Logo and geometry (Journal for research in mathematics education)

ISSN 0883-9530

Journal inrResearch Mathematics Education

Logo

and

Geometry

Monograph Number 10

National Council of Teachers of Mathematics

A MonographSeries of the National Council of Teachersof Mathematics The Journal for Research in Mathematics Education (JRME) Monograph Series is published by the National Council of Teachers of Mathematics as a supplement to the JRME. 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. Any person wishing to submit a manuscript for considerationas a monographshould send four copies of the complete manuscript to the monograph series editor. Manuscripts should be no longer than 200 double-spaced, typewritten pages. The name, affiliations, and qualifications of each contributing authorshould be included with the manuscript. Manuscriptsshouldbe sent to Neil Pateman,Wist Annex 2-223, Department of Teacher Education and Curriculum Studies, University of Hawaii, 1776 UniversityAvenue, Honolulu,HI 96822.

Series Editor ERNA YACKEL, Purdue University Calumet; Hammond,IN 46323 JRME Editorial Panel BETSEY BRENNER, University of California, Santa Barbara;JINFA CAI, Harvard University; MARTA CIVIL, University of Arizona; GUERSHON HAREL, University of California, San Diego; KATHY IVEY, Western CarolinaUniversity, Chair; DAVID KIRSHNER, Louisiana State University; LENA LICON KHISTY,Universityof Illinois at Chicago;CAROLYN KIERAN,Universit6du Qu6bec a Montreal,Board Liaison; RICH LEHRER, University of WisconsinMadison;NORMA PRESMEG, Illinois State University; MARTIN SIMON, Penn State University

Copyright? 2001 by THE NATIONALCOUNCILOF TEACHERSOF MATHEMATICS,INC. 1906 Association Drive, Reston,Virginia 20191-9988 (703) 620-9840; (800) 235-7566; www.nctm.org All rightsreserved Data: Libraryof CongressCataloging-in-Publication ISBN: 0-87353-509-X

The publicationsof the NationalCouncilof Teachersof Mathematicspresenta variety of viewpoints.Theviews expressedor impliedin this publication,unlessotherwisenoted, should not be interpretedas official positions of the Council.

Printedin the United States of America

Table of Contents Acknowledgments ................................................vi LOGOAND GEOMETRY............................ Chapter1 LEARNING.....................1 PROBLEMS IN GEOMETRY BACKGROUND.......................................

1 3

TheoreticalPerspectives.............................3 Research on Logo and MathematicsEducation...........6 ......................10 ORGANIZATION OFTHECHAPTERS

Chapter2

LOGOGEOMETRYCURRICULUMAND METHODOLOGY..................................11 GOALS................. CURRICULUM LOGOGEOMETRY:

14

1. AchievingHigher Levels of GeometricThinking.......14 2. LearningMajor GeometricConceptsand Skills .......14 3. Developing Power and Positive Beliefs in MathematicalProblemSolving and Reasoning........ 15 ANDACTIVITIES CURRICULUM STRUCTURE ...............15

Path Activities .................................... Shapes: Special Paths ..............................17 Motions .........................................19

15

RESEARCH METHODOLOGY ...........................21

Participants ......................................21 Assessmentand Data Collection..................... Chapter3

22

RESULTS .........................................24 TOTALACHIEVEMENT ................................24

K-6 Total ........................................24 4-6 Total ........................................25 SHAPESAND LEVELSOFGEOMETRIC THINKING...........26

Pre-Post Test .....................................26 Unit Tests........................................31 Interview........................................32 ClassroomObservations............................38 ANGLE,ANGLEMEASURE,ANDTURNS..................55

Pre-Post Test .....................................55 Unit Tests........................................61 Interview .......................................

ClassroomObservations............................64 PATHS...................................66

Pre-Post Test .....................................66 Unit Tests........................................67

iii

62

SYMMETRY ........................................ Pre-Post Test ..................................... Unit Tests ....................................... CONGRUENCE .......................................71 Pre-Post Test ..................................... Unit Tests ....................................... MOTIONS................................ GEOMETRIC

67 68 70 71 73 73

Unit Tests ....................................... Interview......................................... ClassroomObservations............................ PROBLEMSOLVINGAND SENSEMAKING:CLASSROOM OBSERVATIONS .....................................

"Tilted"Squares ..................................79 TurtleDeliveries ..................................79 MakingEquilateralTrianglesand RegularPolygons ..... Generalizationof the Angle SumNotion................ Chapter4

DISCUSSION ...

...

...

.....

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

73 75 78 78

84 86 90

TOTALACHIEVEMENT ................................

90

SHAPES AND LEVELS OF GEOMETRICTHINKING ...........

91

ANGLES,ANGLEMEASURE,ANDTURNS .................96 PATHS........................................ ........................................ SYMMETRY ......................................100 CONGRUENCE MOTIONS............................... GEOMETRIC PROBLEMSOLVING.................................

98 99 101 103

ISSUES OF EPISTEMOLOGY,MOTIVATION, AND

IMPLEMENTATION ................................. LOGOGEOMETRY................

Chapter 5

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

STUDENTS' KNOWLEDGE OF GEOMETRY BEFORE INSTRUCTION ......................................

107

NEW DEVELOPMENT AND RESEARCH ON LOGO AND GEOMETRY ........................... TURTLEMATH.....................................

111 111

Design Principles and TurtleMathEnvironment........ Initial Field Test ................................. YearlongField Test ...............................

Chapter 6

103 105

111 115 117

MATHEMATICS TOPICSWITH LEARNINGSPECIFIC TURTLEMATH..................................... IMPLICATIONS .....................................

120 122

IMPLICATIONS FOR THEORY .....................

126

MULTIPLE PATHS TO MULTIPLE TYPES OF KNOWLEDGE .... 126 VAN HIELE MODEL OF LEVELS OF GEOMETRICTHINKING .. 129

iv

NATURE OF THE LEVELS .............................

132

PHASES OF LEARNING ............................... LEARNING GEOMETRY WITH LOGO: A RETROSPECTIVE

139

LOOK ...

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

...........

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

142

References .....................................................147 155

Tables ........................................................

v

Acknowledgments This material is based in part on work supported by the National Science Foundationunder GrantNo. MDR-8651668, "Development of a Logo-Based ElementarySchool GeometryCurriculum."Time to preparethis materialwas also partiallyprovidedby National Science FoundationResearchGrantsNSF MDR8954664, "An Investigation of the Development of Elementary Children's Geometric Thinking in Computerand NoncomputerEnvironments,"and ESIfor Mathematical Thinking, 9730804, "Building Blocks-Foundations 2: to Grade Research-Based Materials Prekindergarten Development."Any opinions, findings, andconclusions or recommendationsexpressedin this publication are those of the authorsand do not necessarily reflect the views of the National Science Foundation. The authorsexpress deep appreciationto the teachersand studentswho participated in the research,including, in Ohio, FranBickum, Robin Fogle, Adrianne Geszler, Linda Hallenbeck, Mary Beth Hazlett, Roger Hosey, Eleanor Jaynes, Sharon Jensen, Annette Marson, Sheri Merriman,Paula Rogerson, Katherine Taras,Betty JaneTaylor,andJoyce Ungarand,in New York,JacquitaAlexander, Philip Bronstein, Bernadette Carnevale, Judy Clarke, Colleen Kelley, David Leibelshon,Sue Lucarelli,PatriciaMacdonald,Alan Miller, CathyRice, Darlene Richardson,KirbySnyder,and Sue Snyder.The authorsalso appreciatethe graduate students who helped conduct and critique this research, especially Kay Johnson-Gentile,HesterLewellen, MichaelMikusa,JamesSchwartz,MaryEllen Terranova,and CarolWilliams.

vi

Logo

and

Geometry by

Douglas H. Clements Universityat Buffalo,State Universityof New York Michael T. Battista KentState University with Julie Sarama Universityat Buffalo,State Universityof New York

Series Editor,Ema Yackel Purdue UniversityCalumet

National Council of Teachersof Mathematics Reston, Virginia

Chapter I

Logo and Geometry Numerous assessments reporta "failureof studentsto learn basic geometric concepts,"especially geometricproblemsolving (Beaton et al., 1996; Carpenter, Corbitt,Kepner,Lindquist,& Reys, 1980; Fey et al., 1984; Kouba et al., 1988; Stevenson,Lee, & Stigler,1986;Stigler,Lee, & Stevenson,1990).Thispoorperformance is due, in part, to the currentelementary and middle school geometry curriculum.Thereis littleopportunityfor the studentsto developgeometricproblem solving, spatialthinking,or geometricconcepts.It is no wonderthat,afterexperiencingsuchanimpoverishedgeometrycurriculumin elementaryandmiddleschool, manyhigh school studentsdo not have the necessarygeometricintuitionandbackgroundfor a formaldeductivegeometrycourse.Thus,a criticalbarrierto students' successfullearningof geometryis the failureof standardelementaryschool geometrycurriculato systematicallyfacilitatestudents'progressionfrombasicintuitions and simpleconceptsto higherlevels of geometricthought. In the Logo GeometryProject,we attemptedto develop a research-basedGrades K-6 geometrycurriculumthataddressedthese deficits. Ourprojectwas one of six funded by the National Science Foundationundera special RFP, "Materialsfor ElementarySchool MathematicsInstruction,"dealing with the use of technology. In this monograph,we discuss a series of studiesthatevaluatethis curriculumand, further,investigatehow elementaryschoolstudentslearngeometricconceptsandhow Logo programmingin turtle graphicsmight affect this learning.We emphasize severalinterrelatedstudiesthatwe conductedin the contextof the Logo Geometry Project,along with relevantfindingsfromour subsequentresearchprovidedfor by two additionalNationalScienceFoundation(NSF)projectsthatbuiltdirectlyon these studies.The remainderof this chapterwill describeproblemsin geometrylearning, ourtheoreticalandresearchbackground,andthe organizationof the monograph. PROBLEMSIN GEOMETRYLEARNING In its CurriculumandEvaluationStandardsforSchoolMathematics,theNational Council of Teachersof Mathematics(NCTM) states that "spatialunderstandings are necessary for interpreting,understanding,and appreciatingour inherently geometricworld"(NCTM1989,p. 48).' Unfortunately,accordingto extensiveeval1 These 1989 Standardswere, of course, those on which the Logo GeometryProjectwas based. The more recentPrinciples and StandardsforSchool Mathematics,or Principles and Standards(National Councilof Teachersof Mathematics,2000), on which Clementsservedas a memberof the writingteam, was not officiallyreleasedat the time of this writing.It shouldbe notedthatthe Principlesand Standards emphasizesgeometryeven more at the earlygrades.

2

Logo and Geometry

uationsof mathematicslearning,elementaryandmiddleschoolstudentsin theUnited Statesarefailing to learnbasic geometricconceptsandgeometricproblemsolving and are woefully underpreparedfor the study of more sophisticatedgeometric concepts and proof, especially when comparedto students from other nations (Carpenteret al., 1980;Fey et al., 1984; Koubaet al., 1988; Stevensonet al., 1986; Stigleret al., 1990). Forinstance,fifth gradersfromJapanandTaiwanscoredmore thantwice as high as studentsfromthe United Stateson a test of geometry(Stigler et al., 1990).Japanesestudentsin bothfirstandfifth gradesalso scoredmuchhigher (andTaiwanesestudentsonly slightlyhigher)thanstudentsfromthe UnitedStates on tests of visualizationand paper folding. Data from the Second International MathematicsStudy(SIMS)showedthatin geometry,eighthandtwelfthgradersfrom the United States scored at or below the 25th internationalpercentile(McKnight, Travers,Crosswhite,& Swafford,1985; McKnight,Travers,& Dossey, 1985). In the recent Third InternationalMathematics and Science Study (TIMSS) assessment, geometry and measurementwere two of the three topics in which studentsfromthe UnitedStatesperformedsignificantlyworsethanthe international average(NationalCenterfor EducationStatistics, 1996). Geometryachievement did not improvemuch from gradeto grade,presumablybecause this contentarea is not emphasizedin elementarygrades.Studentsfromthe UnitedStatesperformed poorly when applyinggeometryandmeasurementknowledgein problem-solving situations(Mullis et al., 1997). As statedin the beginning of this chapter,a primarycause of this poor performance may be the curriculumin the United States. Standardelementary and middle school curriculaemphasize recognizing and naming geometric shapes, writingthe propersymbolismfor simplegeometricconcepts,developingskill with measurementand constructiontools such as a compass and protractor,and using formulasin geometricmeasurement(Porter,1989;Thomas,1982). These curricula consist of a hodgepodgeof unrelatedconcepts with no systematicprogressionto higher levels of thoughtthat are requiredfor sophisticatedconcept development and substantivegeometricproblemsolving. In addition,manyteachersteachonly a portionof the geometrycurriculumthat is availableto them. Porter(1989), for instance, reportedthat in entire districts fourth-gradeand fifth-grade teachers spent "virtuallyno time teachinggeometry"(p. 11). When taught,geometrywas the topic most frequentlyidentifiedas being taughtmerelyfor "exposure";thatis, given only brief, cursorycoverage. The SIMS data for the eighth grade indicate thatteachersratedthe "opportunityto learn"geometrymuchlower thanany other mathematicaltopic (McKnight,Travers,Crosswhiteet al., 1985). At the secondary level, the traditionalemphasis has been on formal proof, despite the fact the studentsareunpreparedto deal with it. As Usiskin (1987) noted: There is no geometrycurriculumat the elementaryschool level. As a result,students enter high school not knowing enough geometry to succeed. There is a geometry curriculumat the secondarylevel, but only abouthalf of the studentsencounterit, and only abouta thirdof these studentsunderstandit (p. 29).

3

Logo and Geometry

BACKGROUND The aim of our projectwas to addressdeficiencies in currentelementaryschool geometryinstructionby developinga reconceptualized,research-basedcurriculum thattakesadvantageof the graphics-basedcomputerprogramminglanguageLogo. To explain this choice, we describeour theoreticalfoundationsas we understood and appliedthem initially and then describehow these foundationssuggest benefits of learninggeometrywith the Logo turtle.In Chapter6, we discuss the implications of our researchfor these theories, including enhancementsand revisions connectionsand of the theoriesthatwe madebothto maintaintheoretical-empirical to betterunderstandstudents'understandingand learningof geometry. TheoreticalPerspectives The two theoreticalperspectives that informedour work were those of Piaget andof the van Hieles. The workof Piaget andInhelder(1967) includedtwo major themes related to geometry. First, the developmentalorganizationof geometric ideas follows a definite order,and this orderis more logical thanhistorical.That is, topological relations (e.g., connectedness, enclosure, and continuity) are constructedfirst, followed by projective(e.g., rectilinearity)andEuclidean(e.g., angularity,parallelism, and distance) relations. At best, this theme has received mixed support(Clements& Battista, 1992b). A varietyof geometricideas appear to develop over time, becoming increasingly integrated and synthesized. Certainly,some Euclidean notions are present at an early age. Originally, these ideas areintuitionsgroundedin such actionsas building,drawing,andperceiving. Children might develop actions that produce curvilinear shapes before they develop those actions thatproducerectilinearshapes. Even young childrenhave basic geometric intuitions that might be productivelybuilt on in the elementary school classroom. The second Piagetian theme is that mental representations of space are constructedthroughthe progressiveorganizationof the student'smotorandinternalizedactions.Thus,any suchrepresentationof space,or scheme2,is not a perceptual "reading"of the spatialenvironmentbut is built up from prioractive manipulation of that environment.This theme is supportedby research(Clements & Battista,1992b). Students'ideas aboutshapesdo not come frompassive looking. Instead,ideasaboutshapesevolve as students'bodies,eyes ... andminds... interact with the environment.In addition,childrenneed to explore shapes extensively to fully understandthem-merely seeing andnamingpicturesof shapesis insufficient. Finally,childrenhaveto explorethe componentsandattributesof shapes.The focus on attributesleads to the next theory. 2 In this paper, we define scheme as a mental network of relationshipsconnecting concepts and processesin specific patterns.Representationis anothertermthatshouldbe clarified.Wheneverthis is not obvious from the context, we will explicitly differentiatebetween internal,mentalrepresentations, such as schemes and externalrepresentations,or inscriptions,such as drawingsor writtensymbols.

4

Logo and Geometry

The second theoretical perspective was that of Pierre and Dina van Hiele. Accordingto theirtheory,studentsprogressthroughlevels of thoughtin geometry (van Hiele, 1986; van Hiele-Geldof, 1984). The van Hiele theory is based on severalassumptions.First,learningis a discontinuousprocesscharacterized by qualitativelydifferentlevels of thinking.These levels progressfroma Gestalt-likelevel the van Hieles call "visual"throughincreasinglysophisticatedlevels of description, analysis,abstraction,andproof.Second,these levels aresequential,invariant, andhierarchical.Progressthroughthe levels is dependenton instruction,not age. Teacherscan "reduce"subjectmatterto a lower level, leading to rote memorization. However, students cannot bypass levels and achieve understanding. Understandingrequiresworkingthroughcertaininstructionalsteps.Third,concepts implicitly understoodat one level become explicitly understoodat the next level. Fourth,each level has its own language;the uninformedteachercan easily misinterpretstudents'understandingof geometricideas. Students at Level 1 (visual) can only recognize shapes as whole images. A given figureis a rectangle,for example,because"itlooks like a door."At this level, studentsdo not thinkaboutthe attributes,or properties,of shapes.Some researchers claim thatthereis an even earlierlevel at which childrendo not reliablydistinguish circles, triangles,andsquaresfromnonexemplarsof those classes. Studentsat this level appearto be unableto formreliablementalimages of these shapes(Clements & Battista,1992b). Otherresearcherspreferto categorizethese childrenmerelyas "beginning"Level 1. Studentsat Level 2 (descriptive/analytic)recognize and characterizeshapesby theirproperties.For instance,a studentmight thinkof a squareas a figurethathas four equal sides and four right angles. Propertiesare establishedexperimentally by observing, measuring,drawing, and model making. Studentsfind that some combinationsof propertiessignala class of figuresandsome do not;thus,the seeds of geometricimplicationare planted.At this level, studentsdo not, however, see relationshipsbetweenclasses of figures.Forinstance,a studentmightbelieve thata figureis not a rectanglebecause it is a square.Many studentsdo not reachLevel 2 until middle school or even high school. can formabstractdefinitions,distinguish Studentsat Level 3 (abstract/relational) between necessaryand sufficient sets of conditionsfor a concept, andunderstand andsometimeseven providelogical argumentsin the geometricdomain.They can classify figures hierarchically(by orderingtheir properties),and they can give informalargumentsto justify theirclassifications(e.g., a squareis identifiedas a rhombusbecauseit can be thoughtof as a "rhombuswith some extraproperties"). Studentscan discover propertiesof classes of figures by informaldeduction.For example, they might deduce that in any quadrilateralthe sum of the angles must be 360? because any quadrilateralcan be decomposedinto two triangles,each of whose angles sum to 180?. As studentsdiscover such properties,they feel a need to organizethe properties.One propertycan signal otherproperties,so definitions can be seen not merely as descriptionsbut as ways of logically organizingproperties. At this level, studentscan now understand,for example, that a squareis a

Logo and Geometry

5

rectangle.This logical organizationof ideas is the firstmanifestationof truededuction. However, the studentsstill do not graspthatlogical deductionis the primary methodfor establishinggeometrictruths. Studentsat Level 4 can establish theoremswithin an axiomatic system. They recognizethe differenceamongundefinedterms,definitions,axioms,andtheorems. They are capable of constructingoriginal proofs. That is, at this level they can producea sequence of statementsthatlogically justifies a conclusion as a consequence of the "givens." The van Hiele theory also includes a progressive,five-step3model of teaching thatmoves studentsfromone level of thinkingto the next. In Step 1 (information), the teacherplaces ideas at the students'disposal. In Step 2 (guided orientation), studentsareactivelyengagedin exploringobjects(e.g., folding,measuring)so that they encounterthe principleconnectionsof the networkof conceptualrelationsthat is to be formed.In Step 3 (explicitation),studentsareguidedto become explicitly awareof theirgeometricconceptualizationsso thatthey can describethese conceptualizationsin their own language and are guided to learn traditionalmathematical language.In Step4 (free orientation),studentssolve problemswhose solutions requirethe synthesis andutilizationof those concepts andrelations.The teacher's role includesthe following:to select appropriatematerialsandgeometricproblems (with multiplesolutionpaths);to give instructionsto permitvariousperformances andto encouragestudentsto reflectandelaborateon these problemsandtheirsolutions;andto introduceterms,concepts,andrelevantproblem-solvingprocessesas required.In Step 5 (integration),teachers encourage studentsto reflect on and consolidatetheirgeometricknowledge, with an increasedemphasison the use of mathematicalstructuresas a frameworkfor consolidationso thateventuallythese consolidatedideas may be placed in the structuralorganizationof formalmathematics.At the completionof Step 5, a new level of thoughtis attainedfor the topic. In general,researchsupportsthe usefulnessof the van Hiele levels for describing the developmentof students'geometricconcepts (Burger& Shaughnessy, 1986; Clements& Battista,1992b;Fuys,Geddes,& Tischler,1988).Thefindingthatmost studentsfrom the United States are not progressingthroughthe levels-but that such progression is possible given better curriculumand teaching-cannot be ignored. For example, most textbooks do not requirestudentsto develop higher levels of thinking as they progress through Grades K-12 (Fuys et al., 1988). Curriculumdevelopers and teachers must enrich the geometry learning of our studentsby going beyondthe traditionalteachingof geometryin the UnitedStates. We took up this challenge with the Logo GeometryProject.We believed thatthe available theory and research were sufficiently developed to guide curriculum developmentin this domain(Clements,in press).

3 The van Hiele model is labeled both stages andphases at differentpoints (van Hiele, 1986); they arecalled stepshereto avoidconfoundingthe variousmeaningsof theseterms.This confoundingwould have been most extremein chapter6.

6

Logo and Geometry

How did we applythe two theoreticalperspectives,of Piagetandthe van Hieles, to this project?From the Piagetian perspective, we took the idea that students constructinitial spatialnotions not from passive viewing but from actions, both andimagined,andfromreflectionson these actions(Piaget& Inhelder, perceptual4 1967). The Logo turtlemoves to createshapes,with studentsgiving mathematical commands to direct the turtle's movements (Clements & Battista, 1992b). becausethesecommandsindicate"tothe turtle"how to makecompoFurthermore, nentsof shapesandhow those componentsmustbe combined,Logo activitiesmay facilitate students' progression to higher levels in the van Hiele hierarchy of geometricthinking.Forexample,withtheconceptof rectangle,studentsareinitially only ableto identifyvisuallypresentedexamples,a Level 1 (orvisual5)activity(e.g., a shape is a rectangle if it "looks like a door"). Writing a sequence of Logo commands,or a procedure,to drawa rectangle"... allows, or obliges, the student to externalizeintuitiveexpectations.Whenthe intuitionis translatedinto a program it becomesmoreobtrusiveandmoreaccessibleto reflection"(Papert,1980a,p. 145). Thatis, studentsmustanalyzethe spatialaspectsof the rectangleandreflecton how they can build it from components.This leads to recognitionof the figure's properties,an exampleof Level 2 (descriptive/analytic) thinking.Takingthisto the next step,if askedto designa rectangleprocedurethattakesthelengthandwidthas inputs, studentsmustconstructa typeof definition.Thus,theybeginto buildintuitiveknowledge aboutthe conceptof defininga rectangle,knowledgethatthey can laterinteactivity.If challengedto draw grateand formalize-a Level 3 (abstract/relational) a squareor a parallelogramwith theirrectangleprocedure,studentsmay logically orderthese figures,anotherLevel 3 activity.Further,Logo workaids in the generation of many examplesof a concept. Overall,then, Logo may have the potential to encouragethe constructionof geometric propertiescharacteristicof Level 2 thinking,somethingthattextbookstypicallydo not do (Fuys et al., 1988). Research on Logo and Mathematics Education

When the Logo Geometry Project began, empirical findings on Logo were ambiguous(Clements,1985). Earlyreviews concludedthattherewere conflicting resultsaboutthe effects of Logo on overallmathematicsachievement.Experiments by Logo's developersgeneratedpositive reports(Papert,Watt, diSessa, & Weir, 1979). In the United Kingdom, low-achieving 11-year-oldboys with 2 years of directedLogo programmingexperienceimprovedto performat the same level as 4 Perceptual is used here, consistentwith Piaget's originalformulation,as meaningphenomenaor experiencesthatdependon sensory input,in contrastto those that are representedmentally(and thus can be "re-presented" imagisticallywithoutsensorysupport).Thus,perceptualshouldnot be confused withthe notionthatwe, with Piaget,reject-that of "immaculateperception"in whichperceivedobjects are immediatelyregisteredin the brain. 5 We use the termvisual throughoutthis manuscriptwhen it promotesconsistencywith the usualvan Hiele interpretation,recognizingthatit would be moreaccurateto use termssuch as spatial (especially for nonsightedindividuals)or imagisticin certainsituations.Oursuggestedrephrasingfor this level is presentedin chapter6.

Logo and Geometry

7

a controlgroupon one generalmathematicstest but fell behindthe controlgroup on anothertest (Howe, O'Shea, & Plane, 1980). Otherstudies showed little positive effect on mathematicsachievement(Akdag, 1985; Pea, 1983), althoughthere were suggestive results for geometric concepts (Lehrer& Smith, 1986; Noss, 1987). Thus, when we began the Logo GeometryProject,it was unclearwhether Logo could have significant positive effects on mathematics achievement. Therefore,our goal was to cast a wide net to examine the effects of a specially designedLogo environmenton elementaryschool students'learningof geometry. Morerecentreviews, conductedafterourprojectwas completed,generallyhave been positive, as illustratedby the following quotationfrom McCoy (1996): turtlegraphicsat theelementary level,is clearlyan particularly Logoprogramming, effectivemediumforprovidingmathematics experiences... whenstudentsareable invariedrepresentations, activeinvolvement withmathematics becomes toexperiment Thisis particularly thebasisfortheirunderstanding. trueingeometry... andtheconcept of variable" (p.443). In the areaof geometry,researchhas focused on concepts of plane figures, especially students'levels of geometricthinkingaboutthose figures; angle and angle measurement;and motion geometry.We discuss each of these in turn. GuidedLogo experiencecan significantlyenhancestudents'concepts of plane figures(Butler& Close, 1989;Clements,1987). Whenaskedto describegeometric shapes,studentswithLogo experiencegive morestatementsoverallandmorestatementsthatexplicitly mentiongeometricpropertiesof shapesthanstudentswith no Logo experience(Clements& Battista,1989, 1990;Lehrer& Smith, 1986). In one study, students were able to apply their knowledge of geometry better than a comparison group, but there was no difference in their knowledge of basic geometricfacts.The researchersconcludedthatthe use of Logo influencedthe way in which students mentally representedtheir knowledge of geometric concepts (Lehrer, Randle, & Sancilio, 1989). In a similar vein, it has been shown that middle school studentsusing Logo move to higherlevels of conceptualizingand (Clements& Battista, beginto integratespatialandsymbolicmentalrepresentations 1988;Hoyles & Noss, 1988). Ninth-gradestudentsmay preferlearningwith Logo. In a studyconductedby Olive (1991), an end-of-semesterdiscussionof theirexperiences in the Logo class revealedthatstudentsbelieved the class to be intriguing and exciting. These studentsreportedbeing less frustratedand more involved in the Logo class than they were in their other classes. They believed that they had learned a lot of mathematics, especially about angles, quadrilaterals,and the Pythagoreantheorem.Finally, they thoughtLogo had helped them with problem solving. However, for these students, success and sophistication in the Logo programmingaspects of the tasks were necessary, but not sufficient, for success and sophisticationwith the geometricconcepts involved in the tasks. Similarly,Logo experienceappearsto affect students'ideas aboutangle significantly. Responses of controlstudentsin one study reflectedcommon usage, such as describingan angle as "a line tilted,"or no knowledgeof angle. In comparison, the Logo studentsindicatedmoremathematicallyorientedconceptualizations,such

8

Logo and Geometry

as "Likewhere a point is. Wheretwo lines come togetherat a point"(Clements& Battista,1989). Severalresearchershave reportedthatLogo experiencehas a positive effecton students'angleconcepts(Clements& Battista,1989;duBoulay, 1986; Frazier,1987;Kieran,1986a;Kieran& Hillel, 1990;Noss, 1987;Olive, Lankenau, & Scally, 1986).However,in some situations,benefitsdo not emergeuntilstudents have more thana yearof Logo experience(Kelly, Kelly, & Miller, 1986-87). Logo experiencesmay also fostersome unintendedconceptionsof anglemeasure. Forexample,studentsmay confuseanglemeasurewiththe amountof rotationalong the path (e.g., the exteriorangle in a polygon) or the degree of rotationfrom the vertical(Clements& Battista,1989). In addition,conceptsgeneratedwhile working with Logo do not replace previously learned concepts of angle measure. For example, students'conceptionsaboutangle measureandthe difficultiesthey have coordinatingthe relationshipsbetween the turtle's rotationand the constructed angle have persistedfor years, especially if not properlyguided by theirteachers (Clements, 1987; Cope & Simmons, 1991; Hoyles & Sutherland,1986; Kieran, 1986a; Kieran, Hillel, & Erlwanger, 1986). In general, however, appropriately of anglemeasure.After designedLogo experienceappearsto facilitateunderstanding with students' of size are more working Logo, concepts angle likely to be matheand abstract & correct, coherent, (Clements Battista,1989;Kieran,1986b; matically Noss, 1987), while showinga progressionfromvan Hiele Level 0 to Level 2 in the span of the Logo instruction(Clements & Battista, 1989). If Logo experiences emphasizethe differencebetweenthe angle of rotationandthe angle formedas the turtletracesa path,confusionsregardingthe measureof rotationand the measure of the angle may be avoided(Clements& Battista,1989; Kieran,1986b). Researchindicates that Logo experiences can also aid the learningof motion geometryandrelatedideas such as symmetry.As they were workingwith a Logo unit on motion geometry, students'movement away from van Hiele Level 0 was slow. Therewas, however,definiteevidenceof a beginningawarenessof the properties of transformations(Olson, Kieren, & Ludwig, 1987). In another study, middle school students achieved a working understandingof transformations andused visual feedbackto correctovergeneralizationswhen workingin a Logo microworld(Edwards,1991). Furthermore,Logo experiencesmay help develop notions of symmetry.Studentsas young as first gradehave been observedusing such mathematicalnotions as symmetry (Kull, 1986). In addition, symmetry concepts are learned by students involved in Logo through middle school (Edwards,1991; Gallou-Dumiel,1989;Olson, 1985). One studentused a specially designed Logo symmetry microworld to learn such concepts and effectively transferredher mathematical understandings to a paper-and-pencil problem (Hoyles & Healy, 1997). Generally,then, studies supportthe use of Logo as a mediumfor learningand for teaching mathematics.6Results especially support Logo as a medium for 6 For more

completerecentreviews, see Clements, 1999; Clements& Sarama,1997; McCoy, 1996.

Logo and Geometry

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learningand teaching geometry (Barker,Merryman,& Bracken, 1988; Butler & Close, 1989; Clements& Meredith,1993; Hoyles & Noss, 1987b;Kynigos, 1991; Miller, Kelly, & Kelly, 1988; Salem, 1989). However, not all research has been positive. First, few studies report that students "master"the mathematical concepts that are the teachers' goals for instruction.Second, some studies show no significantdifferences between Logo groups and control groups (Johnson, 1986). Without teacher guidance, mere "exposure"to Logo often yields little learning (Clements & Meredith, 1993). Third, some studies have shown limited transfer. For example, the scores of students from two ninth-gradeLogo classes did not differ significantly from those of control students on subsequenthigh school geometry grades and tests (Olive, 1991; Olive et al., 1986). One reasonis that studentsdo not always think mathematically,even if the Logo environmentinvites such thinking (Noss & Hoyles, 1992). Forexample, some studentsrely excessively on visual/spatialcues and avoid analytical work (Hillel & Kieran, 1988). This visual approachis not relatedto an abilityto createvisual images but to the role of the visual "data"(i.e., the students' perceptions) of a geometric figure in determiningstudents' Logo constructions. Although helpful initially, this approachinhibits students from arrivingat mathematicalgeneralizationsif overused.Further,thereis little reason for studentsto abandonvisual approachesunless teacherspresenttasks that can only be resolvedusing an analytical,generalized,mathematicalapproach.Finally, dialogue between teacher and students is essential for encouragingpredicting, reflecting, and higher level reasoning. In sum, studies showing the most positive effects involve carefully planned sequences of Logo activities. Appropriateteacher mediation of students' work with those activities is necessary for students to construct geometric concepts successfully. This mediation must help students forge links between Logo and other experiences and between Logo-based procedural knowledge and more traditionalconceptual knowledge (Clements & Battista, 1989; Lehrer& Smith, 1986). Caremust be takenthatsuch links arenot learnedby rote (Hoyles & Noss, 1992). We attemptedto createthe Logo Geometrycurriculumto providejust such a plannedsequenceof activitiesthatincorporatedteachermediation.Ourresearch goals were (a) to evaluate students'learningas they workedwith this curriculum and, in that context, (b) to investigate how elementary school students learn geometric concepts and (c) to describe the role Logo can play in this learning. Because we worked closely with teachers who had to meet state and school curriculumframeworks,the Logo Geometrycurriculumincluded and assessed a wide range of topics. Given the constantstateof changein the field of computersin education,an issue may ariseabouttheperceivedrelevanceof Logo-basedcurricula.Today,Logo plays a proportionatelysmallerrole in schools thanwhenthese datawerebeing collected, especially given the influx of othercomputerprograms.Yet therearefive reasons that the researchreportedhere is significant.First, various versions of Logo are still being sold (and sharewareversions downloaded)and used in classrooms.In

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addition, numerous alternate forms of turtle geometry exist (e.g., "Maps and Movement,"EducationDevelopmentCenter,1993, is but one example of a genre of courseware). These programs, and other navigationally based instructional programs,requirea researchbase. Second, students'work in Logo environments has much to teach us aboutlearningin othercomputerandnoncomputerenvironments that are not necessarily computational.Third, this project representsone vision of institutingthe early Standards(NCTM, 1989) for geometry and thus complementsevaluationsof similar projectsthat primarilyemphasizednumber (Hiebert,1999). Fourth,the assessmentitems (e.g., the pretestitems alone)provide a valuabledescriptionof elementaryschool students'geometricknowledge.Fifth, theresearchhas implicationsfor the developmentof mathematicseducationtheory, particularlythe van Hieles' theoryof geometricthinking. ORGANIZATIONOF THE CHAPTERS This chapterintroducedthe Logo Geometryresearchand developmentproject andits rationale.Chapter2 providesdetails aboutthe development,structure,and content of the curriculumand describes the participantsand methods of our research. Chapter3 presents the results. Chapter4 discusses these results and presentsconclusions and educationalimplicationsof this research.This includes a focus on students' knowledge of geometry before instruction, providing a portraitof the geometricknowledge of 1,605 elementarystudents(GradesK-6) in two states. Chapter5 presentsbrief descriptionsof softwaredevelopmentandresearchthat emerged from the Logo GeometryProjectresearchcorpus.That is, we used our experienceswith the Logo GeometryProjectto develop ourown versionof Logo, "TurtleMath,"in anotherNSF curriculumdevelopmenteffort. The principleson which we based that development, and our research findings, are reviewed in chapter5 by Julie Sarama,who led the researchon TurtleMath. Finally, chapter6 presentsimplicationsof this work for the educationaltheory of geometry understandingand learning. That is, we reflect on the research reportedin chapters3, 4, and 5 to develop an extensive elaborationand revision of the van Hiele theory. Our goal is to ensure that the monographgoes beyond the Logo GeometryProjectto discuss the broadissues of Logo and geometry in mathematicseducation.

Chapter 2

Logo Geometry Curriculum and Methodology The Logo GeometryProjectwas conceptualizedby the codirectors,on the basis of their own and others' research that indicated the potential of Logo turtle graphicsfor developing elementaryschool students'geometriccompetencies. In the past, this potentialoften remainedunrealizedbecause of a lack of curriculum structure. Curriculumdevelopmentwas conductedover four years. Duringthe first year, extantresearchandcurriculain elementaryschool geometrywere reviewed,a new geometrycurriculumwas developed,Logo activitieswere constructedto promote the objectives of thatcurriculum,andmaterialsto guide teachers'use of the Logo curriculumwere developed. This new curriculumwas designedprimarilyto help studentsmove fromintuitivegeometricnotionsto theirformalcounterparts,using Logo as a mediator.For example, Piaget statedthatchildren"canonly 'abstract' the idea of such a relationas equalityon the basis of an actionof equalization,the idea of a straightline fromthe actionof following by handor eye withoutchanging direction,and the idea of an angle from two intersectingmovements"(Piaget & Inhelder, 1967, p. 43). These actions form the intuitive beginning of the angle concept. When studentsintentionallywalk, draw, and discuss angles, they make such intuitionsexplicit. When they programthe Logo turtleto createangles, they must use symbolic code to express these notions and,more important,to quantify them. In this way, Logo mediatesbetween the intuitive and the eventualformalization of the angle concept. In the second year, the Logo-enhancedcurriculumwas field-testedby teachers who were involved in refiningthe materials.Then the materialswere formatively evaluatedand revised. In the thirdyear, more extensive field tests and evaluation were conducted.The field tests of the second and thirdyears constitutethe main researchdata reportedin this monograph.In the fourthyear, the materialswere revised and writtenin final form and disseminatedby the Silver, Burdett& Ginn PublishingCompany(Battista& Clements, 1991). As we have noted, the projectbegan with a comprehensivereview and evaluation of existing elementaryschool geometrycurriculaandrelevantresearch.From bothof these, we createdthe theoreticalfoundationdescribedin chapter1. We also created a scope and sequence of geometric topics that were based on current school curricula.Then, we determinedwhich topics in geometrycurricularemain of centralimportance,which are overemphasized,which are not given adequate attention,andwhich arenot currentlyrepresentedbut shouldbe. Next, we analyzed

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videotapes of childrenworking in Logo environments-observing the students' successes and difficulties in dealing with geometric ideas-to determinewhich ideas mightbe effectively taughtwithina Logo environment.7 In brief,appropriate Logo work can help elementary school and middle school students mentally constructthe types of geometricpropertiesthatthey need to learn,somethingthat textbooksoften fail to do (Fuys et al., 1988). On the basis of this initial work, we constructeda new geometry curriculum sequenceandassociatedLogo activities.This workwill be describedin detaillater in this chapter;for now, we offer an overview. The curriculumis organizedinto threemajorstrands:the Pathstrand,the Shapes strand,andthe Motions strand.In the Pathstrand,the conceptof pathis taughtexplicitly,bothoff computer,through activitiessuch as walkinganddrawing,andon computer,as a recordof the turtle's movements. This concept of path is then used to organize beginning geometric notions. In the Shapes strand, students think about shapes-such as triangles, rectangles,andotherpolygons-as specialtypes of paths.Angle andanglemeasure are included in this strand.The goal of the Motions strandis to use computer graphics to introduce students to geometric transformationsand to help them constructcognitive "buildingblocks," such as mentalrotationof shapes, thatare importantin dealingwith spatialproblems.Conceptsof congruenceand symmetry are explicitly addressedhere as well. To obtainfeedbackon ourcurriculumandits theoreticalrationale,we presented our work to our advisory board, our fellow project directors,and colleagues at several professionalmeetings. Our advisoryboardconsisted of six experts from the areas of mathematics, computer science, and mathematics education or psychology (RichardBrown,KennethCummins,RobertDavis, WallaceFeurzeig, Alan Hoffer, and Grayson Wheatley), as well as two master teachers, Linda HallenbeckandFranBickum.We revised andelaboratedthe initial outlineon the basis of the advice of these groups. We nextobtainedfeedbackfromtwo preliminarypilottests.In thefirst,two graduate assistantsand one experiencedteachertaughtthe geometric ideas to small groupsof studentsfromeach of fourgradelevels-K, 2, 4, and6. Fromvideotapes of the lessons andteachers'dailyjournals,we determinedthe strengthsandweaknesses of the materialsand subsequentlyrevised them. The second pilot test was more elaborate.In this case, forty-two studentsfrom GradesK-6 attended15 daily sessionsof 80 minuteseachin Julyof 1987. An experienced teacher taught the curriculumseparatelyto students at three different graderanges,K-l, 2-4, and5-6. All studentswerepretestedandposttested.In addition, case studieswere conductedfor two to four studentsat each graderange.As each pairworkedon the curriculummaterials,a graduateassistantassignedto that pair observed their work. Each graduateassistant's goal was to ascertain the 7 Fordetaileddiscussionsof deficiencies in the then-currentcurriculaandhow these deficiencies can be amelioratedusing Logo, see BattistaandClements,1986, 1987, 1988a;ClementsandBattista,1986, 1989.

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students'geometricconceptualizationsas the latterworkedthroughthe curriculum materials. Further, individual interviews of about 45 minutes duration were conductedwith each of these case-study students.The purposeof the interviews was to probe the students'geometricideas both before and afterthey experienced the curriculum.Finally, the codirectorsobservedthe teacher'spresentationof the lessons and the individual work of the case-study students. In daily debriefing sessions, the project staff discussed the graduate assistants' case studies, the teacher'sevaluationof the lesson, andthe codirectors'observationsto evaluatethe effectiveness of the curriculumactivities. From this second preliminarypilot test, information such as results of the pretestingandposttesting,interviews,case studies,andotherobservationswas used to revise the materialsagain. The revision created a more complete curriculum including(a) backgroundinformation;(b) lesson plans;(c) studentactivitysheets, homework sheets, and adjunctmaterials;and (d) Logo microworldsand utility procedures. The next step in the developmentof the curriculumwas to trainteachersin the use of the curriculumand to involve them in refiningthe curriculum'sobjectives for each gradelevel. Previously,we had met with the administratorsof five school systems(all workin the firsttwo yearstook placenearKentStateUniversity[KSU] in Ohio).We askedthese administratorsto select 14 experiencedteachersto study, revise, and test the curriculum.The selected teachersparticipatedin a four-week course. Duringthe first 2 weeks of this course, we introducedthe teachersto the curriculumand its rational.During the last 2 weeks, the codirectorsand teachers workedin grade-levelgroups,revising the materialsto make them appropriatefor the separategrades.Eachgroupthenproduceda documentthatdetailedthe objectives andactivitiesappropriatefor theirindividualgradelevel. We synthesizedthe suggestedcurricularrevisionsandgrade-leveldocumentsandproducedgrade-level specific versions of the curriculum. Duringthe following school year(1987-88), the 14 participatingteachersimplementedthis updatedversion of the Logo Geometrycurriculum,devoting approximately 15%of theirinstructionaltime for mathematicsto it. The teachersevaluatedthe materials,bothfor theirinstructionalsoundnessandfor theireffectiveness in teachinggeometry,andsuggestedrevisions.We visitedeach teacher'sclassroom to provideassistance,to assess the extentandmannerof use of the Logo materials, and to talk with studentsabout their work with the curriculum.In addition,we carriedout an extensive assessmentprogramconsisting of paper-and-penciltests andindividualinterviewswith the studentsof the 14 participatingteachers.Finally, the projectteam (ourselvesandtwo graduateassistants)conductedmonthlymeetings to discuss the teachers'evaluationsandrevisions andto discuss how to better teach the curriculum.In the summerfollowing that school year, the teachersmet for 2 weeks to work in their grade-levelgroups.With our guidance,they synthesized theirexperiences and offered suggestions for revisions. Again, we used the resultantdocumentsto makecurricularrevisionsfor the set of materialsappropriate for each gradelevel.

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andMethodology LogoGeometryCurriculum

In the thirdyear,the same 14 teacherstaughtthe curriculumfor the secondtime. In addition,anothergroupof 14 teachersfrom Buffalo, New York, who received only a week of trainingon the materialsduringthe summer,taughtthe curriculum for the first time. Thus, the new set of teachersrepresentedteachersnot involved in the developmentof the materials.Finally, 28 teachercohortswere identifiedby administrators,14 at each site, as "controlteachers"and,along with theirstudents, participatedin the assessment program.A complete descriptionof the research methodologies for the field tests of the second and third years is provided in chapter3. A final summermeetingbroughtthe teachersat each site togetherto make final suggestionsfor revisions of the materials. Finally,in the fourthyearwe beganan analysisof all existing assessmentdatato complementtheteachers'suggestions.We workedwithoureditorsat Silver,Burdett & Ginnto makefinal revisionsandpublishthe materialsas Logo Geometry. LOGOGEOMETRY: CURRICULUMGOALS Logo Geometry(LG) has three major curriculumgoals: (a) achieving higher levels of geometricthinking,(b) learningmajorgeometricconceptsandskills, and (c) developing power and positive beliefs in mathematicalproblem solving and reasoning.These goals providea undergirdingfor ourspecific researchquestions; thatis, they arethreadsthatwe weave throughourpresentationof the results.We discuss each of these goals briefly here as a backgroundfor the next section, which describesthe curriculumstructureand activities. 1. AchievingHigher Levels of GeometricThinking One goal of the curriculumis to help students move toward higher levels of geometric thinking in the van Hiele hierarchy,as described in chapter 1. 2. LearningMajor GeometricConceptsand Skills The concepts and skills of this second goal include, but arenot limitedto, those encounteredin standardcurricula(e.g., shape concepts, but also concepts and skills involving geometric motions and paths). In addition, the curriculum emphasizes the students' development of the related mental processes, such as spatial visualization and estimation, and the abilities prerequisite for formal mathematical thinking, such as conjecturing and convincing. We have also emphasized the thoughtful use of computation and arithmetic operations in problem-solving situations. Consistent with NCTM's first Standards (1989), LG "focuses on the development of relationshipsand understandings;it goes far beyond the common and narrowemphasis on naming and identifying shapes, listing theircharacteristics, and memorizing terms and definitions" (p. 42). This leads us to the discussion of the next major goal.

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3. Developing Power and Positive Beliefs in MathematicalProblem Solving and Reasoning Mathematicsinvolves formingabstractionsin ways thatenableone to solve problems. Thus, in LG, we attemptedto presenteach task as a problemsituationand to have studentsbuildmathematicalideaswithinthe overallcontextof solvingproblems. We attemptedto develop such generalproblem-solvingheuristicsas making tables and looking for patterns.Moreover,in our observationsof the teachersand in the case studies, we realizedhow the tasks were true and significantproblems for the students,even though the tasks do not appearto be problematicto most adults. CURRICULUMSTRUCTUREAND ACTIVITIES The curriculumis dividedinto threestrands:Paths,Shapes,andMotions.In this section, we describe the structureand activities of these three strands,including activities for studentsand teaching strategies. Path Activities Because childrenunderstandbeginning spatialnotions in terms of action, and the mathematicalconcept of path can be thoughtof as a recordof movement, it seemednaturalto emphasizethisconceptin ourbeginningstudyof geometrywithin a Logo environment.For example, having students visually scan the side of a building, run their hands along the edge of a rectangulartable, or walk a straight path might help them abstractthe concept of straightness,but the concept of straightnesscan be broughtto a more explicit level of awarenesswith Logo activities. It is easily demonstratedin Logo that a straightpath is one that has no turning.Thus, we took the position that the concept of path should be explicitly taughtandused as an organizingidea for beginninggeometricnotions. Below we discuss five componentsof the Paths strand. Paths. In the first componentof the Paths strand,studentslearn what a path is and how to identify various kinds of paths. They walk, then describe, several differentinterestingpathsthathave been laid out on the floor with maskingtape, some of which arestraight,some closed, some open, some with bends, andat least one with an arc or other "curvypart." Studentslearn to constructpaths in Logo by giving a sequence of movement commands.For example, a gridlike map appearswith a designatedstartingand ending point for the turtle.The studentis to directthe Logo turtlefrom the startto the finish. Studentsat the GradeK-l level used a special Logo microworldcalled "Singlekey"that allowed them to enter most commandsas a single key press (F forFD 10; RforRT 30). Proceduresfor Paths. In the second componentof the Paths strand,students analyzetherelationshipbetweenLogo commandsandthe componentsof a paththat

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they create. A command such as FD 50 draws a straightpath 50 units long. A commandsuch as RT 90 createsa changein directionalongthepath.Studentslearn to writeproceduresin Logo and, consequently,theirknowledgeof differentkinds of pathsis reinforced.They write proceduresto move the turtlethroughthe maps encounteredin the firstcomponentof thePathsstranddescribedpreviously.In addition,studentsconstructproceduresforpathsthathavesome,butnotall, of thelengths of theirstraightcomponentslabeledin diagrams.Finally,studentswriteprocedures to drawclosed and nonclosed,straightandnonstraight,simpleandnonsimplepaths. Turns.In the thirdcomponentof the Pathsstrand,studentslearnto specify and estimatean amountof a turn.(We believe thatstudentsunderstandthe concept of turnearlierthan the concept of angle. Thus, we introducestudentsto turnsfirst, then, later, we relate turns to angles.) In the context of "playing turtle,"the studentsareintroducedto whole turns(all the way around),half turns,andquarter turnsas a method of specifying an amountof turn.They are then introducedto degrees as the unit of turnthatthe turtleuses. Finally, studentsplay two computer games that provide experience estimating angles and distances in a Logo environment.In the first game, each playergets two chances to turnthe turtletoward a target, then two chances to move the turtle into the bull's-eye. In the second computergame, studentsnavigate the turtlearoundan obstacle and into a target with the fewest commandspossible. Estimationstrategies,such as the use of reference measures,are emphasized. Next, studentshave experienceestimatingthe amountof turnand learnthatan angle is a special kind of path.They predictthe path the Logo turtledrawswhen issued a forwardcommand,then a turn command,then finally anotherforward command, which results in an angle being drawn. The students then use the computerto check theirestimate.This activity is designed to help studentsbegin developingan intuitiveunderstandingof the relationshipbetweenturnsandangles. Path/CommandCorrespondenceand Debugging.In this single lesson, the fourth component of the Paths strand, students learn about two importanttopics in geometricproblem solving within the Logo environment.The first is the correspondencebetween a Logo commandand the componentof a paththat it creates (e.g., a FD 50 commandcreates a straightpath 50 units long; a RT 60 command turnsthe turtle60?). The teacherruns a series of commandsand asks studentsto drawthe path thatwould be createdif one of these commandswas changed.The second topic addressedis the ability to correct errorsin Logo procedures(i.e., "debug"procedures).When students debug their code, they (a) reflect on the correspondence between Logo commands and components of the path these commandsproduce,(b) correctprocedurescontainingerrorsinsteadof startingover, (c) gain an understandingof the natureof theirerrorsand the ways they might be corrected,and (d) learn that this problem-solvingtechniqueof correctingprocedures can be appliedin many differentcontexts. A frameworkis introducedthat provides a general four-step strategyfor correctingone's errorsduringproblem solving. Studentsdebug severalproceduresusing this framework.

