department, ix, 95, 97 science educators, viii, 59, 60, 76, 77, 79, 204 science literacy, 72, 76, 94, 175, 281 science teaching, viii, 59, 60, 65, 77, 78, 81, 88, 90, 91, 93, 121, 204, 213, 218, 219, 220, 221, 222, 248, 272 scientific community, 63, 107, 185

294

Index

scientific knowledge, 16, 74, 76, 77, 85, 88, 92, 152, 153, 156, 164, 165, 186, 191, 196 scientific method, 79, 80, 120, 180, 197, 199, 261 scientific understanding, ix, 63, 96, 118, 171, 203, 208 scores, 230, 235, 252, 264, 266, 267 search, 63, 76, 172, 187, 190, 219 searching, 48, 72, 195 secondary education, 162 secondary schools, 62 secondary students, 209 secondary teachers, 175, 216, 222 selecting, 86, 173, 267 self-actualization, 218 self-awareness, 133, 135, 136, 137, 138 self-concept, 127, 128, 129, 130, 139, 140, 141, 145, 149, 216 self-confidence, 128, 129, 131, 218 self-consciousness, 134 self-control, 135 self-determination theory, 218 self-efficacy, 129, 130, 131, 139, 142, 212, 213, 216, 218, 219, 220 self-esteem, 128, 134, 138, 140 self-evaluations, 145 self-interest, 100 self-knowledge, 134, 135 self-organization, 147 self-perceptions, 128, 132, 136, 147 self-reflection, 134 self-regulation, ix, 126, 131, 132, 134, 135, 136, 137, 138, 141, 142, 143, 145, 147, 148, 149 self-schemata, 130 self-study, 221 self-worth, 134, 143 semiotics, 176 sensory experience, 156 separateness, 131 separation, 41 series, 15, 40, 41, 62, 104, 114, 146, 190, 204, 213, 251 sex, 143, 216 shame, 130, 132, 135, 136, 137, 148 shape, viii, 59, 60, 154 shaping, 31, 153, 162, 168 sharing, 73, 193, 203, 250 shock, 250 sign(s), 7, 25, 29, 36, 41, 44, 45, 259 similarity, 96, 119 sites, xii, 257 skills, xi, 15, 33, 60, 61, 63, 64, 66, 67, 68, 70, 72, 74, 75, 77, 81, 82, 83, 84, 85, 86, 87, 88, 89, 91, 92, 134, 152, 155, 156, 159, 165, 167, 173, 185,

190, 191, 192, 194, 195, 197, 199, 224, 225, 226, 228, 229, 230, 232, 234, 235, 236, 237, 238, 241, 245, 246, 253, 272, 275 soccer, 111 social attitudes, 127 social comparison, 148 social construct, 67, 76, 77, 78, 104, 105, 143, 203, 219, 220, 224 social constructivism, 104, 105, 203, 220 social context, 92, 130, 232, 240 social environment, 131, 134, 137, 144, 173 social factors, 74, 218 social information processing, 141 social learning, 137 social problems, 99 social psychology, 129, 139, 185 social sciences, 188 social skills, 155 socialization, 127 society, vii, viii, xii, 5, 6, 7, 16, 17, 20, 32, 53, 76, 80, 85, 93, 95, 96, 97, 98, 99, 100, 103, 105, 107, 114, 126, 152, 161, 172, 185, 222, 240, 257, 260, 262 sociocultural practices, 90 sociocultural psychology, 54 socioeconomic status, 145 soft money, 278 software, 27, 30, 41, 52, 155, 170, 183 solar collectors, 18 solar system, 164, 168, 264, 265, 267 South Africa, 214 South Korea, 179 Spain, 16 special education, 161 special theory of relativity, 110 species, 152, 159 specific knowledge, 84 specificity, 7, 15, 44, 225 spectrum, 1 speculation, 120 speech, 159, 227, 228, 229, 231, 232, 233, 238 speed, 19, 22, 38, 39, 186, 262 spelling, 33 stability, 130 stages, 22, 44, 46, 53, 69, 70, 71, 127, 164, 180, 208, 281 stakeholder groups, 235 stakeholders, 237 standard of living, 1 standardized testing, 88 standards, viii, 11, 15, 59, 60, 98, 134, 181, 188, 195, 243, 244, 245, 254, 255, 272, 281 stars, 115, 258, 259, 261, 266

