Development of Turn and Turn Measurement Concepts in a Computer ...

Development of Turn and Turn Measurement Concepts in a Computer-Based Instructional Unit Author(s): Douglas H. Clements, Michael T. Battista, Julie Sarama and Sudha Swaminathan Source: Educational Studies in Mathematics, Vol. 30, No. 4 (Jun., 1996), pp. 313-337 Published by: Springer Stable URL: http://www.jstor.org/stable/3482809 . Accessed: 30/04/2014 15:55 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp

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DOUGLAS H. CLEMENTS,MICHAELT. BATITSTA,JULIESARAMA AND SUDHA SWAMINATHAN

DEVELOPMENTOF TURN AND TURN MEASUREMENT CONCEPTSIN A COMPUTER-BASEDINSTRUCTIONALUNIT

ABSTRACT.This study investigatedthe developmentof turnandturnmeasurementconcepts within a computer-basedinstructionalunit. We collected datawithin two contexts, a pilot test of this unit with fourthirdgradersanda field test in two thirdgradeclassrooms.We conductedpaper-and-pencilassessments, interviews, and interpretivecase studies. Turns were less salient for childrenthan 'forward'and 'back' motions. Studentsevinced a progressive constructionof imageryandconceptsrelatedto turns.They gainedexperiencewith physical rotations,especially rotationsof theirown bodies. In parallel,they gainedlimited knowledge of assigning numbersto certainturns,initially by establishingbenchmarks.A - constituteda synthesis of these two domains- turn-as-body-motionand turn-as-number criticaljuncturein learningaboutturnsfor some students.Some commonmisconceptions, such as conceptualizingangle measureas a linear distancebetween two rays, were not in evidence. This supportsthe efficacy andusefulness of instructionalactivitiessuch as those employed.

INTRODUCTION

Numerous studies have addressed students' development of turn, angle,

and turn measurementconcepts, particularlyin computerenvironments. These studies, however, have not focused on the processes of learning.In addition, they have often been narrowin scope; for example, they have studentswork in computermicroworldsdesigned to teach angle and turn concepts only. Thereis a need to studythe developmentof these conceptsin the contextof instructionalunitsthatmoreclosely reflectrecommendations of recentreformdocuments. We are engaged in a large-scalecurriculumdevelopmentproject,funded by the National Science Foundation(NSF), that emphasizesmeaningful mathematicalproblems and depth ratherthan exposure.1One of the geometryunits engages third-gradestudentsin investigationsof geometric paths. We investigated children's learning within this unit, emphasizing theirdeveloping ideas aboutturnsand theirmeasurement. The theoriesof the presentstudyandthe instructionalunit arebased on the notionthatchildren'sinitialconstructionsof spaceemergefromaction, EducationalStudies in Mathematics 30: 313-337, 1996. ? 1996 KluwerAcademicPublishers. Printed in Belgium.


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ratherthan from passive 'copying' of sensory data (Piaget & Inhelder, 1967). An implication is that noncomputerand Logo 'turtle' activities designed to help childrenabstractthe notion of path- a recordor tracing of the movement of a point - provide a useful environmentfor developing theirconceptualizationsof simple two-dimensionalgeometricfigures. Because turns(and angles) are criticalto this view of figures,and because the intrinsicgeometry of paths is closely relatedto real world experiences such as walking,pathactivitiesmay be especially efficaciousin developing students'conceptualizationsof turnand turnmeasurement. Evidence supportsthis hypothesis:Logo activitiesare beneficialto students' development of turnconcepts and turnmeasurement(Clements & Battista, 1989; Kieran, 1986a; Olive et al., 1986). Researchalso indicates that studentshold many differentschemes regardingturnand angle measure. Third graders, for example, frequentlyrelate the size of an angle to the length of the line segments that form its sides, the tilt of the top line segment, the area enclosed by the triangularregion defined by the drawnsides, the length between the sides (from points sometimes,but not always, equidistantfrom the vertex), the proximityof the two sides, or the turnat the vertex (Clements& Battista,1989) Intermediategradestudents often possess one of two schemes. In the '45-90 scheme', oblique lines are associated with 450 turns and horizontaland vertical lines with 90? turns.In the 'protractorscheme', inputsto turnsare based on an image of a protractorin 'standard'position; therefore,no matterwhat the turtle's initial orientation,if the end goal was a headingcorrespondingto 450 on a protractorin canonical position, a 'rt 45' commandmight be issued (Kieranet al., 1986). After working in Logo contexts designed to addressideas of angle and turn,childrendevelop more mathematicallycorrect,coherent,andabstract ideas aboutthese concepts.This may be becausethey use a morepersonally meaningfulturtleperspectivescheme, insteadof, for example,a protractor scheme (Clements and Battista, 1990; Kieran, 1986b; Noss, 1987). Such benefits are educationally significantbecause studentshave considerable difficulty with angle, angle measure, and rotationconcepts and because these concepts are central to the development of geometric knowledge (Clements and Battista, 1992; Krainer,1991; Lindquistand Kouba, 1989; Mantoanet al., 1993; Simmons and Cope, 1990). Ourgoal in the presentstudy was to investigatethirdgrader'sdevelopment of turn and turn measurementconcepts within an instructionalunit on geometricpaths,includingthe role of noncomputerandcomputeractivities in thatdevelopment.Note thatwe use the concept turnas the amount of rotationalong a path (a 'turtle', or differentialgeometricperspective);


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we did not, in our researchor in the curriculumunit, addresstraditional (interior)angle measurementor the relationshipbetween turtleturnsand such angle measurement.

METHOD

Procedure We investigatedthird graders' mathematicalthinkingin the context of a unit of instruction,TurtlePaths, in two different situations. In the first situation,a pilot study,a graduateassistanttaughtthe unit to fourchildren. We conducted interpretivecase studies of each child, with the goal of makingsense of the curriculumactivities as experiencedby the individual student(Gravemeijer,1995). Such interpretivecase studies serve similar researchpurposes as teaching experiments(e.g., Steffe and Cobb, 1988), but are more naturalistic.They also better serve the needs of immediate curriculumdevelopment. In the second situation,a field test, two of the authorstaughtthe unit to two classes the following autumn.Data collection included pre and post interviews,pre andpost paper-and-penciltests, interpretivecase studiesof two students,and whole-class observation. Participants Participantsfor the pilot study were two girls, Anne and Barb, and two boys, Charlesand David, from a ruraltown, all 9 yearsof age. Participants in the field test were students in two third-gradeclasses of inner-city schools, 85%of whom were African-Americanand most of the remainder Caucasian.As was typical for the school, 80%of the studentsqualifiedfor Chapter1 assistancein mathematics.The two case-study studentsfor the field test were Luke and Monica, both 9 years of age. Curriculum In the TurtlePaths unit, studentsexplore paths andthe motions thatcreate them. They walk, describe, discuss, and give commandsto create paths. The main goals regarding turns are to have the students (a) build up images of turn as physical rotation, a change in heading or orientation; (b) distinguish between smaller and largertums (gross comparison);(c) constructand iterateunits of turn;(d) estimateturnmeasuresusing certain units as benchmarks;and(e) recognizethatdifferentphysicalrotationscan yield the same geometriceffect (e.g., rt 210 and lt 150). The unitis


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organized into three main investigations; a description of each session follows. InvestigationI - path and lengths of paths 1 Studentsgive commandsto specify movementsthatcreatepathsand develop language to describe various kinds of paths. The main goal relatedto turnsis to bringto an explicit level of awarenessthephysical motion of rotationand the correspondencebetween this motion and the creationof corners,or bends, in paths. 2 Studentscount steps in a maze to find specifiedpathsandplay an offcomputerMaze Paths game. Problemswith more than one possible solution are emphasized. 3 Studentsplay a 'Get the Toys' game in which they instructthe Logo turtle to get three toys before its batteryruns out of energy. In this game only 900 turns are needed. The goal is for students to write explicit commandsfor turnsand differentiateturnand displacement situations(e.g., as representedin rt vs. fd commands,respectively).