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TurtleDeliveries. In the fifth and final componentof the Paths strand,students apply some of their previously learned skills in a problem-solvingenvironment, and learnaboutthe process of "undoing"(i.e., finding the inverse for) a sequence of actions.They are askedto writeproceduresthatmove the turtlefrom one point on a scene (depictedon the computerscreen) to another(one of threerestaurants) and then returnthe turtleto the startingpoint along the same path. The students must find a pattern for returning to the starting point (undoing the original commands)so thatthey can bringthe turtlehome even if the destinationis off the screen(whichrequiresreplacingthe last commandwith its inverse;e.g., LT 90 for RT 90, etc.). The teacherhelps studentsconnect the idea of "undoinga sequence of turtlecommands"to the relatedidea of "undoinga sequenceof arithmeticoperations"by, for example, relatingit to the following problem:"I am thinkingof a number.If I multiplyit by 2, then add 5, I get 13. Whatis my number?" Shapes: Special Paths Once studentsfirmlygraspthe conceptof path,we move on to havingthemthink about special paths such as squaresand triangles.The goal of this second strand of the LG curriculumis to have studentsview these shapesas pathsandthusbegin analyzing the shapes in terms of their component components and properties. (Thecurriculumdoes notneglectthe "shapesas wholes"perspective.Studentsidentify shapes in the environmentand describeand classify regions, includingfaces of solids, on the attributeof shape.) Squares.In the first componentof the Shapes strand,studentslearnto identify squares,constructthemin Logo, describetheirproperties,anduse themas building blocksto makeothershapes.Studentsfirstidentifyexamplesof squaresin the environment and then plan and write several Logo proceduresto draw squares of varioussizes. The teacherhelps the studentsrelatethe componentsof the squares drawnto the correspondingcomponentsof the Logo procedures.The properties of a squareare discussed. Finally, studentsuse a general squareprocedurewith inputs (variables)to produce creative designs that consist of several congruent squares. Rectangles. In the second component of the Shapes strand,as with squares, studentslearnto identifyrectangles,constructthemin Logo, anddescribetheirproperties. Studentsidentify examples of rectanglesin the environmentand then plan and write several Logo proceduresto drawvariousrectangles.The teacherhelps the studentsrelate the componentsof the rectanglesdrawnto the corresponding componentsof the Logo procedures.The propertiesof a rectangleare discussed. Finally, studentsdefine a rectangleprocedurewith inputs.The studentsare asked if they can make various figures, such as squares,trapezoids,or parallelograms, with theirrectangleprocedure.Additionally,they arechallengedto createdesigns. The characteristicsof squaresandrectangles,andthe relationshipbetweenthe two (i.e., a squareis a special kind of rectangle),are summarized.

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Equilateral Triangles.In the thirdcomponentof the Shapesstrand,studentslearn to identifytrianglesanddescribetheirproperties.In addition,they constructequilateraltrianglesin Logo and use them as building blocks to make other shapes. Studentsidentify trianglesin the environmentand discuss theirproperties.They thenfigureout how to commandthe turtleto makean equilateraltriangle.By using the problem-solvingstrategiesof "guessand check"and "makea table,"students are guided to determine the correct amount of turn in an equilateraltriangle. Again, studentsgeneratecreativedesigns or pictures. RegularPolygons. In the fourthcomponentof the Shapes strand,studentslearn about polygons, regularpolygons, and identifying polygons by their numberof sides. The studentsdiscuss the propertiesof regularpolygons afterengaging in a "guess my rule"activity in which the teachersorts shapes one at a time into two categories(regularandnonregularpolygons). Studentsdeterminethe relationship betweenthe numberof sides in a regularpolygon andthe amountof each turn(i.e., the measureof the exteriorangle). This idea is investigatedin the context of the "ruleof 360";thatis, in orderfor the turtleto turnall the way around,it must turn a total of 360?. Even thoughthe rule of 360 is a powerfuland generalmathematical property,it is accessible to studentsin the elementaryschool grades. ClassifyingAngles. In the fifth componentof the Shapes strand,studentslearn thatan angle can be thoughtof as the union of two rays with a common endpoint, how angles aremeasuredin mathematics,andhow to identifyangles in polygons. One can thinkof an angle as a pathcreatedby a forwardmove, a turn,andanother forwardmove. Up to thispointin the curriculum,only the measureof thisturnalong the pathhas been considered.This lesson emphasizesthe measureof a differentbutrelated-turn, the one thatmoves one side of the angle onto the other.Students use the procedureANGLEthatdrawsa ray,turnsthe turtlethe amountthatis input, and then drawsanotherray. Two computergames aid the studentsin establishingthe relationshipbetween (a) the measureof an angle and the amountof turnbetween the sides of the angle and(b) the amountof turnat a turningpointin a pathandthe amountof turnbetween the sides of the angle thuscreated.In the firstgame, the computerdrawsa ray and, whiletheeyes of player2 areclosed,promptsplayer1 to entera turnmeasure.Because player2's eyes areclosed, only player1 knowsthe amountof turn.The turtlerotates thegivenamountanddrawsanotherray,formingan angle.Player2 guessestheangle measurement.The differencebetweenthe actualturnand the estimatedturnis the score of player 2. In the second game, the computerdraws a path and prompts player 1 to entera turn.Player2 mustenteranotherturnthatheadsthe turtleback towardits startingpoint(i.e.,thesupplementof player1's turn).As partof thiscomponentof the Shapesstrand,the teacherinitiatesa discussionof the usefulnessof 90? and 180? as referenceangles in estimatingangle measureand, finally, initiatesa discussionof angle classification(i.e., as acute,obtuse,andright). InteriorAngles of a Polygon. In this problem-solvingactivity,the sixth component of the Shapes strand,studentsuse Logo andpreviouslydiscussed ideas from

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the Shapes strandto determinethe relationshipbetween the numberof sides of a polygon andthe sum of its vertexangles.Studentsuse the problem-solvingstrategy "look for a pattern." Parallel Lines and Parallelograms. In the seventh component of the Shapes strand,studentslearn aboutparallelismand parallelogramsin both standardand Logo contexts. A concept attainmentlesson by the teacherintroducesparallelogramsandtheirproperties.Afterward,studentsfigureouthow to drawparallellines in Logo and write proceduresfor drawingparallelograms.Studentsmake a table andattemptto determinethe relationshipbetweenadjacentturnsin a parallelogram. Classificationof Quadrilaterals.In the eighth componentof the Shapes strand, students learn about the propertiesof parallelogramsand the hierarchicalrelationshipsbetween parallelogramsandotherquadrilaterals.Given a parallelogram Logo procedurewith inputs, the students are asked if they can make various figures,such as squares,trapezoids,andparallelograms,using only this procedure. Motions The goal of thisthirdandfinal strandis for studentsto developconceptsin motion (transformational)geometry and to help studentsconstructcognitive "building blocks" that are importantin dealing with spatial problems. Davis (1984), for example, describesthe cognitive buildingblocks thatare neededto determinethe areaof a rotatedsquareon a geoboard.In additionto mentalimages of squaresand of the acts of rotatingandtranslatingtriantriangles,he cites mentalrepresentations gles, of putting them together to make other shapes, and even of cutting apart squaresto get triangles.Fundamentalto this strandare the ideas thatthere are an infinite numberof figures congruentto a given figure and that these figures may be relatedby a combinationof geometricmotions (i.e., isometriesof the plane). Introductionto Symmetry.In the first component of the Motions strand,the teacher introducesthe concept of symmetrywith a "guess my rule" activity (in which shapes are sortedinto the categoriesof symmetricand not symmetric).The studentstheninvestigatesymmetrywith Miras,learninghow to determinewhether a shape has a line or lines of symmetry. Mirror. In the second componentof the Motions strand,the studentslearn to predictmirrorimages andto constructsymmetricfigures.They firstpredictmirror images visually,thenthey use a MIRRORprocedurein Logo to check theirpredictions. The MIRRORproceduredraws a dotted symmetryline at the location and headingof the turtleand creates a "mirrorturtle"with thatlocation and heading. For each subsequentcomment the studentsenter, the regularturtleexecutes the commandandthe mirrorturtleexecutes the "symmetriccommand";thatis, directions of left and right are reversed,but all otherpropertiesremain.In this way, a path and the mirrorimage of that path are created. The students then use the MIRRORprocedureto constructsymmetricfigures. Also, studentsexamine the Logo code producedby MIRRORto learnapath perspectivefor constructingmirror

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images; that is, the mirrorimage of a figure defined by a sequence of Logo commandsis defined by that same sequence with the RIGHTsand LEFTs interchanged. This allows studentsto relate their visual notion of symmetrywith the previouslydevelopedknowledgeof paths.Finally,studentsexplorewhatsymmetric picturesand designs they can createwith the MIRRORprocedure. Introduction to Geometric Motions. The goal of this activity, the third compo-

nent of the Motions strand,is for studentsto develop the concept of congruence, to learnaboutthe geometricmotionsof slide, turn,andflip, andto learnthe related concepts of flip lines and turncenters.Priorto using these concepts in Logo, the teacherexplainsthattwo figuresarecongruentif they have the same size andshape (i.e., if and only if one fits on top of the other exactly). Next, the slide, flip, and turn motions are demonstratedon an overhead projector,and the students are asked to label the slide, flip, and turnimages on a correspondingactivity sheet. Studentsuse a transparencyto show the sliding motion for slides and, for turs, the turningmotionandthe turncenter.Studentsuse Mirasto locate the flip (mirror) lines for flips. Motions must be specified by theirmathematicalparameters.For instance,to specify a turn,a studenthas to indicatethe turncenterandthe amount of turn. Logo Motions.Thereareseveralpurposesfor this activity,the fourthcomponent of the Motions strand.Studentslearnhow to performplanarmotions with Logo on the computer.(This is accomplishedvia a MOTIONSmicroworld.)They also begin to acquirethe ability to visualize the results of these motions. In addition, becausestudentshave to instructthe computerto performthese motions,they learn a formalway of specifying slides, flips, and turns.Thus, the computerserves as a transitionaldevice fromphysicalmovementsto moreabstractmathematicalconceptualizations. As such, it helps the students reach a higher level of geometric thinkingin the van Hiele hierarchy. Rocket.In the fifth componentof the Motions strand,the students'spatialvisualization skill and ability to use motions in a problem-solving environmentare developed. Using the computer,the studentsspecify how to move a rocket onto its landing pad with a sequence of slides, flips, and turns.In the process of this activity, the teacherdiscusses with the studentseffective strategiesfor finding a sequence of geometricmotions thatachieves a given goal. For example, students can often "see" that the rocket has to be turned, say, 90? to the right, but they have difficultylocatingthe turncenter.The teacherencouragesthe studentto first turnthe requisiteamountaboutany reasonablepoint, then SLIDEthe rocketonto the pad. (In this way, the studentsare dealing with composition of motions.) In addition, in this activity, students gain furtherexperience with spatial problem solving, and the use of slides, flips, and turnsin a movable environment,when they use the computermotionsto move the componentsof a robotface into proper position. Congruenceand Motions. In the sixth component of the Motions strand,the studentslearnthat anotherway to think aboutcongruenceis that two figures are

21

Logo and Geometry

congruentif, and only if, there is a sequence of slides, flips, or turnsthat moves one onto the other.In doing so, they formalizethe intuitivenotionthattwo figures are the "same size and shape"if one "fits on top of" the other. In this activity, studentsfirstdeterminevisuallywhethertwo figuresarecongruent.Theythencheck this visual estimateon the computerby attemptingto find a sequence of motions thatmoves one figure onto the other. Studentsalso learnthatcongruentfigureshave "corresponding components"that are also congruent.They use the motions commandsto place one figure onto a congruentone, then they recordthe correspondingcomponents.The activityends by asking students to draw conclusions about corresponding components of congruentpolygons. Thus, studentslearn that the correspondingcomponentsof congruentfigures are congruent. Symmetryand Motions. In the seventh componentof the Motions strand,the studentslearn that flips can be used both to create symmetricfigures and to test whether figures are symmetric. In the first part of the activity, students create symmetricfiguresusing flips on the computer.Forexample,studentsflip a triangle aboutone of its sides so thatthe triangle,togetherwithits image,formsa symmetric figure. They drawthe symmetricfigure createdandname the shapeof the outline of thatfigure. In the second partof the activity, studentstest severalfigures to see whetherthey aresymmetric.Using the computer,they determinewhetherthey can flip the figure onto itself. Motions as Flips. In the eighth and final componentof the Motions strand,the studentslearnhow slides andturnscanbe expressedas a compositionof flips. Every slide is equivalentto two consecutiveflips aboutparallellines andconversely.Every turnis equivalentto two consecutiveflips aboutintersectinglines andconversely. The intersectionof these flip lines is the turncenter. This concludes the descriptionof the threestrandsof the LG curriculum:Paths, Shapes, and Motions. Most teachersseparatethe strandsby one or more months acrossa school year.In the next section, we describethe researchmethodologywe used to evaluatethis curriculum. RESEARCHMETHODOLOGY The researchreportedhereemphasizesthe second andthirdyearsof the project. In this section, we describethe teachersand studentswho helped us field-testLG and the assessmentproceduresthatwe used duringthose two years. Participants For the 1987-88 field test, the participantswere 324 studentsin GradesK-7.8 All of the studentswere fromthe classes of LG teachersat Site 1 nearKent,Ohio. 8 LG is for GradesK-6, but one selected teacherwas forced to move to Grade where she elected 7, to teach the Grade6 strandsof LG. This appliedonly to the 1987-88 year.

22

andMethodology LogoGeometryCurriculum

(Therewas no controlgroupfor thisyear'sfield test. ) These teacherswere selected for the projectby school administrators, who basedtheirselectionson the teachers' expertise in teaching mathematicsand their willingness to participate.For the 1988-89 field test, there were two additionalgroups of teachers. First, a new group of LG teachers was added from the Buffalo, New York, area. Second, teachersin bothstateswere matchedwith controlteachersfromthe same gradeand school.Theseteacherswere selectedby theiradministrators becausetheypossessed equivalentexperience, expertise, and mathematicsknowledge to that of the LG teachers.Thesecontrolgroupteacherswereprovideda copy of the goals andobjectives for LG and agreedto teach these objectives.For the 1988-89 year, a total of 1,300 studentsin GradesK-6 were involved in Ohio (Site 1) and New York (Site 2). (Referto Table 1 for the numberof studentparticipantsby site andgrade.)Each teacheridentified three to five studentsto be interviewed;thus, we interviewed approximatelyeight studentsat each gradelevel at each site. The teacherschose studentswhom they judged to be of average mathematicsachievement,to have maintainedgood attendance(no historyof extendedabsences),andto be sufficiently confidentto be interviewedindividually. Assessmentand Data Collection We used multiple assessmenttechniques,including pretestsand posttests that sampledoverallcurriculumobjectives,unittests, structuredinterviewscripts,and classroomobservations.Each of these types of assessmentis briefly discussed in the remainderof this chapter. Pre-Post GeometryAchievementTest.The pre-post(PP) test was intendedas a generaltest of geometricachievementandlevel of thinkingandthusdid not reflect knowledge or formatsthat were uniqueto LG. In this way, a reasonablecomparison could be made between the LG curriculumand traditionalcurricula.In addition, the strengthsand weaknesses of both traditionalinstructionandLG instruction could be identified.Forboth years, the pre-post(PP) test was administeredto all studentstwice: once at the beginning of the school year (October)before any geometryinstructionhad been conductedand, again, at the end of the school year (May) afterall geometryinstructionhad been concluded.The PP test was administeredin two partson consecutivedays at each gradelevel, with only minormodifications (e.g., one item was administeredonly to GradesK-3 students,although a few items were not appropriatefor studentsin lower grades). The items that requireda writtenresponse were administeredindividuallyby the researchstaff to childrenin GradesK-2 (one Grade2 class in 1987-88 readthe items by themselves); students'verbalresponseswere transcribed.Throughoutthis monograph, we use the conventionof labeling items for the two partsof the paper-and-pencil assessment as follows: PP/1.1 indicates the pre-posttest, part 1, item number 1 (where the item numberwas the numberof the item on the test as presentedto students).When an item has several components,anothernumberis added,as in PP/1.4.1, which indicates pre-posttest, part 1, item number4, the first question

LogoandGeometry

23

(congruenceof two circles). To aid the reader,the items are shown when they are discussed in the results. Unit Tests. The unit tests were intended to measure the specific geometric achievementgoals of LG. Teachersadministeredthese tests only to LG students and the items usually differed by grade level (althoughcontiguous grade levels sometimes shareditems). Each unit test was administeredfollowing completion of the correspondingLGunit: Paths,Shapes,andMotions.Because these arecriterion-referencedassessmentswithoutcontrolgroup,pre-post,or grade-levelcomparisons, only brief conclusions from all tests are presented. Interview.LG studentswere individuallyinterviewedthreetimes;beforebeginning the curriculummaterials,afterthe completionof the Shapesstrand(calledthe interiminterviewin this report),and afterthe completionof the entirecurriculum. Controlstudents(1988-89 only) were interviewedat the end of the year only (the postinterview).Duringeach interview,a memberof the projectstaff presentedthe students with geometric problems and probed their geometry knowledge and processes.All interviewsfollowed standardprotocols.These will be describedand illustratedas they are discussed in the results. Case Studies. In selected classrooms, the work of a pair of interview students was studiedintensely. The pairwas videotapedand observedduringalmostevery class session. Videotapeswere transcribedand interpretedin relationto the goals of the LG curriculum. ClassroomObservations.Projectstaff visited classroomson a regularbasis. As they observedthe lessons, they focused theirobservationson interview students. They took field notes that were transcribedafter the session. If the interview studentsworkedon the computerduringa classroom observation,this work was videotapedfor subsequentanalysis. These procedureswere followed duringthe field-testyears,the secondandthird years of the project.The next chapterpresentsthe resultsof these assessments.

Chapter 3

Results The Logo GeometryProjectdevelopeda research-basedGradesK-6 geometry curriculum.This chapterpresentsthe findingsfor bothfield-testyears,the 1987-88 and 1988-89 schoolyears.Resultsforthepaper-and-pencil assessments(PPandunit tests) arepresentedin broadconceptualcategories.Because the PP test was almost the sameacrossgradelevels andwas the only paper-and-pencil test administeredto thecontrolgroup,thepresentation will be organizedaroundpretestandposttestitems, with items and data for these items presented within the body of this section. (Geometricfiguresincludedin anyitems arereducedfor the sakeof space.)Results for the unittests will be discussedas they arerelevant.Given the largenumberand complexityof statistics,summarystatements(markedby summary)are presented for most resultsas a convenienceto the reader.Datafromthe interviewsandclassroom observationsfollow the paper-and-pencildatawithineach broadcategory. TOTALACHIEVEMENT We begin the presentationof the PP datawith several summaryscores. Tables mentionedthroughoutthis chaptercan be foundin the "Tables"sectionthatbegins on page 155. K-6 Total Certainitems were included on the PP tests at every grade level from kindergartento Grade6.9(The tests for Grades4-6 includedadditionalitems, which are discussedin the next section.)The sumof these itemscreatedthe "K-6 Total"score (total possible score, 87; reliability coefficient, r = .85). Table 2 presents the means and standarddeviations for the K-6 Total score by experimentalgroup, grade, and site. A four-way repeatedmeasuresANOVA on the 1988-89 scores revealed main effects for Treatment(F(l, 1055) = 24.87, p < .001), Time (F(1, 1055) = 486.16, p < .001), Grade(F(6, 1055) = 81.95, p < .001), and Site, (F(1, 1055) = 7.05, p < .01), and significant interactionsfor Treatmentx Time (F(1, 1055) = 64.36, p < .001), Time x Grade(F(6, 1055) = 6.38,p < .001), Time x Site (F(1, 1055) = 16.26,p < .001), Gradex Site (F(6, 1055) = 10.75,p < .001), Time x Gradex Site (F(6, 1055) = 9.63, p < .001), and Treatmentx Time x Gradex Site (F(6, 1055) = 5.21, p < .001). All otherinteractionswere not significant. 9 Recall thatone selected teachermoved to Grade7; her studentsreceived the Grade6 assessments.

LogoandGeometry

25

Post hoc Newman-Keuls tests on the Treatmentx Time interactionrevealed that the LG posttest score (M = 71.63) was significantly higher than all other scores (controlposttest, 66.79; controlpretest,60.66; andLG pretest,59.80), and the control posttest score was higher than both pretest scores (p < .01 for each comparison). Posthoc analysesof the Treatmentx Time x Gradex Site interactionfor 1988-89 aresummarizedin Table3. Therearetwo reasonsto attributethisinteractionto differences between individual teachers and classrooms ratherthan to influences of teacherexperience(Site 1 teacherswereteachingLGforthe secondtime).First,there is no interpretabletrendacrossthe gradelevels. Second, in four of the five grades at which the two LG sites differed significantly, a priori observationsrevealed characteristicsof the teachersandtheirpracticesthatwouldexplainthe differences betterthanthe constructof "teacherexperience."Forexample,we hadobservedthat the teachersin certainclassrooms(GradesK and 1 at Site 1;Grade5 at Site 2) were, fromtheirfirstlessons, comfortablewith the LG approach.They statedthatthe LG teachingstrategieswere consistentwith those they use in all topics. In otherclassrooms,we observedthattheteachers(bothat Grade4, Site 2) werenew to computers and the Logo languageand thus had difficultyadaptingto new teachingdemands. These findings and observations, together with a nonsignificant Treatmentx Time x Site interaction(F(1, 1055) = 0.03, p = .87) and the scores from the 1987-88 field test, suggestthatno generalconclusionshouldbe drawnthatstudents whose teachers had two years ratherthan one year of experience teaching LG curriculumperformedsubstantiallydifferentlyon this paper-and-penciltest. In summary,both the LG and controlgroupsmade significantgains on the K-6 Total geometry achievementtest. In addition,the LG group scored significantly higherthanthe controlgroup. 4-6 Total Several items were includedon the paper-and-penciltests only for Grades4 to 6. Thus, the total possible score of the complete paper-and-pencilpretests and posttestsfor these gradeswas 125 points(thisincludesthe 87 pointsof the previous K-6 total). Table 4 presentsmeans and standarddeviationsfor the 4-6 Total test score (reliabilitycoefficient, r = .88). A repeatedmeasuresANOVA revealedmain effects for Treatment(F(1, 527) = 22.24, p < .001), Time (F(1, 527) = 1197.51, p < .001), and Grade(F(2, 527) = 56.53,p < .001), and significantinteractionsfor Treatmentx Time (F(1, 527) = 86.02, p < .001), Gradex Site (F(2, 527) = 34.25, p < .001), Time x Grade(F(2, 527) = 6.10, p < .01), Time x Gradex Site (F(2, 527) = 18.84,p < .001), and Treatmentx Time x Gradex Site (F(2, 527) = 11.96, p < .001). Post hoc analyses on the Treatmentx Time interactionrevealedthatthe LG posttest score (102.13 across grades)was higherthanall otherscores (control posttest, 94.31; control pretest, 82.21; and LG pretest, 81.01), and the control posttest score was higherthanboth pretestscores (p < .01 in each case).

Results

26

Post hoc analyses of the Treatmentx Time x Gradex Site interactionrevealed only minordifferencesbetween gradelevels at the two sites (e.g., only the sixthgradecontrolgroupof Site 1 did not make significantgains; the Site 2 LG score was significantlyhigherthanthe Site 1 LG score for sixth gradeonly). In summary,bothLG andcontrolgroupsmadesignificantgains on the 4-6 Total geometryachievementtest, but the LG groupscored significantlyhigherthanthe controlgroup.Thus, on summaryscores at both grade-levelranges,studentswho worked with the LG curriculumoutperformedstudentswho received traditional instruction. To elaborateeffects revealedby these summaryscores, single items or coherent groupsof items were submittedto repeated-measuresANOVAs or MANOVAs. Forbrevity'ssake,maineffects andinteractionsnot directlyrelevantto the research questions (e.g., Time x Gradeor those involving the sites) will not be discussed; rather,the Treatmentx Time interactionand any other higher orderinteractions (e.g., significant Treatmentx Time x Grade interactions)will be emphasized. Pertinentunit test items will be discussed within each conceptualcategory. SHAPESAND LEVELSOF GEOMETRICTHINKING As describedin chapter2, helpingstudentsachievehigherlevels in the van Hiele hierarchyand reason about two-dimensionalshapes were majorgoals of the LG curriculum.We hypothesizedthatwhile programmingthe turtleto drawgeometric shapesstudentshad to analyzethe spatialaspectsof the shapesandreflecton how thesespatialaspectswererelatedto commandingtheturtleto move.Giventheimportance of these hypotheses,we collected substantialdatarelatedto them, including severalpaper-and-pencilitems, interviewitems, andclassroomobservations. Pre-Post Test Identificationof Triangles, Rectangles, and Squares. Three items dealt with identificationof triangles,rectangles,andsquares(see Figure1 andFigure2). They were designed by Burgerand Shaughnessy(1986) to assess geometric levels of thinkingregardingthese shapes. One point was given for each correctidentification. (The readershould assume this scoring procedureunless otherwise noted.) Therefore,for the triangles,the maximumpossible score was 14; for the rectangles, 15; and for the squares,15. Table 5 presentsthe means and standarddeviations for the responses to these threeitems. To simultaneouslytest differencesbetween the groupsof students,a doubly multivariate repeated-measuresMANOVA was performed. Analyses revealedseveralsignificantomnibusinteractions.The firstwas a Treatmentx Time interaction(Pillais trace,F(3, 1028) = 17.94, p < .001). Two univariatetests were significant,the test for the trianglesitem (F(1, 1030) = 45.80, p < .001) and the test for the rectanglesitem (F(1, 1030) = 21.83, p < .001). Post hoc tests for both items revealed similarresults:The LG posttest scores (11.90 and 10.87 for trian-

27

Logo and Geometry

1.1. Writethe numbersof all the figuresbelow that are triangles.

Figure 1.

gles andrectangles,respectively)werehigherthanall otherscores(controlposttest, 11.05, 10.09; control pretest, 10.29, 9.56; and LG pretest, 10.06, 9.46), and the control posttest scores were higher than both pretest scores (p < .01 for each comparison).Thus, for both items, both groups made significantgains in identifying triangles and rectangles; in addition, the LG group scored significantly higherthanthe controlgroup. For the second significantinteraction,Treatmentx Time x Grade(Pillais trace, F(18, 3090) = 17.94, p < .001), two univariatetests were significant,the test for the trianglesitem (F(6, 1030) = 3.33,p < .01) andthe test for the squaresitem (F(6, 1030) = 3.85, p < .01). For the trianglesitem, the LG groupperformedbetterat the lower grades (e.g., K-2) and higher grades (5-6), when comparedto the control group. Resultsfor the squaresitem were simpler.Post hoc testsrevealedthatthe kindergartenLG group was the only group to make significantpre-postgains. (Means from the 1987-88 field test are consistentwith this finding.) Finally, for the Treatmentx Time x Gradex Site interaction(Pillais trace,F(18, 3090) = 3.02, p < .001), only one univariatetest was significant,the test for the

28

Results

Forthe followingitems, you may use some numberstwice. 1.2. Writethe numbersof all the figuresbelow that are rectangles. 1.3. Writethe numbersof all the figuresbelow thatare squares.

r;

C]A

1SS LPZx0

Figure 2.

trianglesitem (F(6, 1030) = 6.91, p < .001). Post hoc test results are summarized in Table 3. Consonantwith the summaryresults,effects appearto be attributable to differencesbetween teachers.That is, no interpretablepatternemerges that is not best explained by the strong teaching of individuals, such as the Grade 5 teachersat Site 2. In summary,LG studentsperformedbetteroverall,with differencesattributable to the trianglesandrectanglesitems.Forthe trianglesitem,theLGgroupperformed betterrelativeto the controlgroupat the lower andhighergrades.Of all the groups, only the kindergartenLG groupmade significantgains on the squareitems. Particularitems, on which performancedifferedsubstantiallybetween LG and control students, were investigated by examining means for individual figures within items. LG students,especially youngerstudents,gained more thancontrol students,not in theirabilityto identifyfiguresthataretriangles,but in theirability to correctlyidentify as nontrianglesthose figures thatsharespatialcharacteristics with triangles (e.g., the chevron, or deltoid). This was especially salient for the concave quadrilateraland for figures thatappearedto be trianglesbut had curved sides or were not closed. Therewere few differencesbetweenthe groupson triangles with largeaspectratios("skinny")or prototypical(e.g., horizontalbase, right; or equilateral)triangles.

29

Logo and Geometry

For rectangles, large growth was displayed for LG students compared to control students for shapes 2 and 7, both squares. For both of these shapes, the most growth occurred in Grades 4, 5, and 6. Large relative gains were also shown for a right trapezoid (shape 14). Moderate growth was evinced for the nonrectangularparallelograms,especially shapes 5 and 8, in nonstandardorientation. Controlinstructionhad a small negative effect for these shapes. Finally, small differenceswere observedfor the remainingshapes,in some cases the result of a ceiling effect. AttributingProperties to Figures. One item requiredstudentsin Grades2 to 6 to match statementsabout propertiesof figures, A-H, to two classes of figures, squaresand rectangles(see Figure 3). 2.10. Writethe lettersof all statements that describe each type of figure. Some statements can be used to describe morethan one type. Statements: A. Has 4 equal sides F. Has 4 rightangles B. Is a simple path G. Has two long sides and two short C. Is a closed path sides D. Has opposite sides equal H. Requires3 turnsto trace if you E. Has 90-degree angles startat the middleof a side every square every rectangle Figure3. Table 6 presentsmeans and standarddeviations for this item. (The maximum possible scorefor 1988-1989 was 16;for 1987-1988, whichincludedanotherquestion, "everytriangle,"it was 24.) A MANOVA revealedseveral significantinteractions.Firstwas a Treatmentx Time interaction(Pillais trace,F(2, 779) = 38.56, p < .001). Both univariateeffects, for "everysquare"and "everyrectangle,"were significant(F(1, 780) = 70.33,p < .001; F(1, 780) = 23.75,p < .001). There was also a significant Treatmentx Time x Grade interaction(Pillais trace,F(8, 1560) = 3.04, p < .01). Only one univariateeffect was significant,that for rectangles(F(4, 780) = 4.85, p < .01). Means suggest that third-,fourth-,and fifth-gradestudentsimprovedmore in LG classroomsthanin controlclassrooms. These differenceswere less salientfor Grade6 andfor Grade2; the controlgroup made largergains thanthe LG group. In summary,LG studentsimprovedmore than control studentson attributing statementsof geometricpropertiesto the classes of squareand rectangle.For the rectangleconcept, effects tendedto be strongestfor studentsin Grades3-5. Examinationof individualitems indicatedthatLG studentsperformedbetterthan controlstudentson propertiesdealing with paths(e.g., is a simple path;is a closed

30

Results

path) and those dealing with angle measure(e.g., has 90-degree angles; has four right angles). For "oppositesides equal,"LG studentsoutscoredcontrolstudents for bothshapes,the differencebeing smallfor rectanglesbutlargefor squares.The differencesfor the otherpropertieswere small and inconsistent. ClosedFigure:WhatShape?Theitemillustrated in Figure4 askedstudentsin Grades 4 to 6 to considerwhatshapea closedfiguremightbe (totalpossiblescore,5).

2.11. I'mthinkingof a closed figurewithfourstraightsides. Ithas two long sides and two shortsides. The two long sides are the same length. The two shortsides are the same length. Whatshape could I be thinkingof? CircleYes or No. Couldit be a triangle? Yes No Couldit be a square? Yes No Couldit be a rectangle? Yes No Couldit be a parallelogram? Yes No Couldit be a kite? Yes No Figure4.

In contrastto the results for Item 2.10, an ANOVA revealed that differences between the LG and controlgroupson the closed-figureItem 2.11 were insignificant (see Table 7). Disembedding.Disembeddingof geometricfigures was not includedin the LG curriculum.Itemsthatrequireddisembeddingwereincludedbecausethe assessments were designedfor breadthandto revealpossible strengthsof traditionalinstruction or detrimentaleffects of LG. GradesK-1 studentswere presentedwith a disembeddingquestionin one of two forms, as illustratedin Figure 5. (The maximum possiblescore, 10 points,requiredstudentsto identifysquaresas both"squares"and "rectangles.") Table 8 presentsthe means and standarddeviations for the total scores on the GradesK-1 disembeddingquestions.An ANOVA revealed a Treatmentx Time interaction(F(1, 256) = 8.73, p < .01). On this item the LG groupimprovedless thanthe controlgroup. Scores for the more complex disembeddingitem for studentsin Grades 2-6 revealedno significantdifferenceson any maineffect or interaction.In summary, thereis no evidence thatLG improvedstudents'ability to disembedfigures. Lengthand Arithmetic:K-3 Building. Given the integrationof geometry and numberin LG, we expectedthatstudentswould improvein applyingarithmeticto studentswerepresentedwiththe taskillusgeometricproblems.Onlyprimary-grade tratedin Figure6.

31

Logo and Geometry

2.6 Kindergartenversion: Lookat the picture. Trace over all the trianglesin blue. Trace over all the rectangles in red. Trace over all the squares in yellow. 2.6. First-gradeversion: Howmanysquares, rectangles,and triangles can you findin the picture? triangles rectangles squares

/

O

E

A_\m m n

Figure 5.

2.A. Here is a building.The builderswantto builda sign. Howwide must the sign be to fitexactly? 10

SIGN Note: For

Grades

K-,

I measures I I ] I I I

were

as

Ishown; II I I

I I

for I

Grades

2-3,

measures

3 3 Note: ForGrades K-i, measures were as shown;for Grades 2-3, measures were multiplied multipliedby by 10. 10. were Figure 6.

Table 9 presentsthe means and standarddeviationsfor this K-3 Building item, scored 1 or 0. An ANOVA revealeda Treatmentx Time interaction(F(1, 488) = 4.93, p < .05). On this item, the LG groupimprovedmore thanthe controlgroup. Therewas also a significantTreatmentx Time x Gradeinteraction(F(3, 488) = 4.50, p < .01). Kindergartenand third-gradestudentsgained the most relative to theircontrolgroups.In summary,LG studentsscoredsignificantlyhigheron items assessing students'recognitionof the relevanceof arithmeticprocessesin the solution of geometricproblemsandthe accurateapplicationof these processes,notably at kindergartenand Grade3.

32

Results

Unit Tests Recall thatunit tests were designed to measurethe specific geometricachievement goals of Logo Geometry.Therefore, they were administeredonly to LG students,following completionof the correspondingLGunit.As such,theyprovide informationon the degreeto which goals were achieved.Unit tests includeditems on identifyingshapes of variousclasses and on disembeddingshapes. Identificationof Squares, Rectangles, and Triangles. About 75% of Grades K-1 studentscorrectlyidentified squares,rectangles,and triangleswhen shown examplesandnonexamples(e.g., threetriangles,one kite,andone three-sidedshape with a curvedside). Second and thirdgradersperformedwell when askedto identify these figurespresentedwithina largerarrangementof the figures,thoughmost did not identifysquaresas rectangles(accuracyrangingfrom67%to 97%).Fourth gradersperformedsimilarly,with more identifying squaresas rectangles. Most first graders,given the names squares, rectangles, and triangles, identified squaresandrectanglesas havingthe same numberof sides andstatedthatthis numberwas four (69%to 95%). However, they were less able to give the number of turnsin a squareand a trianglewithoutthe benefit of a diagram(58% to 74%). Second graderswere more accuratethanthese youngerstudentsin identifyingthe numberof angles in figuresfor which diagramswere provided(69%to 98%,M = 86%);similarly,thirdandfourthgradersaccuratelyidentifiedthe numberof turns and angles in a largerset of figures (50% to 96%,M = 80%and 50% to 95%,M = 87%,respectively). Identificationof Polygons and Regular Polygons. Intermediate-gradestudents were askedto identifyvariouspolygons with morethanfoursides. Fourthgraders' performancewas mediocre(44%)on a item in whichthey matchedshapesto shape names, including such shapes as nonregularpentagons,hexagons, and octagons. These were not emphasizedby the curriculumor teachers.In contrast,five sixth squares,rectangles, gradersaccurately(81%)identifiedall triangles,quadrilaterals, parallelograms,and hexagons from a set of figures. Secondthroughfourthgraderswere askedto identifyregularpolygonsfroma set of shapes.Second gradersscoredbetween 15%and46% operatingon a set of five figures.Thirdandfourthgradersscoredbetween 17%and22%and21%and41%, respectively,on a set of 12 figures(forcredit,studentshadto makeno errorsin identificationacrossall figures).These same studentsgave an averageof 1.32 and 1.55 (respectively)validreasonswhy regularpolygonsaredifferentfromotherpolygons.. correctlyidentifiedtriangles,squares,andrectanDisembedding.Kindergartners 85%accuracy.Firstgraderscorrectlyidenwith about in embedded pictures gles tified these shapesembeddedin more complex figures with about56% accuracy. Secondgraders,given only the most difficultof the two figuresthatwere presented to first graders(the difficultitem is shown in Figure7), scoredabout38%correct. In summary,this confirms that there is no evidence that LG improvedstudents' ability to disembedfigures.

33

Logo and Geometry

Lookat the figurebelow. Howmanyof each shape can you find?

squares rectangles triangles

Figure 7.

Interview

ProgrammingRectangles. One interview question asked Grades2-6 students, "Howdo you makea squareusing Logo?"(Figure8). The numberof studentswho respondedcorrectly increased from 21% on the pretest to 85% on the posttest. Almost all students(95%) knew that all of the sides of a squarewere the same length. When asked on the posttest how they knew that all of the sides were the same length, 28% of the studentsjustified their answerby referringto the Logo procedure(up from 13%on the pretest)and 38%by arguingthatit wouldn't be a squareif the sides were not all equal (up from 23%).1' On anotherinterviewtask, studentswere asked, "Howis shape 1 differentfrom shape 2?"

Shape 1

Shape 2

33 Shape Shape

Figure 8. 10 Because some questions were not asked of studentswho had not had some previousexposure to Logo, 47 studentswere asked this questionon the pretest;87 on the posttest.

34

Results

The percentageof studentsstatingthatthe side lengths were differentincreased from 3% on the pretestto 27% on the posttest. (Only Site 1 studentswere asked both times.) Students were then asked, "Here is a Logo procedurethat draws shape 1. How would you changethe procedureso thatit woulddrawshape2?"The numberof studentswho gave a correctanswerincreasedfrom 36%on the pretest to 73% on the posttest(studentswho did not know Logo were not askedthe question). Students were then asked, "How would you change your procedurefor shape 2 so thatit would draw shape 3?" There was an increasein the percentage of students whose answers were basically correct (9%, 24%, 23% for pre/interim/postinterviews).There was also an increase(20%, 29%, 35%) in the percentageof studentswho knew thatthe turnsmust be changedbut did not estimate what those changes might be. TriadSorting.The interviewincludeda triadpolygon sortingtaskdesigned(with the help of RichardLehrer)to determinethe van Hiele level of geometricthinking forpolygons.Foreachof nineitems,childrenwereindividuallypresentedwiththree polygons andasked,"Whichtwo aremost alike?Why?"Forexample,one student, presentedwiththe threeshapeslabeled"2"in Figure9, chose the middleandbottom figures, saying thatthey "lookedthe same, except thatthis one [themiddle one] is bent in." She was attendingto the visual aspectsof the shapes;a Level 1 response. After working with LG, she chose top and middle figures, saying that they both had four sides. Thus, she let the overall visual aspect of the figures fade into the background,attendinginsteadto the propertiesof each shape;a Level 2 response. Her conceptual structurehad been reorganizednot only to include, but to give prominenceto, the correspondingproperties of these figures. The illustrationsin Figure9 wereused in the triadtask.Afterwe discussthe variousscoreson this task, we will discuss the variousvan Hiele levels indicatedby the students'responses. TriadDiscriminationScores. The reasonstudentsgave for choosing the pairin a triadas "morealike"was judged accordingto whetheror not the reasondiscriminatedthatpairfromthethirditemin thetriad.Discriminationscoreswerenot calculatedif studentsgave a visual reason.However,if a student'sreasonwas basedon propertiesor a classification,the discriminationscore was coded as follows: 0 What the studentsaid was incorrector did not discriminatethe two shapes chosen from the one not chosen. 1 The studentnamedmore thanone property,not all of which were correct,or discriminatedthe pairfrom the thirdfigure. 2 Thepropertiesthe studentnamedwerecorrectandcorrectlydiscriminatedthe two chosen shapes from the thirdshape. A student's discrimination score was the sum of the discrimination scores received for individual triads. Table 10 shows the average triad discrimination scores of studentsat all gradelevels for each of the threetestingperiods.Averages are for the groups shown. For example, 2.70 is the averagediscriminationscore for Logo students(GradeK) on the postinterview.

35

Logo and Geometry

1.

3.

2.

GX

4.

\

\

5.

6.

8.

9.

\

L 7.

r

Figure 9.

Students in the higher grades had significantly higher discriminationscores than studentsin the lower grades,F(6,197) = 20.47, p < .001. Students'discrimi-

36

Results

nationscores increasedsignificantlyfrom the pretestto the interimto the posttest F(2, 140) =13.49,p < .001, andwere higher,butnot significantlyhigher,thanthose of the controlgroupat the time of the posttest(all grades:Logo 2.66; control2.05). TriadChoicesand Reasons. Students'reasonsfor choosing each triadpairwere classified as visual (Level 1), property-based(Level 2), or classification(Level 3). See Table 11 for descriptionsof the reasons and their levels. The percentagefor each triadis summarizedin Table 12. Overall,fromthe preinterviewto the postinterview, there was a decrease in the averagenumberof studentsat a gradelevel giving visual reasons for their choices (18%, from 86% to 68%) and an increase in the averagegivingpropertiesandclassificationreasons(15%,from 10%to 25%). (The percentagesdo not add to 100 because some studentsoccasionally gave no reasonfor theirchoices.) Althoughnot shown in the tables, the averagedecrease in giving visual reasons for fifth and sixth grades was 18%, and the average increasein giving propertiesand classificationreasonswas 20%. Triad van Hiele levels. Students' van Hiele levels were determinedfrom the reasons that they gave for their triadchoices. To be classified at a given level, a studenthad to give at least five (out of nine) responses at that level. If a student gave five responsesat one level and at least threeat a higherlevel, the studentwas consideredto be in transitionto the next higher level. If a studentwas classified as in transitionfrom Level 1 to Level 2, the studentwas given a score of 1.5. Table 13 shows the percentageof studentsat each gradelevel who were at the variousvan Hiele levels for (1) pretest;(2) interim,thatis, afterLG workwithpaths and shapes;and (3) posttest,thatis, afterthe final unitof LG workthatemphasizes geometricmotionsandrelatedconcepts.For 1988-89, the meanvanHiele levels for shapesfortheLGgroupwerepretest,1.06;interim,1.15;andposttest,1.10.Themean van Hiele level for the controlgroupon the posttestwas 1.04. Althoughthe mean posttestvan Hiele level of the LG groupwas higherthanthatof the controlgroup,it was notsignificantlyhigher.ThemeanvanHiele level of theLGgroupon theinterim test was significantlyhigherthanon the pretest(t(112) = 2.67, p < .01). Thus, we canconcludethat,as measuredby thistask,themeanvanHiele level of theLGgroup was raisedsignificantlyby instructionon pathsandshapesbutthatit declinedby the posttest.(Studentsspent2 weeks on pathsand 2 weeks on shapesfor GradesK-4 and 3 weeks totalon pathsand shapesfor Grades5 and 6.) For 1987-88, the mean van Hiele levels were pretest, .99; interim, 1.05; and posttest, 1.15, a significant increaseF(2,150) = 6.55, p = 002. (Note: Two of the triaditems were differentfor years 1 and 2.) As can be seen fromTable 13, therewas a shift to highervan Hiele levels fromthe pretestto the posttestinterviews. Overall, the percentage of students whose van Hiele levels increased from pretestto posttestwas 32%for the Site 1 LG groupin 1987-88 and 20% and 13% for the Site 1 andSite 2 LG groups,respectively,in 1988-89. Gradelevel increases can be found in Table 14. ReasoningaboutQuadrilaterals.Studentsin Grades4-6 were given the problem illustratedin Figure 10 to assess theirability to reasonaboutgeometricfigures.

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Five teams are involvedin a geometrycompetition,and you are the judge for one of the problems.Each team is toldthata mysteryfigurehas fourstraight sides and is closed. The winneris the team that gives the smallest number of extraclues that willguaranteethatthe mysteryfigureis a rectangle.Here are the clues that each team gave. Whichteam's answer is best? Team A (1) two long sides and two shortsides Team B (1) two long sides and two shortsides Team C

(2) four right (90?) angles (1) four right (90?) angles

Team D

(1) both pairsof opposite sides parallel

Team E

(1) both pairsof opposite sides equal

(2) at least one right (90?) angle

(2) at least one right (90?) angle Figure 10.

As can be seen in Table 15, therewas a dramaticdecreasein the inappropriate choice of Team A (Site 1: 53%, 14%,5%; Site 2: 33%, 12%, 13%).The increase for TeamB (Site 1: 28%,51%,54%;Site 2: 29%,60%,58%) shows an increasing precisionin students'expressionof theirgeometricconceptualizationof rectangle. (It must be noted that most students,on the basis of theirpast experience,do not considera squareto be a rectangle.So, this choice is probablyan increasein precision over Team A.). The increasefrom pretestto interimfor Team C, the correct choice, (Site 1: 8%,20%;Site 2: 8%, 12%)is encouraging.However,the fact that students'performancedecreasedfrominterimto the posttestis consistentwith the decreasein vanHiele levels duringthe sameinterval.Studentsseemedto havemade progress in moving to more formal thinking about shapes during the Shapes module, but they regressedafterwards. Studentswere then asked, "Whichteams selected clues that would definitely makethe mysteryfigurea rectangle?"Forthe Site 1 LG group,therewas a substantial decrease in the numberof studentschoosing Team A, the least precise of the descriptions(see the bottomof Table 15). Therewas a slight increasefor Team B from pretestto posttest.(Note that,for many studentson the pretest,this category containedredundantinformation.)Therewas a substantialincreasefor TeamC on the interim,but it fell somewhatfor the Site 2 groupon the posttest.Forboth sites, therewere noteworthyincreasesin the numbersof studentswho claimed thatthe clues of Team D and Team E were sufficient to guaranteea rectangle. The students'postinterviewreasons for not choosing a team's clues were also analyzed.Almost all of the students'reasonsfor not choosing Team A were valid (Site 1 LG 100%,Site 1 Control96%, Site 2 LG 82%, Site 2 Control77%). The controlstudentswere slightly more likely to drawa counterexample,whereasthe LG studentswere slightly more likely to explain what was wrong with the clues.

Results

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Almost all studentssaidthatTeamB's clues wouldmakea rectangle,so therewere not enough reasons for not choosing it to analyze. There was quite a difference betweenthe reasonsfor not choosingTeamC given by the LGgroupandthe control group.The percentagesof LG and controlstudentswho rejectedTeam C because "Theclues don't say anythingaboutthe side lengths"were 70 and 15, respectively, for Site 1, and 17 and 19 for Site 2. The percentagesof LG and control students who rejectedTeamC because"itcould be a squareor box"were 30 and73, respectively for Site 1, and 83 and63 for Site 2. So, the Site 1 LG studentsrejectedTeam C mostly becauseits clues did not mentionside lengths,whereasthe Site 1 control and Site 2 studentsrejectedit because the clues did not eliminatesquares. There were few differencesbetween the groups for rejectingTeams D and C, with most of the reasons referringto "it said only one angle had to be 90?." However, 16%of the studentswho rejectedTeam E did so because "it could be a square."Both of these reasons are consistent with the overall van Hiele level of the students. Studentsin Grades 4-6 were generally in transitionfrom Level 1 (visual) to Level 2 (properties),so they were likely to abandonthe visual condition for rectangles,namely, "twolong sides andtwo shortsides,"andrequirethat the figure have right angles. However, students still had difficulty with the square/rectangleclassification problem and with the type of logical deduction requiredto deal effectively with the clues of Teams C, D, and E. ClassroomObservations Classroom observations provided data that were useful in supporting and extendingpaper-and-penciland interview data. One majorhypothesis of the LG curriculumwas that as path activities are extendedto more complicatedfigures, studentsanalyze the visual aspects of these figures, including how their components are put together,thus facilitatingthese students'transitionfrom the visual to the descriptive-analyticlevel of geometricthinking.Observationsalso illuminatedthe mechanismsof this transitionas the following episode demonstrates. A class of firstgraderswas investigatingthe conceptof rectangle.The studentshad identifiedrectanglesin the classroomand had built them outof variousmaterials,suchas blocks,tape,clay, andgeoboards.Then they went to the computerlab wherethey were askedto makethe turtle drawrectangles.(Thesestudentswerein the Singlekeyenvironment,in whichtheturtleis givencommandsby typingsinglekeys-F forforward 10 turtlesteps,B for backward,R or L for rightandleft turnsof 30?.) As the activity proceeded,all childrenwere drawingrectanglesin Logo. Afterdrawingseveralrectangleswith verticalsides, one student wantedto be different;he attemptedto drawa rectanglethatwas tilted. He startedby issuingan"R"commandto turntheturtle.He theninstructed the turtleto drawthe first side using five Fs. He pausedfor quitesome timeas he cameto thefirstturn,so one of theresearchersaskedhimhow muchhe hadturnedbefore.He saidthreeRs andhesitatinglytriedthree

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in thiscase.Itworkedto his satisfactionandhe thendrewthesecondside. He hesitatedagain, saying out loud, "Whatturn should I use?" The researchersaid, "How many turnshave you been using?"He quickly issued threerightturns,thenhesitatedagain;"Howfar?Oh, it mustbe the sameas its partner!"Effortlessly,he completedhis rectangle. Even thoughthis studenthadbuiltseveralrectangleswith horizontalandvertical sides, it was not obvious to him thatthe same Logo commandswould work for a tilted rectangle(or, indeed, that there was such a thing as a tilted rectangle).He eventually abstractedthatthe opposite sides must be the same length, but he had not abstractedthe measureof the turnsnor even realizedthatall the turnswere of the same measure.Thus, the LG environmentprovidedhim with the opportunity to analyze and reflect on the propertiesof a rectangle.Note also that because a computerandclassroomenvironmenthadbeencreatedthatencouragedexploration, this studenthad posed a significantchallenge for himself. We observed a similarphenomenonwith second gradersusing standardLogo commands.The studentscould easily give a procedureto make the turtledraw a squarewith its sides verticalandhorizontal.But as soon as they were askedto draw an obliquesquare,theyhadto confronttheirlackof abstractionof therequiredturns. (Like the first graders,they showed evidence of having abstractedthe notion that all sides are the same length). Often,suchinsightsarefacilitatedby discussion.A differentfirst-gradeclass was discussinga groupof quadrilateralsdrawnon the chalkboard,tryingto identifythe rectangles(in preparationfor writinga procedureto drawrectangles).They were focusing on a nonrectangularparallelogram. John: Thisoneis slanted.It can'tbe a rectangle. Slanteddoesn'tmatter.It hastwolongsides,hereandhere,thesamelength Cathy: andtwoshortsides,hereandhere,thesamelength[motioning toindicatepairs of oppositesides]. Eugene: Butit doesn'thavesquarecorers. Thus, the studentswere grapplingwith the propertiesof rectangles. In anotherinstance,a kindergartner explainedthat"rectanglesneed two numbers for thesesides (indicatingone pairof oppositesides withtwo hands)andthese sides, but squaresneedjust one, because every side is the same length."Such conversations reveal substantiveattentionto propertiesof figures. Student work on anotheractivity, "Rectangles:What Can You Draw?" (see Figure 11), demonstratesthe hypothesizedprogressionin van Hiele levels and from nonanalyticalto empiricalto logical thought.In this task, studentsare shown a varietyof quadrilateralsand are askedto determineif each could or could not be drawnwith a Logo rectangleprocedurethattakes two inputs and to explain their reasoning.Manystudents,even in Grades5 and6, oftendidnottrythesquare,saying, "It'sa square,not a rectangle."However,soon they foundthatsome of theirclassmates succeeded in producinga squarewith the rectangleprocedure.This led to involved andpassionatediscussionsregardingthe relationshipbetween these two

40

Results

shapes.As canbe seen,a strengthof thisactivityis thatit requiresstudentsto confront the conflict between the relationships,embodiedwithinthe Logo procedures,and theirown conceptionsof the figures. Withoutworkingon the Logo activities and discussingthem with theirpeers, studentsmay havejust refusedto even consider the possibilityof an inclusive relationshipbetweenthe squareandthe rectangle.