Index statistics, 2, 11, 12, 82, 220, 244, 255, 267 sterile, 3 stimulus, 173 stock, 32 stock exchange, 32 stoichiometry, 69 storage, 55, 91 strategic planning, 72, 277, 281 strategies, x, xii, 16, 41, 45, 66, 67, 68, 71, 86, 87, 90, 104, 132, 135, 139, 175, 181, 186, 190, 191, 192, 201, 208, 209, 210, 211, 212, 214, 215, 217, 221, 241, 245, 251, 253, 258, 267, 271, 275 stratified sampling, 205 stress, xii, 131, 173, 257, 267, 277 string theory, 120 structuring, 14, 17, 19, 164 student achievement, xiii, 16, 271, 275 student behavior, 133 student group, 39, 84, 184, 191 student motivation, x, 201, 205, 209, 210, 212, 213, 215, 217 students, vii, viii, ix, x, xi, xii, 1, 2, 3, 4, 7, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 28, 31, 32, 35, 41, 45, 46, 47, 48, 50, 51, 53, 54, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 90, 92, 93, 95, 97, 114, 116, 119, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 143, 144, 145, 146, 147, 148, 151, 152, 153, 155, 156, 157, 158, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 179, 180, 181, 182, 183, 184, 185, 186, 187, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 201, 202, 203, 204, 205, 206, 207, 208, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 232, 234, 235, 236, 238, 240, 245, 250, 251, 252, 253, 257, 258, 259, 260, 261, 262, 263, 264, 265, 266, 267, 271, 279, 280 subgroups, 84 subitizing, 33 subjectivity, 186 substitutes, 280 subtraction, 37, 57 success rate, 39, 41 supply, 8, 212, 227, 240 surprise, 207, 208, 216, 230, 265 survival, 97 suspense, 216 switching, 34 symbols, 25, 32, 34, 36, 37, 44 symptom, 99 synchronization, 147

295

synthesis, 45, 46, 108, 142, 162 systems, vii, ix, 5, 6, 7, 8, 10, 11, 16, 17, 20, 25, 109, 125, 130, 131, 132, 134, 136, 137, 138, 139, 147, 160, 194, 219, 240, 265

T Taiwan, xii, 239, 240, 241, 254, 255 target population, 156, 162, 167, 171 target variables, 192 targets, x, 76, 151, 157, 165, 274 task difficulty, 130 task performance, 68 task-orientation, 131 taxonomy, x, 127, 151, 153, 157, 160 teacher preparation, 240, 241 teacher training, 14, 16, 31, 54, 172, 174, 240 teachers, vii, x, xii, xiii, 1, 3, 4, 5, 7, 12, 16, 28, 35, 39, 40, 41, 61, 62, 63, 64, 67, 69, 79, 81, 85, 86, 88, 94, 97, 101, 119, 122, 147, 151, 152, 153, 155, 157, 158, 161, 162, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 182, 184, 185, 194, 203, 207, 209, 211, 213, 214, 215, 216, 218, 219, 221, 237, 239, 240, 241, 242, 243, 246, 247, 248, 251, 254, 255, 260, 271, 272, 273, 274, 275, 276, 277, 278, 279, 280, 281 teaching, vii, viii, ix, x, xi, 3, 5, 6, 7, 14, 17, 20, 21, 23, 24, 25, 29, 30, 32, 33, 34, 36, 40, 41, 45, 46, 47, 48, 50, 51, 53, 54, 55, 59, 60, 61, 64, 65, 67, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 91, 94, 95, 100, 101, 103, 104, 112, 117, 119, 139, 140, 145, 146, 147, 153, 155, 159, 170, 171, 172, 173, 176, 190, 195, 196, 197, 198, 201, 202, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 217, 218, 220, 221, 222, 224, 225, 226, 228, 236, 237, 238, 240, 241, 243, 244, 245, 246, 248, 250, 251, 252, 253, 254, 258, 260, 261, 263, 267, 268, 273, 277, 281 teaching strategies, 153, 172, 173, 210, 212 technology, vii, 5, 6, 79, 88, 97, 98, 99, 100, 105, 109, 141, 152, 161, 163, 175, 190, 221, 225, 237, 243, 261, 272 telephone, 229, 230, 275 television, xii, 82, 111, 152, 154, 155, 158, 175, 257, 260 temperature, 9, 74, 80, 190, 262 tension, 77, 83, 116, 248 tertiary education, 238 tertiary sector, 226 test anxiety, 137, 143 test items, 267 textbooks, 1, 2, 3, 18, 21, 96, 115, 118, 163, 168, 261, 262, 264, 267