Investigation II - turns in paths

4 Students turntheir bodies, discuss ways to measureturns,and learn about degrees. They are introducedto anothercomputergame, 'Feed the Turtle', in which they must use their knowledge of turns (other than900) to directthe turtlethroughchannelsof water.Goals include understandingturns as a change in orientationor heading and estimating and measuringturnsusing units of 300 as benchmarks. 5 Studentsplay the Feed the Turtlegame and thus continueto estimate and measureturns.They discuss the natureandpropertiesof triangles and build their own descriptionsof thatclass of geometricfigures. 6 Studentswrite Logo proceduresto draw equilateraltriangles.Goals include finding the turns for this class of figures (the unit does not attemptto develop ideas aboutinteriorangle measure). 7-8 Studentsfindthe missing measuresto completepartially-drawnpaths.

InvestigationIII - paths with the same length: isometricexercises 9-10 Students write as many proceduresas possible to draw a rectangle of a certainoverall length - 200 steps. 11-12 Students write procedures to draw shapes of each of specified perimeters,to design a face on the computer.


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Logo environment A modifiedLogo environment,Geo-Logo,2is an intrinsiccomponentof the instructionalunit. Geo-Logo'sdesign is based on curricularconsiderations and a numberof implicationsfor the learning.and teaching of geometric concepts with turtlegraphics(ClementsandSarama,in press).One critical featureof Geo-Logo is thatstudentsentercommandsin 'immediatemode' in a command center (though they can also enter proceduresin a 'teach' window). Any changes in the commandcenterare reflectedautomatically in the drawing.For example, if in the sequence of commands f 5 0 rt 30 fd 50, a student changes rt 30 to rt 60, the change is immediately reflected in a correspondingchange in the geometric figure. The dynamic link between the commands and the geometry of the figure is critical;the commandsin the commandcenteralways precisely reflectthe geometry of the figure. The environment also includes features specifically designed to aid students'developmentof turnandturnmeasurementconcepts.Followinga turncommand,the turtleturnsslowly on the screeninsteadof immediately appearingat the resultantheading.In addition,a 'TurnRays' optionshows rays during turns. If studentstype rt 120, a ray is drawnto show the turtle's initial heading. Then as the turtle turns, anotherray turns with it, showing the change in heading throughoutthe turn.A ray also marks every 300 of turn. Geo-Logo provides two commands, rtf and ltf, to help beginning students build on previous experience with physical tums ('rtf' standsfor 'rightface' and is equivalentto 'rt 90') and to appreciateturnsqualitativelybeforeencounteringmeasurementin degrees. A line of sight tool displays an arrowemanatingfrom the center of the turtle with additionallines at 30? increments.A turtle turnertool allows the studentto turnan arrowto point in a given heading and then outputs a commandthat would turnthe turtleto face thatheading.Finally,a label turns tool labels all the turns (not the angles) created on the drawing window. Data collection Interpretivecase studies In the pilot test, we performedinterpretivecase studies(Gravemeijer,1995) with all four students.In the field test, we studiedtwo studentsintensely (Luke and Monica). When studentsworkedin pairs,we also observedthe case study student'spartner(Nina was Monica'spartner).A researchersat next to andobservedhercase-studystudentthroughouteach session. If the student's thinking could not be ascertainedthroughpassive observation, the researcherasked the studentto thinkaloud.


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The goal was to observe studentsnaturallyinteractingwith the teacher, peers, and activities. However, if the thinkaloud-strategywas insufficient, the researchersasked questions. Such questionsmay have providedscaffolding for the students'conceptualdevelopment;the questionsarereported in the following section when pertinent.Anotherresearcherobserved the classes as wholes. Interviews

We interviewed pilot students following the completion of the unit to ascertain their thinking about tums and turn measure; we interviewed thirteenstudentsfrom the field test on either the firstor second day of the unit and then again after completion of the unit. The interviewerbegan with a structuredprotocol, but followed students'responsesto probetheir understandings.Individualitems in the interviewwill be describedin the Results section. Paper-and-penciltest We administereda shortpaper-and-penciltest to seven classroomstudents following the pretreatmentinterviewto determineits appropriateness.Studentsappearedto cope with andunderstandthe items adequately;therefore, the test was administeredto the whole class as a posttest. In retrospect, it was unfortunatethat we did not administerthe test to the entire class as a pre-test,even though the paper-and-pencildata do not weigh heavily in the analyses. As with the interview, no total scores or statistics were computer;rather,individualitems were analyzed.These will be described as appropriatein the following section.

RESULTSAND DISCUSSION

In analyzingthe data,six themesemerged.Fourthemeswere relatedto specific conceptualandproceduralknowledge in the domainof turns:concept of tum; right and left directionality(clockwise and counter-clockwise); turn measure, with an emphasis on building benchmarksas units; and combining turns.Two were overarchingthemes:the role of the computer environmentin students' development of turnconcepts, and the dialectical relationshipbetween two cognitive schemes, extrinsicperspectiveand intrinsicperspective,in students'developmentof knowledgeof turns.The extrinsic perspectivescheme involves a fixed, imposed,externalframe of referencesuch as thatused in coordinategeometry.The intrinsicperspective scheme is used when the perspective of the turtleis adopted;such a perspectiveis used in differentialgeometry.


