geometry and spatial reasoning

___________-18_-_________ GEOMETRY AND SPATIAL REASONING Douglas H. Clements STATE UNIVERSITY Of NEW YORK AT BUFfALO

Michael T. Battista KENT STATE UNIVERSITY

Spatial understal'ld lngs are necessary for Interpreting, understanding, and appreciating our inherently geometric world . (National Council of Teachers

of Mathematics, 1989, p. 48) Geometry is gmp!ng space ... that space In which the child lives, breathes

and moves. The space that the child must learn to know. explore, conquer. In order to liye, breathe, and move better in It. (Freudenthal, In Nnional Council of Teachers of Mathematics. 1989, p. 48) Ar islng out of practical activity and man 's need to describe his surroundings. geometric forms were slowly conceptual/led until they took on an abstract meaning of their own. Thus from a prKticaJ theory of earth measure. there de ... elo ped a growing set of relatlom or theorems that culminued in Euclid's Ele ments, the. collection, J)'nthe.sis, and elabon.tiOfl of all this knowledge. (Fehr. 197]. p. ]70) Equations are JUSt the boring part of mathemuics. I attempt to see things In terms of geometry. (Hawking. NuiOflal Research Council, 1989, p. ]5)

draWing, and construction of figu res; (b) Study of the spatial aspects of the physical world: (c) use as a vehicle for represenling nonvisual mathematical concepts and relationships; and Cd) represemation as a formal mathematical system. The first three of these d imensions require the use of Spatial reasoni ng. When the term "school geometry" is used, it almost unl· ver:sally refer:s to Euclidean geometry, e"en though there are numerous approaches to the study of the topiC (for exam· pie, synthetic. analytic, transformational, and vector). The traditional, secondary school version of geometry is axiomatic in narure, elementary school geometry IraditiOnally has emphasized

School geometry is Ihe study of those spatial objects, relationships, and transformations m;u have been formali7.ed (or malhematized) and the axiom:uk mathemaliC"JI systems th:1I have been constructed 10 represent them. Spatia! reasoning, on the exher hand, consists of the set of cognItive processes by which menla! represemations for Spatia! objectS, relationships, and transform31ions are conSlrucled and manipulated. Clearly, geometry and spada! reasoni ng 3re strongly imerrelated, and most mathematiCS eduallors seem to include spatial reasoning as pan of the geometry curriculum. Usiskin (1987), for instance, has desc ribed four dimensions of geomeU)'~ (a) visualization,

The :.Iuthors gratefully acknowledge the helpful commenlS pllJYided by David FU)'5, Brookl)T1 College, and Sh:.lron Senko Mlchlg;ln St:lte UniversJrr. nme to prepare this materi3! was p3111:.l1l1' provided by the National Science found:l(]on under (jr.ult No. MDR-8651668. An)' opinions, tinding5. and conclusions or retommendations e.'l:pressed in thiS publlcation are those of the authors and do 1101 necessarily refleCI the "Jews of the N3tlon:.lJ Science Foundation.

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GEOMETRY AND SPATIAL REASONING

measurement and Informal development of those basic conceptS needed in high school. According 10 Suydam (1985) thert! is a great deal of agreement mar the goals of geometry instruction should be 10 • develop logical thinking abililics; • dellelop spatial intuition about the real world; • impan me knowledge needed to study more malhe mallCS;

",d • teach the reading :md interpretation of mathematical argumentS (p. 481). The National Counal of Teachers of Mathematics (1989) lim/11m Standards [.111s for all students 10

e(ll'-

• identify, describe, compare, mode!, draw, and classify geamelric figures in twO and three dimensions; • develop spatial sense, • e,"plore the effects of transforming, combining, subdiViding, and changing geometr,ic figures; • understand, apply, :lOd deduce propenies of and relationships between geometric figures, including congruence and sJmilarlry; • develop an appreciation of geometry :'IS a means of describing and modeling the physical world; • explore synthetic, transformauonal, and coordinate approaches to geometry, with college·bound srudents OIlsa required 10 develop an undeCSIanding of an 3.xiomatic system through investigating and comparing various geometric systems; and • e,"plore a vector :lpproach to certain aspens of geometry. This chapter contains seven major sections. First, srudents' geometry is bnefly summarized as a back· ground 10 theemire research corpus. Second, research on three m:ljor theoretical perspealves on the developmenr.of geometric thinking-Pilget, the van Hieles, and cognitive sdenceis reviewed. Third, the establishment of wth In geometry is discussed, hlghlighting both theoretical and empirical work. Founh, the relationship between spatial thinking and mathematics, the narure of Spatial reasoning and Imagery, :md altempts 10 reach spatial abilities are considered. The fifth section, representations of geometric Ideas, includes Issues rei:lted to conceprs, diagrams, manipuladves, and computers. Sixth, we examine group and cross-culrural differences. Finally, broad conclusions are drawn from this research corpus. ~rformance in

