The Van Hiele Model of Thinking in Geometry Among Adolescents (Jrme Monographs, Vol 3)

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NationalCouncilof Teachers ofMathematics

A Monograph Series of the National Council of Teachers of Mathematics The JRME monograph series is published by the Editorial Panel as a supplement to the journal. Each monograph has a single theme related to the learning or teaching of mathematics. To be considered for publication, a manuscript should be (a) a set of reports of coordinated studies, (b) a set of articles synthesizing a large body of research, (c) a single treatise that examines a major research issue, or (d) a report of a single research study that is too lengthy to be published as a journal article. Proposals for a monograph may be sent at any time to the monograph series editor. A proposal must contain the following items: 1. An outline of the work with enough detail to permit an evaluation of its significance for mathematics education 2. The names, affiliations, and qualifications of the contributing authors 3. A time line for the development of the monograph If a draft manuscript of no more than 200 double-spaced typewritten pages has already been produced, four copies of it should be enclosed with the proposal. Any other information about the nature of the monograph that might assist the series editor and the Editorial Panel in its review is welcome.

Series Editor FRANK K. LESTER,JR., Indiana University, Bloomington, IN 47405 Associate Editor DIANA LAMBDIN KROLL, Indiana University, Bloomington, IN 47405 Editorial Panel FRANK K. LESTER,JR., Indiana University, Bloomington, IN 47405; Chairman DOUGLAS H. CLEMENTS, Kent State University, Kent, OH 44242 JAMES HIEBERT, University of Delaware, Newark, DE 19716 MIRIAM A. LEIVA, University of North Carolina at Charlotte, Charlotte, NC 28223 J. MICHAEL SHAUGHNESSY, Oregon State University, Corvallis, OR 97331 ALBA G. THOMPSON, Illinois State University, Normal, IL 61761 MARY M. LINDQUIST, Columbus College, Columbus, GA 31993; Board Liaison Proposals for monographs should be sent to Frank K. Lester, Jr. Room 309, Education Building Indiana University Bloomington, IN 47405.

This researchprojectwas supported(1980-83) undera grant(#SED 7920640) from the Research in Science Education(RISE) Programof the NationalScience Foundation. The membersof the ProjectStaff are faculty at BrooklynCollege, City Universityof New York and include: David Fuys, DorothyGeddes, C. James Lovett and RosamondTischler. The materialcontainedin this monographshould not be interpretedas representingthe opinions or policies of the National Science Foundation. Illustrationsanddrawingsin this manuscriptare by RosamondW. Tischler,with the exceptionof the title page and the end page, which come from the doctoralthesis of Dina van Hiele-Geldof (1957/1984).

Copyright ? 1988 by THE NATIONALCOUNCILOF TEACHERSOF MATHEMATICS,INC. 1906 AssociationDrive, Reston, VA 22091 All rightsreserved

Second printing 1995

The publications of the National Council of Teachers of Mathematics present a variety of viewpoints. The views expressedor implied in this publication,unless otherwise noted, should not be interpretedas official positionsof the Council.

Printedin the United States of America

FOREWORD My relationswith BrooklynCollege began in 1980. It was an importantdate for two reasons. First, the Brooklyn College Projecttranslatedinto English some of my writings and those of my late wife, thus making my theory available to a wider audience. Second, it markedthe beginningof the collection of experimental datain the UnitedStatesto supportmy theory. The clinical interviewsconductedby the Brooklyn College Project confirmedmy predictionthat even after pupils had some years of instructionin geometry,theirperformancewould be disappointing. The Project also found that many pupils were able to improve their performance when the instructionwas changedin accordancewith my theory. The van Hiele model contains recommendationsto change textbooks. The BrooklynCollege investigationmade clear thatin the geometrymaterialsin grades K-8 textbooksthe van Hiele levels aremixedup--notsequenced--andbecauseof this the higherlevels are rarelyreached. Frommy own work and that of the Brooklyn College Projectcertainresultsareevident: * We know the shortcomingsof traditionalinstructionandways to improveit. * We know thatinstructionmustbe adjustedto accountfor the differentphases of the learningprocess. * We know thattextbooksmust be rewrittento accountfor the variousphases of the learningprocess. * We know thatinstructionat level 0 can be given at an earlyage andvery often oughtto be given at thatage. Futureinvestigationsandapplicationsof my theoriesin mathematicsandalso in othersubjectsincludethe following: (1) Textbooksof geometrycan be designedin accordancewith the levels. (2) A greatdeal of geometryof level 0 can be given at the primaryschool with childrenof 6-10 years (justlike in the Soviet Union). (3) Investigationscan be startedto learnmore aboutlevel 0 of arithmetic.Such investigationsrelateto childrenof ages 1-6. The methodsneeded to stimulatesuch childrenare quite differentfrom those needed to stimulateolder childrenbecause theiractions, for the most part,are determinedby innermotives.

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(4) An investigationcan be startedto analyze the levels in physics. For this topic the sequencing of the levels is quite complicated. The specialization and mechanizationof modem life is such that level 0 of physics is invisible to a great extent. So, much of level 0 of physicsmustbe providedby instruction.This can be given at the same time andeven coordinatedwith geometryof level 0. (5) I have seen a textbook on economics which takes the levels into account. Fromthe very beginninga readerof this textbookis fascinatedby the sequencingof the material. It is worththinkingaboutthe use of the levels in such othertopics. Fromthe above we may concludethatthe BrooklynCollege investigationshave opened up many new perspectives. I hope that in the future I too will make contributionsin exploringthese new perspectives. March12, 1988 Voorburg,The Netherlands

P. M. van Hiele

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TABLE OF CONTENTS Page ....

Foreword.........................................

iii

List of Tables ..............v.................................

vii

Preface .............

ix

.............................................

Chapter1. Overviewof Project....................................... ResearchObjectives,Methods,Design andAnalyses .........

1 1

Chapter2. The van Hiele Model ...................................... Background............................................ Levels andTheirCharacteristics......................... Development,DocumentationandUse of Level Descriptors . Translationof Writingsof the van Hieles ..................

4 4 5 8 8

....... Chapter3. InstructionalModules .............................. MajorCharacteristicsof Modules ........................ DevelopmentandValidationof the Modules ............... Module1 .............................................. Module2 ........................................ Module3 ..............................................

.

11 11 15 17 29 44

Chapter4. Van Hiele Level Descriptors:DevelopmentandDocumentation.. Formulationof the van Hiele Model....................... Level DescriptorsandSampleStudentResponses ...........58 Level 0 ........................................... Level 1 .......................................... Level2 ........................................... Level3 ........................................... Level4 ........................................... Documentation......................................... AnotherFrameof Referencefor the Levels ...............

58 60 64 69 71 72 77

Chapter5. ClinicalStudy:Interviewswith Sixth GradeSubjects .......... Subjects............................................... Results:An Overview................................... ....... ................. GroupI .... ............... ............................ .............. II Group GroupIII .......................................

78 78 78 82 85 89

v

56 56

99 99 99 101 104 118

Chapter6. ClinicalStudy:Interviewswith NinthGradeSubjects........... Subjects............................................... Results:An Overview ................................... .......... GroupIV .............................. GroupV ......................................... GroupVI ........................................ Chapter7. Discussionof Findingsof ClinicalStudy................... Summaryof Students'Levels of Thinking ................. FactorsAffectingStudents'Performanceon Modules........ Levels of Thinkingon SpecificTasks...................... Retentionof Students'Levels of Thinking ................. Discussionof the InstructionalModules....................

. 133 133 135 139 141 142

Chapter8. ClinicalInterviewswith Preserviceand InserviceTeachers ..... Subjects............................................... Procedure............................................ Teachers'Responsesto SelectedModuleActivities .......... Teachers'Commentson InstructionalModuleActivities ..... Teachers'Identificationof van Hiele ThoughtLevels ........ Implicationsfor TeacherPreparationand ClassroomPractice.

144 144 144 145 151 153 154

Chapter9. Text Analysis ............................................. Goal ................................................. Procedure............................................ Findings ............................................... Text Presentationof ThreeContentStrandsAs Relatedto van Hiele Didactics................................. Implications ...........................................

157 157 157 161

Chapter10. ImplicationsandQuestionsfor FurtherResearch ............. Implicationsaboutthe Levels ............................ Implicationsfor ProjectLevel Descriptorsand Their Use .... Implicationsfor FutureResearch .........................

180 180 183 186

Bibliography.......................................................

192

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172 175

LIST OF TABLES Table

Page

1. AchievementTest Scoresand ModulesCompletedby SixthGraders .....79 2. Sixth Graders'Level of Thinkingon Key ModuleActivities ............

80

3. AchievementTest Scores andModulesCompletedby NinthGraders.....

100

4. NinthGraders'Level of Thinkingon Key ModuleActivities ............

102

5. Percentof Lessons at MaximumLevel 0, 1 or 2 ...........

167

............

6. Percentof Lessonswith ExercisesAll at Level 0 or "Unassignable"......

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168

PREFACE This monograph presents a report of the research project entitled: An Investigationof the Van Hiele Model of Thinkingin GeometryAmong Adolescents, supported under grant number SED 7920640 from the Research in Science EducationProgramof the NationalScience Foundation.The focus of this research was the conductand analysisof six hoursof clinical interviewswith sixth andninth grade students to investigate how they learn geometry in light of the van Hiele model. In Chapter1, an overview of the Projectand its four majorgoals is given. The theoreticalmodel, namely,the van Hiele levels andphaseswithinlevels of thinking, as initially characterizedby van Hiele and others, is describedin Chapter2. The Project's elaboration of these levels, in terms of specific student behaviors, is presentedin a later chaptersince the level descriptorsare most easily understoodif one is familiarwith the context(instructionalmodules)in which they are examined. In Chapter3, the developmentand detaileddescriptionof the Project'sresearch tool, InstructionalModules 1, 2 and 3 (Propertiesof Polygons, Angle Measurement and Angle Sum for Polygons, Area of Polygons) are set forth. Chapter4 follows with the Project'sformulationof the van Hiele model and level descriptorswith sample student responses to questions or activities in the instructionalmodules. Chapter4 also contains documentationof the level descriptorsusing quotations from the writingsof Dina van Hiele-GeldofandPierrevan Hiele. In Chapters5, 6 and 7, the individualperformancesof 32 students (16 sixth graders and 16 ninth graders) during approximately six hours of one-to-one videotaped clinical interviews, using the Project's instructional modules, are analyzed and discussed and the findings are summarized. The performancesof eight preservice and five inservice teachers on selected activities from the instructionalmodulesare analyzedand reportedin Chapter8. An analysisof the geometrystrandin threeUnitedStatesmathematicstextbook series, grades K-8, in light of the van Hiele levels is set forth in Chapter 9. Implicationsof the Project'sstudy--thatis, theoreticalimplicationsaboutthe nature of the van Hiele levels and methods of determiningthem, and implicationsof the study for classsroom practice, teachertrainingand curriculumdesign--as well as questionsfor furtherresearchare discussedin Chapter10. It should be noted that the Projecthas also publisheda monographcontaining translationsof significantworks of the van Hieles in orderto provide the Englishspeakingresearchcommunitywith a resourcethatwill shed more light on the van Hiele model. Among otherwritings, it containsthe complete dissertationof Dina ix

van Hiele-Geldof: The Didactics of Geometryin the Lowest Class of Secondary School. The monographis entitled:English Translationof Selected Writingsof Dinavan Hiele-GeldofandPierreM. van Hiele andis availablethroughEducational ResourcesInformationCenter(ERIC,numberED 287 697).

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VAN HIELE MODEL ~THE OF THINKING IN GEOMETRY AMONG ADOLESCENTS

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CHAPTER 1

OVERVIEW OF THE STUDY This monographis the resultof a three-yearresearchprojectwhich focused on a model of geometrylearningpresentedin 1957 by the Dutch educatorsP. M. van Hiele andhis late wife, Dina van Hiele-Geldof.This model has motivatedconsiderable researchand resultantchangesin geometrycurriculumby Soviet educators,and in recentyears, interesthas been growingin the UnitedStates. This Project,funded by the NationalScience Foundation,Researchin Science EducationProgram, was one of three federally funded investigations of the model during 1980-83. References to the other projects (William Burger, Oregon State University and ZalmanUsiskin, Universityof Chicago)are includedin the bibliography. The van Hiele model identifiesfive levels of thinkingin geometry. According to this model, the learner,assisted by appropriateinstructionalexperiences,passes throughthese levels beginning with recognition of shapes as a whole (level 0), progressing to discovery of properties of figures and informal reasoning about these figures and their properties(levels 1 and 2), and culminatingin a rigorous study of axiomatic geometry (levels 3 and 4). The van Hieles have developed curriculummaterials (in Dutch) based on their model, and others, especially the Soviets, have also appliedit to curriculumdevelopment. Research Objectives, Methods, Design and Analyses The general question that this research addressed is whether the van Hiele model describeshow studentslearngeometry. Therewere four main objectives: (1) To develop and documenta workingmodel of the van Hiele levels, based on severalsourceswhich the Projecthad translatedfromDutch into English. (2) To characterizethe thinkingin geometryof sixth andninthgradersin termsof levels--in particular,at what levels are students?,do they show potentialfor progress within a level or to a higher level?, and what difficulties do they encounter?. (3) To determineif teachersof grades6 and 9 can be trainedto identifyvan Hiele levels of geometrythinkingof studentsand of geometrycurriculummaterials. (4) To analyzecurrentgeometrycurriculumas evidencedby Americantext series (gradesK-8) in light of the van Hiele model. The first objectivewas achieved afteran analysisof van Hiele source material, in particular,Dina van Hiele-Geldofs doctoralthesis (1957/1984) and Pierrevan Hiele's article (1959/1984), "Lapensee de l'enfantet la geometrie,"which were unavailable in English until the Project translatedthem. (See Fuys, Geddes, &

2

Tischler, 1984, EnglishTranslationof Selected Writingsof Dina van Hiele-Geldof andPierreM. van Hiele.) Based on specific quotationsfromthe van Hiele sources, the Project formulateda detailed model of the levels (see Chapter4 for level descriptors).Pierrevan Hiele andtwo othervan Hiele researchers,Alan Hofferand WilliamBurger,examinedthe level descriptorsandvalidatedthemfor each level. The second objectivewas achievedthrougha clinical studythatwas carriedout in several phases. The first involved the development and validation of three modules based on the model and designed for use as a researchtool in clinical interviews. Modules dealt with Propertiesof Quadrilaterals,Angle Relationships for Polygons, and Area of Quadrilaterals.The module on Angle Relationshipswas based on the approaches and materials used by Dina van Hiele-Geldof in her doctoral research which involved a geometry teaching experiment for twelveyear-olds. The modules includedinstructionalactivitiesalong with key assessment tasksthatwere correlatedwith specific level descriptors.Moduleswere pilot tested andrevised along with scriptsfor the interviewers.See Chapter3 for descriptionof contentof modulesandfor sampleactivities. To facilitateanalysisof studentresponsesto tasks in the clinical interviews,the Projectdeveloped protocolforms for each module. These forms, to be completed by reviewersof the videotapes,containednot only check lists andquestionsto assess a student'suse of vocabulary/language, responsesto differenttasks,responsesto key van Hiele level of use questions, response, of materials,andtypes of difficultiesbut also spaces for reviewers'descriptivecommentsabouta student'sattitude,style of learning, non-verbal communication,and preferenceof materials.The modules, together with the protocol forms, were validated by the researcherscited above againstthe Project'slevel descriptors. In the second phase, clinical interviewswere conductedwith 16 sixth graders and 16 ninth graders. In six to eight 45-minute sessions, these subjects worked throughthe modules with an interviewer(a memberof the Projectstaff). Sessions were videotaped. Each subjectreceiveda smallhonorarium. The final phasedealtwith the analysisof the videotapesand synthesisof results for the sixth andninthgraders. This was done in threestages. First,videotapesfor individual subjects were reviewed by one member of the Project staff who completed detailedprotocol forms. The forms were then summarized(1-2 pages for each module) on each student'sperformance. Summary index cards were prepared noting briefly the student's level of thinking (initial and progress), difficulties,language,learningstyle, andmiscellaneous. The next stage involveda review and validationof the initialanalysisof each student'sperformanceby one or more other members of the Project staff. This review included discussing informationrecordedon the protocolforms and viewing again key portionsof the student'svideotapes. In the final stage of the data analysis, one Projectmember reviewed and synthesized results for the sixth gradersand anotherdid the ninth graders. These overallresultswere then discussedand refinedby the Projectstaff.

3

The Project assessed the "entry"level of thinking of students relative to geometry topics that are commonly studied in grades 4-6. This was done mainly through key questions or tasks throughoutModule 1 and at the beginning of Modules2 and 3. These tasks, to which studentscould respondat levels 0, 1, or 2, were presented with little or no promptingfrom the interviewer, who accepted whateverresponse the studentgave. Since, accordingto the van Hieles, level of thinking is determined in part by prior learning experiences, such "static assessments"may not accuratelyassess the student'sability to think in geometryif the student has had little or no learning experiences on the topic involved. Therefore,the Projectalso assessed whatmight be termedthe student's"potential" level by examining the student's responses as the student moved through the instructionin the interviews. This more dynamic form of assessment during a learning experience, as Dina van Hiele-Geldof did in her teaching experiment, enabledthe Projectto examinechangesin a student'sthinking,withina level or to a higherlevel, and also difficultieswhich impededprogress. The thirdobjectivewas achievedthroughone-to-onevideotapedinterviewsby one memberof the Projectstaff with 8 preserviceand 5 inservice teachers. In the first 2-hour session, the teachers worked through selected activities from the instructionalmodules with the interviewer. In the second session the interviewer described the van Hiele model, showed and discussed videotaped segments of students doing selected activities, and evaluated sample geometry curriculum materials(K-8) accordingto van Hiele levels. In a final session, the teacherswere given sample curriculummaterials to evaluate in terms of the van Hiele levels. They were also shown videotaped segments of two students doing geometry activities and asked to discuss the levels of thinking evidenced by the students. Inservice teachers were also asked to comment on and informally evaluate the appropriatenessof the activities in the modules for classroomuse. The preservice and inserviceteachersreceivedhonorariafor participatingin the Project. Concurrently,the fourthresearchobjective,an analysis of the geometrystrand of three widely used commercial textbook series (grades K-8), was initiated in orderto determine: (1) what geometrytopics are taughtby gradelevel in orderto measurethe richnessand continuityof instruction;(2) at what van Hiele level the materialsare at each gradelevel; (3) if the van Hiele level of materialis sequenced by grade level; (4) if there are jumps across van Hiele levels; (5) if the text presentationof geometry topics is consistent with didactic principles of the van Hieles. Data forms were used to collect and record each text's page by page introductionand use of vocabularyat each gradelevel, the aim of each lesson, and the van Hiele level of the expository material, of the exercises, and of the test questionsfor each geometrylesson in the threetext series, gradesK-8. The levels of exposition,exercises andtest questionsof a text lesson were determinedby using the Project-developedlevel descriptors. Completeddataforms were analyzedand summarizedwith comparisonsbeing madeamongthe threetext series.

CHAPTER 2 THE VAN HIELE MODEL Background Experiences of secondary school mathematics teachers indicate that many students encounter difficulties in high school geometry, in particular,in doing formal proofs. What are some causes for these difficulties? During the period from 1930 to 1950, severalSoviet mathematicseducatorsandpsychologistsstudied learningin geometryand triedto answerthis question. Wirszup(1976) reportsthat this very significant research has influenced the improvement in the teaching of geometry only slightly. The truly radical change and far-reachinginnovationsin the Soviet geometrycurriculumhave, in fact, been introduced thanks to Russian research inspired by two Western psychologistsandeducators. (p. 76) The first is Jean Piaget and the second is P. M. van Hiele, a Dutch educator, whose work on the role of intuition in the learning of geometry attractedthe attentionof the Soviets afterhe delivereda paperentitled"Lapensee de l'enfantet la geometrie"at a mathematicseducationconferencein Sevres, Francein 1957. It was published laterin 1959. Frequentreference is made to this paperin the workof A. M. Pyshkalo(1968/1981) as he describesthe Soviet educators'extensiveresearch and experimentationon van Hiele's theory. It is reportedthat the Soviets have substantiallyrevisedtheirgeometrycurriculumon the basis of the van Hiele levels of thinkingin geometry. As experiencedteachersin Montessorisecondaryschools, the van Hieles were greatly concernedaboutthe difficulties their studentsencounteredwith secondary school geometry. They believed thatsecondaryschool geometryinvolves thinking at a relatively high "level" and students have not had sufficient experiences in thinking at prerequisitelower "levels." Their researchwork focused on levels of thinkingin geometryand the role of instructionin helping studentsmove fromone level to the next. In 1957 the van Hieles completedcompaniondissertationsat the University of Utrecht on levels of thinking and the role of insight in learning geometry. Dina van Hiele-Geldofs work (1957/1984) dealt with a didactic experimentaimed at raisinga student'sthoughtlevel, while Pierrevan Hiele (1957) formulatedthe structureof thoughtlevels and principlesdesigned to help students gain insightinto geometry.

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Levels and Their Characteristics Accordingto the van Hieles, the learner,assisted by appropriateinstructional experiences, passes throughthe following five levels, where the learner cannot achieveone level of thinkingwithouthavingpassedthroughthe previouslevels. Level 0: The studentidentifies,names, comparesand operateson geometric figures (e.g., triangles, angles, intersecting or parallel lines) accordingto theirappearance. Level 1: The student analyzes figures in terms of their components and relationshipsamong componentsand discoversproperties/rulesof a class of shapesempirically(e.g., by folding, measuring,using a grid or diagram). Level 2: The studentlogically interrelatespreviously discoveredproperties/ rules by giving or following informalarguments. Level 3: The student proves theorems deductively and establishes interrelationshipsamongnetworksof theorems. Level 4: The studentestablishestheoremsin different postulationalsystems andanalyzes/compares these systems. The van Hieles (1958) noted that learningis a discontinuousprocess and that there are jumps in the learningcurve which reveal the presence of "levels." They observedthatat certainpoints in instruction the learningprocesshas stopped. Lateron it will continueitself as it were. In the meantime,the studentseems to have "matured."The teacherdoes not succeedin explainingthe subject. He seems to speaka languagewhich cannot be understoodby pupils who have not yet reachedthe new level. They might accept the explanationsof the teacher,but the subjecttaught will not sink into theirminds. The pupil himself feels helpless, perhapshe can imitatecertainactions,but he has no view of his own activity until he has reachedthe new level. (1958, p. 75) Overall,the van Hieles made certainobservationsabout the generalnatureof these levels of thinking and their relationship to teaching. P.M. van Hiele (1959/1984) notes that at each level thereappearsin an extrinsicway that which was intrinsicat the precedinglevel. At level 0, figures were in fact determinedby their properties,but someonethinkingat level 0 is not awareof these properties. (p. 246)

6

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Van Hiele (1959) states that the levels are "characterized by differences in objects of thought" (p. 14). For example, at level 0, the objects of thought are geometric figures. At level 1 the student operates on certain objects, namely, classes of figures (which were products of level 0 activities), and discovers properties for these classes. At level 2, these properties become the objects that the student acts upon, yielding logical orderings of these properties. At level 3, the ordering relations become the objects on which the student operates, and at level 4 the objects of thought are the foundation of these ordering relations. LEVEL 0 S

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Van Hiele (1959/1984) also points out that each level has its own linguistic symbols and its own system of relations connecting these symbols. A relation which is "correct"at one level can reveal itself to be incorrect at another. Think, for example, of a relation between a square and a rectangle. Two people who reason at different levels cannot understand each other. Neither can manage to follow the thought processes of the other. (p. 246)

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Languagestructureis a criticalfactor in the movement throughthe van Hiele levels--from global (concrete) structures(level 0), to visual geometric structures (levels 1-2), to abstract structures(levels 3-4). In stressing the importanceof language, van Hiele notes that many failures in teaching geometry result from a languagebarrier--theteacherusing the languageof a higherlevel thanis understood by the student. Progress from one level to the next, asserts van Hiele (1959/1984), is more dependent upon instructionthan on age or biological maturation,and types of instructionalexperiencescan affect progress(or lack of it). It is possible however that certainmethods of teaching do not permitthe attainmentof the higher levels, so that methods of thoughtused at these levels remaininaccessibleto the student.(p. 246) The van Hieles point out thatit is possible to presentmaterialto studentsabove their actual level. For example, students are given properties for rectangles and memorizethem ratherthandiscoveringpropertiesthemselves (level 1), or students just copy a "proof"ratherthancreatingit themselvesor at least supplyingreasonsin the proof (level 2). This results in a "reduction"of the subject matterto a lower level. Accordingto Pierrevan Hiele (1959/1984), progressfrom one level to the next involves five phases: information,guided orientation,explicitation, free orientation, and integration. The phases, which lead to a higher level of thought,are describedas follows with examplesgiven for transitionfromlevel 0 to level 1. Information: The student gets acquaintedwith the working domain (e.g., examinesexamplesandnon-examples). Guided orientation: The studentdoes tasks involving differentrelationsof the network that is to be formed (e.g., folding, measuring, looking for symmetry). Explicitation: The student becomes conscious of the relations, tries to express them in words, and lears technicallanguagewhich accompaniesthe subjectmatter(e.g., expressesideas aboutpropertiesof figures). Free orientation: The studentlearns,by doing more complex tasks, to find his/her own way in the networkof relations(e.g., knowing propertiesof one kind of shape,investigatesthesepropertiesfor a new shape,such as kites). Integration: The student summarizesall that he/she has learned about the subject, then reflects on his/her actions and obtainsan overview of the newly formed network of relations now available (e.g., properties of a figure are summarized).

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In summary,the majorcharacteristicsof the van Hiele "levels"are that (a) the levels are sequential; (b) each level has its own language, set of symbols, and networkof relations;(c) what is implicit at one level becomes explicit at the next level; (d) materialtaught to studentsabove their level is subject to reductionof level; (e) progress from one level to the next is more dependenton instructional experiencethanon age or maturation;and (f) one goes throughvarious "phases"in proceedingfrom one level to the next. Development, Documentation and Use of Level Descriptors The Project'sinitialbrief descriptionsof the van Hiele levels were basedmainly on three articles (van Hiele & van Hiele-Geldof, 1958; van Hiele, 1959/1984; Wirszup, 1976). In order to develop fuller characterizationsof the levels and examples of how they are applied, there was a need to analyze several other van Hiele source documents, in particular, the doctoral dissertation of Dina van Hiele-Geldof (translatedby the Project). As a result of this analysis, the Project developed a more detailedlisting of level descriptors.This listing is presentedin Chapter4 and includes specific examples of studentperformancefor each level descriptor. Since examples are based mainly on the instructionalmodules, the modules are discussed first (Chapter3) before examiningthe level descriptorsin detail (Chapter4). The Projectused its detailed characterizationof the levels in termsof studentbehaviors: (a) in designingthe assessmentand instructionalparts of the threemodules, (b) in analyzingvideo-tapedepisodes of studentsdoing the modules, (c) in analyzing the van Hiele level of textbook materials(exposition, exercises, activities,test items). These descriptorscan also be used to examinethe languageof teachersand studentsduringinstructionand to characterizeclassroom activitiesin geometry. Translation of Writings of the van Hieles At the outset, the Projectendeavoredto collect the majorwritingsof Dina van Hiele-Geldofand Pierrevan Hiele. These were obtainedfrom a varietyof sources includingpublishersin The Netherlandsas well as researchersin the United States (Cilley, 1979; Hoffer, 1981; Mayberry,1983) who were also investigatingthe van Hiele model. Most of the writingswere in Dutch, with a few articlesin Frenchor Germanand two in English. Four majornon-Englishwritings of the van Hieles were translatedinto Englishby the Project. 1. The doctoraldissertationby Dina van Hiele-Geldof(1957/1984) entitled"De Didaktiek van de Meetkundein de Eerste Klas van het V.H.M.O." (The Didacticsof Geometryin the LowestClassof SecondarySchool), 177 pages. 2. The last articlewrittenby Dinavan Hiele-Geldof(1958/1984): "Dedidaktiek

9

van de meetkundeals leerprocesvoor volwassenen"(Didactics of Geometry as a LearningProcessfor Adults), 14 pages. 3. Selected sections of the book by Pierrevan Hiele (1973), Begrip en Inzicht (Understandingand Insight),230 pages. 4. An article by Pierre van Hiele (1959/1984): "La pensee de l'enfant et la geom6trie"(The Child'sThoughtand Geometry),8 pages. Dina van Hiele-Geldof died shortly after completing her dissertation. Thus, except for her dissertation,one articleby her in 1958 and a joint journalarticle (in English) with Pierre (1958), all other writing describing the thought levels in geometryhas been done by Pierrevan Hiele. Dina van Hiele-Geldofs dissertation describesa year-long "didacticexperiment"involving two classes of 12 year-olds which she taught. As she statesin ChapterI of the thesis,the studyinvestigatedthree mainquestions. 1. Is it possible to follow a didacticas a way of presentingmaterialso that the thinkingof the child is developedfrom the lowest level to higher levels in a continuousprocess? 2. Do 12 year-oldsin the lowest class of the secondaryschool have the potential to reason logically about geometric problems and to what extent can this potentialbe developed? 3. To what extent is language operativein the transitionfrom one level to the next? Almost half of the dissertationis a detailed and fascinating log of her teaching experiment. ChaptersmI,IV andX of the thesis describein detailthe subjectmatter covered, methods of presentation,and "classroomconversations"between teacher and students for the didactic experiment. These three chaptersand related ones (XI-XIV) which present analyses of the students'thinkingshould be of particular interestto researchersof the van Hiele levels. The only informationon the levels which was previously availablein English was of a more generalnaturethan that found in this dissertation. The dissertationprovidesspecific examples of students' behaviors at the levels in response to many specific instructionaltasks. The last articlewrittenby Dina van Hiele-Geldof(1958/1984) gives furtherclarificationand insightinto the levels as relatedto a student'sbehaviorandwas recommendedto the Project staff by Pierre van Hiele as being an importantresource document to translateinto English. Pierrevan Hiele wrotemanyarticlessettingforthhis ideas on learning,thought levels, insight and structure. A numberof these articles became the basis of his 1973 book, Begrip en Inzicht(Understandingand Insight). An articlehe wrote in Frenchin 1959, "Lapensee de l'enfantet la geometrie"(The Child'sThoughtand

10 Geometry),which was not includedin Begrip en Inzicht, attractedthe attentionof Soviet psychologistsand educatorswho had long been studyinghow childrenlearn andwere particularlyconcernedaboutstudentdifficultywith geometry. Pierrevan Hiele has also writtenandpublishedgeometrycurriculummaterials basedon the levels for studentsin grades7-12 in The Netherlands.The Projectstaff has examinedthe geometryactivities in these text series, van a tot z (1976-1979). Pierre van Hiele's books, Struktuur (1981), and Structure and Insight (1986) provide furtherclarification of the thought levels and their applicationto other curriculumareas. The Project obtainedthe services of Dr. MargrietVerdonck,a native of The Netherlandsliving in Brooklyn, to do the Project'stranslationwork from Dutch into English. Dr. Verdonck translated Dina van Hiele-Geldofs doctoral dissertation(item 1, above) in its entiretyas well as her last article(item 2, above). The first draftof the translationof the doctoralthesis was sent to Pierrevan Hiele for review. Aside from a few minor suggestionsfor word changes, Dr. van Hiele indicated that he thought it was "a very fine translation." Dr. Verdonck read Begrip en Inzicht (item 3, above) in its entiretyand preparedan English summary of each of the chapters. After conferringwith the Projectstaff, she provideda full English translation of those portions of the text considered importantto the Project'swork. The article by Pierrevan Hiele (1959/1984) in French (item 4, above) had previously been translatedinto English in 1975 by Rosamond W. Tischler,a memberof the Projectstaff. After editing these translations,the Projecthas published, for the benefit of teachers, curriculum developers, and the research community, a monograph entitled:English Translationof Selected Writingsof Dina van Hiele-Geldof and PierreM. van Hiele. The monographcontainsthe completeEnglishtranslationsof items 1, 2 and 4 above. Also includedis an English summaryof Pierrevan Hiele's dissertationentitled:"TheProblemof Insightin Connectionwith School Children's Insightinto the SubjectMatterof Geometry."The monographis availablethrough EducationalResourcesInformationCenter(ERIC,numberED 287 697).

CHAPTER 3 INSTRUCTIONAL MODULES The van Hiele model focuses on the role of instructionin helpingstudentsmove from one thoughtlevel to the next. It was, therefore,necessary for the Projectto develop andvalidateinstructionalmodulesbased on the model and designedfor use as a researchtool in a one-to-oneinstructional/testing setting. This section describes the three modules developed by the Project for the clinical interviews. First the major characteristics of the instructional modules are discussed and then the developmentof the modules is summarized.Finally, a detaileddescriptionof each moduleis given, includinggoals, specific activities,andrationale. Major Characteristics of Modules Content The following geometrytopics are treated. Module1:

Basic geometric concepts (parallelism, angle, congruence, etc.), propertiesof quadrilaterals

Module2:

Angle measurement;angle sums for triangles, quadrilaterals,pentagons; angle relationships in triangles and parallelograms (i.e., exteriorangle, oppositeangles)

Module3:

Area measurement;area of rectangles, triangles, parallelograms, trapezoidsandfigureswhose vertices lie on two parallellines.

Several factors affected this choice of topics. First, a variety of topics were chosen because of the need to assess level of thinking across different topics. Second, topics had to be of such a naturethat they could be presentedat different van Hiele levels--in particular,levels 0, 1, and2. Third,topics shouldbe revelantto the school experiencesof both sixth andninthgraders,yet the topics shouldnot be overly familiar in order to minimize the influence of prior learning on the new learningto take place in the interviews. These topics were comparedwith those normally taught in grades 5-9, as specified by the New York City Mathematics CurriculumGuide (1981). In addition, the content and approach (via tiling patterns, saws/ladders, family trees) in Module 2 were chosen because they embodiedthe instructionalmaterialsused by Dina van Hiele-Geldofin her "teaching experiment"research(1957/1984).

12 Flexibility Since therewould be differencesin the geometryexperienceof sixth and ninth graders,the modules were designed to be used flexibly in the interviews, with optionsfor branchingto instructionat severalpoints,dependingon the responsesof studentsto major assessmentquestions. In the 6-8 hours of interviews, stronger studentsmight progress throughall three modules. Othersrhightwork through Modules 1 and 2, or 1 and 3, requiringconsiderableinstructionalong the way. Additionalprovisionfor contingencywas incorporatedinto the interviewsthrough the module scripts. Althoughthe scriptsguidedthe interviewer,in particularabout key questions and specific instructionalsuggestions, they allowed interviewersto rephrasewordingand to vary instructiondependingon the student'sresponses. Of course, such implementationof the modules depended upon the interviewer's familiaritywith the subjectmatterand spontaneityin questioningand instructing, while following the student'strainof thought. Assessment of Thinking and Key Questions The primarypurpose of the modules was to provide a context for clinical interviews which could shed light on the student'slevel of thinking, cognitive processes (e.g., inductiveand deductivereasoning),and learningdifficulties (e.g., poor geometry vocabulary, perceptual difficulties) that may adversely affect thinkingin geometry. Accordingto van Hiele, two majorfactorsthat determinea student'slevel are abilityandpriorgeometryexperiences. A student'sresponsesto questions about a topic will indeed provide assessment informationabout what a studentcurrentlyknows aboutthattopic. However,this may not yield an accurate assessmentof the student'spotentialto thinkat certainlevels, if the studenthas little or no experiencewith the topic being assessed. In this case, assessmentmust also focus on progress(or lack of progress) thata studentmight make withina level, or possibly to a higherlevel, as a resultof instruction.To this end the threemodules were designedto assess both "entry"level of thinkingabouta topic and "potential" level as reflectedby the student'sperformancein a learningsituationon thattopic. The student's"entry"level of thinkingwas assessed mainlythroughanalysisof his/herresponsesto key questions/activitiesin Module 1 andat the startof Modules 2 and 3. These questions, which allow for responses at different levels, were presentedto the studentwithoutpromptsor hints, althoughthe interviewermight ask follow-up questionsto clarifythe student'sanswers. For example, in Module 1 the studentswere askedto describea rectangleto a friendwho didn'tknow what it was or to sort a mixed-upset of cut-outquadrilateralsinto appropriateboxes (e.g., squares,rectangles,parallelograms,trapezoids)and to tell what they were thinking aboutas they did this. In Module 2, the studentswere asked: "Twoangles of this triangle each are 50 degrees. What does the third angle equal? Why?" If the explanationwas based on "theangle sum is 180 degrees,"the studentwas asked:

13 "Doesthe angle sum of any triangleequal 180 degrees? Explain." Studentscould respondto such questions at differentlevels. For example, to the questionsnoted above for Module2, a studentcould respondat level 0 by measuringthe thirdangle, at level 1 by generalizingthe angle sum on the basis of experimentation,or at level 2 by explaining/provingthis using alternateinterior/corresponding angles (saws and laddersin Module2). The student'sability to thinkaboutspecific concepts in geometryand to make progresswithin levels was assessed in a more dynamicway--throughresponsesto new instructionand to relatedassessmenttasks and key questions. This teaching experimentapproachallows one to look at changesin a student'sthinkingand,most importantly,at difficultiesthatimpedeprogress. This type of assessmentwas done in each module, where after an initial assessment, extensive instruction was providedas needed, with questioningduringinstruction,at summarypoints, and at the end of each modulethrougha final activityinvolvinglevel 1 and 2 thinkingon a new but related topic (e.g., in Module 2 on angle sums, the related topic was exteriorangles of a triangle). Questionsthroughouteach module were correlated with level descriptors (see pages 58-71) to facilitate identification of level of thinkingwhen videotapedinterviewswere analyzed. Phases As stated above, the modules were designed to involve studentsin geometry activities that would reveal their level of thinking. In this sense, the modules embodythe level aspectof the van Hiele model. The modulesalso embodyanother aspect of the van Hiele theory--namely, phases within levels. As described previously in Chapter2, there are five phases: Information,Guided Orientation, Explicitation,Free Orientation,and Integration.Since, accordingto van Hiele, the learningprocess involved in filling in a level or in moving from one level to the next consists of these five phases, the modules were designed to reflect this approach. For example, in an activity on area of parallelogramsin Module 3, the activity opens with informal work with area of parallelogramsto acquaint the student with this topic (Information). Then the student is guided to discover a procedure for finding the areas of parallelograms(Guided Orientation). Next, studentsare asked to express the procedurein words, in particularusing technical terminologysuch as base and height (Explicitation). Next, a variety of problems embodying the concept just learned are presentedto the student for exploration (Free Orientation). Finally, the student summarizes this in a family tree (Integration). The van Hieles appliedthe levels andphases to a teachingexperimentwithina classroom setting over an extended period of time (i.e., one year). The Project neverthelesssoughtto incorporatethe phasesof the learningprocessinto the clinical interviews. Obviouslythe formatandtime constraintsof this study (6-8 hourswith

14 each subject) did not allow for students exchanging newly formed ideas in the explicitationphase or for extendedfree orientationtasks. Informal Introductory Activities The modules were designedto reflectthe spiritof van Hiele levels andphases. Morespecifically,each moduleopenedwith activitiesfeaturingvisual global structures--inModule 1, pictures of cityscapes and other scenes embodying geometry concepts; in Module 2, tiling patternsand floor designs; in Module 3, tangram puzzles, areaof covers of jewelryboxes anda map of lots in downtownBrooklyn. Beginning activities in each module were designed as ice-breakers or change-of-pacefrom previous work. These informalactivities were presentedas games and were meantto providea non-threateningcontextfor beginningthe topic. This type of beginning activity was intendedto be appealing and relevantto the subjects(in this case, innercity adolescents). Initial activities gave the interviewerinformationaboutthe subject'slanguage related to that topic. If non-standardterms were used (e.g., "straight"for right angle), the intervieweracceptedthis responsewithoutinitial correction,introduced alternativestandardterms, and encouragedthe subject to adopt and use standard terms,perhapssayingboth non-standardand standardterms,and graduallyshifting to only standardones. Hands-on Approach Activitieswere in two modes--thefirst, dynamic,involvingthe manipulationof materials(e.g., moving objects, folding, coloring), and the second, more "static," relyingprimarilyon verbalor pictorialinformation(as is done in most textbooks). Special materials were designed for the manipulativemode: cutout cardboard shapesfor sorting;D-stix for makingangles or figures andfor showingparallelism; devices to show angles andto test for congruenceor to measureangles;clearplastic squareinch grids, stripsof squareinches, and tiles for area;cutoutpieces to lead to discovery of area rules; tiles and tiling design sheets for coloring; clear plastic devices to show saws/ladders;propertycards and arrowsto set up in family trees. Project-developedselection sheets, reflecting the static mode, provided students with opportunitiesfor identifying examples and non-examples of concepts and properties. Using these two modes enabledthe researchersto determineif certain subjectswere more successfulin one mode thananother,and whetherthey had any preferencefor a mode or used both simultaneously. Also use of materialscould help less verbal studentsto express their ideas about geometry (e.g., by gesturing with D-stix aboutparallellines andthenaddingverbalresponses).

15 Development and Validation of the Modules The modules were developed, triedout, and validatedin threephases priorto their use in the clinical interviews. The first phase (summerand early fall, 1980) involved the development of initial versions of the modules. After being pilot-tested with 3-4 students,the modules were revised. Statementsof specific goals for each activityin a modulewere clarified. The scriptwas expanded,mainly with suggestionsaboutcontingenciesfor assessmentand instruction. Some special manipulatives were designed and added to the modules (e.g., saws/ladders on transparentplastic to help studentsidentify saws/laddersin grids, strips of square inches to help studentsdiscoverareaof rectanglesas repeatedaddition,L-Squareto assist studentsin findingheightsof figures). In fall, 1980 these revised modules were tried out by an interviewer (a secondaryschool teacher)trainedby Projectstaff. This providedvaluablefeedback about the clarity of the script and appropriatenessof manipulatives from the standpointof a new interviewer. As a result,some minorrevisionof the scriptwas made and key questions were starred in the script. Also, manipulatives were packagedfor easieruse and storage. The second phase of the developmentof the modules occurredin spring, 1981 and focused on the preparationand tryoutof a fourthmodule thatmainly assessed level 2 thinking. Contentof this module extended ideas treatedin Modules 1-3. This module was to be done by studentswho had completedall threemoduleswith evidenceof level 2 thinking. Pilottestingindicatedthatthis was not feasiblebecause of limited interviewtime. As a result,differentpartsof this module were included as final activitiesin Modules 1 (Propertiesof Kites), Module2 (ExteriorAngle of a Triangle), and Module 3 (Midline Area Rule). During the second phase, some minor changes in the scripts were made. Also, materialsfor each module were organizedinto a kit formatthatwas easy to transportandto use duringinterviews. The final phase (summer,1981) dealtmainlywith validationof the modules,in particular,the appropriatenessof key questionsfor assessingthe levels of thinking. Copies of the script and videotapes of clinical interviews were sent to three mathematicseducatorswho were well acquaintedwith the van Hiele model through their own research (William Burger at Oregon State University; Alan Hoffer at University of Oregon; and John Del Grandeat Board of Education,Borough of North York, Ontario,Canada). Their evaluationsindicatedthat the modules were on target. They affirmed that responses to key questions were providing information about levels. The only negative criticisms involved some of the instructiongiven by the interviewers--namely,the interviewerin some instances was leading too much. As a result,interviewerswere instructedto be less directive and to allow students more time to think and respond, especially when guiding studentsto make discoveries. A subsequentvalidationwas doneby Pierrevan Hiele himself, when he conferredwith Project staff for two days in April, 1982. Van

16 Hiele reviewedthe modulescriptsandkey questionsandviewed videotapedepisodes of several studentsrespondingto key questions. It was his view that the modules embodiedthe levels, and that the interviewsyielded informationaboutthe level of thinkingof the subjects. Furthermore,his evaluationof the level of thinking of these students agreed with those of the Project staff, thereby supporting the usefulnessof the module-basedclinicalinterviewsfor assessinglevel of thinking. After the final phase of development and validation, the modules were consideredreadyfor use in the clinical partof the Project. It should be noted that, althoughthe modules went throughconsiderabledevelopmentand revisions, they should not be thoughtof as in final ideal form. In fact, interviewersoften made some adaptationsto the activitiesand script,in particularfor less capablestudents. For example, it was sometimesnecessaryto add extra examples to help a student discovera pattern,to provideextrareview (especiallywhen therewas considerable time between interview session), or to have studentssummarizemore frequently what they had leared. Thus, as interviewersgained experiencewith the modules, they actually provided furtherdevelopment or refinement of the module in the interviewsetting.

17 Module 1 Overview This module concerns classification of two-dimensionalshapes. It serves to assess students'backgroundin topics to be treatedin the latermodules, to provide instructionas needed,andto determinestudents'levels of thinkingaboutshapesand their properties. After an ice-breakingintroductorygame, studentsare invited to talk about geometricalaspects of some pictures. Then a sorting activity leads to explicit considerationof propertiesand inclusion relations. The role of properties in students'conceptionsof shapesis explored,includingthe recognitionof minimal properties. A final classificationactivity is presentedto studentsat a later session, afterthe end of the module,to check on retentionandtransferof thinkingprocesses. The moduleis summarizedin this diagram.

3.3

Propertiesof Classes of Quadrilaterals I

3.4

Relations t Inclusion

> 3.5 Sorting byParallelism

'I

3.6-3.7

3.8

GuessingShapesfrom PartialView/Properties

MinimalProperties 'V

4

Kites: Sorting,Properties,InclusionRelations I

18 Activity 1.

Introductory Activity

This ice-breaking "game"has two goals: to create a relaxed atmospherein which student and interviewer can communicate, and to assess informally the mathematicallanguage(standardandnon-standard) used by the student. ^aPairs of shapes are presented. For the first pair,the interviewersays somethingthatis the same about them and the student says thatis different. For the next pair, the roles are reversed. This is repeatedfor six ~pairs.

A_.*/_\

l---i

1---1_ isomething

A Axn~

~

~

O \\ ~__/

Azsz

o-<>

note: This activityrevealssome interestingcharacteristicsof students,for example: theirinclinationto initiateor to copy ideas;or to gestureand handlematerialsor to look only. The interviewerintroducesno formalvocabulary,insteadusing gestures (fittingan angle on top of another)or commoninformalterms(sides, comers, same length). In this and the next activity, studentsare given as much opportunityas possible to introducetheirown geometricvocabularyspontaneously. Activity 2. Shapes in Pictures This activityassesses students'familiaritywith some basic geometricconcepts, namely: conceptsof shapes:

triangle square rectangle parallelogram

conceptsof components of shapes:

angles,rightangles parallelsides oppositesidesandangles congruentsidesandangles

Moredetailedassessmentis madefor rectangle,rightangle, and parallel.

a

19 Firstthe studentis shown photographsof city environments(San Francisco skyline, buildings) and is asked to find "geometricideas"in them. For shapes identified,they are asked to find other examples in the picture, and to decide on some examples and nonexamples selectedby the interviewer.

.TRlA
The concepts listed above are required for laterwork, and any responserelatedto one of these is pursued in this way. (All other responses are simply praised.) For any of these concepts which are not mentioned, studentsare shown examples and are asked if they recognizethe configurationandcan name it--if not, they are shown "word-and-picture cards."

1 11 n

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A au

cn

I

,,

F

-

WIALc oi tHse A?< Ctt*^-5s

^" t>

/-\G-

Three concepts are explored in greaterdepth as they arise, rectangle, right angle and parallelism. Students are asked to construct exampleswith D-stix, and then to describethe idea to someone who didn'tknow what it was "overthe phone." Selection sheets allow for detailedassessmentof students'graspof these concepts. Studentsare asked "whichof these are _ 's" and "why?" In the course of the session the interviewer keeps track of the student's understanding of the concepts presentedin the word-and-picturecards, and thenproceedswith instructionas needed.

note: While interviewers should not prolong a "guessing game" when correct responsesare unlikely, it is neverthelessimportantthatthe studentsbe given every opportunityto produce their own non-standardvocabularybefore standardterms are introduced. If a non-standardtermis produced,the intervieweruses it together with the standardterm thereafter,until the student seems comfortable with the standardterm. Only three concepts are pursued in depth because it was found to be timeconsuming and a bit boring for the students to discuss more at this point. (In Activity 3, understandingof other concepts is assessed more carefully.) The

20 questioningaboutrectangles,rightanglesandparallelismin this activitydoes allow for responsesat eitherlevel 0 (looking at individualshapes,descriptionsin termsof "lookinglike")or at level 1 (invokingpropertiesof a class of figures). Instructional Branches The module includesinstructionalmaterialfor all of the concepts listed above except for square,rectangle,triangle.It was felt thatstudentswho do not have some level 0 acquaintancewith these threeconcepts were far below expectationsat this level, and would not be able to do enough of the module to make them suitable subjects. Thusany such studentsshouldbe droppedfromthe study. Materialsfor instructionon differentconcepts follow a similarpattern. They begin with referenceto the real world situationsin which the concept arises, then allow for examples to be pointedout. Subjectsthen constructtheirown examples and non-examples, and test their understandingby doing a selection sheet-identifyingexamples and non-examplesand explainingwhy. Finally, studentsare shown a shape and asked what they notice about the shape, allowing them to formulatea propertyusing the concept just discussed (e.g., opposite angles of a parallelogramare congruent). For example, the instructional branch for parallelism begins with consideration of streetson a local streetmap. Sticks arelaid on streets while students discuss the distances . between streets and whether or not streets meet. The term parallel is introduced,and shown with sticks--students then construct

x 1 I / ^ I~ ' Xi lM> . ^wM / X -.= ~ . A*9 I -"~

parallels with sticks. The term is related to

P^>^vis ,9AALf,.L tA.^

.PH^QcL


ll

pictures of other common uses ("parallel bars," "parallel parking") and the idea is on the word-and-picturecard. ^summarized

!

PARALLEL L,NE

^~ <--^+

Then students return to the task of identifying

parallels in the photos, and proceed with the rest of the activity, including in this case selectionsheetsandverbaldescription.

~

?W'^icaof t?s oy?t f ^--?"}

_

~

pafrelvl

?

$\ -C|

D

21

Activity 3. Sorting and Properties of Groups This activity is designedto assess a student'sability to thinkabout shapes (for example, squares) in terms of properties rather than merely by a shape's appearance.Activity 3 has severalpartswhich are describedseparatelybelow. 3.1-3.2 Sorting. These activities assess whethera studentsorts on a "looks like"basis, or by thinkingaboutproperties.