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Conceptof turn Turns and turn commands were not as salient for the children as were forwardor backwardmovementsof the Logo turtle.They were constructed as conceptual objects and discriminatedfrom other concepts only over significant periods of time. For example, in the first lesson, before the teacher introducedthe turtle, she asked the difference between turning along a 'U'- and a 'V'-shaped path on which the childrenhad walked. Luke talked with nearby classmates only about the space in the middle, with no mention of the turningactions. In the Logo environment,Monica and Nina wrote two forwardcommandsin a row to draw an 'L-shaped' path, without an interspersedturn command. Turn commands may be less salient than movement commandsbecause their representationoften involves not a single graphicobject, but rathera relationshipbetween two graphicobjects. Of course, it is true that the turtleperformsa motion for each command,andin Geo-Logo,turnswere slow enoughto be seen. After the action had taken place, however, the recordsof turnactions (bends in paths)were not as salientto studentsas recordsof forwardmovement(line segments). Luke evinced a slow progressionin discriminatingturnsfrom forward or backwardmovements,giving the firstindicationof a dialecticalconflict between two schemes. Following the discussion of the 'U' and 'V', the teacherintroducedLogo commands.Lukegave commandsin the following form: fd, stop, rt, stop. He had apparentlymade sense of the notion of 'turningon a point' and felt the need to explicate the end of the forward motion before the turn took place. Then, the teacher drew a rectangle on the board and asked for Logo commands. Luke's commands were 'fd 7 then right face then go right then left'. Additional observations confirmedour original notion that, if the turtlewas not facing straightup (i.e., 0? heading), Luke became less precise and used an indiscriminate combination of intrinsic and extrinsic schemes. For examples, by 'then rightface', Luke was mentallyturningthe turtle.However, 'thengo right' indicateduse of an absolute,extrinsicscheme (move to an absoluteright). He omittedthe verticalside, andfinishedwith anextrinsicmove-to-the-left scheme. On other occasions, he allowed a single right or left command to fill the dual role of 'turn-and-move'(that is, a rtf command would both rotate the turtle- or just make it 'face' a certainabsolute direction - and then displace it in the new direction),a conception not uncommon for Logo beginners (Clements and Battista, 1991; Singer et al., 1981). These actions were off the computer.The feedback Luke received from the Geo-Logo turtleled him to discriminatebetween movement(f d, bk) and turnsand their relatedcommands.When he entereda turncommand,


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A

B

Fig. 1. Two girls' responseto a requestto drawa path 14 steps in lengthwith two corners.

he was surprisedthe turtledid not move, then reflectedon the discussion and body movementspreviously made in class, to conclude that, 'The rt commanddoesn't turnand move to the right, it just turnsthe turtle'. Tums were also less salient, or phenomenologically'real', to Monica and Nina. They did not tie their notion of turnin a path to theirnotion of corneruntil the second session. Even then, however, the latternotion - a cornerof a closed shape - was more salient (cf. Mitchelmoreand White, 1995). On the Maze Paths activity, they drew a path 'with two corners' as illustratedin Figure 1. The girls indicatedA and B as the two corners. The researcherasked if B was a corner in the path or in the pond. 'Oh! The pond', they replied, and began looking for otherpaths for a solution. Afterwards,they focused only on bendsas indicatinga turnin the path.The act of turningwas still less 'real' to them than were forwardor backward movements,however, as indicatedby their omission of turncommandsin their first attemptsto write Logo code. They also did not take turnsinto account when counting the numberof commandsneeded in the Get the Toys game, claiming - against visually-displayedevidence - that 'turns don't use up the battery'.The researcheraskedthem to keep an eye on the battery as the turtle turned.They agreed it did lose energy, and counted the turns thereafter.The combination of needing to type the Logo turn commandsand seeing the turnsreifiedby the batteryappearedto increase their attentionto turns;turnsand turncommandswere not neglected after this activity. Right and left directionality Some studentsmade efforts to recognize and issue right versus left tums correctly,thoughthey were in the minority.The othersunderstoodthatthis issue was important,but they were inaccurateand used trial and errorto correct mistaken guesses. During the pilot test, the pair of girls and pair of boys differed consistently on determiningwhethera given turn was a left or a right turnon the Get the Toys game. The girls frequentlydid not concern themselves with the issue, but instead entered one or the other,


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then moved the turtle forwardif they happenedto be correct, and back otherwise (recall that this strategywas effective because all turns in this game were 900; the girls consciously used this to their advantage).The boys, in contrast,worked to determinethe correctturndirectionfor each command.Theirbehaviorsindicatedthatthey eithertwistedthemselvesso they were alignedwith the turtle,orprojectedthemselvesmentally('That's my...rightside') andthen 'readoff' the directionfromtheirown bodies. Luke could correctlyidentify his left and righthands.When he played the computergames, however,he frequentlygot confusedaboutthe direction of the turns.It was commonto hear,'Left,no right',followed by, 'Just try it'. Although Luke typically was concernedand serious about being correct (even syntacticalerrors,such as fdd2 0, disturbedhim, possibly due to the errormessage), he usually did not express anxiety about not knowing which way to turnbefore he entereda turncommand.The visual feedback regardingthe directionof the turndid not disturbhim. Luke consistently used a gesture of rotatinghis hand (as if turninga doorknobthis way andthat)for turnsthatwere giving him difficulty.While this supportsthe view thathe had successfully discriminateda tuM from a forwardmovement,it seemed to offerLukelittle assistancein determining the directionof turn(nor the amount,as will be discussedlater). Monica similarlydidnot feel comfortablewithrightandleft distinctions in turningmotions,thoughshe could identifyherleft andrighthands.When the teacherasked the directionof body turns,Monica waitedfor someone else to give the answer. Occasionally, she would say to herself, 'This is my right hand and I am turningthis way', but there were no signs that she successfully applied or internalizedthis strategy.She could state that the turtle was facing 'this way, and it has to turnthis way' but she could not map that on to her knowledge of left and rightwhich may have been exclusively extrinsic. In a similarvein, Monica would usually ask Nina, 'Yourarefacing this way and I want you to face this way. Right face or left face?' Nina turned her body and usually producedthe correct answer.In this way, Monica usually avoided having to find the turns herself. Hoyles and Sutherland (1989/90) similarly observeda situationin which such a relationshipcan effectively block one memberof a pairfrom learningaboutturns(for their student,even over several years). Monica had difficultywith directionof turnuntil the end of the study,especially with a turtleheadingof 1800 (i.e., facing down). She did become more willing, however,to makebody turns to the right, though she still took the class' lead for left turns.Turningher body for the turtle,when she did not rely on Nina, was her only successful