2

6

[9 /7

3

/

WhatCanYouDraw?"activity. Figure11. Figuresforthe"Rectangles:

Otherobservationssupportthe findingthatLG activities,includingthe computer environmentand classroom dialogue, supportstudents' development of higher levels of geometricthinking.After makingthe tilted rectangle(#4 above) by first turningthe turtle, students often say that the nonrectangularparallelogram(#7 above) can also be drawn.They are quite shocked when it does not work. Many who reflecton the difficultyconcludethatthe nonrectangular parallelogramcannot be drawnbecause it does not have 90? turns.But, withoutthe computeractivity, these studentswould not have progressedas far as they did. That is, they would not have confrontedthe conceptual inconsistencies that, in turn, produced the productivereflection.The partialdialogueof a pairof sixth-gradegirls, who were working on this activity, supportthese conclusions. They also illustrateanother valuableaspectof the computerenvironment,providinga way for studentsto inde-

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pendentlytest theirmathematicalideas. Vanessa: I don'tthinkthatyoucandoit. [Shewasthenreadyto go onto nextproblem.] Cheryl: Yes, youcan. Vanessa: Youcan'tbecauseCheryl: Let'stryit. [Thestudentstryit on thecomputerandobservetheresult.] Vanessa:

You can't do it, because the turnsare not 90 [degrees].

Interestingly,Vanessa did not feel that it was necessary to try this example. Actually, she possessed a good reasonfor herbelief, but in the face of conflict, she decided not to shareit. This girl seemed to need to have her theory validatedon the computerbeforeshe was willingto publiclyargueit. Manystudentsat Vanessa's age level arejust beginningto developthe abilityto reasonaboutmathematics.They need affirmationthattheirtentativeconclusionsarecorrectin orderto buildconfidence in this nascentmode of thought.Computerexperimentationallows them to test, refine, andvalidatetheirreasoningandprovidesthem with the opportunityto gain that confidence. Also, as has been previously observed, this episode shows thatstudentsdeterminewhat is trueor not truewith theirown reasoning.They do not use an adultauthorityto resolve conflicts. These actionsfurtherdevelop confidence and habitsof autonomousthinking. In some cases, Logo activities may affect the conceptualizationsof very young children,even when they areengagedin noncomputertasks.In the final interview, studentswere to asked identify all the rectanglesin a collection of figures. At the pretestinterview,most students,fromkindergartento sixthgrade,confusednonrectangularparallelogramsandrectanglesat obliqueorientations.First-graderAndrew was no exception;he includedseveralnonrectangular parallelogramsas rectangles on the pretest.After experiencingthe Logo-basedcurriculum,however, Andrew correctlyidentifieda rectangleat an obliqueorientationandalso correctlydismissed a nonrectangularparallelogram,saying: "That'snot a rectangle;that's slanty." Interviewer: Well,isn'tthisone [indicatesobliquerectangle]slanty,too? Andrew: It's okay.It'sjustturned.Look.If thispencilwas a turtle,youcouldturnit likethis[turnspencilfromobliqueto horizontal Thenit would orientation]. be a regularrectangle. Interviewer: Butwhatif I didthatto thisone [indicatestheparallelogram]? Andrew:

No, even if you turnedit, it wouldn't be a rectangle.It would still have slanty sides ... [indicatesnonrightangle with his fingers] slanty to each other!

Such observationsconfirm that these studentsare developing higher levels of thinking.They areusing Logo as a tool to test out theirideas andthuspromotethat development.This developmentis not instantaneousnortrivial,of course.We also have evidence thatstudentsjust beginningworkin Logo areindeed still "intransition"to Level 2. One interviewedstudent,Luke,who initiallyclaimedthatrectangles possess right angles, retractedthat claim after noticing that several of the shapeshe had selected(nonrectangular parallelograms)were inconsistentwith this after as description.Indeed, selecting "rectangles"all parallelogramsthatwere not Luke was asked rhombuses, by the interviewerwhy he did not choose a trapezoid.

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Results

Luke:

Becausethese two lines arenotparallelto each otherandit doesn'thave a right angle. Interviewer:Whatare all the special things abouta rectangle?You alreadysaid four sides. Luke: A right angle, I think.Yeah, a right angle. Interviewer:So, a rectanglehas to have a right angle? Luke: Yeah. Or four 90? angles. Interviewer:How about [indicatesa nonrectangularparallelogram]? Luke: Well, it doesn't have to have a right angle, but the four sides are parallel. [Pause.] Interviewer:You seem puzzled. Tell me what you are thinking. 'Causethatlooks like a rectangle,but it doesn't have any 90? angles. [Pause.] Luke: I don't thinkthata rectanglehas to have a 90? [angle].... This type of tension between the visual and descriptive/analytic perspectives may indicate the difficulty students have integrating their visually based recognition of figures with their emerging knowledge of the properties of geometric figures. LG provides situations in which such cognitive conflicts frequently emerge as well as a tool (e.g., student-defined procedures for rectangles) for investigating, and often resolving, these conflicts. Other episodes specifically address the notion of hierarchical classification. For example, consider kindergartners Chris and Robbie. They had, as a part of the Singlekey environment, a "Shape" command to draw figures of various sizes. For example, they could type S (for Shape), receive a selection of shapes, type the first letter, (e.g., S for Square), and then receive a prompt to type a number for the length of each side. Use of these Singlekey Shape commands led to interesting discussions about squares and rectangles, as illustrated by the following dialogue: Interviewer:Clearthe screen and show me anothershape. S. I want a square.S again. I want a nine. Chris: What otherkinds of shapes does this thing make? [Pause.]Can you show me Interviewer: any othershape? Chris: [Shows a rectangle.]S. Rectangle.R! Nine. Interviewer:How come it didn't do it yet? Chris: [Witha hint of annoyance]:I don't know! I pushedS then R then 9. Interviewer:It's askingyou for anothernumber.Do you need two numbersfor a rectangle? Chris: Yes [pressesanothernumber].And another9. [Views the screenandlaughs!] Interviewer:Now, what do the two nines mean for the rectangle? I don't know, now! Maybe I'll name this a squarerectangle! Chris: Chris uses his terminology again on subsequent days. He was asked to draw what the turtle would draw if it were given the command S R 5 5. Chris drew a square. Interviewer:Thatlooks like a square. It's both. Chris: Interviewer:How can it be both? Chris: 'Cause 5 and 5 will make a square. Interviewer:But how do you know it is still a rectanglethen? 'Cause these look a little longer and these look a little shorter. Chris:

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Interviewer: [Gives him an accuratesquare]Whataboutthat? A square. Chris: Interviewer:Would it also be a rectangle,or not? No. Chris: Interviewer:Even thoughI made it with the rectanglecommand? It would be a rectanglesquare. Chris: Chris was later asked, "Is a square a special kind of rectangle?" He maintained his language, but revealed that a visual impression was the criterion of choice for him, as the following indicates: A square... if you pushed9 and 9, you would make a square. Chris: Interviewer: So, is a squarea special kind of rectangle? Chris: Yeah, if you pushedboth numbersthe same. Interviewer:How about 10 on two sides and 9 on the other two? Would that make a square?Or a rectangle?Or both? It's both [a squareand a rectangle]. Chris: Interviewer:Is it a square? Yes. Chris: Interviewer:How come it's a square? Chris: 'Cause 9 is close to 10. Interviewer: Some people would say thatit has to be exactly the same on each side to be a square.Who's right? I thinkI am. Chris: The last time this issue was discussed, Chris was drawing different-sized rectangles (a Piagetian task of drawing ever-smaller figures) and got down to a square. Interviewer:Is it still a rectangle? Chris: It's a squarerectangle. Another kindergartner, Robbie, begins in a similar way to Chris. The first time he uses the Shape command, he is momentarily surprised that two inputs are requested. As the following protocol shows, there is evidence that Logo is helping Robbie to think hard about the measures of the sides of rectangles, but it also engenders in Robbie a resistance to double-naming a square as a rectangle. Asked to show a square, Robbie, with a bit of help, pressed S, S, then 1. Asked to make a bigger square, he pressed S, S, 7. Interviewer:Now, what's 7 on that? Robbie: These and these [indicatesall sides]. Interviewer: Show me any othershape you can make. Robbie: [Chooses a rectangle,then 9.] Interviewer:Why does it want you to press anothernumber? Robbie:

[Animated.] Oh! Because it's ... each ... side ... these two [holds arms up verti-

cally, one on each side of himself] sides are the same, and these two [holds arms up horizontally, one above the other, indicating the top and bottom sides] are the same. Interviewer:Oh. So, you said 9 for the firsttwo sides, here andhere, andnow whatareyou going to make the othertwo sides?

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Results

1. [He does so.] Robbie: Interviewer:So, is this nine? Robbie: Yeah. Interviewer:So, where's the one? [Robbieindicatesit.] Interviewer:Okay, show me a square.[Robbiestartsthe keying sequence.] Interviewer:Okay, does this need two numbersor only one? One. Robbie: Interviewer:How come the otherone needs two, and this one only needs one? Robbie: 'Cause these [holds arms up vertically, one on each side of himself] and [holdsarmsup horizontally,one abovethe other,indicatingthe top andbottom sides] ... all of the four sides arethe same [shows all four sides]. Interviewer:Okay,thentrythis [rectangle9 9]. [As theresultshows on the computerscreen] Whatwill you get? Robbie: [Doesn't answeruntil the drawinghas been completed.] It'd make a square [smiles and squirmsin his seat]. Interviewer:Now, is it a rectangle?Because we used the rectanglecommand? Yes. Robbie: Interviewer:Is it a rectangle,or a square,or both?Whatdo you think? Robbie: Both. Interviewer:It's both. Can a rectanglebe a square? Sometimes. Robbie: Interviewer:Sometimes?When is a rectanglea square? When you make a mistake. Robbie: Interviewer:What do you mean?Whatkind of mistake? Whenyou do yourjob on the paper.And you're tryingto make,andyou didn't Robbie: know how to make, a rectangle. [Throughoutthis time, points at the square on the screen, gesturingaroundits perimeter.]And you made it ... and you made it as a square. Interviewer:Oh. Okay. But you made it with the rectanglecommand. Robbie: Umm huh. Yes. [No real engagementwith this issue here.] Interestingly, Robbie intuitively uses equal input to the R (Rectangle) command when he wishes to draw a square. But, he is hesitant to call any figure produced by that command a square, as illustrated below. Interviewer:Try to use the rectanglecommand. This isn't a rectangle. Robbie: Interviewer:Whatis it? A square. Robbie: Interviewer:So, what commandswould you use? S and S and then 5. Robbie: Interviewer:Do you have to give just one number? Yes. Because all these sides would have to be equal. Robbie: Interviewer:What aboutthis? Whatif I put in S R 5 5. Thatwould be a rectanglefor R. Robbie: Interviewer:Right, and then I tell it 5 and 5. R drawswhat he thinksit would be [a square]. Robbie:

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Interviewer:Whatis that shape? A square. Robbie: Interviewer: How did thathappen? Because if I went on the computer,andI thinkI put some numberthe same, I Robbie: did a square,and I wanteda rectangle. Interviewer:Why is that? I can go wrong on the rectangles. Because the rectangle is like a square, Robbie: expect that squaresaren'tlong. [Pause.]But on rectanglesthey are long. Interviewer:What else do you know about a rectangle?What does a shape need to be to be a rectangle? All of the sides aren'tequal.These two [opposite]andthese two [otheroppoRobbie: site] sides have to be equal. Interviewer:How about 10 on two sides and 9 on the other two? Would that make a square? Kind of like a rectangle. Robbie: Interviewer:Would it be a square,too? Robbie: [Pause.]I thinkmay ... [Shakeshead negatively.] It's not a square.'Cause if you make a square,you wouldn't go 10 up, then you tur and it would be 9 this way, and turnand 10 this way. That's not a square. As this episode indicates, the Logo microworlds proved to be evocative in generating thinking about the similarities and differences of squares and rectangles but, for these kindergartners, did not evoke classificatory responses. We next turn to observations in an intermediate-grade classroom. After participating in the paths unit of LG, both Jeremy and Jonathan, fifth graders who were paired with each other, evinced concepts of rectangle and square that reflected this experience with paths, but these concepts were otherwise relatively unelaborated. For example, when Jeremy was asked in an interview, "What is a rectangle?" he replied as follows: It's like kinda squarebut flatterlike that [gestures].It has straightlines and no curves.... It's not a straightpath,becauseit turs. Here,here,here,andhere. It goes on and on. Interviewer:How abouta square?How do the two differ? A rectangleis longer. A rectangleis bigger. Jeremy: Interviewer:Whatif I drawa squarelike this? [Drawsa bigger square.] Jeremy: They can be all differentshapes. [Probablymeans sizes.] And squareshave parallellines. They're all the same. Interviewer. This one's parallelto this one. [Pointingto adjacentsides, believing he does not mean what I mean by parallel.I am rightto believe so.] Yes, they could be lined up [indicatesstacking lines one above the other so Jeremy: that the identicallength is obvious]. Interviewer: Okay, how abouta rectangle? There's only two that are parallel, and two on the other side. [He draws Jeremy: another.]On a squarethey have to be 'cause all the sides are the same size. Jeremy:

Jonathan's response was similar, though he mentioned only side lengths ("two long sides and two short sides") and not "parallelism." Subsequently, both boys

46

Results

wrote proceduresto drawrectanglesand engaged in the "Rectangles:What Can You Draw?"activity, in which they had to decide whethervariousfigures could be drawnwith a rectangleprocedurethat takes the lengths of the sides as input. (See Figure 11.) Two segmentsof Jonathan'sworkwiththis activityaresignificant.The firstdeals with imagistic processes and thinkingaboutpropertiesof rectangles,the second with the rectangle/squarerelationship. Regarding the first, Jonathanhas just successfully used the rectangleprocedureto draw a tilted rectangle(labeled 4 in Figure 11) and is now reflecting on his unsuccessful attemptto make a nonrectangularparallelogram(labeled 7 in Figure 11). Teacher: Couldyouuse differentinputs,oris itjustimpossible? Jonathan: Maybe,if youuseddifferentinputs.[Jonathan typesin a newinitialturn.He staresatthepictureof theparallelogram ontheactivitysheet.]No,youcan't. Becausethelinesareslanted,insteadof a rectanglegoinglikethat.[Hetraces a rectangleovertheparallelogram.] Teacher: Yes, but this one's slanted[indicatesthe tiltedrectangle,labeled4, that Jonathan hadsuccessfullydrawnwiththeLogoprocedure]. Jonathan: Yeah,butthelinesareslanted.Thisone'sstillinthesize[shape]of arectangle. This one [parallelogram]-the thing'sslanted.This thing[rectangle]ain't slanted.It looks slanted,butif you putit back[showsa turnby gesturing, it wouldn'tbe meaningto turnit so thatthesidesareverticalandhorizontal], it wouldn'tbea rectangle. slanted.Anywayyoumovethis[theparallelogram], [Shakeshis head.]So, there'sno way. Duringthe "Rectangles:WhatCan You Draw?"activity,Jonathanused several imageryprocesses, includinggenerating,inspecting,and transforming(Kosslyn, 1983). First,he generatedan image of a rectangle,tracingit on the activity sheet. Aftermakingthe initialturnandtryingto choose inputs,he recognizedthatthe relationshipbetween adjacentsides was not consistentwith the implicit definitionof a rectangle in the Logo procedure. So, his developing conceptual knowledge helpedhim generatea robustrectangleimage. Second,he inspectedthis image and compared it to his image of the parallelogram,noting the differences. Third, Jonathantransformedimages of the figures,mentallyrotatingthem.He notedthat only the nonrectangularparallelogramis "slanted."(Of course, by "slanted," Jonathanmeans nonperpendicular.His emergingknowledge of the propertiesof figures is supportedby his visually based reasoning.) The second significantsegment of Jonathan'sworking with this activity deals specifically with the rectangle/squarerelationship.Shape number3 in Figure 11 is a square. Jonathan: Thisoneis nota rectangle.It's a square.It hasequalsides.Do youwantme to do it withthis?[Indicates thescreen.] Teacher: Canyoudo it withyourrectangleprocedure? Jonathan: No, becausethesidesareequal.So, thatwouldbe a "no." Teacher: So,nomatterwhatyoutried,youcouldn'tmakeit withyourrectangleprocedure? Jonathan: Youcouldn't,no,becausethesidesareequal.

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Teacher: On your rectangleprocedure,what does this first input standfor? Jonathan: The 20? These sides. Teacher: Whatdoes the 40 standfor? Jonathan: Yeah, you could do it. If you put like 40, 40, 40, and 40. [Again, gestures.] Teacher: Okay, try it. Jonathan: So, that would be a square? Teacher: Canyou drawa squarewith yourrectangleprocedure?[Readsfromthe sheet.] Jonathan: You could drawit, but it wouldn't be a rectangle. Teacher: Should I put yes or no? Jonathan: Yes ... you can drawit.

Even with prompting,Jonathanis resistantto call the squarea rectangle.In his conceptualization,one can drawthe squarewith the rectangleprocedure,but that does not "makeit" a rectangle. What influence did this have on Jonathan'sideas about relationshipsamong quadrilaterals?He appeared to understandsomething about the relationship. However,he hadnot fully constructedthe hierarchicalrelationship,as subsequent observationsrevealed. The first observationinvolved class discussion following the "Rectangles:WhatCanYou Draw?"activity.This episode revealsmore about Jonathan'sideas and also illustratesthe ideas of other membersof the class and the natureof the instructionin which all the studentswere engaged. The discussion centered around a homework assignment in which students looked for all possible squaresin an arrayof squares.The identificationof squares proceededwithoutparticularevent, untilJonathanoutlineda rectangleandwanted to count that. That's a rectangle. Well, a rectangleis a square. Davey: Well, if you count a rectangleas a square. Sally: Teacher: How many blocks are across the bottom? Jonathan: Four. Teacher: How many [gesturesvertically]? Jonathan: Three. Teacher: So, is it a square? Jonathan: No. Teacher: Now, Koreen,ask thatquestionyou posed yesterday. Koreen: Well, yesterday I thought you could do any rectangle. You could put any numbersin to make it. And then I found out if you put 50 50 it would make a square. Teacher: Okay, so somebody give me a rule for what you found. All squaresarerectangles,but not all rectanglesare squares.Only if the sides Donny: are all equal. Chorus:

It was possible thatJonathan,like Davey, overgeneralizedthe square/rectangle relationship.(As a side note, the protocol constitutesanotherinstanceof students usingLogo workto help themdiscussideas andconceptualizegeometricrelations.)

Results

48

The class discussion continues, with the teacher emphasizing the hierarchical nature of the square/rectangle relationship. Teacher: Davey: Teacher: Leah: Teacher: Sally: Teacher: Kara: Teacher: Abby: Teacher:

Kristie: Teacher: Koreen: Teacher: Koreen:

Whatare the propertiesof a square? Four sides. Do both of them have four sides? Yea. Right. Fourequal sides with 90? turns. Okay. Only one of those has four equal sides. Which one? The square. Whataboutthe rectangle? Well, the sides aren'tequal... theleft andrightsidesarethe samebut... they're parallel ... but the ones on the top andbottom are longer. That's what we want to get at. Is partof our definitionof a rectanglethatthe vertical and horizontallines are not the same? Is that what we're getting at? What we talked about is that the pair of horizontallines are paralleland the pair of verticallines are parallel.In observation,what we see when we look at what we think of commonly as a rectangleis that they are different.But Koreenprovedto us that ... how many on ourpaperthoughtthey could make numberthree [the square]by using the rectangleprocedure?How? By using the same number. So, a squarecan be a rectangle.A special type of rectangle,but a rectangle nevertheless.But not all rectanglescan go back and be squares. When it says on that test, find all the rectangles.Well, could we find all the squaresand color them in as rectangles? Whatdo you say? Yeah.Becauseyou candraw'em. So, you can do it. So, we wouldn'tbe marked wrong, right?

Discussion turns to a task in which different shapes are to be drawn in different colors. Here, Jonathan stated the inclusive relationship correctly (but there is no evidence that he believed that the converse was not true). Teacher: Kara: Teacher: Koreen: Teacher: Jonathan: Teacher: Jonathan: Gerry:

Should a squarebe green or yellow? Yellow. Only yellow? This one is a squareand also a rectangle.So, what do you do with that? Whatdo you say? Someone else tell me what Koreen's argumentis. withyellow-for-square. I suggestthatwe shouldcoveroverthegreen-for-rectangle Why? Because a squareis a rectangle. No! If they put squaresin yellow, and they know thatsquaresarerectangles, thenwhy would they putthe squaresin yellow? They wouldn'tputthatif they wantedsquaresto be green.

The teacher misunderstood Gerry slightly at the time. Gerry said something closer to "they wouldn't have asked you to color them twice." (So, they might have said green is for "nonsquare rectangles" or something. See below.) Here is an example

Logo and Geometry

49

of students "making meaning" out of the task, including the meaning and intentions they attribute to the people who designed the task. Teacher: Gerry: Lori: Teacher: Corgie: Koreen: Teacher: Koreen:

But you're saying thatmaybe squaresaren'trectangles? No, they are.You could color themgreenfor rectangles.But if they've already colored the rectangles,they wouldn't have to color the squaresyellow. I have a differentquestion.Why can't we call squaresequilateralrectangles? Could we just get rid of the name square? Well, if a little kid goes to nurseryschool, he might not know. A squareclassifies as a bunchof things.An equilateralrectangledoesn't classify as all the things that are square. Give me an example of a square that isn't an equilateralrectangle. [Then explains what "equilateral"means.] Well, like a diamond.

The teacher draws one and has him clarify that he means a diamond with 90? turns. Koreen still maintains that the drawing is not right. Lori: Teacher:

All you have to do is turnit and it would be both a squareand an equilateral rectanglein my definition. Can anyone help Koreen come up with a square that's not an equilateral rectangle?

No one can. They leave it for "something to think about." The issue of "correctness" comes up once more. The teacher shows several quadrilaterals. He asks several students to identify rectangles, which they do. Koreen, unsurprisingly, points to a square. Teacher: Donny: Teacher:

Koreen:

Teacher: Donny: Teacher:

Why is Koreen pointing to the same figure as Donny pointed to [thatis, a square]?Donny? Because it has the same propertiesas a rectangle. Yes, andwe figuredthatout by makinga squarewith ourrectangleprocedure. A polygon is a figure that is made of straightparts ... it is a closed figure;it has straightparts.It is a closed, simple figurethatis madeup of straightparts, and bends. Just like a triangle. When we're in sixth grade,andwe have a mathtest, andit says, "Markthe X on all the rectangles,"and we get the rectangleswrongbecause we X'd out a square,now, who should we come to? [Generallaughter.] You know thatanswer.You tell the teacherwhat? 'Cause a squarehas all the propertiesof a rectangle. Right, and that should convince the teacher.

These discussions illustrate students coming to their own terms with difficult ideas. While these students are still consolidating Level 2 thinking about properties, they are challenged by Level 3 questions. It is questionable, however, whether many had constructed complete and viable knowledge about hierarchies of quadrilaterals. To explore this in more detail, consider the following excerpts from interim individual interviews with Jonathan and Jeremy: Interviewer:Do you rememberwhat Koreenwas saying aboutsquaresand rectangles... ? What would you do ... tell the teacher?

Results

50

Jonathan: I'd markthem all. I would say you could do it, 'cause it's just like a rectangle, it was four sides and90? angles. The differenceis, they ain't the same, all the lines [pointing]. Interviewer:Whatif the teachersaid markall the squares,could you markthe rectangles? Jonathan: No. Interviewer:How come it doesn't go both ways? Jonathan: Maybe it could be a square.I don't really know. Interviewer:How could you find out if thatwas a squareor not? How could you tell? Jonathan: Justtake a guess. Again, Jonathan states the inclusive relationship and now spontaneously provides a property-based rationale. For the converse, he responds correctly at first, but under further questioning, he is unsure of himself and abandons properties (or avoids the issue) by guessing. The transitional nature of his ideas is shown again in the endof-the-year interview. Interviewer:This one is not a rectangle? Jonathan: No. It's a square. Interviewer:And so you can't call it a rectangle. Jonathan: Yeah, you could, because if you did thatrectangleprocedure,and did 40 40, it would be a square.[Jonathanspontaneouslyused this argument,andit was convincing to him-possibly the sole element of his thinking sufficiently convincing to breakthe entrenchedideas.] Interviewer: Okay,but would you put it with these [rectangles]? Jonathan: Yes. As the previous interview shows, Jeremy's ideas were different. After the previously described individual and group work, Jeremy participated in the following individual interview. In contrast to the initial interview on rectangles, Jeremy's conceptualizations here are considerably closer to established mathematical ideas. Interviewer:Whatis a squarein your own words? Jeremy:

A square has four equal sides, four 90? angles, it can be any shape ... uh ...

size. It is, in a way, an equilateralrectangle. Interviewer:A rectangleis what, then?What's the differencebetween thatand a square? The horizontalsides are the same length and the vertical sides are the same Jeremy: length, but not in relationto each other. Interviewer:Could they be the same length? Yes, but usually they are differentlengths. Jeremy: Interviewer:Whatif a kid said"twolong sides andtwo shortsides,"is thata good definition? No, because they have right angles. Jeremy: Interviewer:If we addright angles to the definition? No, you could still be thinkingof differentthings.Manydifferentthings.And Jeremy: no squares. Jeremy's progress is evident. The next segment of the interview even more clearly reveals the cognitive processes Jeremy is using to establish relationships among classes of figures.

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51

Interviewer:Whatdo you thinkof when you thinkof quadrilateral? I thinkof things with four sides, usually rectanglesand squares. Jeremy: Interviewer:Whatelse? Rhomboids. Jeremy: Interviewer:Whatis that? It's a squareturnedsideways. Jeremy: Interviewer:Can it also be differentfrom a square? Yes. [Drawsa parallelogram.The interviewersays thata rhombusmusthave Jeremy: all sides equal and shows Jeremyexamples.] Interviewer: So, are rhomboidssquares? No, 'cause they don't have four right angles. Jeremy: Interviewer:Are squaresrhomboids? Yes. Jeremy: Interviewer:Are ... do you know what parallelogramsare, with opposite sides equal and parallel? Yes. Jeremy: Interviewer: Are rhomboidsparallelograms? Yes. Jeremy: Interviewer:Every one of them? [Looks up at the ceiling for quite some time.] Jeremy: This raises the question of how Jeremy was answering the queries posed to this point. The following interchange suggests an answer: Interviewer:When you look up and thinkaboutthat,what are you thinking? I visualize. Differentrhomboidshapes. Jeremy: Interviewer:Can you draw some? [He does so; they are differentin size, orientation,angle size.] Jeremy: Interviewer: So, how do you check them out to see if they are parallelograms? I couldn't thinkof any thatweren't. Jeremy: Interviewer: So, you figuredthey all were. Yes. Jeremy: Interviewer:Now, are all squaresrectangles? No. Jeremy: Interviewer:Did you visualize? Jeremy: [Very quickly.] No. Interviewer:Oh, so you have a differentway? How did you know thatone? Well, I watchedSquare I TV. Jeremy: Interviewer:Whatdo they say? All squaresare rectangles,but not all rectanglesare squares. Jeremy: Interviewer. How do you know they are right? It says in textbooks. Jeremy: Interviewer:Is there any way you could prove it mathematically? Well, I could tell her the definition of a rectangle and the definition of a Jeremy: square.All rectangleshave four sides, and the opposite sides are parallel. Interviewer:Is thatall?

52

Results

And four rightangles. And squareshave all that. Jeremy: Interviewer:Okay.Do you have to say fourrightangles?Orwould it be enoughto say, "at least one right angle." [Looksat ceiling.] You would need that,'cause thenyou could thinkof someJeremy: thing like this-the opposite sides are parallel,and thatisn't a closed shape. Interviewer:Ah, let's say we agreedthatit had to be closed, simple, quad,opposites sides equalandparallel,andat least one rightangle. [Writesthese down as they are mentioned.]Is thatenough, or would we have to say four? Yeah, this one again. Jeremy: Interviewer:But that's not a quadrilateral,and it's not closed. But it has four sides. Jeremy: Interviewer:But it's not closed. Yeah, you're right.Thatdefinitionwould do. Jeremy: Interviewer:So, that's it. Yeah, but if you said four rightangles, it would be more concise. Jeremy: Jeremy invokes a typical mathematical criterion, conciseness, but applies it to something like the phrasing of a statement. Interviewer:I'm not sure what's more concise. You need four rightangles, because then you could have somethinglike this. Jeremy: Interviewer:Ah, a parallelogram.But thatdoesn't have one right angle. [He drawssome more, thinkssome more.] You're right. Jeremy: Thus, Jeremy appears to use a combination of processes. A basic process is visualization with mental, imagistic transformations, in which he checks the images against his schemes for various shapes. He also applies, and is influenced by, a logical argument concerning properties and statements that he has accepted, and can justify, although it is uncertain to what extent his past or present acceptance is based on his own justification. In the end-of-the-year interview, he clearly and quickly chooses all the rectangles, including the squares, as examples of rectangles. Interviewer:Which of these can be called rectangles? All of these can [choosing all rectangles,includingsquares]. Jeremy: Interviewer:Why? Because they have the propertiesof rectangles. Jeremy: Both Jonathan and Jeremy displayed clear evidence of learning about properties of rectangles and squares, especially considering their conceptualizations at the beginning of the Shapes unit. However, Jonathan struggled throughout his work on the unit with the relationship between those classes of quadrilaterals.Logo-based experiences appear to begin to break down his certainty regarding the exclusivity of the two classes. In addition, on several occasions he states that all squares are rectangles, sometimes providing a Logo- or property-based rationale. However, at one point he appears to accept the converse of the proposition that all squares are rectangles. Later, he denies the converse but admits that he is not really sure. Finally, he does not name a square as a rectangle in the final interview until the interviewer

Logo and Geometry

53

states,"So, you can't call it a rectangle."At thattime he admitsthatone could and providesa Logo-basedargument.Jonathan'sreasoningin this situationwas similar to thatof the kindergartnerswho thoughtthatsquaresandrectanglesusually were separateunless the formerwas producedby a proceduredesignedto drawthe latter. In contrast,Jeremy'sexperiencesled him to constructideas closer to established mathematicalrelationships.He appearsto understandthe rectangle/squarerelationshipwell, givingjustificationsthatarebasedon the propertiesof those figures (althoughjustificationthatrefersto authoritiesis also present).Further,he can apply similarreasoningto otherclasses of quadrilaterals,includingthose whose hierarchical interrelationshipswere not studiedby the class. Like Jonathan,Jeremyused visual processes in this reasoning.A combination of imagistic transformationsand logical argumentsabout shapes and their properties characterizedthe thinkingof both students.In other words, a cyclic effect was observedfor both of them. Work with Logo, which involved thinkingabout properties and the constraints Logo code placed on the properties of figures, affectedJonathan'sreasoning.His visual imagery,in turn,supportedhis thinking aboutthe geometrictaskandthe propertiesof figures.In general,students'concern about the interpretationof correctnessand what will be accepted as correct in differentsituations(e.g., otherclasses with otherteachers)is salient. Logo-based andproperty-based justificationsseem to supportstudents'fledging autonomy,but it is far from certainhow strongtheirindependentthinkingwas at this point. Classroomobservationsalso illustratedhow Logo experiencesaffectedstudents' thinkingaboutthe hierarchicalclassificationof geometricfigures. In the episode below, studentsare analyzinga variablesquareprocedure(SQUARE :X) and are trying to decide how to make a variable rectangle procedure(these and other supportingdatawere first reportedin Battista& Clements, 1992). If we aregoingto makea rectangle,arewe goingto be ableto use the Teacher: same[variable] thatwe usedfora square? procedure Several students: No. Teacher: thinkaboutthat. Whynot?Whycan'twe?I'mgoingtowaitandleteverybody Kristie: Therearetwolongerlineson a rectangle.Theyarelongerthana square. All thelinesarenotequalin a rectangle; theyarein a square.So, if you thinkthat,youcan'tdrawa rectanglewitha squareprocedure. In thesensethatthe 10 orwhateveryouputdownforthesquarereprePaulie: sentsall thesides,whichwouldn'tworkbecauseall thesideswouldbe forit. equal.So you'dhaveto makea newprocedure Youhavementionedthatoppositesidesareparallelandequal.It's the Jennifer: same way with a squareexcept that all sides are equal. So that the two sides thatareparallelare still equal. So, a squarein the sense thatyou're saying is a still a rectangle,but a rectangleis not a square.

Teacher: Jennifer: Teacher: Jennifer:

Canwe buildanyrectanglewiththesquareprocedure? Yes, you can.

CanI builda rectanglewithsidesof 20 and40? No, sorry.Youcan'tbuildeverysinglerectanglewiththesquareprocedure,butyoucanbuildonerectanglewiththesquareprocedure.

54

Results

Now, pairs of studentsmove on to the "Rectangles:What Can You Draw?" activity.As they get to the squareon the sheet,Jennifersays, "It'sa square."Paulie illustrateshis confusionover classificationby saying,"A squarecan be a rectangle, wait. A rectanglecan be a square,but a squarecan't be a rectangle."Jenniferstarts to correcthim, saying, "A squarecan be a rectangle."Paulie interrupts,"Oh,yeah [laughs]." In this episode, all of these studentssee thatthe squareprocedurecannotbe used to make rectangles.Jennifer,however, is the only studentwho seems capable of comprehendingthe mathematicalperspectiveof classifyingsquaresandrectangles. However,hercomment,"inthe sense thatyou're saying,"suggeststhatshe has not yet acceptedthis organizationas her own. The next episode, in which she and her partnerare interviewed,furtherillustratesthat she has not yet adopteda mathematicalorganizationin her classificationof shapes. If I typedin RECT50 51, whatwouldit be? [Doesnotpressthereturnkey, Interviewer: so theprocedure doesnotyetrun.] a Paulie: about Probably square. Jennifer: A rectangle,butit wouldn'tIt wouldbe a rectanglebutsortalikePaulie: butit wouldn'tbe a perfectsquare.[Thepartners Jennifer: Itwouldbe arectangle, press thereturnkey.] Jennifer: Yousee, it's nota perfectsquare. thetopside,orlongerside,withhisfingers.]It'sonlyonestepoff. Paulie: [Measures Even thoughPaulie andJennifersay thatthe 50 51 rectangleis a rectangleandnot a square,theirlanguageseems to indicatetheirbelief in such a thing as an "imperfect square";thatis, we presume,theybelieve in a figurethatlooksjust like a square butdoes not possess the characteristicof havingall sides equal.PaulieandJennifer areclinging to an informal,ratherthanlogical, classificationsystem, one thatstill containsremnantsof theirvisual thinking. Finally, we examine the comments made by sixth-graderKelly duringa class discussion, in which the square/rectangleissue was raised when studentstriedto drawthe squarewith the rectangleprocedure.Kelly asked, "Whydon't you call a rectanglea squarewith unequalsides?"Whenthe teacherrespondedby saying,"A rectangleis a shape which has four rightturnsand opposite sides parallel,"Kelly rejoined, "If you use your definition, then the squareis a rectangle"(Lewellen, 1992). Kelly's comments,like those of fifth-graderJennifer,which were described previously,clearlyindicatean abilityto follow the logic in the mathematicalclassification of squares and rectangles. However, neither student had made that logical networkherown-each girl still clung to the personalnetworkconstructed frompreviousexperiences.As van Hiele says, "Onlyif the usual [as taughtin the classroom] network of relations of the third level has been accepted does the squarehave to be understoodas belongingto the set of rhombuses.This acceptance must be voluntary;it is not possible to force a networkof relationson someone" (van Hiele, 1986, p. 50). For Kelly or Jenniferto move to the next level, each must reorganizeher definitions of shapes in a way that permits a total classification

LogoandGeometry

55

scheme to be constructed.Thatis, the attainmentof Level 3 does not automatically resultfromthe abilityto follow andmakelogicaldeductions;the studentmustutilize this ability to reorganizeher or his knowledge into a new networkof relations.In this network,"Onepropertycan signal otherproperties,so definitionscan be seen not merely as descriptions but as a way of logically organizing properties" (Clements& Battista,1992b).Normally,establishingthis networkentailsmaking sense of and accepting the common mathematicaldefinitions and resultinghierarchies given in the classroom. We will returnto this issue and draw additional conclusions and implicationsin chapter4. ANGLE, ANGLE MEASURE,AND TURNS Rotationsplay a fundamentalrole in forminggeometricfigures in turtlegeometry. Therefore,we expected that Logo experience would provide foundational experiences with rotation.Further,LG explicitly links turtlerotationto concepts of angle and angle measurement.Thus, we hypothesizedthatexperiencewith this curriculumwouldfacilitatethe developmentof the geometricconceptsof angleand angle size. Pre-Post Test test assessedstudents'ideasabout Severalitemson the 1988-89 paper-and-pencil and measurement. Some items addressed the concept of rotationand angle angle the relationshipbetween the turtle'srotationandthe measurementof the resultant angle. Identificationof Angles.One open-endedquestionasked,"Whatis an angle?"We categorizedresponsesto this questionby studentsin all grades, K to 6. Table 16 providespercentagesby categoryof these responses.Frompretestto posttest,the LGresponsesmorefrequentlywere categorizedas reflectingrotationandthe intersection of two lines. There was a correspondingdecreasein the no-interpretableresponsecategoryandthe tilted-linecategory.In comparison,the controlgroupalso showed a decreasein the no-interpretable-response category(althoughless so than its in whereas increase occurred the intersection-of-two-lines LG), greatest category. Most of the othercategoriesfor the controlgroupremainedstable. Another task on the paper-and-penciltest was designed to assess students' understandingof the conceptsof angle (see Figure12). The studentswerepresented with a page of figures and asked to circle all the angles. One point was given for each correctresponse,thatis, a responsein which only angle verticeswere circled, yielding a maximumscore of 14. Means and standarddeviationsare shown in Table 17. An ANOVA on the total scores for this angle identification question revealed no significant treatment effects. Draw an Angle, Draw a Bigger Angle. Two items on the paper-and-penciltest asked studentsto draw an angle and then to draw a bigger angle. Each item was

56

Results

2.5. Circleallthe angles on this page (ifa figurehas morethanone angle, circle each one).

/ Figure 12.

worth a single point. Tables 18 and 19 provide means and standarddeviations for these angle drawingitems. A MANOVA on the total scores revealed several significant interactions.First was a Treatmentx Time interaction(Pillais trace, F(2, 1038) = 5.02, p < .01). There was also a significant Treatmentx Time x Gradeinteraction(Pillais trace, F(12, 2078) = 3.34, p < .001). For the Draw-anAngle item, kindergartenand sixth-gradestudentsimprovedmore in LG than in control classrooms. For the Draw-a-Bigger-Angle item, kindergarten,thirdgrade, and (especially) fourth-grade students improved more in LG than in control classrooms. An examinationof individualresponsesrevealedthatin Grades2-4, most errors occurredwhen drawingtilted lines or geometricshapes(e.g., drawinga rectangle with no sign indicating one or more of its angles). LG instructionhad a strong effect on reducingthe incidence of these errorsat Grades2-3; at Grade4, both LG and controlinstructionreducedthese errors,althoughcontrolinstructiondid so to a lesser degree. At Grade5, a different patternwas observed. Most of the students were able to draw angles even before instruction;instructionhad the effect of changingthe types of correctangles the studentsdrew. Most fifth-grade control studentsshifted from nonprototypicalacute angles to prototypicalacute and prototypical right angles. Most fifth-grade LG students shifted from right angles to acute and nonprototypicalobtuse angles. Shifts in Grade 6 were less pronounced;only control studentsshifted substantially,from prototypicalright angles to prototypicalacute angles.

57

Logo and Geometry

In summary,LG studentssignificantlyoutperformedcontrolstudentsin drawing an angle, althoughthe differencewas slight. The youngest and, to a lesser extent, the oldest LG studentsimprovedthe most relative to control students.Similarly, LG studentsoutperformedcontrolson drawinga largerangle, with kindergarten, third-grade,and (especially) fourth-gradeLG studentsbenefitingthe most. Amountof Turn(Spinner).Three spinnerquestionson the PP test assessed the knowledge of amountof turnfor studentsin Grades2-7 (one of the three similar tasks is illustratedin Figure 13). One point was given for the correctdirectionand turnmeasure(within 20?), for a maximumpossible score of 3 points.

2.4. How many degrees and in which direction (rightor left) should you turn the

spinnerarrowto aim it directlyat the center of the target?

aO

i

*

0

Figure 13.

Table 20 presentsthe means and standarddeviationsfor this item. An ANOVA revealed a Treatment x Time interaction (F(1, 781) = 175.97, p < .001). In summary,

LG studentsat every gradescored substantiallyhigherthancontrolson questions measuringtheirknowledge of degrees of turn.

58

Results

Angle Measure Estimation.One item assessed angle measureestimation (see Figure 14). Onepointwas given for each correctresponse,for a maximumpossible score of 4 points.

2.7. Estimatethe measureof each angle. Markthe box nextto the best estimate.

2

\d

L[30?

L 30?

I 45?

i 45?

i 45?

60? i 90? i 120? i- 160? i 180?

n 60?

-60? i 90? i 120? i 160? L 180?

-

i

30?

i 90? L1120? i 160? i 180?

I

30?

_ 45?

I 60? i 90? i 120? I 160? L1180?

Figure14.

Table 21 presents the means and standarddeviations for these questions. An ANOVA revealedno significantTreatmenteffects. Missing Angle and Side Measures. Two items presentedfigures with missing angle and side measures (see Figure 15). Each response was scored as correct (1 point) or incorrect(0 points), for a maximumpossible score of 7 points. Table22 presentsthe meansandstandarddeviationsfor theseitems.An ANOVA revealeda Treatmentx Time interaction(F(1, 459) = 28.96, p < .001). In summary, LG studentswere more capable of finding missing angle and side measuresthan control students. Angle Measure:Boat. Two items measuredthe ability of studentsin Grades4, 5, and 6 to find angle measuresin problemsituations,both of which were scored 0 or 1. The first, involved a boat's path (Figure 16). Table 23 presentsthe means and standarddeviationsfor the Boat Amount-ofTurn item. An ANOVA revealed a significant Treatment x Time interaction (F(1, 317) = 12.31,p < .01). Therewas also a significantTreatmentx Time x Grade interaction(F(2, 317) = 3.63, p < .05). Of the controlclasses, only the sixth grade made substantialgains. Further,the fourth-gradeLG classes apparentlydid not make greatergains than the fourth-gradecontrol classes.

59

Logo and Geometry

1.8. Findthe missing measures of the angles.

X

90?

righttriangle

800

/ 80?

quadrilateral 1.9. Findthe missing measures of the sides and angles. 50 300

?~

\

X25

A-

parallelogram Figure 15.

2.12. A boat is sailingon a lake, headingtowardits home. Itgoes forward60 yards,turnsright80?,goes forward152 yards,turnsright160?,andgoes forward 173 yards. Itis now back to its originalpositionon the lake. How much does it have to turnto be facingtowardits home again? Answer Howdid you get youranswer? Figure 16.

Results

60

Examinationof individualresponsesrevealedthatthe most common erroneous responsewas 240?, the sum of the two given angles, 80? and 160?.Fourthgraders in both groups,andfifth gradersin the controlgroupwere morelikely to makethis errorafterinstruction.Therewas a reductionin this errorby the LG studentsbut an increaseby the controlstudents.Othercommonerroneousresponses,suchas of 90? and 0?, appearto indicatea lack of understandingof the problem.LG instruction reducedthe incidenceof these errors,whereascontrolinstructionincreasedit. Angle Measure: Wire.The second item thatmeasuredthe abilityof Grades2-6 studentsto find angle measuresused a wire-bendingsituation(see Figure 17).

2.13. 1. A workeris buildinga wire frameforthis piece.

2. Here is the wire that needs to be bent.

J1 400?

^1 40?

3. Howmuchwillthe wirehave to be bent to fitthe cornerexactly?

J1400

Answer Howdidyou get youranswer? Figure 17.

Table 24 presentsthe meansand standarddeviationsfor the wire-bendingitem. An ANOVA revealeda significantTreatmentx Time interaction(F(1, 487) = 4.93, p<.05). A frequent erroneous response was 45?, selected by more LG students than

controlstudents.Even thoughsuch a responsewas given no partialcredit,it probably indicates use of visual estimation and good measurementsense. Another common erroneous response was 140?, a misinterpretationthat the question concernedthe angle of the frameitself ratherthanits supplement.LG instruction reducedthe incidenceof this errorat every gradelevel; controlinstructionreduced this erroronly at Grade6. In summary,LG studentswere morecapablethancontrolstudentsof determining measuresof turnsin two problemsituations("boat"and"wire").Forthe boatitem, substantialgains relativeto controlswere foundfor fifth gradeand sixth gradebut

LogoandGeometry

61

not for fourthgrade.At all gradestested,LG studentsperformedbetterthancontrol studentsfor the less complex wire item. Unit Tests Unit tests were designed to measurethe specific geometric achievementgoals of LG. They included items on identifying turns and angles and on angle measurement. IdentifyingTurnsand Angles. Kindergartnersaccuratelyidentifiedthe number of turnsin illustratedpaths (triangle,rhombus,square;82% to 97%;M = 89%). Recall thatfirst graderswere only moderatelycompetentin giving the numberof turnsin a squareand a trianglewithout the benefit of a diagram(58% to 74%). Second graderswere more accuratethanfirst gradersin identifyingthe numberof angles in figuresfor which diagramswere provided(69%to 98%,M = 86%);similarly,thirdgradersandfourthgradersaccuratelyidentifiedthe numberof turnsand angles in a largerset of figures (50% to 96%,M = 80%;50% to 95%,M = 87%). Measureof Turnsand Angles. These items asked studentsto identify turnsof a given measure,identifythe measureof given turnsandangles, and solve problems involving the totalinteriorandexteriorangles of polygons. Abouttwo thirdsof the first graderswere able to tell "how many small turtleturns(30? in the latterhalf of theirwork with LG's Singlekey) make a quarterturn(90?)"(62% to 77%,M = 69%). Thirdgradersperformedpoorly and fourthgradersmoderatelywhen asked to select the numberof degreesfor anglesof measure30?,45?, 90?, and 120?(given those measures,and 60? and 150?, as choices; 17%to 100%,M = 57%; 43% to 100%,M = 71%). The 90? angles were far more likely to be correctlyidentified by both groups.In a similarvein, fourthgraderssuccessfully identified the right anglesin polygons(only completelycorrectresponsesreceivedcredit;60%to 90%, M = 78%). About 95% identifiedall the rightangles in a squarewith a horizontal base; 89%did so in a squarewith sides at a 45? orientation,with 2.5% identifying no angles and 1.7%identifyingthreeof the fourangles. Approximately81%identified the right angle in a right isosceles triangle(with the right angle at the top), 10%identified all angles (it is unknownhow many did not process the adjective "right"in the directions),5%identifiedthe two acuteangles, and4%identifiedone right and one acute angle. Thirdgraderswere also asked to give the total amountthe turtleturnswhen it makes a square.The rangeof two classroomsindicatesthatmasteryof the idea is possible but must be facilitatedby the teacher(9%to 88%,M = 48%). The lowest performancewas fromstudentsin one classroom,most of whomanswered"4"(i.e., the numberof turnsratherthanthe total amountturned).It may be thatthis teacher inadequatelyemphasizedamountsof turnor thatthe studentsmisunderstoodthe question because of poor reading ability or a misinterpretationby the teacher within the test situation. Fourththroughsixth graderswere given the measureof two angles of a triangle and asked to find the measureof the thirdangle. Performancewas low for fourth

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Results

graders,who were provideda figure (50% to 60%, M = 54%), and also for fifth and sixth graders,who were not given a figure (39% to 63%,M = 49%). Fifth and sixth graderswere asked several additionalquestions about anglemeasurerelationshipsin regularpolygons, includingexteriorangle measuresand sums of interiorangle measures(M = 69%). Unsurprisingly,studentsperformed muchbetteron questionsaboutsquares(85%to 97%)thanon questionsabouttriangles (57%to 64%) andpentagons(28%to 48%). Studentsscoredhigheron questions involving the total amountof turtle-turning(63%for the triangleto 92% for the square)thanon questionsinvolving the sum of the measuresof interiorangles (64% for the triangle, 85% for the square, 48% for the pentagon). However, because directingthe Logo turtle emphasizes only the former,the lack of large differencesindicatesthatLGhelpedstudentsconstructbothof thesetwo constructs. Interview In individualinterviews, studentswere asked to draw angles and then "angles thataredifferent,"to describethe differencesbetweentheirdrawingsof angles, to circle the biggest angle given pairsof figures, andto solve problemsinvolving the amountof turnof a robot. DrawingAngles.Studentswereaskedto "Drawan angle.Now drawanotherangle that is different from your first angle in some way...." Table 25 describes the students'firstdrawnangle;note thatonly two segmentsintersectingat an endpoint were scoredas correct.FortheLGgroups,therewas a declinein incorrectresponses from the pretestto the posttestand an increasein correctresponses.Note thatthe LG groupsscoredbetterthanthe controlgroup.Also, for the correctangles drawn by the LG group,more thantwo thirdswere nonprototypical,with only one third being prototypical(e.g., right angle, "facing"right,horizontalbase). The reverse was the case for the controlgroup.Thus,therearetwo differencesbetweenthe LG group and the control group. First, the LG students drew more correct angles. Second, the LG students'increasedawarenessof the angle concept was richerin thatthey were much more likely to drawnonprototypicalexamplesof angles. Students' Descriptions of Differences between Their Angle Drawings. The second task asked studentsto drawanotherangle that was differentin some way fromthe angletheydrewin the firsttaskandto explainhow theirnew anglediffered from the one before it. Students'responseswere categorized.Table 26 shows the percentageof studentswho gave an explanationin each categoryfor at least one comparisonandwhich was consistentwith theirdrawingsat least some of the time. For the LG group,therewere decreasesfrom pretestto posttestin responsesindicatinga totallack of knowledgeof the conceptof angle,increasesin responsesindicating an intuitiveunderstandingof angle, andincreasesin responsesindicatinga knowledge of the measureof angles or the classificationof angles (by measure). The numberof studentsthatthoughtaboutangle size in terms of tilt, heading,or turningalongthe pathremainedbasicallyconstant,with tilt decliningandthe more sophisticatedheading increasingat two of the three LG sites. The control group

63

Logo and Geometry

hadequalor greaterposttestscoresin the "formal"category,perhapsreflectingtheir curriculum'semphasison classifying angles by measure. Angle Measure.To assess students'understandingof angle measure,they were asked to circle the biggest angle in each of the pairs of angles and in the triangle shown in Figure 18.

1.

2.