C

4

4

In Activity3.1 studentsare shown a collection of cardboardcut-outpolygons and some mats. The interviewer says "These shapes came from several differentboxes but they got all mixed up. This is how someone tried to put them back in groups which belong together." The interviewerthenplaces a couple of pieces on each mat, sortingby numberof sides, and says "Canyou guess wherethis will go? Why? And this? Why? Can you arrangethe rest of the pieces using this idea?" Finally students are asked to describethe way the pieces were sorted, and if they know names for triangle and quadrilateralpiles.

a l

note: The firstsortingactivityis presentedin this "GuessMy Rule"formatbecause it was found that when the challenge to arrangethe pieces was presentedin a less structuredmanner,studentsoften triedto put pieces togetherin a puzzle formatand it was awkwardto establishwhatwas meantby sorting. The open sortprovedto be too time-consuming,althoughmany interestingideas arose. Studentsthinkingat level 0 tend to explainplacementof pieces by phrasessuch as "theylook alike,"while level 1 thinkerstry to find a commoncharacteristicsuch as the numberof sides. In Activity3.2 studentsare showna collection 0____

Do

CD [

-D~ 4<j

-- Zj

/^^_^N

of quadrilaterals. Again students are asked it

"How could we place these into groups of things thatbelong together?"They are asked ftoexplain theirthinkingas they sort. Eventually a sort by square, rectangle, parallelogram, trapezoid,quadrilateralis expected. If studentssort in some otherway, theirthinking is discussed and their work praised,and they are then asked to try it anotherway. If the standardsort is not produced, a "Guess My Rule"formatis followed.

22 note: Again level of thinkingis assessed by the extent to which studentsdescribe placementof pieces in terms of properties. This activity provides a richercontext for sorting than the previous one. Studentsthinkingat level 0 may place pieces together in roughly similar pairs, making judgements by eye, while students thinkingat level 1 might spontaneouslyinvoke properties("I'mputtingall the ones with four right angles here."). The module might have been designed so that studentswould sort pieces into loops (Venn diagrams) but it was decided that it would take too much time to develop this language, and so disjoint mats are used, and inclusion relations are discussedlaterin termsof movingpieces frommat to mat. 3.3 Listing Properties. The next activity serves to assess a student's abilityto characterizethe groupsof shapesin termsof properties,and also, through guidedquestioning,to instructthe studentin this area.

0C DD SQUARE

RECTANGLe

o.e 4 1 [Ther i s. side

[ Allsides

I

o.

A1t.Av5tes.| Ir*t

a^W31es.|

|Thcre aoe 4 sides. ] Offosite si4es I oIr cosirvt .T

The interviewer points to the group of squaresand says "Ifyou were talkingwith your friend over the phone and you wantedto describethesepieces, whatcould you say about them?"For each group of PARALLE shapes there is a name card, and a set of color-coded property cards. (For example, all cardsconcerningparallelismof opposite sides are orange.) As students mention a name or a property,the interviewer places the correspondingcard on the group. Studentsareencouragedto say as much as they can aboutthe squares,thento go on to other shapes. Prompts are given as needed, for example "Is there anything you can say about the sides here? the angles?" If necessary,sticks areplacedon shapes to remind of parallelism, angles. This continues until all properties are listed. Then studentsare asked to look at selection sheets for each shape, on which they identifyexamples and non-examples, and to explaintheirthinking.

23 note: Many studentshave learnednon-standarddefinitions, that is, they include incorrectpropertiesin theirdescriptions(e.g., "Oneside is longer than another"for a rectangle). When possible, these misunderstandingswere corrected--thatis, explanationsand illustrationswere given to show studentsthattwo differentlengths for the sides was not a necessary condition for rectangles; also that the word "rectangle"was derived from "rightangle"--hencethe rectangleonly had to have right angles. These explanationswere not readily accepted by some students (in light of theirpriorschool instruction)and so some of these misunderstandingswere not corrected,given the relatively short amount of time available for instruction duringthe interviews. The responsesof such studentswere noted and subsequent reasoning(basedon theirmisconceptionof a rectangle)was evaluatedaccordingly. Some students listed many propertiesspontaneouslyand systematically,and transferredproperties given for one shape to another. Other students gave few propertiesspontaneously. For these students the instructor'sguidance may have servedto fill in level 1 experiences. A laterassessmentactivity(4) is designedto see if studentshave merely followed along in this activity, or if they can duplicatethe processesinvolved. 3.4 Subclass Relations. This activity assesses if a studentcan identify and explain subclassrelations--forexample,all rectanglesare parallelograms.

D a

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ULRE

RECTANGLE PtaALI

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,

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I [Tni ,,e

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lf

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The interviewer says "Whenyou sorted the first set of shapes, do you rememberthat you had a group of triangles, and one of quadrilaterals or four-sided figures, and five-sided ones, and six-sided ones? Where wouldall these shapeson the tablehave gone?" When studentshave respondedcorrectly,the interviewerpicks up a square, saying "So I could move this squareto the quadrilateralor four-sidedgroup--asquareis a special kind of quadrilateral. What makes it special?" The interviewer then asks: "Can we move this square to the rectangle group? Why? (or Why not?)"

note: Level 0 thinkerstend to find it difficultto acceptplacementof squaresin the rectangle group, since judgmentsabout shapes are made on the basis of "looking like",not on the basis of a propertysuch as "fourrightangles". Howeversome level 1 thinkersalso will not accept this placementof a square--theymay be reasoning correctly,but from an incorrectdefinitionof a rectangle. The following questionis askedin orderto distinguishbetweenthese two situations:

24

67 LD

Someone yesterday said that a square is a special kind of a rectangle. A square is a rectanglewith equal sides. Do you think she could be right? What do you think she said when she put a square in the parallelogram pile? Couldthatbe right? Mightshe have put a rectangle in the parallelogrampile? What wouldshe have said?"

This type of questioning can determine if students can accept the logic of the inclusion relation, even if their own definitions of the shapes dictate non-intersectionof the sets. A studentwho still has difficultyhere will do the next activity(3.5). For studentswho do acceptplacementof a squarewith the rectangles, the interviewcontinuesas follows:

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C

"Wouldit be possible to call EVERYsquarea rectangle? Could I move this rectangleto the squaregroup? Why (or Why not?) Would it be possible to call every rectanglea square? Why?" This is repeatedfor squareand parallelogram. (Thepropertycardsand sortedpieces arethen put away.)

note: Some studentsanswerthese questionsby visually checking the properties listed on each mat one by one. Othersgive a more formal explanation,without looking at the propertycards or the particularshape being considered,saying for example, "squaresare parallelogramsbecause the opposite sides are parallel and oppositeangles are congruent,"indicatingthatthe studentis thinkingof squaresand rectanglesin terms of properties. Of course the presence of the propertycards on the mats here does encouragestudentsto respondin terms of properties. A later activity (4) checks to see if a studentthinksin termsof propertieswhen describing othersets of figures,namelykites. 3.5 Parallelogram Sort. This activity is designed for those studentswhose incorrect definitions of shapes prevent them from giving or following inclusion arguments. This sorting activity helps them to develop the concept of parallelogram.

25

\

Using the same shapes as were used in activities 3.2-3.4 (but without squares and rectangles), the interviewerprovides another "GuessMy Rule"challenge. A few pieces are placed in each of threepiles: no parallelsides, one pair of parallelsides, two pairs of parallel sides. Students are asked to explain their decisions on where the remaining pieces should go. Hints are provided if necessary (placing sticks on shapes, or saying "I was thinkingaboutparallelsides")to help students verbalizethe descriptionof the piles in terms of number of pairs of parallel sides. Then students are asked where the squares and rectangleswill go. Finally, studentsare led to see that anothername for the pile with two and pairsof parallelsides is "parallelograms," they are asked again about the inclusion relationsbetween squareand parallelogram.

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3.6-3.7 Uncovering Shapes. These activities are similar in format, but one is presentedvisually andthe othermoreabstractly.They assess how the student uses partialinformationabouta figure (a partialview or a few properties)to make judgmentsaboutwhatthe figurecould or could not be.

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3.6 The interviewer uncovers a cardboard cut-out in four stages, asking at each stage "Whatcould this be? Could it be anything else? Why? Whatcouldn'tit be? Why?"The process is repeatedfor a second shape if the student seemed to be confused about the directionsthe first time.

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3.7 The interviewer says "Let's try that guessing game again. You will try to figure out what shape I'm thinkingof, only now you won'tsee any of the shape;insteadI'lljust give you some clues about it. I'll show you the clues one by one--after each tell me all you can. What COULD it be? Could it be anythingelse? WhatCOULDN'Tit be?" The interviewerslides a piece of paper down the sheet to uncoverclues one by one. At the end the student is asked "Canyou think of other clues I couldput down? Are you sure?Why?" This processis repeatedfor two sets of clues.

26 note: This activity is presented in a visual form first (3.6) to set the stage for the property-oriented presentation (3.7). Students vary in the type of reasoning demonstrated. In 3.6, some trace possible outlines with their fingers, operating perceptually, at level 0, while others verbalize properties in their answers ("It couldn't be a square because it doesn't have a right angle there.") indicating level 1 thought. Some students are consistent in their answers, realizing that if the shape could not be a rectangle after two properties, it could not be one after more properties are revealed. Others seem to answer with each new clue as if previous answers were unrelated. This activity assesses if a student can deduce 3.8 Minimum Properties. some property from another, within the context of their knowledge. They are encouraged to use a drawing or D-stix to check their thinking. The activity asked a student to select the fewest properties necessary to describe a shape.

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Students are shown a collection of "clue cards" for a square. "Here are a lot of properties of a square that you have talked about. Suppose that you wanted to make up some clues for a square for your friend. Do you think that your friend would need to see ALL of these properties to know that you were thinking of a square? Which cards could you take away? Why? Any others?" When the student has finished, the interviewer takes away first a card which is necessary, then one which is not necessary (if any remain) and asks for each "Could I take this away? Why?" If the explanation is incomplete, the interviewer asks, "Canyou show me why in a drawing?"

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"Now suppose that you wanted to give your friend clues for a parallelogram. What is the smallest number of clues you could give her so that she would know you were thinking of a parallelogram? Why?"

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The interviewer then shows the sheet at left. "Three people were arguing about their descriptions of a parallelogram. These are the only clues that they gave--they were all talking about four-sided figures. Who was correct? Why?"

27 note: It was not expectedthatstudentsin this studywould be able to give complete argumentsabout why some propertiesimplied others. However this activity does allow one to assess the student'srecognitionof the need for such arguments,and the student'sability to search for counter-examplesthroughdrawingor construction. Level 1 thinkersmight respondby saying that "yes, a parallelogramhas all those properties"withoutseeing duplicationor logical relations. Level 2 thinkerswould either reason("You don'tneed 'oppositeangles are equal'because 'all four angles are rightangles'and so they are equal.")or try to test a hypothesis("if the opposite sides areparallel,thenthe oppositesides will alwaysbe equal"). Activity 4. Kites This activity, done in a session following the earlier parts of the module, assesses a student's ability to analyze a new set of figures (kites) in terms of properties,and to recognize inclusion relationshipsinvolving kites. The intention here is not to instructin propertiesof kites, but to observe the student'smethodof approach. Only small hints are given, and studentsare not pressed if they don't respondeasily.

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Students are shown a collection of cut-out shapes arrangedon three cards: "Theseare kites," "Theseare not kites," and "Whichof these are kites?" (The only square included is on the third card.) Students are asked to place the shapes on the thirdcard on the first two cards, and to explain their thinking. If necessary,placementis correctedso that all kites are on the first card. Students are then asked why a rectangularcut-out does not go in the kite pile, and why a square cut-out does. Finally the interviewer asks "How would you describea kite?"and if propertiesarenot yet mentioned, "Whatpropertiesdoes a kite have?"

This activityrevealswhethera studentspontaneouslyformulatespropertiesfor a set of figures (indicatinglevel 1 thought)or tends to rely on a "lookinglike" approach (indicatinglevel 0 thought). The final partof the activity formalizesthe inclusion relations.

28

1

,

cc

IQUADRILATERPALI

Studentsare remindedof earlierdiscussionof how a rectangle is a special kind of quadrilateral. "To show this we sometimes put an arrow like this between the quadrilateraland rectangle card. You could thinkof it as a one-way streetsign." Students are asked if every rectangleis a quadrilateral, and if every quadrilateralis a rectangle, and about which placement of the arrow is correct. They are then shown name cardsfor SQUARE, QUADRILATERAL,and KITE, and some more arrows. "Canyou put arrows down between these cards to show some relationships?" If students don't themselves mentionthe relationship,they are asked "Can a square be a kite? Is every square a kite? Why? (or Why not?)"

note: Since this activity is done withoutpropertycardsbeing in sight, it assesses whetherit is naturalfor a studentto thinkof inclusionin termsof properties. Some students provide fluent explanations of placement of the arrows in terms of sufficientproperties,indicatinglevel 1 or possibly level 2 thought,while othersare not even quite sure about what directionthe arrowshould go, and reason on the basis of a squarecut-outbeing includedin the kite pile.

29 Module 2 Overview This module concernsangle measurementand angle relationshipsin polygons. The sequenceof activitiesis in partmodelledon the workof Dina van Hiele-Geldof, of angle as reportedin her thesis. An initialactivityassesses students'understanding measureand the sum of the angles of a triangle. Subjectsare led to imagineand/or construct tiling patterns from several polygons (squares, rectangles, parallelograms, right triangles, non-right triangles). In the context of these grids they review various geometric concepts which arose in Module 1 (parallel lines, congruentangles, shape identification,etc.), and also discover new patternsand properties,in particular"saws"and "ladders."Propertiesof saws and laddersare used to justify congruenceof angles. This techniqueis then applied to a triangle gridto show thatthe sum of the angles of a triangleis 180 degrees. Once this result is established,studentsare encouragedto relateit to otherfacts (angle sum of other polygons, saw and ladderproperties)and to arrangethese facts in a "familytree." In the last activity, studentsare asked to first discover the relationshipbetween an exteriorangle and the sum of the two opposite interiorones, and then to "prove"it informally,using saws and ladders. The moduleis summarizedin this diagram.

1

Angle Measurement|

2

MakingTilings andGrids

13

Saws andLadders

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Z

4

ColoringAngles

|5

Developing Propertiesfrom Grids

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ExteriorAngle of a Triangle

>1InstructionalBranch

30

Activity 1. Angle Measurement This activity is intended to assess both understandingof and skill in angle measurement,recognitionof how angle measuresof adjacentangles can be added, and understandingof the fact that the angle sum of a triangle is 180 degrees. If studentsexperiencedifficulty with angle measurement,they go to activities in the InstructionalBranch(describedon the next page).

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Studentsare shown a trianglemade of D-stix, and then one side is removed to show a flexible model for angle. They are also shown a telescoping "angle maker" (the arms can slide out). These materialsare used to make two angles, as shown, and studentsare asked "Whichis more open? Which is larger?"By superposition,if necessary,they are led to see thatopennessof an angle stays the same when lengthof sides is changed. They arealso asked to recognizeand constructrightangles. They are then presented with a puzzle involving matching an angle by eye, which motivates more exact measurementto check. First students are asked what they would choose to measure with, then if and how they have learnedto measureangles in school. Students are asked to estimate three angle measurements in degrees,and, if the responseso far is satisfactory,to check the measureswith a protractor.(If studentshave never used a protractor or say they are uncomfortableusing one, acetate angle measureoverlays, as described underthe InstructionalBranch,are provided.) Once students are able to measure angles in some way, they are shown the diagramof two adjacentangles to the left. They are asked to measure the two angles, then to predict the "outside"one (the sum). (In case perception of the angles is a problem, a diagram with color tracingis available.) Studentsare then shown the diagram to the left with three adjacent angles. They are asked to predict angle measurements, after making some measurements.This leads to discussionof the measureof a straightangle.

31

Finally,studentsare shown a triangle,and are asked to measure angles in it (which are simply recordedfor later reference). Then a triangleis shownwith only two measurements marked, and students are asked if they can predictthe measureof the unmarkedone. If they cannot,nothingmore is said aboutangle sum of a triangle at this point. If they can, they are asked to explain their reasoning which probablyinvolves learningaboutangle sum of a trianglein school. Studentsareasked about when they learned this, if they can explain why it is true, and if they learned aboutangle sum of any otherpolygons. Instructional Branch Since students enter this module with a wide range of understandingabout angles, they might enter the instructionalsequence at any point. The instruction follows a developmentalsequence for measurement:first comparisonof angles is discussed, then measuringwith non-standardunits (15 degree wedges), then the meaning of the unit "degree,"and finally use of a measuringdevice. It was found that the protractoras a measuringdevice caused many problemsfor students,and there was not time in this module to remedy them. Thus it was decided to use a simpler device, angle overlays, which suffices for the measuringrequiredin later partsof the module. Studentsare shown an angle and are invitedto make a congruentone with an angle maker. They can check by superposition. They are then challenged to do the same activity, but this time both angles are made on clock faces (unmarked)and so checking cannot be done by direct superposition. Students can check indirectly by moving an angle maker, but realize that this is inaccurate,and that a more reliable way to measure is by placement of wedges as shown. Studentspracticeestimating and measuringwith the wedges, discovering that a right angle measuressix wedges, and a straight angle twelve. It becomes apparent that measuring angles this way is quite imprecise, and that a smaller unit might be useful. Students examine one wedge that is

32 I,s" , fv 7is/

marked off in 15 congruent wedges (each measuring one degree), and are told that angles are in fact usually measured in these

7/

i

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units. They use the fact that there are 15

in an orange wedge to measure

~degrees '

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various angles.

Once they are comfortable

with this process,acetateoverlayswith angles (multiplesof 15 degrees) are presentedas an easierway to measurethanwith the wedges.

note: Since the overlays come in two sheets, one all acute or rightangles and the otherall obtuseor rightangles, studentsare led to compareangles with a rightangle whenmeasuring(althoughthe terminologyacuteandobtuseis not introducedunless the studentbringsit up). In subsequentwork,studentsare given a choice of whatto measure with--the overlays, wedges, or a protractor (the last only if it was previouslyaskedfor andused successfully). Activity 2. Making Tilings and Grids This activityestablishesa "globalvisual structure"(in the van Hieles' terms)in which angle relationshipsof trianglescan be examined. Studentsare encouragedto manipulatetiles if necessaryto establishthis structure,and then to draw the tiling patternsand note families of parallellines. The activityalso providesa contextfor reviewingconceptslearnedin earlieractivities. Studentsare asked "Canyou close your eyes and visualize a floor covered with square tiles? Could you sketch what it would look like?" (Tiles are available in case a student needs them.) The student is then shown a precisely drawn square grid, and is asked "Whatcan you see in it?"

I II

The interviewerthenasks studentsif they have seen any other types of tiling patterns, and shows some pictures of sample floor tilings which use more than one shape. "Whatother shapecould you use if you wantedto use only one shape? Could you use rectangles?"(the latterquestionbeing asked only if the student does not suggest it). "Couldyou draw how they would fit together?" Studentsare led to see how to make a quick drawingusing two familiesof parallellines.

33

/

Students are then given parallelogramtiles, andaskedto use themto make a tiling. Again they are led to first constructand then drawa tiling which contains two families of parallel lines. Next students are given two right triangles (more are available, but having just two at first helps them see the relation to a rectangle). "Haveyou ever seen trianglesused as tiles? Try these ones. Can you describe your method?" When the tiling is complete, studentsare shown a rectanglegrid. "Hereis a precise picture of the rectangle tiling you made. How couldyou put these sticksdownto make it into a triangulartiling? Is that the same pattern you made with the pieces?" Studentsare led to see how a quick drawing could be made using the three families of parallellines. Finally students are given non-righttriangle tiles, and the process just described is repeated. Once the trianglegrid is established from the parallelogram one, students are shown a complete trianglegrid and are asked to identify parallel lines, congruent angles, andvariousshapes. In summary,studentsare asked what kinds of shapes can be used to tile, and what kinds of lines arisein the tilings.

note: Studentsapproachthese tiling tasks in many differentways. Some seem to "see"the overall scheme at first, and workmethodicallyby rows, while othersuse a trial and errormethod for each piece. Some see the relationshipbetween triangle and rectangle/parallelogram grids easily and use it in the construction,while others do not, preferring to place triangles one by one. If students are becoming frustrated,the interviewer can begin a patternto be completed, but as long as studentsremaininterestedin the construction,they are not rushedtowardscreating the standardgrids.

34 Activity 3. Saws and Ladders In this activity a triangle grid is used to identify "saws"and "ladders."The activity assesses a student's ability to see shapes embedded in a grid, and to formulatepropertiesof saws and ladderswithoutinstruction. ~ \\A.~~s

Students look at a right triangle grid (see diagram A to the left), and are asked to identify shapes and lines in it. They are then

B

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givenan acetatesheetwitha ladderdrawnon (diagramB), and asked if they can find it in

fthe grid. They are asked what it looks like, and are shown pictures of "ladder-like"

~objects

g

(diagram C).

Two more ladders

drawnon acetatearepresented(diagramsD and E), to be identified in the grid.

The

instructorthen demonstrateshow on a ladder, one stick can be placed on the "side," and anotherstick slid down it to make the "rungs" 6Q(diagram , F). (Studentsare asked to demonstrate this on another ladder.) Students are shown a pictureof a non-ladder(diagramG): "Do you think this looks like the ladders T,k .... you've seen? Why not? Actually this one is L,,d?, ^ '~ , ~not a ladder. Do you see why?" If students have difficulty forming this concept, a T oce* ,Mi.,? card"is availablewhich shows more \^^ ^^"creature examples and non-examples, and asks the wo,,u,^,,..s ? i c.t G' student to decide on some others. Finally studentsareaskedto describea ladder.

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This sequence is repeatedfor saws: showing an example on acetate to locate in the grid, some "saw-like" pictures,somemoreacetate a and examples, figurewhich is not a saw;then showing the "creature card," and finally

askingfor a descriptionof a saw.

note: This activity allows studentsinitially to develop the concept of ladderand saw at level 0, on a basis of "looks like one". However questioning about why figuresare and arenot laddersor saws, andthe descriptionof these concepts,allows one to assess the student'stendencyto think spontaneouslyin terms of properties. The activity also determines how readily students use standard vocabulary introducedin Module 1.

35 Activity 4.

Coloring Angles

In this activity studentsexaminecongruentangles on a trianglegrid and areled to develop propertiesof parallelism and angle congruence of saws and ladders. They are then asked to apply these properties in a parallelogram grid to formulate/explainthatthe oppositeangles of a parallelogramare congruent. Fi;tishStW. A^,c.y

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Studentsare given coloredmarkingpens anda sheet with a triangle grid on which parts of several saws and laddershave been marked. They are asked to extend these, identifying them as saws or ladders. When this is completed,they areaskedto look at one ladderand color in all angles on it congruentto a given one. (A cardboardcut-outtriangleis available to check congruence.) They then color in congruentangles on each of the otherladders, and are asked "Whatdo you notice about the angles?" Students are led to summarizethe fact that ladders have two sets of congruent angles, which is summarizedon a file cardfor lateruse. Finally, the interviewerasks "Now you've found some featuresof ladders. What else can you say about a ladder?" If students do not respondspontaneously,they are led to summarize the parallelism property of ladders. The same process of formulatingpropertiesis repeatedfor saws. Studentscolor in angles on the saws in the grid, summarizethe property (one set of congruent angles) on a card for later reference, and then discuss parallelism (two families of parallel lines). Finally students are asked to review the special propertiesof a saw.

note: Studentswho are thinkingat level 0 can follow throughmost of this activity by looking at specific examples, but may not be able to summarize fluently. Spontaneousformationof the two propertiesof saw and ladder indicates level 1 thought. The next partof this activity is designedto see if studentscan relatethese properties,and if they spontaneouslyformulateor duplicate"if-then"phrasing.

36

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Students are shown how parallel lines can easily be madeusing opposite sides of a ruler, and how congruent angles can be made by tracing around an angle cut-out. Then the interviewer constructs a ladder using only parallels as follows: "I'm going to make somthingusing only this ruler. FirstI'll make several parallellines, like this--thenI'll draw a line crossing them, like this. What do you think is true about these angles? What could you call what I have made?" Then the interviewer constructs a ladder using only congruentangles by drawinga line and then tracing congruentangles along it. "Whatdo you thinkI could say aboutthese lines? What could you call what I have made?" In summary,the interviewersays: "So you see there are two ways to make a ladder--one using parallel lines, one using angles. If I makeone using parallellines, do you thinkthe angles will always be congruent? And if I makeone with congruentangles,will the lines alwaysbe parallel?" The demonstrationis now repeatedfor a saw. The intervieweruses the ruler to constructa saw with parallellines, and then the studentis asked to constructa saw using only the traced angle. Studentsareled to summarizewhathas been done for saws, and also to summarize againwhatwas done for ladders.

note: If studentsdo not recognize the "if-then"natureof this constructionat this point, they are not questioned further,but if they do, they are asked about the converse. Awarenessof the distinctionbetween a statementand its converse is an indicationof level 2 thinking. Next studentsare asked to apply what they know about saws and laddersto prove informallythatangles are congruent. Students are shown the parallelogramgrid constructedearlier, and are asked to review, in particular,to identify parallel lines, saws andladders.They arethenshown sheets 1 and 2 (see drawings):"Hereare two picturesof a

37 I

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When students can identify the appropriate saw and ladder in sheets 1 and 2, they are shown sheet 3. "Sometimeswe need to use a combinationof saws and laddersto show that angles are congruent." Students are given a chance to try this, and if necessaryare shown how to do it. This process is repeated for sheets4 and5.

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of

part of this grid. If we wanted to color in angles which are congruent, we might use saws andladdersto help. Forexample,on this one, could you use a saw or a laddderto show that these two angles are congruent?" (The summary cards for saw and ladder are availablefor reference.)

X.

Students are then shown the parallelogram grid again,and are askedto color in all angles congruent to a given one. They are encouragedto use saws and laddersto explain the coloring. "Do you notice anythingabout the angles in a parallelogram?"In summary, studentsareaskedto explainwhy the opposite angles of a parallelogramare congruentusing saws and ladders. If they need guidanceto do this, they are then asked to summarize the argument for the other pair of opposite angles. Finally, a summarycardfor opposite angles of a parallelogram is shown, and studentsare asked if (andhow) they have seen this fact before.

note: In orderto be able to reasonthroughsheets 3 and 4, studentsmust be able to applythe transitivepropertyof congruence. Some studentshave difficultywith this unless particularnumbersare assignedto the angles--theydo not seem able to make generalstatements,but can say "Let'ssay this one is 70 degrees,then so is this one, and this one." Other studentshave difficulty more generally in following/giving explanationswith severalsteps. The five sheets on the parallelogramgrid are necessaryto practiceapplication of the saw and ladder propertiesprior to considerationof their use in informal "proofs"(level 2) (such as the one given for congruence of opposite angles of a parallelogram,andthe one to come in the next activity).

38

Activity 5. Developing Properties from Grids This activity is designed to develop angle sum properties of triangles and quadrilateralsusing grids of tilings, and to assess a student'sabilityto explainthese properties. .,

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Studentsare given a triangle grid, on which one triangle has its angles colored in three differentcolors. Studentsare asked to color in congruent angles in appropriatecolors, using saws and laddersto explain. When all angles have beeen colored around a point, studentsare asked "Whatdo the three colors correspondto in the originaltriangle? What

you notice aboutthe colors aroundthis

point?" Then one half is coveredwith a sheet ~of paper. "Whatdoes this tell you about the threedifferentangles together?"The fact that the angle sum of the triangleis 180 degreesis shown summarizedon a card. Students are then asked to verify the fact on another grid, and asked if they think it would work on any

triangle grid.

They also consider the

measurementsthey made of a trianglein

Activity 1, so that at least four instanceshave been verified. Studentsare then asked to explain again how the coloringon the summarycardtells thatthe sum of the angles is 180 degrees, (including why the saws and ladders are saws and ladders). They are asked if they have learned this fact before, and if so, how. Finally, if studentsdid not complete it earlier, they are again shown the sheet from Activity 1 which shows a triangle with only two angle measurementsgiven. note: This is the first time that studentsare asked explicitly about the sum of the angles of a triangle. In Activity 1, it was found thatmany subjectshad learnedthis fact before by rote or by measuringangles and addingthe measures. In Activity 5, it was found thatthis priorknowledgecould interferewith theirreasoningaboutthe saw and ladderargumentbecause they do not feel the need to explain what they alreadyknow. Thus it was decidedto ask aboutpriorlearningof this fact only after it was developedhere.

39

Questioning about use of saws and ladders in other grids or on the summary card allows assessment of whether a student is thinking in terms of a particular diagram,or is able to abstractthe argumentto a more generalsituation. '{J|

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Studentsare then asked to considerthe angle sum of quadrilaterals.They are asked about angle sums while looking at square,rectangle, andparallelogramgrids. They are then asked if the sum of the angles of any quadrilateral will be 360 degrees. To verify, they are given copies of an irregularquadrilateralwith the four angles colored, which can be fitted together. A summary card is presented for angle sum of quadrilaterals. Students are asked if they had learnedthis fact before, and if so, how. By this point, some studentswill have introduced the idea of finding the angle sum of a quadrilateralby dividing it into two triangles. If it has not been suggested, the interviewer leads studentsto discover this method. Once students seem comfortable with this idea, using it on a couple of examples, they are given the diagramshown with two diagonals drawnin: "Someonedividedthis quadrilateral into four triangleslike this, and said that the sum of the angles of each triangle is 180 degrees. Whatwould happenif you do it this way?" Finally,studentsare askedto summarizewhat they've learned about quadrilaterals, and which way they prefer to think about angle sum (tiling patterns or subdivision into triangles).

note: Many students seem to misinterpretthe subdivision of a quadrilateralinto two triangles--theyconfuse measurementof angle with area. The question about four triangles determinesif they really know what is being added, if they can see that an extra 360 degrees is includedwhen the angle sums of all four trianglesare added. The question about which method is preferred distinguishes between studentswho like the strong visual impactof the tiling, and those who appreciate thatthe subdivisionargumentwill work for any quadrilateral.

40

Activity 6. Family Trees This activity is designed to assess the student'sunderstandingof concepts developedpreviouslyin the module, to introducearrowsas a symbol for relations between properties,and to assess the student'sability to interrelatepropertiesand develop a logical hierarchy. Studentsare shown cards from earlieractivities which summarize facts and properties developed. These cardsreferto: |LPOD

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Saw (congruentangles coloredin) Ladder(congruentangles coloredin) Oppositeangles of parallelogramare congruent Straightangle measures180 degrees Angle sum of a triangle Angle sum of a quadrilateral.

SAW

-1les Ofpositet of a. fpour.1Atloyf & I 1

Studentsare asked "Canyou explainwhatthis card means?" If answers are not complete, students are led to review parallelism and angle congruence of saws and ladders, the process of coloring angles in a trianglegrid, and the two techniquesfor finding angle sum of a quadrilateral.

10

, ?,I

The notion of an "ancestor"relationbetween facts is introducedthroughreferenceto facts of arithmetic. "Sometimes you can use informationthat you know to help you learn new things. For example, do you remember when you learnedhow to add numbers? First you had to learn the simple facts such as 2+3=5 and 5+4=9. Then lateryou used these facts to help you lear how to add two-digit numbers, using tens and ones, in problems such as 25+34=59." This relationis pictured with an arrow,as shown. Studentsare asked to put arrows in the diagram relating other types of arithmeticfacts, and to explain their thinking.The completeddiagramis compared to a "familytree."

41 SuvY

of aoA\Ces

of (o tr'i ^a lI is

t 0?.

%,J

The interviewer points to the fact summary cards and says: "Let us return to these geometryfacts we've been discussing. Do you see how one of thesefacts couldbe an ancestor of another?"If necessaryan example is given of how to place an arrowfrom "anglesum of a triangle" to "angle sum of a quadrilateral" cards. Studentsare then asked "Whatcould you say about the angle sum of a five-sided figure, a pentagon? Why?" If a hint is necessary,studentsareprovidedwith sticks to subdivide a pentagon. They may construct either three triangles, or a triangle and a quadrilateral,and if necessary they are led to see how this implies thatthe angle sum of the pentagon can be found as 180+180+180 degrees, or 180+360 degrees. A card is shown summarizingthis fact. Studentsare then askedhow this fact could fit into the family tree startedbefore. Whichever method they use (three triangles or quadrilateral and triangle), the interviewer points out thatit couldbe done the otherway andthat the family tree will then be a bit different. Studentswho have been successfulto thispoint are asked if they can thinkof any other cards that might be put at the bottom of the family trees. Some studentssuggest a cardfor angle sum of hexagons,etc. The interviewer then refers back to the remainingfact cards, more loose arrows,and a scalene triangle grid. "Could we go the other way--could we find some ancestorsof this fact, the angle sum of a triangle is 180 degrees? It may help you to rememberif you look at this trianglein a grid. What did you need to find the angle sum?" If necessary, studentsare helped to recall the use of saws and ladders in this argument. "How would you arrangethe cards?"Studentsare askedto explain their placement of arrowsand cards. The expected family tree is as shown to the left.

42 I5AvJ

a

Finallystudentsareaskedto identifyancestors

LAVDEP

l\

|/|@ /l /5a~k of the fact card for opposite angles of a

parallelogram,and to fit this into the family tree. "Haveyou ever thoughtabouthow facts

s oIf Opposite 'C L, o(L^llo-rom co .5At.

can be related like this before? in geometry? in other mathematics topics? in anything Y]ELg 7else?"

note: The initialintroductionof the languageof ancestorsandfamily trees is put in the context of a familiar topic, arithmetic, to avoid confusion of reasoning difficultieswith difficultiesin geometricconcepts. Some studentstend to interpret the arrowas meaning "I learnedthis first." For this reason,care is taken to show thatdependingon how you thinkaboutit, the pentagonangle sum could fit into the family tree in two differentways. The review of the saw andladderargumentsfor angle sum of a trianglein this activitydeterminesjust how comfortablestudentsare with this type of thinking. Studentswho give the logical relationseasily, and can express them fluently with arrows (explaining why arrows can't be reversed) indicatelevel 2 thinking. Studentswho can only relatethe arrowsto time sequence in whichthe facts were introducedmay be thinkingonly at level 1. Activity 7. Exterior Angle of Triangle In this activity students are challenged to apply previously learned facts (propertiesof saws, ladders, sum of the angles of a triangle) to a new situation (exteriorangle of a triangle),and to provide a logical justificationof an observed fact. This activitydeterminesif studentswho did level 1 or level 2 workbefore can do likewise here in a new situation. The activityis only done if studentshave been successful in formulatingthe angle sum propertyof a triangle,that is, if they are thinkingat least at level 1.

*i Z ^~

~

(45 75', zo')

/0.

\

'/~*

~

Students are given a sheet as shown: "We looked at some propertiesof a triangleearlier, Maybe we can find a new propertyinvolving angles. Could you say anything about the relationship among these three marked angles?" Students are allowed time to think aboutthis. Then they can measureand record the angles. They are then shown a second sheet. "Inthis one, two measures are given. What do you thinkthe third,the exteriorangle, would be?" Again, studentsare given time to think about it. Then they are asked, "Measureto check.

43 Do you notice anything?"Studentsare led, if necessary,to see thatthe sum of the measures of the two interior angles marked is the measureof the exteriorangle marked. When the propertyseems to have been discovered, studentsare askedto summarizeit. They are then asked "Can you explain why thatshouldbe so?" Littlehelp is given at first, since the aim is not to instructin the argument, but to assess the subject's ability to reason throughthis alone. However if necessary, a hint can be given to thinkof saws and ladders, or to provide the dotted line shown, and ask how it relatesto partsof the triangle. Students are led through the argument if they fail to provide the reasoning themselves, for they should not end the module with a sense of frustration. However if they can provide a logical explanationthemselves,they areasked to explainwhatthey say, and if they thinkit is true for any triangle. Finally they are shown another sheet, showing a triangle with the exterior angle drawn in a different position. Students are asked if it is also true for this triangle,andto explainwhy.

/\Q#{#88{~~~~~~~

n

TI.

tAtcvior

Aic.

of

o -trol& Cis c5usl to +two +~^ suw* of tt OphoirQ te

ior 0

a^t\5.

The family tree from the previous activity is then shown, and studentsare asked "Canyou fit this cardinto the family tree thatyou made before? Why did you arrangeit thatway?"

note: Some students provide an algebraic explanation,arguingthat x+y+z = 180, and w+z =

180, so w = x+y. When this happens, students are

xw

congratulated, and asked if they can also do it anotherway, perhapsusing the diagramshown. Studentsreasoningat level 2 are likely to searchfor and use propertiesof saws and laddersto justify the numericalrelationsamongthe angle measures,while those at level 1 may be satisfied to check particularexamples. Level 1 thinkerstend to talk about angles in this activity by attachingvalues to each angle measurement, while level 2 thinkersseem more capableof discussingthe angle measurementsin general.

44 Module 3 Overview This module deals with discovering procedures for finding the area of rectangles,right triangles,parallelograms,trianglesin general,and trapezoids. At level 0, students use tiles or transparentsquare inch grids to find the area of a figure. At level 1, they discoverproceduresfor finding areawhich are generalized for figures of a certaintype (e.g., the area of right trianglesis found by forminga rectangle,finding its area,and takinghalf). At level 2, studentsshow relationships among arearules using family trees and also give informaldeductiveargumentsto justify arearules (e.g., explainingwhy two congruentrighttriangleswhen properly placed must form a rectangle).Hence, for right triangles,the area is one-half the productof its base andheight. (note:Throughoutthis module word descriptorsare used for finding areas of figures instead of symbolic formulas unless students initiatethe use of symbolismthemselves.) The module opens with a change-of-paceactivityusing tangrams.Activity 2 is basically assessment--thusindicatingwhich instructionalactivitiesneed to be done (as shown in the diagramon the next page). Activities 3 to 7 sharea formatwhich blends instructionand assessment. Firststudentsare led to discoverproceduresfor finding the area of a shape, to generalizethe procedure(e.g., "tellhow to find the area of this kind of shape to a friend")and then "explainwhy it works." Several areaproblemsare given and studentsexplain their solutions indicatingwhich rule they used andhow they used it. Finallythe arearulesarerelatedin a family tree. The final activities are optional for students who have done some level 2 thinking in the module. One deals with similarity,use of saws and ladders, and discovery/proofof the midline rule for the areasof figures having theirvertices on two parallellines (Area= midlinex height),thus unifyingthe arearulespreviously developed. The other assesses a student's ability to recognize and explain interrelationshipsamongthe variousfamily trees thathave been constructedduring prioractivities in the threemodules. The module is summarizedin the diagramon the next page.

45

1

Tangrams

2

Assessmentof AreaConcepts--Waysof FindingAreasof Figures Note: ResultsfromActivity2 determined whatactivitywasdonenext. Most studentswentto Activity3 or4.

3

Areaof Rectangles

4

Area of RightTriangles

5

Area of Parallelograms

6

Areaof Triangles

(7

I

Area of Trapezoids 41

8

Areaof FiguresWhose VerticesLie on Two ParallelLines

9 9 FinalAcvity Final Activity on FamilyTree FamilyTrees

Activity 1. Tangrams This activity provides a change of pace from work in Modules 1 and 2. It informally assesses understanding of area and gives some experiences in decomposingshapesinto othershapesin orderto compareareas. Tangram pieces are placed in front of the student with no directives from the interviewer,to see what the studentdoes with the pieces. The interviewerasks if the student has seen thesebefore,andif so, whatwas done with them. After demonstratinghow to make

46

-L

A

1

a simple puzzle, the interviewer asks the student:"Canyou use these two smalltriangles to make this small square? The middle size triangle?This parallelogram?"After making the boat, house, and dragon puzzles, the interviewerdirects the student'sattentionto the bottoms of the puzzles (the shadedparts) and asks: "Whichof these shapeshas a larger bottom part? Or do they have the same space?" The studentis then asked to compare the areaof three shapesjust by looking. The studentcanuse tangrampieces to explain.

i

8/?b

note: This activityprovidessome informationaboutthe student'svisual abilities, in particularwhen they solve the tangrampuzzles. Some studentsworkquicklyand confidentlyand seem to take visual cues from propertiesof the shapes (e.g., sides that match). Others struggle, do puzzles on a "piece by piece basis," become frustrated,and need hints from the interviewer. Studentswho have difficulty with the puzzles also tend to have trouble comparingareas of figures or need to use pieces to do this. Otherstudentsimaginepieces when comparingareas. Activity 2. Assessment of Area Concepts and Ways of Finding Areas of Figures This activityassesses the student'sunderstandingof "area"(as "spaceinside"), measureof area (as "howmany units cover a figure"),and proceduresfor finding the area of rectangles, triangles, parallelograms,and trapezoids. Based on the student'sresponses here, the interviewerbranches to appropriateinstructionin Activities3 to 7.

()

(5) (5) UO?C_
(t)

Square inch tiles, a ruler, and a square inch plastic grid are available to the student. Showing two cardboard pieces, the interviewer says: "Supposeyou wantedto make a little jewelry box for a friend/sister. You want to cover the top with expensive gold paper. You can use these two sizes of boxes. Which top is larger? Which needs more gold paperto cover it? How can you check this?" the interviewersays: "Someoneasks you 'M~Then which top has the greaterarea. Which does? Whatdo you mean by 'area'? Anotherperson says the areaof this cover is 24. Whatdoes he meanby 'theareais 24'?"

47 Then the studentwho uses a rule (e.g., "length times width") is asked to explain why one multiplies here and whether the rule works for other shapes (non-rectangles). This type of assessment is repeated for cut-out right triangles, parallelograms,and trapezoids, if the studentknows a rule. If not, the student uses a gridto figureout the area. note: In this activity the intervieweris carefulnot to provideinstructionon area, but simply to assess whatthe studentknows. Assessmentis not pushedbeyondwhat a studentseems to be able to do. Forexample,if the studentcannotexplainthe area rule for a rectangle (this is the case for many sixth and some ninth graders),the studentis not pressed to explain rules for othershapesand may just be asked to try to find areas of right triangles (using a grid and counting square inches). This activity reveals studentdifficulties with area concepts due to prior learning (e.g., confusion of area and perimeter, mainly for ninth graders). It also provides examples of "reductionof level" (i.e., when students apply area rules by rote withoutthinkingaboutwhy they hold). Activity 3. Rectangles This activity begins with Level 0 experienceson area--namely,countinghow many square inches cover a rectangle. Then it leads students to discover a procedurefor finding the area of a rectangle--multiplyingthe number of square inches in a row by how manyrows are in a rectangle. The procedureis summarized by the arearule "lengthx width." Activities4 and5 builduponthis idea. |

|

| 3|

Having counted squaresto fmd the area of a rectangle,the studentis askedto find a quicker way to do this using strips of squares. "How many squares in a row? How many rows? How many squares in all? Why?" This is repeatedfor several rectangles with prompts if necessary, until the studentformulatesthe area rule "rowstimes numberof squaresin a row." The transitionto "lengthx width" is made by relating area via strips with area determinedby measuring length (rows) and width(squaresin a row). The studentis asked to "describethe rule to a friendover the phone and explain why one multiplies." To solidify understandingof the rule, the studentis asked

48

to find areasof rectanglesand relatedshapes, explainingwhen the rule applies and when it does not. r 7.

i

Now an alternate rule, "base x height," is developed, using an L-square device to measurebase and height of cut-outrectangles held upright. note: After the AssessmentActivity 2, studentscan enterthis activity at different places. Those who do not know any rule for area of a rectangleenter at the very beginning and carefully develop the area rule, usually throughmany examples. Students who know "length x width" but could not explain it may quickly go throughthe developmentof the rule, often catchingon aftera couple of examples. Finally,some studentsknow the rule and explainit, but are unfamiliarwith "basex height." These studentsenter the activity reviewing the rule they know and then develop the alternateform. Activity 4. Right Triangles Studentsare led to discovera procedurefor finding the area of a righttriangle in termsof the areaof a relatedrectangle. After summarizingthis resultas "Area= base x height / 2," they are asked to explain why the rule works. Then they relate the arearules for rectangleand right trianglein a family tree. Finally, finding the area of lots on a map of downtown Brooklyn provides practice on area of a rectangle,square,and righttriangle. This mapis used againin lateractivities. The activityopens with findingthe heightand. base of right triangles, using the L-square device. The studentthen finds the area of a !i ^[ : ^1 couple of cut-out right trianglesby counting squares on an overlay square inch grid. !l ITI Students notice it is hard to count pieces of A secondcopy of the triangleis inches. square availableand studentsare asked if using this would help. Reference to student's prior 6se Wtl5kt A rf of AreK of work with the tangram puzzle is made as The desired procedure is necessary. developedby having the studentfind areasof et S N right triangles systematically--taking 2 congruentcopies, forming a rectangle,using base x height, and taking half. Results for ^I^ ~~~~~~~~~~~~~~~~~~~~~ several examples are recorded in a table. After discovering the method, the student is , .

54 L It

i\

iM H

49

Recta,

asked: "If you were talking to a friend on the telephoneandhe/she had a righttriangle,what would you tell him/herto do to find its area? What measurementsshould be made? What shouldbe done? Explainwhy this works."

It

1 |. riLt itlkt

lori

L/ 'C

1

or

I

//

Then the family tree idea is reintroduced. "Howcouldyou place an arrowto show which of these two rules we used to figure out the other?" I

/1/--I?

Studentsthen find areas of lots on a map of downtown Brooklyn to review area rules learnedso far.

note: Students can respond at different levels in this activity. At level 0, the studentuses a transparentgrid to countsquaresandfigure out areasof triangles. At level 1, studentsdiscoverthe rule "measurelengthandwidthandmultiply,thentake half because two congruenttrianglesmake a rectangle."They say the rule seems to work for any right triangle. At level 2, the student carefully explains why two congruentrighttrianglesform a rectangle. Studentscan do the activity on level 0, then level 1. Some, however, can give careful arguments, when guided and encouragedby the interviewer. For example,the interviewersays: "Well,you said the two righttrianglesmake a rectangle,but how can you be sure? How could you explain that carefully to me?" Here the interviewer'slanguage about what is expectedfocuses the student'sthinkingon level 2, namely,explainingwhy. Activity 5. Parallelograms In this activitystudentsare askedto discoverproceduresfor finding the areaof a parallelogram.Three ways are possible: (a) using a grid and countingsquares, (b) breaking it into two right triangles and a rectangle, (c) cutting off a right triangle and moving it to form a rectangle with the same base and height as the parallelogram.Studentsare guided to discover the thirdmethod,to explain it, and then to relate the parallelogramarea rule to those for rectangleand right triangle via a family tree.

<.

aLL

The activity opens with a discussion of the base and height of cut-out parallelograms, again using the L-square. Then the studentis askedto find the areaof a parallelogramlot on the map, using available materials (overlay grid, ruler, special cut-out pieces) which ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ embodymethodsb andc above. The studentis .

.

.I

50

fre.

of

A.

guided, if necessary, to method c. After trying this method for a couple of parallelograms, the student is asked to summarizethis "to a friend"and also explain why the method works. Then the studentis asked:"Let'sgo back to our family tree. Can you put the parallelogramrule into that tree? Explain."

f

pt grof

lrec.to.It 6--- I CI

P

ef

The studentis also askedto reviewmethodb if he/she discoveredit and to place thatmethod into the family tree. IAreo of

Activity 6. Triangles In this activity students are asked to discover ways to find the area of any triangle. Threeapproachesare shown below. (a)

(b)

Forma rectangle, thentakehalf.

(c)

Subdivideintotwo righttriangles.

Forma parallelogram, then take half.

Instructionleads studentsto discovermethodc. The activityassesses theirabilityto discoverthis method,to explainit, andto relatethis new rule to rulesfor rectangles, righttriangles,andparallelogramsvia family trees. LstA

NKs\^

?I?^^\ '^^^

\>
The activity opens with a review of the area rule for parallelogramsand how it fits into a ~familytree. Then the student returnsto the mmapof Brooklyn to find the area of a triangulargrassy plot. Students can use a grid, ruler,D-stix, and duplicatecopies of the triangle. For whatever approach they use first, students are asked: "Do you think this approach would work for any triangle?

Then students are guided to

approach c. After using it they are asked: "How do you know that the two triangles

51

togetherform a parallelogram? What has to be trueaboutit to be a parallelogram?"After some examples, studentsare again asked "Do you think this method will work for any triangle? Why?" Then studentsare given arearule cardsfor the four shapes discussed so far and asked "How can you arrange them in a family tree? Explainwhy you placedthe arrowsthatway." If the student found the area in other ways, he/she is askedto show the otherway it would fit into a family tree. Activity 7. Trapezoids As in the two previousactivities,this one deals with ways of findingthe areaof a particularshape,namely, a trapezoid. (a)

(c)

(b)

Divide the figure into a rectangle and two right triangles.

Divide the figure into two triangles.

Divide the figure into a parallelogramand a triangle.

Studentsare guided to methodb which can be expressedinformallyas "theareaof the trapezoidequals the sum of the areasof a top and a bottomtriangle;thatis, (top base x height / 2) + (bottombase x height/ 2)." The activity assesses the students' ability to sort accordingto propertiesof a shape, to discover and explain the area rule, and to relateit to otherrulesvia family trees.

'7

The activity opens with a "Guess-My-Rule"

I CI~according '-' r^/ \9 Q\J \1 yO~

to the numberof pairs of parallel sides (i.e., 2, 1, or 0). The studentguesses the ~rrule and then is asked:"Whatmakes a shapea trapezoid?"and "Forwhich of these 3 groups do we know area rules?" This, of course, leads to considerationof finding the area of trapezoids. (note: This activity is the same as the optional activity in Module 1. If the studenthas alreadycompletedthis activity, it is only recalledhere.)

into3 groups sortingof cut-outquadrilaterals

52

"/

The studentis askedto measurethe heightof a trapezoid,and its "base(s)."Then the student identifies trapezoidlots on a map and tries to find a method for finding their areas. The student is encouragedto develop any of the three approachesand is eventually guided to the second approach. "Whatdo you need to measurehere to fmd the areaof the trapezoid? Explain." (For ninth graders with algebra experience, the rule A = 1/2 bh + 1/2 b'h is developed, and they are asked to explain it.)

\ .., ,

RrtLoof Vc-ct* lc

fY \:2~ fro.r o rigttink ofkl

The student is asked:

Imethod \\

on the phone. Then the student is asked to put the trapezoid rule in the family tree and

paeloeloyL'

ft [- o| ft^/ ]Po.t ^

"Do you think this

will work for any trapezoid? Why?" The studentis askedto summarizeby explaining how to find area of a trapezoid to a friend

D=7 [?att os; J \I

explain. If other methods were cited by the student,he/she is askedto put themin a family tree also. The activity closes with considera-

PR,->e of

thirdgroupof the initial sort--generalquadrilaterals. Methods are summarized in the ~familytreeby the student.

-Z troce3o0;

tion of ways to find the areas of shapes in the

LhZ.

Activity 8. Area of Figures Whose Vertices Lie on Two Parallel Lines This activityis for studentswho have done level 2 thinkingin this module. It provides opportunitiesfor studentsto discover a generalprinciplefor finding the areaof figureswhose verticeslie on two parallellines. It assesses a student'sability to learnand apply new concepts (e.g., similarity,ratio,midpoint)and to interrelate previously learnedarea rules using propertiesof parallellines, saws and ladders, and similartriangles. Thus, this activityprovidesan overallassessmentof students' level 1 and2 thinkingrelatedto area.