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strategy.As with most otherstudents,makingmistakeson the directionof tum did not seem importantto Monica. A final example concernsturnsfor rectangles.Monica once statedthat turnsfor rectangleshave to be 'rt 9 0's'. She was asked if she needed to go right all the time and she replied, 'You don't have to, but if you start with left, you'd bettergo left all the way'. Despite this, on the 200 steps activity,Monica and Nina would sometimes turnright for one cornerand left for the next. They correctedtheirmistake,but were so intenton getting the lengths correctthatthey did not pay much attentionto the directionof the turns. Turnmeasure The limited salience of turns, combined with less familiarity with turn measurement,may account for children's slow and uneven development of concepts for turnmeasureas well as for turns.Studentsoften needed a figuralobject (i.e., something physical or graphicalto count) to produce a numberfor a turn measure(Clements et al., in press; Steffe and Cobb, 1988). After measurementin degrees was introducedin the pilot test, the teacherasked the studentsto estimate a certainturn.Charles guessed rt 6 0 (correct),Anne, rt 4 0. Barb laid her pencil, which had nonequallyspaced concentriclines (circling the pencil), across the angle, countedthe lines on the pencil, and said 20. After looking at and listening to others, she said, '50', most likely adopting the group consensus she perceived. In almost all succeeding situations,Barb gave evidence that she needed something figural to use to measure the turn.We infer that she could no re-presentto herself an image of turnsand turningthroughdegrees. On the computer,a correspondingstrategywas to create a figuralunit by iteratinga small turn.In Feed the Turtle,the pilot girls frequentlytrieda turncommand(e.g., 1 t 5 0) and then typed small increments,such as 1 t 10 it 10 it 10 rt 5. Observationsindicated such guessing, followed by successive approximation,was the norm for the pilot students; this occurreddespite the students'introductionto degrees andturningtheir bodies for variousmeasures.For this reason,the game was alteredfor the field test to accept only multiplesof 300. Once students had gained experience with 90? turns, fd 30 this was often used as an input,even when inaccurate.For rt 90 example,Anne andBarbdefineda Logo equilateraltriangle fd 30 as shown, before going to the computer.The teacherasked, rt 9 0 f d 30 'Can you draw what the computerwill draw?' rt


90


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The studensdrewnot only whatthey believedtheircommandsrequired, but also what they needed to create a triangle:They drew the firstturnto be just a bit more than 900; the second, however,was much more, so as to form a triangle. Teacher:That'snot a squarecorner,is it? Barb: No, I'll drawit again. [She drewboth as 90? angles, thencurvedthe sides until the figure closed. Sounding unsure of herself, she said:] Well, I guess that would work. The teacher asked them to try their trianglein Geo-Logo. They tried rt 90, then rt 100 and rt 110, but none created a closed figure. Finally,they triedrt 120 andwere successful.However,none ofthepilot childrenknew on the finalinterviewthatthe turnsfor an equilateraltriangle measured 1200. Thus, these students achieved only a limited knowledge of turnsin this context. The field test students also exhibited a need for figuralobjects and a relianceon 900 turns.In addition,observationsfromLukeofferedus insight into specific processes involved in the developmentof turnconcepts.Luke developed a sense of turn measurementonly tenuously and only after a protractedtime. On the second day of the unit (before any on-computerexperience),he was asked, 'How many rtf's does the turtlehave to do to face backwhere he started?'Luke answered, 'Two'. When asked to explain, he drew two right angles, connecting them to form a rectangle.We infer that his early conception of turn was heavily influenced by school exemplars of right angles, and thereforewas really about a path with a bend. After only one day of 'playing turtle', however, he was startingto build some concept of rotation.This became clear when the interviewerproceededto ask Luke a follow-up question to focus his attentionon the act of turning.Shown a turtledrawnat 0? and asked what it would look like afteryou commanded it to rtf, Luke drew a turtleat a 90? heading.When askedto drawit after a second rtf, however,he drew it at a 0? heading,then correctedhimself and drew it at a 1800 heading. Luke's first estimation of turns other than 900 revealed that he too needed a figural entity to produce a turn measure, and that this was not necessarily based on rotation.Luke worked on an activity with another boy in which a turn was representedas two intersectingsegments and the measure of the turnrequested(Figure2). Luke used the length of the line segments in these representationsto determinethe amountof fun. For example, he said that because one segment was half the size of the line segment indicating 900 on the protractor, the turn was 45?. He did not get

one item correct, althoughfour of the six turns he worked on measured


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Guess Measure

Guess Measure

Guess Measure

Fig. 2. A paper-and-pencilactivity on turnestimation.

900. Luke did not like to 'lose', and,unfortunately,he saw this activityas a competitionbetween himself andhis partner.Therefore,we inferthatthese behaviorsmay not have indicatedhis full competence,butthey do indicate his performanceundermild pressure.Thatis, he fell backto manipulations of numbersandmeasuresof length,with which he felt morecompetent. Additional reasons for Luke's resistanceto using the mental processes proposed by the teacher arose in the next session, during which the researcherrepeatedthe activity with Luke alone. After Luke guessed on the firstfour,the researcherdrewin a rayindicatingthe turtle'sinitialheading, in an attemptto move Luke to a more analytic approach.She asked him if he noticed anythingaboutthe firstfourturns.He indicatedthatsome looked like a 'perfectrightface' and thatthe 300 turn'hada smallerspace between the lines'. By the eighth item, Luke began using language that indicatedan awarenessof motion:'It'sgoing way over' for a 1200 turn(his guess was 150). The researcherstoppeddrawingin the 'line of sight' line segments, following Luke's lead in focusing on the motion.Luke stood up and enacted the turnfor the last two items. Althoughhe still confused left and right,he was more accuratein his estimationsof the measures.He told the researcherthathis partner'got themall rightbecausehe used his body'. Does that help? 'Yes'. Why did Luke resist a helpful strategy?Later,he was overheardsaying thathis partnercheatedby using his body to get the turns correct. Luke's regularteacher had always insisted on completing work sheets in a prescribedmanner.Completinga multiplicationproblem by using repeatedadditionwas considered'wrong'.Figuringout a problem mentally,ratherthanusing the desiredpaper-and-pencilalgorithm,resulted in point deductions.Given this classroomenvironment,it is not surprising thatLuke felt thatusing his body for 'assistance'would be wrong. He felt thathe should have just known the answersupon inspection.



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Luke's resistanceto using such 'body assistance',andhis comfortwith arithmetic,then, delayed his developmentof rotational,turtleperspective, schemes. However,once he was guidedto focus on turns,he beganto adopt such a rotational,turtleperspective,scheme and infusedthe numberswith a quantitativemeaning, albeit only in the turtle context initially. In the following session, Luke was asked to enact various tum commands.For rt 45 he said, '45 is half of 90' and he physically turnedless than 900. For a rt 180 he said that it would be two 90's, and he describeda rt 3 6 0 as a 'big turn'.On papertasks,however,Luketendedto use his finger to measureturns,or to guess, withoutusing any identifiableunit. Though he did indicate a turningmotion, he fell back to his considerablecompetence with arithmeticto producea measure.The numbershe employed in such arithmeticmanipulationswere usuallynot meaningfullyconnectedto quantities,but were numbers'pulled out from' the situation. On the structuredcomputergame, Feed the Turtle,Luke connectedhis numericalwork to quantitiesof turnmore consistently.For example, once he tumed rt 3 0. He remarked,'Ten more'. Thoughthis was inaccurate (it was 300 more;recall the game was restrictedto multiplesof 300, which the researcherstatedto Luke at thatpoint),his notionthatthe turnhadto be greatershowed morecompetencethanhe hadshown on previousactivities. Similarly,a few move later, he said, 'Right...[pause]'The teacherasked, 'Is it more than 90 or less than 90?' More...90...120!' Luke initially had difficultyapplyingthese quantitativeconcepts in the next computeractivity, the less structuredtask of drawingan equilateral triangle. He typed in fd 80 for his first side. For the first turn,he said, 'itf', but changedhis mind andtyped rt 60. Whenthatdidn'thave the desired effect, he said that he should have tried 1 t 6 0, but changed his mind again and typed in rt 9 0. He was still dissatisfiedwith the effects and said, 'No, you have to go over', as he turnedhis handto point to the screen in the area that he wanted the turtleto go. He saw that the turtle neededto turnmoreif the pathwere to continuein the desireddirection.He then synthesizedhis ideas aboutphysicalturnsand turnmeasurementand changed the command to rt 120. Satisfied, he entered fd 80 for the next side andagainfaced adecision regardingthe tum.He triedrt 90 and thenalso enteredrt 8 0, most likely tryingthe side lengthin desperation. He entered fd 80 for the last side andbecame frustratedthathis triangle was not correct.He sat back in his chair and staredat the screen looking dejected.He knew which commandwas wrong,buthe didn'tknow how to fix it. He finally replacedthe rt 8 0 with a rt 3 0, which, concatenated with the rt 9 0, constructedan equilateraltriangle.