STUDENTS' PERFORMANCE IN GEOMETRY According 10 extensive evaluallons of mathematic; learning. eJemen[3ry and middle school srudents in the United Stales are falHng 10 learn basic geometric concepts and geometric problem solving; they are woefully underprepared for the study or more sophisricaled geometric concepts and proof, especially when compared 10SlUdents from Other nations (Carpenter, Corbin, Kepner, lindquist, & Heys, 1980; fey et aL, 1984; Kouba et al.. 1988, Stevenson, lee, & Stigler, 1986: Sligler, Lee, & Steven·

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sao, 1990). For instance, fifth graders from Japan and Tall,V',m scored more than £wice as high as US srudents on a geometry test (Stigler et aI., 1990). Japanese students in both first and fifth grades also scored much higher (and Taiwanese students only slightly higher) than US. students on testS of visualization and paper-folding. Sligier el al. (1990) postulate dmt the laner results may be due both 10 Japanese classrooms' heavy reliance on visual representations for concepts and to expectations that Japanese srudents become competem a[ drawing. Da[a from the Second International MathemalicsStudy(SIMS)showedthat.ln geometry, US 8th and 12th gt3ders scored at the 25th inter· n:llional percentile or below (McKnight, Travers, Crosswhite, & Swafford, 1985; McKnight, Travers, & Dossey, 1985). UslSkin (1987), dtlng data from the 1982 US. Nalional Assessment of Educational Progress (NAEP), reponed that fewer than 10% of 13-year-olds could find the measure of lhe th ird angle for a triangle, given the m~ure of Lhe other £wo angles; o nly 20% could find Lhe length of the hypolenuse or a right triangle given its legs. ( He concluded tllat a greater number of srudents could do the more difficult computatio n because it Is laughtto more srudents.) [n the 1986 NAEP, Koub" et al. (1988) reported srudents' performance at identifying common geometriC figures, such as parallel lines and the diameter of a circle, :lcceptable, but srudents' performance with figures nOI frequendy encountered in elleryday life, such as perpendicular lines and the radius ofa circle, were reported as defiCient Performance dealing with properties of figures, visualization, and applications was poor, For example, only 60% of sevenlh grade srudems could Identify the image of an object reflected through a line; only about 10% of seventh graders could find the area of a square, given tile length of one of Irs sides (56'*' found the area of a rect:lrlgle, given the lengths of 1/5 sides): and less than 10% of seventh graders could Identify which Set of nu mbers could be the lengths of the sides ora trlangJe(even though 66% could do it if segments were gillen). Apparently, srudents can handle some problems much bener If the problem is presented ViSually rather than verbally (Carpenter et al" 1980; Driscoll, 1983b; Kouba el aI., 1988). TIle siruation is even worse :n the high school level. FirSl, only about half of all high school students enroll in :I geometry course. or those e nrolled at the beginning of the school year only 63% were able to correctly identify mangles that were presented along with disrractors (Usiskin, 1987), According to the 1978 NAEP in mathematics, only 64% of the 17-year-olds knew thaI 3 rectangle Is a parallelogram, only 16% could find the area of a region made up of two rectangles, and lust 9% could solve the problem "How many cubic feel of concrete wo uld be needed to pave an area 30 feel long and 20 feet wide with a layer oj inches thick?" Of 17·year-olds that bad a full yea r of high school geometry, only 5796 could calculate the volume of a rectangular solid, 5496 could find the hypotenuse of a righl triangle whose legs were multiples of 3 and 4, and 34% could find the area of a right triangle. Only 52% of enter· ing secondary students could state the area of a square when 1/5 sides Wefe given (Usiskin, 1982). On the 1986 assessme nt, 11th-grade studentS who had n01 taken high school geometry scored at about the same level as seventh graders (Undquist & Kouba, 1989), There were few performance differences In visualization between those srudents who had taken geometry