:

i\

Given a sheet of ruled paper, the student draws several lines in different positions on the paper and is asked what is trueof the segments,formed by the ruled lines, on each of the lines drawn. This leads to the discovery: "If parallellines are equally spaced then they cut off equal pieces on any line crossing them." Other diagrams illustratingexamples and non-examples of this principleareexaminedby the student.

53 Next the student is shown a spider web-like figurewhich containsvarioussets of triangles. In exploringthis figure, the studentdiscovers propertiesof similar triangles--namely,"two trianglesare similarif theirangles agree"and sides of similartriangleshave "corresponding the same ratio." These discoveries are checked by placing cut-out figures on the spiderweb figure.

The studentis guidedto use these discoveries in exploring relationships in figures whose vertices lie on two parallel lines (e.g., rectangle,parallelogram,triangle,trapezoid). The concept of midline is introducedand the studentthen establishes,by means of ladders and similarity,thatthe midline of a triangleis one half of its base. The student is asked: "Wouldthis be true in any triangle? Why? Explain."If guidancehas been necessary,the studentis askedto recapitulatethe argument. The studentis asked: "Couldwe express the area of a triangle in another way, using the midline?" The usual response is "Oh, it is midline x altitude." The student then shows (with guidance as needed), by dividing the trapezoid into two trianglesandby using the midlineof a triangle fact, thatthe midlineof a trapezoidis equalto one half the sum of the bases. From this the studentdiscovers that the area of a trapezoid can also be expressedas midlinex altitude.

a, Tropeo30.d

Mi(ll,e

= -1 Sum

of

?OoLLSt

'w_-~ i

t

(b+b')

The questionis raised:"Couldyou express the area of a rectangle and a parallelogramin a similarway? Explain." This is followed by: "If you wanted to summarizefinding area of these figures, using the midline idea, what would you say?" The studentis usually quite surprisedand delightedby the discovery that midline x altitudeworksin all cases.

54

I\

If porolll ;ies

The studentis then asked to put all the ideas generatedin this activityinto a family treeand to explainthe interrelationships.

ocve

5Ps.cet, IV,lty.

it4y

pC.ros o sa itv.. y c ro50ss

?flta^(L .

,% ll Fo-r f'igvus w;iA vt'iceoS

ipr&lltl

iwtS :

A/---\

t-/-/ z

ft _x

VIAt

Activity 9. Final Activity of Family Trees The goal of this final activity is to assess the student'sability to recognizeand amongpreviouslydevelopedprinciples. point out interrelationships

rto*Xl\ [ 0,

EI

the student is asked to explain each family

Or.f\

tree. The studentis asked: "Canyou see any

_ ight tf.ifkl

Areo.of otllelIoy,,

/ 7

X72 n~ of ARreL tria^k |
The familytreesthatthe studenthas developed in Modules 2 and 3 are displayed (i.e. angle sum for polygons, properties of parallelograms, areas of polygons, areas of figures whose vertices lie on two parallel lines) and

\\to v7 of 1pe30.4 to

/7 .'

relationship among these trees? Do they have any common ancestors? Which? What does

having a common ancestor mean? Explain and/orillustrate." The studentis encouraged traceinterrelationshipsin trees and to note the role of saw/ladder principles. "Do you think the saw and ladder principles have ancestors? What are they?" The student is

helped to recognize that there must be some beginningpoint.

55 LAQDEP,

/ /OOS

Questions are raised and discussed about the possibility of constructing family trees for otherpartsof mathematicsbesides geometry-thatis, arithmeticor algebra?

CHAPTER 4 VAN HIELE LEVEL DESCRIPTORS: DEVELOPMENTAND DOCUMENTATION An overview of the five van Hiele levels, theirproperties,andmovementfrom one level to the next throughfive phaseswas given in Chapter2. This chaptertakes a closer look at the five levels, in particular,at the Project'scharacterizationof the van Hiele model in terms of specific behavioral level descriptors. First, the Project'sformulationof an operationalversion of the van Hiele model is described and examples of student responses are cited for level descriptors. Second, the Project's documentationof the level descriptorsby quotations from van Hiele sourcesis presentedanddiscussed. Formulation of the van Hiele Model The Project'scurrentformulationof the van Hiele model is the result of an evolutionaryprocesswhich involvedthe analysisandreanalysisof van Hiele source materialsand discussions with Pierrevan Hiele and other van Hiele researchers. The Project'sinitial version of the van Hiele model was relatively simplistic and based on Wirszup's (1976) characterizationof the levels, P. M. van Hiele's (1959/1984) own descriptionof the levels and Cilley's(1979) initialresearchon the levels. While this version provided an adequatestartingpoint, it lacked sufficient detail to be an operationalmodel for the developmentof the Project'sinstructional/ assessmentmodulesandfor the assessmentof a student'slevel of thinking.Thus,the Projectneeded to flesh out this skeletalversion. This was done by reviewingother van Hiele sourcematerials--inparticular,Dina van Hiele-Geldofs dissertation. In all, ten originalvan Hiele sourceswereexamined(see page 72 for a listing). In reviewing these sources the Projectsought to locate specific passages that dealtwith the levels. Manysuchpassageswere found,with a majoritycoming from three sources: P. M. van Hiele's "TheChild'sThought and Geometry",Dina van Hiele-Geldofs dissertation,and their joint article, "A Method of Initiation into Geometryin the SecondarySchool." Analysis of these passagesled to revisionand expansion of the initial version of the model and the generaldescriptorsfor each level were recastin behavioralterms. Also, specific behavioraldescriptorsfor each level were formulatedand examples of specific tasks or responses of studentsto activitieswere given for some level descriptors.This second versionwas examined in April 1980 by P. M. van Hiele who agreedwith the Project'sinterpretation of the levels. This second version underwentfurtherchanges as a result of reanalysisof the

57 van Hiele sourcesandcarefuldocumentationof the specific level descriptorsagainst 70 selectedpassagesfrom the sources. This revisedsecondversionwas reviewedby P. M. van Hiele, in May 1981, and also by Alan Hoffer and William Burger,two researcherson the van Hiele levels. Their reviews resulted in minor changes in wording for level 1 and 2 descriptorsand the additionof examples for each level descriptor. These examples served to clarify the Project's interpretationof the levels. Examples for levels 0, 1, and 2 were derived from performancesof 12 year-olds in Dina van Hiele-Geldofs teaching experiment (1957/1984) and performancesof sixth andninthgraderson tasks correlatedwith level descriptorsin the Project's clinical interviews. Students' thinking at levels 3 and 4 were not investigatedeitherin this Projector in Dina's. Examplesareprovidedfor level 3 as illustrationsof thinkingat this level, however they were not directly observed in this study. The Project's current version of the model includes the above mentioned changes and also modificationof the descriptorsfor level 4. While making these revisions, the Project clarified its interpretationof the model and also of the expression "on a level" in relation to a student--namely,the studentconsistently exhibits behaviors for all Project descriptorsfor that level. The descriptorsfor level 0 play a somewhat different role than the descriptorsfor higher levels of thinking. Level 0 is analogousto the groundfloor of a building--itrepresentsthe type of thinkingthat all studentswill initially bring to a new subject. Of course, some studentsmay not be able to do all of the types of actions listed underlevel 0 descriptors, possibly due to lack of experience in the area under study, or to incompleteor erroneouspriorlearning. Such studentsshould not be describedas "notyet at level 0." In the Van Hiele model, thereis no "basement,"no level below level 0. The revised level descriptorsare presentedon the following pages. For each level a generaldescriptorprecedes a list of specific descriptorsand accompanying examples. A discussion of this currentversion in light of results of the clinical interviews is included in Chapter 10 where questions are raised about the appropriatenessof some descriptorsfor levels 1 and 2.

Van Hiele Level Descriptors and Sample Student Responses Level 0:

Student identifies and operates on shapes (e.g., squares, triangles) and other geometric configurations(e.g., lines, angles, grids) according to their appearance.

Level 0 Descriptors

Level 0 Sample StudentResponses

The student 1. identifies instances of a shape by its appearanceas a whole

2.

a. in a simple drawing, diagram or set of cut-outs.

la.

Student identifies squares!in a set of cut-out shap or drawings.

b. in differentpositions.

lb.

Student points out angles, rectangles, and triangl in different positions in a photographor on a pag of diagrams.

c. in a shape or other more complex configurations.

Ic.

Studentpoints to the rightangles in a trapezoid.

constructs,draws,or copies a shape.

2.

Student outlines figures in a grid (e.g., angles parallel lines, ladders).

Student makes figures with D-stix: rectangle parallel lines.

Studentmakes a tiling patter with cut-out triangl and copies the patter (piece by piece) on paper. 3.

names or labels shapes and other geometric configurations and uses standard and/or nonstandard names and labels appropriately.

3.

Studentpoints to angles of a trianglecalling them

Student refers to angles by color (e.g.; the "redan by letter symbols (e.g., "anglesA and B add to ma

4.

comparesand sorts shapes on the basis of theirappearanceas a whole.

4.

Student says "one is a square, the other is a rect when asked to say what is differentabout a cut-ou

Student sorts cutout quads into "squares,rectang "theylook alike." 5.

verbally describes shapes by their appearanceas a whole.

5.

Student describes a rectangle as "looks like a squ as "aslanty rectangle"or angle as "likehands on c

6.

solves routine problems by operating on shapes rather than by using propertieswhich apply in general.

6.

Student uses trial-and-error approach to so tangram puzzles such as making square parallelogrampieces from two small trianglepiec

Student verifies that opposite sides of a rectangle parallelby placing D-stix on edges.

Student uses transparent"angle overlay" to find measure of the thirdangle of a triangle.

Student places square inch tiles on a rectangle counts them to figure out the area of the rectangl 7.

identifies partsof a figure but a. does not analyze a figure in terms of its components.

7a. Student identifies squares by appearanceas a wh taneously introduce"equalsides and right angles

b. does not think of properties as characterizinga class of figures.

7b. Student points to sides of a square and measures but does not generalize equal sides for all square

c. does not make generalizations about shapes or use related language.

7c.

Student does not spontaneously use "all, some, such quantifiers in telling whether all, some, or n shape have a property.

Van Hiele Level Descriptors and Sample Student Responses (continued) Level 1:

Student analyzes figures in terms of their components and relationships between components,establishes propertiesof a class of figures empirically, and uses properties to solve problems.

Level 1 Descriptors

Level 1 Sample Responses

The student 1. identifies and tests relationships among components of figures (e.g., congruence of opposite sides of a parallelogram;congruence of angles in a tiling pattern).

1.

Studentpoints to sides and angles of a figure and s "ithas 4 right angles and all 4 sides are equal."

Student observes that for a parallelogram "th parallel and so are these,"checking with D-stix th or are equally spaced.

2.

recalls and uses appropriatevocabulary for componentsand relationships (e.g., opposite sides, corresponding angles are congruent,diagonals bisect each other).

2.

3.

a. comparestwo shapesaccording to relationships among their components.

3a. Studenttells how a cut-out squareand rectanglea terms of their angles and sides.

b. sorts shapes in different ways according to certain properties, including a sort of all instances of a class from non-instances.

3b. Student makes up a rule for sorting quads--for numberof right angles, or by numberof pairs of

a. interpretsand uses verbaldescription of a figure in terms of its prop-

4a. Student reads property cards "4 sides" and "all s draw a shape with these two propertiesthat is not

4.

erties and uses this descriptionto the figure. draw/construct

b. interpretsverbalor symbolicstate- 4b. When shown a propertycard for "saw,"the s mentsof rulesandappliesthem. scribesa sawanduses it to identifycongruentang

Studentcan explainthe arearule--Area= lengthx andrecognizeswhenit appliesanddoesnotapply 5. discovers properties of specific 5. figures empiricallyand generalizes propertiesfor thatclass of figures.

After coloring in congruentangles in a triang studentnotesthat"thethreeanglesof the triangl sameas thethreeanglesthatmakea straightlinea angle sum of the triangleis 180 degrees." The thinksthis will workfor othertrianglesandtries thisby usinggridsbasedon othertriangles.

After several instances of putting two congue trianglestogetherto forma rectangle,the student you can find the area of a right triangleby m rectangleandtakinghalf its area.

Fromseveral numericalcases, the studentdisco angleof a triangleequalsthe sumof its two nonandbelievesthatthis is trueforanytriangle.

6. a. describes a class of figures 6a. Studentdescribesa squareoverthe telephoneto a sides,4 rightangles,all sidesareequal,andoppos (e.g., parallelograms)in termsof its properties.

b. tells whatshapea figureis, given 6b. Givencertainpropertiesas cluesabouta shape,stu mustbe on thebasisof theproperties. certainproperties.

Van Hiele Level Descriptors and Sample Student Responses (continued) Level 1 Descriptors

Level 1 Sample Responses

7.

identifies which properties used to characterizeone class of figures also apply to another class of figures and compares classes of figures according to their properties.

7.

-laving noted that parallelograms have "oppos student spontaneously adds "oh, so do these squa (pointing to these groups of sorted cutout quads).

8.

discovers properties of an unfamiliar class of figures.

8.

After completing a sort of quads into kites and discovers and verbalizes propertiesthat character

9.

solves geometric problems by using known properties of figures or by insightfulapproaches.

9.

When asked to find some angles in a photograph are lots of angles because there are many triangle each has 3 angles."

Student solves a problem about the line conne centers of two circles of equal radiiand the line co the two points where the circles intersect.The stu a rhombus in the diagram and observes that the perpendicularbecause they are diagonals of the rh

Studentfigures out the angle sum of a quad is 360 the 4 angles arounda point (i.e., 360?) or because into two triangles (180? + 180? = 360?).

Student figures out how to find the area of a ne shape by subdividing or transformingit into shap whose areas he can already determine (e.g., parallelogram into 2 triangles and a rectangle o into a rectangle).

10. formulates and uses generalizations about propertiesof figures (guided by teacher/materialor spontaneously on own) and uses related language (e.g., all, every, none) but a. does not explain how certain properties of a figure are interrelated.

lOa. When shown a parallelogram grid, the student explain how the idea "opposite angles are equal" f from "oppositesides are parallel."

b. does not formulateand use formal lOb. When asked to define a parallelogram,the student but does not identify a set of necessary or a set of su definitions.

c. does not explain subclass relation- lOc. After the studenthas listed the propertiesof all the ships beyond checking specific infamily, the student cannot explain why "all recta stances against given list of propgrams"or why "all squares are kites." erties.

d. does not see a need for proof or lOd. After discovering the principle that the angle sum of logical explanationsof generalizacoloring angles in a triangle grid or by measuring, tions discovered empirically and see any need for giving a deductive argumentto sh does not use related language is valid. (e.g., if-then, because) correctly.

Van Hiele Level Descriptors and Sample Student Responses (continued) Level 2:

Studentformulatesand uses definitions, gives informal argumentsthat orderpreviousl discovered properties,and follows and gives deductive arguments.

Level 2 Descriptors

Level 2 Sample StudentResponses

The student 1. a. identifies different sets of prop- la. erties that characterize a class of figures and tests that these are sufficient.

Student selects properties that characterize a c squares, parallelograms) and tests by drawings D-stix that these propertiesare sufficient.

Student explains that two different sets of proper characterize a class of parallelograms--either "4 sides are parallel"or "4 sides" and "oppositesides a

b. identifies minimumsets of proper- Ib. In describing a square to a friend, the student selec ties thatcan characterizea figure. erties the fewest properties so the friend would b must be a "square." c. formulates and uses a definition for a class of figures. 2.

Ic.

Studentformulatesa definition of a kite and uses it are or are not kites.

gives informal arguments(using diagrams, cutout shapes that are folded, or other materials). a. having drawn a conclusion from given information, justifies the conclusion using logical relationships.

2a. Studentconcludes that "if angle A = angle B and an angle B, then angle A = angle C because they bot angle B."

When asked to explain why angle A = angle B in a grid, the student says "the lines are parallel, and th saw (pointingto it), so angle A equals angle B by a s b. ordersclasses of shapes.

2b. Student responds to the question Is a rectangle a explaining "yes, because they have all the propert and also the special propertyof right angles."

Student uses the propertiesthat characterizekites a why all squaresare kites but not all kites are square c. orders two properties.

2c. Given a list of propertiesof a square,the studentsa equal is not needed because it alreadysays that all fo

Having figured out a rule for the area of a right tria a rectangle, the student summarizesby making a fa ing "you need this thought (rectangle rule) befo rule)."

acute angles in an d. discovers new properties by de- 2d. Student explains that the two minus the rig "180 because 90? to add up triangle duction. leaves 90, and that is what is left for the two acute a

Student deduces that the angle sum for any quad 360? "because the quad can be cut into two trian 180? plus 180? makes 360?." When asked if it is po get 4 x 180? = 720? for the angle sum if the qua divided into 4 triangles (as shown here), the explains that "No, the inside angles are not par quad'sangles. So, if you do 4 x 180? you have to tathe extra angles in the middle, and that gives 720? 360? just as before."

Van Hiele Level Descriptors and Sample Student Responses (continued) Level 2 Descriptors

Level 2 Sample StudentResponses

Studentdiscovers that the angle sum for a pentago by breaking the pentagon into a quad (360?) and a (180?) and says that this will work for any pentago

3.

e. interrelatesseveral properties in a 2e. Student arranges property cards to form a gene family tree. "ancestral" relationships--that is, student expla "straightangle = 180?"are ancestors of "anglesum and how this leads to "anglesum of a quad is 3600." Student tells how the area rule for a parallelogram the area rule for a rectangle and puts this in a famil gives informaldeductivearguments

a. follows a deductive argument and 3a. Student gives reasons for steps in a proof that the a can supply partsof the argument. sum of a triangle equals 180? as the interviewergu the student through the proof.

b. gives a summaryor variation of a 3b. Student is given a parallelogramgrid and asked to deductive argument. a logical explanation why "opposite angles are c gruent." The student is not able to give the explana on his own but does follow the one given by the in viewer for angle A = angle C. Then the stu summarizes the explanation in his own words and explains why angle B = angle D.

Interviewer assists the student through a deduc explanation of why the exterior angle of a tria (angle X) equals angle P + angle Q. Stu summarizes this argument and then gives a comp argument on her own for a variation of this ( angle Y = angle R + angle S).

c. gives deductive arguments on own.

3c. Studentgives explanationon own for "oppositeang are equal."

Studentjustifies why the area of a right triangleis explaining that two congruent right triangles mak put the two trianglestogetherlike this, you get oppo the trianglesare the same size). Angles B and D are angles in the right triangles. Also, angles A and right angles because in a right triangle the two sm angles togethermake 90?. Angle Z is the same as an so Angle Y and Z add up to 90?. So the shape mus rectangle, and the right triangle must be half the a the rectangle." 4.

gives more than one explanation to 4. prove something and justifies these explanationsby using family trees.

Student gives two different explanations why the a equals 180?--eitherby two saws or by a saw and a l are then shown by two different family trees.

Student explains the angle sum of a pentagon equa into three triangles (3 x 180?) or by dividing it into (360? + 180?) and showing each method by a family 5.

informally recognizes difference between a statementand its converse.

5.

In a discussion of saws and ladders, the studentdisc angles are made equal, then the lines are parallel" lines are parallel, then the angles are equal." When same statements, the student realizes "No, in one y lines and make the angles equal, and in the otheryou

6.

identifies and uses strategies or in- 6. sightful reasoningto solve problems.

Given the problem that M is the midpoint of AB in triangle ABC, and MT is parallel to BC, find the ratio of MT to BC, the student uses the strategy of ladder to get congruent angles and hence similar triangles. So since AM:AB as 1:2, then MT:BC as 1:2.

Van Hiele Level Descriptors and Sample Student Responses (continued) Level 2 Descriptors 11

Level 2 Sample StudentResponses

Given two intersectingcircles A and B, not with th same radii, and a common chord CD, show that AB is perpendicularto CD. The studentproves this b establishing that ADBC must be a kite and then th perpendicularity of its diagonals makes AB perpendicularto CD. 7.

recognizes the role of deductive 7. argumentand approachesproblems in a deductivemannerbut a. does not grasp the meaning of deduction in an axiomatic sense (e.g., does not see the need for definitionsand basic assumptions). b. does not formally distinguish between a statementand its converse (e.g., cannot separatethe "Siamese twins"--the statement and its converse). c. does not yet establish interrelationships between networks of theorems.

Student recognizes the role of logical explanatio ments in establishing facts (versus an inductive, em says (after giving a logical explanation) "I know every pentagon is 540? and I don't have to mea student has yet to experience "proof' in an axiom postulates, axioms, definitions) and so is uncert possible "ancestors"to the saw and ladderprinciple

Level 3:

Student establishes, within a postulational system, theorems and interrelationship between networks of theorems.

Level 3 Descriptors

Level 3: Sample StudentResponses

The student

Note: This study was not designed to include an in-d studentsusing level 3 type of thinking. However, proposed student responses which in the Proj indicative of level 3 thinking.

1. recognizes the need for undefined 1. terms, definitions, and basic assumptions (e.g., postulates).

Student gives examples of axioms, postulates, and t plane geometry and describes how they are related

recognizes characteristicsof a formal 2. definition (e.g., necessary and sufficient conditions) and equivalence of definitions.

Student identifies sufficient properties for def parallelogram)and derives other propertiesfrom th

3. proves in an axiomatic setting rela- 3. tionships that were explained informally on level 2.

Studentproves the sum of the angles of a trianglee way (e.g., using the parallel postulate, saws and l about angle addition).

proves relationships between a 4. theorem and related statements (e.g., converse, inverse, contrapositive).

Student proves that if a triangle is isosceles, the congruent, and conversely.

2.

4.

Student proves that two sets of properties are equ shape (e.g., parallelogram).

Using proof by contrapositive, student proves that do not bisect each other. 5.

establishes interrelationships among 5. networks of theorems.

Student recognizes the role of saws and ladders involving propertiesof quadrilateralsand area rule

Van Hiele Level Descriptors and Sample Student Responses (continued) Level 3 Descriptors 6.

compares and contrasts different proofs of theorems.

Level 3: Sample StudentResponses 6.

Student gives proofs via Euclidean geometry geometry (or vector geometry) that the diagona bisect each other and comparesthe two methods of

Student compares alternateproofs of the Pythagor 7.

examines effects of changingan initial 7. definition or postulate in a logical sequence.

Startingwith "Two lines perpendicularto the same student investigates how to prove other parallel line

8.

establishes a general principle that 8. unifies several differenttheorems.

Student proves the following relationship for the vertices lie on two parallellines: area = midline x

9.

creates proofs from simple sets of 9. axioms frequently using a model to supportarguments.

Student gives proofs of theorems in a fmite geomet

10. gives formal deductive argumentsbut 10. Studentdoes not examine independence, consistency does not investigate the axiomatics set of axioms. themselves or compare axiomatic systems.

Level 4:

Studentrigorouslyestablishes theorems in differentpostulationalsystems and analyze comparesthese systems.

Level 4 Descriptors The student 1. rigorouslyestablishes theorems in different axiomatic systems (e.g., Hilbert'sapproach to foundationsof geometry). 2.

compares axiomatic systems (e.g., Euclidean and non-Euclidean geometries); spontaneously explores how changes in axioms affect the resultinggeometry.

3. establishes consistency of a set of axioms, independenceof an axiom, and equivalency of different sets of axioms; creates an axiomatic system for a geometry. 4.

invents generalized methods for solving classes of problems.

5.

searches for the broadestcontext in which a mathematicaltheorem/principlewill apply.

6.

does in-depth study of the subject logic to develop new insights and approaches to logical inference.

72 Documentation As noted above, the Project's current version of the van Hiele model was based largely on analyses of van Hiele sources. Nine major sources were the basis for the documentation reported below. Translations of sources 2, 8 and 9 are included along with source 1 in the Project's English Translation of Selected Writings of Dina van Hiele-Geldof and Pierre M. van Hiele (Fuys, Geddes, & Tischler, 1984). 1.

van Hiele, P. M. (1957). Englishsummary.[Theproblemof insightin connection with school children'sinsight into the subjectmatterof geometry.] (Unpublished doctoraldissertation, Universityof Utrecht).

2.

van Hiele,P. M. (1959). [A child'sthoughtandgeometry.] Bulletinde l'Association des Professeursde Mathematiques de l'Enseignment Public,198, 199-205.

3.

van Hiele, P. M. (1959). Developmentandthe learningprocess. Acta Paedogogica Ultrajectina.Groningen:J. B. Wolters,1-31.

4.

Muusses. andinsight.] Purmerend: van Hiele,P. M. (1973). [Understanding

5.

van Hiele, P. M. (1980, April). Levelsof thinking,how to meet them.how to avoid them. Paperpresentedat the annualmeetingof the NationalCouncilof Teachersof Mathematics, Seattle,WA.

6.

vanHiele,P. M. (1986). Structure andInsight.New York:AcademicPress.

7.

vanHiele,P. M., & vanHiele-Geldof,D. (1958). A methodof initiationintogeometry at secondaryschools. In H. Freudenthal (Ed.),Reporton methodsof initiationinto geometry(pp.67-80). Groningen:J. B. Wolters.

8.

van Hiele-Geldof,D. (1957). [The didacticsof geometryin the lowest class of doctoraldissertation, Universityof Utrecht,1957). secondaryschool.] (Unpublished

9.

van Hiele-Geldof,D. (1958). [Didacticsof geometryas learningprocessfor adults.] N.V. "Excelsior" Antwerp:Drukkerij

note: In this section (pages 72-76) all page references for quotations from sources 1, 2, 8 and 9 are made in terms of pages in the Project's English translation of selected writings of the van Hieles (Fuys, Geddes, & Tischler 1984). About 100 passages in these sources were found to be related to the levels. Others were more specific, perhaps stated in terms of responses of pupils in Dina van Hiele-Geldofs teaching experiment. Several passages repeated ideas in others. In all, 70 quotations were selected for documentation of the levels. A sample of these quotations is given on pages 74-76. Documentation was done by a Project staff member who identified quotations and checked each quote against level descriptors. After the staff member correlated the quotes with level descriptors, another staff member validated the documentation. Differences between raters were identified: (a) quotes that were inappropri-

73 ately matched to a descriptor,and (b) quotes that were not matched initially and shouldhave been. Most differenceswere of the secondtype, and they were resolved by discussingthe passageandpossible descriptors.In most cases, differencesarose from alternativeinterpretationsof or inferences to be drawnfrom the passage, in particular,the more generalpassagesfor levels 2, 3, and4. The documentationyielded 11 quotationsfor level 0, 19 for level 1, 21 for level 2, 11 for level 3, and 8 for level 4. Some quotationsdocumentedmore than one specific descriptor. Many descriptorswere documentedby severalquotations. All but five descriptors(2-la, 2-lb, 2-5, 4-8, 4-9) were documentedby at least one quotation. A complete listing of the 70 quotations and Tables indicating their correlationwith level descriptorsare found on pages 79-105 of the Project'sFinal Report (Fuys, Geddes, & Tischler, 1985) which is available throughEducational ResourcesInformationCenter(ERIC). Results of the documentationsupportthe Project'sinterpretationof the van Hiele model. The largenumberof quotationswas includedin this documentationto facilitatea more completecharacterizationof the levels thanwas providedby other researcherswho did not have available English translationsof several majorvan Hiele sources, most notably, Dina van Hiele-Geldofs thesis. The validation of descriptorsfor levels 0, 1, and 2 is particularlystrong, since most descriptorsare documentedby several quotations. The abundanceof referencesin the sources to levels 0, 1, and 2 is to be expected since the van Hieles were secondary school teachersand thereforechiefly concered aboutteachingandlearningat these levels. This explains, in part, why there are relatively few referencesin their writings to levels 3 and 4. They providenumerousexamples of studentperformanceat levels 0, 1, and 2, but give almost none at levels 3 and 4. Also, they tend to speak in generaltermsaboutthe higherlevels. Projectdocumentationof specific descriptors at level 3 and, in particular,at level 4 is less precise. Hence the Projectregarded these descriptorsas tentative. The Projectstaff engaged in considerablediscussion about the specific descriptorsfor levels 3 and 4. In additionto analyzingthe van Hiele sources, staff membersdrewupon theirexperienceslearninggeometryat the secondary, undergraduate,and graduate levels and their experiences teaching geometryat the secondaryandcollege levels in formulatingthese level descriptors. As noted above five level descriptorswere not correlated with any quotations. These descriptors were kept because they reflected student performances that seemed to fill in a level. For example, 2-la ("identifiesdifferentsets of properties that characterize a class of figures") and 2-lb ("identifies minimum sets of propertiesthat can characaterizea figure")lead to 2-lc ("formulatesand uses a definitionfor a class of figures"). In additionto sheddinglight on the natureof eacihlevel, the quotationsprovide some insight into the overall natureof the van Hiele model. For example, some quotations(see 7 and 12 below) highlightan importantfeatureof the model, namely that at each level there appearsin an extrinsic way that which was intrinsicat the

74 precedinglevel. Others(see 5 and 11 below) supportanotherfeatureof the model-namely, that the levels are characterizedby different "objectsof thought." The importof languageat each level is seen in manyquotations. Almost all of the quotations deal with cognitive aspects of the model--for example,processes such as comparisonof figuresby theirappearance,formulation of a definition,and giving an informalproof. However, a few of the 70 quotations deal with a differentaspectof the model,namelythe intentionof the studentto think in a certainway. For example,one indicatesthatstudentsare at the firstlevel when "theyknow they have to searchfor relations"(source9, page 225). Anothernotes thatat level 2 "purposefuldeductionfinally becomes a habit"(source9, page 231). These quotationssuggest thatthinkingat a level is more thanjust knowing content and performingcertaingeometricprocesses (e.g., giving a proof). It is also being awareof whatis expected,planningpurposefullyto thinkon a level, andmonitoring one'sthinkingas a problemis solved. These aremetacognitiveaspectsof the model. As will be discussed in Chapters 7 and 10, the interviewer-teacherplays an importantrole in helping studentslear the subjectmatterandprocessesand also in becomingawareof expectationsandevaluatingthe qualityof theirown thinking. Quotations Relating to Level 0 1. At the Base Level (level 0) of geometry,figures are judged by their appearance. A child recognizesa rectangleby its formanda rectangleseems different to him than a square. In our researchwhen one has shown a six year old child what is a rhombus,a rectangle, a square, a parallelogram,he is capable of reproducingthese figureswithouterroron a geoboard. Thus the child will not be botheredby the difficulties resulting from drawing figures. At the Base Level, a child does not recognizea parallelogramin the rhombusshape. At this level, the rhombus is not a parallelogram;the rhombus seems to him a completelydifferentthing. (2, p. 245) 2. The importantthing on the basic level is that all the solutions thatpupils are askedto find can be readfrom the structure.The problemsthe pupils have are purely visual, there are no rules. With the structurethe pupils are able to discoverimportantprinciplesof working. With the instructionsgiven to them, the pupils fill out the basic level; that means that they are forming a rich structureon thatlevel. (5, p. 2) Quotations Relating to Level 1 3. At the First Level of Geometry, the figures are holders of their properties. Thata figureis a rectanglemeansthatit has four rightangles, the diagonalsare equal, and the opposite sides are equal. Figures are recognized by their

75 properties. If one tells us thatthe figure drawnon a blackboardhas four right angles, it is a rectangleeven if the figure is drawnbadly. But at this level the propertiesare not yet ordered,so that a squareis not necessarilyidentifiedas being a rectangle. (2, p. 245) 4. A first level is attainedwhen the pupil is able to apply operativeproperties known to him in a figureknownto him. For instance,if a pupil knows thatthe diagonalsof a rhombare perpendicular,afterhaving reachedthe first level he must be able to concludethat,if two equal circles have two points in common, the segmentjoining the centersof the circles areperpendicularto each other. It may be that he does not directly see the rhombin the figure, or he should be able to finish afterhavinghis attentiondrawnto this rhomb. On the otherhand, the pupil not having attained the level, does not see the importanceof the knowledgeof the figurecontainingthe rhomb. (6, p. 41) 5. At this first stage we say that the first level of thinking--the aspect of geometry--hasbeen reached. This implies for example that the pupil who knows the propertiesof the rhomband can name them will also have a basic understandingof the isosceles triangle--semirhomb....At this level a geometric shape is still interpretedas the totalityof its geometricproperties. The pupils are not yet capable of differentiatingthem into definitions and propositions. Logical relations are not yet a fit study-objectfor pupils who are at the first level of thinking. (7, pp. 77-78) Quotations Relating to Level 2 6. At the Second Level, the propertiesare ordered. They are deducedfrom one another:one propertyprecedes or follows anotherproperty. At this level the intrinsicmeaningof deductionis not understoodby the students. The squareis of figurescome recognizedas being a rectanglebecauseat this level defmnitions into play. (2, p. 245) 7.

At this secondlevel of thinkinga childknowshow to reasonin accordancewith a deductive logical system: that is, its argumentsnow show an "intrinsic planning,fullfilling the laws of formal logic." This is not however identical with reasoning"onthe strength"of formallogic. (3, p. 8)

8. A second level is attainedwhen a pupil is able to apply operativelyrelations known to him between figures known to him. Thatmeans thata pupil having attainedthis level is able to apply congruenceof geometricalfigures to prove certainpropertiesof a total geometricalfigure of which the congruentfigures are a part. It means also that the pupil can conclude from the parallelismof lines the equalityof angles. (6, p. 42)

76 9. The childrendiscoveredby reasoningthat the angles of a trianglesum up to 180 degrees, the analogous facts for other polygons, and the interrelation between these facts.... The logical relationswere put into a logical pattern, using the implicationarrow. (7, pp. 71-72) Quotations Relating to Level 3 10. At the ThirdLevel, thinkingis concernedwith the meaningof deduction,with the converse of a theorem, with axiom, with necessary and sufficient conditions. (2, p. 245) 11. The thirdlevel, thatof discernmentin geometry,or the essence of mathematics. The aim of instructionis now to understandwhat is meantby logical ordering (what do we mean by: one property "precedes"another property?). The materialis madeup of geometrictheoremsthemselves. In the orderingof these theorems certain ideas will become apparent,namely: the link between a theoremandits converse,why axioms anddefinitionsare indispensable,when a conditionis necessaryand when sufficient. Studentscan now try to ordernew domainslogically, as for examplewhen they first studythe cylinder.(2, p. 250) 12. At the third level it would be possible to develop an axiomatic system of geometry,but the axiomaticsthemselvesbelong to the fourthlevel. (7, p. 75) Quotations Relating to Level 4 13. Systems of axioms belong to the fourthlevel where in fact one no longer asks the question:what are points, lines, surfaces,etc.? At this fourthlevel, figures are definedonly by symbolsboundby relations. (2, pp. 248-249) 14. Finally we can choose as a subject-matterthe system of propositionsitself. A comparative study of the various deductive systems within the field of geometrical relations is a subject reserved for those, who have reached the fourth level of thinking of geometry. Of these we can say, that they have acquireda scientificinsightinto geometry. (7, p. 80 and 8, p. 211) 15. Only the pupils who have reachedthe scientific insight(fourthlevel) can study the foundationsof the theory. They are able to help with the buildingup of a deductivesystem fromthe foundations.Amongthem,andonly them,one fmds personswho can comparedifferenttheories,who can seek out missing axioms in other geometriesand who can establishthe foundationof a new theoryand build a deductivesystemon it. (8, p. 192)

77

Another Frame of Reference for the Levels In describinglevels of thought,the van Hieles frequentlyused, in sequence,the phrases "aspect of," "essence of," "discernment of" (or "insight into"), and "scientific insight" as applied to thinking in geometry and mathematics. For example,in describinga geometrycourse,P. M. van Hiele (1959/1984) stated: The first part of a geometry course ought to allow the attainmentof the first level of thought,which we will call the aspectof geometry.... The secondpart of the course shouldallow the attainmentof the second level of thought,which we will call the essence of geometryor the aspectof mathematics.... The third partof the course shouldallow the attainmentof the thirdlevel, thatof discernment in geometry,or the essence of mathematics ... If a course could be continued further(which is generallyimpossiblein generaleducation),the fourth level would be attained,thatof discernmentin mathematics.( pp. 249, 250) The van Hieles also used this sameframeof referencein describinglevels of thought for the subject of logic in comparisonto geometry and mathematics. The chart below summarizesthe van Hieles' description of the levels using this frame of reference.

Level 0

GEOMETRY

MATHEMATICS

Level 1

Level 2

Level 3

Level 4

Aspect of Geometry

Essence of Geometry

Insightinto Geometry

Scientific Insight into Theoryof the Subjectof Geometry

Essence of Aspect of Insightinto Mathematics Mathematics Mathematics

LOGIC

Aspect of Logic

Essence of Logic

The following quotations from the writings of Dina van Hiele-Geldof (1958/1984) suggest a possible "fifth"level, and indicate her thinking about a furtherextensionof the chartabove. There has to be a "fifth"level of thinking, that is, insight into the subject logic

....

(p. 231)

The objects of study of a logician are the thinking

operationsof a mathematicalthinker,but his informationhas to be acquiredby means of sensoryperception,thatis, only a mathematicalthinkercan arriveat such a study. (pp. 232-233)

CHAPTER 5 CLINICAL STUDY: INTERVIEWS WITH SIXTH GRADE SUBJECTS This chapter reports the results of the Project's clinical study with sixth graders. Subjectswere interviewedindividuallyin six to eight 45-minutesessions as they workedwith an intervieweron the InstructionalModules. The first section below describes the subjects. In the second section an overview of the results is presented. The subsequentsections presentand discuss resultsin termsof specific behaviorsof individualstudents. In additionto characterizingthe level of thinking of the sixth gradesubjects,these sectionsfocus on the subjects'progress(or lack of it) withinlevels or to higherlevels, and on learningdifficulties. Subjects Subjects for the clinical interviews were selected to reflect the diversity of studentsin New York City publicschools, bothraciallyand in termsof achievement level. There were 16 sixth-gradesubjects--9boys and 7 girls; 12 were minority students(9 Blacks and 3 Hispanics). The 16 were studentsin two large K-6 public schools in Brooklyn (denoted as A and B). School A, located near Brooklyn College, serves a predominantlyminority population with a variety of ethnic backgrounds. School B serves a mainly Hispanic population. All but one sixth gradesubject(Juan)came from school A. The Project'smodules were designed primarilyfor students with average or above average achievement. As indicatedin Table 1, subjects were mainly at or above gradelevel as determinedby theirscoreson mathematicsandreadingsubtests of the Metropolitan Achievement Test (Intermediate, Form L), which were administeredas partof city-wide testing in late Spring. A score of 6.8 or so would be consideredon grade level at that time of testing. Studentsone or more years above gradelevel were classsifiedas high achieversin this study;those one or more years below gradelevel as low achievers. All subjects,except Arthurand Frieda, were interviewedfor eight sessions. Arthurand Friedawere interviewedfor four one-hoursessions; they workedonly on Module 1 and were given extrainstruction andpractice/reviewon basic geometricconcepts(e.g., rightangle, parallelism)and use of these conceptsin propertiesof figures. Results: An Overview As statedearlier,a student'slevel of thinkingwas determinedmainlyby his/her responsesto assessmenttasksin Modules1, 2 and3, in particular,to questionsin key activitiesin these modules. Entriesin Table2 (p. 80) characterizein a generalway each student'sthinkingon key activities. Threetypes of codes were used to describe

79

Table 1 Achievement Test Scores and Modules Completed by Sixth Graders Modules

GradeEquivalencyScores .Stllrint

Andy Norma John Jeffrey Juan Luce David Murielle Gene Frieda Arthur Bruce Ramona Sherry Adam Deanna

R?onlinr

12.9 12.9 12.9 12.0 12.0 9.0 8.8 7.0 6.8 6.8 6.8 6.7 5.9 5.5 5.4 4.5

MAthirmnatlr.e

1

HS HS

x x x x x x x x

HS

HS HS HS HS HS 7.4 7.1 6.8 8.1 7.1 5.4 4.4 5.3

note: X means that studentscompletedthe entiremodule. / means that the module was partiallydone. HS means student scored 9.0 or above.

x x x x x x x x

2

x x x x x x x x

3

x x x

Table 2 Sixth Graders' Level of Thinking on Key Module Activities Group I

d)

cC

Group II

c

t)

?~ tiasic Concepts

gE

.I 0

$

ed g

$

o

.2

=

5

1

1

lp

II

i

S-

u*

(u*

lp

0

0

0

0

0-1

0-1

0

0

0

0-Ig

O-lp

0-lg

0-lg

lp

lp

lp

lp

0-1

0-1

01

0-1

1-2

1-2p

2p

1-2p

1-2p

0-1

0-1

0-1

0-1

0-1

1

1

1

1

O*

--

--

0-1

-2p

1-2p

2s

2p

--

0

0-ig

0-ig

0-lg

g

Subclass Inclusions

0

0

0-1

Uncover Shapes

0

0

0

Minimum Pioperties

0

0*

.

Definitions

--

--

--

--

Kites Properties

0

0

0

0

Subclass

--

-0

~

.

u*

0

Propertiesof Quads

U9

U*

u'

Sorting (Polygons, Quads) 0

bk

u'

u*-1

ir

P ?o

Grou

-

.

0-lg

-

-

-g

0-g 0--lg 0

0

p 2

l

2

1

I

0-1

Ip I

ip

lpps

2p

2p

-

Angle

Measurement

0*

0*

Saw/Ladder

0

0*

0

0-1

0

Is

Is

Is

Is

--

--

Proofsvia Saw/Ladder

2g

-

2g

2p

AngleSum: Triangle

1-2g

Ig

1-2g

2p

AngleSum: Quad,Pentagon

2p

Ig

2p

2s

lp

lp

-

0

0

0

0

Ip

Ig Is

s

g

ip

Exterior Angle

Concept of Area Area: Rectangle

0* 0-1

0-1

0

0

Area: RightTriangle

Ig

Is

Area: Parallelogram

0

0

-

0

.

Area: Any Triangle

i g

Area: Trapezoid Area: MidlineRule Key:

--

O* g p s

unable to respond

weak response respondedwith guidance respondedafter a prompt respondedspontaneously

82 the qualityof a student'sresponsesin termsof the Project'slevel descriptors. (1) Students often did not respond consistently at one level on a task, and the following codes were used in these situations: 0-1 indicatesthatthe student's responses were on an "it looks like" basis as well as being based on some properties. 1-2 indicates that students formulated properties and gave some simple deductive arguments(usually with guidance from the interviewer)but were not able to give argumentsof theirown. (2) Sometimesstudentswere simply unableto respondat all to questions. This is indicatedby a dash (--). At times a student'sthinkingin level 0 was weak;that is, the studenthad incompleteunderstandingof basic concepts (e.g., parallelism, right angle, measure of an angle) and also had little or no facility with relatedterminology. This was coded 0*. (3) Studentsrespondedwith varyingamountsof assistancefrom the interviewer, especiallywhen initiallyrespondingto assessmentquestionsor in instructional situations. At times studentsrespondedwithouthelp to questionsposed by the interviewer. This type of response was coded by a number (e.g.,"l"). Sometimesa studentwould respondspontaneouslywithouteven being askeda question or given any instructions--for example, a student might spontaneouslygive propertiesof a new shape. Othertimes studentsneeded a prompt. For example, the interviewermight prompta studentby asking a generalquestionsuch as "Whatelse do you notice aboutthe parallelograms? Whatdo you notice abouttheirangles?" Still othertimes an interviewermight provideguidance,for example,asking specific questionssuch as: "Whatcan you say about these two opposite angles? . . . And these? . . . What about the

other figures?" Frequentlyguidance was needed in deductive explanations (level 2), with the studentgiving reasonsalong the way andthenbeing askedto summarize the argument. The amount of assistance provided during

questioning is denoted by: s = spontaneous, p = prompt, g = guidance.

As shown in Table 2, these 16 sixth gradersfall roughlyinto threegroups: (I) students who made little or no progress within level 0 or toward level 1; (II) studentsin level 0 who are progressingtowardlevel 1; and (III) studentswho enter with level 1 thinkingand are in varyingstages of transitiontowardlevel 2. These threegroupsare discussedin detailin the following threesections. Group I The thinkingof the students(Sherry,Deanna,Gene) in this groupwas almost uniformlyat level 0, focusing on shapesas a whole and involvinglittle or no analysis of shapesin termsof partsandrelationsbetweenparts. Sherryand Deannawere

83 low achievers, and Gene was about on grade level. The students could identify familiarshapes (square,rectangle,triangle),but did not do this readily in complex configurations (e.g., photographs and tiling patterns). There was also some difficulty with figures in different orientations. Their initial descriptions of "rectangle"were incomplete and poorly stated. Deanna's response was nonverbal--merelya gesturewith her hands of the sides of a rectangle. Sherrysaid "2 as a rectangleand lines wouldbe shortand 2 wouldbe long." Gene identified I for a squaresaid, "Na,that'sa box." They had little knowledge of partsof shapes and some misconceptions. For example, when Gene referredto the sides of a rectangle,he meantonly the two verticalsides. The othertwo sides were the "top" andthe "bottom,"not "sides." All three students had great difficulty with angles. Sherry tended to say "triangle"insteadof angle, even afterextensive instructionin Module 1 and some in Module 2. She may have been seeing an angle as a closed figure, a triangle, indicatingthatshe was relyingheavily on the visual appearanceof the angle andwas not thinkingmore abstractlyin termsof the concept "angle."Or this mightreflecta languagedifficulty,causingher to confuse phoneticallysimilartermssuch as angle and triangle. Sherryand Deannathoughtthat a cutoutparallelogramwith oblique angleshad 4 rightangles. They called obliqueangles rightangles severaltimes. All threedid not recall measuringangles in gradeschool, indicatinga gap in theirlevel 0 experienceswith angles. These studentswere also deficientin theirlevel 0 experiences with shapes. They had not heardof termssuch as parallelogramand quadrilateral,and they had greatdifficultyusing such terms,even shortlyafterinstruction on them. For example,in an activityin which shapeswere graduallyuncoveredand students were asked to name what it could or couldn'tbe, Gene forgot the new words "parallelogram" and "trapezoid"and Sherry said "Oh it couldn'tbe one of these threehardwords"(i.e. parallelogram,trapezoid,quadrilateral). Relations such as parallelism of lines and equality of angles were also not understoodby these students;hence all the instructionalbranchesin Module 1 were done with them. However, even after instruction,which at times was slow, they experienced many difficulties, thus indicatingthat they had not yet learnedthese concepts and the relatedvocabularyin orderto use themeffectively. Forexample, Deannacould makeparallellines with D-stix andtest for parallelismwith D-stix but could not readilyidentify them by eye, nor could she verbalizecorrectlywhy lines were or were not parallel. Sherryand Gene also relied on D-stix to check parallelism and often lapsedback to non-standardlanguage(e.g., "theyare straight")to tell why lines areparallel. These studentswere not able to use the conceptsof parallelismand equalityto describe shapes. In a summary activity on Kites, after six sessions, Deanna mentionedonly "4 sides, 4 angles, andthe sides are different"as thingsto say about kites. When the interviewerprompted,"Lookat the sides, what do you notice?"she said, in a questioning voice, "Equal?Parallel?" She was guessing and seemed

84 incapable of formulating anything but a very simple description of the kites. Difficulties in using geometry vocabularywere compoundedfor these students when doing Activity 3 which requiredthe studentto list propertiesfor types of quadrilaterals.They relied heavily on promptsand guidancefrom the interviewer. Expressionslike "oppositesides are equal"were not partof theirspeechpatternand seemed to be pulled out of them by the interviewer. They did not seem to understandpropertystatements. For example, Sherrythought"oppositesides are parallel"meant some sides are parallel, not necessarily two pair. Non-verbal responses were typical for these students, who seldom respondedwith words or sentenceswhenpointingwould suffice. These studentshad difficulties in attributinga propertyto a group of shapes. Sometimes they would check a propertyfor only one or two shapes in the grour ignoringothers,and then say the propertyheld for the group. This inabilityto test all given examplesin a groupmay indicatecarelessnessor perhapsa lack of level 1 thinkingaboutpropertiesof a class. Othertimes,when they did check each shapein the group,they did this slowly, shapeby shape,andoften with an aid (e.g., D-stix to test parallelism). Certainlytheir lack of prerequisitecompetency with parts of shapes and simple relations (e.g., parallelismof two lines) limited their thinking about a group of shapes. These students respondedincorrectlyor at level 0 on Activities 4 (Uncovering Shapes), 5 (MinimumProperties),and 6 (Kites), which were intendedto assess level 1 andlevel 2 thinking. SherryandGene did some initialworkin Module2, againat level 0, andneeded instructionon angles andanglemeasure. They madesome progressbut it seemedas if the instructionwas not sufficient for studentswho had majordeficiencies with that topic. Gene did tiling and saws/laddersbut with little success. He still used D-stix to check for parallellines. Sherryand Gene struggledin Activity 2 of Module 3 with the comparisonof areastask (5x5 squareversus6x4 rectangle). They respondedby eye. Gene at first said the squarewas bigger (wherethe rectanglehad a base of 4) but then said it was not bigger whenhe turnedthe rectangleso its base was 6. He could find the areaby multiplying "this side times that side" and explained "it'sbetter than counting." Sherryfound the area this way: "I timesed it ... because you can add it ... it would be the same thing"(i.e., 6 + 6 + 6 + 6). She could calculate the area of rectanglesbut did not verbalizea ruleas "lengthtimes width." The above characterizationof these students' thinking indicates that they encounteredmany difficultiesin the modulesand showed little progressin level of thinking. Therewere glaringdeficienciesin theirgeometrybackgrounds.It was as if these studentswere "geometrydeprived",they did not understandbasic geometric concepts and related terminology. All showed a markedinability to talk about shapes,and theirexpressive languagein generalwas weak. In addition,they were unable to use quantifiers such as "all," "some," "none," which are used in

85 characterizingclasses of figures (level 1). Theirperformanceon the modules was mainly at level 0, even after doing all the instructionalbranches in Module 1. Limited improvementin their level 0 visual thinking about shapes, angles, and parallel lines seemed to be facilitated by manipulativesupon which they relied extensively. Group II Students in Group II (Adam, Arthur, Bruce, Frieda, Ramona) exhibited thinkingthat was at times similarto that of studentsin GroupI but at othertimes markedly different, in particularwhen propertiesof shapes were involved. Of these five students, four were approximately at grade level in reading and mathematics,and one, Adam,was about1.5 yearsbelow gradelevel. Two students, Arthurand Frieda,were interviewedlast after several studentinterviewshad been analyzed. They received a modified version of Module 1 over 4 hours. Modificationsstrengthenedthe instructionalnatureof Module 1 and included: (a) availabilityof vocabularyreferencecardswith key termsprintedon them (parallel, right angle, opposite sides, quadrilateral,parallelogram); (b) review of basic concepts at the start of each session; (c) more explicit directions from the interviewer to the student to "tell me carefully .. .," "use these geometry words ...