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On a subsequentattemptto drawan equilateraltriangle,Lukeentered90 forall fd, rt, and lt inputs.The sequenceof commandsconcludedwith 'it 90 it 90'. The researcherasked him to stop enteringcommands so she could copy the commands into her field notes. Luke sat back in his chair and staredat the screen. He noticed thatthe turtlewas facing the oppositedirection.He stated,'I endedup the otherway, too. Isn'tthat 180?' He then observed that the last two commandsdid combine to make 180. This was a significantinsight for Luke;withoutenactingbody turns,Luke projectedhis personalexperienceinto the turtlecontext and connectedhis body rotationand the turtle'srotationto the measureof 1800. Now Luke had a quantityto measureand a benchmarkto use for that measurement; before thattime, he did not possess a unit with which to compareturns. For the remainderof the activities, Luke consistently used 120? for equilateraltriangles and 900 turns when appropriate.For example, Luke chose a trianglefor the nose of his face and enteredrt 12 0 for his first turn(Session 12). Whenthe researcheraskedhim why, Lukedrew a line at a 90? headingand said, 'Because you go up andthat [gesturingto the right at a 900 heading] would be 90 and you need 120 to go down here'. For all other turnsituations,Luke turnedhis body to determinethe directionand amount. Monica and Nina developed greaterknowledge of turn measure. Initially, Monica seemed to associate each right face or left face turn with facing specific walls in the classroom. Soon, however,both she and Nina appearedto conceptualize 900 as a certainamountof turnnot associated with any particularexternal environmentalfeature,(thus not an extrinsic scheme). Often, when walking to the computerlab, they would ask the researcherto give them directions 'like the turtle'.They were consistently accurate about turning 90 to the right or left in this context. Similarly, Monica could easily say where she would be facing afterone, two, or three rt 90's. Deciding turn measures on paper or computerwas more difficult for the girls. Monica typically gesturedto indicatea 900 turnbefore giving an estimate of any turngreaterthan900. Monica's strugglewith the left/right directionof the turnfrequentlyinterferedwithherassessmentof the amount of turn. In addition, even though the teacherand curriculumstressed the motion of turning,Monica and Nina sometimes attendedonly to the final heading of the turn(e.g., turtlestartsat 300, ends at 1800;the turnis seen as rt

18 0?) - an extrinsic feature shared with the 'protractor' scheme

for turns(Kieranet al., 1986). After receiving feedback from the teacher,classmates, and Geo-Logo, Monica changedher strategy,adoptingan intrinsic,rotationalscheme that



327

eventually included two benchmarks.She would swing her arm 900 and then visually gauge whether a turn was equal, greaterthan, or less than 90?. Later, she began to use a second benchmark,commenting, 'That's right 30. It's a small turn'. Other studentsshowed similar signs of using 30? as an iterableunit of turn. For example, one student,shown a 1350 angle, said, 'It's more than90. It's [gesturingwith a turningmotion of the hand]30, 60, 90, 120. About 120'. As with Luke, added lines often interferedwith, ratherthan helped, the two girls' turn estimates. For example, shown a pictureof the turtle beginning at a 900 heading and ending at 2100, Monica estimated a rt 18 0. She was usually accuratewith 1800 turns;perhapsthe horizontalline behindthe turtlemisled her.When tryingto determinethe amountof turn, Nina would frequentlydraw a line perpendicularto the turtle'sheading. She would, however, sometimes mistakenlyuse this line as the beginning of the turn. Undermost circumstances,however,Monica was proficientat estimating turns.For example, when her class solved the firstequilateraltriangle problem together,and after seeing the effect of turnsof 50, 90, 100, and 110, Monica insisted that 1200 was the correctturn,even when most other studentswantedto try 150, 140, 130.... At the computer,however,she initially believed thatthe turtleneeded to turnmoreto make a biggertriangle. After defining several different size triangles and examining their Logo definitions she changed her mind and used a repeat command to make differenttriangles.When asked why she changedthe lengths but not the turnsto make differentequilateraltriangles,Monica declared, 'The turns are the same'. Interviewand paper-and-penciltest Data from the interviews and paper-and-penciltest confirmthat students learned a considerable amount about tums and turn measures, but that theirknowledge was still limited. The firstinterviewtask had two related parts. 1. A robot tums in place 90 degrees to the right every time it turns. How many turnsmust the robot make before it is facing in the same directionas it was when it started?For example, if it startsout facing the frontof the room, how many turnsmustit makebeforeit is facing the front of the room again?(Tell me how you got your answer,even if you just guessed.) 2. Anotherrobotturnsin place 30 degreesto the rightevery time it turns. How many turnsmust this robotmake before it is facing in the same


328


TABLEI Responses to the interview items #1 and #2, on the number of robot turns Answer

1 pre

0 (No Response)

2 post

2

pre

post

2

1

1

2 3

2

1

4

5

10

5

1

1

6

1

2

7

1

1

2

3

2

8

1 1

9

1

10

1

11

1

12 45 215

4

2

2

1 1

directionas it was when it started?(Tell me how you got youranswer, even if you just guessed.) On the post interview, twice as many gave the correct response, even without this help (see Table I). Most other responses for this item on the post interviewcould be labeled 'reasonable'.This was not truefor item 2, where'therewas little movementto correctresponses.On both items, there was some movementtowardmore sophisticatedstrategies(strategies4, 9, 11, and 12 in Table II). This was more apparentfor item 2, indicatingthat studentswere beginning to understandhow to approachthe task, but had insufficientskill or knowledge to complete it accurately. On the pretest(which was administeredfollowing the second session of the unit), Luke firstanswered'two' to item 1 andthen stoodup andcounted to three while turning.He approacheditem 2 quitedifferently.He said that the answer was fifteen because 'thirty divided by two is fifteen'. This supportsthe interpretationthatLuke would operateon availablenumbers (not quantities) to obtain a numerical answer in situations which were


329

IN A COMPUTER-BASED TURNCONCEPTS UNIT

TABLEII Strategiesused on the interviewitems #1 and #2, on the numberof robotturns Code