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and those who had not, -although there were large differences on items requiring knowledge of geometric propenies and on lppliCltions less than 2;% of ll!h-grade srudents correcrly identified which figures h.,d lines of symmetry, whether they had taken geometry o r nor (even [hough symmetry is srudied mroughout elementary and middle school). Even more incri min:\ling of the curriculum is the faCt that only about 30% of high school geomeuT students enrolled in a course for which proof was a go::!1 were able to write proofs or exhibit any undersmnding of the meaning of proof (Senk, 1985: Suydam, 198;). It is no wonder that doing proofs was the least liked mathematics lopic by 17-year-olds on the 1982 NAEP and rhat less than ;0% of the students rated the topic as ImpormnL This depressing picture of students' knowledge of geometry is el:1bor.lled through :t consideratiOn of srudenlS' miscon· ceptiOns, Here are some e."amples (Clements & B:misra, 1989; Fuys, Geddes, & Tischler, 1988; Hoffer, 1983): • :10 angle mUM h:tve one oorlwntal r:Iy • ;J. right :mgie is an angle that points to the right • to be a side of a figure a segment must be venical • a segment is nOt a diagonal if it Is vertical or horlzonmi • II square is nOt a squ:lre if its base is nor horizontal • the only way II figure can be a triangle is if It is equilateral • the height of a triangle or paral1elogr:tm is a side :ldl:lcem to the boISe • the lingle sum of a quadrilateral is the same as ilS area • the Pythagorean theorem Cln be used to caicul:lte the area of a rectangle • if a shape has four sides, then If is a square • the are;! of a q uadrilateral can be Obtained b)' tr:msforming it into a rectangle with the same perimeter

Apparently. much learning of geometric concepts has been rote. Propenies, class inclUSions, relationships, and implications are freq uently not perceh'ed (MaybenT, 1983). A primary cause of this poor performance may be the curriculum, both in what mpies are treated and how they are tre:ued, The major focus of smndurd eiemenlllry and middle school curncob is on recognizing and na ming geometric shapes, ",ri tmg the proper symbolism for simple geometric conceplS, developing skill with measurement and construction tools such as a compass and protractOr, and using formulas In geometric measurement (PoneI', 1989; TIlomas, 1982), TIle.se curricula consist of a hodgepodge of unrel:Jted concepts with no s}'slematic: progression to higher levels of thought-levels requisite for sophistiCned concept development and substantive geometriC problem solvmg. In addition, teachers oftcn do not teach even the impoverished geomeny curriculum that is available to them, Poner, for instance, reponed whole districtS in which Counh- and fifth·grade teachers spent "virtually no time teaching geome[Ty~ (Poner, 1989, p. 11 ). E\'Cn wben mugtll, 8eom~try t'.'as lhe topiC most frequently Identified t\S beIns taught merel y ror "e:(posure"; that is, geometry was g iven only brief, cul1iOry coverage. The SIMS data ror the eighth·grade level indicated th:llteOlchers rated the opponunity to learn geometry much lower than the opportunity to learn any other

10piC (MCKnight. Travers, Crosswhite, & Swafford, 198;). AI. the secondary level. the Ir:tditional emphasis has been o n formal proof, desp1te the r.IC:t the students are unprepared to deal with It. Indeed, as Usiskin (1987) summarizes ' 1lere Is no SCOmel'1· curriCtJluflT ~l tile elementary .school level. As a result, snldeolS enter high school nOi knowing enoogh geometry \TJ succeed There i5 a geometry CtJrriculum ~t the second:1!)' level, bUI

onlr about half of tile Sludenl,.S encounter II. and onlr about ~ third tlle5e studenlS understnnd 11. (p. 29)

or

THE DEVELOPMENT OF GEOMETRIC THINKING Piaget and Inhelder; The Child's Conception of Space