296

Index

theory, xi, 3, 6, 14, 20, 28, 54, 62, 64, 67, 73, 74, 75, 82, 83, 84, 89, 91, 106, 107, 109, 110, 113, 116, 120, 139, 140, 141, 143, 144, 146, 147, 148, 149, 153, 180, 181, 186, 189, 195, 196, 198, 201, 202, 204, 206, 210, 212, 217, 219, 221, 222, 224, 227, 244, 246, 260, 261, 262 thermodynamics, 110 think critically, xii, 63, 257 thinking, ix, xii, 1, 2, 3, 4, 16, 26, 32, 34, 63, 64, 67, 75, 78, 79, 82, 83, 86, 87, 89, 90, 91, 92, 94, 95, 96, 103, 109, 114, 118, 120, 121, 130, 131, 132, 140, 141, 181, 182, 186, 189, 191, 195, 196, 197, 199, 203, 207, 215, 237, 238, 250, 252, 253, 257, 261, 276 Thomas Kuhn, 75 time, xi, 2, 3, 9, 12, 19, 20, 24, 26, 29, 30, 33, 38, 43, 45, 47, 48, 50, 61, 62, 63, 67, 69, 72, 74, 78, 87, 97, 100, 105, 106, 107, 108, 111, 112, 116, 118, 135, 154, 162, 165, 166, 169, 172, 183, 184, 185, 191, 192, 193, 195, 196, 212, 214, 224, 226, 227, 228, 236, 241, 242, 246, 248, 249, 251, 260, 261, 263, 264, 265, 266, 267, 273, 274, 275, 276, 277, 278, 281 TIR, 276 tracking, 193, 276, 278 tradition, 16, 44, 77, 84, 103, 105, 128 traffic, 12, 18, 19 training, vii, xi, 5, 7, 16, 20, 31, 50, 77, 80, 82, 83, 172, 173, 187, 191, 223, 224, 225, 226, 227, 235, 237, 240, 241, 242, 243, 244, 245, 273, 274, 275, 276, 277, 278, 281 training programs, 16, 31 traits, 130, 184 trajectory, 51, 108 transformation, 26, 36, 37, 99, 152, 172, 240 transformations, 46 translation, 8, 10, 26, 43 transmission, 104, 117 trend, 1, 98, 103 trust, 183, 185

U UK, viii, 59, 80, 81, 88, 89, 91, 92, 93, 94, 95, 97, 98, 101, 105, 111, 118, 140, 141, 143, 144, 146, 147, 148, 149 UN, 8 uncertainty, 80, 169, 193 undergraduate, xi, 23, 50, 98, 113, 182, 223, 224, 226, 227, 230, 279 undergraduate education, 227 underlying mechanisms, 192 uniform, 30, 107, 113, 116, 242

United Kingdom, 50, 95 United States, ix, 15, 125 universe, 21, 25, 103, 115, 164, 263, 264, 275 universities, xi, 90, 97, 99, 223, 224, 225, 226, 240, 241, 242, 258, 262, 263, 266 university education, 224, 226 university students, 238 users, vii, 5, 6, 7, 12, 13

V validity, 2, 80, 85, 156, 161, 167, 169, 186 values, 29, 36, 42, 47, 49, 52, 91, 102, 103, 127, 129, 130, 139, 141, 155, 169, 193, 230, 232, 235, 262 variability, 127, 181, 193, 194, 198 variable(s), ix, 24, 26, 28, 29, 36, 38, 39, 40, 42, 43, 52, 69, 72, 77, 81, 115, 116, 125, 126, 129, 130, 131, 138, 147, 180, 181, 186, 188, 190, 191, 192, 193, 195, 196, 210, 219 variance, 192, 242 variation, 73, 80, 127, 130, 131, 134, 135, 137, 194 vector, 120 vehicles, 18, 88, 152 vein, 104 velocity, 22, 29, 30, 106, 262 venue, 224 Venus, 265 Vermont, 274 vessels, 104, 259 vision, 33, 34, 100, 275 Vygotsky, 67, 76, 83, 93, 108, 203, 222

W Wales, 81 war, 105 warrants, 74, 75, 79, 83, 182, 185 weakness, 164 wealth, 15, 99, 173, 186 weapons, 101, 102 weapons of mass destruction, 101, 102 web, 14, 54 websites, 61, 264 well-being, 97, 99 women, 3 workers, 105, 231, 233 working groups, 20 workplace, 226, 227, 235, 238 writing, 2, 8, 13, 24, 28, 29, 33, 35, 36, 37, 40, 41, 42, 48, 49, 53, 71, 99, 175, 207, 213, 219, 231, 233, 237, 238, 264, 267, 280

Index

X XML, 55

yield, 3, 265, 278

Z zeitgeist, 93

Y

297