Using Spatial Imagery in Reasoning Author(s): Michael T. Battista, Douglas H. Clements and Grayson H. Wheatley Source: The Arithmetic Teacher, Vol. 39, No. 3 (NOVEMBER 1991), pp. 18-21 Published by: National Council of Teachers of Mathematics Stable URL: http://www.jstor.org/stable/41195106 Accessed: 08-02-2016 17:22 UTC REFERENCES Linked references are available on JSTOR for this article: http://www.jstor.org/stable/41195106?seq=1&cid=pdf-reference#references_tab_contents You may need to log in to JSTOR to access the linked references.
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RESEARCH INTO PRACTICE
turns,turningthe turtleby increments untilthe headinglooks right.She then makestheturtle andexamines go forward theresults,erasingandreissuingtheturn commandifnecessary.As shecompletes sheannounces thatitis a square, herfigure, even thoughher turnswere not of 90 and shehas used forward students are inves- degrees though second-grade of 30, 5, and 5 forthe last side inputs withLogo. They tigatinggeometry insteadof 30 as forthefirstthreesides. in areworkingin a TEACH environment (She had saidearlierthatsquareshaveall whichtheturtle-movement commands they theirsides equal.) enterarenotonlyexecutedbutsimultaneTeacher:Howdo youknowit'sa square ously recordedin a procedure(Battista inpress;Clements for sure? andClements, 1983-84). earlier had successlessons, they During maneuvered theturtle todrawsquares fully ofvarioussizes.Foreachofthesesquares, turns. Whenasked theyhadused90-degree how theyknewwhethera figurewas a neither student hadmensquare,however, tioned90-degreeturns.
UsingSpatial Imageryin Geometric Reasoning
Classroom Episodes
had Students difficulty tilted recognizing ir
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are thenindividually These twostudents asked to use the turtleto draw a tilted square (theyare shown a pictureof a square whose sides are not verticalor horizontal).Each beginsby turningthe Megan: It's in a tilt.But it's a square becauseifyouturneditthiswayitwould turtleand makingit go forward.Megan usesa trial-and-error approachtofindthe be a square. Prepared byMichaelT. Battista KentStateUniversity Kent,OH 44242 DouglasH. Clements StateUniversity ofNewYorkatBuffalo NY14260 Buffalo, EditedbyGraysonH. Wheatley FloridaStateUniversity FL 32306-3032 Tallahassee, Theresearch on whichthispaperwas basedwas supportedby the NationalScience Foundationunder grantno. MDR-8651668. Any opinions,findings, conclusions,or recommendations expressed are reflect the thoseoftheauthors anddo notnecessarily viewsoftheNationalScienceFoundation.
quitelikesquares.So perhapsbecauseher intention had beento drawa square,she visualizesthatthisfigurecan be turned to looklikea squareinstandard orientation. She does so eventhoughthefiguredrawn does nothave all theproperties thatshe namedforsquares.The students' previously nexttaskis tousetheturtle todrawa tilted rectangle.Coletta,who has discovered thatshe needs90-degreeturnsto drawa square,uses 90s on her firstattemptat makingthetiltedrectangle, reasoningas follows: Coletta:Becausea rectangle isjustlike a squarebutjust longer,and all thesides are straight. but not Well, not straight, tiltedlikethat(makesan acuteanglewith herhands).They'reall likethat(showsa rightanglewithherhands),andso arethe squares. Teacher:Andthat's90 (showinghands at a 90° angle)? puttogether Coletta:Yes. Inotherdiscussions, shehasalso stated thata squareis a rectangle.
Teacher:Does thatmakesensetoyou? Coletta:Itwouldn'ttomy[4-year-old] sister,butitsortofdoes to me. Teacher:How wouldyouexplainitto her? Coletta:We havethesestretchy square Megan's reasoningis clearlyat the bathroom AndI'd tellhertostretch things. visual level in the van Hiele hierarchy itoutand itwouldbe a rectangle. (Battistaand Clements1988; Clements inpress).Shethinks andBattista, ofshapes Hereagain,we see howa student uses as visualwholesrather thanas geometric visual imageryto reasonin a geometric . objectspossessingproperties. According context.First,shereasonsthatrectangles turns visual to van Hiele, forstudentsat thislevel, have90-degree becauseoftheir to it "sort of "Thereis no why,one just sees it" (van similarity squares.Second, a a Hiele 1986,83). As herremarks made sense" that is indicate, square rectangle forMeganthereis a "why."This becausea squarecouldbe stretched intoa however, studenthas reasonedthatthe figurein This be more rectangle. responsemay frontof heris a squareby usingspatial sophisticated thanonemight think, initially her imagery.That is, she has observedthat for Coletta has just demonstrated that are tilted often do not look that and are squares knowledge squares rectangles ARITHMETICTEACHER
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^^^^^^^^^1 Whichfigurescan you draw using a rectangle procedure with two inputs? 3
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Ask students, "What figures can you draw with the rectangle procedure? You may use RT or LT to turnbefore you draw a figure.Record inside eacn figurethe commands that you use to draw it. Ifyou cannot make the figure, put NO inside it and explain why it cannot be drawn." From the activity"Rectangles: What can you draw?" in Logo Geometry (Battistaand Clements, in press).