/

B

/

A

A

B 3.

A

4.

B Figure18.

The total numbercorrectfor all four items was analyzed. LG studentsscored significantlyhigherin judging the size of angles (F(1, 211) = 14.01,p = .000). The meancorrect(out of four)was 2.16 for the LG groupand 1.62 for the controlgroup. For 1988-89, there was also a significanteffect by Grade(F(6, 211) = 43.02, p = .000), Time (F(2, 200) = 55.27, p = .000), andTime x Gradeinteraction(F(12, 200) = 4.78, p = .000). In addition,for 1987-88, there was also a significanteffect by Grade(F(7, 70) = 26.22, p = .000), Time (F(2, 140) = 29.45, p = .000), andTime x Gradeinteraction(F(14, 140) = 2.86, p = .001). Table 27 shows the angle scores by grade.As can be seen from the table, scores increasedlittle in GradesK and 1. However, therewere notableincreasesin the uppergrades.It shouldbe noted that the termangle was not dealt with explicitly in the curriculumfor GradesK and 1. The frequencieswith which studentschose the anglesin the pairsandthe triangle are given in Table 28. The numberof LG studentschoosing the figure with the longer sides (choice B) for pairs 1 and2 declined.The numberof LG studentswho said thatthe angles were the same size in pair3 (the correctchoice) increased.For the triangle,the numberof LG studentschoosing the correctangle, C, increased,

Results

64

whereas the numberof studentswho circled a side of the triangle or the whole triangledecreased. As can be seen in Table29, the percentageof studentswho correctlychose angle A in pair 2 increasedfrom pretestto posttest,with most of the increaseoccurring in Grades3-6. On the posttest,over two thirdsof the studentsin Grades4-6 were not confused by length of sides whenjudging angle size. After each choice of a largerangle, studentswere asked why they made their choice. Table 30 shows the percentageof studentswho gave an explanationin the given categoryfor at least one of the four items andwhich was consistentwith the figures at least some of the time. The numberof LG studentswho made choices thatshowed no understandingof angle size decreased.The numberof LG students of the angleconceptincreased whose explanationshowedan intuitiveunderstanding at Site 2 in 1988-89 and at Site 1 in 1987-88 but decreasedat Site 1 in 1988-89. Explanationsthat showed a formalknowledge of angle measureincreased,most for the Site 1 studentsin 1988-89. Most of the increasein the formalcategorycame from increasesin Grades4, 5, and 6. Turning.Studentswere askedtwo questionsconcerningturning.The first question was, "A robotturnsright 90 degrees (a right angle) every time it turns.How many turnsmust the robotmake before it is facing in the same directionas it was when it started?"The second question was the same as the first, except now the robotturned30? each time. For 1988-89, the LG groupscoredsignificantlyhigher thanthe controlgroupon both items;95% versus74% correct(F(1, 169) = 12.16, p < .001) for the 90? problemand 61% versus 35% correct(F(1, 167) = 11.33, p < .001) for the 30? problem. Classroom Observations

utilized ClassroomobservationsconfirmthatLGactivitiesandteacher-mediation in the LG project are critical in helping studentsbuild more abstractgeometric notions from their initial intuitionsaboutpaths, includingturnor angle measure. For instance, many young students in our Singlekey environmentbelieveddespitehavingobservedtheirteacherdemonstratingthe commands-that pressing L (for "LEFT")would commandthe turtleto move towardthe left regardlessof its heading.Theyhadnotyet constructedtheideaof two qualitativelydifferentkinds of "moves"the turtlemakes: changes in position (FORWARDand BACK) and changesin heading(RIGHTandLEFTturns).Two kindergartengirls were videotapedas they began to constructthis idea. The turtlewas facing "down"(i.e., at a 180?heading). Selene: Darlene: Selene: Darlene: Teacher: Selene:

Let's make it go left. Left is this way [pointsto his left]. [PressesL.] Ohhh [in a disappointedtone]. Whathappened?Whatdid the turtledo? It... turnedleft.

Logo and Geometry

65

Darlene: ButI wanthimto go left! Teacher: Whichwayis theturtleheading? Darlene: Oh,yeah,turnhimandthengo forward! The teacher'squestionshelped the pairto reflect on the results of their actions and to attendto importantfeaturesof the problem.He did not try to directly tell the childrenwhat to do. Laterin the week, the girls seemed to have firmly constructedthe idea of turn along a path.The following exchange was typical. herownhead He'sfacingthisway,yeah.Andwe wanthimto go ... [turning Selene: so thatshe is facingthe sameway as theturtleandthenmotioningwithher hand] ... right!

Darlene:

So, thatwould be ... R. [Tapsher head repeatedlyandemphatically.]

[TypesR.]Now,F forforward! The act of turning their heads to align themselves with the turtle and then motioning may be a sign that studentsreally do "see" a change in directionas a resultof a turningmotion.It is importantto note thatthe studentsdid not construct this criticalandsurprisinglydifficultidea by watchingthe teacher'sdemonstrations or by hearinghis explanations.They used Logo as a tool to experimentwith their own ideas. The teacheronly helped them reflect on theirexperiments. At the conclusion of a maze-like activity (to get from one point on the screento another),the teacherhad severalkindergartenchildrenshow theirwork.He asked questions such as: At what point(s) in your pathwas the turtleturning?Whatcan you tell me aboutthe pathyou just made?How many straightpartsdoes the path have? How many bends? Can you make a path with fewer straightparts?Fewer bends?Were your paths simple paths? Student: Oursareall simple.Butnobody'sis a closedpath. Teacher: Whyis that?Howdo youknowforsure? Student: Becausetheturtlealwaysstartshereandendsin a differentplace,overhere. No matterwhereyougo, thatwouldnevermakea closedpath! We have investigatedwhetherthe pathperspectiveleads to a confusionbetween measuresof the amountof turnalong the path and angle measure.Our observations indicatethatunderproperinstructionalconditions,this need not be so, even for young students. One first-grade boy was asked to draw an angle, then a "bigger angle." He sketched an angle of approximately30?, then a right angle (90?). Spontaneously,he explained that the formerwas a "five turnangle" (i.e., five "R"turnsof 30? each) andthe lattera "three."Finally,he drewangles in which the measure of the turn along the path diminished from 150? "down to" 0?, explaining that"as you lose turns,it gets to be a bigger angle."On prompting,he explainedthat"lose turns"meant"goingfromfive rightsdown to zero."This indicates that notions of angle, rotation, and their measures have been positively affected by work in the LG.

Selene:

In summary,thereis supportfromtheseobservationsfor the notionthatLGassists studentsin buildingon theirinitial intuitionsaboutpathsto constructmore math-

66

Results

ematicallysophisticatedconcepts of geometricobjects, includingangle, rotation, andtheirmeasure.These arequitepositiveresults,especiallyconsideringthatthese notionsoftencause definiteproblemsfor students."Itseems studentshave as much difficultywith angles as theyhave with anyothergeometricconcept.Turnsor rotations are naturalfor young children ... this dynamicway of workingwith angles shouldbe startedearly andcarriedthroughoutthe students'work"(Hoffer, 1988, p. 251). The datapresentedhere confirmthese statementsand suggest thatLG is an effective approachto implementingthe recommendationsto include dynamic work with angles in classrooms. PATHS Logo Geometrytook a pathperspectiveto elementarygeometry.Therefore,we expectedthatLG wouldfacilitatethe developmentof pathconcepts.In this section, we presentthe smallnumberof taskson the PP andunittests thatassessedstudents' ideas aboutpaths. Pre-Post Test

The PP testincludedtwo typesof itemson paths.The firstaskedstudentsto draw pathswith certainproperties.The second askedstudentsfor verbaldescriptionsof shapesto ascertainif and how they would incorporatea pathperspective. Draw a Path. Threeitems presenteddescriptionsof pathsand askedstudentsto draw the paths (see Figure 19). One-halfpoint was given if the path drawnwas missing only one of the requiredcharacteristics;a full pointwas given for a correct response. Table 31 presentsthe meansand standarddeviationsfor the pathsquestions.An ANOVA on the total scores for those questionsadministeredto all students(Items

2.7. Drawone paththat has 4 straight parts, has 4 bends (angles), and is closed. 2.8. Drawone paththat has 3 straight parts, has 2 bends (angles), and is not closed. 2.9. Drawone paththat is closed, but not simple. Figure 19.

67

LogoandGeometry

2.7 and 2.8) revealed a Treatmentx Time interaction(F(1, 1038) = 138.20, p < .001). As expected, LG studentsperformedsignificantlybetter. Therewas also a significantTreatmentx Time x Gradeinteraction(F(3, 488) = 4.50, p < .01). Certaincontrol group classes seemed to make smaller gains than others.Gainsfor the youngerLGclasses tendedto be greaterthanthosefor theolder classes. In summary,LG students outperformedcontrol studentson these path items. There was some indicationthat younger LG studentsmade greatergains in this domain than older LG students, but classes at all grades outperformedcontrol classes. Communication:The Telephone. Item 1.10, administeredto all students in Grade2 and higher, stated,"Pretendyou aretalkingon the telephoneto someone who has never seen a triangle.Writedown whatyou would tell this personto help them make a triangle?"[GradesK-1 studentswere asked, "Whatwould you tell this person ... ?"]. Students' responses were categorized. Table 32 presents the percentagesby category for this item. Differences were small but indicatedthat LG students were more likely to provide complete, correct descriptions of the process of creating a triangle and that they made greaterincreases in the use of sophisticated strategies ("Logo-like commands"or "complex path or components togetherwith Logo"). Unit Tests On the unittestsmeasuringLG students'achievementof LG goals, studentswere asked to identify or drawpathswith specific properties.Kindergartnersidentified pathswith variousproperties,includingbeing closed, straight,andhaving a given numberof turns (50% to 100%;M = 79%). First gradersto fourthgradersdrew paths with such properties(first:77% to 100%,M = 95%, with a single property for each question;2nd: 71% to 100%,M = 94%, with a single property;also 47% to 87%,M = 72%, with multipleproperties;3rd:56% to 100%,M = 87%, with a single property;4th: 55% to 100%, M = 87%, with a single property).Second gradersalso identifiedcertaincomponentsof a path(76%to 100%,M = 90%).Fifth and sixth gradersdecided whetherto attributeeach of three properties-closed, straight,and simple-to nine paths, with credit given only if all threeproperties were correctlyattributedto the path (69% to 78%,M = 72%). Of the single properties, straightandclosed tendedto be masteredby students,whereasthe concepts of possessing a given numberof bends, and especially possessing "curvyparts," tendedto be more difficult.Thus, studentswere moderatelysuccessful in learning paths concepts. SYMMETRY The importantgeometrytopic of symmetryis treatedin two distinctways in LG. First, computertools allow studentsto make symmetricpaths (MIRROR).One

Results

68

turtlewould follow the students' commandsprecisely, the other would create a dynamic"mirrorimage"by turningin the opposite direction(e.g., left 45 instead of right 45). Second, computertools allow students to perform flips; students could then flip shapes to check if they were symmetric.Therefore,we expected thatLG would facilitatethe developmentof symmetryconcepts. Pre-Post Test Draw SymmetryLines. Item 1.5 asked studentsto drawin all lines of symmetry for variousfigures.Studentsat every gradelevel were presentedwith the firstfour figuresin Figure20. The fifth figurewas presentedonly to studentsin Grade3 and above.Eachcorrectline of symmetrywas awardedone point;a pointwas subtracted for each line drawnthatwas not a line of symmetry,such as any line drawnfor the parallelogram.The maximumpossible score was 8 points.

1.5. Drawin all lines of symmetryforthe figuresbelow.


/

Figure 20.

Table 33 presents the means and standarddeviations for the symmetrylinedrawingquestions.An ANOVA on the totalscorefor the firstfourfiguresrevealed a Treatmentx Time interaction(F(1, 1021) = 116.12, p < .001). An ANOVA on the total scoresfor the draw-symmetry-linesquestionsalso revealeda Treatmentx Time interaction(F(1, 609) = 30.98, p < .001). The lower scores for the composite with more items makes sense consideringthat any lines drawnon the parallelogram would be incorrect,lowering the score; perfect performanceon that figure adds zero to the score. Table 34 shows the percentageof correctand incorrectsymmetrylines drawn by each group. Differences between groups varied on the incorrect lines. LG

LogoandGeometry

69

studentsimprovedslightly more on the item (pentagon)thathad only one salient line of symmetry.Both groupsincreasedin the errorsfor the rectangleitem (identifying diagonals),with the increaseof the LG studentsbeing greater.Both groups' errorson the parallelogramdecreased,with the control group decreasingmore. Thus,thereis evidencethatLGstudentsovergeneralizewithregardto diagonallines of symmetryand, to a lesser extent, lines of symmetryfor figures thathave only rotationalsymmetry. With regardto correct lines of symmetrydrawn,there were clear differences between the two groups. LG studentsconsistently improved more than control students,with most of the formerfinding all correctsymmetrylines (e.g., about two thirdsof LG students,comparedto one thirdof the comparisonstudents,finding all four lines of symmetryfor the square). In summary,LG studentsscored significantlyhigher than control studentson drawing all the symmetry lines for given figures. LG students did not differ substantiallyfrom control students in the number of incorrect symmetry lines drawn, but there was a clear, positive effect of LG regardingcorrect lines of symmetrydrawn.This may be due to LG's inclusion of MIRRORand flip procedures. Draw the OtherHalf. As shownin Figure21, Item 1.6 askedstudentsto drawthe other half of a figure to producea symmetricfigure. One point was given if the drawingwas symmetric;thatis, if measuresof lengthsandangleswereapproximately correct.One-halfpointwas given if the figurewas not symmetricbutthe drawnhalf had the same numberof segmentson the oppositeside of the line of symmetry.

1.6.

Drawthe other halfof each figureto producesymmetricfigures.

Figure 21.

Table 35 presentsthe means and standarddeviationsfor the total scores for the Daw-the-Other-Half(to make a symmetricfigure) tasks. An ANOVA revealed a Treatmentx Time interaction(F(1, 609) = 30.98, p < .001). The largest gain by the LG group, comparedto the control, was in the numberof studentsreceiving full credit (23% vs. 15%and 14%vs. 4% increasesfor the two questions).

70

Results

In summary,LG studentsscored significantly higher than control studentson drawing the "otherhalf' of a figure to create a symmetricfigure. There was a tendency for these effects to be particularlystrong for young (kindergarten)LG students.In addition,the effects were largely due to the greaternumberof LG studentsdrawingcompletely correctfigures. Unit Tests Identificationof SymmetryLines. Students were asked to identify whether a dotted line drawn on various figures was a line of symmetry. The students' averagecorrectnesswas generallyhigh for each gradetested;averagecorrectness scoreswere:GradeK (60%to 100%,M= 95%),Grade3 (23%to 100%,M= 89%), Grade4 (44% to 100%,M = 93%), and Grades5-6 (49% to 100%,M = 93%). Across all grades,the most difficult item was a concave pentagon(see Figure22) and a parallelogram,where in each case the line was not a line of symmetry.

< Figure 22.

In anotherproblem,studentswere asked to drawall the lines of symmetryonto given figures.Two pointswere given if all correctlines were drawn.Onepointwas given if half or morecorrectlines andno incorrectlines were drawnor if all correct lines and some incorrectlines were drawn.Average correct scores were moderately high: Grade 1 (0.78 to 1.95, M = 1.55), Grade2 (1.10 to 1.61, M = 1.46), Grade4 (0.58 to 2.00, M = 1.62), and Grades5-6 (0.80 to 1.98, M = 1.64). The most difficult shape administeredto studentsin all gradeswas a square;an equilateral triangle and parallelogram,only administeredto Grades4-6, were even more difficult thanthe square.Certainfigures whose lack of symmetrywas visually salient (e.g., an obtuse triangleadministeredto Grades4-6) were easiest. Studentsin all gradeswere askedto drawthe "otherhalf' of one or more given figures to create a symmetricfigure. The total possible score was 2. Means were as follows: GradeK (M = 0.68), Grade1 (M = 1.14), Grade2 (M = 1.42), Grade3 (0.57 to 1.68, M = 1.00), Grade4 (1.28 to 1.83, M= 1.54), and Grades5-6 (1.76 to 1.93,M = 1.82). These resultssupportthe claim thatLG studentsperformedwell on symmetryitems.

71

Logo and Geometry

CONGRUENCE Congruencewas not a majorfocus of the curriculumor assessment. Only one activity in the LG curriculumemphasizedcongruence:Studentsused geometric motions to determine congruence or lack of congruence in pairs of figures. Congruenceis an importantgeometrictopic, however, andthus some assessment was in order. Pre-Post Test Same Shape and Same Size Item. For six pairs of figures, students were asked

to state whether the figures were the same size and shape, as illustratedbelow. One point was given for each correctidentificationfor each size and shape. The studentswere then to answer the question "How would you show someone that you are right?"Justifications were initially coded. Then, 1 point was given for justificationsthatpertainedto the item andreferredto mathematicalnotions (e.g., geometric motions, measurement, and congruence were given 1 point each; responses in the "look like," "they are the same shape," or "because I said so" categories were scored 0). Thus, the total possible score for identification was 12 points and for justification, 6 points. In Figure 23, the full text is reproduced only for the first pair; on the actual test, this text was repeated following each pair of figures. Table 36 presentsthe means and standarddeviationsfor the questionsthatdealt with congruence.A MANOVA on the total scores for the congruencyquestions, identification(i.e., correctlyidentifying figures as congruentor not) andjustification ("How would you show someone that you are right?"),revealed several significant interactions.First was a Treatmentx Time interaction(Pillais trace, F(2, 1012) = 50.52, p < .001). Effects were more pronouncedfor thejustification score; for each question, LG students outperformed control students on the posttest. The largest difference was for those questions with congruentpairs of figures (the mittens and cone). For identification,only the cone question differentiatedbetween the LG and control groups. There was also a significant Treatmentx Time x Grade interaction on the congruencequestions (Pillais trace,F(12, 2026) = 3.22, p < .001). Only thejustificationscorecontributedsignificantlyto this effect. The resultsof two gradelevels on the justification score appearnoteworthy.First, the mean score of the sixthgrade control students declined. Second, all thirdgrades had low scores on the pretestbut made significantgains on the posttest,especially those in LG. The low pretestscore stems from one LG and one control (cohort)third-gradeclassroom, whose teachers agreed to tell studentsto marka "?"if they were not sure of the answer. In summary,LG studentsperformedbetterthancontrolstudentson congruence items, not in identifying whetherpairs of figures were congruentso much as in justifying their answers.

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Results

1.4. Foreach pairsof figures,tell whetherthey are the same size and shape.

o Same shape? D Yes

D No

Same size?

LYes

Howwouldyou show someone thatyou are right? (The same formatwas repeatedfor each pairof figuresbelow.)

I

I Figure 23.

- No

73

LogoandGeometry Unit Tests

Similaritems were presentedon the unittests. These tests indicateda uniformly high ability to identify pairsof figures as congruentor not in all grades:GradeK (M = 93% to 96%), Grade 1 (M = 92%), Grade2 (M = 94%), Grade3 (M = 91% and96%),Grade4 (M = 97%),andGrades5-6 (M = 96%).Studentsscoredsomewhat lower for five figures as follows: block "L"hexagons, one of which was rotated 180? (administered only in Grades K-l); a pentagon and a hexagon (Grade 1 only); concave pentagons, one of which was rotated90? (Grades2-3); an isosceles and a right trapezoid (one "upside-down"; Grade 3 only); and squares,one of which was rotated45? (Grades3-6). Note that accuracyrose to about 90% across items at the higher grades; our comments about these items being the most difficult were based on within-gradecomparisons and are relative to the overall high performance. GEOMETRICMOTIONS Logo Geometryincluded several activities on geometricmotions. But no items on the PP test directly assessed students' knowledge of motions, because not all controlclassroomstaughtthese competenciesseparately.(We did conducta separate experimental study on this topic in which a comparison group was taught these concepts; this is describedin a succeeding section.) However, several unit tests and interview items for geometric motions are relevant. Unit Tests Kindergarten and first-grade students identified motions to move a train engine into the same orientation as another train engine (on a 2-point scoring system, M = 1.58 and 1.22, respectively). Combinationsof motions were, unsurprisingly,more difficult thanindividualmotions (Grade1 was administeredmore combinations-of-motionsquestions;the lowest score was on a questioninvolving all three motions, slide, flip, and turn). An item, in which motions performed on a block "L"hexagon were to be identified (along with turn centers and flip lines), was administeredto all other grades (an example problem is illustrated in Figure 24). Accuracy was 74%, 69%, 83%, and 91%, for Grades 2, 3, 4, and 5/6 respectively (one third-gradeteachermistakenlydid not have studentsattemptto identify turn centers and flip lines, artificially deflating the third-grade mean). Questionsabouta turncenternot locatedon the figures,abouta turncenterlocated on the figures, and about a flip line for figures separatedby six units were the most difficult. The most difficult motion to identify involved the item illustrated in Figure 24, a 180? rotationwith a turncenter not located on the figures. These difficulties diminished as the grade level increased. Studentsin Grades5 and6 were administeredtwo additionalitems.The firstitem asked them to draw an image given a preimage and a motion (see Figure 25).

74

Results

Circlethe correctmotionthat wouldmove the shaded L onto the unshaded

[L in a single motion.Also draw any turn centers and flip lines. You may use yourL transparencyto locate any turncenters. ?

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Accuracywas 76%.Turnsweremoredifficultthanthe slide or flip, anda rightturn witha turncenterlocatedat the top of the figure,illustratedbelow, was substantially moredifficultthana left turnwith a turncenterlocatedat the bottomof the figure. The seconditem assessinggeometricmotionstestedGrade5-6 students'knowledge of the relationshipsbetweenmotionsandcongruence.Accuracywas moderate

Show how the figurewouldlook if you turnedit LT90 aroundthe circledpoint. ?

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75

Logo and Geometry

(e.g., 67% to 80% for the three items administeredboth years). Most students appearedto understandthebasicrelationships(e.g., 76%accuracyon "iftwo figures are congruent,thereis a sequence offlips thatwill move one onto the other");the most difficult items were those for which studentserroneouslyagreedthat"if two figures are congruent,there is a sequence of slides and turnsthat will move one ontothe other."Thus,most studentsunderstoodthatmotionscan determinecongruence (recall that LG asked studentsto use proceduresthat performedgeometric motions to determinecongruence)and that for congruentfigures, a sequence of motions can be found to move one onto the other. However, about half of the studentsmay not have understoodthat slides and turnsare not sufficient for this purpose.This was not explicitly discussed in the curriculum. Interview Motionsvan Hiele Level Task.Students'van Hiele levels for geometricmotions were assessed with the task, "Moving the Triangle,"describedin Figure 26. To obtain a student's motions level, the following steps were involved. First, the student'sexplanationfor his or heruse of the flip, slide, andturnmotionswas coded. Next, the codes were classifiedinto van Hiele levels. Finally,the averagevan Hiele level for the threemotions was calculated.

"Youand yourfriendare playinga game on the telephone. Hereis whatyour friendsees. [The student was shown a largerversion of the diagrambelow withoutthe unshadedtriangle.]Hereis whatyou see. [Thestudentwas shown a largerversion of the followingdiagram.]You are to tell him/herexactly how to move his/her triangle[gesture to the shaded one] so that it fits onto this triangle[gestureto the unshadedone] exactly. Remember,you are talkingon the telephone, so yourfriendcannot see you or yourpicture.You can use'this trianglepiece [a cutouttriangleis provided]to help you."

Figure 26.

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Results

The motions van Hiele level for the LG group (measuredonly in 1988-89) increasedsignificantlyfrom pretestto posttest (F(1,83) = 67.22, p < .001), with a significantSite x Time interaction(F(1,83) = 8.65, p = .004), with the LG group in Site 1 increasingmore thanthe LG groupin Site 2 (LG:Site 1, .57 to 1.16; Site 2, .67 to .95). On the posttest,therewas also a significanttreatmenteffect in favor of the LG group(F(1,175) = 71.17, p < .001; LG, Site 1, 1.14, Site 2, .82; Control, Site 1, .60, Site 2,.57).1 ShapesversusMotionsLevels.We hypothesizedthatthe students'motionslevel on the preinterviewwouldbe below theirshapeslevel becausethey hadless school experiencestudyinggeometricmotions. Overall,about98%of the studentshad a motions level at or below their shapes level on the pretest,and about 80% of the studentshad a motions level at or below their shapes level at the posttest. These data supportthe notion that students' van Hiele levels are dependent on their previousexperienceswith subdomainsof geometry.Thatis, a student'svan Hiele level is not a global level for all of geometry. Anotherimportantnotioncan be derivedfrom comparingpolygon andmotions van Hiele levels. Overall,65% of the studentsincreasedtheirmotions level from pretestto posttest,29% stayed the same, and only 6% declined. In contrast,15% of the studentsincreasedtheirpolygons level frompretestto posttest,78% stayed the same, and7% declined.Takentogetherwith the observationthatthe students' motionslevels were,forthemostpart,at orbelow theirpolygonlevels on thepretest, this datasuggestsa hypothesis.Thatis, as studentsareintroducedto new geometric material,althoughtheir initial reasoningabout such materialmay be lower than their reasoningaboutgeometricmaterialthey are more familiarwith, their level of thinkingabout the new materialquickly moves up to the level of thinkingon the familiarmaterial.This hypothesis is also supportedby a detailed qualitative study thatis describedin a succeeding section of this chapter. Finally,we returnto a hypothesiswe madeearlierthatstudents'studyof motions at theend of theLogo Geometrycurriculummay havecausedthemto obtaina lower polygonvanHiele level. Forexample,referto the "Triadsorting"under"Interview" in the "ShapesandLevels of GeometricThinking"section of this chapter.In addition,thetriadchoicesandthereasonsforthechoicesof one second-grade,case-study student(shownin Table37) supportthisclaim.Thisstudent'sinteriminterviewshows herfocusingmoreon the componentsof figures,even groupingsome pairstogether using classification reasoning (althoughclearly not reasoning at Level 3). This progress is consistent with our observationsof her in the case studies, and this increased attentiongiven to components of figures continued into the posttest. as Moreover,herresponsesalso changedto reflecther use of visual transformation theprimaryreasonforgroupingtwo figurestogether,butfourof theseresponsesdealt

11Controlstudentsdid not takethe pretest;posttestLGmeans(e.g., 1.16 vs. 1.14) aredifferentbecause only LG studentsfor which there were control studentsfrom matchedclasses were included in the LG/Controlcomparisonanalysis.

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with transformingcomponentsof figures ratherthan whole figures. Indeed, her progressionof responsesfor triad4 show thatshe used geometricmotionsto try to makesenseof theclassificationof a squareas a rectangle.We suspectthatthemotions moresalientin herthinking.(See unitmadereasoningbasedon visualtransformation the Colettaepisode on "stretchysquarebathroomthings"in the final chapter.) Motions Sorting Tasks.Two sortingtasks were used to assess the prominence of the slide, flip, and turnmotion concepts in students.In the first, studentswere shown a stick-figurehorse.They were thenaskedto sortcardsdepictingslide, flip, or turnimages of the horse so thatthe cardsin each groupwere alike in some way. Then they were askedto explaintheirsorting.Studentswere said to have a perfect slide sortingif they put all of the slide congruentimages into one category;similar criteriawere used for perfect turnsortingand perfect flip sorting. Studentswere said to have a perfectmotion sortingif they had perfect slide, flip, and turnsorts. Onlythe slide categoryshoweda significantdifferencebetweenthe LG andcontrol groups (X2(1,225) = 24.73, p < .001), with 53% of LG and no control students having a perfectslide sort.The percentageof LG studentsthathadperfectmotion, slide, flip, and turnsorts on the posttest interview were: 2, 53, 3, 3 for 1988-89; and 0, 14, 6, 0 for 1987-88, respectively. In the secondsortingtask,studentswereaskedto sortthe six tiles shownin Figure 27, which were presented in random arrangements.If they sorted perfectly by motion (e.g., into the columns shown in Figure 27, slides on the left, flips in the middle, turnson the right),this was recorded.If they sortedinto othercategories, such as by shape (e.g., top row versus bottomrow in Figure27), they were asked if they could thinkof anotherway to sort. For 1988-89, 14%of the LG studentsversus 4% of the control studentssorted the tiles perfectly into slide, flip, and turn groups on the posttest (X2(1, 225) = 6.89, p < .009). Six percentof the LG studentssortedthe tiles perfectlyinto slide,

Z\-.

^--%

17V

IVV

"..^

1V

117

q qq

p

VIV

VV

V

VV

q

p b

b

b

Figure 27.

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flip, and turn groups on the posttest in 1987-88. For 1988-89, 18% of the LG and 9% of the control students sorted the slide congruentcards into one group (X2(1, 225) = 3.57, p < .06); 15%of the LG and4% of the controlstudentssorted the flip congruentcards into one group (X2(1, 225) = 8.69, p < .004); and 21% of the LG and 10%of the control students sorted the turncongruentcards as a group (X2(1, 225) = 5.23,p < .03). The percentswere 6%, 7%, and 12%,respectively, in 1987-88. When the studentswere asked to sortthe cardsa second time, the LG group had a slight increase of perfect sorts over the control group (but not significantlyso). On average,the numberof LG studentsmakingperfectsorts the second time was about 10%more than the numberof LG students who did so on the first sort. Overall, the sorting tasks indicated that geometric motions were more salient for the LG students. ClassroomObservations Across all grades, kindergartenthroughsixth, students showed every sign of understandingthe slide, flip, and turn concepts, terminology, and computer commandswithin the context of the curriculum.They became proficientat identifying which motion was called for to move one figure onto the other;further, they seemed to comprehend the relationship between these motions and the determinationof congruence. They even were able to see the inverses of these motions without prior instruction (recall that the curriculum emphasized "undoing,"as when studentsreversedthe commandsto bringthe turtleback along a path). For example, students' competence was displayed when a curriculumcoordinatorfor one school district,who was previouslya mathematicsteacher,observed a second-gradeclass. He observed a problemtwo studentswere contemplating, and suggested that they would have to turn the object and then slide it. The studentslooked at him in puzzlement and one remarked,"Butone flip would do it and that would be faster!" He beat a hasty retreatand thereuponmentionedto the project directorsthat these second graderswere more competent in motion geometry than a class of eighth gradershad been after he had taughtthem a unit on the topic several years before. PROBLEMSOLVINGAND SENSE MAKING: CLASSROOMOBSERVATIONS The maingoals of LG areto develop not only students'geometricsense making but also theirmathematicalproblem-solvingskills. To promotebothgoals, instructionaltaskswerepresentedas problem-solvingactivities,manyof whicharosefrom careful observationand analysis of students'work duringthe field testing of the curriculum.Thatis, duringourfield testing,we foundthatostensiblysimple tasks often presentedrich andchallengingproblem-solvingopportunitiesto students.In this section,we will describeseveraltasksandepisodes thatillustratethe problem-

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solving and sense-makingnatureof the LG curriculum.In these episodes, taken from our in-depthcase studies, one or the other of us (Battistaor Clements) was in the classroom every day as the curriculumwas taught (and is denoted as the "Observer"in the dialogue).We were not thereto teach,only to observe.Ourquestions were aimed at probingstudents'thinking. "Tilted"Squares We will considerfirstthe seeminglysimpletaskof gettingthe Logo turtleto draw a shape in a nonstandardorientation.In one second-gradesession, only two of the seven students reasoned that the procedurethey had developed for drawing a squarewith verticaland horizontalsides could, with the additionof a single turn command,be made to draw a "tilted"square.The other five studentscompleted the tilted squarevisually-choosing a turn,tryingit, visually checking to see if it was correct,and revising their commandsif necessary, often doing this for each of the square'svertices. As we observed this and similarepisodes, we arrivedat two importantconclusions. First,many studentshad abstractedfar less aboutthe propertiesof squares thanwe had first assumed-they clearlyhad not abstractedthatsquareshave four 90? angles. Second, making a tilted squareafter having made a standardsquare was a perfectproblem-solvingtask for these studentsat this time. Indeed,clearly these students still had far to go in their conceptualizationof the propertiesof squares, so working on this problem not only involved the studentsin problem solving, it encouragedthemto makeprogressin theirconceptualizationof squares. Moreover, the task of making a tilted square was intrinsically interesting to students-some even posed it for themselves. And, as we observed students working on this task, we judged that the conceptualizationsneeded to solve this task were, with appropriatetime and guidance,within the students'currentcapabilities. In fact, several weeks after the initial episode, two of the studentswho initiallyapproachedthe problemvisually said thatthe turnsin a tiltedsquarewould be 90. TurtleDeliveries StudentTask:Given the on-screen map shown in Figure 28, write a procedure to teach the turtleto deliver a load of pies to the restaurantRl andto returnhome. The pathhome from the restaurantmust be the same as the pathto the restaurant. Do the same for restaurantsR2 and R3. Studentswere workingin a largegroupwith the teacherwho was using an overhead monitorto display the computerscreen. As a group,the studentshad given commandsto move the turtleto R1. These commandswere enteredinto the Logo editor as follows: TO GO.R1 LT 130 FD 40

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Figure 28.

RT 25 FD 75 RT 90 FD 30 LT 90 FD 10 END This set of commands was run so that all the students could see the result. Becausetheirpathto R1 was incorrect,the studentsdiscussedwhatLogo commands shouldbe changed.The revisedcommandset is shownbelow, with changesin bold type. TO GO.R1 LT 130 FD 40 RT 40 FD 85 RT 90 FD 30 LT 90 FD 10 END After running the revised procedure,the students were satisfied that it was correct.The next partof the task was to get the turtleto returnto its startingposition along the same path. Teacher: Andy:

How are we going to get the turtleback? Everythingthat was a forwardis a backward,and everythingthat is a right is a left, and everythingthatis a left is a right,probably.

Logo and Geometry

Teacher:

So, you'retellingme what?

Steven:

Reverse the stuff.

81

Youreverse,go fromthebottom[of theprocedure] to thetop. Whatcomesfirst? BK 10,RT90,... [So the studentsare correctlygiving the commandsto undo the path.] We'vegot a problemhere,becausehere'stherestaurant. It [theturtle] Andy: goes outof therestaurant [using]BK 10.Then[going]rightwouldbe uphere,so it's goingto get screwedup.

Andy: Teacher: Steven:

Anotherstudent: Now it's going backwards.

The studentsdiscuss Andy's misgiving aboutthe proposedsolution. Andy figures out his mistake as he standsup and acts out the commands;he sees thatyou need a rightturnbecause, in the next command,the turtleis going backwards(he envisioned the turtlegoing forward).The commandsare then enteredinto the computer,and the students see that their proposed solution is correct. Throughoutthis episode, the teacherenteredthe students'commandsinto the computerand moderatedthe discussion. She did not judge the correctnessof the students'ideas; instead,the studentsjudged correctnessby examiningthe results on the computer screen. Judging the validity of one's proposed solutions (as opposed to asking the teacherif the answeris correct)is an essential component of a genuine problem-solvingactivity. The teacherthen sends the studentsto the computersto continue the activity in pairs. Jill and Peter, workingtogetherat a single computer,immediately attemptto get the turtle to the second restaurant,R2. They use the turtle turner and ruler to make measurementson theiractivitysheetandenterthe commandsinto the editor.When they runtheirprocedure,they find that it does not get the turtleto R2. Jill triesto fix theirbuggedprocedureby analyzingthe path it makes and conjecturingabout where their path is erred, ("We need to go 20 more right there.")Peter, in contrast,wants to give commands to the turtle in "immediate mode"so the effects of eachcommandcanbe seen as it is entered. Apparently,he has decided that their originalprocedurehas so many mistakes that starting over will be more efficient. Jill accedes to Peter's strategy, so the students give successive commands to the turtle in immediate mode, recording their commandson paper. The next day, as studentscontinuetheir work on the path for Restaurant2, Jill and Peter entertheirrecordedcommandsinto the Logo editor,runthe resultingprocedure,andobservethatthe path is not quite correct.Interestingly,Jill and Peter express no surpriseor disappointmentwhen theirproceduredoes not work.

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They anticipatedthattheirpathwouldnot be perfectandthatthey wouldhaveto fix it. Jill andPeterthenidentifythe partof thepath that needs to be fixed, carefully locate the corresponding commandin the path's procedure,then change the command. When they try the new path,they are satisfiedthatthe corrected path works. In this episode, Jill and Peterexhibit a numberof processes and characteristics importantfor successful mathematicalproblemsolving. They show confidencein theirabilityto solve the problem,persistencein the face of obstacles,expectations thatinitial attemptsat solving a problemmight fail but that such attemptscan be productivelyaltered(the majorityof studentsin a standardcurriculumsimplygive up if theirinitial attemptat a solutionis incorrect),anduse of metacognitiveskills in decidingwhatapproachto takein fixing the error.Theyeffectivelyuse the importantproblem-solvingsequenceof tryingsomething,reflectingon how it works,then making appropriaterevisions. Finally, because they are in an instructionalenvironmentthatencouragesit, Jill and Peterexhibit intellectualautonomyin solving this problem. Immediatelyupon seeing that their solution is correct,Jill says, "Let'sget himback."Both studentsareexcitedabouttacklingthis new problem;it is not a typicalschool taskthatthey attemptwith littleenthusiasm.As they starton the problem,initially,Jill thinks thatthefirstcommandin gettingtheturtlebackto its startingpoint is thereverseof the firstcommandin theprocedurefor gettingthe turtleto R2. But Petersays thatit must be the reverseof the last command. Jill Observer: Whydoesit haveto be this [thelast]command?[Simultaneously, all of whichseemto havecorrect andPeterofferseveralexplanations, elementsbutareincoherent. Finally,Jillexplains.] We startedfromhere[turtle'sstartingpoint]andendedhere[drawsin Jill: theaira paththatis notstraightbutnota copyof thescreenpatheither (almosta curvedversion)].So we wantto startherethistime[indicates in origR2]so we haveto startwiththelastone[pointsto lastcommand inalprocedure]. Even thoughthe idea of undoinga pathhadbeen discussedthe previousday, Jill andPeterneededto tryit themselvesto makethe idea personallymeaningful.After they had tried it and thoughtaboutit for themselves, both seemed to make sense of the previousday's explanationof reversalandwere able to applyit. Jill's explanationindicatesvery clearlythatshe possessedan abstractconceptualizationof the process of undoinga path.She was able to indicate,by drawingin the air,thatshe possessed a visual abstractimage of a path and could operateon that image; she did not needto go backto the particularpaththatthey weredrawing.Jill was clearly operatingon an abstractionof a path, not only because she imaginedoperations on the path, but also because she loosely drew her "air"path without trying to preservethe exact shapeof the screenpath.(A fifth graderin a differentclass also

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evidenced a clear understandingof the process of undoing.He explainedit to his partneras follows: "Likeif someonewas going to go downthe road,andtheypulled out of your driveway to get somewhere and they end up at a dead end, they wouldn'tjust go forward,even thoughthe first thing they did was go back out of your driveway.") This episodeinvolvingJill andPeterillustratesthe synergisticroles thatstudents' individualandlarge-groupworkplayedin theirgeometricknowledgeconstruction in LG. Evidently,class discussions often offer studentsincompleteblueprintsfor knowledge construction;the actualbuildingof the knowledge structuresdoes not occuruntil individualstudentshave sufficientopportunitiesto engage in personal sense making. LG encourages and supports student sense making as students personally manipulateideas and decide on their validity and exact form. This episode also illustratesthe opportunitythatLG providesfor studentsto form and operateon mathematicalabstractions. Jill and Peter now enter their commandsfor a procedureto get the turtlefrom R2 back to the startingpoint. They try it, find it incorrect,butdo not seem to be able to locate the error.However, even thoughthey are stuckon the problemof fixing theirprocedure, they persist in theirproblem-solvingefforts. Finally, they carefully check their procedurein the editor to see if they have properly reversed all the commands. They find and correct a mistake in a turn command,try the revised procedure,and are excited to find thatit works. As a final challengein "TurtleDeliveries,"studentsaregiven a procedurewhich takes the turtleto OILERS,an "out of town" restaurant;that is, to a location off the screen and out of sight. The challenge:Bring it back. Jonathan: HowcanyougetbackfromOILERS? Is therea map? Kevin:

No.

Mary: Jonathan: Kevin:

Youjustreversethem! Oh,yeah!Withoutevenseeing. That'sweird!Itreallywill work!

Kevin's last commentindicatedhis amazementthata logically-derivedsolution, divorcedfrom visual affirmation,can accomplisha task. This task, in additionto encouragingstudentsto raise their initial "TurtleDeliveries" conceptualizations to a higherlevel of abstraction,also helps studentsappreciatethe powerof abstract mathematicalmodels of phenomena. In sum, what sense-making and problem-solving processes were students involved with in the "TurtleDeliveries" episodes? First, the "TurtleDeliveries" activity representeda significantchallenge for students;it is not a task thatcan be completedin 3 or 4 minutes, as are most school mathematicstasks. Studentshad to develop sufficiently robustmental models of the situationsto contemplatethe requiredturtlepaths. They had to apply theirconcepts of lengths and turns.They hadto maintainthe propercorrespondencebetweensymbols(turtlecommands)and

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geometricentities. They had to test possible problemsolutions, conjectureabout mistakes,and decide on the best strategyfor proceeding(an importantmetacognitive activity). Moreover, both the computer environmentand the classroom culturepromotedand supportedintellectualautonomyin students'mathematical sense makingandproblemsolving. This is in sharpcontrastto traditionalinstruction (Stigler& Hiebert,1999), in which studentssimply attemptto determinewhat the teacherwould like them to say (i.e., using the teacheras the source of mathematicalknowledge ratherthantheirown thought). MakingEquilateralTrianglesand RegularPolygons Student Task: Complete the procedurebelow so that it draws an equilateral triangle. TO EQTRI FD 50 RT FD 50 RT FD 50 RT END At theircomputer,Jill andPeterfirsttry 90 for the turns,then45. They conclude that45 is too small. Peter says thatthe turtlehas to turnfirst to make an equilateraltriangle.When Observerasks whathappenedwhen they tried45, they say it is too big (notethat originallythey thoughtit was too small.) Jill says, "If that's 45 (she motions with her handturningfrom the vertical)and that's 90 (motions), then it would have to be down more (motions about 135)." (Note how Jill is visualizing the turns.She clearly understandsthe turningmotionandhasbecomequiteadeptat estimating turnmeasure.) After they try 135, Peter says the turtle needs to turnmore;Jill less. Jill says she is confused. Peterthen wants to turnthe triangle,so he insertsan initial turncommand into theirprocedure.Jill says thatan initial turnis unnecessary, "Becauseit's a straightline and you turn,it's not going to make any difference."Peterthen claims thatthe procedureneeds first a 45-degreeturn(to changethe orientation),then 145-degreeturns afterthat(to formthe vertices).Jill countersthatall the turnshave to be the same. At this point in the problem-solvingactivity, Jill and Peter are dealing with a numberof conceptualconflicts.Peterthinksthatthe equilateraltrianglecannothave a verticalfirst side. Jill correctlybelieves thatthe first turnis irrelevant.However, Jill gets confused when Petermakes the first turn45 and subsequentturns 145she is thinkingaboutthe requirementthatall the turnsin an equilateraltriangleare

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the same. But she has not made complete sense of the requirement;her understandingof it is easily unraveled.Jill is also tryingto make sense of Peter'sideasthe studentsareproductivelyinteracting.Finally,bothJill andPeterarehavingdifficulty deciding whetheran amountof turnis too much or too little. When Jill and Petertry 145, they conclude thatit is too much of a turn.Also, as a result of seeing theirproceduredrawnon the screen, Peterchanges his mind aboutthe relevanceof the initial turn, saying, "It might just be upside down." [Because Logo allows studentsto test the validityof theirideas, Peterfinally has recognizedhis mistakes.]Jill andPeterthentry 115 andthinkthat it works.But when the observerasks themto hide the turtle,Peter thinks 115 is too much. After some furtherexperimentationin which they discover the relationshipbetween the turninputand the size of the angle the turn creates in the path, Jill and Peter finally arriveat 120?turns,exclaiming simultaneously,"Yeah." Observer: So, whatwasthetotalamountof turning? Peter: 360 degrees.Ohhh[veryexcitedly].We shouldhavejust taken[Jill shrieks,too];we shouldhavedivided3 by 360 [meaning360by 3], and we wouldhavegotten120. Observer:

How come?

Peter: Observer: Peter: Jill:

Youhaveto turn120threetimesto get 360. What'sso specialabout360? It's a fullcircle.Ohhh[groans]. Wewereso stupid.

These two studentswere trulyexcited abouttheirdiscovery.But notice thatthey did not come upon this discovery by logical deduction,nor is it certainthat the observer'squestion,whichis on the students'activitysheet,wouldhaveevokedthis sameflashof insightbeforetheyhadplayedwiththeideasnumerically.Theyneeded to manipulatethe ideas in theirown way, to workout theirconceptualconflicts. Also note thatJill and Peter's problem-solvingefforts were initially hampered by theirconceptualconfusionaboutwhetherthe amountstheyturnedthe turtlewere too much or too little. It was not until they developed an appropriateconceptualization of this geometricconcept thatthey could succeed in theirproblem-solving efforts.Thus,this activitynot only involvedstudentsin significantproblemsolving, butit also helpedthe studentscorrectandsharpenthisimportantgeometricconcept. Indeed, it is highly likely that this experience helped Jill and Peterbetterunderstandthe difficult (for this age student)relationshipbetween the two concepts of turnalong a path and the angle formedby the path. The confusioninitiallyexperiencedby Jill andPetermightbe disturbingto many teachers. But it is actually an importantpart of mathematicallearning, sense making, and problemsolving. Jill and Peter struggledto make sense of what for them were very difficult ideas. It was only throughtheirexperimentationwith the computerandtheirdiscussions with each otherthatthey are able to make sense of

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the ideas for themselves. Thus, overall, this episode is a good illustrationof the constructiveand problem-solvingnatureof the LG lessons, of how mathematical inquiryanddiscoverycan be extremelyexciting for students,andof how students can find and correcttheirown mistakesin the properenvironment. Generalization of the Angle Sum Notion

Even though Jill and Peter made an importantdiscovery-that the turnfor an equilateraltriangle is 360/3-we must still question whether they sufficiently abstractedandgeneralizedthe idea so thatit can be appliedin otherproblems.We get a chance to investigatethis question two days later as the class is attempting to write proceduresfor otherregularpolygons. The studentsare at the computersin pairs. They are to write a procedure,first, for a square,then for a regularpentagon,and so on, using the REPEATcommand.Jill and Peter startwith the regularpentagon. We gottafigurehow muchtherightshaveto be.... Thesquareis 90. Peter: [Theyusea turtleturnerto measuretheturnsfora regularpentagonon theiractivitysheet.]It'salwaysthesameangle.[Jillmeasuresandgets butit doesn'twork.]Well. 55. So, theytryRT55 in theirprocedure, Jill andPeteruse theirturtleturnersto measure,respectively,the screenandthe activitysheet picturesof regularpentagons.[Note again how Jill and Peter persist in their problem-solving endeavors.Peter'sresponse("Well")indicatesthathe is puzzled. But they proceedundaunted.] Whatwaswrongwith55? Observer: I didit likethisandit was at a slant.[Shedidnotturntheturtleturner Jill: to thecorrectinitialheading.] Peter: [Afterfirstsayingit's about63] Wait,I messedup.Almost90.... 85. andviewingtheresult]That'stoo [Aftertryingthisturnonthecomputer little.[Itwasreallytoomuch.] Is thattoo muchortoo little? Observer: Jill: Because,if it was90, it wouldmakea square[tracingthefigureon the screenwithherfinger].I thinkit's toomuch. Jill's reasoning was quite sophisticated.If 85 is too little, then 90 would be moving in the correctdirection.But 90 would give a square.So, 90, andtherefore 85, is too much. (Note that, as with the equilateraltriangle,Jill and Peterare still havingdifficultydecidingwhethera given amountof turnis too muchor too little.) Let'stry80, do youwantto? Jill: Yeah.Maybewe needto,theturnsareallwrong.[Petermumblesin the Peter: background-heseemspuzzled.]If this doesn'twork,it's probably beentoo small. Jill: [Theytry80.]It'scloser. Peter:

Yeah.

Jill:

So, it wouldbe toomuch.

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Peter:

87

[As they go backto the editor]Wait.Divide 5 by 360. You'll get the right answer. [Petermeans divide 360 by 5, a common mistatementfor this age level student.][Afterthey do the division] Same as we forgot to do last time [when they did equilateraltriangles]. [After computing the quotientof 72, enteringit into their procedure,and runningthe procedure.]It worked.

Jill and Peter immediately go on to the hexagon problem. Six divided by 360. [Of course she means 360/6.] Sixty. [Pause.]Oh, so, it goes down every time. Here we go. It's going to work. [When they run their procedure for the regular hexagon, it works correctly].

Jill: Peter: Jill: Peter:

At this time, the classroom teacher, who has been circulating around the classroom, interacting with other pairs of students, arrives to check on Jill and Peter. Whatdid you figure out? Teacher: You have to divide the numberof sides by 360, I mean into 360. [Jill Peter: was explainingalmost simultaneously.] Teacher: Why? Peter: Because there's going to be three angles, and you need to turn 360 degrees, and thatwill tell you how much to turn. Not threeangles. Jill: Peter: Well, I mean how many angles are in the figure? Teacher: Why 360? Peter: Because that's how many degrees it takes to go arounda circle. [As he

saysthis,Jillmovesherfingerin a completecircleon thescreen.]

Jill and Peter now go back into the editor and write a procedure for a regular octagon. They calculate that 45 will be the correct turn. It works, but seeing it drawn on the screen seems anticlimactic to them. Even though Jill and Peter had previously discovered that 360 should be divided by the number of turns for equilateral triangles, they did not generalize this idea beyond that specific situation. They needed, and were provided with, opportuni-

ties forfurtherinvestigation-a chanceto furtherabstractandgeneralizetheirearlier discovery. Also, their first approachto the problem was again to play with the numbers(a good strategywhen facing a new problemthat we certainlypromote in the curriculum).However,Peterseemed to be botheredat one point because he thoughtthey shouldbe able to figure it out ratherthanuse the guess-and-testtechnique. Thatis why he was mumblingto himself. In addition,Jill andPeterwere certainlyreflectingon theiractivity,as indicated, for instance,by Peter's comment,"Oh,so, it goes down every time."More important, Jill and Peter seemed to be moving to a higher level of reasoning.They had not only discoveredthatthe rule of 360 could be used to figure out the amountof turn, but they were also convinced that their method made logical sense. For instance, testing their answer for a regular octagon seemed perfunctory to them;

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the logical structurehad convinced them that their answerhad to work. This is a major accomplishmentfor studentsof this age. In fact, even most high school studentsare "naiveempiricists"andbelieve thatanomaliescan occur even aftera theoremhas been proved. Laterthe teacherhas all the studentsreportanddiscusstheirindividual findings at the computer. Whatdo youthinktheturnwill be fora decagon? Teacher: Many students:

36

Tellus, Peter,aboutyourdiscovery. Youhaveto dividethenumberof sidesof thatfigureby [Hesitatingly] 360 andyou'llget thenumberto turn. Peterwas still not very articulateabouthis discovery,but he clearlyunderstood it. Indeed, it was common for studentsto discover ideas that they had difficulty puttinginto words.Giving studentsthe opportunityto verbalizetheirideasin classroom discussions is an excellent way for them to clarify and consolidate their thinkingandis also consistentwithNCTM'srecommendationsto supportstudents' ability to communicatemathematics. Teacher: Whydo youthinkPetergot 360? It'slikea wholeturn. Andy: Whatdo youmeana "wholeturn"? Teacher: Liketheturtlealwaysendsupin thesameplace.So, dividehowmany Andy: turnsintowhereit standsup. Teacher: Whydo youthinkthatworkswiththeturtlemakinga wholeturn? There's360 degreesin a circle. Cary: Teacher: Peter:

Several days later,Jill and Peter are doing the "Sumof the Vertex Angles in a Polygon"activity, as is describedbelow. StudentTask: Each proceduredrawsa polygon. For every polygon, figure out and recordthe measuresof all of its vertex angles. You can check you answerfor the first triangleby typing Ti, turningthe amountthatyou answered,then giving the FD 60 command.If your answer was correct,the turtle should be at the top vertex of the triangle.Try to discover a relationshipbetween the numberof sides of a polygon andthe sum of its vertex angles. HINT: Thinkaboutthe TotalTurtle TurnTheorem. SUM OF VERTEX NUMBER MEASURESOF OF SIDES VERTEXANGLES ANGLES POLYGON EXAMPLE: TO T1 FD 60 RT 90 ...................................... FD 80 RT 143 .....................................