" This last modificationwas intendedto make the studentmore awareof the kind of languagethatwas expected. Indirectlyit also meantthatthe expectationwas "to describecarefully,"which is one aspectof level 1 thinking. It shouldbe noted that modifications (b) and (c) were also made when Adam was interviewed since he neededextrainstructionand review. The students in this group identified shapes, at least familiar ones such as rectanglesand triangles,in photographsand other complex configurations. Some showed an ability to recognize instantly a collection of specific shapes. Bruce pointed out a "whole row of triangles"in a picture ratherthan identifying one trianglethenlooking for another. Some studentshad orientationdifficulties. Adam called I a right angle but not /\ because it didn't"go straight."He thoughta rectanglehad only 1 rightangle,but thenturninghis head to view the figurenoted it did have 4. He also orientedcutoutfigures the same way (base horizontal)when sortingthem or discussingthem. The geometric languageof all five studentswas richerand more precise than that of studentsin GroupI, althoughit tended to be as informaland non-standard initially. They had heardtermssuch as hexagon,pyramid,parallel,rightangle, and 90?. However,most could not describevery well whatthese termsmeant. Ramona said a rightangle was "linesthatgo acrossandup"andat anothertime "linesthatare straight." These students gave initial descriptionsof a rectangle that were more complete than those of Group I subjects, but still using mainly non-standard vocabularyand relatedto concretemodels or specific instancesof a rectangle. For example,Friedasaid,

86 [referringto a D-stix rectangle she made] Sides are longer. . . they'reeven [showing 2 sides are equal length and directlyabove each other]... and these are straight[gesturingthatthe sides meet at rightangles]. Whereas students in Group I completed Module 1 without improving their initial descriptionsvery much, GroupII studentsdid improve. Friedadescribeda rectanglein the fourthsession: "Thetop and bottomhave straightparallellines ... top andbottomarethe same size andthe two sides arethe same size ... It has 4 right angles ... it has opposite angles ... and they are equal [prompted] ... it has opposite

sides andthey are the same size." She then said a parallelogramwith obliqueangles was not a rectanglebecause "it'sslanted,"but added "becauseit doesn'thave right angles"when the intervieweraskedherto "saythis moreexactly." This tendencyto lapse into using non-standardlanguage,even after instructionon terminology,and to respondmore exactly aftera directivefrom the interviewerwas characteristicof these students. This was particularlythe case in Activity 3 when students first described properties of groups of quadrilaterals. They needed considerable promptingandguidance,and this activitywas basicallya learningexperiencerather thanassessmentfor them. Arthurand Friedafrequentlyreferredto the vocabulary referencecards while doing this. All studentswere able to give propertiesusing familiarconcepts (4 sides, 4 angles, right angles, sides parallel)but had difficulty with propertiesinvolvingoppositesides andangles. Frequentlythey said "theshape had oppositeangles"and no more (i.e., omittingwhat was trueaboutthe angles). It seemed as if they needed more level 0 experience in identifying, comparing,and measuringsides and angles of figures. These studentsmade progressin describing shapesin termsof properties(level 1), althoughthis was mainly limitedto familiar shapes (squares,rectangles). For example,considerAdam'sdescriptionof squares andrectanglesat the beginningof the thirdsession afterpropertieshadbeen listed in session2. Squares have 4 even sides. The sides are equal. Squareshave right angles. They have parallellines [He pointedto shapesin the squaregroupas partof his explanation.]. These rectangles have all right angles, have opposite sides. They'reparallel [prompted],all 4 not the same, 2 even, 2 other even. These [parallelograms]have 4 sides and [afterguidance] opposite sides are parallel andequal. Bruce and Arthurgave similar descriptions of these three types of shapes. Arthur'sresponsefor parallelogramswas quite interesting. He was the only student of all 16 to say thathe "forgotwhatthatshapelookedlike." He said this for parallelograms but when shown cutouts from the sorting in Activity 3, he immediately responded: "Oh, yeah ... 4 sides ... 4 angles ... sides are parallel." He seemed to

have "mentalimages" for the familar shapes "square"and "rectangle"and could verbalizetheirproperties,but he lackedthis for other,less familiar,shapes. Activity 3-6 (Uncovering Shapes) also provided more practice on using

87 properties. Students frequentlyused propertiesin telling why the hidden figure could or couldn't be a certain shape but also lapsed into level 0 explanations. Similarfindingswere obtainedin Activity 6, Kites. These studentstendedat first to give level 0 "it looks like" descriptionsof kites and with some guidance to use properties. Adam gave "4 sides and 4 angles"for kites. Then when promptedto look at the sides, he pointedto equal ones but could not expressthis verbally. When askedaboutangles, he said, "norightangles,"which was correctfor all but a square in the kite set. Whendirectedto look for equal angles, Adamsaid, "therearenone." The interviewerreviewed the meaning of equal angles and then Adam noted, "oh, yeah, these are equal,"pointingto equal angles in some kites. The responsesof the otherfour studentsin GroupII were similarto this. They were not able to analyze these unfamiliarshapes in termsof propertieswithouthelp. Progresshere was slow but at times striking. For example, Frieda spent about 10 minutes on the kite activity. After she was promptedto look at partsof the shapes,she identifiedequal sides, learned and then correctly used new vocabulary ("adjacentsides"), and graduallyformeda verbaldescriptionof kites. Her final descriptionof kites was "a pair of opposite angles are equal and two pairs of adjacentsides are equal." At the end of her last session (fourth)she was askedwhatshe had learned. She said, "about shapes ... how they are formed ... theirnames." Then when asked abouthow to name shapes, she replied, "Idon'tknow ... [pause]oh, thinkaboutthe things they have in it... the sides and angles"[pointingto a D-stix model for a kite]. She was makingprogressinto level 1, albeit slowly and not withoutlapses to level 0. Other studentsalso made limitedprogress. While showing evidence of some level 1 thinkingaboutpropertiesof shapes, GroupII studentsdid not logically interrelatepropertiesand give deductiveexplanations(level 2). All were able to explainsimplesubclassinclusionssuch as "squares are quadrilaterals"but none explained other inclusions such as "squares are parallelogramsandkites." The studentshad not yet formeddefinitionson which to base subclassinclusions. An interestingapproachto subclassinclusionwas

givenby Adam. In thekiteactivityAdammadethis

KITE

family tree and correctly explained easy inclusions. When asked "Cansquaresbe kites?"he said, "oh,yes" I-I an arrow show to but SQUARE "because QUAD this) (putting explained there was a square in the kite group." He was not X-justifying the subclass inclusion by a logical / RECT explanation (level 2) but rathermerely naming the squareas a kite on the basis of its appearance(level 0). The students did not logically relate properties in Activity 3-8 (Minimum Properties),althoughsome studentssaid 4 sides meant4 angles. Adamdid this and also said "oppositesides areequal"was not neededin describinga squarebecauseof the clue "all sides are equal." This thinking seemed to be about number relationships (If 4 are ...

then 2 are .. .), not geometric ones, and hence did not

88 indicategeometricreasoningat level 2. The absence of deductiveexplanationwas also evidencedby the studentswho did Activity 4 in Module 2, which requiredthe use of saws/laddersin deductivearguments. As indicated in Table 2, responses of Adam and Ramona in Module 2 and Adam, Ramonaand Bruce in Module3 were mainly at level 0, and, with guidance, at level 1. AdamandRamonaenteredModule2 with little knowledgeof anglesand

measurement. Adam said a right angle was "90?"but had no idea what this meant.

To him, "90?"was just a name he associated by rote with "rightangle." He had orientation difficulties with angles: not recognizing right angles in different positions, needingto turn a page with drawnangles to check for angles congruentto a given one, andnot identifyingcongruentangles in saws. He said angles b marked"a"are equal but did not agree that angles b marked"b"were equal. Both Adam and Ramona had difficulty estimating the measure of angles in degrees. Ramonaprogressed slowly from almost no concept of angle to a limited understandingof angles and measurement,but did not work with angleseasily in gridsand saws/ladders. In Module 3, Ramonaand Bruce understoodarea to mean how many squares are needed to cover a shape, althoughBruce initiallyconfused areawith perimeter. Ramona could find area only by counting squares (level 0). Bruce's thinking revealedreductionin level. He multipliedto find the area of the 6 by 4 rectangle saying "that'sthe only way to do it"to explainwhy he multiplied. For a trianglehe multipliedall threesides! In summary,the studentsin GroupII began in level 0 much like those in Group I, but made progress within level 0 (learningbasic concepts) and towardlevel 1. They used newly learnedconcepts to describeshapes and formulatepropertiesfor some classes of shapes, in particular,familarones such as square and rectangle. Difficulties were encountered on unfamiliar shapes (e.g., parallelograms and trapezoids). However,progresswas markedby frequentinstabilityand oscillation between level 0 and 1. These studentsseemed to be in transitionfrom level 0 to 1. While they began to think about classes of shapes in terms of properties(level 1), they showed little or no evidence of logically relatingproperties(level 2). None saw a need to explainwhy. They all tendedto be moreverbalthanthose in GroupI, althoughmost had difficulty expressinggeometricideas, especially with standard geometric vocabulary. The students frequentlyused manipulativedevices (e.g., D-stix) in checking propertiesor when answering. These students, like those in Group I, had a poor backgroundin geometry, reportingthat they had done little geometryin grades5-6. They tendedto respondmore easily in the interviewsthan studentsin GroupI and also were less dependenton the interviewerfor feedback and reinforcement. Their progress was slow, and they seemed to profit from carefully paced instruction that required them to work concretely while they verbalizedresultsin newly learnedterminology.

89 Group III The third group of sixth graders consisted of eight students (John, Luce, Norma, Juan, Murielle,Jeffrey, David, and Andy) all of whom had high achievement in mathematics(high school gradeequivalencyscores). As indicatedin Table 1, studentsin GroupIII exhibitedthinkingrelatedto descriptorsfor levels 1 and 2, with level 1 often occurringon entry assessment tasks and level 2 occurringas a resultof instruction(i.e., "potentiallevel"). The eight seemed to subdivideinto two groups:Luce, John, and Norma who needed additionalexperiences which helped them to be successful on all level 1 descriptorsyet made some limited progress towardlevel 2, and the other five studentswho were more fluent and confidentin level 1 andmade considerableprogresstowardslevel 2. Below, resultsare reported on these eight students'workon each of the threemodules. Module 1: Analysis. All eight students readily identified shapes in photographsand were familiarwith basic geometricconceptsand terms. Most had learnedaboutangle, rightangle, parallel. Whennew ideas were introducedstudents seemed to approachlearning them confidently. For example, when asked if she knew the meaning of "oppositeangles," Murielle responded,"I never heard that wordbeforebut I thinkI know what it means...." Murielle,Luce andJuanneeded to review the meaning of paralleland John and Luce reviewed right angles. Luce noted that she "didright angles last year but forgot." Interestingly,later, in giving propertiesof squares,Luce did not cite "allangles are rightangles,"indicatingextra review might have been useful for her before the listing of properties. Most of the students had no trouble rememberinggeometric terminology. Murielle,who scoredonly on gradelevel in reading,showedsigns of weaknesswith standardterms. She often lapsed into using terms such as "slanty"for non-right angle and "straight"for parallel, even after standardterms had been introduced. However, her use of non-standardterms was correct and precise. Juan also had some initial difficulties with new terms,but the interviewer,sensing this, addeda review of terms before Juan discussed propertiesof shapes. This seemed to help Juansolidify his familaritywith those terms,andhe didn'thave difficultywith them in the discussionof properties. Orientation of shapes was not a problem for these students in Module 1. However, the orientationof a shape did influence how some students named it. Normainitiallythought /\was not a rightangle unless turnedthis way I . Juan called O a diamondbut said "it was a parallelogramif it was like this 7 ." Sometimesstudentsrotatedcutoutshapesor turnedtheirheads to view shapesfrom a preferredorientation. All eight students seemed comfortablewith the idea of shapes having parts: sides and angles. When asked to identify angles in a picture,Jeffrey responded: "Oh,yes, I just foundtriangles[a whole row] so therearelots of anglesbecauseeach

90

trianglehas threeangles"(level 1). Respondingin a similarway by using properties of a shape,Juanfound severalrightangles by noting "Ijust found squaresand they have rightangles." These studentsgave initialdescriptionsof shapes(i.e., rectangles)in Activity2 thatwere informaland similarto those given by studentsin GroupII. The Group Il descriptionswere usuallymore completeandelaborate.However,these students did not use precise geometric language to describe rectangles,even though they knew termssuch as paralleland rightangle. Responses of these studentsto several key assessmentactivities in Module 1 (Sortingand PropertyCards,UncoveringShapes,and Kites) indicatedthatthey all exhibitedlevel 1 thinking. It shouldbe notedthatthese tasks,while intendedmainly for assessment,providedthese studentswith learningexperiencesthathelped them sharpentheiruse of geometriclanguageand strengthentheirassociationof a shape to formally statedproperties. While these studentssortedby propertiesof shapes, theirexpressivelanguagewas informal. This tendencyto describesorts informally persisted in a later assessment sort on kites, even after formal terms had been introduced.However,in Activity 4-2, Kites, a promptfrom the interviewerusually resultedin moreprecise language. Muriellesaid: "theyhave four sides ... two pair of adjacent sides are congruent . . . all four sides slant . . . both sides are

symmetrical." For these studentsthe descriptionof quadrilateralsvia propertycards(Activity 3-3) went quiteeasily withoutmuchguidance,except whenpropertiesinvolvednew termssuch as "oppositeangles are equal." But aftera promptfrom the interviewer (i.e., "whatdo you notice about the angles . . . opposite angles"),studentsusually gave appropriateresponses and tended to check spontaneouslywhetherthis new propertyheld for other groups. They all seemed to realize that the objectivehere was giving propertiesfor each of the groups. The studentsverified by eye whether a propertyheld, ratherthan by relying on D-stix or other manipulativesto check. Their ability to do this quickly by eye seemed to enable them to check rapidly whethera propertyheld for all figuresin a groupas opposedto studentsin GroupsI andII who tendedto check figureby figure. The studentstendedto use propertyexpressionssuch as "notall anglesare right angles"or "oppositesides are not parallel"when telling why certainshapes could not arise in Activity 3-6, UncoveringShapes. At times, however, they lapsed into level 0 explanations(e.g., it doesn't look like a rectangle)and informallanguage (e.g., sides are slanty),mainlyin the firstpartof this activitywherethe stimuluswas a cutout shapebeing graduallyuncovered. The visual natureof the task may have evoked level 0 responses. In the second partof this activity where propertycards were uncovered, studentsused "propertycard"language to match the format in whichthe taskwas presented.

91

Resultsfrom Activity 3-8 (Minimumpropertiesneededto define a squareanda parallelogram)indicatea mix of level 1 and 2 thinkingfor several students. When asked for the fewest propertiesfor a parallelogram,Andy said, "4 sides ... opposite sides are parallel"and noted "you need the four sides, otherwise it could be a hexagon." He recalledthis from a previousactivityaboutparallelsides andopposite sides of a hexagon. He seemed carefulabout selecting necessaryconditionshere. Luce initially said all seven propertieswere needed--aresult she probablyrecalled from Activity 3-3 where she listed properties that characterizeparallelograms (level 1). But upon questioningaboutindividualproperties,she eliminatedsome: "fourangles" because there are "foursides," "oppositesides equal"because "all sides are equal"(see level 2 descriptor2-2c). David and Normaalso did not have the fewest number of propertiesfor a square;they included "opposite sides are parallel"along with the properties (four sides, all right angles, all sides equal) which the other five students chose. Jeffrey, John and Juan all used the same strategyto choose the fewest properties--selectingpropertiesto eliminate shapes they didn'twant (level 1). For example,Jeffreysaid, "foursides gives any quad... opposite sides parallel takes away all others except squares, rectangles, and parallelograms... all right angles leaves out parallelogramsand I need one more ... all sides equal." Murielle and Andy respondeddeductively (level 2). After thinkingsilently for abouta minutebefore answeringthe interviewer,they selected the threebasic propertiesand thenexplained: [Murielle] Obviously,if it has four sides it has four angles ... because it says "all right angles." Once you know that you don'tneed to know the opposite

angles are congruent....

Because it already says all sides are equal, you don't

need "oppositesides areequal."

[Andy] Four sides means four angles ... all sides equal means opposite sides are equal so we don'tneed that... rightangles are equal so oppositeangles are equal. Thus, while most of the eight arrived at a minimum set of properties for square,the qualityof thinkingto do this differed. Andy and Muriellespontaneously proceededto eliminatepropertiesdeductively(level 2) and the othersproceededby addingpropertiestill a squarewas characterized(level 1). It should be noted that when questioned a bit later, Jeffrey, David and Luce did explain how some propertiesresultedbecause of others. Student responses to questioningabout subclass inclusions indicatedthat all eight could readilyexplain simple inclusions. For example, squaresare quadsbecause they have four sides. There was some initial confusion, however, about inclusion involving certainshapes and quads. Luce thoughtthat squareswere not

quads because "it's got other properties ... ," thinking that quads were shapes with

only one property,"foursides," as suggested by the sorting in Activity 3-3. The interviewer was easily able to help the student correct this misconception.

92 However, for several students confusion about inclusion relations among other shapes persistedeven after some interventionby the interviewerto show that one shapewas a special type of anothershape (e.g., squaresare special parallelograms). Muriellecontendedthata squarewas not a parallelogrambecause "it'snot slanty... andhas rightangles." David said thata squarewas not a "regular" rectanglebecause "itssides are all equal and rectangleshave sides with two differentlengths"(which was David's initial definitionof rectangle). Even when some studentsagreedwith the interviewerabout such inclusions, they later lapsed into incorrectstatements. For example, John agreedthat squaresand rectanglesare parallelograms,but later in the Kites activity revised that, indicatingparallelogramshave "no right angles" and then, after a promptfrom the interviewer,correctedthis. A main check on subclass inclusion thinkingwas done in Activity 4-2 (Kites) where studentswere asked to make a family tree to orderthe shapes kite, square, The arrowhere meant"is a special." rectangle, parallelogram,and quadrilateral. Correct responses ranged from very exact Q,P argumentsusing definitions(level 2) by Jeffrey and Andy to informal explanations from Norma and Juan (level 1). David and Luce were

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Module 2. The responses of these eight students;in Module 2 provided further insights into their capacity to discover properties of figures such as saws/ladders(level 1) and to follow/give deductive argumentsto orderproperties (level 2). Unlike studentsin GroupsI andII, these subjectsall had experiencewith angles and angle measurement. They easily identifiedangles in figures, although Norma initially thought right angles had to be oriented a certain way and Andy seemed to have some perceptual difficulty in identifying right angles. They estimated the measures of angles well, and several used a strategy to estimate angles--for example David, who put his thumbon a obtuse angle to show a right angleandacuteangle andaddedhis estimateof the acuteangleto 90?. All studentswere quick to discover propertiesof saws and ladders(level 1). Propertieswere sometimesinformallystatedand often involvedextraneousaspects, such as in a ladderthe "rungshave to be equal length"(Murielle). The extraneous propertiesmay have been cited for saws/laddersbecausemost of the exampleshad those characteristicsandbecausethe students,who were quickto note such features, gave all the properties they noticed when describing a figure such as a saw or ladder. The studentsassociated"parallellines"and "equalangles"with saw andladder. Herethey were dealingwith a statementandits converse(e.g., the statement,"iftwo lines are parallel, then alterate interiorangles are equal," and its converse, "if alternateinterior angles are equal, then the two lines are parallel."). Dina van

93 Hiele-Geldofreferredto these two as "Siamesetwins." All eight studentshad some difficultyin separatingthese Siamesetwins. Severaldid not distinguishbetweenthe statementand its converse--a result similar to that found by Dina in her teaching experimentwith 12 year-olds. Otherstudentseventuallydid make this distinction, but only afterinitiallyassociating"parallellines -, equalangles"with saw/ladder. The studentsencounteredinformaldeductive argumentsin Module 2--first in chainingargumentsusing combinationsof saws andladders,then in informalproofs thatthe oppositeangles of a parallelogramareequal andlaterin proofs for the angle sum of a triangleand the exteriorangle of a triangle. None of the eight studentsspontaneouslyproduced an argumentchainingsaws/laddersto explainwhy angle 1 = angle 2 (Activity 3). However, given only the promptto "trytwo steps," Andy arguedthat angle 1 = angle x by a ladderand angle 2 = angle x by a saw so angle 1 = angle 2. Andy colored in equal angles as he explainedratherthanusing terminology such as "angle 1= angle 2."

2

David, Jeffrey,Murielle,andJuaninitiallytriedwithoutsuccess to use just saws or just ladders, and responded "I don't know how" or "it can't be done." After being given some guidance,they realized what was expected and how to give an argument. As David said, "Oh,now I see it!" Murielleeven seemed to generalize the approachin these arguments:"If one angle is congruentto some angle and anotherangle is congruentto that same one, thatmakes the other two congruent." However,threeof the eight (Norma,John,andJuan)had difficultyhere andneeded guidancefrom the interviewer,who asked them to give reasonswhy certainangles were equal. One student,Luce, did not seem to catch on at all. But in her final session she correctlyused saws/laddersin the angle sum of a triangleexplanation. Of the eight students, five (Jeffrey, Juan, Murielle, Andy, and David) presented deductive arguments why opposite angles are equal for a parallelogram,using saws and ladders. Andy and David both gave clear and careful explanations-basically that angle a = angle c by a ladder and angleb = angle c by a saw, so since angleb andangle a bothequalangle c, angleb = angle a. Jeffreyand Juangave similarargumentsafterhaving been guided through it first by the interviewer.These were examplesinvolvingdeductiveargumentsthat logically interrelateproperties(see level 2 descriptors2-3a, 2-3b, 2-3c). Now we look at how studentsapproachedthe angle sum of a triangle. Norma enteredModule2 knowingthis; she recalledthatshe had "heardthis in fifth grade." However,she did not seem to be sureof it, for she askedthe interviewera few times

94 "does the triangle equal 180??" even after the interviewerhad guided her througha saw-ladderexplanationof this. Later when the interviewerexplainedwhy the angle sum of a quad is equalto 360?, Normawas puzzledabouthow the two triangleswhich looked so differentand unequalcould both have an angle sum of 180?. Perhaps she was thinkingabout the areas of these trianglesratherthan their angle sums. Nevertheless, Norma exitedfrom Module2 showingthatshe was beginningto realizethat"explainingwhy"was what was expectedin the interviews. She readthe card"Anglesum for a triangle= 180?" and immediatelyadded "because.. ." She had to be helped throughthe proof but the expectationto explain why was clear to her. Throughoutthe previous sessions Norma had strongly resisted attempts to have her explain, since in her view "explaining"was not an expectationin school mathematics. As will be discussed later, a similar belief seemed to affect the performanceon the modules of ninth graderswho were studentsin highly procedure-oriented algebracourses. The other seven studentsshowed varying degrees of progress towardlevel 2 thinkingrelativeto angle sums. John,Luce, JuanandMuriellediscoveredthe angle sum for triangles by coloring in angles in a triangulargrid and verifying equal angles (same color) by saws/ladders.Andy, David, andJeffrey knew the angle sum for a trianglefrom school, so they were asked to verify it by a new approach(i.e., not by measuringbut by a deductiveargument). The studentstended to leave out partsof the deductiveargument. Murielleand David summarizedthatthe colored top angles (a, b, and c) are the same as those inside the triangleand theirsum equals 180?-but neglected to be precise about why certain d angles (e.g., a and d) are colored the same. However, they both gave "saw" as a reason when questionedby the interviewer. All studentsexcept Norma establishedby informaldeductiveargumentsthe angle sum for

quads and pentagons. They easily showed the angle sum interrelationships via a family tree.

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There were some interestingvariationsin how differentstudentsfound these angle sums. Luce proceededinductively,notingthat"theangle sum for all quadsso far (squares,rectangles)equaled360o." She then divideda quad into two triangles to explain anotherway. Murielle explained that the angle sum for squares and rectanglesequals360?because"4x 90? = 360?" andthenclaimedthatthe anglesin a parallelogram"sumto 360?"because"itis like a slantyrectangle."Next she madea tiling using quads and observedthatthattherewere four colors of angles arounda point, and so she concluded that the sum is 360?. Finally she explained this by subdivisionof the quadinto two triangles.

95 David was asked aboutwhat comes next afterthe angle sum for quad= 360?. Reasoninginductively,he speculated"pentagonshave a sum of 720?, thatis double 360?." David measuredangles in two pentagons"tocheck,"obtainingsums of 543? and 535?! He then explained by subdividinginto 3 triangles and reconciled the discrepancybetween the 540? and his two measurementssaying, "the measurements were off a bit." When questionedabouthis preferencein approach,David, who was initiallyweddedto experimentation by measuringto show the angle sum of a triangle,now voted for the subdivisionexplanation. The interviewer'sdiscussion with David about a measuringapproachthat leads to discoveries (level 1) and a deductive approach(level 2) seemed very appropriatehere for him. It seemed to help him begin to realize the role of the deductive method and reinforced his understandingof the new expectationto explainthingscarefully. As indicated above, the sixth graders showed progress toward level 2 by following and/or giving informal deductive arguments about angle sums for polygons. Progress, although limited for some students, was also seen by their performanceon the activityon the exteriorangle of a triangle,which was presented as an assessment task in a session after completion of Activity 6. The students (including Norma) quickly discovered that the measureof an exterior angle of a triangleequals the sum of the measuresof the remoteinteriorangles, usually after measuringangles in two examples. Some studentsdevelopeda proofon theirown, even discoveringwhereto place the auxilaryline, while otherswere guidedthrough an explanation. Most of these studentshad no troubleshowing how this new fact was interrelatedto previous ideas by constructinga family tree, thus beginning to build a networkof theorems. Module 3. The intent of this module was to examine furtherthe students' thinking--inparticular,at level 1 (discoveringarea rules) and level 2 (explaining area rules and logically interrelatingthem)--in a context different from that of angle measurement. Of the eight students,two (Murielleand Juan)did not do the modulebecause they spent the 8 sessions on Modules 1 and 2. One otherstudent, Luce, did only the initial assessmentpartof Module 3. Five studentsspent from 1-1/2 to 3 sessions on it, developing area rules for rectangles, right triangles, parallelograms,any triangle,and even trapezoidsand any quadrilateral.Of these six, all had higher than level 0 understandingof area as "covering space" and interpreteda certain area as how many squarescould cover the shape. They all knew the area rule for rectangle,and appliedit to find how many squareinches of gold were needed to cover the top of a jewelry box (Activity 2). Two (Normaand Luce) could not explain why the area rule worked. Luce noted: "Theytaughtme like that . . you got to figure out what to multiply first... use a rule ...." This rote learningof a rule is an instanceof reductionof level. The four otherstudents explainedwhy the rule workedby specific examples (e.g., "thereare four rows of six in the rectangle,so multiply."). None knew a rule for righttriangles,although Jeffrey and Andy spontaneouslyformulateda methodwhen given a righttriangle: put two trianglestogetherto makea rectangle,thentakehalf its area.

96 In Activity 2, which was designed to assess understandingof area and area rules, some unexpectedlevel 1 thinkingaboutpropertiesof solids was evidenced. Two students(Norma,David) treatedthe jewelry box much like they did the cutout quadrilateralsin Module 1 and noted properties. When asked to describe the box and if it had any properties,Norma,pointingto partsof the box, responded: "Four sides [i.e., faces] ... opposite sides are congruent ... they have the same area ...

these [top edges] are parallel." David gave similar propertieswhen questioned. object,not the class of rectangular Althoughfocused on a specific three-dimensional solids, theirthinkingindicatesthatthey were able to identifyrelationshipsbetween partsof solid figures (level 1). As described below, Jeffrey, Norma, and John made progress in level 1 on area, whereas Andy and David progressedeven furtherby being guided to give careful deductive arguments to establish area rules for right triangle and parallelogram.The differencein progressis probablynot due to differencesin the abilities of these studentsbut ratherto differentexpectationsset by the interviewer who, for Andy and David, augmentedthe scriptwith more detailedexplanationsand askedthemto give carefularguments. Aided by the instruction,Jeffrey, John, and Norma developed proceduresfor finding areas of shapes. After finding the areas of two pairs of cutout congruent right triangles, Norma formulateda method: take two congruentright triangles, put them together to make a rectangle,then divideby two. Studentswere asked to report these methods "over a telephone to a , friend."For a parallelogram,John said "takeit and cut off two trianglesat ends, put them togetherand then find area of the two squares [or rectangles]." Laterhe formulatedanotherapproach,cut off the triangleat one end, move it to otherend; "measure the bottomand how far up, thenmultiply." Jeffrey and John came up with imaginativesolutions to problems of finding areas. For a parallelogram,Jeffrey said, "Makea rectanglearoundit and subtract off two triangles."Johnfound fourmethodsfor trapezoids: (1) make a parallelogramandsubtracta triangle (2) make a parallelogramand adda triangle (3) make two trianglesanda rectangle

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(4) make two triangles. However, all this work involved only level 1 thinking--mainly, solving problemsusing propertiesof shapes. This was not reallydeductivelevel 2 thinking

97 becausethese studentsdid not explainwhy certainrulesmustbe true.They observed thattwo cutoutcongruentrighttrianglesform a rectangleand said the arearulewas "length x width divided by two,"without explainingwhy the two trianglesmust make a rectangle. This lack of deductive thinking also characterized the family trees / | created by these students to show how area rules are related. Their trees seemed to be A osl IARE OF a visual format for a Lr E C:? summarizing simply procedure.This tree by Jeffrey shows how he / used a triangleand a rectangleto find the areaof R o any parallelogram. The relationship was not but generalized inductively here, deductively. Andy andDavid made level 1 discoveriesof areaproceduresas did the students above. But they made inroadsinto level 2 whenjustifying arearules that they had discovered. David gave the following argumentwhen asked why 2 right triangles give a rectangle. He explained that ABCD mustbe a rectangleby showingall the angles are right angles. The argumentbegan ^ with the observationthat angle A and angle C B are right angles because of the right triangles. 3 Now to get angle B and angle D to be right 2 angles, he first said, "Well, angle 2 is 30? so angle 4 is 60? and they make 90?." Here he was

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using specific valuesfor anglesto explain. The interviewerpressed David about why angle 2 and angle 4 form a right angle. He explainedthat in a righttriangle"thetwo angles equal 90? because 180?minus 90? leaves 90?, so angle 2 + angle 4 = 90?. Since the two triangles are the same, angle 4

is the same as angle 1, so angle 2 + angle 1 = 90?, a right angle, and likewise for angle B."

Andy gave a similarargumentwhich he summarizedby a family tree. At first he couldnot explainwhy angle2 + angle4 made90?. At the followingsession Andy said that he had figured it out "while eating dinner." He reasonedthat the acuteanglesadd 4 suf orA to 90? in a right triangle. The proof Andy gave

was correct; he did need some guidance to explain all the details. The interviewer discussed these careful argumentswith Andy and David, who used the words "clinchit" and "be technical," respectively, to characterize rI^li , their explanations. This was an attemptby the ocF^

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98 gress towardlevel 2. Moreexperiencein giving careful("technical")explanations and discussingthe quality of theirexplanationswould no doubthave helped these two students to give careful informal argumentsmore fluently and consistently, therebyenablingthemto reachlevel 2. In summary,the GroupIII studentsexhibitedevidence of level 1 thinkingfrom the outset,althoughmost neededto fill in or review some conceptsat level 0. They also needed to become more fluent with terminologyused at level 1 in describing shapesin termsof properties.Most studentsprogressedtowardlevel 2 by following andthen summarizingarguments,while a few progressedfartherandbeganto give argumentsmore independently and with more details. They tended to equate "proof" with generalization by examples (i.e., inductive approach) and only graduallydid some acquirea sensitivityto a deductiveapproachto geometry(level 2). Even when they followed/gavedeductiveexplanations,studentsdid not yet seem sureof the power of theirarguments.Thatis, althoughthey gave simple deductive proofs, they did not yet clearly see the need for careful deductive explanations. They were quiteverbalandtendedto expressthemselvesconfidently. They seemed more reflective about the problems they were doing and also about their own thinking.

CHAPTER 6 CLINICAL STUDY: INTERVIEWS WITH NINTH GRADE SUBJECTS Results of the interviews with ninth grade subjects are reportedbelow. First the subjects are described,then in subsequentsections results in terms of specific behaviorsof individualstudentsarediscussed. In additionto discussingthe level of thinkingof ninthgraders,discussionsof relationshipsbetweenlevel of thinkingand school achievement, learning style, language, learning difficulties, and thinking processes arepresented. Subjects Subjects for the clinical interviews were selected to reflect the diversity of studentsin New York Citypublicschools,both raciallyandin termsof achievement level. Therewere 16 ninth graders--5boys and 11 girls; 13 were minoritystudents (10 Blacks, 1 Hispanic and 2 Orientals). The 16 were enrolled in two public secondaryschools--onejuniorhigh school andone high school. The Project's modules were designed primarily for work with students of average or above average achievement. Table 3 presentsthe readingand mathematics achievementscores for each studentbased on the student'sperformanceon the Metropolitan Achievement Tests, Advanced 1, Form L which were administeredas part of city-wide testing in late Spring. Three students(Pete, Pat and Barth)were enrolled in a ninth gradefundamentalsof mathematicsclass; one student(Madeline)was in a three-semesterelementaryalgebrasequence;all other students were enrolled in a regular two-semester elementary algebra sequence. Table 3 also show the amountof work completedby each studenton the Project's instructionalmodulesin approximatelysix to eight hoursof clinical interviews. As statedearlier,a student'slevel of thinkingwas determinedmainlyby his/her responsesto assessmenttasksin Modules1, 2 and3, in particularto questionsin key activities in these modules. Entriesin Table 4 characterizein a general way each student'slevel of thinkingon key activities. The codes used in this table to describe the quality of a student'sresponse at a particularlevel are the same as those used previously in Table 2. When reviewing the performanceof the ninth grade subjects, one shouldkeep in mind thatthereare severalinformalgeometryunits in the New York City MathematicsCurriculumfor grades7 and 8 and presumablythese subjectshave had some instructionin geometrythroughthese curriculumunits. Results: An Overview As a resultof the analysisof the videotapesandin a mannersimilarto thatused for the sixth graders,the Project staff assigned the ninth gradesubjectsto three

100

Table 3 Achievement Test Scores and Modules Completed by Ninth Graders Test Scores PRArtin

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101

groups on the basis of their performanceson key activities in the modules. A detailedanalysisof the responsesand level of thinkingof the subjectsin GroupIV, V andVI is presentedbelow. Group IV The thinkingof the two studentsin this group(PatandPete, whose achievement scores were at least one year below gradelevel) was to a large degree at level 0 as they did the activities in Module 1. They seldom analyzedshapes in termsof their partsor theirproperties. Identifyingshapes in differentorientationsor in complex configurations(photograph)was a problemfor these students. Because of theirlack of familiarity with basic geometric concepts and language, they were guided throughall the InstructionalBranches(parallel,angle, rightangle, opposite angles, oppositesides, congruent,angle measurement.. .) of the modules. Pete commented about the word "rectangle"(after instructionwas given), "Oh,that'sa new word." He had a similarcommentaboutthe word "parallel."Pete had difficulty learning andrememberingnew ideas andcontinuallyconfused"rectangle"and "rightangle." Not until the fourthor fifth interviewsession did Pete begin to use new words such as parallelogramand rectangleanduse themcorrectly. Pat had similar difficulties with language. The word "side"proved to be a major stumbling block for her. Her concept of "side" of a shape was a vertical segment. Because of Pat'slack of clarityon the concept of "side,"she had considerabledifficulty doing the first sort of polygons. Pete also needed many examples before he caught on to the initial sort of polygons, but later carefully explainedthathe had sortedthemby the numberof sides. These two students'initialdescriptionsof squareandrectanglewere incomplete and poorly stated. In the quadrilateralsort, Pat describeda squareas "looks fat, looks like boxes, sides are not longer,they are shortandsmall;straightall the way." She describeda rectangleas: "longerthana square,has four sides, aboutthe sameas a squarebut longer,"and a parallelogramas: "slantedon sides, a rectangleis not slanted." She sortedthe shapes into threepiles--rectangle,parallelogram,square-mainly on the basis of "pointyness."Consequently,trapezoidand otherquadswere placed ratherrandomlyin her threepiles. Both studentsneededmuch guidanceand prompting in Activity 3, Properties of Quadrilaterals. They had difficulty attributinga propertyto a groupof shapes--sometimescheckinga propertyfor only one or two shapes in the group. This failureto test all given examples in a group suggestsa lack of level 1 thinkingaboutpropertiesof a class of figures. Neitherof the studentsfully graspedsubclassrelationships,althoughPatargued correctly (fromher incorrectdefinitionof parallelogram)thata rectangle cannot go in the parallelogrampile because "ithas rightangles and parallelogramsdon't." Throughout this activity on subclass relations, Pete argued from a visual perspective, never referringto nor basinghis argumentson, the propertieswhich

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104 were listed on the table in frontof him. Forexample,a rectanglecould not go in the parallelogrampile "unlessyou cut it... take a triangleandadd it to it." Similar to the GroupI sixth grade students,both of these studentsresponded incorrectlyor at level 0 (i.e., it looks like) on Activities 4 (Uncover Shapes), 5 (Minimum Properties), 6 (Kites) which were intended to assess level 1 and 2 thinking. Neither studentunderstoodthe directionalnatureof the "is a special" arrowin the task of arranginga tree with the words "square,""kite,"and "quadrilateral";both said it didn'tmatterwhich way the arrowwas placed. Orientationof shapesalso continuedto be a problemfor them. Interestingly,the strategyadopted by both studentswas "guess"--Patfrequentlysaid "Idon'tknow, just guessing"and Pete commented"Alwaystake a guess, it might be right."However, their guessing thaneducatedguessing. tendedto be random--withoutthought--rather Pat did some initial work in Module2, againat level 0, and needed instruction on angles and angle measurement.After a numberof examples, she became quite proficientat estimatingthe size of angles in termsof cut-outwedges; she then chose to measureangles with a protractorand after some confusion did reasonablywell with a little guidance. Pete did four of the activitiesin Module 3 on area. Despite some initial confusion about perimeterand area and some random guessing of answers,some of which were correct(andhe commented"luckyguess"),he showed considerableinsight in solving problems involving finding the area of irregular shapes (rectangles with pieces cut out) and the surface area of an open box. Difficulties with vocabularypreventedPete from giving clear explanations. His approach to finding areas of shapes was always visual and physical. As he explained,"Youcould cut--takeoff triangleandput it on--won'tchange." Pete was also able to explain some arearules reasonablywell using physical models. In the last two sessions, Pete beganto use some of the new geometryvocabularycorrectly andconsistentlyandwas able to explainnot only the rulefor findingthe areaof any trianglebut also why the rule works, thus exhibiting some level 1-2 behaviorson areatasks. The above characterization of these two ninthgradestudentsindicatesthe same glaringdeficienciesin theirgeometrybackgroundas was noted in the GroupI sixth gradestudents. Little school experiencewith geometrycoupledwith languageand memory difficulties resulted for the most part in level 0 performance. These students, especially Pat, seldom realized that they could figure things out in mathematicsby thinking about them--rather,they utilized a strategy of random guessing which they evidently felt had worked for them before. While these students made some progress filling in level 1 as a result of instruction and experiencewith the activities,progresswas limited. Group V As a result of the analysis of the videotapes,the Projectassigned seven ninth

105 grade subjects to Group V (Barbie, Barth, Beth, Doreen, Madeline, Elena and Jorge). Performancesof these seven students were markedlydifferent from the studentsin GroupIV althoughat times some of the studentsin this groupexhibited level 0 thinking to justify a response. Table 4 provides a characterizationin a generalway of the level of thinkingon key activitiesof each studentin this group. The responsesto the activitiesin Modules1, 2 and 3 of the studentsin this groupare analyzedbelow. Module 1. The students,for the most part, readily identified shapes such as rectangles,squares,and trianglesin photographsand othercomplex configurations. Also mentionedby some studentsin this groupwere parallelogramsand trapezoids. Some students had orientation difficulties and needed to turn a figure before decidingon its shape. In describinga rectangle,all the studentscited the propertyof "4 sides with two equal longer sides and two equal shortersides." Only Barbie indicatedthe need for rightangles, althoughDoreen and Elena said it was "almost like a square." After giving the descriptionof a rectangle,subjectswere asked to constructa rectanglefrom a set of D-stix. Jorge and Madelineselected 4 sticks of the same length. Madelinequickly realizedher figure was a squareand remadethe figure with differentsticks. When the D-stix constructionof a rectanglemade by each subjectwas alteredslightly by the interviewerso that it became a figure with non-rightangles (i.e., a parallelogram),Jorgeand Beth still thoughtthe figure was a rectangle. Barthcalled it "adiamond"and Doreensaid it was "aslantedrectangle." Otherssaid, "No,for a rectanglethe sides have to be straight." Parallel lines were described by all the subjects as "lines that never meet." Doreenthoughtthe lines also hadto be equalin length. InitiallyElenaandBeth used the word "straight"in lieu of the word "parallel."Right angles were describedas "squarecomers"by Elena, Barbie,and Doreen;Barbiealso spoke of "right"angles and "left"angles. Madelinedescribeda rightangle as an angle with "straightsides" and is "over30? up to 90?." Othersdescribedrightanglesas "90?."JorgeandBeth, although saying that right angles contained90?, were unable to distinguishright angles from obtuse and acute angles, or to identify right angles in different orientations. After instruction,Beth was able to identify right angles correctly although she frequently called them "righttriangles." Jorge, on the other hand, despiteextensive instructionanddespitehavinga cut-outmodel for checkingto see if an angle was a rightangle, continuedto have greatdifficulty in identifyingright angles. This difficulty persisted throughoutthe eight interview sessions, thus suggestinga visual perceptionproblem. Most of the students recognized or were familiar with basic geometry vocabulary(angle, parallel, diagonal, congruent,opposite sides... ) and needed only a brief review. The geometriclanguageof all seven subjectsin this groupwas richerand more precise than that of GroupIV studentsalthough,as noted above, initiallyit tendedto be informaland non-standard.Theiruse of precise vocabulary in giving descriptionsof figurestendedto improveas they workedthroughthe first

106 module;perhapsthey beganto realizethatthis is whatwas expectedof them. In doing the quadrilateralsort (Activity 3), six of the seven subjects did a systematic sort into piles of "rectangles,squares, parallelograms,others", with Madeline, Beth and Barbie then separatingthe "others"pile into "trapezoid"and "miscellaneous."Jorgedid the sorthesitantlyandexplainedhis sort as "Iput sets of similar figures in each group"and "theshapes that are left over are not similar." Actually, some of the shapes in his sets were similar(in the technicalsense of the word) and some were not. When the studentswere asked to list the propertiesof the shapes in each pile, most respondedquickly and correctlyfor squaresand rectangles. Promptswere needed to obtain all the propertiesof a parallelogramand trapezoid,particularly concerningthe parallelismof the sides. When a new propertywas noted, such as "opposite angles are equal" (in a parallelogram), Doreen and Barbie said spontaneously:"Oh,you could say that aboutthe squareand rectanglepiles also." When asked: Are these properties true for all parallelograms or just these parallelograms?,the students in this group were very thoughtfuland Madeline's answer was typical: "Hm. . . all parallelograms. . . yeah, all." On the selection

sheets, most studentsspontaneouslyreferredto propertiesin informallyjustifying their classification of figures. Fluent studentresponses indicatedthat they were analyzingfigures in terms of their componentsand relationshipsbetween components and establishingpropertiesof a class of figures (i.e., necessarypropertiesof a conceptwere establishedempirically),thusexhibitinglevel 1 thinking. In Activity 3-4 (InclusionRelations),students'priorlearninginfluencedtheir answersto the questions:Could a squarebe placed in the rectanglepile? Why or Why not? Madeline, Doreen, Barth and Elena gave negative replies with Elena adding, "All rectangles are long; that's what they taught me," and the others indicatinga need for two longer and two shortersides. After discussionsbased on using only the propertieslisted in frontof them,these four studentsagreedthatthey could think of a squareas a "special"rectangle. Jorge gave no reply and afterhe was guidedto check out each of the propertiesof a rectangleas being applicableto a square,he decided "Yes." Beth and Barbie gave affirmativereplies with Barbie saying: "Letme look at the properties... Yes... Justbased on these propertiesyou could put a squarein the rectanglepile; but if you had the propertyof one longer andone shorterside, you could not." In responseto the questions,Can a rectanglebe placed in the squarepile? Why or why not?, all studentsjustified theirnegative replies by citing a property(e.g., "allthe sides arenot even"). By now the studentsrecognizedwhatwas expectedand so theiranswersto successivequestionsrelatingto placinga squareor a rectanglein the parallelogrampile, were carefully given--justifying their answers by citing propertiesof the figures. Beth tendedto interchangethe idea of "havingone more property"and "missing a property"as she explained why a figure was not in a

107 particularpile. For example, she said: "A rectangleis not a parallelogrambecause it is missing a property--ithas four right angles." Beth's difficulty in mixing up these ideas was not fully resolvedandpersistedin lateractivities. The subjectsenjoyedthe two activitiesin UncoveringShapes(Activities3-6, 7) via clues and all based their reasoningon propertiesof shapes (level 1) although therewas an occasional"itlooks like"at the beginning. Jorgeneeded guidanceand had to be continually reminded to consider the clues sequentially and not individually. Beth had difficultyconceptualizingthe meaningof "atleast"on clue cards,such as "hasat least one rightangle." Exceptfor Jorge,who consideredall propertiesas necessary,all the studentsin doing the Minimum Propertiesactivity (Activity 3-8) for a square immediately removed"oppositesides are equal"and "oppositeangles are equal"as unnecessary. They reasoned that the descriptors"all sides are equal"and "all angles are right angles"covered the opposite sides and opposite angles. They also removed "four angles"since "if it has four sides, it must have four angles." Most of the students were uncertainabout the necessity of the descriptor"oppositesides are parallel." Theirgeneralfeeling was thatperhapsit was not necessary,but all chose to leave it in theirlist of necessarydescriptorsfor a square. In doing the same activity, but selecting necessary descriptors for a parallelogram,Jorge was the only one who initially selected the minimumnumber of necessaryproperties(i.e., four sides, opposite sides are parallel). In additionto Jorge's descriptors,Beth and Barbie thought that "oppositesides are equal" and "oppositeangles are equal"were necessary. When askedto verify theirselections, the studentsdrew figures using the differentproperties,and then decided thattheir additionalpropertieswere unnecessary. As seen in the above studentresponsesto the tasks in the Minimum Propertiesactivity, there is evidence of some level 2 behaviors. The final assessmentactivity,Kites, in this module providedan opportunityto see if students described properties of a class of figures and if they saw interrelationships amongfigures(level 1). Jorgewas unresponsiveandneededto be guided to look at the specific propertiesof a kite. Initially Madeline and Beth describeda kite as "diamondshapewith fourpoints,two sides equal (pointing)and othertwo sides equal (pointing)." Madelineplaced the squarein the non-kitepile explainingthat "it doesn'tlook diamondshape"(level 0). The interviewerrotated the square45?. Madelinesaid, "Oh,it is diamondshape,it has foursides, it has four points,and two pairsof adjacentsides areequal." She thenproceededto changethe squareto the kite pile, thusmakingher decision on the basis of properties(level 1). Beth put the squareon the kite pile but had difficultyverbalizingwhy. She talked vaguely about propertiesbut seemed to judge on a "looks like" basis and needed guidanceto completeher explanation. Doreen,Elenaand Barbiedescribedthe kite as having four sides, two pairs of equal sides, and then focused on the diagonals.