Strategies

1 pre

0 1 2 3 4 5 6 7 8 9 10 11 12

No response/Idon't know Just guessed Spatiallyindicateddirections Draws or indicatesrobotwalking a path with bends (e.g., a square) Arithmeticcalculation-useof whole turn (e.g., 360) notion Arithmeticcalculation-90/30 = 3, 3X 4= 12 Arithmetic/numbermanipulation-other (uninterpretablesemantics) Other ? 1 (off by one) no classifiablerationale Draws an accuratediagramto indicate successive directions Refers to degrees (e.g., '90?' without interpretablemeaning) Turnsbody Uses mentalimage with verbalmediation

2 post

2 2 2

pre

post

2 1 1 2

3 1

1 1 1

2

2

1

1 1

1 1

1

1

1 1

2 5

5 1

1

4 I

unfamiliar or difficult. On the post-test, Luke answered '4' for item 1 withoutturninghis body. Afterquestioning,he was able to drawa diagram of the robotturning,indicatingthathe wouldbe facingin the same direction after four turns.He was very confidentwith his answer.He immediately stood up to answer item 2 and counted to eight as he turneduntil he was facing the directionin which he had started. On the pre-test, Monica turned all aroundto solve item 1, counting each 90? turn,before answering'four'. On the post-test,she quickly gave the correctanswer.On the pre-testfor item 2, she turnedwhile counting, and answered'seven'. On the post-testinterview,she was more reflective, and startedcountingwhile turningmore accurately,but afterturning1800, quickly turnedthe remainder,and concluded, 'eight'.


330

DOUGLASH. CLEMENTS ETAL.

Fig. 3. A paper-and-penciltest item on turns.

The paper-and-penciltest included an item on turnsillustratedin Figure 3. This item shows substantiallearningof turns.On the pre-test,there were one close response (within 100) and 20 not close responses.On the post-test there was to a total of 12 correct responses, 7 close responses, and 2 responses that were not close. Over all field-test children,average scores (one point for a close response, zero otherwise)on the post-testfor the three turnswere 0.72, 0.43, and 0.66. On the post-testonly, Monicawas accuratein two of the threeturns.She used her pencil to estimate the directionand amountof tum. Luke would not attemptthese items on the pre-test;on the post-test,he drew staticlines and then estimated turn commands that were in the correctdirectionbut missed being 'close' (e.g., 200? insteadof 180?). Combiningturns Meaningful combinations of turns would be a strong indication of the constructionof turn measure. Luke did not combine turn commands as frequentlyas he combined length commands(e.g., changing fd 80 fd 20 to a single fd 1200), and gave more of an indication that what he was combining was not rotationsper se, but numberswithoutquantitative grounding.In one instance, he examined the last three commandshe had entered, and, while the researcherwas writing notes (which was, again, slowing Luke's pace of work)askedif he could changethe two rt 9 0's to a rt 180. The researcherasked him if he can combine all three (rt 90 rt 90 lt 30) andhe replied, '210', addingthe threenumbersmentally.



331

He tried it and noticed that the turtle was not at the same heading. 'I'll have to leave it'. The researcherprompted, 'Stand up and try it'. Luke did so, acting out each turnin the correctdirection.He then said, 'Right 150...you have to subtractthe 30'. This supportsour contentionthatLuke often operatedon numbers(only; i.e., his initialmentalsum of 210) rather than on quantities and that he could operate on both simultaneouslyif asked to performbody turns,and thus focus on the physical rotation. In the Feed the Turtlegame, Monica and Nina usually estimatedturn measuresaccuratelyandso rarelyhadthe need to combineturncommands. Otherchildrenin the class would combineturncommands,especially after promptingby the teacher,but theirhesitationto do so and theirinsistence on checking the results may indicate that these combinationswere not operations on quantities for them. For example, the teacher asked one pairto combine 'rt 90 rt 30'. They replacedthe commandswith rt 12 0, but carefully checked that the turtle 'was heading the correctway'. We conjecture that, because rt 30 did not have a strong quantitative meaning for them, the combinationof the two commandswould also be a vague conceptual object. In this domain the computerproved valuable, a topic to which we now turn. Role of the computer The Geo-Logocomputerenvironmentplayeda significantrole in children's learningabout turnand turnmeasure.First and foremost,turtlegeometry constitutesa context in which turnsandtheirmeasurearecritical.Children talked to each other aboutthe slow turningof Geo-Logo'sturtleand thus, may have helped them construct turns as conceptual objects and build dynamic imagery for rotations. The feedbackgeneratedby the computerwas instrumentalin motivating children to reflect on their notions of turn measurement.For example, Luke thought that entering fd 120 would close his equilateraltriangle (when rt 12 0 fd 100 was needed). The power of the mediumwas in offering feedback that was differentin kind from the feedbackofferedby computer-assistedinstruction.Geo-Logoprovidednonevaluativefeedback consisting of the enactmentof his own ideas. Luke was quite surprisedthat his initial commanddid not work and was motivatedto find out why. Geo-Logo feedback consists largely of the graphicresults of running Logo code precisely, without human interpretation.This was important to the pilot girls who, seeing the differencebetween their own interpretations of their Logo code for an equilateraltriangle and the Logo turtle's implementationof that code, immediatelyexaminedand alteredthe code. Giving commandsto a noninterpretiveagent, with thoroughspecification


332


anddetail,has been identifiedas an importantadvantageof computeruse in facilitatingthe learningof mathematics(Johnson-Gentileet al., 1994). The Geo-LogoenvironmentallowedLukeandothersto easily trydifferent tums. Comparedto regularLogo, the commandsthatare the same stay the same and do not have to be re-entered;more important,those thatare changed are not executed as an additionalcommandaffecting the graphics; instead,the figurechanges to reflectthe changein the commands.This helped studentsencode contrastsbetween differentturncommands. Further,many students would use Geo-Logo to explore iterationsof turns (rt 90 rt 90; or rt 30 rt 30 rt 30), and thus build a (superordinate)unit for measuringrotation. As stated previously, when combining inputs to turn commands, students were less confident about the results than when combining inputs to fd and bk commands (cf. Clements et al., in press). Most studentsinitially gave no indicationthat they were combining rotationalunits, per se (i.e., quantities).Ratherthey appearedto operate on numbers (sans quantitativegrounding),use the resultof thatnumericaloperationas the inputto a new turncommand,and check the turtle'sresultantfinalheading.The benefitof the computeris that they received this immediatefeedbackand thus began to give quantitative meaning to this combining operation.