3 90? 37

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Logo and Geometry

FD 100 RT 127 .....................................

53?

END Total TurtleTurning 360? TO T2

FD 71 LT 135 ....................................

FD 50 LT 90 .....................................

FD 50 LT

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

END Total TurtleTurning As Jill and Peter startto measureturns,the observerasks them what their plan is. The studentsexplain that they are going to measure the turns to find out what to use in their procedure. ThenJill says to the observer,"Isthereanotherway?"-to which he replies that they can figure it out without using their turtle turners.Jill then says, "Use 360." Thus,using logical reasoning,as opposedto empiricalmethods,was well within Jill's grasp.However, as in the previousactivities, therewas a naturalinclination for studentsto startwithan empiricalapproach.Jill wouldnothavethoughtof trying to use logical reasoninginitially without the observer'sintervention;it was as if his questionaboutthe solutionstrategywas a metacognitivepromptfor herto think about and evaluate possible strategies before implementing them. Schoenfeld (1992) talks about similar needs for studentsin college! By regularlyrepeating promptssuch as these, teacherscan help studentsdevelop importantmetacognitive problem-solvingskills. But it shouldbe emphasizedthatsuchpromptswill not be useful to studentsif they arenot given the opportunityoperateat theirown level. That is, students must be cognitively "ready"to make personal sense of such prompts.

Chapter 4

Discussion The Logo Geometry Project attempted to ameliorate problems in geometry teaching and learning by developing a research-based Grades K-6 geometry curriculum. This chapter discusses the results of evaluating this curriculum with 1,624 students and their teachers and, more widely, discusses the results of investigating how elementary school students learn geometric concepts and what role Logo programming in turtle graphics played in students' learning of geometry and spatial sense. TOTAL ACHIEVEMENT Across Grades K-6, Logo Geometry students scored significantly higher than control students, making about double the gains of the control groups on the prepost paper-and-pencil assessment (e.g., the K-6 Total Scores). These findings are significant for two reasons. First, in designing these assessments, we took the conservative stance of testing in a paper-and-pencil mode-the format in which the control students did their geometry work. That is, LG students had to transfer any knowledge developed in (and thus potentially linked to) the computer context to paper-and-pencil items. We also avoided paper-and-pencil items connected directly to the LG curriculum. Second, the curriculum is a relatively short intervention, lasting only six weeks. We expected that students who received rich geometry instruction for a greater part of the year, and more significantly, for several consecutive years, would perform even better. Examination of analysis of variance interactions did not provide convincing evidence that students whose teachers had two years of experience teaching the LG curriculum performed differently on paper-and-pencil tests than students whose teacher had one year of experience. Similarly, LG students in Grades 4-6 outperformed control students on the composite score, which included all items administered to students in these grade levels (Table 4, 4-6 Total Scores). A conjecture emerges in comparing these two composite scores. Recall that the K-6 Total composite score included only items that were administered to all students, whereas the 4-6 Total score also included these items plus the more difficult items only administered to students at higher grade levels. Although LG students in Grades 4-6 appear to perform moderately better on the former composite, they outperform the control group at every grade and site on the latter (see Table 3). In the extreme example, Site 2 Grade 4 LG students did not make significant gains and performed significantly lower than the Site 1 control group on the former score but made significant gains and outper-

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formedboth controlgroupson the latterscore. Thus, intermediate-gradestudents may have benefitted more from the LG curriculumin areas tappedby the more demanding,higherlevel items includedonly on the 4-6 Total test. SHAPES AND LEVELSOF GEOMETRICTHINKING Three items, adopted from previous van Hiele-based research, assessed the identificationof triangles,rectangles,and squareswithina large set of distractors. LG studentsperformedbetteroverall, with gains attributableto the triangleand rectangleitems (of all the groups, only the kindergartenLG group made significant gains on the squareitems). The particularitems on which LG studentsperformedbetterthan the controls were investigated. LG students, especially younger students, gained more than controls-not in theirability to identify figures thatare triangles-in theirability to correctlyidentifynontrianglesthatsharevisualcharacteristics withtriangles(e.g., a chevron).This was especially salientfor the concave quadrilateralandfor figures that appearedto be trianglesbut had curved sides or were not closed. Patternsof responsesfor these figures show a movementfromunsophisticated,holistic visual responses to responses that were based on a more accurateconsiderationof the components and propertiesof the figures. LG may enhance students' triangle scheme with imagistic constraints,based on turtlemovements with appropriate movement-basedmentalmodels. For example, imagistic constraintsfor triangles may include three forwardmovementsthat create a sequence of threeconnected straightpathsthatform a figure thatends where it starts(andthus is closed). Such constraintsmay promptcorrespondingreflectionson the componentsandproperties of triangles;for example, that three line segments form a closed figure. The lack of differences between the groups for identificationof triangles,even those in nonstandardorientations,argues that the positive effects of LG may not lie in the developmentof basic visual prototypesfor triangles.Because differenceswere noted on the ability to correctlyidentify nontrianglesthatsharedvisual characteristics with triangles,the positive effects of LG were moreprobablyin the enhancement of conceptualknowledge. Thatis, as in Reif's (1987) model of interpreting concepts, visual prototypesareused, but in unfamiliarsituationsor when possible inconsistenciesarise, they arechecked with more formal,rule-based,knowledge. This lattertype of rule-based,conceptualknowledge may be what studentsin LG classroomsdeveloped. This is consistentwith othereffects. Forexample,the smallereffects for squares would be expected in that the class of triangleshas membersthat are less easily assimilatedto a small numberof visual prototypesthandoes the class of squares. Forrectangles,largegrowthwas displayedfor LG students,comparedto control students,for the two squares.For both of these shapes, the most strikinggrowth occurredin Grades4, 5, and6. This correspondsto theLGinstructionat those grade levels, which attemptedto help studentsunderstand(via properties)the hierarchical classificationconceptof squaresas specialcases of rectangles.Similarly,moderate

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effects on nonrectangularparallelograms,especially those in nonstandardorientation,indicatedthatLGinstructionencouragedchildrento evaluateshapesby properties ratherthanby visual appearances. Unit tests for LG students (Grades K-4) confirmed results for the triangle, rectangle,and squareitems and suggestedthatonly at Grade4 did studentsbegin identifyingsquaresas rectanglesspontaneously.Further,only studentsin Grades 5 and 6 performedwell on items involving regular and nonregularpolygons, althoughthis may havebeen due to a lack of emphasison namingandworkingwith such figures in the lower grades. LG studentsimprovedmore than control studentson attributingstatementsof geometric propertiesto the classes of squaresand rectangles.For the rectangle concept, effects tendedto be strongestfor studentsin Grades3-5 and weakestfor Grade 2. It may be that students younger than Grade 3 have more difficulty abstractingcertainpropertiesfrom their work with LG. Over the grades,LG had the strongesteffect, comparedto the controlinstruction,on the propertiesdealing with pathsandwith angle measure.LG studentsoutperformedcontrolstudentson the property"oppositesides equal";however,this differencewas largefor squares but small for rectangles.Thus, the differencesbetween the groupsdo not relateto familiaritywith the property,per se, but to understandingthis property'sconsistency with the propertyof "allsides equalin length."LG studentscould applyboth propertiesto the class of squares.This abilitydemonstratesflexible consideration of multiplepropertiesthatmay lay groundworkfor laterhierarchicalclassification. By synthesizingthe resultswithregardto hierarchicalclassification,we conclude thatimportantprecursorsforhierarchicalclassificationaredevelopedby LGinstruction.Still, studentshavedifficultyintegratingthesepreliminaryideasintoa propertybasedhierarchicalclassificationsystem.In otherwords,most studentsarelearning to thinkaboutthese shapesat Level 2 but arenot sufficientlyknowledgeableabout the propertiesof these shapesto constructviable Level 3 systems of relationships. Why, then, did the first rectangleitem seem to indicatethatLG instructionhelped substantialnumbersof studentsidentifyfiguresin a mannerconsistentwiththehierarchicalrelationship?WhenLG studentssucceededin identifyingsquaresas rectangles on the rectanglesitem, they could have done so by asking themselves if a "Rectangle"procedurecould have drawneach of the given shapes.These students may have been able to use their experience with Logo proceduresto overcome of thesefigureswhile stillnotbeingable previouslyheldexclusiveconceptualizations to deducerelationshipsfrommoreformallystatedproperties. Observationrevealed that the Logo microworlds proved to be evocative in generatingthinkingaboutsquaresandrectanglesfor young students.Some kindergartnersformedtheirown conceptin responseto theirworkwith the microworlds. For example, the "squarerectangle" phrase, which was generated by several studentsbesides Chris,was appropriatedby the teacherof thatclass. This concept was appliedonly in certainsituations.Squareswere still squares,and rectangles, rectangles,unless you formed a squarewhile working with procedures-on the computeror in drawing-that were designed to producerectangles.The concept

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was stronglyvisual in nature,and no logical classification,per se, such as classinclusion processes, should be inferred.The creation,application,and discussion of the concept, however, was arguablya valuable intellectualexercise. Even for students such as Robbie, who rejected such naming, the disequilibriumthat the microworldengenderedwas a valuablestimulusfor reflection. Observationsof fifth graders (Jonathanand Jeremy) are consistent with the conjecturethatstudentsheld multipleconceptions,some of which wereevoked and appliedonly in certainsituations.Jonathan'sreasoningsometimes was similarto thatof kindergartnerswho thoughtthat squaresand rectanglesusually were separateunless the formerwas producedby a proceduredesignedto drawthe latter.At othertimes,Jonathanwouldrelatetheseclasses of shapes,butonly whenprompted by a teacherto use property-basedreasoning.This indicatesthatJonathanhad not achievedLevel 3. However,the Logo environmenthelpedhim andotherclassmates generatealternativeconceptionsthatmay conflict with ways of thinkingthatwere previouslyestablished.Thesealternativeconceptionsmayeventuallyserveas a catalyst for the developmentof higherlevels of geometricthinking.The creationof such concepts supportsthe hypothesisthatcertainLogo experiencesmay lead to situation-specific concepts and behaviorsthatare consistentwith hierarchicalclassification,withoutbeingbuilton property-based relationshipsas describedby vanHiele. In contrast,fifth-graderJeremyconstructedideas muchcloser to establishedmathematicalrelationships.He gave justificationsthat were based on the propertiesof squaresand rectangles.Further,he applied similarreasoningto other classes of werenot studied includingthosewhosehierarchicalinterrelationships quadrilaterals, to Grade by the class. In sum,it appearsthatLG canhelp studentsfromkindergarten 5 visualize, reflect on, and discuss geometricfigures and relationships.Although most studentswill begin to learn aboutproperties,even achieving strongLevel 2 thinking,othersmay even constructhierarchicalrelationshipsthatindicateLevel 3 geometricthinking.Slow movementto suchthinkingis reasonable,consideringthat partitioningshapesinto separateclasses is not only the societalnormbutis as logically consistentas hierarchicalclassification(de Villiers, 1994). Thus, resistance to hierarchicalclassificationdoes not necessarilyalign with the logic of the inclusionbutoftenalignswiththemeaningof the activity,bothlinguisticandfunctionallinguisticin the sense of correctlyinterpretingthelanguageusedfor class inclusions andfunctional in the sense of understandingwhy it is more useful thana partition classification(de Villiers, 1987, 1994). The triadtaskof theinterview,whichaskedstudentsto choose two of threeshapes that were most alike, was also designed to elicit students' level of geometric thinking.Acrossthe nine items,thepatternwas forLG studentsto move fromvisual to propertyor classificatory responses. However, the predominanceof visually basedreasoningat all administrationsimplies thatthis movementwas in the beginning stages. The movementis consistentwith whatwe predictedin ourtheoretical analysis.For severalproblems,therewas also an apparentshiftfromgiving (visual) componentsas reasons to using visual/spatialtransformationsto justify answers (cf. Lehrer, Jenkins, & Osana, 1998). As we argued earlier, the motions unit,

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which studentsstudiedbetween the interimandposttestinterview,made students more aware of how one shape could be moved or visually changed to look like another.Although students'choices of figures changed little from the pretestto posttest, the reasons for those choices changed, sometimes dramatically.And as was expected,these shiftsin level of thinkingoccurredin olderstudentsmoreoften thanin youngerstudents. Even though noteworthyprogresswas made on many of the triaditems, there were other items for which visually salient similaritiesattractedthe majorityof students.The advantageof viewing figuresin termsof propertiesor classifications shouldbe mademoreevidentto studentsto encouragethemto abandontheirvisual focus. Indeed,perhapsin many mathematicalsituations,studentsdid thinkof the figure in termsof properties.However, they probablysaw no largeradvantageto employing a logical-mathematicalapproachto the triadtask. So, they were easily swayed to choose the pairsthat matchedbest visually. Finally, it must be kept in mindthatourtriadclassificationsystem,becauseit emphasizespropertiesandclassificationof staticfigures, is biased againstvisual transformationsas evidence of sophisticatedgeometricthought.However,thinkingaboutfiguresin termsof such transformationsis sometimes at the height of geometricinsight. In summary,it is importantto note that, using a first draft of the LG curriculum,the triad data suggestthatthereis a progressionin levels of thinking,thattakesconsiderabletime, as measuredby a far transfertask. Also, ourfindingsindicatethatstudentsimprovedon a taskthatrequiredcomplex However,as measuredby the triad reasoningaboutthe definitionsof quadrilaterals. task, students'performancedecreasedfrom interimto the posttest,consistentwith the decrease in van Hiele levels. Thus, studentsmade progressmoving to more formalthinkingaboutshapesduringthe shapesmodulebutregressedslightlyafterwards.Morethanlikely, these findingsaredependenton the van Hiele level of the students.Studentsin Grades4-6 were generallyin transitionfromLevel 1 (visual) to Level 2 (properties).Therefore,they were likely to abandonthe visual condition of "two long sides and two shortsides"for rectanglesand requireinsteadthatthe clasfigureshaverightangles.Still, studentshaddifficultywiththe square/rectangle sification problem, and they grappledwith using the type of logical deduction requiredto deal with the variousaspectsof this complex item effectively. Classroomobservationsindicatethat,within the context of the LG curriculum, studentslearnedto analyzethe visualaspectsof these figuresandhow theircomponents are put together,thus facilitatinga transitionfrom the visual to the descriptive-analyticlevel of geometricthinking.The Logo environmentandtasksprovided studentswith opportunitiesto analyzeandreflect on the propertiesof two-dimensional shapes.Also, the need to buildLogo proceduresallowed studentsto develop understandable,implicit definitionsof these shapes. Further,the computerenvironment provided students affirmationthat the conclusions that they reached throughlogical reasoningwere indeedcorrect,whichhelpedthembuildconfidence in this nascent mode of thought. Thus, the computerexperimentationafforded studentsthe opportunityto gain confidence. In some cases, such as Jonathan's

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buddingLevel 2 awareness,we see both of these advantagesat work. His initial explorationswith Logo and his "runningthroughthe proceduredefinition in his head" contributedto his emerging sense of the propertiesof shapes and of his certaintythata given shape was not a memberof the class of rectangles. Overall,resultsprovidesupportfor the hypothesisthatLG is effective in developing schemesfor basic geometricfiguresthatimproveperformanceon van Hielebased assessments.This is consistentwith otherstudiesindicatingthatthe appropriateuse of Logo helps studentsbegin to make the transitionfrom Levels 0 and 1 to Level 2 of geometricthought(Butler& Close, 1989;Clements,1987;Clements & Battista,1992b;Hughes& Macleod, 1986). Also, as predicted,Logo experience encouragesstudentsto view anddescribegeometricobjectsin termsof the actions or proceduresused to constructthem(Clements& Battista,1989). Consistentwith van Hiele's predictions,however, most studentsdid not move from one level to the next in a few weeks of instruction. The ability to write a computerprogramto drawshapes also increased,though even relatively simple tasks remaineddifficult for many students.Thus, learning Logo programminghad a facilitative effect on students' geometric conceptualizations. However, severalcaveats are also in order.For example, some degreeof learningfor some studentsmay have been independentof theirlearningof turtle geometry.In addition,it mightbe thatthe LG curriculumwas too short(recallthat no class experiencedthe approachfor more thanseveral weeks of a single school year) to supportcomplete learning of the relevant turtle geometry, and thus its effects on geometry learningwere foreshortened.That is, as others from a spectrumof theoreticalperspectiveshave statedfor the case of intellectualskills, one shouldnot expect large transferwithoutmasteryof the initial domainfrom which transferis predicted(Andersonet al., 1993; Clements & Merriman,1988; Klahr & Carver,1988;Kurland,Pea, Clement,& Mawby, 1986).Nevertheless,classroom observationsrevealthatthe environmentsofferedstudentsopportunitiesto analyze andreflect on the propertiesof geometricfigures.The act of writingLogo code to drawshapesandthenconfrontingthe conflictsbetweenthe relationshipsembodied within the Logo proceduresand students'intuitionsaboutfigureswas particularly influentialin students'learning. Two typesof shapeproblems,disembedding(GradesK-l) andthebuildingquestion (GradesK-3), were differentin nature.The first,disembedding,was not a goal of LG.Resultsindicatedthatalthoughkindergartners could accuratelyandcomprethe basic of hensively identify shapes square,rectangle,and triangleembeddedin a little more than a thirdof the secondgraderscould do so when these pictures,only were embedded in shapes complex geometricfigures.As expected,overall,results indicatedno beneficial effect of LG on this ability, but no evidence of benefit of the traditionalcurriculumoccurredeither. Second, work with LG significantly improvedprimary-gradestudents'recognition of the relevance of arithmeticprocesses in the solution of geometricproblems, and it improvedtheirability to apply these processes accurately(e.g., in the the K-3 Building question). Such gains, however, may more likely be observed

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when studentsarecomfortablewith the numbers(measures)involved.LG students in GradesK-l madegreatergainsthancontrolstudentsusing single-digitnumbers. Conversely,only Grade3 LG studentsmade substantialgains comparedto control studentsfor two-digit numbers.The notion that they must be "comfortablewith the numbersinvolved"is conjecturaland shouldbe evaluatedin futureresearch. ANGLES, ANGLE MEASURE,AND TURNS Considering the critical role that turtle rotations play in forming geometric figures,it was expectedthatLG would facilitatethe developmentof the geometric concepts of angle and angle size. LG studentsdid not identify angles from a set of figures better than control students did. (Unit tests indicated that LG students couldidentifythe numberof turnsandanglesin geometricfiguresaccurately;above 80%.) However, LG students did significantly outperformcontrol students in drawingan angle, even thoughthis differencewas slight. The youngest and, to a lesser extent,the oldestLG studentsimprovedthe most relativeto controlstudents. On the interviews,LG studentsdrew more correctangles thanthe controlgroup. The lattertendedto drawshapesandtiltedlines. In addition,LGstudents'increased awarenessof the angle conceptappearedricherin thatthey were muchmorelikely to drawnonprototypicalexamples of angles. LG students'descriptionsof an angle emphasizedthe concepts of rotationand, to a lesser extent, bending more than those of control studentsdid. Similarly,in the interviews, the LG students' descriptionsof differencesbetween their angle drawingsdemonstratedthe learningof boththe conceptof angleandof the measure of angles for the purposeof classification. Similarly,LG studentsoutperformedcontrolson drawinga largerangle. Here, kindergarten,third-grade,and especially fourth-gradeLG students seemed to benefit the most. The LG studentsat every gradescored substantiallyhigherthan controlson questionsthat measuredknowledge aboutthe amountof turn.Again, the interview results were consistent with this finding. Our results indicate that measure of turn is a concept that can be learned by students in any grade of elementaryschool. LG studentsoutperformedcontrolstudentsin judging the size of angles at Grades2 andhigher.These tasksinvolved difficult distractors,which demanded,for example,thatstudentsdifferentiatebetween greaterangle measure and greaterside length. Most of the increasein the formalcategoriesof explanation occurredamong intermediate-gradestudents. Manyof thesebenefitswerehypothesizedto resultfromanemphasison paths.We havealso investigatedpossiblecriticismsof usingthepathperspective.Forexample, studyingpathscouldleadto a confusionbetweenmeasuresof theamountof turnalong the pathand the measureof the angle. Ourresearchto dateindicatesthatthis need not be so, even for young students,underproperinstructionalconditions. Therewas no differencebetweenLG andcontrolgroupson angle measureestimation.In a similarvein, performanceof LG studentswas moderateon unit-test itemsof this type.In comparison,differencesbetweengroupswere evidenton tasks

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thatasked studentsto applyknowledge of angle measurein problem-solvingsituations.LG studentswere farmorecapablethancontrolstudentsof findingmissing angle and side measuresand of determiningmeasuresof turns.(Althoughperformance was just over 50% correct,the unit-testitems provideconfirmation.)Also, LG studentswere substantiallymorecapableof determiningthe measureof angles in wordproblems(forthe boatitem, substantialgainsrelativeto controlswerefound for fifth and sixth gradersbut not for fourthgraders;studentsin all gradesfrom 2 to 6 performedbetterthancontrolsfor the moretractablewireitem).Finally,knowledge of the total amountof turtleturningfor polygons (shown on unit-testitems) was somewhat greaterthan knowledge of the sum of the interiorangles of polygons. This is reasonable,given thatdirectingthe Logo turtleemphasizesonly the former.Yet the lack of largedifferencesbetweenthe scoresof LG studentson items involving internal and external angles indicates that LG effectively facilitated students'developmentof both of these two concepts, with no detrimentto either. Classroomobservationsreveal thatpersonalexperiencesin LG (e.g., as opposed to only demonstrations)areessentialin students'development,anddifferentiation, of these two concepts. In summary,comparedto the traditionalcurriculum,LGhadonly moderatepositive effects on the students'abilityto identify anddrawangles or to estimateangle size. These resultsareconsistentwith those of Kelly (1986-87), who reportedthat benefits may not emerge until aftermore thana year of Logo experience.The LG curriculum that is the focus of this monograph had large facilitative effects, however,on students'conceptualdifferentiationbetweenturnsandchangesof position, on the elaborationand sophisticationof knowledgeof angle and its measure, on knowledge of amountof turn,and on the applicationof knowledge of turnand angle measurein problem-solvingsituations(cf. Clements & Battista, 1989; du Boulay, 1986; Frazier, 1987; Kieran, 1986a; Kieran& Hillel, 1990; Olive et al., 1986). Probablybecause LG explicitly taughtthe relationshipbetween amountof turn and measure of angle produced, students gained knowledge of both these measuresas well as theirrelationship. Integrationof informationfromall datasourcesallows us to hypothesizethatthe developmentof the students'notionof the turnalong a pathmoves throughseveral steps. We conjecture(albeit post hoc) the following seven increasinglysophisticated, conceptualconstructions. 1. Studentsconstructnotions for two distincttypes of changes of state-change in heading(causedby the RIGHTand LEFTcommands)and change in unidirectional displacement(caused by the FORWARDand BACK commands).Indeed, young studentstend not to make any distinctionbetween these two-often, they believe thata RIGHTcommandactuallymoves the turtleto a location to the right of its presentlocation. 2. Studentsseem to view the turncommands(RT andLT) merely as changes in heading;they give little attentionto the actual turningmotion. They often issue repeatedturncommandsuntil the turtleis facing the desireddirection.

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3. To students,the turnmotionanddirectionbecome salient.They recognizethat if theyhaveturnedthe turtlethe wrongway, theycan compensatefor thisby turning the turtlein the oppositedirection.However,studentsact as if they believe thatthe directionof a turnthatmoves the turtleto a desiredheadingdependsonly on the finaldestinationandnot on the turtle'scurrentheading.Forexample,to turna turtle to a headingof 90? ("east"),these studentswould issue a RIGHTcommand,even if the turtlewere at a headingof 180?. 4. Students recognize that the direction of a turn that moves the turtle to a desiredheadingis dependenton the turtle'sheading,but they areunableto determine this directionwithoutphysical aid. 5. Studentscan determinethe directionof a turnby modeling with theirbodies. However, they seem unableto do so internally. 6. Studentsmentallyprojectthemselvesinto the place of the turtle.This is often accompaniedby some physical movement-the turnof the head or upperbody. 7. Studentsexamine the turtle's present heading and desired headings. From these, the studentsdeterminethe directionof turnby directlyapprehendinga rotational vector. In using this approach,studentstake a more extrinsic (ratherthan intrinsic)perspective. In a follow-up study,we foundthatabove-average,fourth-gradestudentsdid not show the initialdifficultieswith turns.However,we did confirmthe constructions of Steps 4 to 7 again (Clements& Burs, 2000). PATHS The LGcurriculumwas designedto develop andbuild on ideas aboutgeometric paths.Therefore,it is not surprisingthat studentsin this curriculumsubstantially outscoredcontrolstudentson pathitems. Therewas some indicationthatyounger LG studentsmadegreatergains in this domainthanolderLG students,but students at all gradesoutperformedcontrolstudents.Unit tests confirmedthese results,with accuracyin drawingpathsto fit given descriptionsrangingfrom72%to 95%,with most averagesin the 80%to 90% range.Thus, the targetedpathconcepts seemed to have been learnedwell by studentsin all grades. Items that requiredinterpretingor producingLogo turtlecommandsto match given paths were more difficult, with accuracyvarying widely dependingon the complexity of the task and gradelevel (Grade5-6 classes averagedin the 75% to 85% range).The complexity of mappinggeometricfigures to correspondingLG commandsandvice versa,especiallyin the staticandless familiarmediumof paper and pencil, should not be underestimated.Nevertheless, more LG studentsthan control studentswere able to use what they had learnedto provide complete and accuratedescriptionsof the process of creatinga two-dimensionalfigure. thatit is naturalfor at least some studentsto view Also, observationssubstantiated shapesin termsof paths.Forexample,when askedto tell whethertwo figureswere the "sameshapeand same size" andthen askedto explain what they would do "to

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show someone that you are right,"many young studentsthat were interviewed respondedin a path-orientedmanner.Shown two congruentshapes resembling mittens (one the flip image of the other), one kindergartnersaid, "I would show them all aroundlike this (tracesthe shapeswith the fingers of each hand,simultaneously tracing correspondingcomponents).They're the same, but they're just facing differentways." SYMMETRY Logo Geometrystudents scored significantly higher than control students on posttestmeasuresof symmetry,bothon tasksaskingthemto drawall the symmetry lines for given figuresandthose thataskedthemto drawthe "otherhalf' of a figure to createa symmetricfigure. For the latter,therewas a tendencyfor these effects to be particularlystrong for young (kindergarten)LG students.In addition,the results were largely due to a greaternumberof LG studentsdrawingcompletely correctfigures. LG studentsdid not differ substantiallyfrom control studentsin the numberof incorrectsymmetrylines drawn. Indeed, evidence of overgeneralizationby LG students regardingdiagonal lines of symmetry and, to a lesser extent, lines of symmetryfor figures that have only rotationalsymmetryshould be addressedin instruction. In contrast, there was a clear and decisive positive effect of LG regardingcorrectlines of symmetrydrawn. Unit-testitemsconfirmedtheseresults.LG studentswerehighlyaccuratein identifying whethera dotted line drawnon a figure was a line of symmetry(89% to 95%). Across GradesK-6, the most difficult items involved concave pentagons andparallelograms,wherein each case the line shown was not a line of symmetry. Studentsperformedsomewhatless accuratelywhen asked to drawall the lines of symmetryonto given figures (grade-level means on a two-point scoring system rangedfrom 1.46 to 1.64). The most difficult figures were those with more than two lines of symmetry(includingall figuresthatwere also those with obliquelines of symmetry) and the parallelogram, possibly due to the latter's rotational symmetry. Easiest were certain figures whose lack of symmetry was visually salient(e.g., an obtusetrianglefor Grade4-6 students).Finally,studentswereasked to drawthe "otherhalf' of one or more given figures to createa symmetricfigure. This was a difficult task for kindergartnersand moderatelyhardfor those in the primarygrades.Indeed,only the scores in Grades5-6 were high. Most students, however, had the correctnumberof segmentsplaced approximatelyin the correct position; achieving a high score requiredno substantivedistortion. At every grade in each school, the mathematics curriculum emphasized symmetry.Therefore,studentsabovethe kindergartenlevel hadreceivedincreasing amountsof experiencewith symmetrybeforethe studybegan,andcontrolchildren studied the topic extensively during the course of the study. Nevertheless, LG studentsmade significantlygreatergains, which we attributeto severaldays of LG explorations.These explorations allowed them to build a measurablystronger

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conceptualizationof symmetricrelations.WritingLG commandsfor the creation of symmetricfigures, testing symmetryby flipping figures using LG commands, and discussing these actions apparentlyencouragedstudentsto build richer and more generalimages of symmetricrelations(possibly to the extent of overgeneralizationin some instances)andto reflecton the constructionof symmetricfigures and abstractthe propertiesof symmetry.Studentshad to abstractand externally representtheiractions in a more explicit and precise fashion for the LG activities thanthey did in otheractivities,suchas the freehanddrawingof symmetricfigures. These findings supporta related,laterstudy,which addeda special comparison group (Johnson-Gentile,Clements, & Battista, 1994). In the 1994 study, the LG group used the same curriculumas the LG studentsof the study reportedin this monographand similarlyfor the controlgroup.A new treatmentgroupcreatedin 1994, which used manipulativesand paper-and-pencilversions of the LG activities on motions and symmetry,matchedas closely as possible. This was designed to ascertainthe specific contributionsof the computer,per se. These noncomputer experimentalstudents,for example, used paperand pencil and Mira for drawing and checking insteadof using the LG MIRRORprogram. Interviewsrevealed that both experimentalgroups, especially the LG group, performedat a higherlevel of geometricthinkingthandid the controlgroup.Both experimentalgroups outperformedthe control group on immediateand delayed posttests.Althoughthe two experimentalgroupsdid not significantlydifferon the immediate posttest, the LG group significantly outperformedthe noncomputer experimental group on the delayed posttest. Investigation of individual items suggested specific effects on symmetryconcepts. LG groupscores increasedand noncomputerexperimentalgroupscores decreasedon tasksinvolving figuresthat withno lines of symmetry. hadmorethanone line of symmetryor on parallelograms LG students appearedmore likely to identify horizontal and oblique lines of symmetryandless likely thanstudentsin the othertwo groupsto be misled by the rotationalsymmetryof the parallelogram.This suggests thatdifferencesbetween the groups were on those items thatrequiredstudentsto resist applyingintuitive visual thinkingand to apply analyticalthinkingin a comprehensivemanner. CONGRUENCE LG studentsperformedbetterthancontrolstudentson congruenceitems, not so muchby identifyingwhetherpairsof figureswere congruentas by justifying their answers.Largegainson thejustificationscore supportthe hypothesisthatLGhelps students conceptualize or formulateand communicatereasons for their mathematicaldecisions. This suggests a movementaway fromvan Hiele Level 1, visual thinking,towardLevel 2, descriptive/analyticthinking.Interestingly,the largest differencewas for those questionswith congruentpairsof figures.This difference suggests thatLG helped studentsmore on questionsin which describinga difference (e.g., "thisone is bigger")could not be used as a springboardto justification. Instead,a method (e.g., using geometricmotions) was needed to describe a way

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to "show" congruence. However, because these questions used more complex figuresandmotions,this conjecturemustbe evaluatedin futureresearch.Unit tests confirmedmore than 90% accuracyon identificationof congruentand noncongruentpairsof figuresfor the LG groupin all grades.The most difficultitems were those thatrequiredmentaltransformationsof figures(rotationsor reflections).But the impact of these difficult items decreased substantiallyin the intermediate grades. GEOMETRICMOTIONS Because of the lack of systematicinstructionof motiongeometryin most control classrooms,the pre-posttest in this study did not include items directlyassessing this areaof knowledge. Unit tests revealedthatstudentsin GradesK-l identified single motions accurately,with many identifying sequences of motions as well. Studentsin Grades2-6 identifiedsingle motionsandflip lines andturncenterswith increasingaccuracy(74%to 91%).The most difficultmotionto identifywas a 180? rotationwith a turncenter not located on the figures. More difficult than identifying motionswas identifyingturncenters(especiallythose not on the figures)and flip lines for figures that were widely separated.These difficulties diminishedas the gradelevel increased.Studentsin Grades5 and 6 were administeredtwo additional types of tasks. First,they were requiredto accuratelydrawan image from a preimageanda motion.The most difficultitems were those involving turns,especially those with the turncenter located at the top of the figure. The second type of task assessed knowledge of the relationshipsbetweenmotions andcongruence. Most students understood that motions can determine congruence and that a sequenceof motionscanbe foundto move one congruentfigureontoanother.About half, however, may not have understoodthatslides andturnsarenot sufficientfor this purpose.Before this conjecturecan be accepted,moreresearchis needed,since some studentssimply may have misinterpretedthe item (e.g., as saying, "If two figures are congruent,there is a sequence of slides and turnsthat will move one ontothe other,acceptingthatslides and turnsare the requisitemotions").Interviews revealed that geometric motions were more salient for the LG students. They tended to sort by flips, slides, and turns in a more spontaneous,more frequent, manner. In addition,the van Hiele level for motions increasedmore for the LG students, especially those from Site 1, thanfor the controlstudents.Also, as predicted,these levels were equalto, or lower than,the van Hiele level for shapesfor most students. For some reason, their thinkingaboutmotions may have been more naturalthan theirthinkingaboutshapes.Do these studentspossess high spatialability?Or, are these studentswho hadtheirshapeslevel declinefrominterimto theposttest?These studentsoperatedat the higher level and they scored higher on motions because they were tested immediatelyafter the Motions strand.Therefore,forgettingdid not really affect theirmotions performance.Finally, datasuggest thatthe studyof motions at the end of the LG curriculummay have caused some studentsto obtain

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a lowervan Hiele level on the shapes(triad)task,possiblybecausethe motionsunit madereasoningon the basis of visual transformationmore salient.This possibility shouldbe tested in futureresearch.If true,it may suggest thata differentorderor natureof the presentationof thesetopicsis warranted.In anycase, thesedataconstitute anotherindicationthatthe students'movementfromthe visual to the descriptive/analyticallevel was tentativeand fragile ratherthanstrong. Because the controlgroupsin the mainstudydid not have comprehensiveexperience with motions, we conducted a separatestudy to check these results, as described previously (Johnson-Gentileet al., 1994). Two experimentalgroups received the LG lessons on motions. However, one used manipulativesandpaper andpencilto duplicateas closely as possiblethe computeractivitiesof theLGgroup. These noncomputerexperimentalstudentsworkedwith figures identicalto those fromthe LG ROCKETprogrambutthey workedwith figuresthathadbeen duplicatedon acetate.To turntheirrockets,they placed a pencil point at the turncenter and used a protractorto determinethe amountof turn.For this 1994 study, the controlgroupreceived traditionaltextbookgeometryinstruction. The threegroupsdid not differ significantlyon pretestgeometryachievement, but both experimentalgroups significantly outperformedthe control group on immediateand delayed geometric motions posttests. In addition,the LG and the noncomputer experimental students substantially outperformed the control studentson tasks measuringlevels of geometric thinking.This substantiatesthe conjecture that the curriculumis effective. Comparing the two experimental groups,LG studentswere more likely to forgo visual strategiesand use Level 2 descriptive/analyticstrategiesfor solving geometric problems. The LG group's scores significantlyincreasedfromthe immediateto the delayedposttest,whereas those scores of the noncomputerexperimentalstudentssignificantly decreased, resulting in a significant difference in favor of the LG group on the delayed posttest. In a similar vein, scores of the LG group increased on items requiring the identificationof motions and their properties(i.e., turncenters). Althoughcautionmust be used in interpretingthese post hoc analyses, they do suggest thatsignificantdifferenceson the delayed test can be attributedto the LG students'constructionof higher-level conceptualstructures;that is, LG students analyzed components and propertiesof geometric motions (van Hiele Level 2). Thus, this separate study revealed no evidence that use of the LG motions curriculumhelped childrenperformtasks at Level 1 of geometricthinkingbetter thancontrolinstruction;thatis, such use did not appearto facilitatethe ability to draw symmetricfigures, drawfigures given specified motions, or apply motions to problemssuch as sortingtasks.Thereis, however,evidence thatthe use of Logo facilitatedstudents'transitionto Level 2 thinking.LG studentswere more likely than students in the other two groups to describe the properties of geometric motions and symmetryconstructsand more likely to solve problems involving these constructsat a more abstractand analyticallevel. This adds to the body of evidence supporting the hypothesis that Logo experiences can help students become cognizantof theirmathematicalintuitionsandfacilitatethe transitionfrom

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visual to descriptive/analyticgeometric thinkingnot only in the domain of twodimensional figures but also in the domain of motion geometry (Clements & Battista, 1989, 1990, 1992b; Edwards, 1991; Gallou-Dumiel, 1989; Lehrer & Smith, 1986; Olive et al., 1986; Olson et al., 1987). This is significant because, even in secondaryschool, studentsmay be at the lowest two van Hiele levels in transformationgeometry, showing limited use of precise language and little knowledge of properties(Soon & Flake, 1989). PROBLEMSOLVING Both the "Turtle Deliveries" and "EquilateralTriangle/Regular Polygon" episodes illustrateimportantcomponentsof the environmentthatwe have triedto create in our curriculum.Studentsare constantlytryingto make sense of mathematical ideas. They are constantly reasoning and reflecting on that reasoning. They areconstantlyinvolvedin mathematicalproblemsolving.Indeed,we feel that these episodes portraystudentswho areactively involved in instructionthateffectively promotesthe developmentof problem-solvingability.Silver andThompson (1984) state thatevidence for effective instructionin problemsolving should not be restrictedto counting the numberof problems that students solve correctly. Instructioncan also be judged as effective in this regard"if it producesstudents who spendmoretime in analyzinga problem'sstatement,searchfor alternatesolutions to a problem,have greaterconfidence in their ability to solve problems,or show increasedwillingness to attemptdifficultproblemsthatmay takea long time to solve. In fact, the benefitsof suchinstructionmay be moreimportantthanmerely increasingthe numberof correct solutions to a narrowdomain of problems"(p. 541). We believe thatthe episodes of students'activitywith ourcurriculumabove clearlydemonstratethe type of instructionalenvironmentscalled for by Silver and Thompsonand the NCTM Standards. ISSUES OF EPISTEMOLOGY,MOTIVATION,AND IMPLEMENTATION One consequence of LG's problem-centered,exploratoryapproachis that we observe progressin the areasof reasoningand mathematicalepistemology. That is, students'beliefs aboutandmethodsfor establishingmathematicaltruthbecome moresophisticated.Earlyin the curriculum,studentswho were askedtojustifytheir reasoning or responses often said, "That's what our teacher told us last year." Younger students are mute or say, "Itjust is" or "It looks like it." Later in the curriculum,they move towardempiricismby saying, "Look,I can show you," or "That'sa square... it's just tilted. I can prove it by measuringthe angles." Finally, some of the older studentsbegin to reason on the basis of the network of relations between the propertiesof figures, to use van Hiele's terms. They begin to organizepropositions,to stringthemtogetherin sequences.Forexample, they reason that a "tiltedsquare"is a squarebecause the procedureused to draw it incorporatesthe definingpropertiesof a square-or thatevery squaremustbe a

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Discussion

rectanglebecauseevery squarecan be producedwith a rectangleprocedurejust by makingthe two inputsfor the sides the same. Thus, observationsof studentsshow that they are moving from (a) nonanalytical or authoritarian-basednotions of knowledge, to (b) more autonomous, empiricalbases for establishingtruth,and finally to (c) knowledge as reasoning aboutmathematicsas a system. Also, we believe evaluationresults supportthe claim thatthatLG is consonant with the NationalResearchCouncil's (1989, p. 6) recommendationsthat"students actuallyconstructtheirown understandingbased on new experiencesthatenlarge the intellectualframeworkin which ideas can be created." Another importantpoint we wish to mention is that all of the students in the project have been motivated by the curriculum.They enjoy working with the computer. They like playing with the new mathematicalconcepts. One group of students made large paths on the playground with chalk, then talked about them as they were viewed from their third-storyclassroom. In one inner-city school, the students talked for weeks about paths to different locations within the school. "I can get to the office by going left 90, forward200, and left 90." In this same class, the studentswere also highly motivated and competent with the Motions strand. One day, the teacher was extremely pleased with the students' performance.She emphasized to them the value of working cooperatively. But, more important,she talkedabouthow they had succeededon a difficult task and how valuable it is to persist in problem solving; she emphasized to the studentsthatthey could solve problemsif they persisted.This teacherfelt, as do many teachers in the inner cities, that such students desperately need to be successful in mathematics and other academic pursuits. Success in solving problems in such courses as mathematicsbuilds self-confidence. Many of these teachersfeel thatinner-citystudentsget very little encouragementat home. And this particularteacher believed that our curriculumhad provided a vehicle for studentsto learn how to be confident, which is more importantto learn thanthe mathematicscontent. In additionto our specific findings regardingLG students'achievementof the curriculumgoals, our research has demonstratedthat achieving these goals is more likely if both curriculaandinstructionalapproacheshave four generalcharacteristics,as follows. They should: 1. Systematicallyfacilitatestudents'progressionfrombasicintuitionsandsimple concepts to higher levels of geometric thought.Studentsshould "putthe pieces together"in a profoundsense. 2. Rely on substantiveexplorationof ostensiblysimpleproblemsto promotedeep and lasting conceptual learning. Students should search for relationships and meaning. 3. Help studentsdevelop the ability to move away from authoritarianbases of knowing("theteachertold me")to autonomousbases ("Iknow becauseI can build it, measureit ..." and ultimatelyto the ability to "reasonit out").

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4. Allow childrento constructgeometricideas actively.Geometriclearningexperiences should stress action in threerelatedways. * Includephysicalactivity-such as childrenwalking,drawing,andtracingshapes; * Emphasizegeometricshapes as active entities-either formedby the action of the turtlemoving along a path,or transformedby motions; * Include reflective activity-in planningprocedures,reflecting on those procedures, posing and solving problems, and consideringthe ideas of oneself and others. Many researchquestions remainto be answered.Because the teachershelped develop the curriculumandbecause the numberof sites was limited, our analyses could not adequatelyaddressquestionssuch as the following. How do the hierarchically-leveled variables, including site, years of experience, teacher, and curriculum,interact?Are there significant gender differences and interactions (Noss, 1988)?Regardingthe effectsof teacherexperience,we havemixedevidence. Overall, students from Site 1 LG and Site 2 LG classrooms did not differ in achievement,althoughthe Site 1 LG teachershad more experience with the LG curriculum.However,thereis a suggestionthatsuchteachereffects may have been presenton tasks thatrequiredcomplex geometricreasoning. On a taskneedingsuchcomplexreasoningaboutthe definitionsof quadrilaterals, Site 1 LG students maintainedtheir advantagecomparedto the Site 1 control students as the complexity of the task increased; however, the corresponding advantagefor Site 2 studentsdiminished.Studentsof teachersless experiencedwith the curriculummade smallergains, especially on morecomplex, high-level tasks. Anotherway of statingthis is thatstudentswhose teachershad more thana single experienceteachingthe curriculummademore substantialgains on complex tasks. Thus,ourfindingsfor teacherexperienceareneithersimplenorcertain.Individual teacherabilities appearto be more importantthan one versus two year's experience teachingthe curriculum,althoughthereis a suggestion that such experience may correlatewith studentgains on complex tasks. LOGOGEOMETRY Overarchingpositive effects of LG can be summarizedsimply. One main effect was in increasing students' ability to describe, define, justify, and generalize geometricideas. Students'greaterexplicationand elaborationof geometricideas withinthe Logo environmentsappearto facilitatetheirprogressionto higherlevels of geometricthinking.Forexample,in programmingthe Logo turtle,thereis a need to analyze and reflect on the componentsand propertiesof geometricshapes and to make relationshipsexplicit. In addition,the necessity of building Logo procedures encourages studentsto build understandable,implicit definitions of these shapes.Studentsconstructmoreviableknowledgebecausetheyareconstantlyusing graphicalmanifestationsof theirthinkingto test the viability of theirideas. There is also supportfor the linkageof symbolic andvisual externalrepresentations.The

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Discussion

students'use of Logo to negotiatebetween the two has been identifiedas a main sourceof learning(cf. Hoyles & Noss, 1987a).Finally,the programminglanguage permitsstudentsto outline and then elaborateon and correcttheirideas. In a similarvein, therewas a positive effect on the students'flexible considerationof multiplegeometricproperties.As we previouslyhypothesized(Clements & Battista,1994), computerenvironmentscan allow the manipulationof specific screen objects in ways that assist studentsto view them as geometric(ratherthan visual/spatial)and to recognize them as representativesof a class of geometric objects. The power of the computeris that studentssimultaneouslyconfrontthe specific andconcretewith the abstractandgeneralized(as representedby the Logo code). Studentstreata figure both as having characteristicsof a single shape and as one instanceof many such figures. (This is closely relatedto the notionof "situated abstraction,"Noss & Hoyles, 1992.) Logo environmentsappearto demandandthus facilitateprecisionin geometric thinking.In contrast,thereis imprecisionwhenstudentsworkwithpaperandpencil, andthey can be distractedby the actualeffort of drawing.The need in Logo environmentsfor more complete, exact, and abstractexplicationmay accountfor the students'creationof richerconcepts.Thatis, by usingLogo studentshaveto specify steps with thoroughspecificationanddetailto a noninterpretiveagent.The results of these commands can be observed, reflected on, and corrected;the computer serves as an explicative agent. In noncomputermanipulativeenvironments,a studentcan makeintuitivemovementsandcorrectionswithoutexplicit awareness of mathematicalobjects and actions.For example,even young childrencan move puzzlepieces intoplace withoutconsciousawarenessof the geometricmotionsthat can describethese physical movements.In noncomputerenvironments,attempts aresometimesmadeto promotesuch awareness.Still, descriptionsof motionstend to be generated from, and interpretedby, the physical action of the students (Johnson-Gentileet al., 1994). This interpretationof the resultsis consonantwith previous researchindicatingprolongedretentionand continuousconstructionof early Logo-basedschemes for geometricconcepts (Clements, 1987). Finally,computerandclassroomenvironmentsthatpromotea problem-solving approachto education,as LG does, appearto have benefitsfor the developmentof both mathematicalconcepts and processes (e.g., reasoning,connecting,problem solving, communicating,and representing).They seem especially beneficial for developing studentcompetence in solving complex problems.Further,because studentstest ideas for themselves, computerscan aid them to move from naive to empiricalto logical thinkingand encouragethem to make and test conjectures. In summary,LG places as muchemphasison the spiritof mathematics-exploration,investigation,criticalthinking,andproblemsolving-as it does on geometric ideas. We believe thatit has the potentialto develop valid mathematicalthinking in students.As a final example of this, considerfirst-graderAndrew.At the final interview, he was quite sure of himself. When asked to explain something he thought clearly evident, Andrew would always preface his remarks with an emphatic"Look!"On one item, he was asked,"Pretendyou aretalkingon the tele-

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phone to someone who has never seen a triangle.Whatwould you tell this person to help them make a triangle?"(PP1.10, Table 32). Andrew: I'd ask,"Haveyouseena diamond?" Interviewer: Let'ssaythattheysaid"yes." Andrew: Well,cutouta triangle.[Pause.]No, I madea mistake. Interviewer:How? Andrew: They have never seen a triangle.Well, cut it off in the middle. Fold it in the middle, on top of the otherhalf, then tape it down, and you'll have a triangle.

Thenhangit on thewallso you'llknowwhata triangleis! Whatif theysaidtheyhadn'tseena diamond? Interviewer: Andrew: Makea slantedlineover,thenanotherslanttheotherwaydown,thenanother slantedlineup,thenanotherslantedlineto thebeginning. Interviewer:[Thinkshe is tryingto describea triangle]What? Andrew:

[Repeatsthe directions.Then ...] That's a diamond.Now, do what I told you before!

Andrew had done what mathematiciansare so fond of doing: He had reduced the problemto one that was alreadysolved! At the end, he asked, "Will this test be on my reportcard?'Cause I'm doing really good!" Throughoutthe interview, it was apparentthatAndrew was sure of his own reasoningand knowledge from his experience.AlthoughAndrewis nottypicalof studentsin ourproject,it is importantto note thatstudentssuch as Andrewmay laterbecome mathematicians,scientists, and engineers. Andrew had been reflecting greatly on the ideas in the curriculumand relished the opportunityto discuss them so thathe could demonstratethe results of his thought. STUDENTS' KNOWLEDGEOF GEOMETRYBEFOREINSTRUCTION Examiningassessmentitems before any instruction(besides traditionalinstructionfrompreviousgrades;i.e., PP pretestscores)providesa portraitof thegeometric knowledge of 1,605 elementaryschool studentsin two states (because there was only one classroom,we omit the seventh-gradedatafrom this discussion). These datahave alreadybeen presentedin tables;in this section, we focus on particular findings. Consistentwith previousresearchreviewedin chapter1, students'performance aftertraditionalinstructionwas not high nor was much learnedfrom one gradeto the next.Considerstudents'scoreson the triangleidentificationtask(possiblescore, 14): 9.02, 9.03, 9.70, 10.07, 10.28, 11.47, and 11.34, for K-6, respectively.'2 Correspondingscores for the rectangletask (possible score, 15) were 9.47, 9.15, 9.12, 9.76, 9.67, 10.21, and 10.18; for the squaretask (possible score, 15), 12.10, 13.54, 13.13, 13.44, 13.28, 13.77, and 13.66. These are shape-recognitiontasks. Besides not displayingmasteryat any gradelevel (except for the simplestcase of

12 Recall thatthese are the

averagesof the pretestscores from the originaltables.