108 Doreennoticedthatone diagonalcut it into two congruenttriangleswhen she folded the paperkite. Elenamade the same observationandadded"if the otherdiagonalis drawn,the trianglesare not congruent."Barbie, referringto the diagonals,spoke of "theframe of the kite"and said they must cross at a T, at right angles." Barbie said, "a squareis a non-kitebecause it does not have propertiesof all otherkites." She checked out her kite propertiesagainstthe square: "Ithas four sides, the frame fits, the adjacentsides areequal." She then said somewhattentatively,"Itcouldbe ... a squareis a special kind of kite." Barbie'sexplanationshows that she was thinkingof propertiesof a class of figures(level 1). When askedto show the interrelationships among quadrilaterals,kites, and squaresusing "is a special" arrows, as shown in the adjacent diagram, Jorge placed arrows 1 and 2. When asked if he could put K33 iQE anotherarrowto show a relationof squareandkite, he arrow 3. Elena Madeline, Doreen, correctlyplaced and Beth placed arrows 1 and 2 correctly. However, Madeline initially placed arrow3 in reverseposition;Doreen chose not to add arrow3 showing she was still uncertainabout the relationship;arrow3 was confusing to Elena and she did not know how to place it; Beth, still showing uncertainty,reversed arrow 3 several times. Barbie revised her arrowsmany times saying to herself: "Kiteis a special quad,squareis a specialquad--rightanglesandall sides equal,... a kite is a special

I

type of square ...

no ...

a square is a special type of kite because it has the

properties of the rest of the kites" and finally she placed the arrows correctly. Barbie seemed to have a visual image of a kite and the squaredid not match that image althoughthe squaredid have "thepropertiesof the rest of the kites." Module 2. The responses of the seven students in this group to Module 2 activities provide further insights into their capacity to discover properties of figures, such as saws/ladders and angle sums of polygons (level 1) and to follow/give informaldeductivearguments(level 2). As ninth graders,the students had school experience in learning about and measuringangles in the seventh/eighthgrades. All were able to identifythe larger angle of a pair,recognizingthatthe size of the raysformingthe angles did not affect its measure, with Barbie remarking:"the length of the sides is not important." Exceptfor Jorge,the studentswere quite good at estimatingthe size of acute,right and obtuse angles. Jorge'sinabilityto identify a right angle consistentlypersisted even aftercompletingthe InstructionalBranchon Angle Measurement. Right angles in differentorientationswere troublesome to some students. Some turnedthe angle or the page to look at it; Barth mutteredsomething about a "left angle"but would not elaborate.Referringto the diagramat the right, Madelinesaid, "x is not an angle

Ix

109

because the right side of the angle always has to be flat [horizontal]." This misconceptionapparentlyhad arisenfromher experiencein measuringangles with a protractor, where she had been cautioned "to place the protractor on the horizontalray." In orderto determinetheirfamiliaritywith angle relationshipsin a triangle,the studentswere asked to estimate and record the measuresof the three angles of a triangle and then to check their estimates with the transparentangle overlays (or with a protractor). Jorge, Beth and Barbie spontaneouslyfound the sum of their angles. Whenaskedwhy they did that,they said thatthey remembereddoing thatin school. When the otherswere askedaboutthe angle sum, they vaguely remembered having learneda rule aboutthe angle sum for a triangle. None of the studentscould rememberor explain why this rule was true. After tiling with squares, rectangles, parallelograms, right triangles and trianglesand seeing the associatedgrids, the studentsrecognized that their tilings and the grids were formed by two or three sets of parallel lines. From this they learnedto look for and trace other configurationsin the grids, in particular,saws and ladders and sets of congruentangles. Students needed varying amounts of experiencewith saws and laddersin differentpositions before feeling comfortable with them. Afterexplorationof two methodsfor constructingboth a saw and a ladder(i.e., first by parallel lines, second by congruent angles), the students were asked to describe what was done. None of the studentsin this group spontaneouslyused "if-then"language (level 2); rather,they tended to use a proceduralapproachand said, "well, you startedwith parallellines here and ended with congruentangles" and "the other was the reverse, startwith congruentangles and end with parallel lines." When the interviewerexpressed the relationshipsin "if-then"language,all the students agreed with the statements. When considering the statements, "if parallellines, then congruentangles"and "if congruentangles, then parallellines," several students (Jorge, Doreen, Elena) said they were "the same", showing an inability,as Dina van Hiele-Geldofdescribedit, to distinguishbetweenthe "Siamese twins"(statementand its converse). Five problemsin Activity 4 requiredstudentsto show thatparticularangles in a grid were congruentusing saw/ladderprinciples and the culminatingassessment task was equivalent to giving an informal proof of "opposite angles of a parallelogramare congruent."The purposeof this activity, done over two sessions, was to see if studentscould give or follow informaldeductivearguments(level 2). All studentscompletedproblem 1 (involving 1 ladder)and problem2 (involving 1 saw) correctly. The next threeproblemscould have been done in several ways (2

ladders, saw and ladder, . . . ) and students had considerable difficulty finding

appropriateladders/saws and determiningwhat strategy to use. Figure-ground problemsand orientationproblemswere particularlyevident. Madelineand Barbie

110

initially insistedon tracingin many possible ladders(saws) in the grid--makingthe diagramso complex that they became confused. They apparentlyfelt the need to "see"all the saws and laddersbefore looking for a strategythat would help them relate a particularpair of angles. Jorge, Elena and Beth also needed considerable guidance in planning and finding an appropriatestrategy. To justify their conclusionof congruentangles,studentsgave an informalstatementof the transitive property--Jorge'sanswer is typical: "AngleA = angle X by ladderand angle B = angle X by saw so they are all equal." When askedwhich angles were equal,Jorge and the otherstudentseitherrepeatedtheirstatementsor said "A = X = B." During the second session on this activity,the studentswere more successfulin identifying congruentangles via saws/laddersand tendedto name the two particularangles in their conclusions but still needed occasionalpromptingin their argumentsto help themkeep theirideas in focus. All the studentswere able to follow argumentsand alternativesolutions suggestedby the interviewerand also to repeator summarize the interviewer's arguments, although to Beth this was a struggle. After one interviewsession, Beth, a very serious student,confidedthat she neededextrahelp with saws and laddersas she did not fully graspthe ideas. This help was given. It became clear that solving problemsby reasoningand finding a strategywas a new conceptfor this studentwho thoughtof mathematicssolely as memorization.On the final assessmenttask in this activity, only Jorge was able to give independentlyan informal deductive argument to show that both pairs of opposite angles in a parallelogramare congruent; the other students were able to give parts of the argumentor to follow and then summarizethe interviewer'sargument. Activity 5 was designedto have the studentsdiscover why the angle sum of a triangle is 180?. The previous activities had given the students considerable experience in identifyingcongruentangles via saws/ladders. AlthoughJorge and Elenacorrectlyexplainedthe coloringof theirangles on the gridand explainedthat the sum of the threeangles thatformeda straightangle were "thesame as the three angles in the triangle,"they both were unconvincedthatthis angle sum relationship would always be truein every triangle. Afterrepeatingthe activity on at least two different triangular grids and finding the same results, Elena said it would "probablywork"on anothergridbut Jorgerefusedto generalize. When questioned on whetherhe thoughtthe angle sum for a trianglewas more or less than 180?,he said "probablyless or more"but did not seem to have any particularidea in mind. Interestingly,he always used the fact that the angle sum was 180? in solving the numericalproblemsin the activity; when asked why he could do that, he replied, "Theangle sum is 180 in this triangle."Jorgepersistedin his viewpointthroughout the remaininginterviews. Since the interviewersuspectedthathe mightbe thinking about spherical geometry or non-Euclidean geometry, several Project staff members(in an effort to gain insight into his thinking)discussed these angle sum ideas with him but these ideas were not meaningfulto him. He said he "justhad a feeling thatthe angle sum for some trianglewas not 180?";apparentlythe informal deductive argumenthe had given was not convincing to him. The remaining studentsgave initial argumentssimilarto those given by Jorge and Elena but then

111

they were convinced about the generality of their own arguments and readily showed it was trueon differentgridsandhence was truefor every triangle"because you could do the same thing." The students quickly suggested and explained why the angle sum for a square/rectangle was 360? (four right angles). The angle sum for both a parallelogramand a quadrilateralwas discovered (throughcoloring angles on a grid, tiling, or dividing into two triangles)by all students,but Madelinehad greater confidencein her conclusionwhen she measuredthe angles to find the sum thanby her informaldeductiveargument. Beth, when confronted with the possibility of dividing the quad into four trianglesand gettingan angle sum of 720? insteadof 360? said, "Perhapsboth could

be right . . . depends on the method you use." All the other students needed

guidanceto find the fallacy in the argumentthatthe angle sum of a quadis 720? by dividing it into four triangles. Except for Elena, all were able to follow and then summarizethe explanation.

In discoveringthe angle sum of a pentagon,the approachof Jorge, Madeline and Doreenwas to divide a pentagoninto threetrianglesobtainingan angle sum of 540?;Barthand Elena dividedit into a quadanda triangle,obtainingthe same sum (360 + 180); Beth andBarbieinitiallysuggested720? for the angle sum with Barbie dividing a pentagoninto two quads (360 + 360). Madelinewas totally convinced with her discovery and explanation,stating "anyfigure with five sides, the angle sum is 540?; you do not have to measure."(Herewe see a movementfromlevel 1to level 2 thinking.) The students readily placed the card for the angle sum for a pentagonin theirfamily trees (Activity 6-3) correctly. In finding the ancestorsto the angle sum for a triangle,studentsreviewedhow they found the angle sum. Barbieand Doreen immediatelyassembledtheirfamily trees correctlyand explainedthem well; Madelineand Jorge took a bit more time and needed only one promptto assemble and explain the tree correctly;Beth and Elenaneededmore guidanceto completethe task successfully. All studentsseemed interested in the-family tree approach to showing interrelationships. While informally discussing the idea of family trees with some of the students, the interviewer asked if saw and ladder have ancestors, Madeline and Barbie immediately suggested parallel lines and angles as ancestors; Jorge and Beth proposedparallellines. All the studentsagreedtheremustbe some beginningpoint in an ancestraltree. Discussion broadenedto consider the possibility of having family trees in otherareas of mathematics. Several studentsquickly proposedthat additionwas an ancestorto multiplicationand then countingand numberwould be ancestorsto addition. The fimalassessmenttask,the exteriorangle of a triangle(Activity7-3), for this modulewas done by four studentsin this group. In this activitystudentswere faced

112 with an open question of finding a possible relationship among the three angles indicated in the figure. The four students, after studying the figure, chose to measure the angles in each of three examples to see what they could discover. Jorge and Barbie automatically found the sum of angles a, b and c, "justto see what they add just curious." At first Madeline thought all the angles were equal but up to ... after her first set of measurements she noticed that angle c equaled the sum of angles a and b. She hypothesized that this relationship was true and checked it out in the next two examples. She verbalized her "theory"and said it was "positively" true in all triangles. When asked if she had to measure it in the next triangle, her "no"reply was immediate and emphatic. When asked for a logical argument, Madeline gave no response except for a big sigh. Several prompts were given such as: "Is any part of angle c related to angle a or angle b?" She provided several of the steps in the argument and then gave a review of the complete argument including the reasons for positioning the auxiliary line that was needed. In order to determine if the student understood the deductive argument given, a second problem involving the same principle but in a different orientation was presented. When Madeline was given the second example, she placed the auxiliary line correctly, carefully explaining how and why, and then proceeded to give a complete informal deductive argument (level 2) with no assistance. When asked to place her newly discovered principle in her family tree, she explained her correct placement well. She seemed pleased with her accomplishment and said, "I think I understand" and was now convinced that one could reach conclusions about angles "without measuring." Jorge, Beth and Barbie needed more guidance than Madeline in developing their informal arguments for the first problem. In the second problem, Beth gave a good argument with no assistance and then stated what principle she had established and explained how to place it in her family tree. Jorge, on the other hand, could provide only part of the explanation without guidance and then placed the exterior angle of a triangle card incorrectly (coming from the angle sum of a quad) in his family tree. Barbie, when given the second problem, had the impression that all the angles equaled 60? and the following dialogue ensued between the interviewer (I) and the student (S): I: S: I: S:

Does it matter what size the angles are? Yes, you can't be sure unless you know the measurement of them. You don't have any faith in it without measuring? Right. [to herself] Show these two angles put together equals this one. [She uses a D-stix with no particularstrategy in mind and no success.] I: [prompting] Think about what we did in the previous problem. S: Oh. [And then Barbie, without any assistance, placed a D-stix correctly and proceeded to give a good argument justifying her statements with properties of parallel lines, saws and ladders, congruent angles and concluded that "two opposite interior angles put together equals the exterior angle."]

113 I: S: I: S: I: S:

Are you convincedthatthis is truein every triangle? Yes. You don'thave to measure? [reluctantly]No, you don'thave to measure. Wouldyou like to measurethese? [brightly] I definitelywould!

Barbie happily measured and said: "just curious to see how much each angle measured." Barbie's extensive prior experience in arriving at conclusions by measuring overshadowed any real sense of present accomplishment in using informal deductive argumentsto arrive at new conclusions. However, her work with family trees--showingand explaininginterrelationships among ideas--showed quick comprehensionand clarity. Module 3. As noted previously, the intent of this module was to examine further the student'sthinking at level 1 (discovering area rules) and at level 2 (explainingarea rules and logically interrelatingthem) in a context differentfrom that of angle measurement. Because of the total time restrictionof six hours of clinical interviewsfor each subject,only five of the studentsin this groupcompleted all (Doreen,Barth,Elena)or part(Beth andBarbie)of this module. Activity 2, designedto assess a student'sconceptof area,begins with a problem of covering the top of a jewelry box with gold foil--one top is a square(5 x 5), and the otheris a rectangle(6 x 4). When asked which was larger, Doreen, Barthand Elenaall replied"theyarethe same"andjustifiedtheirresponsesas follows: Doreen:

Barth: Elena:

Each angle is 90?, so they should be the same.

[afterphysically overlayingthe shapes] One has a bigger width and the othera biggerlength. The areasareequalbecausethey areboth20.

Two commonmisunderstandings areevidentin the responsesof Doreenand Elena: (1) thinkingof areaas relatedto angle sums and (2) confusingareawith perimeter. Beth and Barbieboth said the squarewas largerand Beth replied:"Thediameterof the squareis bigger"while Barbie'sdecision was made on the basis of physically overlaying the shapes. Doreen and Elena describedarea of a figure as the "space inside,"while Barthsaid, "Thenumberof tiles representthe size." Beth thoughtof areaas "thewidth and length aroundit" and Barbiesaid, "Itis length times width, that's the formula for area." When questioned why that was the formula, she replied "you could add also 2L + 2W or count squares." (Note the perimeter interferenceagainin the responsesof Beth andBarbie.) All the studentshad learned the rule (length times width) for the area of a rectangleand could apply the rule correctlyin numericalexamples but none could explain why the rule works. This would be an instance of what Pierre van Hiele calls "reductionof level" (i.e., learninga principleby rote withoutunderstandingit).

114 The students were all quite successful at solving problems which involved rectangularfigures in differentconfigurationsor with pieces cut out. They usually could explain more than one way of solving the problems. In one problem, the studentswere askedto findhow muchgold foil they wouldneed to coverthe sides of a jewelry box (6 x 4 x 3); they had done the top of the box in the opening activity. Barth immediately took a ruler, measured the height of the box, found the perimeterof the base of the box, multipliedto find the resultand then gave a clear explanation of why his procedure was correct. Barth's solution was visually reinforcedfor him when he saw thatthejewelry box was so constructedthatit could be laid out as a networkandhe could see the height andperimeterof the base of the box as the widthandlengthof a rectanglein the network. To solve the problemof covering the sides of a jewelry box, Beth found the productof the length andheight, thenmultipliedby 4 and said, "4 x 18 is the area." The interviewer turned the box 90? on its base and asked the student to do the problemagain. This time she found "4 x 12 is the area"and seeing it was not the same as before said, "I'mtrying to recallsomething... square18... square12. ..." The interviewerguidedher to thinklogically aboutthe problemwith questions such as: Whatis the areaof the firstside?the secondside? She did thesecalculations on paperand said, "It's60 cubic squareinches ... not reallya cube ... 60 inches." The interviewer (I) asked the student (S) to reconsider the 4 x 3 side and the following dialogueensued: I: S: I: S: I:

Whatis the areaof this side? Areais 12 inches. Will a piece of string12 incheslong cover the side? Thatwould be too long. Whatwill cover it?

S:

Square inches ...

tiles.

I: Whenwouldwe use the ideaof cubicinches? S: Whenyou fill the box. The memorizedfacts relatingto perimeter,areaandvolume were ratherjumbledin Beth'sthinking. Barbie also had some difficulty with the jewelry box problem. Her solution was to "findthe amountof squareinches in the first side and find the amountof squareinches in the second side, then just multiplythem." When the interviewer seemed a bit puzzled, Barbiequickly revisedher answerto "findamountof inches in length and amount of inches in width and then just multiply them." The interviewerrestatedthe problemand Barbie said, "Oh,you mean volume." She arrivedat the correctsolutionafterbeing given help similarto thatgiven to Beth. Activity 4 was structuredto have studentsdiscoverthe areaof a righttriangle by finding a pattern. Only one of the students(Barbie)knew the rule. Students

115 were given four differentpairs of congruentrighttriangleswith which they formed squaresor rectangles. A fifth pair had the legs noted as b and h. For each pair, using a transparentgrid overlay to find the areas, the studentsthen recordedin a chartthe length of the base and height of the triangleand of the rectangleand the areasof the rectangleand triangle. The patternwas easily found by most students with Doreen commenting:"Areaof a righttriangleis always one-half of its square or rectangle." Beth also recognizedthe patternbut when she reachedthe general case in her chart where the area of the rectanglewas b x h, she said tentatively: "Areaof triangleis b ... but you still need the height." The interviewer,puzzled by this response,pursuedthe following line of questioning: I:

If we continuedthe chartand the area of a rectanglewas 50, what would the areaof the righttrianglebe? S: 25 I: And if the area of a rectanglewas 100, what would the area of the right trianglebe? S:

50

I:

And if the area of a rectanglewas p x q, what would the area of the right trianglebe?

S: p...

probably...

or possibly q.

Beth's inability to handle the general case and her conception of taking half of somethingby literallycuttingthe symbolic expressionin half vertically (i.e., p I q) was a bit startlingconsideringshe was just successfullycompletingone year'sstudy of elementaryalgebra. Students were given practice in finding areas of right triangles using their newly discovered principle. Initially several students (Doreen, Elena, Beth and Barbie)had difficulty measuringthe height of the triangle;they tendedto measure an adjacentside ratherthanthe height. When Doreen explainedthe rule, she said, "Measurethe bottom,then measurethe side of the triangle,multiplyand divide by two." Instructionwas given in the use of a L-squareto clarify and alleviate this difficulty. AlthoughBarbie knew the rule for finding the area of a right triangle, she also used the measureof an adjacentside as the height. When finding the areas of righttrianglesthe concept of perimetercontinuedto intrudefor several students (Beth, Barbie and Elena). At one point Barbie was asked the differencebetween areaand perimeterand she replied,"Areaincludes the sides and what is inside but perimeteris what is outside." The students were then asked to begin a family tree for area. Elena's explanationof her arrowbetween rectangleand righttrianglewas more procedural thanlogical: "Tofind this (righttriangle),I have to find this (rectangle)first."Beth explainedthe placementof her arrow(which was backwards):"A rectangleis half of whata righttriangleis." Askedto reconsiderher statement,she thoughtand then repeatedthe same statement.She was guidedto look at herpatternsheetandexplain

116 the pattern. After her explanation,she reversedher arrow. Before Beth and Barbie completed their last clinical interview, they were asked: "If we put two congruent right triangles together, will we always get a rectangle?"This questionwas posed in orderto assess the student'slevel of thinking in termsof the qualityof the explanationgiven. Barbiereplied, "By puttingthem together,you can just tell" (level 0). Beth responded,"If you put them together correctly,the opposite sides will be congruent"(level 1). When asked whathad to be trueto be a rectangle,both studentscited properties--oppositesides paralleland four right angles (level 1). In tryingto justify that the opposite sides are parallel, bothstudentssuggestedusing D-stix andplacingthemin such a way thatthey would not meet. When pressed to give a careful argumentto justify the parallel lines formedby placingthe two righttrianglestogether,both studentsneededguidanceto thinkof applyingsaw/ladderprinciples. In the explanation,Barbiekept reasoning fromthe converse,showingher inabilityto separatethe "Siamesetwins." (It should be noted that the studentswere given considerableexperience using "if lines are parallel,then alternate-interior angles are congruent"in the module tasks but they had only one brief experienceusing the converse. Until the studentshave greater experiencewith the converse, it is probablyunrealisticto expect them to "separate the twins.") Whenaskedto show thatthe angleswererightangles,Barbiemeasured with the transparentoverlay angle tester and said: "These two angles when put togethermake 90?--it'sobvious." The interviewerremarked,"Supposeit measured 89? or 91??" Barbiereplied,"Roundit off." The studentwas pressedto give a more carefulargumentusing the angle sum of a triangle. However, in the middle of her explanation, she again reverted to it "looking like" a right angle and used a protractorto measure it. With furtherguidance, she was able to establish right angles by means of a deductiveargumentbut the quality of her response showed mainly level 1 thinkingwith no consistent movement towardlevel 2. Beth was guided to see how one pair of angles formed a right angle and then was able to explain why the otherpair also formeda right angle. Basically Beth was able to follow argumentsand providepartsof argumentsbut she had not yet reachedthe stage of spontaneouslyor independentlythinkingof or initiatinginformaldeductive arguments. Doreen,Barthand Elena completedthe next threeactivitiesfor finding areaof a parallelogram,a triangleand a trapezoidwith varyingdegrees of success. Elena suggested three different approachesfor finding the area of a parallelogram-convertingit to a rectangle,dividingit into a rectangleand two righttriangles,and using an transparentoverlay grid. She had some difficulty describingand finding the height; the concept of the height makinga right angle with the base was a bit elusive. In explaining her family tree, she tended to do it from a time line frameworkratherthanby logical relationships(i.e., "Wedid this first, next we did this, and then we did this."). She easily developeda rulefor findingthe areaof any triangle,using two congruenttrianglesto forma parallelogramandnoted that"you would use b x h and divide in half and that would work for any triangle." Elena

117

thoughtof four differentways of finding the area of a trapezoid--divideit into two triangles, into a rectangle and two right triangles, into a parallelogram and a triangle, or build a parallelogramaroundit and subtractthe extra triangle. She successfully developed a rule for finding the area of a trapezoid (with little assistance) using her first method and simplified it algebraically. Then she immediately wanted to check out her new rule by a numerical example. She spontaneouslyput her new rule card in her family tree and explainedit well. She also made correctrevisions in her family tree when consideringthe possible use of her other approaches for finding the area of a trapezoid. The quality of her responsesin these activitiesindicatedgood progresstowardlevel 2 type of thinking. Doreenneeded some promptingwhen she persistedin thinkingof the height of a parallelogramand of a triangleas the lengthof a side adjacentto the base andalso when she arguedthat the area of a parallelogramwas 360?. She also thoughtof differentapproaches(much the same as Elena) for finding the area of a trapezoid and was successful in organizingand explainingher family tree. Her explanations varied from level 0 (e.g., "you know its a trapezoid by its shape, it looks like triangleshave been choppedoff'), to level 1 whereshe madejudgmentson the basis of propertiesof a class of figures. Barthclearly explainedhis idea of convertinga parallelograminto a rectanglein orderto find its area. He tendedto use the words "lengthand width"in talkingabout areasof parallelogramsand trianglesso he too measured an adjacent side as the height. Barth called trapezoids "half

parallelograms . . . well, not really half." He found the area of a trapezoid by the

method of two triangles and with a promptdeveloped a rule: "Takea ruler and measurebase andheightof one triangleandtakehalf, thenmeasurebase andheight of the othertriangleand take half, and add these together." AlthoughBarthwas a ninth grade student,he was not studying algebra,so his rule was not refined any further. Fromthe above analysisof the GroupV students'performanceson Modules 1, 2 and 3, it is apparentthat theirlanguageis richerand more fluent than thatof the studentsin GroupIV. In generalthey tendedto be at least level 1 thinkersdespite occasionallapses to level 0 type responsesandsome studentsin the groupexhibited progresstowardlevel 2 thinking. Barth,Elenaand Doreenbecamemore consistent in their level 1 thinking and clarified a numberof their misconceptions as they worked through the modules. Jorge's apparent perceptual difficulties (e.g., inability to consistently identify right angles in figures) along with his non-acceptanceof some generalprincipleshe developed (e.g., angle sum for any triangle)impededhis progress,allowingcompletionof only two modules;however, he seemed to have momentsof good insight. On the otherhand,Madeline,who also only completed two modules, worked slowly and thoughtfully filling in some conceptsat level 0 or 1, andmade clearstridestowardlevel 2 thinking. She moved froman almosttotal relianceon a visual approachor measuringto an understanding of the role of deductionin solving problems,thus makingmeasuringunnecessary. Beth and Barbie, while in the high ability group, performedmainly at level 1

118 althoughboth gave evidence of some progresstowardlevel 2 thinking. A strong subjectby both studentsseemed to opinionof mathematicsas a procedure-oriented impede their willingness to deal with different approaches to a problem or to explorenew mathematicalideas. Group VI As a result of the analysis of the videotapes,the Projectassigned seven ninth graders to Group VI (Alice, Carol, Kathy, Samantha, Linda, Mau and Ling). Performancesof these studentswere differentfrom those in GroupIV and Group V, in particular,most exhibitedmore consistentlevel 1 thinkingwith evidence of some level 2 thinking,andworkedmorerapidlyandconfidently. All seven students in this groupcompletedModules1, 2 and 3 with some Extensionsin the six hoursof clinical interviews. (While Kathy'sresponses in many instances, as will be seen below, would place her in GroupV, she workedmore rapidlythanthe studentsin Group V, consequently completed more activities, and towards the end she exhibitedlevel 1 thinkingmoreconsistentlywith evidence of some level 2 thinking. In orderto compareher responsesto those activities with other studentswho also did these extendedactivities,she has been includedin GroupVI.) See Table4 for a characterizationin a general way of each student's level of thinking on key activities. The responsesto the activitiesin the modulesof the studentsin this group areanalyzedbelow. Module 1. In responding to tasks in the opening activity, all the students showed that they were familiarwith basic geometry concepts. For the most part they used standardterminologyin describinggeometricfigures. While not always precise, fluency and spontaneityseemed to characterizethe languagethey used in speakingaboutgeometryideas. In describinga rectangle,Carol'sinitialanswerwas typical:"foursides, two are equal and the othertwo are equal, all sides parallel,all four right angles, closed." Linda'sdescriptionof parallellines was "twolines that nevermeet, are the same distanceapart--theyare like in a plane." Kathyspoke of a right angle as formed by "two lines--one going down, one going across and they meet to form an angle of 90?"while Alice said, "Itis 90?, is like an L or a comer of a square." The quickness and systematicapproachused by the studentsin this group to completethe first sort (sortingpolygons by numberof sides) and theirspontaneous introductionof the numberof sides to describetheirsortcontraststrikinglywith the performancesof some of the studentsin GroupIV and GroupV. Kathyand Carol were a bit concernedabouttwo concave shapes which they said had "threeangles and four sides and looked triangular."When asked to point out the angles in the concave shapes, Kathysaid (pointingto the reflex angles) "theseare funny angles" and immediatelydecided the shapes did belong in the quadrilateralset. Ling also wondered momentarily whether the sides forming the reflex angle should be thought of as "one side--one bent side" or two sides; he decided on two sides.

119

Again, in the second sort (sortinga set of quadnlaterals),the studentsworked quickly and systematically. Ling began by sorting into two sets: "figures with parallel sides and those with non-parallelsides." Then, he changed to three sets: "figures with right angles, parallelogramsand figures with no parallel sides." Finally, he arrivedat four sets: "square,rectangle,parallelogramand figures with no parallel sides." Alice, Carol and Linda also sorted into four sets of "squares, rectangles, parallelogramsand others"with Carol thinking of parallelogramsas "rectangleswhich are lopsided,not perfect." Mau,Kathyand Samanthasortedinto five sets: "squares, rectangles, parallelograms, trapezoids and others" and as Samanthadid the sort she spontaneouslyasked,"Areall squaresrectanglesor are all rectanglessquares?"It was suggestedthatshe mightbe able to answerthatquestion afterdoing some of the upcomingactivities. All the studentsreadilylisted propertiesof the shapes(Activity 3-3) in each of their piles. Upon noting a new propertyfor one pile, they tended to check their otherpiles quicklyto see if the propertyapplied. Few promptswere needed. Kathy tended to say "notslanty"ratherthannoting right angles for a rectangle. The fluency of the students'responsesindicatedthatthey were analyzingfiguresin termsof theircomponentsandrelationshipsbetweencomponentsandestablishingproperties for a class of figures. Samantha,on examiningher list of propertiesfor squaresand rectangles, thought back to her question and spontaneouslyconcluded: "Oh, all squaresare rectanglesbecausethey have all the propertiesof a rectangleand all the sides areequal." She also explainedclearlywhy all rectanglescould not be squares. When the studentswere asked in Activity 3-4 (InclusionRelations)if a square could be placed in the rectanglepile, Alice, Ling andLindaimmediatelysaid: "No, because a rectanglehas two longer and two shortersides and a squarehas all sides equal." They were reasoning correctly from their incorrect,previously learned, definitionof a rectangle. When asked if the squarecould go in the rectanglepile if only the propertieslisted were considered,they immediatelyresponded:"Yes, but the rectangle could not go in the squarepile." Linda also arguedthat "a square could not go in the parallelogrampile because the sides are not slanty and a parallelogramdoes not have right angles." This reasoning was consistent with Linda's unlisted property for a parallelogram--"itis slanty, that is, has oblique angles." (Although the description and propertiesof a parallelogramhad been discussed with Lindapreviouslyand she had acceptedthe idea that rectanglesand squares were types of parallelogramsand had explained why, she nevertheless continuedto think of parallelogram,rectangleand squareas disjointsets. Perhaps the quadrilateralsort done in Activity 3-2 in some way reinforcedthis idea for her.) It is interestingto note Linda'slanguage--sheconsistentlyused the word "because" in her explanationsof subclassrelations,she tendedto use quantifierssuch as "all" and "some",and she explainedher ideas with statementssuch as "asquarehas four sides, that'swhy it's a quadrilateral."Otherstudentsin this group also tended to spontaneouslyjustify theirresponses. Mauwas particularlythoughtfulandfluentin his explanationof subclassrelationssaying,for example,"Aparallelogramis a quad

120 because it has four sides, is not a trapezoidbecause they have only one pair of parallelsides, is not a rectanglebecause they need rightangles, and is not a square because they need rightangles and all sides equal,but you could put a rectanglein the parallelogrampile because... [herehe listed all the properties]." The Uncovering Shapes and Uncovering Clues activities (Activity 3-6, 3-7) were easily done by the students who referredto propertiesof figures to justify their conclusions. Mau gave very quick logical answers,citing precisely the right propertythatwas needed to acceptor refutea given figure'sinclusionin a category. Linda was also quick to respond to each clue saying, for example, "Squareand rectangleare possible because of the rightangles, but now squareis eliminated,all sides arenot going to be equalsince one side is alreadylongerthananotherside." In decidingon the minimumclues (Activity3-8) neededto describea rectangle, Alice selected "foursides, all angles are rightangles, opposite sides are congruent" and ruledthe otherclues out by saying, "Ifit has four sides, then it has four angles; if all angles are rightangles, their oppositeangles are equal;and if opposites sides are congruent, then they are parallel." Carol selected three clues to describe a square--"fourangles, all angles are right angles, all sides are congruent"and said, pointing to the remainingthree possible clues, "theseare not necessary, they just come along--if you have four angles, you have to have four sides; if all angles are right angles, then opposite angles are congruent;if all sides are congruent,then opposite sides are congruent." (Note the "if-then"language of both Alice and Carol.) Carol'skeen reasoningalso was evident in her question: "Don'twe need anotherclue thatsays the figureis closed or is thatjust assumed?" As she pondered her own question and drew the sketch at the right, she decidedto add "foursides"as a necessaryclue if yourdon't assume the figure is closed. Mau, in selecting minimum clues to describea squaresaid, "Foursides, all angles are rightangles, all sides congruent."When asked if the clue "foursides" could be omitted, he immediatelysaid "No" and drew the figureat the rightto justify his conclusion. Ling and Samanthadid the minimumpropertiesactivitylater(afterModule2). They were asked, "Whatwould be the minimumpropertieswe would need to give to a friend to be sure he/she would know we were thinkingof a parallelogram?" Both studentsreplied,"Foursides, opposite sides parallel."They were then asked, "If we only use your two properties,would we be able to show that all the other propertiesof a parallelogramare true?"The studentsthoughtthis was possible and easily showed that the opposite angles of a parallelogramwere congruentusing a saw/ladderargument. Then by drawingin a diagonal, they establishedwith very little guidancethat the triangleswere congruentand so concludedthat the opposite sides of a parallelogramare congruent. In reflecting on what he had done, Ling asked, "So all we need in orderto know that a quad is a parallelogramis that the opposite sides are parallel?" The interviewer replied, "Yes, does that seem

121 reasonableto you?" Ling said, "No,I thinkthereare some exceptions." To explore his doubts, Ling made several attempts to make careful drawings, even measuringthe sides. Although he was unsuccessfulin making a drawing which was a counterexample,he

remainedunconvinced,still thinking that it was

/

i , possible for the sides to be unequal if you only start with the fact that the sides areparallel. Herewe have an interestingexample of a studentwho was able to give a clear cogent deductive argumentto establish a fact but who remainedunconvincedof the truthof the fact he had established. Ling and Samanthawere then asked to give the minimum propertiesneeded to be sure a quadrilateralwas a rectangle. Their responseswere similar:"Isa quad ... a parallelogramwith four rightangles." To explorewhether four right angles were needed, they used a D-stix model of a parallelogramand focused on one angle as they moved the model to differentpositions. Ling noted that as one angle becamea rightangle, the othersdid did too ("they are all attached").With a few prompts, B / the studentsargued,"If A is a right angle, then B is a / rightangle [oppositeangles of a parallelogramare con/ / gruent],but A + B = 180?,so C + D = 180 [anglesum in a quad is 360?], so C = 90? and D = 90? [oppositeangles of a parallelogramare congruent]." Samantha therefore concluded that it was enough to say that "a rectangleis a parallelogramwith one rightangle." Ling did not thinkthatone right angle was enoughdespitehis logical argument.He continuedto thinkthatone could somehow have both pairs of opposite sides paralleland still have the bottom side shorterthan the top (referringback to his drawings)and so he concludedthat one would need more than one right angle. In spite of efforts to help him understand what he had achieved by his argument,he remainedunimpressedandunconvinced of the truthof the statementhe had establishedthrougha logical argument. In the last assessment activity (Kites) in this module, the students'responses basically referredto propertieswhen asked to describe a class of shapes (kites). Carolsaid, "Oppositeangles are congruent[one pair],adjacentsides congruent[two pairs], a diagonalgives two congruenttriangles,[foldinga papercut-out]diagonals meet inside and are perpendicularlines." Alice stated,"Itlooks like a kite, has one pair of congruentangles, two longer congruentsides and two shortercongruent sides." Whenaskedaboutthe rhombus(whichwas in herpile of kites), she removed it since it did not fit her description("twolongerandtwo shortercongruentsides"). When askedif a squarewas a kite, she thoughtthatwas possible andput it in the kite pile. She then consideredthe rhombus,put it back in the kite pile, and changedher description to "four sides, one pair of opposite angles congruent, two pairs of adjacentsides congruent."Kathyalso did some paperfoldingof the model andnoted that "congruenttriangles are formed by a diagonal and some kites have parallel lines." She also said that "a squareis a kite and it could be three dimensional"-probablythinking of a "realworld"kite. Linda'sresponse to this kite activity in terms of symmetryis of interest:"Itis diamondshaped,has two pairs of congruent

122 sides so it will fly more evenly in the sky--it'sbalanced."Lindasaid, "A squareis a non-kite." Her basic argumentwas "it wouldn'tfly well--not pointed enough." Hence her final descriptionof a kite was "foursides, two pairs of adjacentsides congruent,cannotbe a solid shape, it must be pointed enough to flow in the air." Her strong perception of a "flying"kite made it difficult for Linda to accept the possibilityof consideringa squareas a kite. The above sample responses indicate that some students in this group were makingprogresstowardlevel 2 thinkinginsofaras they were able to identifysets of properties to characterizea class of shapes (level descriptor 2-la). They also quickly and correctly assembled their "is a special" arrow diagramsto show the interrelationshipsof square, kite and quadrilateral. Mau and Kathy went even further;they explained and showed the interrelationsof all the members of the quadrilateralfamily (see diagrambelow). Parallelogram -Trapezoid --

Rectangle --

Quadrilateral --

Square

Kite

Module 2. Group VI students were all quite proficient in estimating and measuringthe size of angles (manychoosing to use a protractor).All knew thatthe angle sum for a trianglewas 180? (having leared it in seventh/eighthgrades)but none could explain this angle sum fact. Carolthoughtit had somethingto do with "beingpart of a circle" while Ling said he had "triedit for a few triangles and it works." Mau confided thathe had "alwayswonderedwhy it was true"but did not know why. For the most part, the students had no difficulty doing the various tilings, recognizing sets of parallellines and congruentangles in their tilings, identifying some saws/laddersin grids and describingthem in termsof theirproperties. Kathy needed extended experience in identifying and recognizing saws and ladders in various positions in grids and then had major difficulties in finding congruent angles in the grids. There appearedto be few figure-groundor orientationproblems for the other studentsin this group. Alice had such keen visual memorythat after studying the outline of a saw or ladder on a transparentoverlay, she could immediatelyplace it correctlyon a grid. Some of these students seemed able to separate "the Siamese twins" (statementand converse). Alice, Mau, Linda and Samanthaused "if-then"language. For example, Alice commented, "If you start with parallellines, then you end with congruentangles and [pointing]if you start with congruentangles, then you end with parallellines--andthese are different." In Activities 4-4 and 4-5, the studentswere requiredto give logical arguments using saws/laddersto show why certainangles were congruent. Each of the first two problemsrequiresonly one saw or one ladderand posed no difficulty for the

123 students. The interviewerexplainedthatsometimesit is necessaryto find a "missing link"when trying to show that two angles are equal, and the interviewerreviewed the use of the transitiveproperty(which the students had all learned in school). None of the studentsindependentlysolved the thirdproblem(requiringtwo saws, ladder and saw, or two ladders), but all were able to follow and summarizethe interviewer'sexplanation (model solution) and in some instances to suggest an alternate method for solving the problem. Alice gave precise arguments (identifying congruent angles via saws/ladders and then using the transitive property)in the next two problemsandalso in the final assessmentactivity,showing thatboth pairsof oppositeangles in a parallelogramare congruent. In the courseof a discussion,Ling thoughtthatthe sum of a pairof consecutive interior angles of a parallelogramwas 180?. He was encouragedto try to establishit logically. The interviewersuggestedadding an "x" to the adjoining diagram. After a thoughtful moment,Ling said, "B + X =180? by straightangle, / A = X by a ladder, so B + A = 1800." He concluded that

the same argumentwould hold going all aroundthe angles of the parallelogram. First by coloring in congruentangles in a grid via saws/ladders(Activity 5-3) and thenby informallogical arguments,the studentsbeganto explainwhy the angle sum for a trianglewas 180?. InitiallyAlice arguedin reverse,statingthat"theangle sum of a triangleis 180? and thereare two of each of the triangle'sangles at each vertex of the triangulargrid, so the sum of the six angles about a point is 3600." With a little prompting, she recognized a straight angle in her grid and then explained why the three angles of the triangleequaledthe three angles makingup the straightangle and so the sum was 180?. MauandKathyneededsimilarprompts. Lindaand Samanthagave similarargumentsnotingthe straightangle andjustifying theiruse of saws/laddersby pointingout the sets of parallellines in the grid. All the studentsarguedthat the angle sum for squares/rectangleswas 360? on the basis of four right angles. Alice and Caroldiscoveredthat the angle sum for a parallelogramand a quadrilateralwas also 360?, justifying their decisions on the basis of coloring angles in a parallelogramgrid and tiling with quadrilateralsand then later explaining their result on the basis of dividing the figures into two triangles(180? + 180?). Mau,Ling and Samanthaimmediatelyexplainedthe angle sum for a parallelogramand for any quadrilateralon the basis of subdivisioninto two triangles. Samanthaand Ling wonderedaloud whether the angle sum for a concave quadrilateralwas also 360?. They explored this idea, justified their conclusionby drawingin a diagonalto createtwo triangles,and seemedpleasedthat the same approachworked. Kathyand Lindainitially said they did not know what the angle sum of a parallelogrammightbe. "Doesit have to be a set amount?"asked Linda. With guidance,coloring and tiling, they arrivedat a correctconclusionand were able to justify theirresult.

124 When confronted with the possibility of dividing a quadrilateralinto four triangles and obtaining720? as a possible angle sum, Alice, Mau, Samanthaand Carol(aftersome thought)pointedout the "extraangles being countedin the center, so you must subtract 360?."

The angle sum of 540? for a pentagon was discovered and justified easily

throughsubdividing the pentagon into three triangles or a quadrilateraland one triangleby all the studentsin this groupexcept Kathy,who said the angle sum "is greaterthan360?--maybe4000." With a promptto thinkaboutwhat we did to find the angle sum in a quadrilateral,Kathyquickly subdividedthe pentagoninto three trianglesandnoted thatthis approachwould work in all pentagons. Alice and Ling both asked what the angle sum of a hexagon would be. They were encouragedto explore this idea and they quickly explainedhow every hexagon could be divided into a quadrilateraland two trianglesso the sum would be 720?. It is interestingto note that many of the students in this group spontaneouslywondered about and wantedto exploreextensionsof the ideas with whichthey were working. Activity 6-4 (FamilyTrees) was designedto assess whetherthe studentscould developedin this module (i.e., saws/ladders, logically explain the interrelationships straight angle, angle sum of polygons, congruence of opposite angles in a parallelogram). All the students in this group built, without any false starts, a family tree for angle sums of polygons and explainedthe ancestorsfrom a logical standpoint(level 2) and not from a proceduralor time-line perspective. Therewas a brief hesitation on their part as they wondered where "opposite angles of a parallelogramare congruent"would fit in their family trees. Some needed the prompt, "How did you establish that this was true?" Then, almost instantly recognizingthe ancestors,they placed the statement in theirfamily trees correctly. Activity 7-3 (Exterior Angle of a Triangle), one of the possible extension assessmentactivities,was done by threestudentsin this group. Alice did not at first see any relationship among the three marked angles (exterior angle and two non-adjacentinteriorangles), but after measuringthese angles in three different triangles, she noticed a pattern:"the two interior angles add up to the exterior angle." In attemptingto give a logical explanationfor her theory, Alice needed a prompt,"Wouldit be possible to divideup the exteriorangle in some way?" Almost immediatelyshe recognizedthe possibilityof a saw, drewin a line throughthe vertex of the exteriorangle parallelto one side of the tri, angle and said, "A = B by a saw. E = F by a ladder. Since A = B, E = F and exterior angle = B + F, then exteriorangle = A + E." She also gave a similarargument for the second problemin this activity,explaining carefullyhow she was placingthe auxilaryline parallel /E to one side in orderto forma saw andladder. Kathy'sresponseto the threemarkedangles was "mustbe the same ... no, the

125 exterior angle is probablybigger ... looks like these two [pointing to the two interiorangles] would fit in the exteriorangle." She checkedout her hypothesison three cases and when she found it worked,she exclaimed, "Oh,I got it right! I got it!" Kathyneeded guidanceto put togethera logical argumentbut she had a clear view of her aim and gave some of the steps in the argument(e.g., noting the ladder and congruentangles). Afterwardsshe was able to summarizewhat was done and restateher argumentcoherently;she was makingprogresstowardslevel 2 thinking. Linda'sinitial responseto the threemarkedangles was "theexteriorangle and the interiorangle are the same"but this was quicklyrevisedto "theexteriorangle is the same as the two interioranglestogetherbecauseif you subtractthemfrom 180?, they are the same"(an insightfulalgebraicsolution). Then Lindathoughtshe might explain it by saws/ladders;her aim was to get the two interiorangles equal to two parts of the exterior angle but she needed one or two promptsto find the correct ladder. After completingher argumentandrestatingher conclusion,she was asked, Is it always true? Linda: "Yes--probably always." Interviewer: "We didn't measure, Would you have more faith in it if you measured?"Linda:"No." In the second problem,Linda identifiedthe exteriorangle and two non-adjacentinterior angles, turnedthe paperfor orientation,thoughtof a saw, drew in an auxiliaryline (statinghow and why) and proceededto give a clear logical argumentwithoutany assistance. The above instancesindicatethat these studentsare exhibiting level 2 behaviorsin justifyingnewly discoveredpropertiesby deduction. Furtherevidence of their level 2 thinkingis shown in theircorrectplacementof the new principlein their previously constructedfamily trees and their thoughtfullogical explanations of the relationshipof the new principle(exteriorangle of a triangle)to some of the otherideas shown in theirtrees. Module 3. In the introductoryTangramactivity, Alice and Linda explained that the parallelogram, rectangle and trapezoid (all made from the squareand two small right triangles) were equal in area because "all quads are equal, all have 360?." After some \ / discussion, both studentssaid, "theytake up the same amountof space"and then recognizedthatareadid not relate to angle measure. Kathy, on the other hand, insisted, after comparingthe rectangleand the trapezoid,that the trapezoid"takes up more space because it's longer." The interviewer(I) put two righttriangleTangrampieces togetherin differentconfigurationsandthe following dialogueensued. I:

Are you saying thatthe two trianglesin figureA take up more space thanthe two trianglesin figureB? S: Yes, they arecoming to a point in A andfit together inB. I: Could I move the pieces of the trapezoidso they would fit on the rectangle?

S: Neatly?

FigureA

126 I: Yes. S: No, no way. I: [demonstrates-- moving the left triangle over and fittingit on the rightside] Are they the same? S:

I guess so.

I: You are not sure? S: No, not sure.

Figu B

The interviewertriedhardto have the studentsee they were equivalentin area,but to no avail. The studentwas not convinced,since "byeye" one seems bigger and she said, "butone is longer." It would appearthat the studenthad some perceptual difficulties or had a problemwith conservationof area. The other studentshad no difficultywith this introductoryactivity. In comparingthe sizes of the tops of two jewelry boxes (Activity 2-1), Mau, Ling, and Carolthoughtthey were the same size. At first Alice and Kathythought the rectanglewas bigger ("it'slonger,"said Kathy)and then decidedthe squarewas bigger. From the outset Linda and Samanthathought the square was bigger. Decisions were made by overlayingthe figures,using a transparentoverlay grid or using a ruler. Responsesto tasks assessing the student'sknowledgeof the areaof a rectangleindicatedthatAlice, Kathy,Ling andLindaknew the rule (lengthx width) and could use it in solving problemsbut could not explain why it worked. Typical responseswere "that'sthe principle"(Ling) and "it'sa rule"(Linda). Whenpressed for some explanation of the rule, most said, "well, you could count" and Kathy added,"oryou could add the four sides." In this task all the studentsdescribedthe area of a figure as the "numberof squaresinside." When the alternateform of the rule "lengthx width"was introducedas "basex height,"Ling said, "Ithought'base' and 'height'were only used with triangles." Mau, Samanthaand Carol easily explainedwhy the area of a rectanglerule works. Samanthasaid (abouta 4 x 6 rectangle), "You'rereally addingfour sixes, multiplicationis a shortcutfor addition." Finding the area of the outside of the jewelry box did not present major problems for most of the students;Alice's solution was typical: "Findthe area of two sides and multiplyby two." Linda'sfirst responsewas "add2L and 2W." The problem was restatedand she said, "Oh,you want the area ... [mumbledabout perimeterand 'perimeteris area outside'] . . . add area and perimeter,something like that, I guess." The interviewersaid, "Let'sfind the area of this face." Linda quickly used a ruler,found the areaof each face correctly,then found the sum and announced,"Thearea is 72 squareinches." In spite of several discussions on the meaning of the units, Kathy tended to give her answers to area tasks in "inches." However, when prompted(e.g., "Do you mean inches?"or "Whyis the answerin inches?"),she correctedher answersto "squareinches." It is interestingto note that uncertainty about area units and confusion of area with angle sum and with perimeterwas also apparentin the thinkingof some of the studentsin this groupas it was with studentsin GroupV.