CONCLUSIONS

Turns are less salient for children than the motion from one position to another (i.e., fd commands). The data from this study, consistent with those frompreviousresearch(ClementsandBattista,1992; Kieran,1986b; Mantoanet al., 1993; Mitchelmoreand White, 1995; Simmons and Cope, 1990), indicate that studentsomitted turncommandsmore often thanforward and back commands, showed more difficulty estimatingturn measures, and were less proficientcombining turn commands than forward and back commands(cf. Clements et al., in press). Thereare severalpossible reasons for this. Graphically,turns and the result of tums are often represented,in diagramsand computerdrawings,by two connected line segments.A turncan be viewed as the relationshipbetweenthe headingsof these segments, a sophisticatedconcept. The turningmotion itself usually does not leave a trace (and the 'line of sight' ray must be reconstructed from memory).Also, turnsareusually not clearlydistinguishedfrom other movements in most real-worldcontexts. For instance,when we turna car or bicycle, we are also moving forward. The dataalso indicate thatchildrenmust internalizephysical rotations, connectthis internalizedvisual/kinestheticimageryto measuresvia bench-



333

marks,and finally impose these dynamic,quantitativementaltransformations onto static figures (thus coordinatingintrinsicand extrinsicperspectives). Our observationsprovide descriptionsof how children might do this. Luke evinced a progressivebuildingof imageryandconcepts related to turns.He gainedexperiencewith physicalrotations,especially rotations of his own body, and relied heavily on these experiences. In parallel,he gained limited knowledge of assigningnumbersto certaintums. First,he builtup landmarksin the domainof degrees,althoughhe faced a hurdlein that his static 'squarecorner' depictionof a 900 turnkept him from focusing on physical motions. Luke focused on the beginning state and ending states only, ratherthan on the transition(cf. Piagetiantheory, Clements and Battista, 1992). Moreover,the beginning and ending states were representedby line segments,which then for Luke were more salient thanthe relationshipbetween the headingsof two connectedsegments, or the transformationthatwould move the turtlefrom one segment'sheading to that of the other. Luke's attitudesandbeliefs aboutmathematicsandmathematicseducation influencedhis choice of strategieswhen workingon activitiesthroughout the unit. He liked to use arithmeticbecause he could be exact and sure that he was correct. He felt that mathematicsin general was about finding a correct answer quickly by following a single prescribedprocedure or 'just knowing'. Some tasks in this unit were inconsistentwith Luke's beliefs in that they could be solved in a variety of ways, so he graspedat anythingto find a single correctanswer.Even in the turtleenvironment,he focused more on such performancethanon making sense of the situation to solve problemsand refrainedfrom employing helpful, even necessary strategies. Luke's resistance to using body turns had a deleteriouseffect on his estimateof turnsat the computerand, to an even greaterdegree, on paper. Moreover,his tendency to impose a static graphicor numericalrepresentationon the situationbecause of his seriousconcernfor correctnessinterfered with developing a dynamic, rotation-basedscheme for turns.With guidance from adults and feedback from the computer,Luke startedto connect the two ideas of turn-as-body-motionand turn-as-number. Under stress, however,he sometimeslost thatconnection,usually falling back to operatingon numberswithouta correspondingimage of a turningmotion. Further,this connection and the resultantidea of turnmeasurementwere tenuous, even at the conclusion of the unit of instruction.Nevertheless, Luke's development of imagery and his initial connection to numerical ideas suggest thata synthesisof these two domainsmay be a criticalphase in learningaboutturns.


334


In many aspects, Monica developed greaterknowledge of turns than did Luke, althoughshe still has difficultywith tums in which she and the turtlewere at or nearlyat oppositeheadings.She built internalframeworks for 900 turns.Turnsmore or less than900 were estimatedbasedon another unit, 30? (300 was an iterableunit of turnfor otherstudentsas well). We conclude that several mentalprocesses arenecessaryfor conceptualizing a turnand its measure.First, childrenhave to maintaina recordof mental images of both the initial heading and final heading of an object, using a frame of reference to fix these headings (probablyan externalor internalizedvertical and horizontal framework;note that children need to leam to differentiateand coordinateintrinsicand extrinsic perspective schemes). Second, they have to re-presentthe activity of rotationof the object from the former to the latter (imaging the rotation),and compare that image to one or more iterationsof an internalizedimage of a unit of turn - for our children, units of 900 or 300 - or partition the re-presented

turninto these units. While many students had much to learn about rotationmeasure, the common phenomenon of assuming a connection between rotation and length measures, as in conceptualizing angle measure as the distance between two rays (Clements and Battista, 1989, 1990; Krainer, 1991), was not in evidence. Further,each studentobservedwas makingvaluable cognitive constructionsconcerning rotation.This supportsusefulness of the specific on- and off-computeractivities employed. Finally,evidence also suggests thatdealingwithmultipleideasincreased students'difficultiesin learningabouttum measurement.MonicaandNina, for example,did not seem to be able to attendto tums when workingon difficultlength relationshipsin the 200 steps activity.Similarly,in the Get the Toys game, manydid not worryaboutleft versusrightdirectionality.Thus, increased demands on processing may explain some of these difficulties with more complex tasks (Boulton-Lewis, 1987; Hiebert,1981).

IMPLICATIONS

Tums and tum measurewere not ascendantin these third-gradestudents' thinkingaboutgeometric figures.However,they did make significantdiscoveries and gains in this domain.If turnswere to be integratedthroughout the K-6 mathematicscurriculum,difficulties may diminish and benefits increase. Our results imply that static representationsof turns,no matter how cleverly designed,may not only be inadequatefor children'slearning, but may delay their development of dynamic concepts of tum and turn measurementby limiting their constructionof notions of physical motion



335

and the integrationof this notion with geometric figures and numerical measures.Instead,teachersneed to emphasizephysicalmotions (especially turningone's own body, especially for this age student),recordsof the transforrnation (e.g., by holdingout one armto maintainthe initialheading, then moving the other arm throughthe turn),reflectionon these motions (recall, in contrast, the apparentuselessness of the students' nonreflective hand twitching, cf. Hoyles and Sutherland,1989), and connections between such physical activities andotherdynamicactivitiessuch as those using computerandpaper.Further,studentsshouldbe encouragedto return to the physical motions as an aid duringsuch computerand paperactivities (despite their not uncommonresistance).Emphasisthroughoutthese activities shouldbe given to the establishmentof perceptuallysalientunits of tum (e.g., 900, 180?, 300 or 450, etc., whetherinventedby studentsor suggested by the teacher3)and theirrelationships.Finally,in concertwith otherresearchers(Hoyles and Sutherland,1989), we have foundthatactivities such as those featuredin the curriculum,featuringwell-definedgoals (often with pictorial representations),are critical in developing concepts of turnand tum measurement. The difficultyof rightandleft directionalityis familiarto manyteachers and researchers,with some suggesting thatthis raises the age at which use of Logo is appropriate.Many studentsin this study understoodthe idea of left and right directionality,but used strategies(e.g., enteringeither 1 t or rt, then moving the turtleforwardif that happenedto be correct,and back otherwise)to circumventthinkingaboutit. Combinedwith previous research (Clements and Battista, 1989, 1990; Weaver, 1991), these data indicateto us thatworkwith the turtlecan facilitateleft-rightdirectionality for some childrenfrom kindergartento sixth grade,but thatsome students will grow in competence for years. Further,nothing is lost, and much is gained,from providingthese latterstudentswith simple aids to distinguish between left andright.Geo-Logoprovidessuch aids bothdirectly,through measurementtools, and indirectly,though ease of correction.To better facilitatethe developmentnot only of directionalitybut also turnmeasure, we areplanningadditionalmodificationsto makeinitialturnmotionsmore salient.