similarin havingangles made by 90degreeturns.Thus,she mayhave understood at an intuitivelevel thatall recfromone antanglescouldbe generated ones otherbycertain legaltransformations, thatpreserve90-degreeangles.That is, herconceptualknowledgemayhave influencedherimagery.ContrastColetta's withthatofMegan,whose useofimagery withher contrasted visualtransformation verbal statement.We have additional evidence thatColetta's thinkingabout than shapeswas basedmoreonproperties was more So Coletta's imagery Megan's. byproperty-based likelytobe influenced considerations.
on the at thepictureoftheparallelogram the Because can't. No, sheet.) you activity lines are slanted,insteadof a rectangle over goinglikethat.(He tracesa rectangle theparallelogram.) Teacher: Yes, but thisone's slanted labeled4, thetiltedrectangle, (indicating drawnwiththe thatRyanhadsuccessfully Logo procedure). Ryan: Yeah, butthelines are slanted This one's (meaningnonperpendicular). This stillinthesize (shape)ofa rectangle. ' s slanted. - thething in- one(parallelogram) Teacher: Could you use different Itlooks isn'tslanted. Thisthing(rectangle) puts,or is itjustimpossible? a turn back if it but different if used slanted, (shows youput inputs. Ryan:Maybe you the so that to turn it He stares turn. in initial a new meaning bygesturing, (Ryantypes
Anotherexampleof visual reasoning occursas a fifth graderworkswithLogo inpress). andClements, (Battista Geometry can draw he whether is Ryan deciding witha rectangle variousfigures procedure thattakesthelengthsofthesidesas input. to usedtheprocedure He has successfully drawa tiltedrectangle(labeled4 infig.1) on hisunsuccessful andis nowreflecting make a to paralnonrectangular attempt lelogram(labeled7).
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1991 NOVEMBER
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Change the Logo procedure to make figuresВ and C.
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Logo procedureforshape A: TO FIGURE FD10 RT90 FD100 RT90 FD10 RT90 FD100 RT90 END itwouldn't sidesarevertical andhorizontal), be slanted.Anyway you move this(the itwouldn'tbe a rectangle parallelogram), (shakinghishead). So there'sno way. Ryanhas usedthreeofthefourclasses ofimageprocessesdefinedbyKosslynanimage,inspecting animage, generating and transforming an image(1983). First, aftermakingtheinitialturnandtrying to choosetheinputs,he recognizesthatthe betweenadjacentsidesis not relationship ofa consistent withtheimplicit definition in He then the rectangle Logo procedure. generatesan imageof a rectanglethatis consistent withthisdefinition, tracingit on theactivity sheet.Second,he inspects thisimageandcomparesitwiththeparallelogram,notingthedifferences. inthe Third,toelaboratethedifference "slantiness" ofthefigures, Ryanmentally transforms hisimagesoftherectangle and theparallelogram. His assertionthatonly
We needtodo morethanfocuson properties. therectangle couldbe rotatedto looklike a rectangleis the capstoneof his argument.His emergingknowledgeof the of figuresis supportedby his properties visualreasoning. In sum, we see a reciprocaleffect. WorkwithLogo involving about thinking of has refined Ryan's properties figures visual reasoning.In turn,his visual reasoninghas supportedhis analysisof the offigures. properties Finally,anepisodewitha kindergarten
student illustrates another natural rolethat in imagery plays geometric problemsolvJohn is asked to use ing. Logo todrawan that has three bendsin it. He open path drawsa "box" inwhichthebeginning and do not touch. He endingpoints explains thathe "justthoughtof all thepathswe made. One of the paths had the right number."Johnvisuallyreviewssolutions tosimilarproblems, analyzingeachtosee if it meets the constraints for the new problem. Theseepisodesareconsistent withthe that (Battista1990) hypothesis spatialvisualizationmaybe a moreimportant factor ingeometric for students problemsolving who are at thevisual,ratherthanhigher, levelsofgeometric Thatis notto thinking. that visual is unimsay,however, imagery at levels of portant higher thought. Many studentsthatwe have classifiedat the level (the level imdescriptive-analytic mediatelyfollowingthevisuallevel) are inhibitedin theirreasoningand problem solvingbecause theirspatialimageryis notcoordinatedwiththeirknowledgeof properties.For instance,manyof these studentsknow the propertiesof a rectangle but have difficulty recognizing whether a figurein a nonstandard spatial orientation(such as shape 7 in*fig. 1) possesses these properties.Moreover, becausestudents atall levelsoftenreason byconsidering specificimages,theirreacan soning go astrayiftheymakea mistake in generating, or transinspecting, those In sum, visual forming images. when imagery properly developedcan makea substantial contribution at all levels of geometricthinking.Thus, in we shouldnotfocus teachinggeometry, on of figuresand relasolely properties them. We shouldalso tionshipsamong students vivid help develop imagesand coordinatetheseimageswiththeirconceptualknowledge.