127 In Activity 4 (Right Triangle), all the students knew a rule (1/2 b x h) for finding the areaof a righttrianglebut some needed to complete a set of sequenced tasks to help them discover why the rule was true. Carol gained insight into the meaningof the rule when she discovereda patternwhen constructingandmeasuring four rectangles made with four differentpairs of congruenttriangles. She said, "Thetriangle is half of a rectangleor square;since two congruentright triangles form a rectangle,the area of the trianglehas to be half the area of the rectangle." Linda and Kathy both said, "the right triangle is half a square or rectangle"and proceededto explain how to find the areaof the triangle:"Youfind half the length andhalf the widthandmultiplythem." This was checkedout numericallyin several examples using a overlay grid and found to be incorrect. The two studentsneeded guidanceto overcomethis misconception. All the studentsbegan theirareafamily treescorrectlyand Mau'sexplanationwas typical:"thisthought[pointingto the area of a right triangle card] came from this one [pointing to the area of a rectangle card]"and "since two congruentright triangles form a rectangle, the area of the trianglewas half the rectangle." Only Mau and Samanthaknew rules for finding the area of a parallelogram (Activity 5) and the area of any triangle (Activity 6) and could explain the rules (giving informal deductive arguments). Samantha,using a grid overlay, said, "Changethe parallelograminto a rectangle by cutting off a right triangle and moving it to the other end; then the parallelogramand rectangleare equal, so the area is base times height." They also explainedthe relationshipsof the parallelogram and triangleto otherpartsof theirfamily tree diagrams. Lindasaid the area of a parallelogramwas b x h but used the rule as the productof two adjacentsides. Since the other studentsin this groupindicatedthatthey did not know how to find the area of a parallelogram,instructionwas given on the meaning of "base"and "height"as relatedto a parallelogramandany triangle,andthe studentslearnedhow to use an L-squarefor finding the measureof the height. Lindathen said, "Oh,so move the trianglecut off by the heightto the otherside andthe size of this rectangle is the same as the parallelogrambut the shape is different." The other students exploredvariousways of findingthe areaof a parallelogram.Alice used a grid and after several numericalexamples hypothesized the area was " also b x h, like a rectangle." Then she showed how to converta parallelograminto a rectangle(by cutting off a triangleon the left and moving it to the right side.) Carol initially thought of cutting the parallelograminto a rectangleand two right triangles and finding the areas of the threepieces. Ling said, after a moment'sthought,"Ihave two ways of doing it--builda rectanglearoundit and then subtractthe two triangles or divide the parallelogram into two triangles" (here he was using his prior knowledge for finding the areaof any triangle). The studentshad no difficulty in addingthe parallelogramto theirarea family trees and explaining how it was related to the rectangle and/or right triangle dependingon the method(s) they had used. All the studentswere creativein their approaches to discovering the area of any triangle: "divide it into two right

128 triangles,""takea duplicatetriangleand form a parallelogram,""builda rectangle aroundit and subtractthe areas of the other two triangles,""use a grid." Most students suggested two or three different ways and showed that their methods worked in acute and obtuse triangles. Kathy,having divided the triangleinto two righttriangles,said, "areaof the firsttriangleis 1/2 b x h andthe second is 1/2 b x h, so the whole triangleis b x h." Guidancewas necessary to help her see that her resultwas incorrectand to assist her in derivingthe correctresult. The placement of the areaof a trianglecardin the family trees was correctlydone andjustifiedby all the studentsexcept Linda whose explanationwas: "A triangle equals 180?, a rectangleor parallelogramequals 360?, thereforethe triangleis half." (Note how angle sums again reappearin an areaargumenteven thoughLindahadjust finished giving a logical explanation relating the area of any triangle to the area of a rectanglebuiltaroundit.) The studentswere askedto explainhow they could be sure thata parallelogram was formed when they put two congruenttrianglestogether. Linda describedthe positioning of the triangles "so the edges are not going to come together... the sides are parallel." Asked how one might explain thatthe sides were parallel,she spontaneouslysaid, "makea saw"andpointedto two equal angles. With a prompt she recalled, "needset of parallellines, then equal angles and if equal angles, then parallellines." The following dialogue ensued when the interviewer,tracingher line of reasoning,asked:how do we know these angles [pointing]areequal? S: I: S: I:

Becauseyou have parallellines. But you aretryingto show these lines areparallel. [laughs]If you flip it over... [promptsby moving--notflipping--onetriangleon top of the other]What can you say aboutthe angles? S: They are equal. I: Why? S: Because the trianglesare congruent. I: So now whatdoes thatshow? S: The lines areparallelby a saw. Lindawas then asked to explainwhy the otherpairof sides were parallel. She had difficulty selecting the correct angles to show the sides were parallel; once they were identified,she againengagedin circularreasoning. However,she was able to follow the logical argumentof the interviewerand to provide some of the needed steps along the way. With guidance,she began to see the differencebetween the statementand its converse, recognizedwhen each shouldbe used and then gave a complete correctsummaryof the argument. Asked if it will always be truethat if two congruent triangles are placed together the resulting figure will be a parallelogram, she replied immediately, "well, if they are properly placed." Carol'sdifficultieswith this problemwere very similarto Linda's. Alice, Ling and Samantha,in doing the same problem,recognizedthat they had to show the sides

129 were parallel. They needed only the prompt,"Howdo we show lines are parallel?," to make themthinkof a saw or ladderby congruentangles, andthen they gave good explanations. Following this, they easily gave logical argumentsto show the other pair of opposite sides were parallel. Evidently these three students had little difficulty separating the "twins" and could use the statement or converse in appropriatesituations. Althoughnone of the studentsknew how to find the area of a trapezoidfrom previous experience, they all were spontaneous in describing different possible approaches(Activity 7) for trying to discover a rule. Initially they all suggested subdividinga trapezoidinto a rectangleandtwo righttriangles;then they thoughtof subdividingit into two triangles,or into a parallelogramand a triangle. Some also considered building a rectangle around the trapezoid and subtracting the two triangles. The studentsexplored the method of subdivision into two triangles in detail and discoveredandexplaineda rulefor findingthe areaof a trapezoid. Most of the studentsreadilyexplainedwhy the heights of the trianglesand the trapezoid were equivalent: "theparallellines are the same distanceapart."All were firstyear algebra students, so when they arrivedat the rule 1/2 b x h + 1/2 b' x h , some rewroteit in factoredform spontaneouslywhile othersneeded a prompt. Alice and Mau immediatelywanted confirmationof their newly discovered rule by using a numericalexampleto check it out. The studentsexplainedtheirplacementsof the areaof a trapezoidcardin their family trees clearly and made correctadjustmentsof the arrowsas they explained possible altemateapproachesfor findingthe rule. Kathygave good explanations;it appearedthat she had begun to be more comfortablewith the idea of a family tree and she respondedmore fluently,explainingwhy cardswere put into the family tree a certainway and discussing altemateways. As the studentsworked throughthe activities in this module, they exhibited many level 2 behaviors--givinginformal arguments,using insightfulreasoningto solve problems,following deductivearguments and supplyingpartsof the argument,and giving more thanone correctexplanationor approachto a problemandthen relatingthese approachesin a family tree. All the students except Alice (who had already completed six hours of interviews, including Activity 9 which will be discussed later) did an extended activity (Activity 8) to Module 3 in which they endeavored to discover a more general area rule for figures which have all of their vertices on two parallellines. Introductorytasks to this activity developed ideas of similartrianglesand the ratio of their correspondingsides and also the notion of a midline. The studentsthen spontaneouslyused "ladder"argumentsto explain why a trianglecut off by a line segment joining the midpoints of two sides of a triangle is similar to the given triangleand recognizedthatthis was true for any triangle. On the basis of similar triangles, students discovered and explained that the midline of a triangle was one half the lengthof the base. As a consequence

130 of this discovery,the students went on to explainthatthe areaof a trianglecouldbe thoughtof as midlinex height. All the students needed some prompts in discovering the lengths of the midline of a trapezoidin relationto the bases. Initiallysome guessedthe midline was one-half the bottom base or tried to think about similar trapezoids. But with the prompt,"let'sput in a diagonal of the trapezoidand look at the pieces of the midline," all the students quickly saw two sets of similartrianglesand discoveredthe relationof the midline to the bases. Ling, Lindaand Samanthaapproachedthe problemthroughnumerical examples first and then gave a more general explanationof why the median of a trapezoidwas one-half the sum of the bases. In startingher explanation,Linda questioned,"ShouldI tell you the reasonwhy?" She had begun to thinkof the need for justifying each step; she proceeded to give a very careful explanation. After theirdiscoveries,the students,rememberingthe aim of this activity, spontaneously rewrotethe rule for the areaof a trapezoidas midlinex height. Samanthaexpressed surprisethat the rule was the same as the one for the triangle. Mau was delighted with his result but added, pointing to the rectangle and parallelogram,"it'snot going to be midline x height for them." Note the quality of thinkingdone by this student--alwaysthinkingahead,questioningwhetheran idea can be extended. When asked aboutthe length of the midline of a rectangle,Mau instantlyhad insight into the whole problem and exclaimed (in surprise):"Oh, it is midline x height ... midlineto base is 1:1." The otherstudentsneededa little moretime andexploration to arrive at the same result. The studentswere asked if their new rule could be applied to a given kite. Answers were similar;for example, Samanthasaid, "No, becausethe comers arenot on parallellines." One of the culminatingtasksin this activitywas to assemblea family tree of all the ideas needed to deduce the area rule of midline x height. On the whole, the students assembled and explained (with little or no guidance) most of the interrelationshipsshown in their trees correctly. The placement of the similar triangles principle as an ancestorwas a bit troublesomefor some of the students untilthey were promptedto reanalyzehow they had establishedthatthe midlineof a trianglewas half the base. At thatpoint, Linda interjected,"Andyou also need a ladder as an ancestor to this." She was very thoughtful about constructingher family tree andgave a preciseexplanationfor each of the interrelationships. Since Kathy had made a few false startsin earlieractivities but now seemed more comfortablewith manyof the ideas, her responsesto anothertask, designedto assess furtherthe level of her thinking,were examinedcarefully. She was askedto explain why the opposite sides of a quadrilateralare congruent, given that the oppositesides (bothpairs)areparallel. Kathyspontaneouslydrewin a diagonaland the following dialogueensued:

131 S: The two trianglesare congruent. 7 I: Why? S: We could measure... no ... with two saws, angle a = angle b, angle c = angle d, and the thirdangles are equal by what'sleft from 180? b ... so the trianglesare similar. I: [prompting]The trianglesare not congruentyet... thinkaboutthe sides. S: [tries folding one triangle over the diagonal to see if the sides are congruent] Oh, the ratioof the sides [pointingto the commonside] is 1:1, so the ratioof the othersides is 1:1, so the sides arecongruent. The above argumentalong with the family tree (shown below) that Kathy then assembled (and carefully explained) for this problem shows evidence of level 2 thinking. sidesof a quadareparallel | lopposite [simrnilar triangles

|

| [oppositesidesof a quadarecongruent Linda and Mau did the same problem but experienced some difficulties. Both needed a prompt(suggestion of drawinga diagonal) but then Linda immediately gave a good argumentto show why the triangleswere similar;however, she had difficulty explaining that the ratio of the sides was 1:1 and so needed guidanceto show that the sides were congruent. Mau needed guidanceto complete the logical argumentsince he wanted to cut out the shapes and fit them together in orderto show congruence. Doing the same problem, Ling and Samanthaimmediately thoughtof drawingin a diagonal as Kathyhad done and then gave careful logical explanationswith no assistance. At this point, Mau and Carol had completed their six hours of interviews. Alice, Kathy, Ling, Samantha and Linda were able to complete the last task (Activity 9) which was to compare several of the family trees which they had assembled (e.g., propertiesof a parallelogram,angle sums of polygons, areas of triangles and quads, areas of figures with all vertices on two parallel lines). In examining her family trees for ancestors,Alice noted that saws and ladderswere needed in orderto deducethe congruenceof oppositeangles of a parallelogram,the angle sum for polygons, the exterior angle of a triangle relationship. She also recalled using saws to show that two congruenttrianglesform a parallelogramin obtainingthe areaof a trianglerule;here she was showing insight into how several of her family trees had the same ancestors. To the question"Wouldsaw and ladder have ancestors?,"she immediatelyreplied,"parallellines and angles." Tracingback

132 furthershe said, "Iguess they have ancestorstoo, but I don'tknow what they are." She agreed there had to be some beginning. In noting that saw/ladder was a common ancestorin several trees, Kathy said, "Saws and laddersseem to help in everything."Asked if saws andladdershave ancestors,she replied,"Theylook like they come fromparallellines andangles." Kathyalso explainedin the areatreethat the rectanglewas the mainancestorandhow the otherswere derivedfromit. When askedif she saw a relationshipbetweentrees,she responded,"Yes,we need to know all this stuff aboutarea [pointingto the areatree] before we could find this midline rule"and she spontaneouslyplaced an arrowbetween the two trees. Linda,Ling and Samanthaalso noted the role of saws andladdersin severaltrees andexplained interrelationsbetween theirtrees. The interviewer briefly explored the idea of an axiomatic (postulational) system with each studentand askedif theremightbe "familytrees"in otherpartsof mathematicsbesides geometry, for example, arithmetic. Thinkingof arithmetic, Alice said, "Additionis an ancestorto multiplication."She was not familiarwith an axiomatic system; she indicated that she had heard the words "axiom" and "postulate"but commented,"I don't rememberwhat they are." To illustrate,the solutionof the equation3(x+2)=18 was considered.Alice explainedeach step of the solution and gave appropriatereasons (i.e., distributive property, subtraction axiom, division axiom). She then recognizedthat axioms were familiarto her as "rulesof algebra." The idea of a theoremwas discussed and she recognizedfrom her family trees how certain principles (theorems) has been deduced from definitions and postulates. There was a similar dialogue about family trees with Linda. She spontaneouslysaid, "Inorderto do algebra,you had to know arithmetic, so arithmeticis an ancestorto algebra." She gave an explanation(similarto that given by Alice) for the solution of an equation. She also noted, "Inorder to do multiplication,you had to know additionand numbers." Linda thought she had heardof the words "axiom","postulate","theorem"but said, "Idon'tknow if there is any differencebetween an axiom and a theorem."The responsesof Carol,Ling, andSamanthato this discussionwerequitesimilarto thosegiven by Alice andLinda with Samantharemarking,"Inever thoughtof algebralike that,but it makessense." The analysisabove indicatesthatmost of the studentsin GroupVI, while filling in some concepts at level 1, also made significantprogresstowardlevel 2. They workedmore rapidlyand consistently,and expressedthemselvesmore confidently, thanthe studentsin GroupsIV andV. They werequickto suggest alternateways of approachinga problemand appearedto enjoy exploring new ideas and concepts. Since reasoningby deductionin geometryand giving carefulargumentswere new approachesfor the students,some neededmoreexperiencein this type of activityin order to fully appreciatethe power of their arguments. However, this group of students exhibited most of the level 2 descriptorsin their work on the module activities. They were not only thoughtfuland inventive about the problemsthey were doingbut were also reflectiveabouttheirown thinking.

CHAPTER 7 DISCUSSION OF FINDINGS OF CLIN4ICALSTUDY The analyses of the sixth and ninth graders'clinical interviews have been presented in the two preceding chapters. The purpose of this chapter is to summarizeanddiscussthe findingsof the clinicalstudy. The clinical study indicates that the van Hiele model provides a reasonable structurefor describing students'geometry learning. The analyses of the videotaped clinical interviews provide insight and informationnot only on students' levels of thinkingin geometrybut also on factors affecting students'performance on the instructionalmodules (e.g., language, visual perception,misconceptions, prior learning, students'thinking processes and learning styles). These will be discussed following a brief summary of the levels of thinking of the students describedin Chapters5 and 6. In addition,students'levels of thinkingon specific tasks,retentionof levels of thinking,andaspectsof the instructionalmoduleswill be examined. Summary of Students' Levels of Thinking Sixth Grade Subjects Analyses of the videotapedinterviews indicatedthat the sixth gradersin the studyfell roughlyinto threegroups. Group I. Threeof the 16 were strictlylevel 0 thinkers. They began at level 0 and for the most part, remainedat level 0, even after instruction. Their thinking showed a lack of analysis of shapes in termsof theirparts,lack of familiaritywith basic geometric concepts and terminology, and poor language (vocabularyand grammar)both generallyandin mathematics,especiallyexpressivelanguage.These students frequently forgot terms and concepts even shortly after they had been introducedby the interviewer. All had a weak backgroundin school geometryand also difficultywith arithmeticconceptsand skills. In fact, these threeseemed to be "geometrydeprived." They showed little knowledge of basic geometric concepts andlanguage,and they reportedhavingseldomstudiedgeometryin gradeschool. Group II. Five of the 16 began in level 0, much like the threestudentsabove, but made progresswith level 0 (learningbasic conceptsand terms)and into level 1 (usingthese conceptsto describeshapesandto formulatepropertiesfor some classes of shapes, in particular,familiarones such as squares;rectangles). However, they had difficulty characterizingless familiarshapes (e.g., parallelograms)in termsof properties. Their progresswas markedby oscillationbetween level 0 and level 1. Carefulinstructionand frequentreview of conceptsand termswas neededto sustain

134 theirprogress. While they beganto thinkaboutshapesin termsof properties(level 1), they did not try to relatepropertiesin a logical ordering(level 2). One of these five was below gradelevel, the otherfour were on gradelevel. They tendedto be more verbal than the three "level 0 thinkers,"but they had difficulty expressing themselves using standard geometric terms and often used manipulatives in checkingproperties(e.g., placingD-stix on sides of a shapeto show parallelism)or in explaining. These students also had a weak backgroundin geometry. They tendedto respondmore easily to the interviewerthanthe three studentsabove and also were less dependenton the interviewerfor feedbackandreinforcement. Group m1. Eight of the 16 studentsshowed thinkingat levels 0 and 1 at the startof Module 1, althoughmost had to fill in or review some concepts(rightangle, oppositesides and angles) at level 0. They also neededto become more fluent with level 1 languagefor describingshapes in terms of properties(e.g., "oppositesides are parallel"). These students progressedtoward level 2 by following and then summarizingarguments,for example, why the opposite angles of a parallelogram are equal via saws and ladders. A few progressed farther and began to give explanations (or simple proofs) more independentlyand with more details and rigor. Initially, however, most students equated "proof' with generalizationby examples (i.e., inductive reasoning) and only graduallyafter experiencing some deductiveexplanationsin Modules 2 and 3 did they seem to acquiresome insight into an informaldeductive approach. Some students,however, did not yet seem sure of the power of their deductiveargumentseven though they could follow an argumentor give one on theirown. They did not yet see the need for such deductive arguments. They were not at level 2, that is, they did not approachproblems deductively nor did they appreciatethe role of deduction in geometry. These studentswere all above grade level in achievement. They were quite verbal and tendedto express themselvesconfidently. They also seemed more reflective about the questionsandproblemsin the modulesandabouttheirown thinking. Ninth Grade Subjects Since the seventh and eighth grademathematicscurriculumhas large units on informalgeometry,it was expected thatmost ninth graderswould have a stronger background in basic geometric concepts than sixth graders. Analyses of the videotapedinterviewsindicatedthatthe ninthgradesubjectsfell into threegroups. Group IV. The two ninthgradelevel 0 thinkershad characteristicssimilarto those describedabove for the sixth gradelevel 0 thinkers. Their decision-making about shapes and properties was always on a "looks like" basis. Particularly noticeable was their limited vocabularyand poor language (i.e., their inability to expressan idea clearlyin a completesentence). Group V. The enteringlevel of seven ninthgraderswas assessed as level 0 in transitionto level 1. Most had to fill in or review some basic geometricconcepts.

135 They thoughtof certainshapes (triangles,rectanglesand squares)in terms of their propertiesbut they had less or no knowledge of parallelogramsand trapezoids. They used or learnedlevel 1 language for describingshapes and their properties, only occasionally reverting to level 0 type explanations. In order to justify conclusions, they frequentlyresortedto an inductive approach(i.e., measuringa numberof specific cases). In some instancespriorlearningand/ormisconceptions interferedwith progress (e.g., "a right angle points to the right"; "one ray of an angle must be horizontal"; "a square cannot be a rectangle"). As with the comparablegroupof sixth graders,some of these studentsprogressedtowardlevel 2 by following and/or summarizing arguments and by trying to relate some propertiesby a logical ordering. Group VI: The remaininggroup of seven ninth graders(whose entry level was assessed as level 1) needed very little review of basic concepts and used appropriatebut sometimesnon-standardlanguageto describefigures. They readily explained subclass relationsand learnedto relatepropertiesin a logical ordering. They not only followed argumentsbut provided simple deductive explanations, therebyshowing characteristicsof level 2 thinking. In addition,some were able to formulatedefinitionsandjustify necessaryand sufficientconditionsin given tasks. All of the studentsin this groupwere able to successfullycompleteat least some of the final optional assessment activities on Modules 2 and 3. They spontaneously wonderedabout and wanted to explore extensions of ideas with which they were working--the"whatif' phenomenon. Factors Affecting Students' Performance on Modules There were some striking similarities in the performanceof sixth and ninth graderson the modules. Yet, as mightbe expected,becauseof age, maturation,and greaterexperiencewith geometryconcepts,therewere also significantdifferences. Factorsaffecting students'performances(i.e., limiting theirprogresswithin a level or to a higherlevel of thinking)aredescribedbelow. Language Although geometry is accepted as a strand in the K-8 school mathematics curriculum,it was interestingto note that some studentsappearedto be "geometry deprived"in terms of their vocabulary, some used non-standardlanguage, and others used standardlanguage, although sometimes imprecisely. Examples of studentvocabularyfrequentlyused for basic geometricconceptsare given below.

136 GeometricConcept

StudentVocabulary

angle rightangle parallellines lines perpendicular diagonal side perimeter area rectangle equal

point,vertex,triangle straight,righttriangle straightlines, horizontallines straight,verticallines slantedline straight,vertical area,distancearound,volume perimeter,space,volume box, long square even, same, similar

The influence of everydaylanguageand experienceis seen in the students'use of "straight"when speaking about parallel lines, perpendicularlines, sides or right angles. This may be an importantclue for teachingsuch terminology--namely,that a carefuldistinctionbe made between commonusage of a word and mathematical usage. The preferenceof some studentsfor using the gestaltof closed finite regions ratherthanopen infinite space is seen in theirconsistentuse of the words "triangle" and "righttriangle"when referringto "angle"and "rightangle." As expected, ninth graders tended to be more familiar with geometry terminologythansixth graders. At the outset of an interviewmost students'mathematicallanguagewas ratherimprecise. As the interviewprogressed,most students (guidedby the interviewer)began to use termsmore accurately. Some had trouble rememberingnew terms indicatingthat a vocabularyreferenceboard would be a valuable addition to the clinical interview setting. Initially, some students respondedto questionsby pointingor using one wordor shortphrasesshowinglack of fluency or expressive language or perhaps reflecting prior experience in mathematicsclasses wherethattype of responsewas all thatwas expected. Students who were in transitionto level 1 neededwell-defineddirectiveswhich focused their attentionon appropriatelanguageto describepartsof shapesand sets of shapes. Although most studentswere able to make simple informal deductions,few spontaneouslyused a logical type of language. They did not exhibitlanguageneeded

for level 1 such as the use of quantifiers (e.g., "all these rectangles have . . .").and conditional statements (e.g., "if the shape has . . ., then it is a . . ."). Gregory and

Osbome (1975) point out that the interplayof logic and language--asin the use of "some,""all,""neither,""nor"--isa vulnerableand confusing areafor adolescents. Studentswho were level 1 thinkersor progressedtowardlevel 2 picked up ideas quickly, rememberedterminologyand used it appropriately.They became more fluent in talking about geometry as they moved throughthe modules. They also displayed the ability to reason well, both inductively and deductively, as was reflectedby theiruse of languageassociatedwith levels 1 and 2 (e.g., "all,""any," "because,""Ican explain,""itfollows that,""itis truebecause,""Ican prove that").

137 Studiesby Shaughnessyand Burger(1985) and Mayberry(1983) reportsimilar use of impreciselanguageand languagedifficultieson the partof studentsK-8 and pre-service teachers. Shaughnessy and Burger (1985) note the importance of languageat differentlevels of thought. They indicatethat"studentsmay have vastly differentgeometricconceptsin mind thanwe think"(p. 425). This underscoresthe need to assess the meanings that students attach to geometric terms during the instructionalprocess. The resultsof the Project'sclinical study clearly supportthe contentionthat language structureis a criticalfactor in movementthroughthe van Hiele levels of thought.

Visual Perception, Misconceptionsand Prior Learning Perceptual difficulties, including orientation and figure-ground problems, were evidentin the performanceof some subjects. Burgerand Shaughnessy(1986) also reportthatthe turningor moving of figuresto more customarypositionsby the students helped them identify such properties as right angles, parallel lines, congruentfigures. Students'past experienceswith these figures (in textbooksor in teacher illustrations)may have been limited to specific orientations. This study found, as did Fisher (1978), that students'formationof certaingeometricconcepts tendedto be biased in favor of uprightfigures. Misconceptions and prior learning, in addition to language and perceptual difficulties, impededthe progressof many students. Examples of misconceptions include: - An anglemusthave one horizontalray. - A rightangle is an angle thatpointsto the right;some angles were also called left angles. - To be a side of a figure the segment must be vertical (e.g., a rectangle in standardpositionhas two sides anda top andbottom). - A segmentis not a diagonalif it is verticalor horizontal. - The heightof a triangleor parallelogramis the side adjacentto the base. - The angle sum of a quadrilateral is the same as the areaof the quadrilateral. Vinner and Hershkowitz (1980) studied images which students attach to certain geometricconcepts. They found,as did this Project,thatsome studentswho know a correct verbal description of a concept but also have a special visual image associatedtightlywith the concept(e.g., a leg of a righttrianglemust be horizontal), have difficultyapplyingthe verbaldescriptioncorrectly. Examplesof interferenceof priorlearninginclude: - Having learned that a rectangle has two congruent long sides and two congruentshort sides, studentscould not accept the subclass relationship thata squarewas a specialtype of rectangle.

138 - Vaguely remembering having learneda geometric fact, some students triedto recall the fact andapply it to a totallyunrelatedsituation(e.g., "use a2 + b2 = c2 to find the area of a rectangle.").

- Havinglearnedor memorizedrulesrelatingto perimeter,areaandvolume of figures, many studentswere totally confused aboutthese concepts and the units of measure(e.g.,"cubicsquareinches"). Wirzup(1976) points out thatthe van Hiele model indicatesthatmaturationin geometry is a process of apprenticeship. The above examples of student misconceptionsand interferenceof priorlearningfocus on a need to structurethe apprenticeshipprocessmore carefullyfor studentsin gradesK-8. Student Views, Thinking Processes and Learning Styles In addition to problems caused by misconceptions, interference of prior learningandperceptualdifficulties,a student'sview of mathematicsand of learning mathematicscannotbe overlooked. At times students'progresswas impededby theirexpectationthatmathematicsis a subjectto be recalledor memorized,not one involvingdiscoveryor reasoning. At the beginning,many of the ninthgradersused a ratheralgorithmicor procedure-orientedapproachto tasks. Their usual reply, when askedfor an explanation,was "that'sa rule." This was also truefor some sixth graders. The idea thatone could stop andthinkabouta geometryproblem,explore it and find a solution withoutusing a rule was new to many students. Gradually those studentswho made progresstowardlevel 2 realized that explanationswere expected and began spontaneouslyoffering reasonsor giving argumentsto justify theirstatements. As with any groupof learners,some subjectslackedconfidence,gave up easily, needed constantreassurance,were reluctantto take risks while others were more self-assured,were persistent,enjoyed independentexplorationand were willing to make conjectures. Some were impulsive, others reflective. Two ninth graders, while lackingconfidence,constantlyused a strategyof randomguessing in problem solving situationsbecause"youmightbe right." All subjectsmade extensive use of the concretematerialsprovidedto explore relationships,discover patterns,or confirmhypotheses. The use of manipulatives andotherconcretematerialsallowedthe studentsto try out theirideas, look at them, be reflective, and modify them. The visual approachseemed not only to maintain student interest but also to assist students in creating definitions and new conjectures,in gaininginsightinto new relationshipsand interrelationships. Some studentsquickly arrivedat generalizationsafterexaminingonly one or two instancesand laterhad to revise theirconclusions;otherswere more cautious, carefulandthoughtfulin formulatinggeneralizations.Some reasonedby analogyto arriveat conclusions--sometimescorrectlyand sometimes incorrectly(e.g., "Since

139

the sum of the angles of a triangle(180?)is one-halfthe sum of the angles of a quadrilateral(360?), then the areaof a triangleis one-halfthe areaof a quadrilateral."). After arrivingat generalizationsby inductionor deduction,studentsfrequently chose to make a drawingor to measure,thus checking their result in at least one example. The students'desireto put in values to makeit a specificexampleindicates their need to examine a concreteexample ratherthanjust thinkingof the abstract idea; the particularhelped to clarify the generalconcept. Since the deductiveprocess of obtainingconclusionswas a new (anduntried)processfor them,the students felt the need to verify their resultsby methodswith which they were familiar. As they gainedmore experiencein using deduction,some studentsbegan to appreciate the power of theirargumentsandno longerresortedto measuringas a check. As has been seen in the precedingchapters,many studentsmade considerable progress in moving to a higher level of thinking while a few made little or no progress. Possiblefactorsthatmay explainthis lack of progressinclude: - lack of prerequisite knowledge

- poorvocabulary/lackof precisionof language - unresponsivenessto directivesandgiven signals - lack of realizationof whatwas expectedof them - lack of experiencein reasoning/explaining - insufficientor inappropriateactivitiesto promoteprogress - insufficienttime to assimilatenew conceptsandexperiences

- rote learning attitude

- not reflectiveabouttheirown thinking Those studentswho made the most progressto a higherlevel of thinkingtendedto be systematic and flexible in their approach,were willing to accept changes in definitions and reason from a new basis, recognized when they were wrong and reflected on why that could be, and spontaneouslythought of alternateways of solving problems. Levels of Thinking on Specific Tasks Initial activities in each of the three instructionalmodules were designed to assess students'level of thinking. A question to be asked is why did some sixth graders and ninth graders engage in mainly level 0 thinking on some "entry" assessmenttasks? One possible explanationis students'lack of experiencein doing geometryin school. A second explanation,as will be seen in Chapter9, is thateven when geometry was studied, the text material probably did little to encourage higher levels of thinking. A thirdexplanationis that the "entry"assessmenttasks which involved cut-outfiguresor diagramsand can be respondedto at levels 0, 1 or 2, are done most naturallyat level 0 which matches the format in which they are

140 presented. Several of this Project'stasks for measuringentrylevel were similarto those used by Burger (1982) who conducted two non-instructional45-minute interviewswith studentsin gradesK-12. Burgerfoundmainly level 0 thinkingfor subjectsin gradesK-8, just as this Projectdid for most assessmentsof entry level. However,in this study,most students,afterworkingthroughsome activitiesin the instructionalmodules, performedat a higher level on "potential"assessmenttasks thanon "entry"tasks,especiallystudentswho progressedtowardlevel 2. Accordingto the van Hieles (1958), it is possible to teach materialto students above their actuallevel resultingin what he calls "reductionof level." Therewas evidence of this in clinical interviews with both sixth and ninth gradersduring discussionsof certaintopics (e.g., angle sum of a triangle,arearules for rectangles and triangles). Frequently,studentsknew the rules by rote (no level assigned) and could apply rules in problems (level 1) but were unable to explain why the rules were true (level 2). Three specific ideas (subclass inclusions, Siamese twins, and proof) were difficult for many sixth gradersand some ninth graders. The activity of sorting quadrilaterals(Activity 3-3) into separategroups of figures may have led some studentsto a misconception,thatis, to "see"squares,rectanglesandparallelograms as separategroupsratherthanrecognizingsubclassrelationshipson the basis of sets of properties. The difficulty of seeing the differencebetween the Siamese twins (statement and its converse)probablycan be alleviatedto some degreeby giving studentsmore experience in physically constructingthe two situations and in providing more settingswhich involve eitherthe statementor its converse. Time constraintsin these interviews did not allow for extended "apprenticeship" with this concept or with formal deduction. As was seen in Chapters5 and 6, a few sixth gradersand some ninth graders(GroupVI) afterlimited instructionand experiencewere able to give careful deductive arguments("proofs")although frequentlynot appreciatingthe power of theirlevel 2 arguments. Since the three modules focused on differentconcepts--propertiesof figures,

angle sums and area--an assessment of a student's level of thinking across concepts could be made. For both the sixth and ninth graders who completed two or three of the modules, it was evident that the highest level of thinking attained on one concept remained consistent across other concepts. In beginning a new concept, students frequently lapsed to level 0 thinking but were quickly able to move to the higher level of thinking they had reached on a prior concept. This evidence supports the van Hiele contention that levels of thinking remain stable across concepts.

141 Retention of Students' Levels of Thinking The results of this clinical study are based on a student'sperformanceover a relativelyshortperiodof time--usuallythreeweeks for the eight interviewsessions. Whetherthese resultsare more permanentover a longerperiodof time and to what extent students retain what they had learned were not researchquestions of this Project. However, the Projectdid explore thembriefly by reinterviewingone sixth graderand one ninth graderone year after their initial interviews. This interview included tasks on rhombuses--sorting, finding properties and formulating a definition, subclass inclusions, and establishingpropertiesof figures (e.g., angle sum, parallelismof sides, equalityof angles). The sixth grader,David, in GroupHI (now completing seventh grade), readily discoveredpropertiesof a rhombusand gave different definitions for it (level 1). He also explained carefully subclass inclusionsfor the rhombuswith quadrilaterals andparallelograms.He also was able to prove thatthe sum of the measuresof the angles of a triangleis 180? using saw/ laddersandto prove thatthe sulmof the measuresof the anglesof a rhombusis 360?. The ninth grader, Linda, in Group VI (now in tenth grade), who was just completing one year of high school geometry,quickly discoveredpropertiesof a rhombus and gave a definition for it. Her definition included more than was necessary. She was asked if all the propertiesmentioned were necessary in the definition. Her reply: "Wellthey are all true." She was remindedthat a definition should contain only the minimumpropertiesneeded. She thoughta moment and then correctlyrevised her definitionwith the comment:"Ohyeah, then you could prove the rest." She thenproceededto prove the otherpropertieson the basis of her revised definition. She explained subclass inclusionsfor the quadrilateralfamily, gave carefulproofs for the angle sum of a triangleand of a quadrilateral.She noted thatin her geometryclass, they had used alternateinteriorangles and corresponding angles of parallel lines, but "they are the same as saws and ladders." When the interviewerexplored the student'sunderstandingof a postulationalor axiomatic system (see level 3 descriptors),the responsesindicatedthatthe student,aftera year of study of high school geometry,continuedto do thinkingcharacteristicof level 2 with some aspectsof level 3--thusher thoughtlevel in geometrymightbe described as in transitionto level 3. Both studentshad done well in the original set of interviews,having filled in level 1 and andmade considerableprogresstowardlevel 2. Thus, thesetwo students did not lapse backto a lower level; they retainedmuchof whatthey had leared both in terms of content and of what was expected of them (e.g., discover properties, give proofs and explanations).This questionof retentionin relationto the levels of thinkingshouldbe researchedfurtherwith a largerrepresentativesample.

142 Discussion of the Instructional Modules As noted earlier,the instructionalmodules were designed as a researchtool to assess level of thinking and were structuredto embody van Hiele's five phases (information, guided orientation, explicitation, free orientation, integration) requiredfor transitionfrom one level to the next. A numberof the activities were modeled after those used by Dina van Hiele-Geldof (1957/1984) in her year-long "teachingexperiment."The Projectfound certaintechniquesand tasks particularly effective for developingand/orassessingstudentthinking. The intervieweralong with the instructionalmaterialsplayed a special role in helping studentsto progress within a level or to a higher level. The interviewer provided instruction designed to move students to a higher level. Also, the interviewerguided studentresponsesthroughquestioningand directivesaboutthe quality of responses, thus helping students to learn the rules of the game. For example,studentsneeded to lear to observerelationshipsbetweenpartsof a figure and to make generalizations(level 1) or to give deductive explanations(level 2). Studentson a given level realizethis, but studentsin transitionneed guidanceabout can use a meta-languageaboutthinkingto expectations,andthe interviewer-teacher communicatesuchexpectationsto the student. Studentsrespondedfavorablyto initial introductionof concepts in real world settings (e.g., photographs,maps, tiling). Completionof tasks in which they were given examples and non-examples (e.g., kites) or sets of clues for shapes (e.g., square,parallelogram)enabledstudentsto develop definitions. Game formatsused in some tasks were effective. For example, in the task requiringminideductions (e.g., saws and ladders),studentswere placed in the role of detectiveschargedwith finding the missing link. The culminating"familytree"tasks in the modules were good assessmentactivities. To determineif studentscould summarizewhattheyhad leared and if they had reflected on their actions, the techniquerequiringthem to "explainthis to a friendover the telephone"was particularlyuseful. The instructional materials in Module 1 were designed to review topics normallycovered in grades4-8, not to develop them for weak students. Also, the interviewscheduledid not permittime neededto developtopics carefullywith these students. Additionalresearchis needed to determinewhetherothermaterialsand extendedinstructionwouldenablelow abilitystudentsto makeprogressinto level 1. The availability to the students of a wide variety of visual materials and manipulativesto select fromand to use was an importantfeatureof the instructional modules. The use of materials and/ormanipulativesby the students frequently helped them bridge a language gap--allowing them to "explain"their ideas by demonstratingwith concrete objects or drawings. The sorting, tiling, and family tree activities allowed the studentsto move pieces (shapes, tiles, arrows)as they were thinking, to discover patterns, to visualize relationships, to make

143 modifications,and to reflecton whatthey were doing. A uniquecharacteristicof this researchwas its focus on a teachingexperiment approachin the spiritof Dina van Hiele-Geldof. Throughclinical interviewsit was possible to monitor systematically the students'progress through the carefully designed instructionalsequences. This Project'sapproachto assessing a student's level in a learningcontextis similarin spiritto the methodof assessinga "potential" student'slearningpotentialrecommendedby Vygotsky (1962). He points out that "studyingchild thought apart from the influence of instruction, as Piaget did, excludes a very importantsourceof changeandbars the researcherfromposing the question of the interactionof development and instructionpeculiar to each age level" (p. 117). In this research,the conceptsstudentswere learning,the processes they were applying, the errorsthey were making, and the level of thinking they were demonstratingwere carefullyassessedby meansof the students'performances on activities in the instructionalmodules. The detaileddescriptionsin Chapters5 and 6 of individualstudents'performancesdocumentchanges in students'thought levels in geometry as a result of instruction. The Project'sinstructionalmodules were an effective researchtool for this intensivestudy of individualstudents'levels of thinkingin light of the van Hiele model. Futureresearchusing aspects of these instructionalmodulesin teachingexperimentswith groups(or classes) of studentsis neededandwill be discussedin Chapter10.

CHAPTER 8 CLINICAL INTERVIEWS WITH PRESERVICE AND INSERVICE TEACHERS One of the goals of the Project was to determineif preservice and inservice teacherscould learnto identifyvan Hiele levels of thinkingin geometryof sixth and ninthgradestudentsandto identifythe levels of thinkingrequiredby the exposition and exercises in the geometry strandin various text materialsfor grades K-8. In orderto have the teachersrecognizethe level of thinkingrequiredby text materials and identify the thought levels of the students' responses to Project activities (recordedon videotape),the Projectbelieved that it was importantfor the teachers to have first hand experience doing some of the same module activities as the students. This provided an opportunitynot only to explore the teachers'level of thinkingin geometry but also to have the teacherscomment on the suitabilityof some of the Project'sinstructionalmaterialsand activitiesfor use in the classroom. Presentedbelow are a descriptionof the subjectsand proceduresused followed by four sections: (a) Teachers' Responses to Module Activities; (b) Teachers' Commentson InstructionalModule Activities; (c) Teachers'Identificationof the Van Hiele Thought Levels; and (d) Implications for Teacher Preparationand ClassroomPractice. Subjects There were 13 subjects--8preservice and 5 inservice teachers. Of the eight preservice teachers (undergraduatestudents at Brooklyn College), six were elementaryeducationmajors,one was an early childhoodeducationmajor,and one was a mathematicsmajorpreparingto teach at the secondaryschool level. Five of the elementaryeducationmajorshad completedone year of high school geometry and one had only studiedhigh school geometryfor two monthsbefore transferring to a commercialcurriculum. The early childhoodmajorhad studiedno geometry in high school. Among the five inservice teachers, two were currentlyteaching sixth grade, one was teaching seventh grade in a middle school and two were teachingninthgrade,all in Brooklynpublicschools. Two were teachersof some of the sixth and ninth grade students involved in the Project'sclinical study. The inservice teachersranged in teaching experience from one to eight years with an averageof 4 yearsof teaching. Procedure Each subject spent approximatelyfive to six hours over four sessions with an interviewer. At the outset,the subjects(eight preserviceteachersand five inservice teachers) were given a brief background on the work of the van Hieles, an

145 explanation of the van Hiele thought levels (including level descriptors)and an overview of the Projectand its goals. In approximatelythree hours of one-to-one clinical interviews, the subjects completed selected activities from the Project's InstructionalModules and informallyevaluatedthe tasks in termsof suitabilityfor classroom use. In a follow-up one-hour session, videotaped segments of five studentsdoing some of these activities were shown to the subjects. Each segment was followed by identificationand discussionof the student'sresponsesin light of the van Hiele level descriptors. Samplepages on geometryfrom text materialsfor grades 3-8 were also examined and discussed in terms of level of exposition and level of thoughtneeded for the studentto respondto the exercises correctly. At a later one-hour assessment session, subjects were shown four other videotaped segmentsof studentsdoing activitiesfromthe InstructionalModulesand were asked to discuss each student'sresponses in terms of level of thought. They were also askedto review a set of 10 samplegeometrytext pages (grades3-8) and to describe the level of exposition and the level of thinkingneeded to respondto the exercises correctly. Teachers' Responses to Selected Module Activities The subjectsworkedthroughthe selectedactivitiesin the InstructionalModules at differentratesdependingon the amountof review needed. All of the preservice teachersfinished the designatedactivities in Modules 1 and 2 and four completed some or all of the selected activitiesin Module3. The inserviceteacherscompleted all of the selected activities in Modules 1, 2 and 3 except for one teacherwho was unable to complete the last module. Described below are some of the teachers' responsesto module activitieswhich characterizetheirthinkingin geometry. Module 1 All the preserviceteachers(except for the two who had not had a high school geometry course) and all the inservice teachers were quite fluent in their use of standardgeometry vocabulary. There was some uncertaintyamong six preservice teachersand one inserviceteacheraboutwhatpropertiesshouldbe includedin their descriptionof a rectangle(e.g., "Ithink it has to have two longer and two shorter sides"; "Don'tthe sides have to be straight?";"I'mnot sure if it has to have right angles";"Idon'trememberwhethera rectangleis a squareor a squareis a rectangle because a squarehas all sides equal."). Also this same groupthoughtof a diagonal of a figure as "somethingwhich bisects or divides the figure in half." In describing parallel lines, one of the two subjects who had not studied high school geometry said: "linesthatlook like sticks, aboutthe same size (I guess it doesn'treallymatter), next to each other,but thereis an even space between them." The othersaid: "kind of like a mirror, lines facing each other, beside each other." This last subject thought "side of a figure"meant a vertical segment. Whenever instructionwas needed,all the teachersrespondedeagerlyandquickly.

146 In Activities 3.1 (Sorting Polygons), 3.2 (Sorting Quadrilaterals)and 3.3 (Propertiesof Classes of Quadrilaterals),most subjectssortedquickly, confidently and systematically. Many properties of the various quadrilateralswere given spontaneously.Some subjectswere carefulandcautiousin makingjudgmentsabout shapes and prefaced their statementswith phrases such as: "Accordingto what is

given, then .. ." or "It appears that .. ." or "Opposite angles appear to be equal."

Some promptingwas occasionally necessary to elicit propertiesof parallelismof sides or congruenceof oppositeangles.

Some interestingresponses(shownbelow) were spontaneouslygiven by several preservice(P) and inservice(I) teachersin Activity 3.4 (InclusionRelations)which included questions such as: Could the squarego in the rectanglepile? Could the rectanglego in the squarepile? Couldthe squarego in the parallelogrampile? P: [pointing to array of shapes and properties] As we go along, the less

specific ones embrace the more specific ones ... hm ... so these are special ... so, yes ... according to my definition every square is a rectangle, a special kind . . . [pause] . . . that's great! [The subject's apparent pleasure

was an indicationof some new insightgainedaboutthese relationships.]

P: WhenI was learningthata squarewas a rectanglebut a rectanglewas not a square,I wonderedwhy. This is a lot easier way of doing it; it makes it clearer. I:

[looking at the lists of properties]A rectanglehas all the propertiesof a parallelogram,so a rectangleis a special parallelogram... so rightangles are necessaryfor a rectangleandthatwas whatwas missing in my original description of a rectangle . . . I see now. [Subject was being reflective

abouther own thinking.]

I:

[after checking propertieslisted] A square could go in rectanglepile; a

square could go in parallelogram pile ... [long pause] ... I never thought of that . . . I just discovered something! [Evidence of new insight into

relationshipspreviouslylearned--theAHA! phenomena.]

The activities involving uncoveringshapes/clues(Activities 3.6 and 3.7) were done quicklyandeasily by the teachers,many of whom spontaneouslyjustifiedtheir statementson the basis of propertiesof figures. One teachercommented:"These uncovering activities are really good; they are fun." The Minimum Properties (Activity 3.8) was quite challenging for most teachers. Some explained why properties not selected were not necessary. One preservice teacher drew illustrations of counterexamplesto justify the need for certain properties. One inservice teacher, after selecting a set of minimum properties, spontaneously proceededto deduce the otherremainingpossible propertiesand then pointed out thatthe originalminimumset constituteda definitionof the figure. Anothernoted

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that "differentsets of minimum properties allow for the possibility of having differentdefinitionsof a figure." Module 2 In the openingactivity,Angle Measurement,most of the teachersestimatedthe size of the angles well. One of the preserviceteacherswho tendedto overestimate the measuresof the angles commented:"No wonderI can'tparkmy car." She was reflecting on how her tendency to overestimate impacted on another activity involving angles. An inservice teacherthoughtthat the measure of an angle--its openness--shouldbe foundby using the lengthof a line segmentplacedbetweenthe rays. All but two of the preserviceteachersknew that the angle sum for a triangle was 180? and three were able to give a deductive argumentto establish it; all the inserviceteachersknew this fact andtwo gave carefulproofs. Sampleresponsesare given below: P: A circle has 360?. I don'tknow why a trianglehas 180?... they musthave shown me, but I don'tremember... I thinkit has somethingto do with the areaof the figure. I:

A trianglehas 180?. I really don'tknow why--I was just told that if you take the three angles in any triangle and measure them, they add up to 180?--it'ssomethingwe accept.

The Tiling activity was done quickly and systematically (using various strategies)by all the teachers. After an introductionto the Saw andLadderactivity, many of the teachersneeded guidanceand practicein identifyingsaws and ladders and congruentangles in complex grids. There were a few instancesof orientation or figure-ground difficulties. The miniproofs involving saws and ladders in Activity 4 were challenging to some of the teachers with one preservice teacher commenting: "These are like little proofs from tenth grade which I could never

figure out ...

but I like these" and an inservice teacher commenting: "These are

making me think hard." In this activity several of the teachers gave very clear deductiveargumentsand spontaneouslysuggestedand explainedalternatemethods for solving the problems. For the two preserviceteacherswho had no high school experience in doing this type of task, the activity was presentedas a game--"To prove angles equal game"--withtwo rules (saw andladder)and a strategyof finding the missing link when needed. By the fourth game (problem), one teacher was enthusiaticand successful and completed the remaininggames (problems)easily with great delight; the other teacher was impulsive about putting in all saws and ladders in each diagrambefore really thinkingabout the problem and so became confusedand frustrated.(Laterthis teacherbecamemore successful as she realized the need to first think about the problem and plan ahead.) After giving an explanation of the relation of the game to doing simple deductive proofs in geometry,the first teacherremarked:"Thisis refreshingthinking... I see that."

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After coloring in congruent angles by saw and ladder on a triangulargrid (Activity 5), many of the teacherswere quick to recognize the importanceof this activity in termsof "readingfrom the grid"andfinding the angle sum for a triangle. Among the spontaneousteacherremarkswere the following: P: Oh, those [pointingto angles forminga straightangle] are the threeangles of the triangle ... oh, that's wonderful!

I:

... hm...

there's your proof for the angle sum of a triangle is 180?.

All the teachers discovered or proved the angle sum for a quadrilateralis 360? either by tiling with congruentquadrilateralsand readingthe grid or by dividing the quadrilateralinto two triangles. All but three of the teachers were able to explainthe fallacy in thinkingthatthe angle sum of a quadrilateral equals720? when two diagonalsare drawnand four trianglesare formed. Most of the teachers were able to assemble and explain the Family Trees (Activity 6) for the angle sum of a triangleandfor the equalityof oppositeangles of a parallelogramwith little or no guidance. Typicalcommentsincluded: P: This shows how you arebuildingon simplerfacts. I:

This shows the use of definitionsandpostulatesin the buildingof a family tree.

In the final assessment activity for Module 2, Activity 7, ExteriorAngle of a Triangle, two of the inservice teachers knew the relationshipand the remaining preserviceand inserviceteachersguessed the relationship"byeye" or discoveredit by measuringthe angles in several triangles. Four of the teachersgave algebraic proofs to establish the discoveredrelationship. Guidancewas needed in orderfor some of the teachersto thinkof constructingan auxiliaryline to be used to developa geometricproof. With the suggestion of an auxiliaryline, most were able to give deductive argumentsusing saws and laddersto establish the relationship. Some typical remarksincluded: P: I liked this ... I learneda lot. You get a chanceto thinkabouthow things are related . .. you learn to respect saws and ladders (I never thought I

would) ... you get a chanceto thinkthingsout first.

I:

Saws and ladders show this relationshipin such a concreteway--I love it ... you can visualize it and it makes it easier to understand.

The teachers were able to place the newly derived principle in their family trees correctlygiving appropriateexplanations.

149 Discussion with the teachersabout a postulational(axiomatic) system and its elements (undefined terms, definitions, postulates or axioms, theorems) and the meaning of proof showed that only the three teachers with a substantial mathematicalbackgroundhad an understandingof these concepts. Even one of these threeteachersacknowledgedthatshe was not sureof the differencebetweena definitionanda postulate--"That's my weakness,"she commented. Module 3 Nine teachers (five preserviceand four inservice) completed activities in this module on area. In the initial assessmentactivity (Activity 2), the concept of area andthe areaof a rectangleseemedto be familiarto all the teachers. All were able to solve problems involving areas of rectanglesand all but one teacherwere able to explain why the area rule for a rectangle worked. However, two preservice teachersand one inservice teacherwere not clear on the type of unit to use in area measure. The answergiven for the areaof a rectangle6 inches by 4 inches was 24 inches althoughthey had covered the rectanglewith a squareinch grid or with 24 square-inchtiles. All but one indicatedthatthe areaof a righttrianglewas half the areaof a rectanglesince they could "see"that if two congruentrighttriangleswere placed togetherappropriately,they would form a rectangle. Only one preservice teacherandthe inservicemathematicsteacherswere able to give a carefuldeductive argumentto establishthis fact; the otherswere able to give partsof the explanation or to follow the argumentgiven by the interviewer. In this initialassessment,most of the preservice teacherswere uncertainhow to find the area of a parallelogram and suggestedfindingthe productof two adjacentsides; all of the inserviceteachers proposed converting the parallelogram into a rectangle or dividing the parallelograminto a rectangleand two righttrianglesin orderto find its area. In the succeeding activities in this module, area rules for geometric figures were discoveredby the subjects. Whatcharacterizedthe teachers'(bothpreservice and inservice) approachesto these activities were theirspontaneoussuggestionsof several alternateways for findingthe areaof each of the figures. They appearedto enjoy thinkingabout differentways of doing each activity. Two of the preservice teachersthoughtthatusing angle sum relationshipswouldhelp in findingthe areaof a triangle;this was similarto statementsmade by some of the sixth and ninthgrade subjects. The teachers were particularlyresponsive to the activities involved in buildingthe family tree for area. Studyingthe family tree she had constructed,one preservice teacher commented:"I understandnow how the formulas came about and how they are related." Anothersaid: "Thefamily tree helps you understand relationships."

Teachersrespondedfavorablyboth to the materialsused in developingthe area concepts (tiles, transparentsquare grids, L-square for finding heights, cut-out shapes, models, .. .) and to the hands-onapproach. One inservice teacherwas so pleased with the Project'smodel which converteda parallelogram Co into a rectangle, C

150 she made a replica of the model and used it with her sixth grade class. Latershe reportedhow much easier it was to develop the concept of area of a parallelogram with her class using the Project'sapproachthanthe way she had taughtit in the past. In using a model which convertsa trapezoidinto a triangle,one teachercommented: "Oh,that'snice--thebase of the triangleis madeup of the two bases of the trapezoid, so the areais one-halfthe sum of the bases times the heightjust like the formulawe derivedbefore--that'sbeautiful!" On the same task, anotherteacher commented: "Thisis perfect!... it's so clear... I know this will stick in my head." The teachers were very thoughtfulabout the discoveries they were making. After a discussion of the meaning of height of a figure, one preservice teacher, reflecting back on some previous work, commented: "That'swhat I was doing wrongbefore ... I shouldhave found the height insteadof measuringthe adjacent side." Most teachers were not satisfied with finding the area of a figure for a specific numericalcase, they searchedfor patters, triedto find a generalizationand looked for interrelationshipsand extensions. Having discoveredthatthe areaof a trapezoid could be expressed as "midline times altitude," one teacher paused, reflectedand then said: "Hm... the same principlewould apply to a parallelogram ... and to a triangle... that'sgreat!" Anotherteachermakingthe same discovery commented: "Oh boy, they are all the same--midline times altitude . . . it's logical

but I never thought of it." The teachers with strong mathematicalbackgrounds naturallyand spontaneouslyjustified each step of theirexplanations,giving careful deductive arguments, citing definitions and postulates, . . . (i.e., showing evidence

of many of the Project'slevel 3 descriptors).

In summary,responses to Module 1 activities show that, in terms of level of thinkingin geometry,two subjectsenteredat level 0 while all othersubjectsentered at level 1 or higher. With instruction,those at level 0 attainedlevel 1 and one of these subjects also progressedtowardlevel 2. As might be expected, most of the preservice and inservice teachers exhibited considerablefluency of language in describingand explaininggeometricideas. It is of interestto note that some of the errorsor misconceptionsof the teacherswere the same as those of the sixth or ninth gradesubjects,for example,thinking"sides"refersonly to verticalsegments,using the phrase "straight lines" when referring to parallel lines, thinking that a parallelogramhas to have oblique angles, and thinking that the angle sum of a polygon relatesto area. The teachers made frequent comparisons between the module method of presentation and their previous learning and ideas. Several of the activities providedopportunitiesfor the teachersto gain insight(the AHA! phenomenon)into previously studied mathematicalconcepts which they had memorized but never really understood. The spontaneity of the responses and the quality of the explanationsgiven by most of the subjectsby the end of Module3 indicatedthatthey were exhibitingmany of the characteristicsof the Project'slevel 2 descriptors.The two ninth grade teachers (licensed mathematics teachers) and the preservice

151 secondary school teacher (mathematicsmajor), who immediately justified their statementsby proving their conclusions (by giving ratherrigorousargumentsthat used definitions, postulates, and other proven facts), were exhibiting some of the level 3 descriptorslisted by the Project. In a study investigating the van Hiele levels of geometric thought in undergraduatepreservice teachers,Mayberry(1983) found throughtwo one-hour interviewsinvolving geometrytasks thatpreserviceteachersoften did not perceive properties of figures (level 1), frequently did not perceive class inclusions, relationshipsand implications(level 2), and found deducingrelevantfacts from a given statement to be very difficult. While the goals of this study and that of Mayberryare quite different,it is interestingto note thatthe threehours of clinical interviewsprovidedpreserviceteachersin this study resultedin teachersexhibiting many level 2 characteristics. In workingthroughthe instructionalmodules, therewere instancesof teachers seeing new relationships in previously learned (and not really understood) structuresas well as indicationsthat teacherswere consciously reflecting on their own thinking. Fromthis, it would appearthatthe teacherswere, as the van Hieles mightsay, gaininginsightinto the subjectof geometry. Teachers' Comments on Instructional Module Activities Teachers were asked to comment on module activities in relationto possible classroom use. In general their reactions were positive and enthusiastic. Many indicatedthey wantedto use these activitieswith theirstudents. Some commentson selected activitiesare reportedbelow. Module 1 Activity2: Shapesin Pictures - The use of photographs, maps, picture of parallel bars, ...

and concrete

materials(D-stix, angle testers,...) is good motivation;the studentswill see geometryideas used in reallife.