ACKNOWLEDGEMENT

The authorswould like to thankMaryLindquist,Leslie Steffe, andGrayson Wheatley and three anonymous referees for their helpful comments on early draftsof this manuscript.


336

ETAL. H. CLEMENTS DOUGLAS NOTES

'Investigationsin Number,Data, and Space:An ElementaryMathematicsCurriculum'is a cooperativeprojectbetween the Universityof Buffalo, Kent StateUniversity,andTERC. National Science FoundationResearchgrantnumberMDR-9050210. Time to preparethis materialwas also partiallyprovidedby 'An Investigationof the Developmentof Elementary Children'sGeometric Thinkingin Computerand NoncomputerEnvironments',National Science FoundationResearch grant number ESI-8954664. Any opinions, findings, and conclusions or recommendationsexpressedin this publicationare those of the authorsand do not necessarilyreflect the views of the NationalScience Foundation. (Clementsand Meredith,1994), the stand-aloneversion, 2 Geo-LogoTmand TurtleMathTm copyright, Douglas H. Clements and Julie SaramaMeredith.Development system copyright, Logo ComputerSystems, Inc. All rightreserved. 3 We use degrees as the unit of measurementfor communicationpurposes;obviously, studentsmight invent theirown nonstandardunits.

REFERENCES Boulton-Lewis, G.M.: 1987, 'Recent cognitive theories appliedto sequentiallength measuring knowledge in young children', British Journal of Educational Psychology 57, 330-342. Clements, D.H. and Battista, M.T.: 1989, 'Learning of geometric concepts in a Logo environment',Journalfor Researchin MathematicsEducation20, 450-467. Clements, D.H. and Battista, M.T.: 1990, 'The effects of Logo on children's conceptualizations of angle and polygons', Journal of Research in MathematicsEducation21, 356-371. Clements, D.H. and Battista,M.T.: 1991, The Developmentof a Logo-Based Elementary School GeometryCurriculum(Final Report:NSF GrantNo.: MDR-8651668). Buffalo, NY/Kent, OH: State Universityof New Yorkat Buffalo/KentState University. Clements, D.H. and Battista, M.T.: 1992, 'Geometry and spatial reasonsing', In: D.A. Grouws (Ed.), Handbookof Researchon MathematicsTeachingand Learning,pp. 420464. New York:Macmillan. Clements, D.H., Battista, M.T., Sarama,J., Swaminathan,S. and McMillen, S.: in press, 'Students' developmentof length measurementconcepts in a Logo-basedunit on geometric paths', Journalfor Research in MathematicsEducation. Clements, D.H. andMeredith,J.S.: 1994, TurtleMath.Montreal,Quebec:Logo Computer Systems, Inc. (LCSI). Clements, D.H. and Sarama,J., in press, 'Design of a Logo environmentfor elementary geometry', Jounal of MathematicalBehavior. Gravemeijer,K.P.E.: 1995, Developing Realistic MathematicsInstruction,Utrecht, The Netherlands:FreudenthalInstitute. Hiebert,J.: 1981, 'Cognitive developmentand learninglinear measurement',Journalfor Research in MathematicsEducation12, 197-211. Hoyles, C. and Sutherland,R.: 1989, Logo Mathematicsin the Classroom,London,England:Routledge. Johnson-Gentile,K., Clements, D.H. and Battista,M.T.: 1994, 'The effects of computer and noncomputerenvironmenton students' conceptualizationsof geometric motions', Journal of EducationalComputingResearch 11(2), 212-140.



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Kieran,C.: 1986a, 'Logo and the notion of angle among fourthand sixth gradechildren', Proceedings of PME 10, pp. 99-104. London,England:City University. Kieran, C.: 1986b, 'Turns and angles: What develops in Logo?' In: G. Lappan (Ed.), Proceedings of the EighthAnnual PME-NA,Lansing,MI: MichiganStateUniversity. Kieran,C., Hillel, J. andErlwanger,S.: 1986, 'Perceptualandanalyticalschemasin solving structuredturtle-geometrytasks', In: C. Hoyles, R. Noss and R. Sutherland(Eds.), Proceedings of the Second Logo and MathematicsEducatorsConference,pp. 154-161. London,England:University of London. Krainer,K.: 1991, 'Consequencesof a low level of actingandreflectingin geometryleaming - Findings of interviews on the concept of angle', In: F. Furinghetti(Ed.), Proceedings of the FifteenthAnnual Conferenceof the InternationalGroupfor the Psychology of MathematicsEducation, Vol. II, pp. 254-261. Assisi, Italy: ProgramCommittee, 15th PME Conference. Krajewski,L.: 1989/90, 'The team approachand Logo', J. Computingin ChildhoodEducation 1(2), 57-63. Lindquist,M.M. and Kouba, V.L.: 1989, 'Geometry',In: M.M. Lindquist(Ed.), Results from the Fourth MathematicsAssessment of the National Assessment of Educational Progress, pp. 44-54. Reston, VA: National Council of Teachersof Mathematics. Mantoan,M.T.E.,Barrella,F.M.F.andPrado,M.E.B.B.: 1993, 'Understandinganglenotion in a Logo context', In: N. Estes and M. Thomas (Eds.), Rethinkingthe Roles of Technology in Education,Vol. 2, pp. 981-983. Cambridge,MA: MassachuseetsInstituteof Technology. Mitchelmore,M. and White, P.: 1995, 'Developmentof angle conceptby abstractionfrom situatedknowledge', Paperpresentedat the AmericanEducationalResearchAssociation, San Francisco,CA. Noss, R.: 1987, 'Children'slearning of geometricalconcepts throughLogo', Journalfor Research in MathematicsEducation18, 343-362. Olive, J., Lankenau,C.A. and Scally, S.P.: 1986, Teachingand UnderstandingGeometric RelationshipsthroughLogo: Phase II. InterimReport:TheAtlanta-EmoryLogo Project. Atlanta,GA: Emory University. Piaget,J. andInhelder,B.: 1967, TheChild'sConceptionof Space,New York:W.W.Norton. Simmons, M. and Cope, P.: 1990, 'Fragileknowledge of angle in turtlegeometry',Educational Studies in Mathematics21, 375-382. Singer, D.G., Singer, J.L. and Zuckerman,D.M.: 1981, TeachingTelevision.New York: The Dial Press. Steffe, L.P. and Cobb, P.: 1988, Constructionof ArithemeticalMeanings and Strategies. New York:Springer-Verlag. Weaver,C.L.: 1991, 'Youngchildrenlearngeometricandspatialconceptsusing Logo with a screen turtleand a floor turtle',UnpublishedDoctoralDissertation,StateUniversityof New Yorkat Buffalo.

State Universityof New Yorkat Buffalo, 593 Baldy Hall, N.Y 14260, 400 USA.


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