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Discussion

squares,for which near90%correctnessscores arearguablyclose), these elementaryschool studentsdid not improvesubstantiallyacross the gradesfor any item. We learn more by examining individual items, with triangles providing the most fecund case. Virtuallyall studentscorrectlyidentifiedthe quadrilateralas a could idennontriangle.Most students,especially those olderthankindergartners, in standard standard 1 in orientations Item PP/1.1; see tify triangles (e.g., Shape in 6 a were nonstandard orientation and identified 8) Figure 1). Triangles (Shapes somewhatless accurately,with students'scoresfor these figuresincreasingslightly with age. Althoughmost studentsrespondedaccuratelyon these geometricfigures, it remainsquestionablewhetherseven years of schooling should be necessaryto reach acceptable,but not perfect,levels of performance. Performancewas lower on the figures with convex curvilinearsides (Shapes 3 and7), andtherewas little sign of scores increasingwith grade.On the two figures with concave curvilinearsides (5 and 14) performancewas quite low. (The relatively high performanceof the kindergartenclasses on both of these figures is curious.)An inferenceis thattraditionalgeometrycurriculado notprovideadequate experienceswith such nonexamplesnor with discussionsof the necessaryproperties of triangles(e.g., straightsides). More consistentprogressionwith age is seen for the concave quadrilateral(Shape 9). Performancewas quite low for primarygradestudentsbutincreasedto moderatelyhigh levels for upperintermediate-grade students.On the two open shapes (Shapes4 and 13), studentsachieved moderate accuracy.Both masteryandgrade-relatedincreaseswere evidentfor Shape 13;the needto continuean obliqueline segmentover a greaterdistancemay haveled fewer studentsto "close"this figure.Provisionfor open "triangular" figures anddiscussions of closure are probably neglected in traditionalcurricula, even though researchhas establishedthatthis propertyis salient even to young children. Students'abilityto correctlyidentify Shape 11, an obtuse triangleat a nonstandardorientation,improvedwith grade,but even some older studentscontinuedto misclassify this figure. The nonsimple figures (10 and 12) may have confused students.Performancewas relativelylow anddevelopmentaltrendswere not clear. Again, these resultsconfirmone majorconclusion:Traditionalgeometrycurricula are deficient. They do not provide a sufficient range of systematically varied exemplars and nonexemplars.Perhapsmore significantly, they do not provide studentswith an adequateexperienceof discussing such figures andmakingdecisions to classify them on the basis of a considerationof theirproperties. In a similarvein, studentshad difficulty attributingpropertiesto figures, with scoresof 5.36, 4.21, 4.37, 6.23, and6.13 for squaresand4.45, 4.13, 4.05, 5.46, and 5.08 for rectangles,for Grades2 to 6, respectively(theseareaveragesof the pretest scoresin Table6). Again,this shows moderateperformanceandsmallgainsat best. Only a small minorityof intermediate-gradestudentscould identify what shape a closed figurecouldbe when given a set of properties(exceptfor the obviouschoice; see Table 7). Turningto a critical geometricobject, the angle, we find that 29% of students could not provide any descriptionof "angle"and 13%describedangle as a tilted

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line. On a simple angle-identificationmeasure,scores out of a possible 14 were 7.57, 7.75, 8.17, 7.52, 9.17, 11.06, and 11.23 for GradesK to 6, respectively.For the single item of drawingan angle, scored0 or 1, the respectivemeanswere 0.08, 0.20,0.11,0.18,0.50,0.89, and0.84. In Grades2-, mosterrorsin drawingan angle involved drawinggeometric shapes and tilted lines. Thus, a majorityof students could identify angles and a majorityof intermediate-gradestudentscould drawan angle,butmanystill hadfew ideas, or only nonmathematicalideas, aboutthe angle concept. For the single "drawinga bigger angle"item, meansfor GradesK-6 were 0.02, 0.02,0.02,0.02,0.16,0.43, and0.51, respectively.Forthe "spinner"item assessing the estimationof the amountof turn,meansfor Grades2-6 were (out of a possible 3), 0.08, 0.00, 0.21, 0.66, and 0.68, respectively. For multiple-choiceestimation of the measuresof drawnangles (out of 4), means were 1.50, 1.65, 2.36, and 2.94 for Grades3 to 6, respectively.Forcalculatingthe measuresof missing angles and sides (out of 7), means were 1.69, 2.37, and 2.53 for Grades4 to 6, respectively. (Almost no primary-grade students could use arithmetic processes to solve geometricproblemsaccurately;see Table9. Onlya smallminoritycouldsolve more complex angle-measureproblems,such as the "boat"and"wire"tasks.)Together, these means indicate that students in Grades 5 and 6 are learning about angle measurement,but they answered only half to three fourths of simple problems correctlyand evinced much lower competenceon more complex problems. On drawinglines of symmetry,the students' scores (out of a possible 8) were 0.49, 0.30, 1.34, 2.36, 3.15, 4.42, and 3.85 for Grades K-6, respectively. On drawingthe otherhalf of a figure,means(out of a possible 2) were 0.01, 0.18, 0.52, 0.90, 1.04, 1.43, and 1.35 for Grades K-6, respectively. For symmetry, then, growth is more evident than for most other areas, althoughperformanceof even the oldest childrenhas much room for improvement.The decreasefrom Grades5 to 6 has no clear explanationand shouldbe checked in futureresearch. Scores on identifyingcongruentfigures (out of a possible 12) were 7.56, 8.01, 8.49, 8.29, 8.32, 8.52, and 8.58 for GradesK-6, respectively;forjustifications,the scores (out of 6) were 2.71, 3.51, 3.92, 1.18, 3.77, 4.52, and 4.58. Thus, students show moderatecompetence in identifying congruentfigures but do not improve substantiallyfrom kindergartento sixth grade.Studentsdo show improvementin justifications. In summary,consistent with nationaland internationalassessments and other studies (Beaton et al., 1996; Carpenteret al., 1980; Fey et al., 1984; Koubaet al., 1988; Lehreret al., 1998; Stevenson et al., 1986; Stigler et al., 1990), the performance in geometryof studentswho were given traditionalinstructionwas low to moderate.Studentsarenot learningmuch geometryfrom one elementarygradeto the next. Results reportedhere confirm those findings and extend them across a rangeof topics and types of problemsin elementaryschool geometry.Traditional geometry curriculaare deficient (cf. Clements & Battista, 1992b; Fuys et al., 1988;Lehreret al., 1998). Resultspresentedhereconfirmthis deficiency andindicate thatsuchcurriculado notprovide(a) a sufficientrangeof systematicallyvaried

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exemplarsandnonexemplars,(b) adequateexperiencediscussinggeometricobjects and theirproperties,or (c) rich experiencesthatinclude geometricreasoningand geometricproblemsolving.Thereis a needfor more,different,andenhancedgeometry educationin every gradefrom kindergartento sixth.

Chapter 5

New Development and Research on Logo and Geometry by Julie Sarama

Althoughthe researchconductedwith the Logo GeometryProjecttook yearsto complete, many of the lessons learnedduringthe projectspawnedlater software development and research. This chapter discusses these activities and related empiricalresearch.For brevity's sake, this chapteronly overviews these results, and thus the level of specificity and "grainsize" are differentfrom the previous chapters;referencesare providedto articlesthat provide full detail for the interested reader. TURTLEMATH The authorsof this reportwere involved in anotherNationalScience Foundation project,following Logo Geometry,thatdeveloped the geometryand softwarefor the Investigations in Number, Data, and Space curriculumwith colleagues at TERC.Douglas ClementsandJulie Sarama(previouslyJulie Meredith)createda new version of Logo thatbuilt upon the LG microworldsand research.This new version, called "Geo-Logo"in the context of the Investigationscurriculum,was developed as a stand-aloneproductcalled "TurtleMath"(Clements& Meredith, 1994). In the chapter,we describethe developmentof TurtleMath,andthe results of several studies on its use in learningmathematics.We begin with a description of the design principles. Design Principles and the TurtleMath Environment In the development of Turtle Math (Clements & Sarama, 1995), we (Julie Sarama and Douglas H. Clements) abstractedfive principles from the extant research, including Logo Geometry. Using these principles we made specific design decisions. Figure 29 is an illustrationof the Turtle Math environment, followed by a discussion of each of the principles. 1. Encourageconstructionof theabstractfromthe visualand intuitive.The nature of the interactionin certainenvironmentspromotesthe connection of two types of externalrepresentations,formal symbolic and dynamic visual, supportingthe constructionof mathematicalstrategiesandconceptionsfrominitialintuitionsand visual approaches.TurtleMathsupportssuch connectionsin four ways. Consider

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New Development and Research on Logo and Geometry

Pull down to choose menus.

I*

File Edit Font Windows

D Command Center

fd 40 square

Type commands in the Command Center. Press the RETURN or ENTER key to run them. Or, change a command and press RETURN or ENTER to run that change.

.

_

Click to choose

Options

Helpl

, a tool.

Untitled ii.ici:_ (Drawing)

The turtle draws in the Drawing window.

:

i

Teach

to square fd 20 rt

90

fd rt fd rt

20 90 20 90

rt 90 end Put defined procedures in the Teach window. Use the Teach tool to define a procedure using the commands in the Command Center or enter a procedure here. Change a procedure and click on the Command Center to run the changed procedure.

Figure 29.

the difficult topic of turns and angle measure. First, Turtle Math provides two commands, "rtf" (stands for right face) and "ltf," to help beginning students build on previous experience and appreciate turns qualitatively before encountering measurement in degrees. Second, Turtle Math provides measurement tools. The "line of sight" tool (fifth icon from the left in Figure 29) displays rays in 30? increments at the turtle's heading and position. An on-screen protractor called the turtle turner (sixth from the left in Figure 29) provides an exact measurement for any turn or angle, as shown in Figure 30 below. Third, the "label turns" tool (eighth icon from the left in Figure 29) allows students to view the measures of all turns on the

A command for this turn is 'rt 60'.

Figure 30.

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screen.Anotherway TurtleMathsupportsthe growthof the abstractfromthe visual lies in its overall structure,describedin the following section that deals with the second principle. 2. Maintainclose ties betweenexternalrepresentations.The natureof programming createsthe need to makerelationshipsbetween symbolic andvisual external representationsexplicit. But studentsmay lose the connection between the two. Turtle Math encourages immediate mode programming;that is, students type commands into the command center and see them immediately enacted on the screen. A tool (farthestleft on the toolbar)transformsa set of commandsinto a procedurein the "Teach"window with one step.Further,any changeof commands in eitherlocation, once accepted,is reflectedautomaticallyin the drawing.In this way, the commandwindow code constituteswhat we call a prolepticprocedure. The Logo code in the commandwindow standshalfwaybetweentraditionalimmediate mode recordsandprocedurescreatedin an editor,helping link the symbolic and visual representations.The structureof the Command Center-long and narrowto the side of the graphicsscreen insteadof the traditionalshortbut wide placement below this screen-permits the immediate inspection of more commands,which facilitatesconnectingsymbolic and visual representationsand patternsearching.Moreover,the admissibilityof the code to modificationencourages experimentationandexplorationandscaffolds workingwith trueprocedures. Another issue is the directionalityof the symbol-visual connection. One of Logo's mainstrengthshas beenits supportof linkagesbetweenvisualandsymbolic representations.One of its limitations has been in the lack of bidirectionality between these modes (Noss & Hoyles, 1992). That is, one creates or modifies symbolic code to producevisual effects, but not the reverse. TurtleMathprovides a "drawcommands"tool (ninthicon fromleft in Figure29) thatallows the student to use the mouse to turnandmove the turtle,with correspondingLogo commands created automatically.A related, "change shape" tool (tenth icon from left in Figure 29) allows a student to manipulatefigures on the screen and watch the commands that generated them change automatically. (See Figure 31 for an example of these tools in operation.) Althoughthe modificationof the figure resultsin the modificationof the code, the code shouldideally be easy to modify as well. This leads to the thirdprinciple. 3. Facilitate examinationand modificationof code. Although computershave the potentialto help encourageanalyticthinking,researchindicatesthattheremay be little reason for students to abandonvisual approachesin favor of analytic approachesunless they are providedwith supportfor such analysis and are challenged with tasks whose resolution requires an analytical approach.Several of TurtleMath's tools providesuch support.The CommandCenterinterfacealready described helps students examine the sequence of commands reflected on the graphics screen, helps them modify these commands, and allows them to see immediateresultsfromthis modification.Tools for editingcode (in additionto the "teach"tool alreadydiscussed) allow erasing(the second and thirdtools from the

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New Developmentand Researchon Logo and Geometry

Using the "draw commands" tool, a student mightcontinuouslycreate firsta turn(RTor LT)command. Then a movement(FDor BK)command, and so on, untilshe clicks outside the window. Using the "changeshape"tool allows a student to manipulate figures on the screen and watch the commands that generatedthem change automatically.

LT 45

+ *

LT 45 FD 100

FD 50 RT 90 FD 75 RT 90

FD 50 RT 90

FD 75 RP 90

Inthe example shown here, the student has first manipulated a side of a rectangle. As she slides the side, the commands change automatically and dynamically,as shown.

She then manipulatesa vertex. In this case, differentcommandsare altered.

FD 50 RT 70 FD RT FD RT FD RT

80 110 50 70 80 110

FD RT FD RT FD RT FD RT

50 70 80 110 77 90 75 90

Figure 31.

left erase one commandor all commands in the commandwindow, again with dynamiclinks to the geometricfigure)andinspectingandchangingcommands(the "walkingfeet"icon-second fromthe rightin Figure29-is the "step"tool, which allows studentsto "walkthrough"any sequence of commands;each commandis simultaneouslyhighlightedand executed). 4. Encourage procedural thinking. Turtle Math provides a Command Center

window as a prolepticprocedure,tools to "walkthrough"the definitionof a procedure, and automaticconnectionsbetween changes in proceduresand changes in drawings.

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5. Allow a domain in which to explore diverse areas of mathematicsusing a consistentmetaphor.The benefitof havinga consistentway to talkandthinkabout differentareas of mathematicsis that it can serve as a way for studentsto bring coherenceto a variedmathematicscurriculum. Initial Field Test The results of the first field test, involving several third-gradeclassrooms,are summarizedhere for four of the five principles (for full details, see Clements & Sarama,1995). Insufficientdatawere collected for the fifth principle. 1. TurtleMath appearedto encourageconstructionof abstractconcepts from visual intuitions.For example, intuitive commands,such as "rightface," helped studentsappreciateturnsqualitativelyat first. Then, restrictingturnsto multiples of 30 andprovidingmeasurementandlabelingtools scaffoldedstudents'construction of measurementin degrees. Further,there was no evidence of confusion betweenturnandmovementcommands,thoughsuchconfusionhas been frequently reportedin previousresearch. 2. Turtle Math helped children establish connections between externalrepresentationsof mathematicalideas. The CommandCenter'sshapeand the dynamic link between the Logo code and the graphicdrawingenhancedstudents'propensity andabilityto connectsymbolicandvisualrepresentationsandto connectthose representationsto numberconcepts, especially when combinedwith the "energy level" featureof severalactivities.Childrenwere highly motivatedto reflecton the commandsandtheirprecisegeometriceffects, perhapsleadingto an understanding of both the advantagesand disadvantagesof algorithmicsolutions. Overall, the structureof TurtleMath,especially the CommandCenter,playeda significantrole in encouragingchildrento accuratelyestimatelengths,synthesizepartsinto wholes, apply arithmetic in situations that were meaningful to them, and reflect on path/commandcorrespondences.In the Logo Geometry Project, we found that path/commandcorrespondencehad to be taughtexplicitly, and so an activity was developedfor this purpose.WithTurtleMath,no specific, separateinstructionwas given on this idea. The structureof TurtleMathandthe associatedactivitiesalone resulted in all children using the idea in problem-solving activities, with no observedinstancesof confusion regardingthe idea. 3. The ease of editing TurtleMath code and the dynamiclinking featurefacilitated students'reflectionon, and modificationof, theirLogo code, which helped them experimentwith differentproblemsolutions (e.g., amountof turn).In some cases, however, these featuresmay have allowed randomtrial and errorwithout reflection.Ease of examinationand modificationled to spontaneousexplorations. On one occasion, the teacherof one of the third-gradeclassroomssaw an opportunity to engage students in the mathematicalactivity of problem posing when discussing students'proceduresfor equilateraltriangles.The class was discussing how the numberschangedfor makingbigger and smallertriangles.

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New Developmentand Researchon Logo and Geometry

Teacher: Great.We got the turtleto drawbiggerandsmallerequilateraltriangles.Who can summarizehow we did it? Maurice: We changedall the forwardnumberswith a differentnumber-but all the same. But the turnshad to stay 120, 'cause they're all the same in equilateraltriangles. Chris: We didn't make the biggest triangle. Teacher: Whatdo you mean? Chris: What's the biggest one you could make? Teacher: Whatdo people think? Ronald: Let's try 300. The class did. Maurice: It didn't fit on the screen.All we see is an angle. Teacher: Where's the rest? Beth: Off here [gesturing].Remember,we've got it set so thatthe turtledoesn't wrap aroundthe screen. Let's try 900! TA: The student typing made a mistake and changed the commands to FD 3900. Teacher: Whoa! Keep it! Before you try it ... class ... tell me what it will look like! It'll be bigger. Totally off the screen!You won't see it at all! RN: Jereb: No, two lines will still be there,but they'll be way far apart. The children were surprised when the drawing turned out ... the same! Teacher: Is thatwhat you predicted? Ronald: No! We made a mistake. Maurice: Oh, I get it. It's right.It's ... just fartheroff the screen.See, it goes way off, there, like aboutpast the ceiling. The teacher challenged them to explore this and other problems they could think of. Jereb: I'm going to find the smallest equilateraltriangle. Ronald: We're going to try to get all the sizes inside one another. This episode also supports the notion that Turtle Math's structure supports the development of abstract ideas from students initial visual intuitions (principle 1). The Teach window's omnipresence (there is no need to invoke an editor or a "flip side" to view procedures stored in it), shape, and dynamic link to the geometry of the figures it produced were vital features in encouraging students to engage in exploratory mathematical activity. 4. Turtle Math supported the creation, revision, and use of procedures. For most students using regular Logo, procedure definition is limited to saving work (Hillel, 1992). Initially, this was true for the students in the study; however, the procedure definition soon transmuted or transformed into higher levels of procedural use, including explorations with procedures. Although fully supported by Turtle Math, the use of procedures as a programming technique was not expected in this novice third-grade class. However, all students defined procedures from the beginning and used subprocedures in defining superprocedures with understanding. Thus, the

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Turtle Math structure scaffolded immediate mode into procedural planning processes. YearlongField Test In this study,TurtleMathwas testedthrougha yearlonguse in two fourth-grade classrooms (Sarama,1995). Data were collected by using both in-class observations andteachingexperimentsandwere then analyzedusing qualitativemethods. The natureof the students'mathematicalexplorationswith TurtleMathprovided additionalinsightinto how the softwarepromotedmathematicallearningthatwas consistentwith the design principles,which were operationalizedinto goal behaviors. The four Process Standards(connections,communication,problemsolving, and reasoning)of the NCTM (1989) were similarlyoperationalized.Eventually, two overarchingthemes emerged across the design principles: motivation and facilitation. The TurtleMath environmenthad a positive effect on students'motivationto engage in most goal behaviors defined by the design principles and the NCTM Process Standards.Of those relatedto the five design principles,TurtleMathhad a neutraleffect on students' use of procedures(Design Principle4); motivated studentsto establishconnectionsbetweenrepresentations,to reflecton andmodify their code, and to use a consistent metaphorto think aboutmathematics(Design Principles2, 3, and 5); and motivatedand requiredstudentsto constructabstract concepts from visual intuitions (Design Principle 1). Of the four main NCTM Process Standards,Turtle Math motivated studentsto make connections and to communicate(thelatteronly partially)andmotivatedandrequiredstudentsto solve problems and reason mathematically.The nature of the motivation is that the studentsarerewardedintrinsicallyfor engagingin these behaviors.Thatis, the software does not explicitly tell them to engage in these behaviors,nor does it offer reinforcement.Instead,the studentsengage in these behaviorsbecause doing so helps them meet the needs of the situation. TurtleMathsoftwarefacilitatedthe behaviorsspecified by the five design principles and four NCTM Process Standards.Facilitationwas furtherclassified into three subcategories:mechanicalfacilitation,conceptualfacilitation,and facilitation of students' applicationand extension of mathematicalideas to situations beyond the Turtle Math environmentand activities, per se. After analyzing the dataandcomparingthe resultsto the initial field tests, it became apparentthatthe software facilitated the goal behaviors related to the principles and standards; however, again, this facilitationvaried across goals and subcategories. Principle 1. TurtleMath facilitatedstudents'constructionof the abstractfrom the visual. Thatis, the level of students'abstractionincreasedwith the amountof time they worked with the concepts in TurtleMath. This was most evident with the conceptof length, in which they internalizedthe gridprovidedby TurtleMath. In the first activity, these studentsused this grid to begin constructinga flexible mentalrulerthathelpedthemestimatevariouslengthswhen using differentscales.

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In the domainof turns,althoughthe studentsdid not use Itf and rtf, they built up knowledgeof turnsover time usingbenchmarksof 90? in theirestimating.Because of the lack of instruction,the tools thatwere designed to facilitatethe abstraction of turnconcepts were not used to a greatextent, except for the line-of-sight tool. Indeed,use of the line-of-sight tool, in additionto participationin the turnactivities, was efficacious in facilitatingthe abstractionof turnmeasure.If the students areto extendthese abstractconceptsandbe able to applythemto situationsoutside the computerenvironment,they need as many opportunitiesto do so as possible. Thus, it appearsthat extensive use of Turtle Math over time is necessary for studentsto generalizetheirabstractedconcepts. Principle 2. The structureof the CommandCenterconnects the Logo code to the graphicswindow, therebylinkingthese externalrepresentations.Teachersand instructionalactivities, however, are importantcatalystsin orderfor the students to consciously establishthatconnection.Because the studentsin this studydid not engage in the same type of activities as the studentswho were partof the initial field tests, they were slower to make the connection between the code and the graphics.Use of TurtleMathover time, however,did facilitatestudents'conscious correawarenessof theseconnections.Oncetheybecamefacile withpath-command spondence,the studentsextended such relationsto off-computersituationsnaturally. Thereafter,they could discuss a set of commandswrittenon a chalkboard and decide what graphicfigure the commandswould produceandthen talk about alteringcertaincommandsand getting a differentgraphiceffect. Principle 3. The environmentsupportedthe students in the mechanisms of examining and modifying Logo code. The long CommandCenter allowed the studentsto see severalcommandsat one time. The ease of using merelythe delete key or the erase-onetool facilitatedsimpleediting.Althoughthe steptool was rarely used, as the studentsbecame moreproficientat programming,the structureof the CommandCenteralone, with its dynamiclink to the graphicswindow, supported their debugging and geometricexplorations.This structurebecame so naturalto the studentsthat they were able to modify code in off-computersituations,even thoughthey were unableto immediatelysee the resultsof theirchanges.Thus, the softwaresupportedthe conceptualexaminationand modificationof code. Principle 4. The goal of proceduralthinkingwas facilitatedat the mechanical level. If the studentswere askedto define a procedure,they simply clicked on the teach tool and did so. However, the studentswere thinkingprocedurallyonly at the lowest level (e.g., in orderto save work).The single class session in which the students did begin to use proceduresmore conceptuallyfollowed the teacher's encouragingand modelingthe use of the procedureandthen initiatingan activity thatnecessitatedwritingprocedures.Overall,however,studentswere not observed usingproceduralthinkingin a generalizedsense at all duringthe courseof thisyear. Thus, unless the teacherspecifically and explicitly taughtand modeledthe use of proceduralthinking,the featuresof the environmentthatwere intendedto support such thinkingwere not utilized.

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Principle5. Given an entireschool yearto workin the TurtleMathenvironment, studentsused the turtlemetaphorto thinkaboutthe situationon the computereven when the situationwent beyond typical turtlegeometry.They made connections betweenmathematicaltopicsthatwereusuallysegmentedby the school curriculum when they worked on Turtle Math projects that requiredknowledge of several differentconcepts(e.g., coordinates,turnmeasure,lengthestimation,fractions,and decimals).Also, the turtlemetaphorwas appliedwithoutteachermediationto those situationsoutsidethe computerenvironmentthatevoked the students'experience on computer.Duringthe course of this study, these off-computersituationswere limited to geometricalsettings. The software similarlyfacilitatedthe four mathematicalprocesses of problem solving, communication,reasoning,andconnectionsthatarerecommendedby the NCTM Standards.The following paragraphsbriefly discuss how this facilitation occurred. ProblemSolving. TurtleMathfacilitatedproblemsolving at a conceptuallevel in thatthe studentsneededto hone theirproblem-solvingskills to solve the variety of problemsthey encounteredduringthe year.In otherwords,they triedtheirsolution, evaluatedthe results, reevaluatedtheir initial strategy,monitoredhow well thingswere going in general,andretestedfinal solutionsto be surethe originaltask was completed. The experienceof solving problemson the computerhad a positive effect on the way the studentssolved problemswhen they were not on the computer.One of the teachers,Mrs. Smith, noted thather students,especially the betterstudents,were far less likely to give up on problemsafter they had workedin TurtleMath. She noted that these studentsinitially startedthe year accustomedto being "good at math,"but then they had difficulty working throughnovel problems.In orderto minimize this difficulty, Mrs. Smith presented her students with off-computer problem-solvingsituationsthroughoutthe year. Still, she became convinced that it was the computer that "took (their problem-solving competencies) one step further." Communication.TurtleMathfacilitatedmathematicalcommunicationbecause it gave the studentsa languagewithwhichto expresstheirmathematicalideas.Also, because the studentsworkedin pairs,communicatingabouttheirideas became an essential element of completinga project. Reasoning. Similarly,the natureof TurtleMath's noninterpretivefeedback,as well as the structureof its CommandCenter,facilitatedmathematicalreasoning. This reasoningalso was observedduringoff-computergeometrylessons. Most of the teachersinvolved in this researchprojectattributedthe developmentof mathematicalreasoningto workingin the TurtleMathenvironment. Connections.The NCTM Standardregardingconnections, which is similar to our Design Principle5 was also facilitatedby TurtleMath; that is, the students workedon projectsthatrequiredthem to synthesize concepts andprocesses from

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several domainsof mathematics.These connections seemed to be maintainedin off-computerdiscussions about those projectsas well. The teacherscommented that the students pointed out connections and rememberedthe activities they workedon more thanthe teachersdid themselves! In summary,TurtleMatheffectively transformedthese fourth-gradeclassrooms intomathematicallearningenvironmentsof the type recommendedby the NCTM's Standards.

LEARNINGSPECIFICMATHEMATICSTOPICSWITH"TURTLEMATH" In three other studies, we (Sarama,Clements, and colleagues) investigatedthe developmentof specific mathematicsknowledgeandcompetenciesin TurtleMath environments.Here we will summarizebriefly the results of each study, then discuss the role of TurtleMath and the implicationsfor teachingby considering the threestudies as a whole. One study examined students' development of linear measure concepts (Clements,Battista,Sarama,Swaminathan,& McMillen,1997).We observedthree levels of strategiesfor solving lengthproblems,andwe hypothesizedthatstudents pass throughthese levels duringtheir developmentof length concepts. Students, operatingat the first level, did not segment lengths and also did not connect the numberfor the measurewith the length of the line segment. Rather,they applied general strategies, such as visual guessing of measures and naive guessing of numbersor arithmeticoperations.These studentstendedto be identifiedby their teacheras those with low mathematicalability. Strategiesat the secondlevel weremostcommonamongthirdgraders.Theydrew hashmarks,dots,orline segmentsto partitionor segment(i.e., notmaintainingequal length components)lengths.Two factorscould have causedthis. A turtlestep is a smallunit(1 mm or less on most monitors);moreover,students'experienceis probably such thatobjects 100 unitsin length aresubstantialin size. These factorsmay have made the abstractionof the turtle step difficult for those who wished to assign numbersin a meaningful,quantitativemanner.Therefore,these students markedoff lengths in units that made sense to them, usually units of 10. They needed to have such perceptibleunits to quantifythe length. Studentswho used strategiesat the thirdlevel, like those at the first level, did not use figurativepartitioning(or had ceased using partitioningat some point). However, they did use quantitativeconcepts in discussing the problems, drew proportionalfigures, and sighted along line segments to assign them a length measure.Therefore,we assumethatthey hadwhatSteffe calls "interiorized"units of lengthandhaddevelopeda measurementsense thatthey could impose mentally onto figures.These observationssubstantiateSteffe's argumentthatthese students had createdan abstractunit of length (Steffe, 1991). This is not a staticimage, but ratherit is an interiorizationof the processof moving (visuallyor physically)along an object, segmentingit, and countingthe segments. When consecutive units are

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considereda unitaryobject, the studenthas constructeda "conceptualruler"that can be projectedonto unsegmentedobjects (Steffe, 1991). We hypothesizedthatonce studentsat the firstlevel have had sufficientphysical measurementexperienceiteratingandpartitioningintounits,theyconstructschemes thatallow them to partitionunsegmentedlengths. Such second-level schemes are figurative;thatis, studentsneedto use physicalactionto createperceptualpartitions. In solving problems,these partitioningschemes develop to include the constraint that equal intervals must be maintained. This constraintleads, in turn, to the constructionof an anticipatoryscheme,becausethe equal-intervalconstraintcanbe realizedmost efficientlywhen it is done in imagery,in anticipation,withoutforcing perceptualmarkings.At this point, third-levelstrategiesemerge. The second study investigatedthe developmentof turn-and-turnmeasurement concepts (Clements, Battista, Sarama,& Swaminathan,1996). Turns were less salient for students than "forward"and "back"motions. Still, they evinced a progressiveconstructionof imageryandconceptsrelatedto turns.In addition,they gainedexperiencewith physicalrotations,especiallyrotationsof theirown bodies. In parallel,they gained limited knowledge of assigning numbersto certainturns, initially by establishingbenchmarks.For some students,a synthesis of these two domains-turn-as-body-motion and turn-as-number-constituteda crucialjuncturein learningaboutturns.Some common conceptions,such as conceptualizing anglemeasureas a lineardistancebetweentwo rays,werenot in evidence(although they arereportedfrequently,e.g., Clements,1987;Cope & Simmons,1991;Hoyles & Sutherland,1989; Kieran,1986a;Kieranet al., 1986). Ourresultssupportedthe efficacy and usefulness of the instructionalactivities. In a follow-up study, we found that above-averagefourth-grademathematics studentsdid not show the same initial difficulties with turns.In fact, they synthesized andintegratedthe two schemes,turn-as-body-movement andturn-as-number. Furtheranalysis has revealed a process of psychological curtailmentin which studentsgraduallyreplace full rotationsof their bodies with smallerrotationsof an arm,hand,or finger,andthey eventuallyinternalizedthese processes(Clements & Burs, 2000). The thirdstudy focused on the developmentof concepts of geometric figures (Clements,Sarama,& Battista,1998). Workwith TurtleMathfacilitatedstudents' conceptualizationof the propertiesof geometric shapes as well as the students' connection between these geometric properties, measurements, and number conceptions. Both the researchersand teachers were remindedof the depth and long-term development of many ostensibly simple ideas, such as "shape"and "triangle."Many students' ideas remained a combination of those developed throughthe class work andfrompreviouslydeveloped concept images. However, certainideas in the unit, such as that of a simple, closed path and turtlemotions (FD and BK) and turns, seemed to be significant organizing principles for the students. Our results supportthe notion that students should become explicitly aware of these principles and then of other propertiesof geometric figures that are less immediately salient to them, such as equality of turnsand angles. They

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need to apply these ideas to figures in orderto determinewhetheror not the latter are membersof specific geometric categories. IMPLICATIONS There are several implicationsthat can be drawnfrom these studies regarding the design of mathematicscurriculaand particularlyof geometricsoftwareactivities. In general, we found that computerenvironmentssuch as TurtleMath can encourage and supportthe development and use of mathematicalconcepts and processes.In fact, benefitscan be attributedto specific characteristicsof the Turtle Mathcomputerenvironment. Firstandmost fundamentally,turtlegeometryconstitutesa contextin which the componentsand propertiesof geometric shapes are critical. For example, in our studiesstudentshadto entercommandsthatspecified the type of component(e.g., line segment or angle) and its measure.In this way, TurtleMath allows students to distinguishbetweenattributesthatareandarenot definingpropertiesof shapes, such as orientation. Second,the computerprovidesintrinsic,nonevaluative,andmeaningfulfeedback thatis instrumentalin motivatingstudentsto reflect on ideas aboutshape,length, andturns.In manyinstancesin ourstudies,computerfeedbackled studentsto relate side lengthsor angle measuresto the propertiesof geometricfigures.TurtleMath feedback consists largely of the graphicresults of runningLogo code precisely, For example,many studentswould use TurtleMath withouthumaninterpretation. to exploreiterationsof turns(RT 90 RT 90; or RT 30 RT 30 RT 30) andthusbuild a (superordinate) unitfor measuringrotation.Studentswereless confidentaboutthe resultsof combininginputsto turncommandsthancombininginputsto FD andBK commands.Initially,moststudentsgave no indicationthattheywerecombiningrotational units, per se (i.e., quantities).Ratherthey appearedto operateon numbers (withoutquantitativegrounding),use the resultof thatnumericaloperationas the inputto a new turncommand,and check the resultantfinal headingof the turtle. Because studentsreceivedthis immediatefeedbackfrom the computer,they were able to give quantitativemeaningto this combiningoperation. Third, the computer context is motivating. Several students, categorized by theirteachersas the "lowest ability"studentsin mathematics,often solved paperand-pencilproblemswith visually-basedguessing. However,these same students deliberatelyused arithmeticwhen workingin TurtleMath. Teachershad to stop one pairof boys who were fightingbetweenthemselvesover who would get to do the arithmeticto computethe missing length for a rectangle. A fourth advantageof Turtle Math is its flexibility and dynamic connections between the symbolic and graphic.The geometricsetting providedboth motivations and models for students'thinkingabout geometry, numberand arithmetic operations, and their interaction.The motivations included game settings and activities that allowed studentsto createtheirown pictures.The models incorporatedlengthandrotationas settingsfor buildinga sense of bothnumbersandoper-

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ations on numbers,with measuringand labeling tools supportingsuch construction. Conversely,the numericalaspectsof the measuresprovideda contextin which studentshadto attendto certainpropertiesof geometricforms.The measuresmade such properties(e.g., opposite sides equal) more concrete and meaningfulto the students.The dynamiclinks between these two domains, structuredin the Turtle Math environment(e.g., a change in code was automaticallyreflectedin a corresponding change in the geometric figure), facilitated students' constructionof connectionsbetween theirown numberand spatialschemes. Fifth,tools based on researchandfine tunedthroughfield testingcan encourage students'mathematization.For example, TurtleMath's slow turnswere noted by studentsand helped them build dynamicimageryfor rotations.The measurement and labeling tools encouragedthem to eschew guessing when they were required to use such mathematicalprocesses as measurementand analysis. Unlike regular Logo, commandsin TurtleMath thatare the same stay the same and do not have to be reentered.Moreimportant,those thatarechangedarenot executedas an additional commandaffecting the graphics;instead, the figure changes to reflect the change in the commands.This featurehelped studentsencode contrastsbetween differentcommands.Therefore,these studies,then,addto the literatureindicating that enrichingthe primitives and tools available to studentsfacilitates students' constructionof geometricnotions andincreasestheiruse of analytical,ratherthan visual, approaches (Clements & Battista, 1992a; Clements & Sarama, 1995; Kynigos, 1992, 1993). As we mentionedbefore, thereare also implicationsthatcan be drawnfrom the studies regardingcurriculumdesign. In the following paragraphs,we discuss seven implications that result from the studies described in this chapter.First, althoughstudentsuse some of the aforementionedtools "naturally,"they mustbe encouragedto use others.For example, althoughthe step tool is useful for debugging or tracinglarge procedures,studentsneed to be taughthow to use this tool when workingon the computer.A shortlesson showing how the step tool works will not resultin studentsusing this tool to its fullest potential.If the teacherdoes not continue to model the use of it and other tools, these tools may be forgotten afteran initial exploration. Second, curriculaand teachers should encourage connections throughoutthe mathematicscurriculum.Formanystudents,connectionsbetweengeometricforms and numericalideas are tenuousat best, even in situationsdesigned to emphasize and develop these connections. Lack of linkages limits the growth of number sense, geometricknowledge, and problem-solvingability. Studyingmore geometry may amelioratethis situation.The TurtleMath tasks provide experiencesin which thereis a mutuallybeneficialsynthesisof the schemesof turtle-basedpaths, geometric properties,and arithmetic.Our studies show that those studentswho connected ideas had more powerful and flexible solution strategiesfor solving spatialproblems. Third,descriptionsof students'learningshouldbe specific. The van Hiele levels may providea broadframework,but they are inadequatealone (see chapter6 and

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cf. Lehreret al., 1998). For example, students struggle to learn specific length conceptsandskills. At the firstandthirdlevels of solving lengthproblems,students in our studies, did not use hash marks to segment or partitionline segments. Teachersshouldtake special care to observehow studentsworkingat these levels interpretthe task, because they need to engage in activities of differenttypes. At the thirdlevel, studentsmay be challenged with more complex missing lengths tasks.In addition,studentsat the firstlevel need to engage in partitioninganditeratinglengths,continuallytyingthe resultsof thatactivityto theircountingschemes. Tasks in which applyingonly numericalschemes is ineffective may be especially useful. In the domain of turnsand angles, studentsneed to maintaina recordof mental images of both the initial heading and final heading of an object, using a frameof referenceto fix these headings(probablyan externalizedor internalized verticaland horizontalframework).They have to re-presentthe motoractivity of rotatingthe object from the former heading to the latter, and compare that representedimageto one ormoreiterationsof aninternalizedimageof a unitof turnunits of 90? or 30? for our students-or partitionthe re-presentedturninto these units. Otherimplicationsfor curriculumdesign takethe form of caveatsfor the developmentof computeractivitiesandtools. The fourthimplicationis thatanycomputer environmentmay focus on certainmathematicalaspects to the neglect of others. The Turtle Math activities are no exception. The presence and measure of turns/angles,the measuresof sides, and so on, arepropertiesthatstudentsabstract from their Turtle Math work and apply in other situations. However, implicit propertiesof the turtle-generatedshapes, such as straight(vs. curved) lines may not be as readily internalized(recall thatin the Logo Geometryenvironment,the teacherused Logo to bring such featuresto explicit awareness).Thatis, the turtle always drew straight lines; students did not have to explicitly decide which commandto use to make it do so. Fifth, specific, seemingly immaterial,detailsof activitiesactuallycan be essential. Sometimes, we were too cavalierin makingcertainchanges thatwe thought would "simplify"a screenor activity.For example, an early navigationalactivity of Geo-Logo includedan on-screen"battery"whose energydecreasedwith every commandthe studentissued. The batterymotivatedthe studentsto examine code, connectthe commandsto the turtle'spath,andcombinecommandsand,therefore, lengths.In a laterfield test,the batterywas omittedfor simplicity'ssake.This omission was a mistake.Althoughthe studentsin the studyeventuallyachievedthe goal behaviors,the processwas slowerandless complete.As anotherexample,the coordinateactivity"SunkenShips,"which was changedfroman earlierversion,should not have had the distanceclues based on taxicabgeometryremovedfrom it. The newer versionof this activity,which merelyused "nearand far"clues, causedthe studentsto guess ratherthanapply mathematicalanalysis. Sixth, activitiesmustbe carefullydesignedto encouragemathematicalanalysis. Observationsconfirmpreviousfindingsthatmanystudentsof this age initiallyrely on visual cues only insteadof such analyticalwork as using dynamicimageryand

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arithmeticto find precise mathematicaland programmingrelations within the geometryof the figure(Clements& Battista,1992b;Hillel & Kieran,1988). There is littlereasonfor studentsto abandonlimitedsurface-levelvisualapproachesunless they arechallengedwith tasksin which resolutionrequiresan analyticalapproach. Seventh, the final implication, regarding curriculumdesign, maintains that philosophic consistency is more than an academic concern. Constructivism,the drivingtheorybehindthe design principles,the software,andNCTM's Standards, was also an importantpart of the teaching philosophy of the more successful teachers.In this study,some teachersheld a certainphilosophyaboutcomputeruse. They wanted the studentsto "master"a concept fully before the lattercould use the computerandthenrequiredthese studentsto complete shortertasksthatcould be quickly checked for correctness.This philosophy caused the studentsto have less time on the computerand gave them less opportunityto creatively explore mathematics.Therefore,a consistentphilosophyin pedagogyandthe manycomponents of curriculummaterialsis criticalfor reform. In conclusion, the use of previousresearchon Logo (includingLG) in deriving design principlesand then the installationof the design principlesin TurtleMath proved to be a positive effort in helping studentsexhibit desirablemathematical behaviors.However,curriculumandsoftwaredesignis an iterativeprocess.Further insight into the students' learning of specific concepts could and should have directimplicationson yet newer versions of Logo.

Chapter 6

Implications for Theory We based our research on a synthesis of Piaget's and the van Hieles' theories. We made modifications to the van Hiele theory in two publications, suggesting a level lower than Level 1 in one (Clements & Battista, 1992b) and postulating substantive alterations to common interpretationsof the van Hiele theory in the other (Clements, 1992). In this postscript, we summarize and update these modifications and demonstrate how findings from Logo Geometry support them. According to the theory of Pierre and Dina van Hiele, students progress through levels of thought in geometry (van Hiele, 1959, 1986; van Hiele-Geldof, 1984). Thinking develops from a level driven by visual patterns through increasingly sophisticated levels of description, analysis, abstraction, and proof (Clements & Battista, 1992b). We suggest that it may not be viable to conceptualize a purely visual level (Level 1 in the van Hiele hierarchy), followed and replaced by a purely verbal descriptive level (Level 2) of geometric thinking (and so on), which is a perspective not infrequently taken in some discussions of the van Hiele theory (note that van Hiele and we have never really characterized Level 2 as purely verbal; "descriptive" can certainly include visual aspects). Instead, we suggest that different types of reasoning-those characterizing different levels-can coexist in an individual and develop simultaneously but at different rates and along different paths. Each path leads to slightly different combinations of multiple types of knowledge. This view implies a different conceptualization of levels of geometric thinking and of students' development of these levels. MULTIPLE PATHS TO MULTIPLE TYPES OF KNOWLEDGE From the perspective that we suggest, levels do not consist of unadulterated knowledge of one type only. This view is consistent with recent literature from Piagetian and cognitive traditions (e.g., Minsky, 1986; Siegler, 1996; Snyder & Feldman, 1984) as well as with reinterpretations of the van Hiele theory, including Clements (1992), which we are elaborating here, and others (Gutierrez, Jaime, & Fortuny, 1991; Lehrer et al., 1998; Pegg & Davey, 1998). These interpretations reject the assumption that one level of geometric knowledge, such as visual knowledge, exists in an individual. Instead, for example, children at both Levels 1 and 2 possess triangle schemes that include visual/imagistic and nonvisual or verbal declarative knowledge ("knowing what") about shapes. Children who were described as "geometry deprived"3 (Fuys et al., 1988) have a number of imagistic prototypes for triangles, such as an equilateral triangle and a right triangle, both with a horizontal base (Hershkowitz et al., 1990; Vinner & Hershkowitz, 1980).

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These prototypesare not absolutelyrigid,but they have constraints.For example, the morethe lengthsof the legs of a righttrianglediffer,the less likely thattriangle will be assimilatedto thatprototype.Such prototypescan be thoughtof as having multivariatedistributionsof possible values (e.g., for the relationshipsbetweenthe side lengthsandfor the degreeof the base's rotationfromthe horizontal)in which, the nearerto the mode of the distributionthe perceivedfigureis (e.g., equalfor the side lengths and 0? rotationof the base), the more likely it will be assimilatedto thatprototype(paralleldistributedprocessing,or PDP, networksmodel this type of system, see Clements & Battista, 1992b; McClelland, Rumelhart, & the PDP ResearchGroup, 1986). Researchhas substantiatedthatchildrenpossess severaldifferentprototypesfor figures (e.g., a verticallyand a horizontally-orientedrectangle)withoutaccepting the "middle"case (e.g., an obliquely-orientedrectangle).In one study, subjects studieda preponderanceof rectangleswith extreme values and few intermediate values of variablessuch as size (Neumann, 1977). Subjects were presentedwith test items andaskedto ratetheirconfidencethattheyhadalreadystudiedthatparticularitem. Interestingly,the subjectsratedthe extremes(e.g., largeor smallrectangles) much higher than items created by using the mean of these values (e.g., middle-sized rectangles),showing that they extractedmultiplefoci of centrality, therebycreatingseveralvisualprototypes.Thus,theydid not cognitively"average" whattheyhadstudied.Thisfindingis consistentwithstudieson the vanHiele theory (Burger& Shaughnessy,1986; Clements& Battista,1992b;Fuys et al., 1988), as arethe datapresentedin this monograph(e.g., recall students'performanceon the shape identificationtasks). Again, let us consider"geometrydeprived"children'sknowledge.We suggested thatthey have a numberof constrainedimagisticprototypesfor triangles.They also possess verbaldeclarativeknowledgethatmay includethe nametriangleanda few statementsof components,suchas threesides andpossiblythreecomers. However, these statementsarenot constrainedfurther;for example,therearefew limitations placed on the natureof these sides (e.g., they might be curved)and comers. Consistentwith our suggestionthatchildrenpossess multipletypes of geometric knowledge, we suggest that the children's knowledge of geometry might be enhancedin differentways. First,theirimagisticprototypesmightbe vastly elaboratedby the presentationof a varietyof exemplars,throughthe systematicvariationof irrelevantandrelevantattributes(e.g., throughdynamicmediasuch as the 13Fuys et al. use this phraseto describechildrenwith extremelylimited andrigid school experience with geometry.Of course,all studentsreceive substantialdaily experiencewith spatialforms,andtheir ideas about geometric objects, such as angles, are strongly influenced by this experience and the common languagepatternsused to communicateaboutit (Clements& Battista,1989). Althoughthese in-school and out-of-school, socially constituted experiences are critical to ecological validity (Bronfenbenner,1989), theirfull considerationlies outsidethe scope of this chapter.Thatis, as an elaborationof extantwork on the van Hiele theory,this chaptersharesthe focus of thatwork on students' knowledge; however, this does not diminish the criticalityof the instantiationof that knowledge in differentcontexts.

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computer). Children must actively attend to the exemplars and connect them throughverbal labeling. Such enhancementis accomplished mainly through a usually unconsciousvisual inductionprocess. Second,throughthe presentationof particulartasksandengagementin dialogue aboutthem,students'verbaldeclarativeknowledgemightbe refinedto extend,elaborate, and constraintheir visual knowledge. The LG research,for example, indicates thatcertainexperienceswith the Logo computerlanguageextend students' conceptionsof the "threesides" of a triangleto include the constraintsthat these sides mustbe straightandthatthey all must interconnect(i.e., thata trianglemust be a closed figure). Students,even those clearlynot above Level 1 thinking,could use Logo termsto discuss these constraintsand applythem in classificationtasks, therebyrejectingshapes with curvedsides and the wrong numberof sides-even those shapes that sharevisual characteristicswith triangles(e.g., a chevron).An alternativeinterpretationof these experiences might be that the Logo activities merely helped children construct more varied visual prototypes for triangles. However,evidencedoes not supportthis, becausealthoughthe Logo studentswere not more accuratein identifying triangles,they did accuratelyidentify nontriangles thathad visual aspectsof triangularity.4To accomplishsuch correctidentification in the instanceof closure, elaboratedconceptionsmust mitigatethe effects of gestaltvisual tendencies,giving precedenceto declarativecriteriain figureclassificationtasks.For example,morechildrenin ourLogo classrooms,comparedto the controls, rejectedfigures createdby omitting a small segment of a prototypical triangle('1.), justifying theirdecision with ideas such as the following statement: "Itlooks like a triangle,but it can't be because triangleshave to be closed all the way." This approachrequiresmore conscious reflection,in which abstraction, induction, generalization, and short deductive chains of reasoning are combined. We hypothesize that each of these two paths towardenhancingknowledge of geometrycan be followed separatelyor together.If separately,knowledge of one type can "substitute"for knowledgeof anothertype on certaintasks,withincertain limits (includinga performancedecrement,in accuracyor speedof execution).For example,a richandvariedexposureto variousexemplarscould allow near-perfect performanceon trianglediscriminationtasks such as those discussed previously. Performancewould sufferonly if a figurewerepresentedthatfell outsidethe range of any of the multipleimagisticprototypesdevelopedby the childrenor if the task demandedreasoningbasedon propertiesnot supportedby such imagistic-oriented schemes (e.g., calculations regarding angle relationships), in which case the schemes would be inadequateto the task. 14Not all typesof Logo experiencenecessarilybuildgreaterverbaldeclarativeknowledge;some types of Logo experiencemight simply help studentsrefinetheirvisual images. Varioustypes of experience may includereflectionon turtlemovementsor involvementin class discussionsaboutturtlemovements and shapes.The formermightrefine students'visual imagery;the lattermightbuildup relevantverbal declarativeknowledge. In the LG research,these two effects are confounded.

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Similarly,well-developedverbaldeclarativeknowledge, even in the absenceof exposure to various exemplarscould allow perfect performanceon such simple discriminationtasks. However, performancewould suffer (at least in speed of execution) because a chain of reasoningwould be requiredfor every figure that couldnotbe immediatelyassimilatedintothe (hypothesized-limited) rangeof imagistic prototypes.In each of these instances,the characteristicsof the figures would have to be "readoff' the figureandcomparedto thosethatareheld in verbaldeclarof the class. This typeof verbalknowledgewould ativeknowledgeas characteristics more sophisticatedanalysis of geometric figures. Withoutthe imagistic support knowledge,however,the rangeandflexibility of the applicationof thisknowledge would be limited. Researchon the learningof geometry shows that studentslimit theirconceptualizations to studied exemplars and often consider common but nondefining features(e.g., an altitudeof a triangleis alwayslocatedinside the triangle)as essential featuresof the concept (Burger& Shaughnessy, 1986; Fisher, 1978; Fuys et al., 1988; Kabanova-Meller,1970; Zykova, 1969). Further,students,who know a correctverbaldefinitionof a conceptbutalso have a limitedvisual prototypeassociated with it, may have difficulty applying the verbal description correctly (Clements & Battista, 1989; Hershkowitzet al., 1990; Vinner & Hershkowitz, 1980). This is consistentwith the theoryof simultaneousandsequentialprocessing. In this view, pictorially-presentedmaterials may be more likely to evoke the visual/holistic/simultaneous materialswould processing,whereasaurally-presented be more likely to evoke verbal/sequential/linear The processing. type of presentation may interactwith individualdifferencesin abilities in these two domains. These hypotheseshave two implications.First,at least threepathsto the elaborationof geometricknowledgearepossible:enhancementof imagisticknowledge, enhancementof verbaldeclarativeknowledge,or enhancementof both (including interrelatingthe two). All threewould lead to increasedperformanceon low-level geometryitems foundon most mathematicsachievementtests. Similarly,all three may lead to performanceon certaintasks that could be interpretedas indicating growth towardhigher levels of thinking.Enhancingonly imagistic knowledge, however, would not permitsophisticatedverbalconceptualreasoning.Enhancing only verbal declarativeknowledge would allow such reasoning(albeit not cause or ensureit) butwouldplace constraintson its applicationandgeneralizability(thus making it unlikely to be applied generally). Only a simultaneousand integrated enhancementof bothtypes of knowledgewouldencouragethe developmentof rich androbustgeometricknowledge-knowledge thatnot only would supportfuture achievementin geometrybut also wouldallow the applicationof geometricknowledge in a wide range of mathematicaland nonmathematicaldomains. VAN HIELEMODELOF LEVELSOF GEOMETRICTHINKING These hypotheses also have implicationsfor the van Hiele model of levels of geometricthinking.Conceptualizinggeometricgrowthas being strictlyvisual,then

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strictlydescriptive/analytic,thenlogical, and so on, is neithercompletelyaccurate noroptimalforeducationaltheoryandpractice.(Nordoes strictlinearitynecessarily reflectvan Hiele's view, thoughhis theoryis oftendiscussedin this form.)A synergistic approachhas more validity.In such a view, verbaldeclarativeknowledgeis a componentof thinkingat Level 1 (and,to a morelimitedextent,at Level 0, as will be addressedin the following paragraph),andenhancingverbaldeclarativeknowledge solidifies geometricachievementat these levels (van Hiele, 1986, p. 83-86, allows for such a role). There is a range of such verbalknowledge components. As previously described, these include simple statements, such as "has three sides," and ostensibly palpableand simple symbols, such as the word "triangle." More than merely a label, a term such as triangle serves as a nexus aboutwhich images andverbaldeclarativeideascoalesce. In this sense, we agreewith Vygotsky (1934/1986), who said, Theprimordial wordbynomeanscouldbereducedto a meresignof theconcept.Such a wordis rathera picture,image,mentalsketchof theconcept.Itis a workof artindeed. Thatis whysucha wordhasa "complex" character andmaydenotea number of objects belongingto onecomplex"(p. 133). Thus, a word has several zones of unequalstability;in the context of geometric forms, these zones include the hypothesized prototypes determinedby multivariatedistributions.The zones aretied to certaincontexts,often so tightlythatthey connect automatically.For example, a studentwho had workedwith Logo turtle geometryin first gradewas asked aboutthe numberof angles in a trianglein his third-gradeyear.He queried,"Whatdo you mean 'angle,' ... corers?" (Clements, 1987). Later,he was asked how he had drawna trianglewith the turtletwo years previously.He replied, WestartedandthenI putleftandI madeit turnso it wouldbe likethat[rotating hand]. ThenI madeit go forwardso it wouldgo likethatandthenI madeit turnon anangle. An angle [shoutingand laughing]!A turn!A turn ... the same thing!"