Activities 3.1, 3.2, 3.3: SortingPolygons, Quadrilaterals;Propertiesof Classes of Quadrilaterals - Sorting is a great idea. I never thought of using it in math but it ties in beautifullywith what I do in languagearts--classifyingwords by sound,by endings, ... and I also develop classificationschemes in science and social studieswith the children.

152 - This is a terrific way of having children learn about the properties of figures. I'm going to try it with my class. [The teacherreportedlater that she had introducedthis activity in her sixth grade class and "thechildren loved it. They could see how the figures were related to each other. I'm going to do more lessons like this." She broughtsamples of the students' workto show how they developedtheirideas and recordedtheirfindings.] Activities 3.6, 3.7: GuessingShapesfromPartialView/Properties - The uncovering shapes is enjoyable. In the uncovering clues [properties], you have to think about it a bit more--it'sa little more difficult--butit's a good activity to follow-up after the children have learned about the propertiesof figures. The activity shows if you understandit or not because you are thinkingthatmuch harder. Activity 3.8: MinimumProperties - This is a tough activity--a challenge--childrenmight get frustrated. My brightkids [sixth graders]would be stretchedbut they could do it; it would get them to think. How could you do this activity with a whole class? [Discussion of this question includedan examinationof the five van Hiele phasesin the workof Dina van Hiele-Geldofwith seventhgraders.] Activity 4: Kites--Sorting,Properties,InclusionRelations - I thinkstudentswouldlike this activity. It'sa nice final assessment. - I like this. The activity would really force students to look at properties. The one way streetarrowsare good. They reinforceyour understandingof the propertiesand theirinterrelationships. Module 2 Activity4: Miniproofsusing saws andladders - Grids are a good method to use. It's a very effective way to have students deduce conclusions. Using saws and ladders in a grid is a nice informal approachto deductioninsteadof using formalconditionalstatementsat the beginning.

Activity 5: Coloring;Angle Sums for Polygons - Now I see why thattheoremis true--thisis a good way of showing children thatthe threeanglesaddup to 180?.

153 - I like this for children. By coloring angles and readingfrom the grid, you have establisheda principleof geometrywhich I hadjust memorized. Activities 6 and7: FamilyTrees;ExteriorAngle of a Triangle - I had not thoughtof interconnectionsamongtheorems. I never knew whatI was doing in 10th grade--but this is a good way to have students see interconnections.This is definitelyhelpful. - This family tree activity would definitely help 10th grade geometry students--theyknow isolated facts--theyhave no connectives for the ideas. This is a very visual way of showingthe interconnections. Module 3 The teachers'commentson Module 3 activitiescenteredon the approachesused in developingthe arearulesandthe use of the "familytree"idea. - We just throwformulasout at children. . . they need this type of hands-on experience.

- The family tree is helpful to show area interrelationships.It gives students insightinto the whole mathematicalsystem. - It is importantfor teachers to help studentsorganize the ideas they learn. The family tree is an extremelyuseful way to help studentsdo this. Teachers' Identification of the van Hiele Thought Levels After the teachershad completedthe clinical interviews, the interviewerand the teachersdiscussed the van Hiele levels of thinkingand the Project'slevel descriptorswere discussedin detail. Videotapedsegmentsof five studentsdoing some of the activities in the InstructionalModules were viewed and students'responses were analyzed in light of these level descriptors. Wherethere were differencesof opinion aboutthe level of a student'sthinking,these were resolved by reanalyzing studentresponses and referringto level descriptors. This analysis and discussion enabled teachers to justify their decisions by citing appropriatelevel descriptors. Some teachers were reflective of their own experiences doing the activities in the InstuctionalModulesandquestionedandcommentedon theirown level of thinking. Sample pages from the geometry strand of several commercial texts (see Chapter9 for description)for grades 3-8 were analyzed by the teachers and the interviewer. Discussion focused on the level(s) of thoughtneededfor the studentto respondto the expositorytext materialandto the exercise section. Hereagainwhen

154 differencesof opinionarose,these were examinedandresolvedby referencesto the level descriptors. In addition, the teacher'scommentaryfor each of the sample pages was reviewedto see how the suggestedmethodologyandextensionsrelatedto levels of thought. Inserviceteacherscommentedon theirown experienceswith text materials and expressed an interest in analyzing their current and future text selections in the context of the van Hiele model. The inservice teacherswere also reflective about their own style of teaching and method of questioning in the classrom--openly asking, "I wonder if the way I teach encourages higher order thinkingskills." This follow-up session showedthatall the preserviceand inservice teacherswere learningto identifyvan Hiele levels of thinkingin studentresponses andlevels requiredby the expositionandexercises in text materials. In the final session the teachersviewed videotapedsegments of four students doing activities from the InstructionalModules. Each teacher recorded (on an Assessment Record Sheet) his/her assessment of the level of thought of each student'sresponses giving some justificationfor the decision (usually citing some level descriptors). The teacherswere also asked to review a set of 10 sample text pages, differentfrom those discussed in the priorsession. Each teacheranalyzed andrecordedwith some justificationtwo aspectsof each samplepage: (1) the level of the expository section, and (2) the level of thought needed for the student to respondto the exercises. An analysis of the teachers'responses relatingto assessing students'level of thinking on videotaped segments when compared to assessments of Project evaluators (staff and consultants) showed an 87% agreement. When a similar comparisonwas made for assessing the level of text exposition and text exercises, the agreement between the teachers and Project evaluators was 78% and 84% respectively. Therewas no patter of commonerrorsin identifyinglevels in either situation. These results indicatethatpreserviceand inserviceteacherscan learnto identifyvan Hiele levels of thinkingin studentresponsesand in text materials. Implications for Teacher Preparation and Classroom Practice In the clinical interviews, many teachers indicated that much of their prior learningof geometryhad been by memorizationand rote. It was their view that teachereducationand staff developmentprogramsand mathematicscoursesshould providepreserviceand inserviceteacherswith appropriateexperiencesto help them gain insight into concepts and make orderlyprogressto higher levels of thinking. They indicatedthatdoing the Projectactivitieshad given themsuch experiences. A primary goal of teacher preparationprograms is to have mathematics teachers develop an appreciationof the deductivenatureof mathematics. A program,such as thatdevelopedby Musserand Burger(1988) at OregonState University, which incorporatesthe van Hiele approach--itslevels and phases--shouldprovidea good model for teachersto use in theirown teachingof geometry.

155 The teachersemphasizedthe majorrole that the concreterepresentationalmaterials (providedin the Project'sactivities)had played in helping them think intuitively aboutconcepts and in assisting them in gaining insight (the AHA phenomenon). They were unanimousin theirenthusiasticendorsementof the hands-onvisual concreteapproachto developinggeometricconceptsfor studentsin grades6-9. The teachersrecognizedand appreciatedthe structureof the Projectactivities which led the student to higher levels of thought, (i.e., materials that can lead studentsto experience topics implicitly at one level and then explicitly at the next higher level). They found the "family tree" approach, while challenging, an effective way of developing thinkingabout interrelationships.The teachers also recognizedthe essentialrole of languagein each of the levels. The teachers in this study consideredtheir experience in learning about and using the van Hiele levels to evaluate student performanceand text material an importantaspect of teacherpreparation. They explained that they could now use this model to identify and analyze thoughtlevels in theirown classroomquestioning, in students'responses, and in curriculummaterials. They believed that a knowledgeof the van Hiele level hierarchywould assist them in planningactivities, and in writing exercises or questions requiringdifferentlevels of thought. They recommended that this type of experience be included in teacher preparation programs. Several findings of the Projectlead to practicalimplicationsfor the classroom teaching of geometry. One finding was that many sixth gradersreporteddoing relatively little geometry in school. In fact, several seemed to be "geometry deprived." This suggests thatgeometryis simply not being given due emphasisin the classroom. It is one of the step-childrenof the mathematicscurriculumwhich tends to be dominatedby the computationalstrands. Studentcommentsdocument this regretablesituation. For example, "Geometryis what a substituteteacherdid" or "Geometry... we usually do thatin June." The responsibilityfor covering the prescribed geometry curriculumis the teacher's,but it is shared by the school curriculumcoordinators,mathematicssupervisors,and the school principalwho need to oversee implementationof this curriculum. Anotherfinding was thatstudents'learningoften involved a reductionof level. Forexample,most studentsin this studyknew arearulesonly by rote. Some triedto recall (ratherthan think out) what their teacherhad told them about the subclass relationshipbetween squaresand rectangles. Thus, when geometrywas taught,it appearedto be mainly at a recallor knowledgelevel. Thereare severalreasonswhy teaching only for recall or rote learning should be avoided. First, such teaching prevents students from engaging in appropriatethinking about geometry topics. For example, students are simply not learning much geometry if they memorize relationshipssuch as "all squaresare rectangles"and "areaof a rectangleis base times height,"withouttryingto explain them, at least intuitively. Second, students

156 tend to forget or confuse memorizedinformationand are often unable to apply it, especially in non-routinesituations. Third, reductionof level conveys the metacognitivemessage thatlearing geometryis just a matterof memorization.This, in turn,preventsstudentsfrom even wonderingif propertiesare true,and if so, why. Teachersin gradesK-4, which deal traditionallywith level 0 thinking,should providechildrenwith a varietyof experiencesthatlay the groundworkfor thinking about shapes in terms of properties. One way is to encourage children to use appropriategeometric language in expressing their ideas. Children should be exposed to challenging, interesting, constructive geometry experiences which naturallyelicit such language. Examplescan be found in the literatureon tangrams, tessellations,symmetry,constructionsof geometricpuzzles, etc. Teachersshould accept children'snon-standardlanguagein initial lessons (e.g., "even"for parallel; "corer" for angle) but gradually wean them from it to more precise language, especially when non-standardlanguage can lead to confusion. The notion that geometry is something to explore, discover, and explain ratherthan to memorize can be developedeven at level 0. To developlevel 1 thinkingchildrenmustbe given many opportunitiesto work with collections of shapes--to discover propertiesof classes of figures,to exploresubclassrelationsand to formulategeneralizations. These suggestions also apply for teachers in grades 5-9, which treat level 1 thinkingandeven level 2 for some topics such as angle sums, area,andpropertiesof shapes and definitions. Teachers here should also be careful about the use of quantifiers(all, some) which are needed for level 1 work. Gregory and Osborne (1975) found a clear correlation between the frequency of seventh grade mathematics teachers' use of conditional statements (e.g., "if-then")and their students'understandingof logical statements. They point out, "Studentsneed modeling from teachersas well as ample opportunitiesto use logic and language" (p. 37). Havingstudentsexplaintheirchoice of "always,sometimes,or never true" for a propertyascribedto a shapeis an effective way to do this. Asking studentsto tell how they would check this deals with the methodologyof level 1 (i.e., inductive thinkingbased on a class of shapes)and can focus theirattentionon metacognitive issues of level 1, namely, what is expected (i.e., to discover relationships, properties)and how to solve the problem(i.e., by testing various cases). Similar recommendationsapplyfor level 2--forexample,askingstudentsto explainwhy the angle sum of a rectangle is 360? (not just asking for the sum) and then having students think about the explanations given. These suggestions point out the importanceof teacher questioning in directing the students' thinking. Raising appropriatequestions,allowinga sufficientresponse-timeanddiscussingthe quality of the answersaremethodsthattake into accountlevel of thinking. The above discussionindicatesa need for preserviceand inservicepreparation in contentandpedagogyrelatedto the van Hiele model. Suchpreparationwill assist teachers in developing geometry lessons that encourage students to think at progressivelyhigher levels.

CHAPTER 9 TEXT ANALYSIS Goal One goal of the Projectwas to study Americangeometrycurriculum(K-8), as evidencedby text series, in the light of the van Hiele model. Questionsinvestigated by the Projectwere: (1) Whatgeometrytopics are taughtby gradelevel? Does the selection of topics indicatecontinuityof instructionandrichnessof geometryexperience? (2) At what van Hiele level are geometry curriculummaterials at each grade level? (3) Is the van Hiele level of materialsequencedby gradelevel? (4) Are therejumps across van Hiele levels, eitherwithin a gradeor from grade to grade? (5) Is the text presentationof geometrytopics consistentwith didacticprinciples of the van Hieles? (In particular,how does text presentationof content strandscoveredin the Project'smodulesrelateto van Hiele'stheoryandto the resultsof the clinical study?) Procedure The Project selected three commercial textbook series (K-8), published in 1980-81, for study. Criteriafor selection were frequencyof use both in the United States (as reportedin the Science EducationDatabook, Directoratefor Science Education, National Science Foundation, 1980), and in local Brooklyn school districtsfrom which studentswere to be drawnfor the clinical study (as reportedby mathematics coordinators). In general, geometry materials intended for the average studentwere reviewed, althoughactivities for an enrichedprogramwere also noted. The text series selected offer a varietyof supplementarymaterials,for example workbooks or duplicating masters for reteaching, extra practice, enrichment, laboratory explorations. Using different selections from these materials, teachers can structurevery different types of learning experiences for their students. However, each series claims that the students'text and teacher's edition used alone can provide a complete mathematicsprogram. Since many teachersand studentssee no moreof a series thanthe students'text, it was decidedto analyzeonly this and the teacher'sedition.

158 Geometryarises in the students'text in four types of formats:in full lessons (usually double pages, but sometimes single ones) accompaniedin the teacher's edition by lesson plans, complete with objectives; in full page activities without accompanying objectives, labelled as laboratoryactivities or as recreations;in smallerinserts(puzzles,brainteasers,challenges)in lessons in othercontentstrands to provide a change of pace; and incidentally,for example, in a lesson on word problems. The teacher'seditionprovidescommentsandteachingsuggestionsfor all of these formats,and also overviews or backgroundfor geometrymaterial,some supplementaryworksheets for practice or enrichment, and sometimes special sections of activities, projects,and challenges,which include ways to enhancethe geometrystrand. All of the above-mentionedaspectsof both students'and teacher's editionwere examined. In orderto addressthe first questionabove, the Projectundertookan analysis of the type and extent of geometryvocabularyused at each gradelevel. This study was similarto Soviet researchon the applicationsof the van Hiele model to their curriculum, which included a listing of geometry vocabulary by grade level, leadingto a measureof the richnessof geometrypresentedand the consistencyof a spiralapproach. To answerquestions(2)-(4) above, an attemptwas made to assign a level to a text page. This attempt led to recognitionof how differentaspects of a page can have differentlevels. It is not the text page which has a level of thinking,but rather the studentreadingit or the teacherteachingit. It appeareduseful to considerboth the minimum level of thought with which a student could complete the page (correctlyrespondto all questions),and the maximumlevel at which the exposition was written (the level of thinking requiredto completely understandall of the exposition). These levels may differfor a given text page:for example,when in the exposition shapes are defined in terms of properties(level 1), but in the exercises studentsare asked only to name figures, which they can do with no referenceto properties,simply "by eye" (level 0). For some exercises it is not appropriateto assign a level of thinkingin geometry. For example,many lessons in the geometry strand concern the application of formulas, and it is often the case that related exercises can be done correctlywith minimalgeometricthought--thatis, by relying on algebraicor arithmeticprocedures. Also, many lessons concerntechniquesof directmeasurement,ratherthangeometricthinking. Thus when van Hiele levels of geometric thinking are assigned to text pages, some pages fit into a category "unassignablelevel of thinking." The two levels of thoughtrequiredby students("minimum"and "maximum") sometimesdiffer fromthe van Hiele levels of two additionalaspectsof the text:the teacher'sedition (aims, notes on teaching and guiding discussion, supplementary activities, etc.) and the tests includedin the text. Lesson aims were classified as being concernedwith: (a) developmentand practiceof vocabulary("identify"and "name"--ingeneral level 0); (b) developing concepts (classifying, formulating

159 properties--ingeneral level 1); (c) developingrelationshipsamong concepts and properties (in general level 2); (d) constructions(e.g., ruler and compass); (e) developing and practicing techniques of direct measurement; (f) practicing and applyingformulas;(g) explorationor recreation; and (h) testing. The suggestions in the teacher'seditionmay promotea much richerexperiencefor studentsthancan possibly be given by a few text pages, and guidelinesfor class discussionmay allow for development of a higher level of thinkingthan the text page. Test questions usually are drawn from the exercises in the lessons of a text, but they do not necessarily reflect all levels of thinkingarisingin the exercises (probablybecause the higher level questions are often labelled as optional, or only for the most able students). In orderto answerquestion(5) above, concerningthe consistencybetweentext presentationsand the van Hieles' didactics,threegeometrycontentstrands(chosen for relationto the contentof the Project'sinstructionalmodules) were reviewed in detail for the three series. The results are summarizedin a later section of this chapter. The Projectconsiderednot only how the text series could be describedin terms of van Hiele levels, but also how the van Hiele model can influence evaluationof textbooks. Examplesof questionssuggestedby the van Hiele model follow. (a) For studentsat level 0, whose thinkingaboutshapesis global, "byeye," and for whom descriptionsof shapes in terms of propertiesare inappropriate, is the visual informationgiven in the text adequate? For example, if all diagramsof squareshave one side horizontal,then this characteristicwill be includedin a level 0 student'sidea of a squareregardlessof whatthe text page says. The Project examined diagramsfor level 0 introductionof shapes for features such as inclusion of multiple orientations and non-examples. Also noted were instructionalproceduresappropriatefor the level 0 studentaccordingto van Hiele--such as physical manipulation andpresentationof conceptsin real worldsettings. (b) A crucialaspectof the progressionfromlevel 0 to level 1 is the recognition of propertiesas characteristicsof classes of figures (not tied to only one specific figure). Do the text materialsencouragethis recognition? Forexample, are studentsled to discover/formulatepropertieson theirown, and on the basis of manyexamples? If a propertyof a shapeis developedon the basis of only one example,it is not clearthatlevel 1 thoughtis involved. (c) Van Hiele claims that implicit in each level are thoughtprocesses which becomeexplicitat the next level (see Chapter2, p. 6). Thusone can choose teaching materials for a level 0 student which allow the implicit developmentof relationswhich will be formalizedlater. Howeverchoices of instructional materials have differing potential for such implicit

160 experiences. The Project decided to look for materials in the texts which point out such potential to the teacher, where it exists but is not noted, and where opportunities are missed. Data Forms and Their Analysis Two types of data sheets were used. Data Sheet A contained a listing of geometry vocabulary used in grades K-8, and provided space for noting use of vocabulary at each grade level. Data Sheet B was designed to be used for each grade level. For each full text page containing a geometry lesson or activity the following aspects were noted. ITEMS ON DATA SHEET B etc.) Pagenumber(andwhetheroptional,forenrichment, Aim(usingclassification describedabove) andused introduced Vocabulary Maximumlevelof exposition Minimumlevel:percentof exercisesatlevel0, level 1, level2, orof "undetermined level" Levelof testquestions(whenapplicable) referenceto "realworld,"suggestionof physicalmanipulation; for correctness, Diagrams: lessonsconcernedwithidentification of shapes,inclusionof non-examples andmultiple orientations Arethesediscovered/formulated thebasisof by thestudent?--on Properties/Relationships: howmanyinstances?Areseveralproperties of thesameshapeconsidered?Are properties logicallyordered? on logicalsequencingof concepts;on Spacewas also providedfor comments,in particular of definitions,statements, gapsin levels;on reductionof level;on correctness/completeness diagrams;on whetherstudentsareaskedto explain;on thoughtprocessescalledfor (visual memory,verbal recall, patternfinding, discovery, logical ordering,analogy, applying formulas,etc.). Notation was made of ways in which the notes in the teacher's edition suggest different entries in the above data sheet than the student's page alone. For example, the proposed lesson plan might suggest a higher level of thinking than the text exposition. (Record was made only of material in instructional notes, not in background material for the teacher.) The back of the sheet was used to note any other occurrences of geometry ideas in other parts of the text, either incidentally, or in small set-off sections (e.g., puzzles, riddles, and brainteasers), and also to note by grade additional geometry resources in the teacher's edition.

161 Pierrevan Hiele examinedan initialformof the datacollection sheets and their use on samplepages duringhis visit to the Projectin 1982, andapprovedtheiruse in analyzingtexts. This techniqueof examiningtexts was also discussedwith Project consultants(JohnDel Grande,KathleenHart)andotherresearchersof the van Hiele model at professionalmeetings. While one researchercompleted all data sheets, other Project staff verified sample sheets and notations for particularpages which exemplified common characteristicsof the texts. In case of disagreementon a text page, a third staff memberwas consultedand in most cases, afterdiscussion,all staff membersagreed on assignmentsof level. The contentsof the completedDataSheets A were comparedfor the threeseries by tabulatingextent and reoccurrencesof geometryvocabularyused. A count was made for each text of the numberof pages primarilydevoted to geometry(including those measurementtopics commonlyincludedin the "geometrystrand,"such as perimeter,area, volume, angle measure,but excluding the developmentof direct measurementof length). Items on Data Sheet B were tabulatedby gradelevel. For example,for each grade,the following countsof lessons were madeby text series: - by categories of aims;

- by maximumlevel on the textpage; - by minimumlevel of thinkingrequiredto complete 100 or 75 percentof the exercises suggestedfor averagestudents; - by gaps in level amongpartsof the text page; - by incorporationof physicalmanipulationandreal world images. These datawere comparedfor the threetext series to determinesimilaritiesand differencesamong the series--inextent of geometryinstruction,level, and style of presentation. By summarizingcommentson thoughtprocesses called for, further characterization of the texts was obtained. Findings Description of Text Series The three text series examined share many features, yet also differed in importantcharacteristics. To provide a context for the later discussion, a brief descriptionof each series is given here. Series A offers a rich variety of geometry experiences, devoting the greatest percentof pages to geometrictopics, and introducingmany concepts which arenot mentioned in the other two series (for example, the four color map problem, pentominoes, center of gravity of a triangle, curves of constant width). The

162 presentationof many topics includes photographsor diagrams of manipulative objects, and a greatdeal of physical manipulationis suggested. The text includes manymathlab activities,geometricpuzzlers,as well as incidentaluses of geometry in a varietyof real world contexts. Howeverexpositiontends to remainat level 0. Students are not often asked to go beyond direct perceptualexperience to make generalities,andmanyopportunitiesto relategeometricfacts logically aremissed. Series B has the least variety of geometric topics. Physical manipulationis suggestedin generalonly in the teacher'sedition,and so unless the teacherprovides supplementarymaterials, students may not get a visual image of the concrete models. Illustrationsoften seem to play a decorative role, ratherthan relating significantly to the content. Some illustrationsare of "realworld" settings, but seldom are they used in the examples in a non-trivialway. Exposition is often at level 1, but studentsare seldom askedto formulateideas, and most exercises are at level 0 or are of unassignable level in geometry, throughoutthe series. (An exception to this generality is provided by some of the masters in the teacher's edition intended for "enrichment",which often provide level 1 experiences, especially in the uppergrades.) Logical relationshipsare not exploredin general. There are fewer opportunitiesfor geometric exploration unrelatedto acquiring skills (of measurementor formulaapplication)thanin the othertwo series. Finally, thereare severalcontenterrorsin the answersprovidedin the teacher'sedition (for example,in identifyinga squareas a non-rectangle),which may cause a problemfor teachersrelying on this resource. Series C sharesmany of the characteristicsof Series A (presenceof real world applications,suggestion of manipulativesin the studenttext, exploratorylessons), but lacks the rich variety of geometry topics covered. However lessons tend to requirea higherlevel of thinkingfromthe studentsthanthe othertwo series. In the upper grades especially, this series provides level 1 or even level 2 exposition, sometimes with exercises to match. This was the only series surveyed to ask studentsfrequentlyto explainan answer. Results of Analysis of Data Sheet A Geometry Vocabulary. When a count was made for each grade level, the numberof topics coveredby at least one of the series increasedsteadilyfrom 11 for Kindergartento 121 in eighth grade. (A total of 152 topics were identifiedin all of the texts reviewed.) The summaryof Data Sheet A showed that there is general consistencyon scope andsequenceof majortopics--thatis, those thatarecoveredon the summarizingtests, especially identification of shapes, and measurementof perimeter,area, volume, and angle. Below are listed topics on which the series essentiallyagreed(i.e., therewas no more thanone omissionby one series over the grades indicated,althoughthe topic may have been introducedearlierby just one series).

163 Identification of shapes: Square,Triangle,Circle(K-8) Rectangle(1-8) andTriangular Parallelogram, Pentagon,Hexagon,Octagon,Polygon,Rectangular prism(4-8) Rhombus,Equilateral Quadrilateral, triangle,Cone,Cylinder(5-8) Scalene,Isoscelestriangle,Regularpolygon(6-8) Trapezoid(7-8) Measurement: Perimeter(4-8) Areaby counting(2-6) Areaby formula:rectangle,triangle(5-8); circle(6-8); parallelogram (7-8); trapezoid(8) Surfacearea(6-8) Volumeby counting(3-6) Volumeby formula:rectangular prism(5-8); cylinder(7-8) Angle: Meaning(4-8) Rightangle(4-8) withprotractor, Acute,obtuseangle,Measuring Anglesumof a triangle(5-8) (8) Anglesumof a quadrilateral There is much more variation in scope and sequence of topics related to parts of shapes, their properties, and relations among properties. Below are listed some such topics, together with the initial grade of introduction for the three series (in order, series A, B, C). A hyphen "-"indicates that the topic was not covered in any grade. Polygons Segment(2,6,6);Vertex(5,1,2);Side (4,1,2); Diagonal(6,8,8) Circles Radius(3,3,4);Diameter(4,3,4);Arc (4,5,7); Circumference (6,6,6);Pi (6,7,6) Congruence: of sides(7,7,3);of angles(7,7,5); of figures(3,5,3) Similarity: of figures(3,6,5) Scaledrawing(7,7,7) Indirectmeasurement (6,5,7) Triangles Congruenttriangles(8,8,8) Theorem(7,7,8) Pythagorean ratios(8,8,8) Trigonometric Symmetry: of figures(3,1,3);Lineof symmetry(4,3,1) Lines Parallelin a plane(5,4,4);in space(6,7,-) Alternateinterior,exterior,corresponding angles(7,8,8) Intersect(5,4,4); Perpendicular (5,5,5);Skew(6,7,-) Polyhedra Vertex,Edge,Face(3,6,5) Euler'sformula(5,-,8)

164 There is considerablevariationin the treatmentof transformationalgeometry. Series B covers slides, turns and flips consistently in grades 4-7, while series A touches on one or two a year in these gradesand all threein grade 8, and series C only mentionsslides andflips briefly in grade7. Solid geometryreceives similarly varied treatment--inthis case series A has the most complete coverage, including identificationof generalprismsand pyramidsand the five regularsolids. Series C omits namingthe regularpolyhedra,and series B omits all of these solid geometry topics. Ruler and compass constructionsstartin grade5 in series B, in grade6 in series A, and in grade 7 in series C. By grade 8, the series agree on major constructionscovered. SeriesA providesmuchmoreexposureto unusualgeometrytopics andSeriesB much less, with Series C being in the middle. This was demonstratedby analyzing the summaryfor Data Sheet A as follows. A count was made of each topic/grade level entryin which only one seriesappeared.These topics were: kite, convex/concave,inside/outside/on,straightangles, skew lines, rotational symmetry, cross-sections of polyhedra, icosadedron, dodecahedron, octahedron,tetrahedron,explanationof angle sum of a triangle,angle sum of general polygons, spiral, golden rectangle,parabola,estimation of irregular areas Series A appearedin 11 such entries, series B in 1, and series C in 5. About 80%of the topics identifiedare coveredby all series in some gradelevel, and more thanhalf of the ones omittedby at least one series are "recreational". In general, once a topic is introduced,it is reviewed each successive year, except for the "exploratory"topics (such as cross-sections of polyhedra,optical illusions, mazes, spirals,rotationalsymmetry). This was demonstratedby a count for each series of the numberof topics introducedbut not returnedto within two years. Series A had 14 such topics, seriesB had 2, and Series C had 7. These numbers reflectthe inclusionof morerecreationaltopics in series A and C. However,in this respecta count as simple as the one on Data Sheet A does not revealthe whole story. For example,polyhedra(in particular,prisms and pyramids)are introduced in grades 4-5, and for the remainingyears surveyed continue to be mentioned. Early work focuses on classifying and naming them, and maybe counting faces, edges andvertices. But by the latergrades,the polyhedramay only be mentionedas a context for applyinga formulafor surfaceareaor volume by substitutingvalues into algebraicexpressions (in some cases, the two-dimensionalplans of the solids are provided), and thus studentsmay not be requiredto increase their geometric understandingof polyhedraor their ability to visualize them. Anotherexample is providedby symmetry. While this topic is mentionedin five successive gradesin two series, and in eight successive gradesin the other,for the most partthe level of thinking remains the same, and first grade questions remarkablyresemble fifth grade questions on this topic. Thus, in these cases, repeatedoccurrencesof the

165 topics may represent a "circular"rather than a "spiral"curriculum. A more meaningful descriptionof spiralling of curriculumis obtainedby examining the level of thinkingrequiredtogetherwith the topic count. Results of Analysis of Data Sheet B Number of Pages Devoted to Geometry. The series do vary considerably in the numberof pages devotedto geometry.Even withoutcountingthe many recreationalpages, series A has considerablymore than the average number of geometry pages in four of the grades 3-8. Series B, on the other hand, is consistentlybelow averagefor seven of the grades1-8, and containsno recreational pages. Series C lies in the middle,with a highernumberof pages in grades2-4. The series also differ in the amountof geometryarising incidentallyin the texts. All series include, as a regularfeature of the text page, corers devoted to puzzles, brainteasers,problems, etc. In series A and C these are frequently geometric. Series B includesmanyfewer in geometry. Aims of Geometry Lessons. The objectives provided in the teacher's editionsof text seriesaregenerallytied to test items. Van Hiele statesthat"ifthe test problems cover a limited field of acquirements,algorithmicskill will suffice to solve them. It is a fairly simple matterto teach a child a numberof manipulative structuresthat enable him to reduce a high-level problem to a lower level of thinking"(van Hiele, 1957/1984, p. 241). (Van Hiele refers to this process as "reductionof level.") Thus, he continues,some types of insight "cannotadequately be assessed by means of test-papers."For this reason,one might not expect either text objectives or tests to indicate thinkingat level 1 or 2, and this expectationis confirmed in the series examined. (Results of the analysis of test questions are describedlater.) The instructionalaims of lessons in the grades K-3 text series examined are predominantly "identify" or "name" (development and practice of vocabulary--level0), or "measure"(directmeasurement,such as counting square units--"unassignable"level), and this patternis followed in all three series. In grades 4-6 there is, in addition,an increasein aims of the type "draw"(e.g., ruler and compass constructions--donealgorithmically, thus level 0 or unassignable level), or "findthe measureby applyinga formula"(unassignablelevel). As noted above, one dramaticdifference among series is in the inclusion of geometry activities for which no learningobjective is given and/orfor which the most appropriateaim appears to be open exploration or recreation. Series A includesan averageof eight of these per grade,Series C includesan averageof two per grade, while Series B offers only two in all seven grades. The lessons which were classified in this way do not lead studentsto formulatepropertiesin general, and thus are at level 0. (However,in series C the lessons aremore likely to relateto otherpartsof the geometrystrand,andto have potentialfor level 1 thinking.) In all

166 of the series in GradesK-6 only nine lessons were foundwhose statedaims were to formulateproperties(level 1), and of these, five were intendedonly for the most advancedstudents. The fact thatthe statedobjectives(and,as will be discussedlater,the relatedtest questions)are so overwhelminglyat level 0 or of unassignablelevel in gradesK-6 does not necessarilyindicatethatelementaryschool studentsare exposed to no level 1 thinking. For example, in a lesson where studentsfill in a chartcontainingnumbers of faces, edges and verticesof polyhedra,the statedobjectiveof the lesson is to identify or count these aspects, and all testing is on this level. However, students may well be askedto notice patters or generalcharacteristicsof shapesin the exposition of the text page, or in the lesson plan recommendedin the teacher'sedition. No text page in gradesK-6 of the series surveyedhas as an aim for studentsto relatepropertieslogically. The developmentof logical reasoningdoes not appearto be an importantaim in teachingelementaryschool geometry. The sixth gradetext of Series B providessupportof this statement. In the introductionto the chapteron geometry,the teacher'sedition points out that special activities relatedto "Logical Reasoning"are includedthroughoutthe series, and that in this chapterthereis one which turnsout to be unrelatedto geometry. In the such activity, a "brainteaser" same text, a page in the geometrysection labelled "Be Reasonable"turs out not to refer to reasoning about geometry, but rather to checking reasonableness of numericalresultswhen applyingformulasfor areaand perimeter. The curriculumin grades 7 and 8 offers many opportunitiesfor developing generalpropertiesand informallogical relationships,but most of these are missed, and "reductionof level" is common. The characteristicsnoted above for gradesK-6 hold true for grades 7-8. Series A again containsby far the most recreationalor exploratoryactivities (over twenty related to geometry), while series B contains none and series C contains two. (However series B, in particular,does provide some experiences of this sort through"enrichment"masters in the teacher'sedition.) The aims of over half of the geometrylessons in each series concernapplication of formulas, or ruler and compass constructions. Direct measurementtechniques play a much less importantrole thanin earliergrades,but a largenumberof lessons are still devoted to identifying and naming geometric figures. Series A containedsix lessons with aims relatedto level 1 thinking(developingpropertiesor relationships);series B and C both containthree. Howeverthis count of aims does not entirely reflect the level of thinkingin the text presentation,because even in lessons wherepropertiesor relationshipsare developed,the statedaim might be to applya formulaor to measure. (Thishappensoften in series C in the uppergrades.) Van Hiele Levels of Text Materials. As discussed above, one can assign levels to variouspartsof the text page. Table 5 (on the next page) summarizesthe "maximum"level of text pages, that is the level of thinkingrequiredto understand all of the exposition on the text page. For each grade level andeach series,the

167 Table 5 Percent of Lessons at Maximum Level 0, 1 or 2 Grade K 1 2 3 4 5 6 7 8

0 100 100 100 95 96 71 80 58 66

Series A 1 2 0 0 0 0 0 0 0 5 0 4 0 29 0 20 5 37 20 14

Series B 0 1 100 0 100 0 100 0 88 12 71 29 65 35 55 45 67 33 29 68

2 0 0 0 0 0 0 0 0 3

Series C 0 1 2 0 100 0 0 100 0 0 100 0 0 92 8 0 75 25 85 15 0 0 60 40 0 43 57 19 63 18

percent of pages with "maximum" level 0, 1 or 2 is shown. Assignment of level followed the Project's level descriptors. If discussion of aspects of a figure involved only what could be seen on the page, level 0 was assigned; however, if statements referred to a class of figures, level 1 was assigned. Level 2 was assigned only if some logical relationship between geometric properties was discussed. Lessons concerning techniques of direct measurement, or giving directions for a construction with no attempt at generality or explanation are here classified under level 0. This table indicates that the exposition in the three text series follows a fairly consistent pattern. Level of thinking required to follow the lessons increases gradually. Series A tends to include more entirely level 0 lessons throughout the grades (in particular, the large numbers of recreational activities). As compared to the work of students (grades 7 or 8) reported on in Dina van Hiele-Geldofs thesis, little level 2 thought is required here. Table 5 does not necessarily reflect the level of thinking of students using the text, who may complete exercises correctly without following the text. As long as a student is getting correct answers to exercises, it is probable that neither student nor teacher will worry about understanding the expository parts of the text. Thus Table 5 should be compared to Table 6 (on the next page), which shows the minimum level of thinking required to do the exercises. Entries show the percent of pages at each grade level which can be done with geometric thinking of level 0 or of "unassignable" level. For each series there are three columns. The first column shows the percent of exercises at level 0 or "unassignable" when only the exercises recommended for the average student are considered. The second column shows the corresponding percent when exercises for enrichment, or for the above-average student are also counted in. In the third column, the figures in parentheses are the percent of pages where at least 75% of the exercises for average students can be done with the lowest level of thinking. From Table 6 one can see that a level 0 student could do very well in geometry as evaluated by the exercises in series B.

168 Table 6 Percent of Lessons with Exercises All at Level 0 or "Unassignable" Gade

K 1 2 3 4 5 6 7 8

Series A Av. Enr. 75%

Av.

100 100 100 95 100 94 66 92 89

100 100 100 100 100 88 100 100 100

100 100 100 95 100 94 80 92 89

(100) (100) (83) (97) (91)

Series C

SeriesB 7 100 100 100 100 86 88 (94) 95 93 93

Enr 75%

Enr. 100 100 100 100 80 96 100 78 78

100 100 100 100 70 96 96 52 59

(95) (96) (83) (81)

Note: Entriesin columnsheadedAv. includeonlyexercisesintendedfortheaveragestudent. Entriesin columnsheadedEnr.includealso "enrichment" exercises,andexercisesintended for aboveaveragestudents. Entriesin columnsheaded75% show the percentof lessons whereat least 75%of the exercisesforaveragestudentscanbe doneatlevel 0 or "unassignable". Series A and C both require a certain amount of level 1 thinking. (The only exercises which required level 2 thinking are in series C in grades 4 and 7, and concern inclusion relations.) A level 0 student could do well even in the "enriched program" in series A and B. Only in series C would such a student frequently be given exercises which required a higher level of thinking. Analysis of Test Questions. A third table which should be included at this point is one showing the level of test questions by grade level. This table would be a very simple one! Almost no test questions were found in the three series surveyed which could not be done with level 0 thinking, or else with rote learning of a formula (e.g., substituting values into an algebraic expression for area) or a procedure (i.e., ruler and compass constructions). The only exception was in the grade 8 text of series C. Here 24% of a mid-chapter test consisted of level 1 questions relating to properties of shapes. The end-of-chapter test contained about 5% level 1 questions, but in a diluted true-false format. No level 1 geometry questions appeared in the end-of-book test. One feature all of the grade K-6 texts share is the scarcity of questions that require a sentence for an answer. Many questions require only a yes or no answer, or identification of a figure (e.g., "List the letters of figures which show parallel lines."). If students are asked to name a figure, there is very often a list of names provided to choose from. This is especially true in the test questions. This characteristic is related to the lack of level 1 thinking in the exercises, because without explanation, it is often easy to make judgments "by eye," at level 0.

169 Summary: Van Hiele Levels of Texts. In response to question (2) concerningthe van Hiele level of geometrycurriculummaterials,it was found that the text series examineddiffer somewhat,but sharea generalpattern. Thereis very little level 2 thinkingexhibited in the texts, startingonly in grades 7-8. There is some level 1 thinking,from grade3 on. However, averagestudentsdo not need to think above level 0 for almost all of their geometryexperiencethroughgrade 8, if one judges by the minimumlevel requiredto completeexercises and test questions relatedto geometry. As for the sequencingof geometrymaterialsby level of thinking(question(3)), materialin the exposition of the texts is in general sequenced by level. But this question is not relevantto considerationof exercises and tests as there is so little progressionhere beyond level 0. Question (4) concernedjumps across van Hiele levels. The most significant jumpsfound were withina text page. Frequentlyexpositionis at a higherlevel than exercises. If studentscan completeexercises and tests on a topic at level 0, even if the exposition is at a higher level, it is increasingly likely as students progress throughthe grades that there will be a significant jump in level from their own previousexperiencein geometryto what is requiredin the text exposition. Students will presumablyencounterdifficulty with a secondaryschool geometry course at level 2 if they can successfullycompletegrade8 with level 0 thinking. The next sections consider aspects of texts relevantto question (5)--whether text presentationis consistentwith didacticprinciplesof the van Hieles. Characteristics of Diagrams. Various characteristics of text diagrams were noted in the study, including correctness,referenceto "realworld"models, and suggestion of physical manipulation. It is especially appropriateto examine these aspects of text presentation for students thinking at level 0, who are particularlydependenton visual impressions, and it is apparentfrom the above tablesthata greatdeal of elementaryschool instructioninvolves level 0 thinking. Van Hiele emphasizesthe importanceof a "globalstructure,"the involvement of studentsin theirsurroundingsin a way which helps them to form geometricconcepts. All of the texts include illustrationswhich are meant to refer to common objects. However,often objectsare takenout of context,or objects in the drawings look distorted to show neat geometric shapes (e.g., trees which are equilateral trianglesor circles). Sometimesthe artworkcould evoke many geometricconcepts, but is not referredto in more thana trivialway in the exposition(e.g., a pictureof a quilt with congruentand similar triangles, but with no reference to this aspect). Therewas not much differenceamong the series: the averagepercentof lessons at each grade level for which significantreal world images were noted was 24% for series A, 17%for series B and 20% for series C. In many cases the teacher'seditions suggesteduse of the students'environmentfor developinggeometryconcepts.

170 Occasionally the attempt to include visual references to the "real world" backfires. For example, a lesson on symmetry in the grade 2 text of series B concernedstudentsflying kites. Studentswere asked to identify symmetrickites. One example drawnwas a kite made of a series of circlesjoined in parallelplanes, certainlysymmetricin its usual flying position. However,the drawingshowed it in a curvedposition,so the kite was identifiedas unsymmetric(which would surelybe confusingto a studentwho had actuallyseen such a kite). The grade 8 text of this series provided a similar example, in which the text identified a two dimensional while a similarreal cup would have a plane of pictureof a cup as "unsymmetric," Such confusion a three-dimensional object and the between symmetry. two-dimensionaldrawingof an objectis a commondifficultywhen texts use visual representationsof the real world. In series B, there are also examples of the confusion of a concept with its symbolic representation in a diagram. Third grade teachers are directed to "emphasizethat a line has arrowsat both ends"(as opposed to discussing this as a characteristicof the symbolfor a line). The accompanyinglesson, in which students aretaughta four step procedureto make a line segment(marktwo points,label one, label the other, use a ruler to connect the points), provides an example of an extremelyproceduralapproachto a concept,ratherthanan intuitiveone. This type of lesson might suggest reductionof level, where conceptformationis reducedto a matterof makingandrecognizingmarkson paper. The van Hieles also emphasizethe importanceof having studentsphysically manipulategeometricmodels. Of course the printedpage cannotprovide a model itself, but artworkor directionscan stronglysuggest physical manipulation.Series A used such artworkor directionsmuch more frequentlythanthe othertwo series. The averagepercentof lessons, acrossgradelevels, in which physicalmanipulation was strongly suggested on the text page was 52%. For series B and C the figures were 21%and 19%respectively. Howeverthe teacher'seditionwould often suggest a manipulativematerialwhen the text page did not. (In both series, when these suggestions were also counted, the above averages were closer to the figure for series A.) The materialssuggested includedpaper (folding and cutting), mirrors, rulers, geoboards, tracing paper, blocks and sand for volume measurement, protractor, compass and straightedge. Seldom used were materials such as geostrips,variousshapesof tiles, or collections of shapesto be sorted,which allow studentsto explore, constructand arriveat their own discoveries of propertiesof figuresor of classes of figures,thuspreparingfor level 1 thought. The diagrams in level 0 lessons dealing with identification of shapes were examinedwith regardto the inclusion of non-examplesas well as examples. (For example, when rectanglesare first mentioned,can studentsassume that they need only have "twolong sides and two shortsides,"or are any examples without right angles identifiedas not being rectangles?) All threeseries have examples of such lessons where misconceptions could easily be formed due to omission of

171 non-examples; however, series C has more pages which include non-examples. Sometimesstudentsareaskedto makedecisionsaboutexamplesin exerciseswithout information being provided in the exposition. Lessons concerned with shape identification were also examined with respect to the inclusion of multiple orientationsor other variationsof a figure. Again, all series containedexamples wheremisconceptionscould easily be formed. Series C containedthe most lessons where multiple orientationswere introduced. Some common examples of lessons lackingvariationson a conceptwere: - squares(or rectangles,or triangles)whereone side is always horizontal; - triangleswhich are alwaysacute,andhave no very small angles; - rightangles or righttriangleswhere one side is always horizontal; - parallel lines always representedby line segments of the same length, or formingsides of a rectangle; - polygons which are alwaysconvex; - lines of symmetrywhich are always horizontalor vertical; - prismsandpyramidswith bases alwayshorizontal; - in applicationsof the area formulafor triangles,bases always horizontal, altitudesdrawnin, vertically; - trapezoidsalways drawnwith parallelsides horizontal. In its clinical study,the Projectnotedexamplesof students'faultyformationof conceptssuch as these,perhapsbasedon similartext presentations. Formation of Properties and Relationships. In general, only one instancewas discussed when a propertyor relationshipwas introduced,especially in series B. This is a significantpoint in the considerationof a text in the light of van Hiele level theory. To reach level 1, students must recognize properties as characteristicsof classes of objects,ratherthanas attributesof a particularexample. If a propertyis discussed in the context of one diagram,it is probablethat a gap in communicationwill occur--thatis, the text authormight see the one example as a general instance (at level 1 or even 2) but the studentmight see it as the object of studyitself (at level 0). For most importantproperties, the text states the property, and then asks studentsto verify it in one, or frequentlymore,examples. Sometimesthe studentis asked to fill in blanks relatedto a figure (e.g., for a rectangle:length, width, area), and the text will then state the relationshipand then ask for application of the

172 formula. The exceptions to this patterntend to be in the "recreational" geometry topics--thosewhich do not appearon tests, and which are labeled as optional (for example, Euler'sformulafor polyhedra). The K-8 text series examinedcontainedvery few examplesof level 2 thinking, whetherexplicit or implicit. Therewere very few examplesof relationshipsamong properties. Also therewere few examplesof methodsof presentationwherelevel 2 reasoningis possible implicitly. The topic of rulerand compass constructionsis especially interestingin this regard,because it can provide such a rich area for developing relationships. The texts surveyedcompletely miss this potential,both in the studentpages and in the teacher'sguide. The individualconstructionsare in generaltaughtas a sequenceof stepsto follow, with no logical connections. Studentsarenot askedto considerwhat line segmentsmustbe congruentas a resultof constructionby circles, andhence are unlikely to recognize that certainshapes (isosceles triangles,rhombuses)underlie the constructions, and that their properties are related to the results of the construction. For example, the propertiesof the diagonalsof a rhombusrelate to the constructions of perpendicularbisector of a segment, angle bisector, and perpendicularto a line througha point. (Of course such propertiesof a rhombus are not developed in the text series surveyed.) Seldom are constructionsrelated even to other constructions. A mathematicallytalented student may form these relationshipsindependently,but texts, even in the teacher'sguide, give little help. Text Presentation of Three Content Strands, as Related to van Hiele Didactics One furtheraspectof the text analysiswas a detailedexaminationof the content strandscorrespondingto the three instructionalmodules developedby the Project. For each strand,the text presentationswere consideredin the light of the Project's researchinto the van Hieles' own instructionalprocedures,and those developed in the Project'smaterials. The threestrandswill be discussedseparatelybelow. Properties and Relationships among Polygons It is interestingto examinehow text series handlethe relationbetween squares andrectangles,so often misunderstoodby upperelementaryschool students. Some series scrupulouslyavoid situationswherethe issue arisesin earlygrades:if students are asked to identify rectanglesin a Grade 1 text in series C, no squareswill be shown. This continuesuntilthe text has presentedsquaresandrectanglesin termsof properties(grades 3-4). Of course if squares are omitted from all examples of rectangles, it is entirely naturalfor students to form the incorrect definition of rectangleobservedso often in the clinical study;that is, rectanglesmust have "two long sides and two short sides" in additionto the standardproperties. Series B is

173 careless in this regard, providing situations where a square is definitely not identifiedas a rectangle,as describedabove. In an activityin the grade2 text of this series, studentsare directedto color a pictureso that all squaresare one color, all rectanglesanother. In this case the artworkprecludesrecognitionof a squareas a special type of rectangle,or even of a rectangleas being composedof two adjacent squares (how should one color this?). For contrast, series C contains a similar activity in the grade 2 text, but here only triangles,circles and rectanglesare used, andso thereis no conflict. Series C was the only one to attempta full andcorrectformationof the concept of rectangle in Kindergarten,when in a story it is pointed out that squares are indeedrectangles. But this approachis not followed in grades 1-3 in this series. By grade7 all of the seriesexpect studentsto be able to identifysquaresas specialtypes of rectangles. Yet, in general, studentsmay never have achieved level 1 thinking aboutthese shapes, since they need never respondin exercises at a level above "it looks like"(level 0) or by counting/measuring aspectsof an individualfigure. The text treatmentof inclusion relations sometimes suggests that different minds were at work producingdifferentaspectsof the series. An exampleof this is providedby the Grade3 text of series B in materialconcerningthe relationbetween squaresand rectangles. First, there are tests of Basic Skills in a multiple choice formatat variouspartsof the text. One questionasks "Whichis a rectangle?"and a squareincluded among the possible answers is markedincorrect. (A non-square rectangleis also included.) In a laterlesson, studentsare askedto measureandcount sides of a squareand rectangle,and the text states definitions. The accompanying lesson plan instructsthe teacherto "besure studentsunderstandthatevery squareis a rectanglebecauseit fits the definitionof a rectangle,"and, in addition,teachersare told to writethis fact on the board. Howeverin the exercisesfor this lesson students are asked to "writetriangle,rectangleor square"to identify given figures, and the answers provided indicate that multiple answers are not intended. This lesson suggests a reductionin level, for studentsmay memorizea sentencewhich they do not really understand, and they are not expected to interpret or apply it in subsequentexercises. Development of Angle Measurement and Angle Relations for Polygons The three series are consistent in providing some informal work with angle (recognizingright angles "byeye" or superpositionof a squarecorer, comparing angles with right angles) in the grade 4 texts, and in introducingmeasurementof angle with a protractorin grade5. No series providesmany examples of congruent angles which appearquite different (for example, in length of sides), and so one might say that the level 0 developmentof this concept is lacking. (In fact in her thesis, Dina van Hiele-Geldofsuggeststhat"angle"is a difficultabstractionto form at level 0.) Use of a protractoris taughtby rote, and nowhere(in eithertext pages or lessons suggestedby the teacher'seditions) are studentstold what a "degree"is,