Instructional Implications
The foregoingdiscussionsuggeststwo recommendations forteaching: 1. Recognize thata greatmanystudentsuse visual imageryto reason.Respectthismodeofthinking. Helpstudents use ittoanalyzeandconvincethemselves ofthetruth ofgeometric ideas. TEACHER ARITHMETIC
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workin pairs, 2. Discuss visual reasoningwithstu- commands.Have students dents.Askquestionsthatmighthelpstu- onereadingthecommandsandoneacting dentsincorporate conceptualknowledge as the turtle,followingthe commands witha metric ruler(FORWARD 50 means intotheirvisual-reasoning processes.For Students her bathroomforward 50 Coletta after instance, mm)anda protractor. gave thatsquaresarerectangles, taketurnsactingas theturtleon paper. tilejustification theteachermighthavedrawna trapezoid References I canstretch thisfigure andasked,"I think MichaelT. "SpatialVisualization andGenBattista, Is this intoa rectangle. figurea geometric derDifferences in HighSchoolGeometry." Journal for Researchin MathematicsEducation2 1 rectangle?"This question would have (January1990):47-60. Colettato clarifythenatureof prompted Battista,MichaelT., and Douglas H. Clements."A when her"stretch" transformation. Also, SchoolGeomeCase fora Logo-basedElementary discussinga conceptsuch as thatof a triangle,make sure to use a varietyof different examples,suchas manydifferentsizes and shapesof trianglesin nonThisapproachwill orientations. standard evokedynamic imageswhen helpstudents When students aboutconcepts. reasoning with mathematical familiar havebecome transformations, theycan be encouraged mentaltransformations their to describe mathematical terms. using
Arithmetic Teacher36 (NovemtryCurriculum." ber 1988):11-17. - .LogoGeometry. N.J.:SilverBurdett Morristown, & Ginn.In press. Clements,Douglas H. "SupportingYoung Children'sLogo Programming." Teacher Computing 11 (December1983-January 1984):24-30. "GeClements, DouglasH., andMichaelT. Battista. ometryand SpatialReasoning."In Handbookof Researchon Mathematics Teaching,editedbyD. A. Grouws. Reston,Va.: NationalCouncil of TeachersofMathematics andMacmillan.Inpress. Kosslyn,StephenM. Ghostsin theMind'sMachine. New York:W. W. Norton& Co., 1983. vanHiele,PierreM. Structure and Insight.Orlando, Fla.: AcademicPress,1986. W
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Many of thetasksthatwe have presentedto studentsin our researchand curriculum-development projects can students' coordination of visual promote and Some imagery conceptual knowledge. follow. our students (In examples project, ingradesK-l useda single-key versionof in 2-4 students used TEACH, Logo, grades and studentsin grades5-6 used an enhancedversionof regularLogo [Battista andClements,in press].) 1. Askstudents todrawa square Activity in Logo. Then ask themto drawa tilted (Grades square.Do thesameforrectangles. 1-4) 2. Give studentsa procedure Activity RECT thatdrawsa rectanglewhenthe twodimensions fortherectangle aregiven as input.Askthemwhichfigures (as infig. can draw with RECT. (Grades 1) they 2-6) thethree 3. Showstudents Activity figures infigure 2, twoat a time.Askhowfigure A is different fromfigure B. (Themajority of studentswill give a global response, suchas, "It's biggerorlonger,"notrefertosidelengths.)Givethe ringspecifically thatdrawsfigstudents a Logo procedure ureA and letthemtestiton a computer. Then ask themhow theycan altertheir procedureto drawfigureB. Repeatwith В andС (Grades2-6) figures Note: If you do nothave access to a computer,you can still use Logo-like
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