This notionof multipletypes of often compartmentalizedknowledgeis consistent with otherresearcherswho find thatLevel 1 involves multiplementaloperations (Lehreret al., 1998). In a similarvein, imagisticknowledgeshouldnotbe associatedsolely with Level 1 thinking.Schemes for geometricforms are abstractionsfrom active kinesthetic and visual experiencesat all levels (Fischbein, 1987; Piaget & Inhelder,1967). There are two implications of our synergistic approach and the data that supportsit for models of geometric thinking.First, we have arguedthat there is evidence for a level more basic than van Hiele Level 1; Level 0, which we call pre-representational,or pre-schematic(Clements & Battista, 1992b). Research from the van Hiele perspective has identified students who fail to demonstrate thinking characteristicof Level 1-including students at the high school and college level; in these studies,categorizationin a level below Level 1 was reliably distinct from categorizationinto Level 1; it was also stable and predictive of achievement (Mayberry, 1983; Senk, 1989; Usiskin, 1982). Further,Piagetian

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researchsuggests thatgeometricobjects arefirstconstructedby the studenton the perceptualplane and only laterare reconstructedon the representationalor imaginal plane (Piaget & Inhelder, 1967). Therefore,at Level 0, children may have constructedmanygeometricobjectsat the perceptualplane.They often can match and identify many shapes (most reliably,prototypes).However, they attendonly to a propersubsetof a shape'svisualcharacteristicsandareunableto identifymany common shapes. In tactile contexts, they can distinguishbetween figures thatare curvilinear(e.g., a circle) and those that are rectilinear(e.g., a square)but they cannotdistinguishamongfigureswithinthoseclasses.Evenin visualcontexts,these childrenmay not be able to constructan image of shapes or a re-presentationof the image. They are unable to rotateshapes and place them into part-wholerelationships(Wheatley& Cobb, 1990). Thus,beforeLevel 1, studentslack the ability to constructand manipulatere-presentablevisual images of geometricfigures. Second,at levels of thinkinghigherthanLevel 1, geometricknowledgecontinues to includenonverbal,imagisticcomponents;thatis, every mentalgeometricobject includes one or more image schemes-recurrent, dynamicpatternsof kinesthetic andvisual actions (Johnson,1987). Thus, imagisticknowledgeis not left untransformed and merely "pushedinto the background"by higher levels of thinking.'5 Imagery has a number of psychological layers, from more primitive to more sophisticated(eachof which connectto a differentlevel of geometricthinking)that play different(but always crucial)roles in thinking,dependingon which layer is activated.Thus,even at the highestlevels, geometricrelationshipsareintertwined with images, thoughthese may be abstractimages. Further,imagistic knowledge is used by some childrenas the basis for visually based reasoning.For example, this reasoningmay even involve an early form of Level 3 relationalreasoning dealing with figure classification. In the following dialogue, second-gradeLG student,Coletta, statedthata squareis a rectangle. Teacher: Does thatmake sense to you? Coletta: It wouldn't to my [4-year-old]sister, but it sortof does to me. Teacher: How would you explain it to her? Coletta: We have these stretchysquarebathroomthings. And I'd tell her to stretchit out and it would be a rectangle.

Here Colettauses imageryto reasoninformallyaboutthe relationshipsbetween figures-a squareis a rectanglebecause it could be stretchedinto a rectangle.She may have accepted certain"legal"transformations(e.g., those that preserve90? 15Earlyworkby van Hiele, andthe workof his interpreters,espouse this view most directly.In more recent writings,van Hiele (1986, p. 42) alteredhis position of valuing higher levels more so thanthe visual level, thoughhis idea of the psychological role of imageryis more obscure.Statements,such as "However,afterthe explicationof thosepropertiesandrelationsby an analysisor discussion,the symbol loses the characterof image, acquiresa verbalcontent, and thus becomes more useful for operations of thinking"(p. 61) and "Theimage has fallen into the background"(p. 62), maintainthe notion of the decreasingimportof images, but others,such as "Ifabstractstructuresaredevelopedfromvisual structuresin which images play an importantpart,a portionof the rich(andunexpressed)contentswill probably be structuredtoo" (p. 67), suggest more agreementwith the basic ideas expressedhere.

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angles,whichshe previouslyhadassociatedwithbothrectanglesandsquares).This illustratesa process of reciprocalshapingoccurringbetween her imageryand her verbaldeclarativeknowledge (for otherexamplesfromLG, recall Jonathan'sand Jeremy'suse of imagistic processes detailedin chapter3). In summary,imagistic knowledge strengthensgeometricthinkingby supportingmore robustand generally applicablereasoning.This pertainsto every level of thought;for example,high school studentswere reportedto use both Level 1, visually based reasoning,and Level 3, analytical reasoning, simultaneously in locus tasks (Hershkowitz & Dreyfus, 1991). To repeat,ourargumentdoes not meanthattheimages do not changewith development.The essence of Level 2 thinking,for example, lies in the integrationand synthesisof propertiesof shapes, not merely in the recognitionof shapes.At this level, childrenhave transcendedthe perceptualand have constructedthe properties to be singularmental geometric objects that can be acted on and not merely descriptionsof visual perceptionsor images (cf. Steffe & Cobb, 1988). Ideally, however, these objects are not solely "words or pictures"(Davis, 1984) but a synthesis of verbal declarative and rich imagistic knowledge, each interacting with andsupportingthe other.Thus,the questionshouldnot be whethergeometric thinkingis visual or not visual. Rather,the questionis whetherimageryis limited to unanalyzed,global visual patternsor if it includes flexible, dynamic,abstract, manipulableimagisticknowledge(Clements,Swaminathan,Hannibal,& Sarama, 1999). This lattertype of knowledge,andthe concurrentdevelopmentof activeand reflectivevisualizationthatacts on figures,in additionto drawings,is a viable goal at all levels of thinking.Therefore,we believe that Level 1 should be called the syncreticlevel, ratherthanthe visuallevel, signifyinga globalcombinationwithout analysis (e.g., analysis of the propertiesof figures). In transitionbetweenLevel 1 andLevel 2, studentslearnto recognize,describe, andmanipulatenot only individualshapesbut also theircomponentsand,eventually, their properties.Note that studentsrecognize and describethe propertiesof individualfigureswithoutnecessarilyascribingthosepropertiesto all of the shapes in that class. Knowledge of propertiesof isolated shapes ("this rectangle")or classes of shapes (rectangles)eventually coalesces into knowledge of properties of two-dimensionalshapes and a generaldispositionto make sense of novel twodimensionalsituationsby analyzingproperties.Figures32 and33 illustratethe two contrastingconceptualizationsof geometriclevels of thinking.They encapsulate argumentsto this point and set the stage for argumentsdevelopedin the following section. NATURE OF THE LEVELS What is to be said about the nature,especially the hierarchicalnature,of the levels? First, they are indeed levels, ratherthan stages. Karmiloff-Smith(1984) defined a stage as a substantiveperiod of time characterizedby cognition across a variety of domains qualitativelydifferentfrom that of both the precedingand

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Time Figure 32. Hypothesized linear view of the levels of geometric thinking. Note: In this view, each level ripensto fruition,engenderingthe beginningof the next level-which incorporatesand subordinatesthe earlierlevel-and finally fades away.

Shading indicates dominance

(D

a

E 0

0

Time Figure 33. Hypothesized synthetic view of the levels of geometric thinking. Note: In the synergisticview espoused here, types of knowledge develop simultaneously.Although syncretic knowledge is dominantin the early years (darkershading indicates dominanceof a particularlevel of thinking),descriptiveknowledge is presentand interactswith it, thoughweakly (symbolized by the small double arrowat the left). Syncreticknowledge, descriptiveverbalknowledge, and to a lesserextentat first,abstractsymbolicknowledgegrow simultaneously,as do theirconnections.When abstractknowledge begins ascendance,connectionsamong all types are establishedand strengthened (indicatedby thickerarrows).The unconsciousprobabilitiesof instantiationassociatedwith each level are indicatedonly by the intensity of shading.Not shown but similarin progressionare the executive processes that also develop over time. The executive processes serve to integrate these types of reasoningand, more important,to determinethe level of reasoningthatwill be appliedto a particular situationor task.When only this dominanceis representedin a graph(see Figure34), we see a version of the "overlappingwaves" metaphor(Siegler, 1996). (Note thatFigures 32 and 33 and the accompanying analysis were originallypublishedin Clements, 1992.)

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a) c C

E 0

0

Time Figure 34.

succeeding stages. Similarly, a level is a period of time of qualitativelydistinct Thus, studentsareat Level 1 when their cognition,but withina specific domain.16 overall cognitive organization and processing disposes them to think about geometricshapesprimarilyin termsof visualwholes;they areat Level 2 whentheir overallcognitiveorganizationdisposesthemto thinkaboutshapesin termsof their propertiesandwhenthe studentshave the abilityto do so for the shapeswith which they have worked. The following two episodes illustratethe differencebetween Levels 1 and 2 with studentsworkingin LG. A second graderwas examiningher attemptto drawa tilted squarein a version of Logo that allowed studentsto erase commandsand their graphiceffects. This studentused a trial-and-errorapproach,giving one commandat a time, erasinga commandif the partof the figure the commanddrew did not look right.Because she did not use 90? turns, the figure this students produced was only a crude approximationof a square.But she concluded that it was a square,reasoningas follows. Howdo youknowit's a squareforsure? Interviewer: Michelle: It's in a tilt.Butit's a squarebecauseif youturnedit thisway,it wouldbe a square. Michelle did not referto properties;her decision was based on her thinkingthat the figurewill look like a squarewhen turned.Contrastthis visual responseto that 16Thereareother of levels. Forinstance,they arehorizontalmarkerson a verticalscale interpretations of measurementwith no temporalcomponent(Glaserfeld& Kelley, 1982). We use Karmiloff-Smith's interpretationhere.

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of two fifth graders who had drawn a tilted square by using equal forward commandsand 90? turns. Interviewer:Is it a square?

Sammy:

Yes, a sidewayssquare.

Interviewer: How do you know?

It hasequaledgesandequalturns. Thoughthese studentswere dealingwith the same taskin similarenvironments, the basis for their decision making and figure identificationwas different. The second graderconstructedher figure visually andused visual means to identify it. Her justification was disconnected from the process she used to construct the figure but was still visual. In fact, she had statedearlierthatsquareshave all sides equal.But, althoughshe triedto implementthis equalityin herLogo procedureby makingthe inputsto the forwardcommandsall 30, she abandonedit by addingtwo forward5 commandsfor the last side, in orderfor the figure to close. The fifth graders,on the otherhand,knew the propertiesof a squareand knew how to use Logo to imbuetheirfigurewith those properties.Theyjustifiedtheiridentification by referringto those properties. An additionalinsight can be gleaned from the first episode. According to van Hiele, for studentsat the visual level, "Thereis no why, onejust sees it" (van Hiele 1986, p. 83). He continuesby saying, Withoutthe existenceof a networkof relations,reasoningis impossible.The first speaker[a studentat Level 1] didnotarriveat hisjudgmentby meansof reasoning. He sawtherhombandthatwassufficient.Norwouldit be possibleto havehisjudgmentwithdrawn by meansof reasoning(p. 110). Sammy:

As Michelle's remarksindicate,however,therewas a "why"for her. She reasoned thatthe figurein frontof her was a squareby performinga visualtransformation on it andby comparingthetransformed imageto herimageof a square.Itis notas simple as van Hiele's descriptionmightsuggest;Michelledid not simplysay "It'sa square because it looks like a square."Some studentsat the visual level may utilize only simplevisualcomparisonsto makejudgments.Butthisepisodeshowsus thatothers, who areperhapsmore advancedbut still at the visual level, use more sophisticated visualthinking.Note,however,thelimitationsof herimagery.She visuallyconcludes thatthe figureshe drewcouldbe turnedto look like a squarein standardorientation. She does this even thoughthe figuredrawndoes not have all the propertiesthatshe previouslynamedfor squares(i.e., thatall the sides areequal).Therefore,hervisual imagerywas not stronglyconnectedto her verbal,property-basedknowledge. Neitherstagesnorlevels arerecurrent(Karmiloff-Smith,1984);thatis, a student at stage (level) n does not returnto stage (or level) n - 1. Van Hiele (1987) claims that the levels of thinkingapply to both geometry and othermathematicaltopics such as arithmetic,but he also claims that childrenusually are at differentlevels within these differenttopics. Further,most empiricalstudies suggest thaton differentsubtopicsof geometry, people exhibitbehaviorsthatareindicativeof differentlevels (Denis, 1987;Mason,

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1989; Mayberry, 1983) and sometimes favor reasoning at different levels on different tasks (Burger & Shaughnessy, 1986; Clements & Battista, 1992b). Assessment assigning people to a single level forces a partialloss of the richness of students'responses.In severalLG interviewtasks,codes suchas "1.5"wereused to moreaccuratelyindicatestudents'responses(cf. Gutierrez,Jaime,Shaughnessy, & Burger, 1991). The finding of distributedresponsesacross levels suggests that the domainat which levels of thinkingoperatemay be, at most, subtopicswithin the broadtopic of geometry,althougha level reachedin one subtopicmay catalyze learning to think at that level in a different subtopic. (Recall our elaborations regardinglevels of thinkingfor polygons and motions in chapter3.) Summarizing,there are three reasons to considerreconceptualizingthe nature of levels of geometricthinking.First,the numberof levels (andeven the mapping betweenthese differentinterpretations)is ambiguous.Second, therearereportsof strongerandweakerperformanceat certainlevels. Third,people's performanceis reportedto be spreadacrosslevels. Thus,the questionemerges:"Howwide a band canbe permittedbeforethepresentnotionof hierarchicaldependencymustbe reformulated?"(Clements& Battista,1992b). Further,therearefew signs of discontinuitiesbetweenlevels thatarepostulatedas fundamentalto the theory(van Hiele, 1986, p. 49). Also, growthin small incrementsis observed more frequentlythan large jumps in learning (Burger & Shaughnessy, 1986; Fuys et al., 1988; Lunkenbein,1983;Usiskin, 1982). Consistentwith these results,we takethe position thatlearningis incremental(see Figure33) ratherthanintermittentandtumultuous (i.e., occurringmainly between stable levels). Also, it might be arguedthat the levels hierarchyis more logical thanpsychological in nature(Clements& Battista,1992b).Therearetwo theoreticalrationales for this argument.The first arguesthattheremay be no otherway thatbehaviors at each level could possibly be sequencedon logical grounds;if thatis so, thenthe hypothesizedsequenceis not explanatory(or even particularlydescriptive)in any psychologically meaningful way and does not need empirical verification.The second arguesthatthe sequencemay be an artifactof externalandarbitraryeducational engineering.We discuss each of these rationales,in turn,in the following paragraphs. First, the set of levels might constitute what Brainerd (1978) has called a "measurementsequence."He postulatesthatoften a behaviorsequenceis not "in the organism"butrather"inthe tests";thatis, the sequenceresultsfromdefinitional connections between the behaviors being measured.A measurementsequence occurs when tasks, measuringbehaviorsat each stage or level, requirethe knowledge and proceduresof the precedinglevels as well as additionalknowledge or procedures.Thus, invariantsequences must, by logical necessity, be "found"by empiricalstudiesbecauseit is not possible to devise valid tests thatdo not measure earlieritems. This is undoubtedlya valid concernin some instances.However, it is not certainthatwe can a prioriand logically determinethat a task requiresthe knowledge and proceduresof the precedinglevels. In some cases, for example, learningsimple numberconcepts like countingand cardinalitybefore learningto

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multiply, such a determinationmight be made. However, Brainerdgives as an example"learningto addbeforelearninghow to multiply"(p. 177);in this instance, it is possible that the a prioriprerequisitestatus, "logical"to most adults, would not apply to every studentin every situation.The mathematicseducationresearch literatureindicatesthat some "logical prerequisites"do not always precedethose abilities for which they are purportedlyprerequisite(e.g., Hiebert, 1981). The applicabilityof this criticismmay vary,dependingon the contiguousstages one considers.Forexample,logically orderingproperties(Level 3) or constructing proofs dealing with those properties(Level 4) would seem to requireawareness of those properties(Level 2) by logical necessity. In this instance, Brainerd's aforementionedmeasurementsequence criticism might be applicable.However, it is not clear if logically orderingclasses of figures requiressyncretic (Level 1) knowledge in the same way. This leads to the second theoreticalcriticism. The second theoreticalrationaleis thatthe levels might resultfrom curriculum sequences. That is, the empiricalresults, that level n appearsbefore level n + 1, may be an artifactof the curriculumsequenceexperienced.Thereis some support for this notion.Forexample,even young childrencan be instructedin a nonvisual, descriptivemanner,leading to higherachievementon certainconcepts. In particular,relationshipsfor geometricfigures thatdo not have a small numberof visual prototypes(templates)anddo involve relativelysimple deductiveargument,such as quadrilaterals,can be taughtto first graders(Kay, 1987). In a similarvein, de Villiers (1987) reportedthat hierarchicalclass inclusion and deductive thinking coulddevelopindependentlyin eighth-gradeandninth-gradestudents,andthathierarchical thinking depends more on teaching strategy than on van Hiele level. Resequencing the curriculumand therefore, ostensibly, students' learning of geometricideas appearto weaken the criticismthatthe levels constitutemeasurement,ratherthandevelopment,sequences.However,by the sametoken,suchresequencing,raisesthe possibilitythatthe orderof emergenceof the types of thinking at each level is neitherlogically (measurement)nor developmentallyconstrained. Despiteempiricalfindingsthatappeartroublingto thetheoryanddespitethe arguments that the levels may be the result of measurementor curriculumsequences ratherthanpsychological sequences,we believe it is too earlyto demotethe levels of geometric thinking to a framework made up only of nonorderedtypes of thinking.Otherevidence on the hierarchicalorderingis fairly consistent (Denis, 1987; Gutierrez& Jaime, 1988; Mayberry,1983), althoughthere are exceptions (Mason, 1989). In addition,the LG dataindicatethat,presentedwith appropriate geometrictasks and problem-solvingtools (in this case, modified Logo environments), young children, who initially respond at the syncretic level, construct ideas aboutpropertiesof geometric figures and come to appreciatethe power of using this knowledge in the solution of problems. We hypothesizethat,even thoughall types of thinkingdo grow in tandemto a degree,a criticalmass of ideas fromeach level mustbe constructedbeforethinking characteristicof the subsequentlevel becomes ascendantin the student'sorientation toward geometric problems (see Figure 33). Thus, while multiple levels

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develop incrementallyand simultaneously,one level maintainsa privilegedstatus (Minsky, 1986, describesa similarhypothesisaboutstages of development).This statusis achievedby both intentionsandinterschematicrelationshipsthatyield an internally-computedprobabilityof successful responsefor each level of thinking thatis appliedto the given situation.Successful applicationleads to the increasing use of a particularlevel. However,underconditionsof increasedtask complexity, stress,or failure,thisprobabilitylevel decreasesandan earlierlevel servesas a fallback position (Hershkowitz& Dreyfus, 1991; Siegler, 1986). No level of thinkingis deletedfrommemory(Davis, 1984). Thatis, re-recording a mental representationat a more explicit level does not erase the earlierrepresentation. (Indeed, replacing or deleting any successful representationswould eliminate "fallback"strategies that arguably are essential when new, untested knowledge is being formed, cf. Minsky, 1986.) The schemes that constitutethe representationareinstantiateddynamically(i.e., theirformis interactivelyshaped by the situationalcontext, includingthe task demands);and they can be modified at this time (or new schemes can be formed), although schemes that are basic subschemes of many other schemes become increasinglyresistantto alteration becauseof the disruptiveeffect alterationwouldhaveon cognitivefunctioning.The syncretic(visual)level has a specialstatusin thisregard.Due to dailylife in a spatial world, this level receives constantapplication.Consequently,it is maintainedas a separatesystem, andthe probabilityof its instantiation(constructionof a mental model of the world in a particularsituation) also is consistently maintained. Therefore,this level is an interpretiveanddefaultbehaviorfor most people in most situationsthatthey interpretto be nonmathematical.When the situationrequires an interpretationfrom a mathematicalperspective,however, a schematicshift is at the ascenlikely to occurthatleads to a rapidinterpretation(or reinterpretation) dantlevel of the person involved. This situationposes a problemfor postulatinglevels of geometricthinking.That is, levels aretheoreticallynonrecurrent(Karmiloff-Smith,1984);however,people not only can, but frequentlydo, "return"to earlierlevels of geometricthinkingin certaincontexts.In the LGresearch,this is illustratedmost clearlyin the triadtask. (See also Lehrer(1998); Pegg (1998) providesdifferentexamples.)Therefore,we postulatethe constructof nongeneticlevels. Nongenetic levels have two special characteristics.First, progress through nongeneticlevels is determinedmoreby social influences,andspecificallyinstruction, thanby age-linkeddevelopment.(At this point,this only implies thatprogression does not occurby necessity with time but demands,in addition,instructional intervention.Certainlevels may develop undermaturationalconstraints;further researchis needed on this issue.) Second, althougheach higher nongeneticlevel builds on the knowledge thatconstituteslower levels, its nongeneticnaturedoes not precludethe instantiationand applicationof earlierlevels in certaincontexts (not necessarily limited to especially demandingor stressfulcontexts). For each level, thereexists a probabilityof evokingeachof numerousdifferentsets of circumstances.However,thisprocessis codeterminedby consciousmetacognitivecontrol,

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and this controlincreases as one moves up throughthe levels. Therefore,people have increasingchoice to overridethe defaultprobabilities. The use of different nongenetic levels is environmentallyadaptive;thus, the adjective"higher"shouldbe understoodas a higherlevel of abstractionandgenerality, withoutthe implicationof eitherinherentsuperiorityor the abandonmentof lower levels as a consequence of the developmentof higher levels of thinking. Nevertheless,the levels would constituteveridicalqualitativechangesin behavior, especially in regardto the constructionof mathematicalschemes (i.e., construction of geometricobjects) out of action. PHASES OF LEARNING Some researchershave maintainedthatgeometricknowledgeis initiallylearned in conscious, declarative form (Anderson, 1983). Proceduralizationgradually replaces the original interpretiveapplicationwith productionsthat performthe behaviordirectly.For example, afterinitially learningand verballyrehearsingthe geometric side-angle-side congruencerule and then figuring out how it applies, studentsbuild a productionthatdirectlyrecognizes the application.This declarative-to-proceduralsequencemay accuratelyportraycertaintypes of learning,especially with older students in traditional educational environments. We posit, however,thatan inversesequencebetterdepictsmuchlearningof geometry,especially at the lower levels. In this view, students'initial knowledge is proceduralized and schematicized at a low level to which there is no conscious access. Gradually,this knowledgeis mentallyre-representedso thatit becomes available, as data,to othercomponentsof the cognitive system and eventuallyto consciousness. (The "doingto understanding"sequence also might occur in laterphases of learning,basedon the interactionbetween symbolic andvisual modes of thinking, on the partiallayersof discriminationthatareconstructed,andon the way in which the computer acts as cognitive scaffolding for the learner,cf. Hoyles & Noss, 1987a.) This view is consistentwith the work of Karmiloff-Smith(1984; 1986; 1990), who postulates a repeating three-phasecycle of representationalredescription. Phases,in contrastto structurallyconstitutedstagesor levels, arerecurrent,general (across-domain) processes that people work through during development or learning. At Phase 1, the building of mental representationsis predominantly drivenby interactionsof the students'goal-directedschemeswith the environment. The students'goal is behavioralsuccess, or the reachingof thatgoal, which sometimesis evaluatedby consistencywith adultresponsesandfeedback.17 Such success 17This sentence could be misconstruedto mean that studentsare controlled the externalenviby ronment.We assume studentsarealways sense-makingbeings; however,duringthis phase in building a representation,they are actively makingsense of theirsocial andphysical environments,ratherthan theirrepresentations,of which they are not yet conscious. This active sense makingis critical,in that it allows the studentsto differentiatebetween environmentsthatdo and do not assist goal attainment.

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leads to the recordingof isolated correspondencesbetween environmentalsituations that do or do not allow or aid the attainmentof the goal in a form inaccessible to the system (i.e., input-outputcorrespondencesalong with theircontextsin compiled form). In this form, any relationshipbetween bits of knowledge are, at best, implicit. When behavioralsuccess is achieved, a Phase 2 metaprocess(a procedurethat operateson internalknowledgestructures)evaluatesthe knowledgebase. Now, the forms goal is notto behavesuccessfully,butto gaincontrolovertherepresentational (Vygotsky, 1934/1986, similarlystatedthatthe developmentof thoughtcannotbe derivedfrom the failureof thoughtandpostulateda genetic predispositionto gain control of mental representations).The first operationof Phase 2 is to re-record Phase 1 representations in a form which can be accessed, though not yet consciously. The implicit representationsare analyzedinto semanticizedcomponents, linking them into a simplifiedbut growing networkstructurethatis predicated on the usefulness of the initial correspondencesto goal attainment.The second operationis to form relationshipsbetween bits of knowledge. These two new operationsplace demandson cognitive processing, which togetherwith the in new situ(over)simplifiedstructureandthe need to test the mentalrepresentation ations, often leads to new "errors"-ostensibly a step backwardto Phase 1 from an adult'sperspective-that maskthe progressin explicatingrepresentationsof the domain.Finally,these two operationsconstitutean internalizationof relationships andprocesses thatwere previouslyonly implicit (cf. Steffe & Cobb, 1988). is achieved,studentsdevelop Phase 3 control Once successfulre-representation mechanismsthatintegrateandbalanceconsiderationof the externalenvironment andthe new internalrepresentationalconnectionsforged in Phase 2. At the end of Phase3, theseconnectionsarere-recordedagainin abstractsymbolicform,the first form accessible to conscious thought.Now, performanceimproves beyond that which was achieved at eitherof the two earlierphases. At the syncreticlevel of geometricthinking,studentsareimplicitlyrecognizing the propertiesof shapes. For example, their schemes for squaresand for rectangles both containpatternsfor rightangles. But these arepatternsin spatialsubsystems thatemerge when instantiated;they are not conceptualobjects (mentalentities that can be manipulatedor scrutinized,Davis, 1984). Right angles are not representedexplicitly and, therefore,no relationshipis formedbetween them. In general,this type of representationexplainshow operationat one level can presuppose knowledgefromthe succeedinglevel, withoutallowing access to knowledge at the higher level because the form of such knowledge is proceduralizedor schematizedbehaviorallyand is, therefore,inaccessible to the rest of the cognitive system. Whenshapesaredealtwith successfullyon the level of behavior,metaprocesses re-recordthe mental representations,creatinga mental geometric object for the visual image of the right angles and a link between these mental objects for differentcases of rightangles,includinglinksbetweenthose in rectanglesandthose in squares.Because studentsin Phase 2 are seeking controlover theirrepresenta-

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tions of these geometricforms, some increasesin "errors"(from an adult's point of view) may occur (e.g., maintainingthe notion that a nonrectangularparallelogramis a rectangle"becauseyou're only looking at it from the side"). Eventually,the propertiesbecome conceptualobjects thatare availableas data to conscious processes.Thatis, visual featuresbecome sentientin isolationandare linkedto a verballabel;andthe studentbecomes capableof reflectingon the visual featuresandthuscan explicitlyrecognizethe shape'sproperties.At first,however, the flexibility of applicationis limited. Indeed,studentscan be expected not only to thinkat differentlevels for differenttopics (Clements& Battista,1992b)butalso to think at differentphases for differenttopics. Problem solving and discussion involving the geometric objects help build connections between the constructed knowledge (e.g., of right angles) and other similarlyaccessible knowledge (e.g., of parallelismandside lengthsof rectanglesandsquares).Eventually,connections arebuiltbetweenpropertiesof figures such as rectanglesand squaresandof properties of othergeometricobjects. In this manner,students'geometricknowledge can become increasinglyabstract,coherent,andintegratedbecauseit is freedfrom the constraintsof compiled, and thus inflexible, mental representations.At each level, the degree of integrationincreasesas the connectionsspangreaternumbers of geometric,and eventually,nongeometrictopics (cf. Gutierrez& Jaime, 1988). Here again, instructionhas a stronginfluence. Ideally, it encouragesunification; however, the instructionof isolated bits of knowledge at low levels retardssuch development.Unfortunately,the latteris pervasive in both curriculummaterials (Fuys et al., 1988) and teaching (Clements & Battista, 1992b; Porter, 1989; Thomas, 1982). To place instructionalimplicationsin a differentlight, it is only afterthe third phase that students become explicitly aware of their geometric conceptualizations; therefore,it is afterPhase 3 thatthe last threeinstructionalsteps in the van Hiele model (explicitation,free orientation,andintegration)can begin.18An implication is that short-circuitingthis developmental sequence (e.g., by beginning withexplicitation)is a seriouspedagogicalmistake.Deprivedof the initialconstruction of their own mental geometric objects and relationships(images), students constructPhase 1 (behaviorally"correct")verbalresponseson the basis of "rules withoutreason"(Skemp, 1976). A more viable goal is the constructionof mathematicalmeaningfrom actions on geometricobjects and subsequentreflectionson those actions. In summary,developmentof geometricpropertiesas conceptualobjects leads to pervasiveLevel 2 thinking.This developmentrepresentsa reconstructionon the abstract/conscious/verbal plane of those geometricconceptualizationsthatPiaget and Inhelder(1967) hypothesizedwere first constructedon the perceptualplane 18The firsttwo steps, informationandguidedorientation,of the van Hiele model's five-step instructional sequence would be, with some modification,consonantwith the three phases of development describedhere. This topic is relatedto, but differentfrom, the topics addressedhere and will not be examined.

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and then reconstructedon the representational/imaginalplane. Thus, Level 2 thinkingrequireswhatPiagetcalledthe constructionof articulatedmentalimagery, which develops most fully throughthe combinedenhancementof both imagistic andverbaldeclarativeknowledge,as previouslydiscussed.We posit thatthe same recurrentphases explainthe reconstructionon each new plane (level) but thatthe role of social interactionandinstructionincreasesin importancewith higherlevels. Finally,the theoryof reiteratedphasesof re-representation appliesequallyto children and adults, regardless of their overall stage of cognitive development (Karmiloff-Smith,1990). This is consonantwith reportsof low van Hiele levels among high school students and preservice teachers (Burger & Shaughnessy, 1986; Denis, 1987; Gutierrez& Jaime, 1988; Mayberry,1983; Senk, 1989). LEARNINGGEOMETRYWITHLOGO:A RETROSPECTIVELOOK Ourtheoreticalanalysisreturnsus full circle to ajustificationfor the use of Logo or othersimilarcomputerenvironmentsfor learninggeometry.Drawinga geometric figure on paper, for example, is for most people a highly proceduralizedand compiledprocess. Such a procedureis alwaysrunin its entirety.This is especially the sequentialinstructionsthat truefor youngchildren,who have notre-represented alter the drawing procedure in any they implicitly follow. Then, they cannot much less substantivemanner(Karmiloff-Smith,1990), consciously reflect on it. In creatinga Logo procedureto drawthe figure, however, studentsmust analyze the visual aspects of the figure and theirmovementsin drawingit, thus requiring themto reflecton how the componentsareput together.As (Papert,1980a) states, "[When]intuitionis translatedinto a program,it becomesmoreobtrusiveandmore accessible to reflection"(p. 145). In our words, Logo programmingprovidesthe motivation,context, and language to supportthe re-recordingof Phase 1 proceduralizedand compiled drawing processes into explicit mental representations accessible to consciousness. Challengingstudentsto altertheir paper-and-pencil drawingtechniquesmay also be beneficial. For instance,they might be asked to draw a figure in a differentsize or orientationor to drawall the componentsof a figurebutin a differentorder(e.g., all the corers first).However,in Logo students have to specify detailed,specific steps to an noninterpretiveagent. The resultsof thesecommands,andthe commandsthemselves,canbe observed,reflectedon, and corrected;Logo serves as an explicativeagent.As may be recalled,an exampleof this was whenfifth-graderJonathanstruggledto decidewhetherhe coulddraweach of a groupof figureswith a rectangleprocedure,whichtook the lengthsof the sides as input.Both Logo and the activity helped Jonathanwork throughthis problem. In one case, he did not even have to try his second solution.When deciding what inputsto use, he recognizedthatthe relationshipbetweenadjacentsides themselves was consonant or not (in this instance, not) with the implicit definition of a rectanglein the Logo procedure.His first attemptusing Logo and his "running through the procedurein his head" contributedto this emergent certainty. (In chapter5, we discussed the imagistic processes Jonathanused.)

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Similarly,studentsin a Site 1 LGclassroomhadexploredthe notionthata square was a rectangle-a specialtype of rectangle.They thenhadcreateda parallelogram with Logo. One of the studentscame up to his teacherthe next day and said that he was thinkingaboutparallelogramsat home. "Is a rectanglea special parallelogram?"he asked. "Why do you say so?" "Because it's just like the rectangle procedureif it had 90? turns."This conversationshows thatthe studenthad used his Logo experiencesto extendhis thinkingaboutrelationshipsbetweenpolygons. More research, however, is needed before firm conclusions can be drawn regardingourreconceptualizationof levels of geometricthinking.In particular,the types of thinkingthatdefine each level mustbe furtherclarifiedto adequatelytest the theory of levels of geometric thinking against alternativetheories such as measurementsequences and nonordered,curriculum-drivensequences. Further, questions need to be asked, such as the following. Why exactly does level n + 1 develop afterlevel n? Whatare the psychological functionsof thinkingprocesses at level n thatlead both to increasedmasteryof the environmentand to the eventual constructionof level n + 1 (i.e., in what ways are level n thinkingprocesses necessaryandbeneficial,ratherthanmerelydeficient)?How well does knowledge at level n have to be constructedbeforeknowledgeat level n + 1 knowledgebegins its ascendance,andwhy? Whatlevels of thinkingaremathematicallyuseful for the studentandnot merelyusefulas analyticaltools for the teacherorresearchers?What internaland externalevents mediatesequencedgrowththroughthe levels? Finally,movementsto richerimagisticandverbaldeclarativeknowledgeconstitutedifferent,albeitmutuallyreinforcing,pathsfromLevels 0 and 1 to higherlevels of geometric thinking. As was argued previously, both types of knowledge (including shortdeductive chains right from the beginning) and their interaction ideally shouldbe facilitatedby geometryinstruction.One of the mainimplications of the precedingcritiquefor educationalpracticeis thateven acceptingthe veracity of the levels in one form or another,teachersshould expect and encourageuse of differenttypes of thinking-visual, descriptive,andlogical-at every level, albeit in differentratiosandwith differentforce. Forexample,earlylogical thinkingmay consist of shortdeductivechains thatare instantiatedonly as concretestatements ratherthanas abstract,generalizedstatements. To summarize,Logo programmingcan help studentsconstructelaborateknowledge networks(ratherthan mechanicalchains of rules and terms) for geometric topics. Thereareseveraluniquecharacteristicsof Logo thatcan facilitatestudents' learning,which have been developedin this monographor in previousreviews and discussions (Clements & Meredith, 1993; Clements & Sarama, 1997; Noss & Hoyles, 1992).Note thatclaimsor conjectureshavenotbeenincludedhere.Instead, the following list constituteshypothesesthathave received empiricalsupport. The first set of characteristicsinvolve the Logo environment,per se. The nature of various Logo environments,especially those enhanced to aid mathematical learning,providescertaineducationaladvantages. * The commands and structureof the Logo language can be consistent with mathematicalsymbols andstructuresin ways thatarepedagogicallyuseful. For

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*

*

*

*

*

*

* *

*

Implicationsfor Theory

example, turtlegeometrycommandssuch as FD andRT focus students'attention on criticalaspects of figures. Drawingwith Logo's turtlegraphicsandcreatingrunnablecode aremeaningful andinterestingtasksfor students;these tasksthenmotivatethe use andlearning of geometricand othermathematicalideas (Ainley, 1997). The turtleworldinvolves measuresthatarevisible, quantifiable,andformalizable, helping to connect spatial and numericthinking(Clements et al., 1996; Clementset al., 1997; Noss & Hoyles, 1992). Logo can encouragethe manipulationof screen objects in ways that facilitate students'thinkingof them as mathematicalobjects and thus as representatives of a class. In this way, Logo can evoke more abstractgeometricthinking. Logo can promote the connection of symbolic with visual representations, therefore, supportingthe constructionof mathematicalstrategies and ideas from initial intuitionsand visual approaches(Clements& Sarama,1995; Noss & Hoyles, 1992). Thus, it can serve as a transitionaldevice from physical movements to more abstractmathematicalconceptualizations.It can make mathematics more concrete while simultaneously supportingthe students' formalizationof actions algebraicallyas a computerprogram(Hoyles, 1993). Logo can permit students to "outline"or "sketch"their ideas or problemsolvingplansandthenelaborateandcorrectthem.Studentscan workon specific cases as they work "topdown"(Hoyles & Noss, 1987a) or as they thinkabout generalizations.Interactionwith the computercan directattentionto key points in the mathematicalproblemsolving. Studentscan use Logo's symbolic code to reducethe cognitive load of a task, dealingwith one relationshipat a time andthensynthesizingthemall into a final procedure(Hoyles & Noss, 1987a). Logo can be a medium for the expression of mathematicalideas (Hoyles, Healy, & Pozzi, 1992). It can provide tools and motivationto make relationships explicit. Especially with appropriatetools and structure(Clements & Sarama,1995), it can reducethe "cost"of changingthe expressionof ideas or experimentingwith parameters(Kynigos, 1993). Thisexteralization of ideascan also serveas mediafor communicationof mathematicalrelationships(Hoyles, 1993). Students'thoughtsare capturedin their commandsand can be reenactedand revised. Theiractions,and the turtle'sactionsthatthey command,can become objects of mathematizationand reflection(Clements& Sarama,1997; Kieren, 1992; Noss & Hoyles, 1992). Studentscan also returnto test out ideas. Thatis, they can alwaysreturnto "run"theircode andthusreturnto action-orientedways of thinkingas they extend higherorderideas (Kieren, 1992). Logo can help documentstudentactions, leading to meaningfulmathematical symbolization.Studentscan build a symbolic mathematicallanguage that is based on this Logo documentation.

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* Logo can provide scaffolding for mathematicalanalysis; that is, a symbolic representationon the computercan allow the studentto erectscaffoldingaround the solutionof a problem,andsubsequentlyattendto only those elementsof the solution for which details need to be filled in (Noss & Hoyles, 1992). * Logo providesfeedback.Logo's feedbackis distinctfrom"rightor wrong"feedback associatedwith CAI materials;instead,it is an instantiationof students' expression of geometric ideas. With extended environments,such as Turtle Math, studentsreceive symbolic feedback when they manipulatea graphics object and graphic feedback when they edit their symbolic code (Sarama, 1995). * Logo canrequireandfacilitateprecisionandexactnessin mathematicalthinking. * Logo can provide a window to students'mathematicalthinking(Noss, Healy, & Hoyles, 1997; Weir, 1987). Such an environmentprovidesa fruitfulsetting for teacherswilling to work with, and listen to, students. * Logo can help students construct more viable knowledge because they are constantly evaluating a graphical manifestation of their thinking. Further, studentsmoreablyapplymathematicalknowledgein problem-solvingsituations. * Because studentsmay test ideas for themselves on the computer,Logo environmentsaid studentsin moving from naive to empiricalto logical thinking. These environments also encourage students to make and test conjectures. Thus, Logo facilitates students' development of autonomy in learning (as opposedto seekingauthoritativeopinion)andpositivebeliefs aboutthe creation of mathematicalideas. * Logo can help structurestudents'play to encouragesymbolic and mathematical characteristicsof exploratorymathematicalactivity (Hoyles, 1993). Or, as Papert(1993) states,"Thecomputersimply,but very significantly,enlargesthe rangeof opportunitiesto engage as a bricoleuror bricoleuse in activities with scientific and mathematicalcontent"(p. 145). * Even if used for less exploratorypurposes,Logo allows studentsto searchfor relationshipsthatseem beyond theirgraspat thatmoment;they can try a range of possibilities on the computer(Battista& Clements, 1986; Hoyles & Noss, 1989), whereason paperthey often become stuck. Consistent across studies is the finding of the critical role played by both the curriculum,in which Logo is embedded,including the tools and structureof the version of Logo itself, and the teacher.The second set of characteristicsbelow, is a summaryof the benefitsof a soundcurriculumfollowed by suggestionsthatmay help provideeffective instruction. * Curriculamust find the rightlevel of representationfor students.For example, it may not be an efficient use of time to have studentswrite Logo procedures to performgeometricmotions,buttheycan use tools availablein enhancedLogo environmentsto do so (Battista& Clements,1988b;Clements& Sarama,1996; du Boulay, 1986).

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* Carefullyplannedandresearchedsequencesof Logo activitieslead to a greater likelihood of learning. * By being intrinsicallyrewarding,these activities shouldencourageanalysis. * Versions of Logo with tools specifically designed to enhance mathematical activity lead to a greaterlikelihood of mathematicallearning than versions withoutsuch tools (Clements& Sarama,1995; Kynigos, 1993; Sarama,1995). * Extendedexperience with Logo may be importantto realizing its maximum benefits. * The best use of Logo may involve full integration into the mathematics curriculum. Too much school mathematics involves exercises devoid of meaning. Logo is an environmentin which studentsuse mathematicsmeaningfully to achieve theirown purposes.The Logo languageis a formalsymbolization thatstudentscan invoke, manipulate,and understand(Hoyles & Noss, 1987b). Using Logo in this way can help fulfill the early vision of "teaching students to be mathematiciansversus teaching about mathematics"(Papert, 1980b, p. 177). Effectiveteachermediationrequiresmultipleactions.Teachersmustbe involved in planningandoverseeingthe Logo experiencesto ensurethatstudentsreflecton and understandthe mathematicalconcepts. Teachers might use the following approaches. * Focus students'attentionon particularaspects of theirexperiences. * Educe informal language and help connect this informal language to Logo representationsand to formal mathematicallanguage for the mathematical concepts. * Suggest pathsto pursue. * Facilitatedisequilibriumby using computerfeedbackas a catalyst. * Encouragemathematicalanalysis. * Continuallyconnect the ideas developed by the studentsto those from other contexts, such as real-worldsituations.

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Tables Table 1 Numberof StudentParticipantsby Treatment,Site, and Grade Control(1988-89) LG (1988-89) Site 2 Site 1 Site 2 Site 1 Grade 26 34 K 25 37 1 44 45 43 47 50 44 2 53 52

LG (1987-88) Site 1 19 45 54

3

26

24

30

23

12

4 5 6

82 48 73

48 48 51

76 55 68

44 47 57

78 48 49

7 19 Total 351 305 348 296 324 Note. Because the main comparisons for the Pre-Post paper-and-penciltest involved the LG and control groups participatingduring 1988-89, these columns are presentedfirst in the tables within this section.

Table 2 Means and StandardDeviationsfor Grades K-6 TotalScores LG (1988-89) Control(1988-89) Site 1

Site 2

Site 1

LG (1987-88)

Site 2

Site 1

Grade Pre Post Pre Post Pre Pre Post Post Pre Post K 48.89 70.65 54.16 59.49 52.77 55.34 57.18 60.94 43.66 67.03 (9.30) (6.12) (6.13) (15.61) (4.86) (6.51) (5.34) (6.80) (8.14) (6.84) 1

2

3 4

52.65

73.76

56.60

67.70

56.21

68.63

58.35

64.54

48.84

64.45

(5.36) (7.36) (6.40) (15.43) (3.86) (9.93) (4.89) (6.52) (6.19) (5.58)

57.29

(9.51) 58.74 (5.21) 61.14 (9.00)

68.81

50.44

69.02

(14.99) 65.67 (4.43) 74.48 (8.09)

(13.96) 54.02 (7.74) 60.14 (10.88)

(6.68) 68.35 (7.16) 65.51 (22.87)

77.60

64.46

77.96

56.53

(5.91) 52.78 (6.14) 59.89 (8.97)

67.93

52.70

56.80

(9.04) 55.55 (8.37) 70.97 (14.51)

(12.57) 55.65 (9.70) 62.81 (6.13)

(11.77) 59.41 (19.64) 60.54 (17.25)

76.89

66.51

72.50

49.29

65.36

(6.64) (7.98) 51.67 (3.74) 55.46 68.97 (7.95) (6.24)

5

71.28

6

(8.43) (6.35) (7.00) (6.51) (6.86) (6.37) (7.73) (7.85) (7.07) (14.88) 68.96 75.00 71.67 77.96 68.62 69.66 67.74 73.91 60.53 65.89 (9.76) (6.90) (6.36) (12.89) (9.97) (10.77) (9.01) (7.15) (7.80) (11.02)

7

71.20

67.92

68.24 (12.75)

71.35

75.69 (4.68)

Notes.All dataexceptthelasttwocolumnsarefromthe 1988-89fieldtest,whichincludedmatched andcontrolgroups.Thus,forthefirstfourcolumns,LGteachersatSite1hadtwoyears experimental whereasthoseat Site2 hadonlyone yearexperience. Datain experiencewiththeLGcurriculum, thelasttwocolumns,fromthe1987-88fieldtest,camefroma similartest,butthereweredifferences in items thataffect the score. Missing datawas mistakenlynot collected by the third-gradeteacher.