174 thatis, thereis no developmentof the conceptof a unit of measurefor angle. SeriesA andC introducethe sum of anglesof a trianglein grade5, andSeriesB does this in grade6. All developmentof this propertyis by experiment(except for the grade8 treatmentin Series C), usuallyby one or both of the following methods: measuringangles with a protractorand adding;or by cuttingout a triangle,tearing off the corers, and arrangingthem on a straightline. (Since measurementof a straightangle is omitted before this latter experiment,it is especially likely that studentswill learn the propertyby rote.) Usually the experimentis performedno more than twice, allowing little opportunityfor generalization. The objectives given in the texts for the lessons concernedwith angle sum of trianglesare for the most part"tofind a missing angle of a triangle"or "tomeasureangles of a triangle," andit is only these skills thatare tested. In all series,many opportunitiesare missed for level 1 development,since the text usually formulatespropertiesand does not requirethe studentto try a numberof examples. In grades7 and 8 of Series A, the angle sum of a quadrilateral is developed by measuring and paper-tearing experimentsimmediatelyafterthe correspondingfact for trianglesis reviewed,but no logical connection is drawn. In grade 8 of both Series A and B, the lesson on angle sums of polygons is placed before one on angle relationshipswhen parallel lines are cut by a transversal,so no logical use can be made of these relationships. Series B does require an advanced eighth grader to apply the procedure of subdividinga polygon into trianglesto find angle sum of a decagon,but emphasisis on the numericalaspects,andstudentsmay easily do this task while confusingangle with areameasure(as was foundin severalclinical interviewswith ninthgraders). Forthis contentstrand,series C offeredthe highestlevel of thinking. While the is includedonly as an additionalactivityin the teacher's angle sum of a quadrilateral edition,the grade7 text does have some level 1 or 2 questionsfor advancedstudents related to angle sum of a triangle. For example, "Can a right triangle be equilateral?"The grade 8 text provides a level 2 exposition of the angle sum of a triangleusing angle relationshipsfor parallellines and a transversal,and a level 1 activityof findinga patternafterfilling in a chartfor angle measuresof polygons of variednumbersof sides. However,test questionsresemblethose in the otherseries, namely,all level 0. The developmentof this topic in the threetext series differs significantlyfrom the developmentrecommendedby the van Hieles. First, the van Hieles propose presentationof the materialin a structure--"When proof is given in the right way, the result is first readfrom a structure;afterwardsproof is given by arrangingthe elements of the structurein a logical way" (See Dina van Hiele-Geldofs thesis, Chapter 14). An example of this is given in the technique used by Dina van Hiele-Geldof, and in this Project'smodules, where triangles are presentedin the structureof a grid, and the proof is readfrom the grid using the languageof "saws" and "ladders."The experimentaltechniquesused to establishangle sum in the text seriesprovideno such structure,and so the proof when finally given is not basedon

175 priorimplicitexperience. The van Hieles explicitly recommendagainstthis heavy relianceon measuringor cuttingandgluing as techniquesin learninggeometry. Development of Measurement of Area Measurementof area by counting square units is introducedby Series C in Kindergarten,by series A in second grade,andby series B in thirdgrade. All series continueto provideexamplesof findingareaby counting(for example, "howmany little squarescover this shape?")in subsequentgrades. Grade5 is the standardtime to intoduceformulasfor areaof a rectangleand righttriangle,and grade7 for area of a parallelogram. The three series agree on the way of introducingthe area formulas. After much experienceof countingnon-rectangularshapes made up of squareunits, students count a few rectangles,and then are shown the formulato summarizethe result. The meaning of multiplicationis not explicitly referredto; ratherthe exposition jumps from a "fourrows of five" type of language to the formula. (The resultsof the Project'sclinical study indicatethat some studentsdo not relateareameasurementof a rectangleto a model for multiplication,and hence do not really understandwhy they should multiply, beyond "it gives the right answer.") Thereafter,exercises involve substitutingvalues into the formula. The same patternis followed for area of a triangle and parallelogram. The exposition may include a level 1 developmentof the formulas,but the exercises involve only applicationof the formula. In most cases the altitudeof the triangleis provided(with no excess numbersincluded--exceptfor four examples in grade8 of series C) and so studentsneed never think of the meaningof altitude,and will be likely to make errors in finding the altitude if it is not provided. The Project's clinical study confirms this. Studentsare never asked to explain the formula,or even a procedure for finding area. While the exposition may explain why the formulas give the correct area, the exercises do not encourage thinking about logical relationshipsbetween the formulas, nor are alternateways to derive the formulasconsidered. In summary,this topic provides many examples of reductionof level since studentscan easily do all the exercises correctlyby memorizingsome facts thatthey may not understand. Objectives related to area seem to be predominantly applicationof formulas,and seldom includedevelopmentof the meaningof area,or explanationof the formulasor relationshipsamongthem. Implications The considerationof levels of thinkingin the contextof geometrytext materials is a timely one. Data from the Second InternationalMathematics Study (a crossnationalanalysis) indicatethatthe performanceof Americanjunior and seniorhigh school studentsin mathematicsis mediocre--slightlyabove averageon computation,

176 but "well below averagein answeringmore sophisticatedquestions, such as word problems"(Reportedin The New YorkTimes, September23, 1984, page 30). This suggests thatstudentslack higherorderthinkingskills, which may be relatedto the van Hiele levels of thinking. The average Americanachievementin geometry is "exceededby 75 percentof othercountries."Since the text is an importantinstructional tool in Americanclassrooms,this text analysis might have implicationsfor ways in which to improvethis unsatisfactoryperformanceof Americanstudents. The results of this text analysis have implicationsin three general areas:for further research into curriculum materials, for the design of text and other curriculummaterials,and for classroompractice. Implications for Further Research into Curriculum Materials This study focused on the geometry strand in the text series. It would be interesting to find out if the characteristicsfound in this strand held for other strandsin the mathematicstext series. In particular,is the level of thinkingabout numericaltopics similarlylow, and is reductionof level as common? This studywas limitedto threemajorcommercialtext series publishedin 1984. Differences were found among the three series. Furtherresearchmight indicateif the range of characteristicsof these series are typical for other commercialseries. Have calls for changein texts in recentyearsproducedany changesin the aspectsof texts examinedin this study? Are therenow any commonlyused text series which involve more level 1 and 2 thinking,and which are more consistent with the van Hiele model thanthe ones surveyed? In particular,it wouldbe interestingto look at some of the more innovative,thoughless frequentlyused, Americantext series. It would also be valuableto look beyond text materials,to the many other resources for teaching geometry available to teachers, for example, sets of activity cards, enrichment masters, teaching guides to accompany commercially available geometry manipulativematerials. Foreign text series might also be examined. Most interestingwould be to examine the texts writtenby Pierrevan Hiele himself (availableonly in Dutch at present),and also Soviet texts which were revisedbased on van Hiele principles. Implications for Design of Text and Other Curriculum Materials Perhapsthe most significantimplicationsof this study of text series lie in the area of suggestions for design and revision of geometrytext materials. Textbook authorsare, of course,undersevere restrictionswhen it comes to writinggeometry material.Only a set numberof pages can be devotedto the geometrystrand,andthe natureof the printedpage makes certain types of manipulativeexplorationsand discoverylearing difficult. A text page can only suggest, but not dictate,use of a manipulativematerial. Even if an authorwould like to have studentsdiscoverand formulateproperties,thereis a need to have a text summarizeresults,and students

177 may easily learnhow easy it is to look aheadfor the answerto a challenge. Perhaps these considerationssuggest that teachersshould look beyond a text for geometry instructionalmaterials,to activity cards,or sequencedworksheets. However, even within the context of a textbook, this text analysis does provide suggestions for textbook authors who wish to develop curriculum materials which are more consistentwith the van Hiele model of thinkingin geometry. First,the teacher'sguide might be more explicit in identifyingvan Hiele levels of partsof the text, and in helpingteachersplan instructionto fill in levels and lead to a higher level of thinking. Textbookauthorsmight considerusing the structure of van Hiele's "phases"as a guide in planninginstruction. More attention should be given to selection of visual examples in lessons involving level 0 thought. In particular,care shouldbe takenthatstudentsnot form incorrectconcepts (such as those listed on page 158) based on a too limitedrangeof examples. Also more care could be takento build up and use the student'sglobal structurein which these conceptsarise. This text analysis noted a deficiency in level 1 thinking, especially in the exercises and tests provided. More opportunityshould be providedfor studentsto advance to and use level 1 thinking. In particular,students should be led to formulatefor themselves, on the basis of many examples, propertiesof classes of figures. Exercises and tests should reflect this type of activity, so that studentsdo not become accustomed to reduction of the level of thinking, and so that they develop a new view of whatthey arelearningabout(thatis, notjust namingfigures, but observingand comparingcharacteristics,and looking for generalproperties). When level 0 understandingof geometryis being developed, presentationcan be designed in ways which incorporateimplicitlythe types of propertieswhich will be developed by level 1 thinkersexplicitly. For example, level 0 thinkerscan be asked to constructsquaresfrom a collection of sticks of variouslengths. While this task can be done at level 0, purely "byeye," therealso arises implicitlythe basis of propertiesof a square,since studentsmust select sticks of equal length, and must adjustthe corers to make rightangles. (This sort of approachmight be contrasted with one which dependson a student'sidentificationof a squarefroma set of figures on the printedpage.) In the uppergrades,wheremorepotentialfor level 1 thinking was evident in the text exposition (if not in the exercises), text writersmight look ahead to the ultimate goal of level 2 thinking, in selecting geometry experiences which incorporateimplicitly (for level 1 thinkers)relationshipswhich might be formulatedexplicitly lateron (by level 2 thinkers). Examplesof how this might be done for the topic of angle sum of a polygon are provided in Dina van HieleGeldofs thesis, and were discussedhere in Chapter3. The clinical study conductedby the Projectindicatedthat students'inabilityto advancein level of thinkingmay be relatedto theirdeficienciesin language--bothin

178 knowledge of geometryvocabulary,and abilityto use it precisely and consistently. The text analysis indicates that studentsdo not receive much help in developing languageability from theirtexts. A suggestionfor text writersmight be to include more questions that requireuse of expressive language and spontaneousrecall of geometryvocabulary(e.g., "describethe sides of this figure"ratherthan "identify which sides are parallel"),and also questionsthat requireformulationof thoughts into sentences(e.g., "explainwhy... "). Text authorsare underpressurenot to exceed certainreadinglevels for each grade. It is possible thatthe readinglevel criteriaappliedto texts rule out some of the language structurerequired for higher level thought (for example, use of quantifiers,and "if. . . then" constructions). If this is the case, a serious issue arises--whetherreadinglevel criteriashouldbe allowed to so influencethe level of thinking in geometry exhibited in the text. Text authorsmight consider ways in which language can be modified to reducereadinglevel but not to reducelevel of thinking. Implications for Classroom Strategies The resultsof this text analysishave importantimplicationsfor classroomuse of textbooks. First,teachersshouldbecome awareof the potentialgaps in level in partsof the text page, andof the low level of thinkingwith whichmost exercisesand tests can be completed. They should be especially alert to the possibility of reduction of level. If teachers have a choice of texts, they might consider the alternativeswith these points in mind. But if there is no choice, teachers can develop classroom strategies to help students to get as much as possible from availablematerials. Some suggestedstrategiesaregiven below. (a) Do not rely on a text to fill in the levels. Use texts as a follow-up to more exploratoryactivitiesin geometry. (b) Encourage students to talk about geometric concepts, and to develop expressivelanguage. Thatis, ask themto describea figure,ratherthanjust to select a namefor it froma list. (c) To help studentsfill in level 0 understandingof geometric concepts, the teacher should be alert to possible misconceptionsformed as a result of limited visual examples. Textbookpresentationscan be supplementedby manymanipulativemodels andexamplesin the environment. (d) To help studentsprogressto level 1 thought,the teachercan supplementthe one or few examplesin the text developmentof a propertyby encouraging students to test many examples, with drawings or manipulatives, to determineif propertiesare trueor false.

179 (e) To help students progress to level 2 thought, the teacher can raise the level required in many routine exercises by asking "Why?", or "Explain your answer." (f) The teacher can revise or supplement tests to reflect higher levels of thinking. The textbook is an important tool, and even a guide, especially for beginning teachers. However the texts surveyed are lacking as instructional materials for helping students develop higher levels of thinking. The teacher is the key to effective classroom use of the text, and the teacher must supplement and modify existing texts in order to help students fill in and progress through the levels.

CHAPTER 10 IMPLICATIONS AND QUESTIONS FOR FURTHER RESEARCH This Projectwas one of several undertakenduringthe early 1980's thatbroke groundin the United Stateson researchrelatedto the van Hiele model. The Project addressedsome general questions about the model but in turnraised many more questions, as might be expected for an initial investigation of the model. This chapter discusses implications of the study and presents questions for further research. First, there are theoretical implications about the model itself, in particular,the featuresof the levels, the Project'slevel descriptors,and the meaning of "a student'slevel of thinking." Then, suggestionsare given for futureresearch, some directlyrelatedto this studyandothersset in broaderresearchcontexts. Implications about the Levels This study examinedthe validity of the van Hiele model for characterizingthe thinking in geometry of sixth and ninth graders. Results discussed in Chapter7 about students'levels of thinking and factors affecting their levels relate to this generalissue. In addition,resultshave theoreticalimplicationsaboutthe fourmajor characteristics of the levels: their hierarchial or fixed-sequence nature, discontinuitybetweenlevels, languageaspectsof each level, and the implicit-explicit natureof thinkingat adjacentlevels. Results supportthe fixed-sequenceaspect of the levels, at least for levels 0, 1, and 2 with which this study dealt. Studentswho performedsuccessfully at level 1 or 2 also performedsuccessfully at lower levels. Similarsupportfor the hierarchialnatureof the levels has been evidencedfor K-12 students(Burger& Shaughnessy,1986), high school students(Denis, 1987;Usiskin, 1982), and pre-serviceelementaryteachers(Mayberry,1983). Findingsof this study also bear on the questionof whethera student'slevel of thinking is consistent across different topics. In general, the highest level of thinkingattainedby a studenton one topic was also attainedon othertopics. This is not to say, however,thata studentdidnot need to begin at level 0 or level 1for a new topic after having performedat level 1 or above on other topics. In fact, many students who entered Module 1 showing some level 1 thinking for some topics needed to do instructionalbranchesin the modules on othertopics, filling in their level 0 and 1 thinking. While these students'"entry"level of thinkingmay have varied across topics, dependingmainly on theirprevious school experiences with those topics, theirlevel of thinkingafterinstructionseemed to be consistentacross topics. Similarresultsaboutvariationin "entry"level across topics were reported by Mayberry(1983) for pre-serviceteacherson seven differenttopics andby Denis (1987) who assessed PuertoRicanhigh school studentson four topics in interviews using tests developedby Mayberry.Whetherthese differencesacrosstopics persist afterinstructionwas not investigatedby Mayberryor Denis.

181 The van Hieles claimedthatthereis discontinuitybetweenlevels. Resultsof this studyaremixed on this point. Performancesof some studentsindicatethatthey are at a plateau for a level and cannotprogressto the next level. For example, some sixth graders(GroupI) appearedto be level 0 thinkerswho thought of shapes in terms of their appearanceas a whole and were unable to analyze them in terms of theirparts. Also some sixth and ninthgraders(GroupsII and IV) showedprogress to level 1,but the jump to giving informalarguments(level 2) seemed beyond their capabilites at that time. Some were unable to follow and give argumentsand, perhapsmore importantly,did not see any need for such arguments. Results for other studentssuggest that movementbetween levels proceeds in small steps. Some sixth gradesseemed to be in transitionbetween levels 0 and 1, dealing with familiarshapesin termsof propertiesbut then lapsing to level 0 when confrontedwith unfamiliarshapes. Shaughnessyand Burger (1985) also found studentsin transition. For severalstudents,"if conflict occurredbetweenthe visual and the analytic levels of reasoning(levels 0 and 1), the visual usually won" (p. 423). Transitionin thinkingwas also observedby Lunkenbein(1980) in a teaching experiment on polyhedra with 10 and 11 year-olds. Students' thinking about geometricobjects "changedgraduallyfrom a more global perceptiondeterminedby appearanceto a more and more detailedvisual descriptionin terms of properties" (p. 174). Change seemed to take place throughan "oscillatingprocess"--invan Hiele terms--betweenlevels 0 and 1 until objectsbecamebearersof properties. Transitionwas also observed between levels 1 and 2 for some sixth graders (GroupIII) and most ninth graders(GroupsV and VI). They dealt with shapes in terms of properties and sometimes gave informal argumentsrelating properties such as simple subclass inclusionsand chainingargumentsinvolving saws/ladders. However, they did not consistently give deductive explanations;sometimes they lapsed into explaining by examples. Also, some were able to follow or give deductiveargumentsbut did so with little convictionabouttheirnecessity. BurgerandShaughnessy(1986) also detected transitionbetweenlevels 1 and2. Some students"oscillatedfromone level to anotheron the same taskunder probing from the interviewer"(p. 45). Some showed "flashes of level 2 reasoning ... usuallyas a resultof probing... [but]left to theirown devices, seemed to preferthe relativesafety of level 1 reasoningand tendedto avoid deduction,even they knew it was available"(p. 45). They also noted that observationsof though transitional thinking may suggest that the levels are "more continuous in nature than their discrete description would lead one to believe" (p. 45). However, these observationsmay not reflect continuityin learningbut rathercontinuityin teaching (Hoffer, 1983). Although the progress of subjects in this study appearsin some ways to be continuous, it may in fact be discontinuous. That is, under guidance of the interviewer, the students made incrementalprogress in learning and using new

182 concepts and in makingjudgmentssuch as testing if propertiesapply to unfamiliar shapes. But, at the same time, a gap still existed in their ability to initiate these processes spontaneously. Understandingand self-initiation of higher levels of thinkingmay come suddenly--inan "Aha,now I can see what I'm supposedto do here"experience. Herethereseems to be a change in how studentsthinkaboutand approach a problem in geometry--a change in their metacognition: their understandingaboutthe natureof the task and theirbelief aboutwhatconstitutesan appropriateresponse. The assertionof the van Hieles thateach level has its own languageis supported by this study. For example, "rectangle"meant different things to students on differentlevels. For a few sixth graders,it meanta shapethatlooked like Oor ] (level 0). Most studentswere able to speak descriptivelyaboutrectanglesin terms of properties(level 1). Many (GroupsIII, V,VI) were able to fit rectangleinto a logical context (level 2). For example,some explainedwhy a rectangleis a special parallelogramor why the sum of the measuresof its angles is 360?. Shaughnessy and Burger (1985) also noted the importanceof language at differentlevels. As discussed in Chapter7, students encounteredvarious difficulties with language (e.g., recalling geometric terminology, using expressive language) and had misconceptions about geometric concepts. Their inability to use quantifiersand logical language greatly limited their progresswithin a level or to a higher level. However, the effectiveness of special techniques used in the interviews (e.g., vocabularyor propertycards,visual family trees representinglogical relationships) suggestsinstructioncan be designedto overcomesome of these languagedifficulties andto promotethinkingat higherlevels. In additionto identifyingdifferencesin languageinvolving geometrycontent and logical relationships, results suggest that students use language involving metacognitionaboutthe qualityof thinkingand expectationsat variouslevels. At level 1, such languagemight be "Oh,I see a pattern"or "Letme see if that always works";for level 2, "I should prove this, right?"or "Ihave to clinch it." The van Hieles do not make explicit referenceto such language, althoughthey indirectly touch on metacognitive aspects of thinking. For example, at level 1, the student "purposefullylooks for relations" or at level 2, "purposefuldeduction finally becomes a habit of thinking"(van Hiele-Geldof, 1958/1984, p. 231). Findingsof this study suggest that metacognitive language should be incorporatedmore explicitly into the level descriptorsof the van Hiele model. (See discussion in a subsequentsection). A fourthcharacteristicof the model is thatlearningon a level involves making explicit what was learned implicitly in the preceding level. This feature was supportedduringthe interviews.For example, some studentscolored equal angles in a grid using an angle tester to check for equal angles (level 0) and, in a later activity,they quicklycheckedby eye, notingpatternsof equal angles (e.g., via saws or ladders). Otherstudentsimplicitly learnedthatthe measuresof the angles of a

183 trianglesum to 180? by using saws and laddersto color equal angles in a triangle grid (level 1) and later by logically orderingpropertiesin a family tree, including deductiveexplanationsfor the ordering(level 2). Of course,these resultsmay have occurred simply because of the design of the modules which embodied this implicit-explicit feature. That materials can lead students to experience topics implicitly and then explicitly at a higher level is a noteworthy finding with importantimplicationsfor designing curriculum. Accordingto findings from text analyses (Chapter9), textbook materialsrarelyincorporatethis feature into their presentationof geometry material. Moreover,at times, material is presentedin ways thatcan impedeprogresstowarda higherlevel, in particularfrom level 1 to 2. Schoenfeld (1986) contends that students develop an inappropriateseparationof empiricalmathematicsanddeductionas a directresultof instruction."Thedialectic interplayof induction and deduction"(p. 242) is lacking, or, in van Hiele terms, level 1thinkingdoes not lead in a naturalway to level 2. In summary,results of this study supportthe four major features of the van Hiele model and suggest, as indicatedabove, possible modificationof two features, discontinuityandlanguage,to includeaspectsof metacognition. It shouldbe noted that results also supportP. M. van Hiele's (1986, 1987) recent characterizationof the model in terms of three levels: visual, descriptive,and theoretical. Here the Projectcorrelateslevel 0 to the visual, level 1to the descriptive,andlevels 2-4 to the theoretical,with level 2 involving informaldeductions,level 3 being axiomaticand formal,and level 4 involving axiomaticsystems and logic. Discussions with Pierre van Hiele in June, 1987 indicate that he agrees with this interpretation.That the three-level model may not be as sufficiently refined to characterizethinking in geometryas the originalfive-level model is supportedby this Project'sfindingsthat students progressed toward level 2 (informal theoretical) but with no sign of axiomaticthinkingandresultsfromBurgerandShaughnessy(1986) which revealed some axiomaticthinkingof college-level students. Implications for Project Level Descriptors and Their Use The Project'slevel descriptors(see pages 58-71) are generally validated by quotationsfrom van Hiele sources (see Chapter4), by discussions with Pierrevan Hiele, and by the resultsof the clinical interviews. Furthersupportfor these level descriptorscomes from comparingthemwith the 28 "level indicators"identifiedby Burgerand Shaughnessy(1986) which closely matchor complementthose used in this study. However,the Project'suse of the descriptorsto characterizethe levels of students'responsesto variousassessmenttasks suggests thatsome descriptorsneed to be modifiedandthatsome new ones shouldbe addedas discussedbelow. In Module1 studentswereaskedto identifyangles in a picture. Most responded at level 0 (descriptor0-lc) by simplypointingto or tracingsome angles. However, one studentrapidlyidentifiedmanyangles, saying thatthereare lots of trianglesand

184 each has threeangles--a responseusing propertiesof trianglesfor which therewas no correspondinglevel 1 descriptor. This suggests that a modified version of descriptor0-lc be added to level 1 such as "identifiesinstances of a shape using propertiesof relatedfigures." Responses of some students on class inclusion tasks were matched with descriptor2-2b (gives informalarguments:ordersclasses of shapes). For example, some studentsexplained that squareshad all the propertiesof parallelogramsand some extra ones, so were "specialkinds of parallelograms."Othersbegan with an informal definition of parallelogram and explained why squares had those properties so "must be" parallelograms too. However, other students gave responsesat levels 0 and1. One saidthatsquaresarekites becausehe saw one square sortedin the kite collectionandso concludedthatsquaresarekites, with no appealto properties(level 0). Several said that squareswere parallelogramsafter checking thatcut-outsquareshad the propertieslisted beside a set of cut-outparallelograms. This is level 1 thinkingbecauseit is based on empiricalobservations,not deduction. Since descriptor2-2b did not cover these othercases, modifiedversions of it might be added to the listing (e.g., 1-6c: establishes subclass inclusions by empirically testingif examplesof a shapehave propertiesof anothershape). "Orderingpropertiesof shapes"is usually associatedwith level 2 thinking,as indicatedby descriptor2-2c. A studentwho used saws and laddersto explain why opposite sides are equal for parallelogramswas judged to exhibit 2-2c thinking. However,not all responsesto tasksinvolving orderingof propertieswere at level 2. For example,some studentsdrewquadrilateralswith rightangles andobservedthat opposite sides are equal and parallel. They did not deduce one propertyfrom another(level 2), but ratherestablishedthe relationshipempiricially. This suggests the need for addinga level 1 variationof descriptor2-2c. Descriptor2-2e (gives informalarguments:interrelatesseveralpropertiesin a family tree) was used when students made family trees where arrows meant inferences (i.e., knowing A was true meant B must be true). However, other students created trees where the arrow signified a time-sequence for steps in an explanationor a visual summaryfor relationshipsbetween properties. This latter use of a family tree calls for a new level 1 descriptor such as "to summarize empiricallyestablishedrelationshipsbetweenpropertiesby a family tree." Finally, student responses to tasks calling for informal deductive arguments (descriptor2-3c) indicatedthatmodificationsof 2-3c are needed. Some responses were not at level 2. For example, one studentexplained the area rule for a right triangleby putting togetherpairs of congruentright triangles and observing that "thetwo trianglesmake a rectangleso take half." This explanationwas interpreted as a variationof descriptor1-5 (discoverspropertiesof a figure). Responseswith a deductiveflavor (level 2) variedin detail andprecisionand indicatedthattherecan be a range of explanations for descriptor2-3c, from informal ones that involve

185 some deduction along with some explanation based on observations, to more "technical"argumentsin which details are carefullydeduced,althoughaxiomatics andformaldefinitions(level 3) arenot considered. Analyses of students'comments about their thinking during the interviews suggest thatsome metacognitivelevel descriptorsbe addedto the Project'scognitive descriptors. It is clear thatthe van Hiele theory includesmetacognitiveaspects of thinking (e.g., insight, purposefully looking for relations). In retrospect, the Projectshouldhave includedexplicit level descriptorson metacognition.Two main facets of metacognitionare the student'sknowledge of cognition (e.g., about the nature of tasks or strategies) and regulation of cognition such as planning and monitoring (Garofalo & Lester, 1985). A student'scomment to the interviewer that"Oh,you probablywantme to see whathappenswithhexagons"(havingfound angle sums for triangles, quads, and pentagons) suggests awareness of the expectationto discoverpatternsor make generalizations(level 1). Dina van HieleGeldof might have had in mind such awarenessof expectationon level 1 when she wrote: "Pupilsare at the first level. Because of this they know they have to search for relations"(1958/1984, p. 225). A student'scomment, "I need to clinch it," indicates level 2 metacognitionabout the goal of giving a careful explanationor proof, which is akin to van Hiele-Geldofs "purposefuldeduction"(1958/1984, p. 231). Studentcomments aboutmonitoringtheir thinkingand planningwere also observed (e.g., at level 2, "Oh, I need to prove this part and then I've got it."). These types of descriptorscan be relatedto P. M. van Hiele's (1957, 1973, 1986) notion of "insight." He states that "insightis recognized as such, if a person acts adequatelyand intentionallyin a new situation"(1986, p. 159). To have insight, Hofferwrites (1983), "studentsunderstandwhatthey are doing, why they are doing it, and when to do it" (p. 205). As indicatedabove, additionof descriptorsat each level to include aspects of metacognition seems warranted. Similar recommendationshave been made for the inclusionof metacognitionin models of mathematicalthinking,such as in problemsolving (Silver, 1985). The additionof metacognitive descriptorsto the Project'scognitive level descriptorssupportsthe contention that no process model of mathematicalthinking is complete unless it makesexplicit provisionfor metacognitiveaspectsof thinking. A word of cautionshouldbe given aboutthe interpretation of level descriptors and theiruse. First, descriptorscan be misinterpreted.The inclusion of examples of studentperformancefor each descriptorcan preventmisinterpretation. Crowley (1987) used examples from this Projectand from Burger's(1982) to illustratevan Hiele levels. Second, as illustrated throughout this section, use of the level descriptorsto characterizea student'sthinkingfor a task requiresthatjudgmentbe based on the quality of the student'sexplanation,not simply on the answer itself. Many questions can be answeredcorrectlyon differentlevels, so researchersand teachersshould be carefulto assess the "why"of a student'sresponse. Agreement on interpretationand application of level descriptors is critical for cumulative researchon the levels.

186 Implications for Future Research This section offers suggestions for further research on the model, some growing directly from results of this study about students' levels of thinking, methodologyfor assessingit, and instructionalapproachesto fosterhigherlevels of thinking. Otherrecommendationsareproposedin more generalsettings. Research on Students' Levels of Thinking Several questions for future research arise from results of the clinical interviews,both involving studentswho made little or no progress at level 0 and those who made significantprogresswithin level 1 and even towardslevel 2. One main recommendationis for replicationof the Project'sinterviewswith comparable samplesof sixth andninth gradestudents. Questionsto be addressedinclude: Are comparableresults obtainedfor studentsat the threelevels of achievement? Do studentsthinkat a higherlevel in instructional/assessment interviewsthanindicated other studies such as those of Usiskin and (1982) by Burger (1982)? What characteristicsof the levels are supported?Do students'levels vary acrossdifferent topics? Follow-up of studentsshould be done to assess permanencyof level on a topic over a period of time. Usiskin (1982) exploredthe predictivevalidity of the levels for geometryachievement. Additionalwork of this sort is needed involving correlationof level of thinkingwith subsequentachievement. Assessmentof level of thinkingmight also be done with youngerchildren(grades 1-5), in particular,to explorethe originsand growthof thinkingat levels 0 and 1. Replicationmight also be done with olderstudentsjust priorto theirstudyof geometry,perhapsto identify "studentsat risk"with respectto level of thinkingneededfor tenthgradegeometry. Findingsthata majorityof sixth andninthgraderswere able to attainlevel 1and even make progresstowardlevel 2 suggestvarioustypes of furtherresearch. First, reanalysesof videotapesin this studymight examinethe role of metacognitionin a student'sprogress,especially duringstudent-interviewerinteractionson key tasks. Did more directivesaboutexpectationsin tasks and more feedbackaboutthinking result in greater progress in level of thinking? Another line of research is to examinewhetherprogressto higherlevels for a topic can be fosteredin the natural settingof a classroom(e.g., a whole-classteachingexperiment).Such a studymight deal with one of the Project's modules or could evolve into a curriculum development and evaluation project which would examine changes in levels of thinking for several topics. This type of research parallels that in Dina van Hiele-Geldofs doctoral dissertation. Grades seven and eight, which have been targeted for development of activities that promote higher levels of thinking (Shaughnessy& Burger, 1985; Prevost, 1985), are particularlysuited for such a project. Follow-up of these studentsthroughhigh school geometry would be a naturalextensionof this type of research. Results of the clinical interviewsfor below-gradelevel sixth gradersshowed

187 both low "entry"level and "potential"level of thinking. However, the modules were not designedfor studentswith severe deficienciesin geometrybackgroundand in language. The questionariseswhetherthese studentscan fill in thinkingat level 0 and progress toward level 1, given more time during interviews, different instructionalmethods,andmodules thatgive greateremphasisto certainvan Hiele phases. Results of this study indicatesome areasfor improvement:more extended development of new topics, careful review between sessions, options in communication (verbal and non-verbal) and aids to reduce memory demands involving new termsduringPhase 3 (Explicitation),and more opportunityto apply newly learned ideas duringPhase 5 (Integration). Researchmust not neglect the slow learner. Most researchon the van Hiele model has dealt with topics in plane geometry. Further research might explore levels of thinking in arithmetic, algebra, or three-dimensionalgeometry. Levels of thinkinghave been appliedin other subject areassuch as economics andchemistry(ten Voorde, 1979). Van Hiele (1986, 1987) discussedthe levels in relationto othermathematicstopics and other subjectareas. His currentwork (see Foreword)focuses on applicationsof the levels in arithmetic, physics and otherareas. Following van Hiele's lead, researchersmight investigate the levels as a more generalmodel of thinking. Research Related to Methodology Methodologicalaspects of this study suggest the need for furtherdevelopment of ways to assess students'levels of thinking. A significantcontributionto van Hiele researchwould be the developmentof an easy-to-useinstrumentfor clinical assessment of a student's"entry"level, perhapsbased on a synthesis and refinementof some activitiesin this studyandin Burger's(1982). Such an "assessmentkit"should include a key for scoringcommonresponsesat appropriatelevels which would aid researchersin more reliablyanduniformlymeasuring"entry"level of thinking. The feasibility of assessing level of thinkingby a paper-and-pencilinstrument should also be explored. Constructionof items that can elicit responsesat various levels and thatcan be scoredreliablywould be valuableto researchersas well as to evaluators(e.g., state-leveltesting) and classroomteachers. This study'sreview of test items in textbooksindicatedthatalmostall were at level 0 (or reductionof level, recall). Availabilityof items for levels 1and 2 would supporttext-basedteachingat those levels. Because students in this study often responded correctly but at differentlevels to a question, it may be difficult to determinea student'slevel by means of answersto multiple-choiceitems. Researchis needed to establishthat a correctanswer to an item does in fact reflect a certainlevel hypothesizedfor that item. Items that require students to explain "why" (by drawings or written explanations)might accuratelyassess level. Initialwork on such open-endeditems and on a level-related scoring schema has been done at the secondary level (De Villiers & Njisane, 1987). However, in this Project's study younger students,

188 especially those in grade 6, had difficulty talking about geometry, in particular, "tellingwhy." Writingan explanationwould no doubtbe more difficult for them, so that tests that demandwrittenexplanationsmay be inappropriatefor assessing theirlevel of thinking. level merits The use of dynamicassessmentto determinea student's"potential" furtherresearch. Query:Is it possible to assess potentiallevel in less time thanthe six to eight sessions in this study? Design and validationof assessmenttasks and specification of the interviewer's role in them require substantive research. However,effective measuresof "entry"level might correlatehighly with measures of "potential"level and hence suffice for many researchand classroom practice situations.Researchcorrelatingthe two types of assessmentsis needed. Research on Curriculum and Instruction in Geometry The van Hieles formulatedthe levels in response to analyses of their own teachingof geometry. Thus,it is not surprisingthatresultsof researchon the levels shouldhave implicationsfor researchon classroomteachingof geometry--namely, curriculumand instruction. Findings in this study show that geometry was a neglected partof the school mathematicsexperiencesof many students,and what was taughtwas often taughtrotely or requiredminimalstudentexplanation. These results combined with findings about the students' potential for level 1 and 2 thinkingand aboutthe paucity of textbookmaterialat those levels underscorethe need for researchleading to improvedcurriculumand practice. Currenttextbook series shouldbe analyzedaccordingto van Hiele levels. Researchon innovativetext series might be undertakensuch as Joyce's (1984) study of a unified mathematics program(grade 7) which was found to foster thinking that leads to level 2 and whose sequence followed van Hiele phases. The level of thinking of geometry curriculumin other countries also merits examination, in particular,the Soviet Unionwhereduringthe 1960'sextensivechangebasedon van Hiele levels was made in the geometrycurriculum(Pyshkalo,1968/1981;Wirszup,1976). An analysisof some Soviet text materialon geometryfor grades 1-6 is in progressby one of this Project'sstaff members. Researchmight also examine samples of curriculummaterialsthat reflect the van Hiele levels andprovideactivitiesfor apprenticeshipleading up to level 3. In particular,the following materialsmightbe studied: Hoffer's(1979) geometrytext DeVillier's (1985, 1986) material on Boolean algebra, the Geometric Supposer software (Schwartz & Yerushalmy, 1985) and materials (Chakerian,Crabill, & Stein, 1987; Yerushalmy,Chazan,& Gordon,1987). Materialsthatprovidea rich basis for thinkingat levels 0 to 2 shouldalso be examined,such as non-traditional text series and supplementarycurriculummaterials in the United States, Dutch materialsof van Hiele (1976-79) and Goddijn(1980) and materialsby Del Grande (1982). As discussed in Chapter9, there are other lines of van Hiele research relatedto curriculumdevelopmentandevaluation.

189 Researchon instructionin geometry can be built on successful methods and materials in the Project'smodules. Researchmight explore the effectiveness of specific techniquesfor fostering thinkingon a level such as finding shapes in the environment(level 0), uncoveringclues (level 1), and building family trees (level 2). Such "instructionalmicro-strategies"(Van Patten, Chao, & Reigeluth, 1986) might be incorporated into longer, unit-type "macro-strategies"such as in a teachingexperimentbased on the van Hiele phases. Anotherdirectionfor research is the examinationof the geometricthinkingin teacher-studentinteractionsduring classroomlessons, perhapsusing an observationschedulebasedon level descriptors. Identificationof characteristicsof "expert"teachersof geometry (i.e., those who foster higher orderthinking)is anotherrelatedtopic for research. Findings about the interviewer'srole in fostering a student'sthinkingin this study suggest that a teacher'scognitive and metacognitiveteaching moves be examined in relationto students'levels of thinking. As mentionedin Chapter8, thereare also possibilities for relatedresearchin teachereducationsuch as examiningthe effects of preparing teachersto identifythe level of a student'sthinkingor text material. Research in Other Settings The van Hiele model might be investigatedin more generalresearchsettingsin developmentalandcognitivepsychology. One directionis to explorethe model in a Piagetiancontext. P. M. van Hiele (1959, 1986) statedthatan importantpartof "the roots of his work"can be foundin the theoriesof Piaget,but he also notedthatthere are many important"disagreements."The question whether the theories of van Hiele and Piaget belong to the same "researchprogram"was addressedby Orton (1987) who comparedassumptionsof theirtheories. Otherquestionsaboutthe two theories arise quite naturally. For example, what is the relationship between Piagetianstages and van Hiele levels? Exploringthis question,Denis (1987) found that only 36% of studentswho had taken high school geometry had reachedthe formal operationalstage, with most of them attainingonly level 2. She also found significantdifferencesin van Hiele level betweenstudentsat the concreteandformal operationalstages. Furtherresearchmight replicatethis study with studentsprior to their study of geometry. Similar research might also be done with younger pre-operationalandconcreteoperationalstudentsandat levels 0 and 1. Anotherdirectionfor futureresearchon the van Hiele model is suggested by Soviet researchwhich, duringthe 1950's and 1960's, was largely concernedwith didactics, both specific and general. Clinical interviews and long-term "teaching experiments"were contexts for investigatingmathematicalthinking, in particular concept learning and problem solving in geometry (e.g., Zykova, 1969a, 1969b; Yakimanskaya,1971)). A synthesisof this Soviet researchon geometryas it relates to the van Hiele model could lead to a betterunderstandingof the van Hiele model. Also, research on the model from a Vygotskian perspective is suggested by correspondencesbetween notions of Vygotsky (1962) and aspects of the van Hiele model. One correspondence is between Vygotsky's "zone of proximal

190

development," and the Project's "potential" level of thinking. A second correspondenceis between Vygotsky's approachfor assessing a student'szone-namely within the influence of instructionand with die assistanceof an adult--and the Project'sapproachfor assessing "potential" level via its instructional-assessment interviews.

Finally, various themes and ideas from cognitive psychology provide a rich context for research related to the levels. One direction discussed previously involves a student's metacognition and level of thinking, perhaps examined systematicallywithin a theoreticalframeworksuch as that of Stemberg (1984). Another direction is exemplified by the research of Lehrer, Guckenberg and Sancilio (in press) who suggest how the van Hiele model can serve as a vehicle for research involving constructs in cognitive psychology. Their currentteaching experiment in a Logo-based instructional setting focuses on fourth graders' developmentof declarative(e.g., a verbalproposition)andprocedural(e.g., a Logo program)interpretationsof geometricconcepts and of their "pre-proof'thinkingat levels 0, 1, and 2. Initial results illustratehow these interpretationsdevelop in a dynamic, interrelated way through small-group work on "carefully crafted activities"relatedto van Hiele levels. Findingsalso suggest parallelsbetween the developmentof children'sorganizationof geometricknowledge from a cognitive science perspective and the first three van Hiele levels. Research on using a computer-basedintelligent tutoring system to teach students how to construct geometryproofs (Anderson,Boyle, & Reiser, 1985) also uses a geometrycontext for investigation of cognitive theory, in particular,Anderson'sACT* model for cognition. In van Hiele terms this researchdeals with formal deductive thinking (level 3) and suggests the possibility of futureresearchon similartutorialsto teach cognitive skills at other levels. Future research might apply cognitive process analysisto performanceof studentson key assessmenttasksused in this studyor in Burger's (1982). This could lead to a more detailed cognitive process-based description of thinking of each level, which in turn could be used to explain characteristicsof the van Hiele model and to design instructionto foster higher levels of thinking. As suggestedabove, the van Hiele model is an appropriateobject of study in a varietyof cognitiveresearchcontexts. On the one hand,featuresof the model (e.g., the levels and their characteristics, the interplay between instruction and development) can be investigated from more general perspectives, further clarifyingthe model. On the otherhand,the model itself can serve as a contextfor research in an area, providing a source of content-specific verification of more general theories. Evidence of both types of researchon the van Hiele model are alreadyseen, for example at a 1987 Conferenceon the Learningand Teachingof Geometry (Senk, in press) which brought together mathematics educators, cognitive psychologists,teachers,geometers,and P. M. van Hiele himself to share ideas aboutresearchon geometry,in particular,the van Hiele model.

191 Duringthe pastten yearsthe van Hiele modelhas emergedas an importanttopic for researchin mathematicseducation. However,researchon it is still limited. The van Hieles developedthe model to impacton theirteaching. This Projectattempted to shed light on the model andits usefulnessas a paradigmfor examiningthe level of geometricthinkingof adolescents,of theirteachers,and of mathematicstextboook materials. It is hoped thatthe Project'swork will stimulatefurtherresearchon the van Hiele model by mathematicseducation "levelists,"by researchspecialists in otherfields, and also by teams of these researcherswho throughcollaborationcan bring their respective expertise to bear on questions of mutual interest. This researchis needed since the model, with its emphasison developing successively higher thought levels, appearsto signal directionand potential for improvingthe teachingof mathematics.

192 BIBLIOGRAPHY Anderson, J. R., Boyle, C. F., & Reiser, B. J. (1985). Intelligent tutoring systems. Science,

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193 Garofalo, J. & Lester, F. K. (1985). Metacognition, cognitive monitoring, and mathematical performance. Journalfor Researchin MathematicsEducation, 16, 163-176. Goddijn, A. (1980). Shadow and depth. Utrecht,The Netherlands:Vakgroep OW & OC. Gregory, J. W., & Osborne, A. R. (1975). Logical reasoning ability and teacher verbal behavior within the mathematicsclassroom. Journalfor Researchin MathematicsEducation,6, 26-36. Hoffer, A. (1979). Geometry. Menlo Park,CA: Addison Wesley. Hoffer, A. (1981). Geometry is more than proof. MathematicsTeacher, 74, 11-18. Hoffer, A. (1983). Van Hiele-based research. In R. Lesh and M. Landau. (Eds.), Acquisition of mathematicsconcepts and processes (pp. 205-228). New York: Academic Press. Joyce, J. (1984). An analysis of geometrymaterialin the Unified MathematicsProgram(grade7) according to van Hiele levels. Unpublishedmaster'sthesis, Brooklyn College, Brooklyn, NY. Lehrer, R., Guckenberg, T., & Sancilio, L. (in press). Influences of Logo on children's intellectual development. In R. E. Mayer (Ed.), Teaching and learningcomputerprogramming: Multiple researchperspectives. Hillsdale, NJ: Erlbaum. Lunkenbein, D. (1980). Observations concerning the child's concept of space and its consequences for the teaching of geometry to younger children. [Summary]. Proceedings of the FourthInternationalCongress on MathematicalEducation (pp. 172-174). Boston: Birkhauser. Mayberry, J. (1983). The van Hiele levels of geometric thought in undergraduatepreservice teachers. Journalfor Researchin MathematicsEducation, 14. 58-69. Musser, G., & Burger, W. F. (1988). Mathematics for elementary teachers: A contemporary approach. New York: Macmillan. National Science Foundation. (1980). Science education databook. Washington, D. C.: U. S. GovernmentPrintingOffice. Orton, R. (1987, June). Do van Hiele levels and Piaget belong to the same research program? Paper presented at the Conference on Learningand Teaching Geometry, Syracuse University, Syracuse, NY. Prevost, F. J. (1985). Geometry in the junior high school. MathematicsTeacher, 78, 411-418. Pyshkalo, A.M. (1981). Geometry in grades 1-4 (Problems in the formation of geometric conceptions in pupils in the primary grades). ( A. Hoffer (Ed.) and I. Wirszup, University of Chicago (Trans.)). Chicago: University of Chicago. (Original document in Russian. Geometriya v I-IV klassakh (Problemy formirovaniya geometricheskikh predstavienii a mladshikh shkol' nikov). Moscow: Prosveshchenie Publishing House, 1968.) Schoenfeld, A. (1986). On having and using geometric knowledge. In J. Hiebert (Ed.), Conceptualand proceduralknowledge (pp. 225-264). Hillsdale, NJ: Erlbaum. Schwartz, J., & Yerushalmy, M. (1985). Geometric Supposer software. Boston: Educational Development Center. Senk, S. (in press). Report on the Conference on Learning and Teaching Geometry: Issues for Research and Practice. Syracuse, NY: Syracuse University.

194 Shaughnessy, J. M., & Burger, W. F. (1985). Spadework prior to deduction in geometry. MathematicsTeacher, 17, 419-428. Silver, E. A. (1985). Researchon teachingmathematicalproblem solving: Some underrepresented themes and needed directions. In E. A. Silver (Ed.), Teaching and learning mathematical problem solving: Multiple researchperspectives (pp. 247-266). Hillsdale, NJ: Erlbaum. Stemberg, R. J. (1984). Mechanisms of cognitive development: A componential approach. In R. J. Stemberg (Ed.), Mechanisms of cognitive development (pp. 163-186). New York: W. H. Freeman. ten Voorde, H. (1979). Verbalizing and understanding.[Summary]. EuropeanJournalof Science Education, 1, 469-471. (Original document in Dutch, doctoral dissertation, University of Amsterdam, 1977). Usiskin, Z. (1982). Van Hiele levels and achievement in secondary school geometry. (Final reportof the Cognitive Development and Achievementin SecondarySchool GeometryProject.) Chicago: University of Chicago. (ERICDocument ReproductionService No. ED 220 288). van Hiele, P. M. (1957). De problematick van het inzicht gedmonstreed van het inzicht van schodkindren in meetkundeleerstof. [The problem of insight in connection with school children's insight into the subject matter of geometry.] (Unpublished doctoral dissertation, University of Utrecht, 1957). van Hiele, P. M. (1959). Development and the learning process. Acta Paedogogica Ultrajectina (pp. 1-31). Groningen: J. B. Wolters. van Hiele, P. M. (1973). Begrip en inzicht. [Understanding and insight.] Purmerend, The Netherlands:Muusses. van Hiele, P. M. (1976, 1977, 1978, 1979). van a tot z. [From a to z.] (Text series for secondary school students). Purmerend,The Netherlands:Muusses. van Hiele, P. M. (1980, April). Levels of thinking, how to meet them. how to avoid them. Paper presented at the presession meeting of the Special InterestGroup for Research in Mathematics Education,National Council of Teachersof Mathematics,Seattle,WA. van Hiele, P. M. (1981). Struktuur.[Structure.] Purmerend,The Netherlands:Muusses. van Hiele, P. M. (1984). English summary. [The problem of insight in connection with school children's insight into the subject matter of geometry.] (Unpublished doctoral dissertation, University of Utrecht, 1957) In D. Fuys, D. Geddes, & R. Tischler (Eds.), English translation of selected writings of Dina van Hiele-Geldof and P. M. van Hiele (pp. 237-241). Brooklyn: Brooklyn College. van Hiele, P. M. (1984). A child's thought and geometry. In D. Fuys, D. Geddes, & R. Tischler (Eds.), English translationof selected writings of Dina van Hiele-Geldof and P. M. van Hiele (pp. 243-252). Brooklyn: Brooklyn College. (Original document in French. La pensee de 1'enfantet la geometrie, Bulletin de l'Association des Professeurs de Mathematiques de l'EnseignmentPublic. 1959, 198, 199-205) van Hiele, P. M. (1986). Structureand insight. New York: Academic Press.

195 van Hiele, P. M. (1987, June). Finding levels in geometry by using the levels in arithmetic. Paper presented at the Conference on Learningand Teaching Geometry, Syracuse University, Syracuse, NY. van Hiele, P. M., & van Hiele-Geldof, D. (1958). A method of initiation into geometry at secondary schools. In H. Freudenthal(Ed.), Reporton methods of initiationinto geometry (pp. 67-80). Groningen: J. B. Wolters. van Hiele-Geldof, D. (1984). The didactics of geometry in the lowest class of secondary school. In D. Fuys, D. Geddes, & R. Tischler, English translation of selected writings of Dina van Hiele-Geldof and P. M. van Hiele (pp. 1-214). Brooklyn: Brooklyn College. (Original document in Dutch. De didaktiek van de meetkunde in de eerste klas van het V. H. M. O., Unpublisheddoctoral dissertation,University of Utrecht, 1957). van Hiele-Geldof, D. (1984). Didactics of geometry as a learning process for adults. In D. Fuys, D. Geddes, & R. Tischler (Eds.), English translation of selected writings of Dina van Hiele-Geldof and P. M. van Hiele (pp. 215-233). Brooklyn: Brooklyn College. (Original documentin Dutch. De didaktiekvan de meetkundeals leerprocesvoor volwassenen. Antwerp: Drukkerij"Excelsior"N.V., 1958) Van Patten, J., Chao, C., & Reigeluth, C. M. (1986). A review of strategies for sequencing and synthesizing instruction. Review of EducationalResearch, 56, 437-471. Vinner, S., & Herskowitz, R. (1980). Concept images and common paths in development of some simple geometric concepts. [Summary]. Proceedings of the Fourth International Conference for the Psychology of Mathematics Education (pp. 177-180). Berkeley, CA: University of California. Vygotsky, L. S. (1962). Thought and language. Cambridge,MA: M. I. T. Press. Wirszup, I. (1976). Breakthroughin the psychology of learning and teaching geometry. In J. L. Martin (Ed.), Space and geometry: Papersfrom a researchworkshop (pp. 75-97). Columbus, Ohio: ERIC/SMEAC. Yakimanskaya, I. S. (1971). The development of spatial concepts and their role in mastery of elementary geometric knowledge. In J. Kilpatrickand I. Wirszup (Eds.), Soviet studies in the psychology of learning and teaching mathematics (Vol. I) (pp. 145-168). Palo Alto, CA: School MathematicsStudy Group. Yerushalmy, M., Chazan, D., & Gordon, M. (1987). Guided inquiry and technology: A year long study of children and teachers using the Geometric Supposer: ECT final report. Boston: EducationDevelopment Center. Zykova, V. I. (1969a). The psychology of sixth-gradepupils' mastery of geometric concepts. In J. Kilpatrick and I. Wirszup (Eds.), Soviet studies in the psychology of learning and teaching mathematics (Vol.I) (pp. 149-188). Palo Alto, CA: School MathematicsStudy Group. Zykova, V. I. (1969b). Operating with concepts when solving geometric problems. In J. Kilpatrick and I. Wirszup (Eds.), Soviet studies in the psychology of learning and teaching mathematics(Vol. I) (pp. 93-148). Palo Alto, CA: School MathematicsStudy Group.

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