156

Tables

Table 3 Treatmentx Timex Grade x Site Interactionsfor 88-89: Post Hoc Analyses K-6 Total 4-6 Total Triangles(Items 1.1-1.3) LG Control Control LG Control LG * 1 2 1 1 2 1 2 2 1 2 2 Grade K 1 2 3 4 5 6

+++ +++ + ++ + + +

+ + ++ ++ ++ +

+ + + +

+

+ +

++ ++ +++

++ ++ ++

+ +

+ + +

++ ++ + ++ ++ ++

+ ++ ++ + ++

+ + + + +

*Note. l=Sitel. 2=Site2. frompretestto posttest Key: + = Significantly improved ++ = Inaddition,significantly othertreatment (eitherLGorcontrol)groupat outperformed thatsite all othergroups +++= Inaddition,significantly outperformed

Table 4 Means and StandardDeviationsfor 4-6 TotalScores Control(1988-89) LG (1988-89) Site 2 Site 1 Site 2 Site 1 Pre Post Pre Post Pre Post Post Grade Pre 78.79 4 86.96 72.78 97.50 73.23 95.06 72.01 91.80 (12.76) (11.87) (10.90) (11.64) (10.92) (11.40) (8.74) (9.62) 5 92.48 105.81 79.54 106.33 90.01 100.39 83.74 94.96 (12.25) (9.42) (9.89) (9.68) (10.63) (10.50) (11.55) (12.87) 6 87.48 100.81 93.61 108.01 86.43 90.12 87.70 99.25 (16.15) (11.12) (9.43) (9.91) (15.45) (16.58) (11.07) (10.25) 7

LG (1987-88) Site 1 Pre Post 79.07 109.02 (12.86) (10.61) 105.10 122.34 (11.78) (8.91) 91.55 106.31 (13.61) (12.41) 111.39 119.64 (13.18) (6.64)

157

Logo and Geometry

Table 5 Means and StandardDeviationsfor Writingthe Numbersof All the Figures ThatAre Triangles,Rectangles, or Squares(ItemsPP/1.1-1.3) Control(1988-89) LG (1987-88) LG (1988-89) Site 1 Site 2 Site 1 Site 2 Site 1 Post Pre Post Pre Post Pre Post Pre Post Grade Pre K 1 2 3 4 5 6 7 K 1 2 3 4 5 6 7 K 1 2 3 4 5 6 7

8.39 (2.79) 8.82 (1.82) 10.28 (2.20) 10.39 (1.70) 11.00 (1.98) 11.59 (1.80) 11.37 (2.19)

11.40 (1.93) 11.95 (1.85) 11.94 (1.98) 11.48 (1.62) 12.14 (2.14) 12.24 (2.12) 12.61 (1.69)

9.30 (2.21) 9.04 (2.16) 9.53 (2.43) 9.96 (1.97) 10.00 (2.30) 11.00 (2.23) 12.27 (1.58)

10.74 (1.90) 11.31 (1.93) 11.50 (1.76) 9.43 (2.04) 12.00 (1.79) 12.78 (1.55) 13.00 (1.46)

Triangles 10.38 9.84 (1.60) (2.34) 9.53 10.92 (1.64) (2.97) 10.47 11.23 (2.16) (3.04) 9.03 9.33 (1.94) (1.68) 10.20 11.89 (2.16) (2.06) 11.61 12.16 (2.07) (1.81) 11.51 11.26 (2.02) (2.25)

8.91 (1.80) 9.38 (1.91) 9.38 (2.33) 10.87 (1.63) 10.24 (2.07) 11.50 (1.89) 11.14 (2.05)

10.56 (1.94) 9.59 (1.89) 9.63 (2.33) 10.62 (1.80) 11.10 (1.88) 11.67 (1.93) 12.36 (1.63)

8.11 (2.35) 8.38 (2.10) 8.81 1.99 10.08 (1.56) 9.95 (2.11) 11.63 (2.06) 10.41 (2.39) 12.05 (1.54)

10.90 (2.27) 12.86 (1.37) 11.90 (1.96) 11.94 (1.80)

8.91 (2.98) 8.41 (2.38) 9.60 (2.41) 10.43 (1.56) 9.78 (2.33) 10.64 (3.00) 10.19 (2.35)

10.05 (1.67) 10.31 (2.87) 10.78 (2.55) 9.78 (1.51) 11.09 (2.98) 11.17 (2.96) 10.48 (2.82)

9.59 (2.18) 9.18 (2.28) 8.58 (2.15) 9.75 (2.47) 9.14 (2.64) 9.54 (2.33) 9.84 (2.59)

Rectangles 10.80 10.12 10.11 (2.00) (1.63) (1.59) 10.83 9.44 10.68 (2.29) (1.18) (3.03) 9.72 8.69 10.15 (2.51) (2.31) (2.19) 10.00 9.30 8.33 (2.80) (1.34) (1.49) 11.28 9.75 11.11 (2.64) (2.08) (2.56) 11.91 9.71 10.78 (2.99) (2.18) (2.87) 12.11 10.49 10.51 (3.00) (2.58) (2.58)

9.85 (1.44) 9.17 (1.99) 8.78 (2.34) 10.00 (2.04) 9.24 (2.12) 9.84 (2.93) 9.39 (2.66)

10.85 (1.74) 8.74 (2.21) 8.51 (2.55) 9.81 (2.34) 8.97 (1.95) 10.02 (2.54) 10.84 (2.88)

8.89 (1.88) 9.56 (1.69) 9.96 (1.68) 9.33 (2.27) 10.41 (1.73) 11.33 (1.81) 11.00 (1.37) 11.00 (1.60)

10.88 (1.45) 10.22 (1.44) 10.95 (1.86) 11.08 (2.06) 12.70 (1.99) 10.85 (1.97) 12.44 (2.43)

11.57 (4.28) 13.66 (1.48) 13.51 (1.53) 13.39 (1.27) 13.20 (1.35) 13.82 (2.06) 13.96 (1.04)

14.20 (1.28) 14.05 (1.10) 13.25 (2.36) 12.09 (2.19) 13.58 (1.89) 13.93 (1.99) 13.41 (2.14)

13.03 (2.81) 13.62 (1.03) 13.11 (1.57) 13.42 (1.56) 13.30 (1.75) 13.70 (1.71) 13.80 (1.21)

13.88 (1.07) 13.74 (1.58) 14.04 (1.40) 13.35 (1.72) 13.14 (2.51) 14.02 (1.63) 14.27 (1.35)

Squares 12.65 13.00 (1.70) (1.63) 13.81 13.92 (1.18) (1.15) 13.33 13.60 (1.49) (1.50) 12.67 12.57 (2.31) (1.94) 13.58 14.14 (1.80) (1.40) 13.67 14.04 (1.31) (.89) 13.88 13.79 (1.32) (1.39)

13.53 (1.08) 13.72 (1.31) 13.18 (1.95) 13.65 (1.23) 13.10 (1.71) 13.66 (1.31) 13.41 (1.22)

13.59 (.99) 13.70 (1.15) 13.56 (1.50) 13.95 (1.12) 13.44 (1.39) 13.89 (1.42) 13.62 (1.98)

9.74 (4.50) 12.87 (2.78) 12.52 (2.64) 14.08 (.79) 13.22 (2.11) 13.98 (1.25) 13.24 (1.92) 13.63 (1.34)

14.00 (1.37) 13.37 (2.15) 13.39 (2.06) 13.80 (1.63) 14.12 (1.61) 13.08 (1.84) 14.56 (.98)

10.71 (1.72) 10.15 (1.99) 11.32 (1.97) -

Tables

158

Table 6 Means and StandardDeviationsfor IdentifyingStatementsThatDescribe Each Typeof Figure (ItemPP/2.10) LG (1988-89) Control(1988-89) LG (1987-88) Site 1 Site 2 Site 1 Site 2 Site 1 Grade Pre Post Pre Post Pre Post Pre Post Pre Post 2 3 4 5 6 7 2 3 4 5 6 7

Squares 5.51 4.98 (1.27) (1.20) 3.50 2.95 (1.57) (.95) 5.86 3.96 (1.80) (1.46) 5.96 6.86 (1.46) (1.06) 5.42 6.14 (2.05) (1.42)

4.83 (1.74) 4.30 (1.82) 3.88 (1.93) 6.91 (1.18) 5.74 (2.16)

6.69 (1.12) 6.30 (1.58) 6.81 (1.39) 7.66 (.57) 7.27 (1.19)

5.38 (1.43) 3.75 (1.73) 4.11 (1.98) 4.67 (1.90) 6.69 (1.42)

4.30 (1.23) 3.57 (1.16) 3.57 (1.28) 5.78 (1.49) 4.64 (1.55)

4.12 (2.05) 4.96 (1.22) 5.90 (1.37) 6.49 (.78) 5.89 (1.28)

Rectangles 4.27 4.71 5.33 4.02 4.64 3.75 (1.53) (1.33) (1.52) (1.33) (1.41) (1.12) 4.30 3.75 5.57 3.70 3.27 3.45 (1.39) (.99) (1.15) (1.03) (1.06) (1.34) 3.61 5.43 3.68 5.14 4.07 5.13 (1.38) (1.34) (1.31) (1.36) (1.47) (1.49) 4.22 6.50 5.04 5.59 4.34 5.24 (1.47) (.91) (1.60) (1.38) (1.54) (1.52) 5.43 6.16 4.46 5.14 4.60 5.86 (1.54) (1.21) (1.69) (1.61) (1.70) (1.48)

6.54 (1.38) 6.78 (.90) 6.98 (1.19) 7.48 (1.05) 7.42 (1.10)

4.85 (1.35) 3.18 (1.26) 4.91 (1.70) 5.59 (1.80) 6.31 (1.60)

5.24 (1.18) 5.05 (1.50) 6.13 (1.70) 6.62 (1.67) 6.98 (1.60)

6.77 8.66 (1.61) (1.02) 6.33 (1.87) 9.22 5.00 (2.08) (.82) 9.58 8.02 (1.62) (1.01) 6.47 9.21 (2.33) (.93) 9.61 8.68 (1.67) (.78) 5.93 (1.67) 6.17 (1.75) 5.33 (1.72) 7.90 (1.75) 6.27 (1.96) 8.21 (1.58)

7.68 (1.88) 8.96 (1.65) 9.15 (1.35) 8.51 (1.68) 9.50 (.99)

Table 7 Means and StandardDeviationsfor "WhatShape CouldI Be ThinkingOf?" (Item PP/2.11) LG (1987-88) Control(1988-89) LG (1988-89) Site 1 Site 2 Site 1 Site 2 Site 1 Post Post Pre Pre Post Pre Pre Post Post Grade Pre 3.24 3.41 2.77 2.84 3.28 3.14 3.87 4 2.80 2.62 2.66 (1.14) (1.02) (.82) (1.12) (.99) (1.11) (.84) (.86) (1.02) (.72) 4.00 3.24 3.37 3.87 3.36 2.98 3.67 2.94 5 3.23 3.36 (1.17) (1.01) (1.20) (1.09) (1.00) (1.05) (1.00) (.99) (.74) (.93) 3.14 3.52 2.71 3.32 3.77 3.21 3.86 3.16 6 3.12 3.65 (1.04) (1.19) (1.05) (.99) (1.32) (1.17) (1.04) (.99) (.80) (.84) 3.79 4.13 7 (.71)

(1.1)

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159

Table 8 Means and StandardDeviationsfor Disembedding,K-l (ItemPP/2.6) LG (1988-89) Control(1988-89) LG (1987-88) Site 1 Site 2 Site 1 Site 2 Site 1 Grade Pre Post Pre Post Pre Post Pre Post Pre Post K 5.17 6.70 5.43 6.54 2.85 6.42 5.56 6.53 (1.87) (.73) (1.01) (.74) (2.11) (1.02) (1.08) (.96) 1 6.73 7.79 6.13 6.69 6.70 8.18 6.32 6.87 (.90) (1.42) (1.01) (.92) (.89) (1.45) (.59) (.34)

Table 9 Means and StandardDeviationsfor K-3 Building(ItemPP/2.A) LG (1988-89) Control(1988-89) LG (1987-88) Site 1 Site 2 Site 1 Site 2 Site 1 Grade Pre Post Pre Post Pre Post Pre Post Pre Post K .04 .05 .00 .05 .04 .11 .09 .00 .00 .00 (.21) (.23) (.00) (.23) (.20) (.32) (.29) (.00) (.00) (.00) 1 .18 .36 .02 .09 .07 .22 .09 .09 .00 .00 (.39) (.49) (.15) (.29) (.26) (.42) (.28) (.29) (.00) (.00) 2 .06 .24 .07 .22 .04 .11 .03 .46 .04 .13 (.25) (.43) (.25) (.42) (.20) (.31) (.16) (.50) (.19) (.34) 3 .00 .43 .00 .32 .03 .00 .04 .20 .08 (.00) (.51) (.00) (.48) (.18) (.00) (.21) (.41) (.29)

Table 10 TriadDiscriminationScores LG (87-88) Grade Pre Int K .20 1.10 1 1.00 1.00 2 .63 1.50 3 .67 .33 4 1.07 2.47 5 1.82 3.55 6 1.60 4.00 7 4.88 6.50

Post 2.70 1.30 .88 1.00 1.53 3.64 4.10 7.25

Pre .00 .00 .13 .50 .21 1.61 3.95

LG (88-89) Int .14 1.38 1.13 2.63 1.84 5.50 5.76

Post .00 .56 .88 1.50 1.74 4.39 6.76

Control(88-89) Post (only) .29 .25 .25 .57 1.70 3.94 5.30

Tables

160

Table 11 Descriptionsof Reasonsfor ChoosingEach TriadPair Classifiedby van Hiele Level VISUAL

This is given if student'sreasonfits one of these.

both objects look like a window, door, or box studentinterpretsdrawingsas picturesof 3D objects or such objects in perspective(e.g., This looks like the top of a box and this one the side of the box. This trapezoidlooks like what you would get it you looked at a rectanglefrom here). both have slantedsides (or, nonslantedsides) slantedsides both are "pointy"or have "sharpcorers" sharpangles both would look the same if I moved this line over here (includes deformation deformation) both are the same size (big, little, fat, skinny,long, short) size both look like they are the same shape same shape both look like (some specific geometricand abstractshape) look like shajpe eristic both sharea visual characteristicor specific component sharecharact' both look the same or look alike look same both are the same, but one is moved, shrunk,stretched,flipped, motions turned,slid (or they would be the same if one was moved) look like obje:ct 3D projectior

P EXPPRO] same # sides same # angle,s side length parallelsides

This is given if student'sreasonfits one of these

both have same numberof sides both have the same numberof angles (or turnsor bends) both have x sides the same length or not the same length both have a pairof opposite sides parallel both have (or do not have) rightangles (or 90? turns) rightangles otherspecific referencesto angle-relatedproperties(e.g., having to angle do with relationshipsbetween measureof angles, such as "theseboth have acute angles") both are or are not symmetric symmetry n both are congruentor similar congruent/sir mentionedhaving same numberof sides and same numberof angles same #sides/ same #angYles with no interveningmentionof anotherproperty names anotherproperty otherproperty EXPCLASS

This is given if student'sreasonfits one of these

classify

they both are parallelograms,rectangles,scalene, obtuse, isosceles, and right,etc. (Furtherjustificationby the studentmust not contradict this level.)

| OTHER other

This is given if student'sreasonfits no othercategory any otherreason

I

Logo and Geometry Table 12 Percent of StudentsChoosing Categories of Reasonsfor Triads 4 1 2 3 6 Triad 5 7 Propertiesor classificationreasons 16 9 6 18 11 3 9 pretest 28 42 12 16 21 20 posttest 42 increase 26 *18 24 10 10 10 11 Visual reasons 84 86 90 80 95 85 85 pretest 65 78 44 80 71 77 posttest 55 decrease 29 22 12 36 15 14 9

161

8

9

Mean

13 27 15

9 14 5

10 25 15

82 63 19

86 76 10

86 68 18

* preandpostmaynotgiveexactincreaseordecreasebecauseallfiguresarerounded Note.Becausethetriaditemsweredifferentforthefirstyear,theseitemsareanalyzedonlyforthe 1988-89data.

Table 13

Percentof Studentsat Each van Hiele Levelfor Pretest (1), Interim(2), and Posttest (3) Interview Grade

K Level* 1 2 3 Logo Site 1 1987-88 0

10

0.5 1 1.5 2 2.5 3

20 90 50 60 20

30

20

1

1 2

3

1

2 2

3

1

3 2

3

4 2

3

7 87 73 7 27

87 13

91 73 9 27

29 71

22 56

75

20 20 40

22

25

40

1

1

5 2

10

10 10 10 80 90 80 10

13 88 88 88 13 13

17 100 83 100

Logo Site 1 1988-89 0 100

0.5

33 100

1 1.5

67

50 50

67 33 100

100 100

75

50

25

50

100 100

2 2.5 3

20

Logo Site 2 1988-89 0 50 0.5

1 1.5 2 2.5 3

20 40

100 100 50

25 29

33

71 67

50 25

17 17

67

67

33

50 50

67 100

80

20

33

100

20 20

33 33

20

33

40

20 100

20 60

* Fractionalvalues indicatestudentsin transitionto the next higher level. Note. For each gradeat each site, this table presentsthe percentof studentsclassified at each level of the van Hi 1987-88, kindergarten,at the pretest(interview 1 on this table), 10%were classified at Level 0, 90% at Level Level 0, 60% at Level 1, and 20%in transitionto Level 2 (i.e., Level 1.5).

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Table 14 Percent of Studentsby Grade Level Whosevan Hiele Level IncreasedfromPre- to PostInterview Grade LG(87-88) LG(88-89 Site 1) LG(88-89 Site 2) K 30.0 0 0 1 30.0 0 0 2 0 25.0 0 3 25.0 0 16.7 4 12.5 8.3 14.3 5 36.4 44.4 25.0 6 50.0 46.2 37.5 7 85.7

Table 15 Percent of Responsesfor ReasoningAbout Quadrilaterals Team Pre Interim Post Team choices Site 1 A 52.8 14.3 5.4 B 27.8 51.4 54.1 C 8.3 20.0 16.2 D 2.8 5.7 2.7 E 2.8 2.9 0.0 A&C B, D&E B&D D&E B&E Site 2 A B C D E Site I A B C D E Site 2 A B C D E

2.9 2.8 2.8 2.9

5.4 5.4 5.4 5.4

Control

21.9 43.8 15.6 3.1 6.3 3.1

6.3 33.3 29.2 8.3 20.8 8.3

12.0 60.0 12.0 8.0 8.0

12.5 58.3 8.3 12.5 8.3

12.5 54.2 12.5 8.3 12.5

Studentchoices for teams thatwould make a rectangle 63.9 91.7 13.9 25.0 22.2

22.9 88.6 28.6 34.3 25.7

13.5 97.3 27.0 45.9 37.8

28.1 90.6 18.8 25.0 31.3

54.2 83.3 12.5 37.5 41.7

50.0 96.0 36.0 54.2 58.3

54.2 95.8 25.0 54.2 62.5

45.8 83.3 33.3 75.0 58.3

164

Tables

Table 16 Percent by Categoryin Reponseto "WhatIs an Angle?" (ItemPP/2.1) Control(1988-89) I LG (1988-89) Pre Pre Post Post Category 14 22 38 38 No interpretableresponse 6 7 9 6 Shape or partof a shape 4 2 3 2 Line/segment/side(e.g., of a square) 4 11 10 8 Tilted line 2 1 0 2 Orientation/heading 2 1 2 19 Rotation/turn 4 3 5 5 Somethingthatforms a squarecomer 8 20 8 15 Intersectionof two lines/segments 7 4 9 5 Union of two lines/segments 2 1 1 3 Looks like object (e.g., a half of a boat) 4 3 4 7 Somethingthatbends 4 4 4 3 A thing with degrees (measured) 0 0 1 0 Special path (as in curriculum) 1 1 1 2 A line thatis not straight 0 0 1 0 Somethingthatis closed or not closed 15 14 11 9 Other

Table 17 Means and StandardDeviationsfor "CircleAll the Angles" (ItemPP/2.5) LG (1987-88) Control(1988-89) LG (1988-89) Site 1 Site 2 Site I Site 2 Site 1 Post Pre Post Post Pre Pre Post Pre Post Grade Pre 11.71 8.15 6.95 8.68 7.47 6.71 7.15 7.54 K 7.52 11.05 (1.95) (1.54) (2.52) (3.11) (1.85) (2.99) (2.36) (2.86) (2.30) (1.76) 9.56 7.49 10.76 8.87 10.74 7.04 9.98 1 7.20 10.95 8.13 (2.02) (2.26) (2.55) (2.11) (1.92) (2.63) (2.17) (1.83) (2.49) (2.69) 7.18 11.03 8.59 8.63 11.00 8.33 9.91 2 8.53 10.65 8.16 (2.22) (2.57) (2.29) (2.21) (2.12) (1.99) (2.14) (2.38) (2.23) (2.39) 7.32 11.20 7.08 8.68 7.17 10.52 7.07 9.43 3 8.96 (1.58) (2.39) (3.03) (1.90) (2.86) (2.77) (2.12) (1.15) (1.88) 4 9.19 11.14 8.98 11.78 9.17 11.98 9.82 11.28 8.67 11.63 (2.77) (2.05) (2.47) (1.33) (2.63) (1.38) (2.30) (1.47) (2.67) (1.72) 11.31 12.07 10.57 11.11 11.33 12.45 10.80 11.64 11.31 12.35 5 (2.07) (1.44) (2.12) (2.09) (2.31) (1.32) (2.21) (1.72) (2.00) (1.05) 10.97 11.52 11.96 12.16 11.30 11.86 11.45 12.16 10.49 11.55 6 (2.78) (2.04) (1.29) (1.83) (2.61) (2.80) (2.32) (2.03) (2.27) (2.23) 12.21 12.72 7 (1.36) (1.53)

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Table 18 Means and StandardDeviationsfor "Drawan Angle" (ItemPP/2.2) LG (1987-88) LG (1988-89) Control(1988-89) Site 2 Site 1 Site 2 Site 1 Site 1 Pre Post Pre Post Pre Post Post Post Pre Grade Pre .14 .23 .11 .24 .32 .00 .82 K .75 .04 .00 (.00) (.44) (.35) (.43) (.20) (.32) (.43) (.47) (.00) (.39) .14 .82 .34 .78 .07 .54 1 .14 .74 .33 .60 (.35) (.44) (.48) (.50) (.35) (.39) (.48) (.42) (.25) (.50) .00 .11 .53 .76 .27 .72 .06 .72 .24 2 .13 (.34) (.43) (.45) (.46) (.24) (.45) (.00) (.43) (.31) (.51) .74 .87 .20 .45 .27 .85 .00 3 .22 .21 (.42) (.45) (.41) (.34) (.41) (.51) (.46) (.37) (.00) 4 .88 .50 .46 .88 .66 1.00 .88 .51 .98 .36 (.50) (.33) (.51) (.16) (.50) (.33) (.48) (.00) (.48) (.33) 5 1.00 .85 .89 .86 .98 .91 .87 .92 1.00 .89 (.32) (.00) (.36) (.31) (.35) (.14) (.29) (.34) (.28) (.00) 6 .70 .94 1.00 .86 1.00 .96 .89 .85 .69 .94 (.46) (.32) (.24) (.00) (.36) (.35) (.00) (.20) (.47) (.25) 7 1.00 1.00 (.00) (.00)

Table 19 Means and StandardDeviationsfor "Drawa Bigger Angle" (ItemPP/2.3) LG (1988-89) Control(1988-89) LG (1987-88) Site 1 Site 2 Site 1 Site 2 Site 1 Grade Pre Post Pre Pre Post Post Pre Post Pre Post K .00 .20 .00 .06 .04 .00 .06 .06 .00 .00 (.00) (.41) (.00) (.24) (.20) (.00) (.24) (.24) (.00) (.00) 1 .00 .08 .04 .12 .00 .11 .06 .26 .00 .00 (.00) (.27) (.21) (.33) (.00) (.31) (.25) (.44) (.00) (.00) 2 .02 .18 .04 .26 .04 .38 .00 .07 .02 .03 (.15) (.39) (.21) (.44) (.20) (.49) (.00) (.26) (.13) (.16) .22 3 .04 .04 .00 .23 .39 .00 .10 .00 (.21) (.42) (.20) (.50) (.00) (.43) (.00) (.31) (.00) 4 .20 .11 .16 .59 .70 .29 .23 .46 .09 .39 (.40) (.50) (.32) (.46) (.37) (.46) (.42) (.51) (.29) (.49) 5 .60 .28 .83 .74 .43 .76 .50 .80 .33 .03 (.50) (.38) (.46) (.44) (.50) (.43) (.51) (.40) (.48) (.16) 6 .47 .68 .42 .60 .73 .91 .65 .27 .88 .72 (.50) (.47) (.45) (.29) (.50) (.49) (.48) (.33) (.45) (.45) 7 .42 .94 (.00) (.24)

166

Tables

Table 20 Means and StandardDeviationsfor the Spinneror Amountof Turn(ItemPP/2.4) LG (1988-89) Control(1988-89) LG (1987-88) Site 1 Site 2 Site 1 Site 2 Site 1 Pre Grade Pre Post Post Pre Post Pre Post Pre Post 2 .00 1.45 .12 .09 1.33 .64 .18 .17 .00 1.39 (.00) (.83) (.29) (.82) (.44) (.85) (.45) (.44) (.00) (.86) 3 .00 1.22 .00 2.09 .00 .18 .00 1.35 (.00) (.80) (.00) (.85) (.00) (.50) (.00) (.59) 4 .18 1.76 .25 .26 .16 .21 1.82 1.65 .55 .92 (.44) (1.09) (.65) (.92) (.60) (.84) (.57) (.98) (.57) (.86) 5 1.07 2.14 2.26 .65 1.04 .45 .73 2.23 .39 .89 (1.18) (.81) (.80) (.77) (1.00) (1.00) (.66) (.78) (.98) (.97) 2.42 1.18 6 .83 2.00 .90 .80 1.37 .55 .33 2.02 (1.06) (.96) (.96) (.78) (1.04) (1.5) (.79) (1.01) (.72) (1.01) .84 2.22 7 (.83) (1.17)

Table 21 Means and StandardDeviationsfor "Estimatethe Measureof Each Angle" (Item PP/I. 7) LG (1988-89) Control(1988-89) LG (1987-88) Site 2 Site 1 Site 2 Site 1 Site 1 Post Post Pre Post Pre Post Pre Post Pre Grade Pre 1.50 1.75 1.25 1.41 2.05 1.82 2.43 3 (.96) (.91) (.62) (.96) (1.24) (1.30) (1.16) 1.77 2.73 1.71 2.94 1.71 3.08 1.62 2.67 4 1.46 2.83 (1.01) (1.14) (1.21) (1.00) (.97) (1.03) (1.07) (1.17) 1.18) (.98) 2.37 3.19 3.11 2.32 3.08 2.40 3.56 2.65 3.51 2.08 5 (1.27) (.81) (1.11) (.97) (1.02) (1.17) (1.38) (.76) 1.33) (1.01) 3.27 2.59 3.27 3.22 3.50 2.66 3.13 3.70 6 2.95 2.95 (1.24) (.99) (1.03) (.85) (1.22) (1.22) (.99) (.61) 1.27) (.82) 3.79 3.89 7 (.63) (.32)

Table 22 Means and StandardDeviationsfor "Findthe Missing Measures of the Angles and of the Sides" (ItemsPP/1.8-1.9) Control(1988-89) LG (1987-88) LG (1988-89) Site 1 Site 2 Site 2 Site 1 Site 1 Post Pre Post Pre Post Pre Pre Post Pre Post Grade 2.44 1.60 2.73 1.80 2.43 1.32 2.56 1.56 4 1.55 2.68 (1.38) (1.43) (1.28) (1.66) (1.37) (1.34) (1.47) (.97) (1.30) (1.32) 2.67 3.53 2.31 4.30 5 2.63 4.83 2.16 4.52 3.90 2.10 (1.33) (1.95) (1.52) (1.67) (1.14) (1.67) (1.62) (2.05) (1.46) (1.70) 4.62 2.24 4.52 2.96 2.41 3.03 2.55 6 2.49 3.86 4.93 (1.69) (1.81) (1.40) (1.93) (1.67) (2.22) (1.25) (1.75) (1.46) (1.89) 3.32 6.06 7 (1.46) (1.66)

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Table 23 Means and StandardDeviationsfor "AngleMeasure:Boat Amount-of-Turn"(Item PP/2.12) LG (1988-89) Control(1988-89) Site 1 Site 2 Site 1 Site 2 Pre Post Pre Pre Post Grade Pre Post Post 4 .05 .10 .05 .05 .00 .05 .03 .04 (.00) (.19) (.22) (.23) (.22) (.30) (.22) (.17) .12 5 .29 .00 .23 .05 .05 .15 .06 (.00) (.43) (.37) (.24) (.33) (.46) (.22) (.21) 6 .12 .29 .09 .48 .13 .10 .07 .26 (.29) (.51) (.33) (.46) (.33) (.30) (.25) (.44)

Table 24 Means and StandardDeviationsfor "AngleMeasure:Bending Wire"(ItemPP/2.13) LG (1988-89) Control(1988-89) Site 1 Site 2 Site 1 Site 2 Grade Pre Post Pre Post Pre Post Pre Post 2 .00 .49 .00 .26 .00 .02 .03 .06 (.00) (.51) (.00) (.44) (.00) (.15) (.16) (.24) 3 .00 .14 .00 .48 .00 .00 .21 .00 (.00) (.35) (.00) (.51) (.00) (.00) (.00) (.43) 4 .28 .42 .15 .08 .04 .14 .08 .03 (.36) (.45) (.28) (.50) (.20) (.35) (.17) (.28) 5 .26 .56 .13 .53 .16 .13 .12 .32 (.44) (.50) (.34) (.51) (.34) (.37) (.33) (.47) 6 .10 .44 .22 .86 .19 .15 .20 .66 (.30) (.50) (.42) (.35) (.40) (.36) (.40) (.48)

Tables

168

Table 25 Classificationsof Students'Angle Drawings (Percent of Students) LG Site 1 (88-89) Site 2 (88-89) Pre Int Post Pre Int Post None 20.3 4.9 0 11.1 3.6 0 3.1 0 1.6 0 0 1.9 Segment 9.4 11.5 7.9 11.1 9.1 5.6 Shape Tiltedseg. 15.6 8.2 11.1 20.4 7.3 9.3 Intersection 0 0 0 0 0 0 0 0 0 0 0 0 Heading 0 0 1.6 0 0 0 Object 0 Arrow 0 0 0 0 0 Arc 0 0 1.6 0 3.7 0 Total Incor. 48.4 24.6 23.8 46.3 20.0 16.8 Proto.right 9.4 11.5 7.9 18.5 20.0 22.2 Proto.acute 7.8 6.6 3.2 9.3 5.5 1.9 Proto.obtuse 0 0 3.2 1.9 1.8 3.7 subtotal 17.2 18.1 14.3 29.7 27.3 27.8 12.5 23.0 15.9 0 10.9 9.3 Right Acute 17.2 27.9 36.5 16.7 32.7 44.4 0 5.5 1.9 Obtuse 3.1 4.9 7.9 0 0 1.6 0 1.8 0 Straight subtotal 32.8 55.8 61.9 16.7 50.9 55.6 Total correct 50.0 73.9 76.2 46.4 78.2 83.4 1.6 1.6 0 7.4 1.8 0 Other

Control (88-89) Site 1 88 Site 1 Site 2 Pre Int Post Post Post 13 11.1 9.4 12.5 2 1 0 0 3.7 1.9 9 12.5 6 11.1 3.8 5 7.4 13.2 25 9 1 0 0 0.0 0.0 0 0 0 0.0 0.0 2 0 1 0.0 1.9 0 0 0 0.0 0.0 1 1.2 0 1.9 1.9 52 31.2 18 35.2 32.1 9 18.8 17 22.2 22.6 7 6.3 4 18.5 18.9 1 0 3 3.7 3.8 17 25.1 24 44.4 45.3 8.7 13 9 0.0 5.7 20 32.5 40 16.7 11.3 2 2.5 5 3.7 0.0 0 0 0 0.0 1.9 31 43.7 58 20.4 17.9 48 64.8 64.2 68.8 82 0 0 0 0.0 5.7

169

Logo and Geometry

Table 26 Percents of Classificationsof Students'Responses to "HowIs ThisAngle DifferentFrom This One?" Control LG (88-89) Site 1 Site 2 Site 1 (87-88) Site 2 (88-89) Site 1 (88-89) Pre Int Post Pre Int Post Pre Int Post Post Post 1.2 0.0 0.0 10.9 0.0 2.0 0.0 0.0 None 21.9 4.9 0.0 Lengthof 3.6 5.7 1.6 8.2 14.5 12.2 5.5 5.6 14.8 17.4 25.9 sides 32.9 21.3 21.0 24.5 5.4 11.1 30.8 22.5 22.2 14.5 13.2 Shape Numberof 6.2 8.7 4.9 9.1 0.0 6.2 8.2 1.6 10.2 3.6 1.9 sides No under62.6 42.6 37.1 48.9 14.5 18.6 53.0 48.6 53.0 38.1 18.9 standing 9.1 11.3 17.2 16.4 8.1 16.3 10.9 13.0 24.7 8.7 8.6 Tilt 34.4 29.5 30.6 22.4 34.5 33.3 22.3 26.2 37.0 29.1 22.6 Heading 0.0 1.8 1.9 0.0 3.7 2.5 1.9 2.0 Turnon path 0.0 4.9 4.8 51.6 50.8 43.5 40.7 47.2 48.2 47.0 38.6 48.1 38.2 35.8 Related Size (unelab.) 25.0 21.3 41.9 20.4 32.8 27.8 12.3 17.5 39.5 23.6 26.4 Wider 9.4 23.0 12.9 12.2 14.5 29.6 19.8 22.5 24.7 20.0 13.2 Sharper pt.

1.6

0.0

1.6

10.2

36.0 44.3 56.4 42.8 0.0 Rotation/rays 1.6 3.3 3.2 14.1 32.8 19.4 10.2 Measure 9.4 21.3 32.3 8.1 Classify 25.1 57.4 54.9 18.3 Formal 0.0 0.0 0.0 0.0 Deform 6.1 Other 12.5 9.8 12.9 termsaresummaries. Note.Underscored Intuitive

0.0

3.8

47.3 1.8 16.3 23.7 41.8 0.0 14.9

61.2 0.0 25.9 29.6 55.5 0.0 22.6

-

-

-

32.1 0.0 9.9 6.2 16.1 2.4 6.1

40.0 2.5 35.0 18.7 56.2 0.0 11.2

64.2 0.0 22.2 29.6 51.8 1.2 9.9

0.0

1.9

43.6 1.8 18.2 36.3 56.3 0.0 7.3

41.5 0.0 24.5 32.1 56.6 0.0 24.5

Table 27 Angle Scores by Grade (Out of a Possible 4) LG (87-88) LG (88-89) Control(88-89) Grade Pre Post Pre Int Int Post Post K .21 .30 .40 .60 .07 .21 .00 .00 .44 1 .31 .50 .50 .56 .50 2 .25 1.25 1.50 .38 1.19 1.31 .13 3 .00 .33 .17 .70 1.90 2.20 .64 4 .87 2.73 2.33 .74 2.84 2.63 2.25 2.18 5 2.82 2.82 1.78 3.44 3.33 3.06 6 2.50 3.00 3.50 2.86 3.48 3.67 3.15 7 3.38 3.75 3.88

170

Tables

Table 28 Choicesfor the Question "WhichAngle Is Bigger?" (Percent of Students) Site 1 89 Pre Int Post Pair 1 IDK* A B Same Pair 2 IDK A B Same Pair 3 IDK A B Same

Site 2 89 Pre Int Post

Site 1 88 Pre Int Post

Control89 post Site 1 Site 2

0.0 0.0 0.0 0.0 0.0 1.6 0.0 0.0 0.0 40.7 62.5 61.7 39.1 67.2 66.7 22.2 58.2 50.0 59.3 37.5 38.3 59.4 32.8 31.7 77.8 40.0 48.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.8 1.9

3.6 0.0 47.3 49.1 47.3 50.9 1.8 0.0

0.0 0.0 0.0 0.0 0.0 0.0 1.9 0.0 0.0 29.6 56.3 55.6 26.6 54.1 55.6 14.8 49.1 48.1 70.4 43.8 44.4 71.9 45.9 44.4 83.3 49.1 48.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.8 3.7

1.8 0.0 40.0 41.5 58.2 58.5 0.0 0.0

1.2 0.0 0.0 0.0 0.0 0.0 1.9 1.8 0.0 51.9 41.2 39.5 48.4 24.6 27.0 53.7 38.2 40.7 30.9 40.0 33.3 25.0 21.3 25.4 14.8 18.2 11.1 16.0 18.8 27.2 26.6 54.1 47.6 29.6 40.0 46.3

0.0 3.8 45.5 35.8 14.5 30.2 40.0 28.3

Triangle IDK 3.7 8.7 1.2 1.6 1.6 4.8 0.0 1.8 1.9 7.3 3.8 A 7.4 3.7 4.9 18.8 19.7 9.5 9.3 18.2 18.5 14.5 17.0 B 1.2 1.2 2.5 3.1 4.9 1.6 0.0 9.1 5.6 3.6 11.3 35.8 61.2 58.0 34.4 54.1 58.7 18.5 38.2 57.4 C 41.8 35.8 sideAB 27.2 17.5 9.9 29.7 13.1 19.0 42.6 18.2 9.3 23.6 26.4 side BC 2.5 0.0 0.0 0.0 0.0 0.0 0.0 1.8 0.0 0.0 0.0 side AC 0.0 0.0 0.0 0.0 0.0 0.0 1.9 0.0 0.0 1.8 0.0 whole figure 22.2 7.5 23.5 10.9 4.9 1.6 24.1 10.9 7.4 7.3 3.8 * Foreachpairof angles,thefigureon theleftis "A"andtheoneon therightis "B."IDK= "Idon't know."

Table 29 Choicesfor the Question "WhichAngle Is Bigger?" by Grade (88-89 LG),Angle Pair 2 Pretest Posttest Interim A B A Choice B A B K 0 0 100 7.1 92.9 92.9 81.3 100 0 100 12.5 1 0 2 0 100 25 75 25 75 3 0 100 50 50 40 60 4 10 90 70 30 68.4 31.6 81 5 55.6 5.3 14.3 38.9 89.5 6 72.7 27.3 95.2 4.8 95.2 4.8

Logo and Geometry

171

Table 30 Classificationsof Students'Responses to "WhyDid YouSay ThisAngle Was Bigger?" (Percent of Students) Control89 Site 1 88 Site 1 89 Site 2 89 post Site 1 Site 2 Pre Int Post Pre Int Post Pre Int Post 0.0 0.0 0.0 2.5 1.2 0.0 0.0 0.0 1.6 3.3 0.0 None Lengthof 56.3 39.3 34.9 66.7 38.2 27.8 69.1 45.0 50.6 54.6 39.6 sides 5.5 3.8 6.2 7.5 2.5 9.3 0.0 0.0 7.8 1.6 3.2 Shape Numberof 1.2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 3.1 0.0 0.0 sides No under68.8 44.2 38.1 76.0 38.2 27.8 75.3 55.0 55.5 60.1 43.4 standing 1.8 0.0 7.4 0.0 0.0 0.0 0.0 1.9 Tilt 14.1 0.0 1.6 1.8 1.9 7.4 1.2 2.5 1.9 1.8 1.9 1.6 0.0 3.2 Heading 1.8 1.9 1.2 3.7 3.7 0.0 1.8 3.7 Turnon path 3.2 14.8 7.9 5.4 3.8 1.9 3.6 7.5 16.0 4.9 6.2 18.9 14.8 12.7 Related Size (unelab.) 31.2 19.7 23.8 22.3 7.2 13.0 13.6 12.4 22.2 21.8 11.3 Wider 57.8 57.4 52.4 24.1 52.7 50.0 49.4 57.5 59.2 49.1 50.9 -3.6 3.8 3.8 0.0 3.8 3.1 1.6 1.6 Sharperpt. 92.1 78.7 77.8 50.2 59.9 66.8 63.0 69.9 81.4 74.5 66.0 Intuitive 1.8 0.0 1.9 0.0 0.0 0.0 3.7 2.5 Rotation/rays 0.0 4.9 7.9 8.6 16.2 20.9 20.0 9.4 5.6 10.9 14.8 17.2 24.6 22.2 Measure 3.7 7.5 17.3 14.5 7.5 1.9 5.5 5.6 4.7 6.6 23.8 Classify 9.4 16.4 20.4 12.3 27.4 40.7 36.3 16.9 Formal 21.9 36.1 53.9 3.8 0.0 0.0 2.4 3.7 0.0 0.0 0.0 Deform 0.0 0.0 0.0 5.5 11.6 4.0 7.3 20.7 3.7 5.0 2.4 9.4 11.5 4.8 Other Note.Underscored termsaresummaries.

Table 31 Means and StandardDeviationsfor "Drawa Path" (ItemsPP/2.7-2.9) LG (1988-89) Control(1988-89) LG (1987-88) Site 1 Site 2 Site 2 Site 1 Site 1 Post Pre Pre Grade Pre Post Post Pre Post Pre Post 2 .71 2.16 .26 .44 .41 2.09 1.74 .52 .29 2.05 (.61) (.89) (.51) (.83) (.67) (1.04) (.73) (.58) (.70) (.72) 3 .35 2.09 .25 2.22 1.02 .45 1.13 .88 .75 (.63) (1.08) (.61) (.80) (.84) (.98) (.62) (.86) (.84) 4 .74 2.12 .75 2.11 .50 1.01 .61 1.23 .90 2.26 (.92) (.76) (1.01) (.84) (.73) (1.02) (.95) (1.02) (.89) (.82) 5 2.44 2.01 .77 2.59 1.46 1.86 .61 1.12 1.76 2.44 (.88) (.48) (1.07) (.62) (.89) (.81) (.84) (.98) (.93) (.52) 6 1.30 2.61 1.19 2.37 .91 1.38 .85 1.32 1.43 2.11 (1.05) (.50) (.92) (.80) (.89) (1.11) (1.01) (.89) (.85) (.90) 7 2.08 1.92 (.56) (.43)

Tables

172

Table 32 Percents by Categoryfor "Telephone"(ItemPP/1.10) LG (1988-89) Pre Post Category No response Differentfigure Componentsonly Correct,but unwarrantedassumptions or inexact measures Correct,complete, procedure Indistinguishable Specific example Other Simple path (turn,go up) Components(lines, points) Logo-like commands Complexpathor components+ Logo

Response 6 28 2 3 54 50 12 29 4 13 Language 7 19 7 3 1 3 7 5 62 60 10 0 5 12

Control(1988-89) Pre Post 28 3 51

9 4 56

16 3

23 8

19 6 4 5 59 0 7

10 5 4 3 75 0 3

Table 33 Means and StandardDeviationsfor "DrawingSymmetryLines" (ItemPP/1.5) Control(1988-89) LG (1987-88) LG (1988-89) Site 2 Site 1 Site 2 Site 1 Site 1 Pre Post Pre Post Post Pre Post Post Pre Grade Pre Total first four items .82 1.16 3.94 .08 .05 .47 .46 2.91 K .30 3.89 (.70) (1.82) (1.02) (1.50) (.39) (.23) (.90) (1.00) (1.34) (1.52) 3.07 .09 5.51 3.05 .34 .93 4.81 .09 1 .07 5.62 (.46) (1.21) (1.16) (1.85) (.61) (1.06) (1.18) (2.36) (.42) (1.50) 1.29 4.92 4.33 1.43 3.63 5.84 .47 2 2.16 1.36 5.59 (1.75) (2.48) (2.20) (1.96) (1.42) (1.65) (1.77) (.86) (1.70) (2.45) 2.17 2.78 5.10 1.87 2.48 2.42 5.39 3 2.57 5.26 (1.20) (1.25) (1.79) (1.31) (1.57) (1.60) (1.31) (1.89) (2.17) 2.68 6.65 6.44 3.71 3.76 5.58 4 6.63 3.66 2.92 2.80 (2.22) (1.48) (1.78) (1.92) (2.09) (1.56) (1.15) (1.50) (2.06) (1.25) 7.26 3.18 5.69 5.90 7.15 7.37 2.07 5.84 5.11 6.90 5 (2.78) (1.54) (2.47) (1.09) (1.65) (1.17) (1.78) (1.41) (1.77) (1.00) 6.15 4.29 5.50 2.88 6.43 3.84 4.12 4.16 6 6.09 4.09 (2.80) (1.36) (1.69) (1.74) (3.09) (2.72) (1.44) (1.74) (2.58) (1.95) 8.00 6.16 7 (1.89) (.00) Total of all questions 1.58 1.87 4.00 1.17 1.71 3.00 1.75 3.96 1.74 3 (1.05) (1.45) (1.45) (1.55) (1.21) (1.35) (1.29) (1.84) (1.73) 5.82 2.01 2.69 2.82 2.24 5.91 4.47 2.45 4 2.20 5.71 (2.00) (2.01) (1.56) (2.13) (1.80) (1.96) (.87) (1.25) (1.81) (1.97) 5.25 6.98 2.16 3.87 6.63 4.86 1.41 6.59 4.32 6.17 5 (2.72) (2.06) (1.69) (1.50) (1.79) (1.65) (1.27) (2.05) (2.11) (1.34) 5.25 4.06 2.17 2.98 3.13 3.35 4.86 3.50 5.25 2.94 6 (2.74) (1.92) (1.43) (2.58) (2.81) (2.54) (1.51) (1.96) (2.08) (2.51) 7.67 5.32 7 (2.00) (.69)

173

Logo and Geometry

Table 34 Percentage of Correctand IncorrectSymmetryLines Drawn (ItemPP/1.5.) LG (1988-89) Control(1988-89) CorrectLines IncorrectLines IncorrectLines CorrectLines Lines Pre Drawn Post Pre Pre Post Post Pre Post Heart 0 40 40 3 92 13 85 85 93 1 7 60 97 60 87 11 13 6 2 1 1 1 3 1 1 1 4 1 1 5-9 1 1 Rectangle 0 1 2 4 5-9 incorrectly Pentagon 0 1 2 3 4 5-9 Square 0 1 2 3 4 5-9

41 40 18

4.8 552

37 4t3 18 1 12

Parallelogram 0 1

2 3 4

2 19 79

10 90

3 15 17 2 63

78 10 8 2 2

62

76 18 2 1 0 3

91 7 0 1

90 6 1 0 2 1

98

31 55 9 1 4

39 29 22 2 8

2

35 1

44 40 16

48 52

14 42 44

23 77

1

1 4

38 44 9 1 9

11 43 12 1 32

77 11 6 4 2

75 6 18

76 16 3 1 1 4

84 14 1

89 6 1 1 2 1

95 3 1

19 65 11 2 3

34 43 15 2 6

1

1

1

174

Tables

Table 35 Means and StandardDeviationsfor "Drawthe OtherHalf" (ItemPP/1.6) Control(1988-89) LG (1988-89) LG (1987-88) Site 2 Site 1 Site 2 Site 1 Site 1 Pre Pre Post Pre Post Post Pre Post Post Grade Pre .13 .00 .97 K .00 .34 .00 .00 .03 .00 .25 (.00) (.41) (.00) (.48) (.00) (.00) (.12) (.35) (.00) (.37) .19 .12 .88 .64 1 .21 .89 .08 .88 .31 .89 (.22) (.57) (.55) (.69) (.43) (.69) (.46) (.73) (.29) (.52) .45 .95 .67 1.16 .40 .58 .75 2 .95 .52 .95 (.57) (.65) (.59) (.52) (.58) (.60) (.53) (.60) (.58) (.49) .67 1.57 1.45 .96 1.43 .47 3 .52 .72 .98 (.51) (.52) (.91) (.55) (.47) (.68) (.51) (.69) (.50) 1.42 1.03 1.50 1.33 4 1.11 1.53 .97 1.37 .75 1.50 (.65) (.53) (.71) (.51) (.65) (.38) (.71) (.65) (.62) (.46) 1.66 1.24 1.65 1.45 1.56 1.51 1.53 1.78 5 1.33 1.61 (.65) (.52) (.67) (.36) (.48) (.48) (.62) (.44) (.47) (.42) 1.36 1.77 1.05 1.04 1.29 1.63 1.81 1.93 6 1.24 1.60 (.58) (.45) (.48) (.23) (.62) (.38) (.53) (.41) (.56) (.47) 1.47 1.67 7 (.42) (.38)

Logo and Geometry

175

Table 36 Means and StandardDeviationsfor "Congruency-Are Figures the Same Size and Shape?" (ItemPP/1.4) LG (1987-88) Control(1988-89) LG (1988-89) Site 2 Site 1 Site 2 Site 1 Site 1 Post Pre Post Pre Post Pre Pre Post Post Grade Pre Identification 4.53 5.76 8.85 9.00 7.73 8.94 9.80 8.24 9.43 K 8.43 (1.24) (.89) (1.34) (1.09) (1.43) (1.00) (1.33) (1.17) (1.17) (.56) 4.71 5.71 8.72 9.32 8.67 9.34 1 9.67 9.00 9.93 8.93 (1.21) (1.08) (1.29) (1.03) (1.21) (1.05) (1.14) (.80) (.94) (.56) 9.10 4.95 5.68 9.48 9.60 9.57 9.19 2 9.66 9.34 9.38 (1.29) (1.53) (1.24) (1.05) (1.25) (1.30) (1.03) (1.36) (.70) (.57) 4.50 8.76 9.70 9.76 9.13 9.48 8.83 3 9.30 8.83 (1.29) (1.85) (.95) (.85) (1.39) (2.17) (1.15) (.70) (1.45) 9.38 9.05 5.13 5.78 4 8.84 8.97 8.99 9.33 9.26 9.53 (1.46) (.94) (1.31) (1.84) (1.16) (.91) (1.01) (1.34) (1.24) (.51) 9.07 9.49 5.54 5.65 9.76 9.35 9.67 9.53 9.55 5 9.09 (1.41) (.69) (.74) (1.06) (.90) (.65) (1.09) (.94) (.97) (.75) 9.21 9.68 5.22 5.33 9.57 9.71 9.70 9.33 9.29 6 9.47 (1.27) (1.15) (.91) (1.00) (1.73) (1.02) (.97) (.87) (1.09) (1.06) 5.89 5.83 7 (.32) (.38) Justification 4.21 2.47 3.71 K 5.00 3.05 4.05 3.27 3.17 3.68 1.09 (1.38) (1.34) (2.22) (1.87) (2.13) (1.72) (2.38) (1.27) (1.61) (1.36) 1 2.41 4.07 4.50 4.22 3.40 4.22 3.44 5.08 4.96 4.61 (2.05) (1.22) (1.33) (1.36) (1.98) (2.00) (1.44) (1.13) (1.68) (1.49) 2 4.02 3.88 5.14 4.33 4.64 4.17 3.21 4.24 5.41 4.29 (2.03) (1.06) (1.66) (1.11) (1.78) (1.42) (1.75) (1.64) (1.79) (1.44) .32 3 1.96 5.26 .29 5.13 2.13 2.81 2.19 2.50 (2.36) (1.48) (.55) (1.36) (1.93) (2.27) (1.29) (2.29) (2.43) 4 4.40 4.44 3.83 5.46 3.52 4.48 4.41 2.67 4.12 3.56 (2.25) (1.20) (1.47) (1.98) (2.34) (2.15) (1.68) (2.02) (1.81) (1.27) 5 5.55 4.67 4.75 5.46 4.94 5.29 5.22 3.85 4.39 4.93 (2.01) (1.11) (1.73) (1.33) (1.82) (1.49) (1.38) (1.20) (2.24) (1.39) 6 5.52 4.76 4.72 4.83 5.80 4.79 4.89 4.28 3.61 3.06 (1.88) (.76) (1.17) (1.19) (1.74) (1.80) (1.80) (2.01) (1.87) (1.34) 7 4.53 4.39 (1.50) (1.09)

Tables

176

Table 37

Grade 2 Student'sTriadChoices (Reasons) Triad Pre Int Post

1

2

3

4

5

6

7

8

9

3(11) (cl) (cl)

3(11) 3(11) 3(vp)

3(11) 1(pr) 2(vtp)

3(11) 3(cl) 3(vt)

3(vt) 3(sz) 3(vt)

1(vt) 1(vp) 1(vtp)

2(vp) 2(sz) 2(sz)

1(vt) 2(vtp) 2(vtp)

2(11) 3(vt) 3(vtp)

11= looks like, Level 1 sz = size, Level 1 vp = visual components,Level 1

vt = visual transformation,Level 1 vtp = visual transformationof components,Level 1 pr = properties,Level 2 cl = classification,Level 3

Exampleof visual transformation: Triad5 (post). B and C. Because they are both the same thing except they are turneda differentway. Examplesof visual transformationof components: Triad3 (post). A and C. They'd be the same if this line were a little bit straighteron the top. Triad8 (interim).A and C. Because A looks like C if these were just pushedin like that. Triad4 (pre). B and C. Because the squareis exactly like a rectangleexcept it's not as spreadout as a rectangleis. Triad4 (interim).B and C. These two, B and C. Because C is really a rectangle.I don't know why but my teacherjust told us that. Triad4 (post). B and C. Because it's sortalike a squarebecause if you went like thatit would be a square. Triad6 (pre). A and C. No, I mean A and B. Because this one is sortalike this if you turnit like that way, the way this one is facing. Triad6 (interim).A and B. Because this one looks like it's just a little bit bigger and the bottomline is just slanteda little and the otherlittle line [left side] here. Triad6 (post). A and B. Because this sortalooks like [it] if the line [left side] were straight,[then] the line on the bottomwere straight.

Table 38

Numberof Componentsof MotionsMentioned,as Indicatorsof Level Student says:

# Components attended to:

Student is at least at Level:

Slides Move it to the right Slide it to the right Slide 3 spaces Move it to the right 3 spaces Slide it to the right 3 spaces

1 out of 1 out of 1 out of 2 out of 2 out of

2 2 2 2 2

1 1 1 2 2

1 out of 1 out of 1 out of 2 out of

3 3 3 3

1 1 1 1

3 out of 3

2

Turns Turn it to the right Turn it 90 degrees Turn it halfway Turn it right 90 degrees Turn it right 90 degrees using this point as turn center

Flips

Turn it over (flip) Flip it over Flip it so sharp point goes from left to right side Flip over the vertical side

0 out of 1 0 out of 0 out of 1 1 out of 1

1

1 1 1 2

Logo and Geometry Table 39 Level Determinationsof Case-StudyStudentsBefore and AfterMotions Unit (From Lewellen, 1992, p. 131) Motions Levels Polygons Levels Grade Interim / Final Student Interim / Final 6 1 1.5 1 1.5 Kurt 1 John 6 1 1 2 6 2 2 1 Kelly Allison 6 1.5 1.5 1 1.5 4 1 1.5 1 Jerome 1.5 4 1 David 1 1 4 1 Sandra 1 1 1 4 Helen 1 1 1 1

177

Back Matter Source: Journal for Research in Mathematics Education. Monograph, Vol. 10, Logo and Geometry (2001) Published by: National Council of Teachers of Mathematics Stable URL: http://www.jstor.org/stable/749925 Accessed: 30/06/2009 18:09 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=nctm. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit organization founded in 1995 to build trusted digital archives for scholarship. We work with the scholarly community to preserve their work and the materials they rely upon, and to build a common research platform that promotes the discovery and use of these resources. For more information about JSTOR, please contact [email protected].

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