nuo Major Tbemes. Two major themes of Piager 3fld Inhelder's ( 1967) influential theory on children's conception of space will be discussed. First, representallons of sJXlce 3re constructed through the progressive organiZ3t10n of the child·s moror and internalized actions, resulting in operational systems. Therefore, the representatiOn of sp3ce is nOl a perceptual -re3ding off" of the spacial environment, but is the build·up from prior active manipulatiOn of th:1I envlronmcnt, Second, the progressive organizatiOn of geometric ideas follows a definite order, and this order Is more logical th:m hIStorical In th;lt initially topo logic:tl relations (for example, connectedness, enclosure, and continuity) are constructed. and lmer projective (rOOiline:rrity) 3nd Euclidean (:lIlgulariry, par.dlelism, and distance) rel:lIions. This has been termed the topological primacy thesis. It is important to reiterate thai P!;lget and Inhelder were diSCUSsing the child·s ability to represent space. lhey m3intlin that perce ptual space. is construCted early in the sensorimOtor period. Nevertheless, perceptual space precludes me deve.lopmen! of representatiOnal. or conceptual. space. in that its developmeO! also embodies the topological primacy thesis, It, too, is constructed rJ(hcr than exiStent fmm me OUtset of development. Representational space, in addition, reflectS propenies of logical operation:lJ thoughL 7bpologica l Primacy and Constnlctivism

H.vnc £\10!r:am· pie, children mlghlleam to coordinate certain actions thm produce culVilinear shapes before coordinating those mat produce rectillnear ones (Martin, 19700), although theoretical e.'(pianatlons of s uch sequences are lacking. Observed bcl;: of synchronv bern-een perceptual and conceplOal abilities suppons Piagel's consll"U([ivist position (Rosser et aI., 1964), bUi speCific cogni· tive constructions gener.lIly h:lve nOl been idemified Such research should a lso e.-;;plore the deform:nlons that d lildrcn do accept in their represenCllion of figures. For ex· ample. some children ecru-.lle an ~almost clo:sed~ figure with a closed l'arianL Previously une:~plored factors such as lal1guage, schooling, and the Immediate SOCial culture also dem:lnd :lltenlion ( Darke, 1982), Piagetian imerprel:lliOnS of students' per· formance on tusks tOO often emphasize logical fallures :It the e:\-pense of uncovering the developmem of Ideas not yet dilTerenti:ued and integrated; thus, new approaches are ....-ar· ramed. In any case, it appcilrs that certain Eudlde;tn notions are present :u an early age (Rosser et al.. 1984: Rosser et :11., 1988) and, contrary 10 Piaga and Inhelder and interpreters (peel, 1959), even preschool chUdJ'en Ill:IY be able to work with cenain Euclidean ideas. Similarly, results regarding Piaga and lnhelder's constructS of projective and Eudide-Jn space reveal dlat young children ha\le basic competencies in esl:lblishing spatial frame'NOrks that could be effcoively bull! upon in the classroom. How, ever, we should probably e.-;;peCt, in students of all ages, 0 gen· era! Gestalt lendenC}' 1O....~lrd S~'1l1fnetry and simplicity, for t',..'(ample. in matching and reproductiOn [;J.Sks, Research is needed 10 identify instruction fociJilaling the construction and selection of increasingly sophisticated reference systems for o rganizing sp:l1ial information Researchers ha\'e tended nOt to d iscuss Piaga and Inhelder'S second maior theme: Children's representation of sjXlce is nO! a perreplllal "reading off" of their spatial environment, but is con· SlfUCled from prior aaiye manipulatio n of thOt environment This is surprising, in that most of Piaget and Inhelder's results do suppon this hypothesis al least implicifi)' (results from one study, Wheatley & Cobb, 1m, provide direct support, as discussed in a succeeding section). More anlcul:lled research from a constructivist position is needed. In this regard, we tum brlenv 10 other work consistent With this critical Idea fischbein (]987) argues that people's intuition of space is not inn:lle and not reducible to a conglomerate of sensori:ll Images, Space represenwtions constitute a complex s}'Stem of conceptions-although not necessarily formuloted e1>:phcidywhich exceed the dma ilt hand and the dom:J.in of pert-eption In general. Subjective space is an imerprel.1t10n qf reali~, nOt a reproduction of it It is shnped by :lnd exceeds experience. Consistent with Piagct's constructivism, Fischbeln'S theory further e lobor:ues the n:lture of the intuition of space. To begin, Imuition is not merel y :l renection of objectiYely given space propenieSi ramer, it is a ~highly complex system of e.xpecra·

tions, and programs of ;!([ion, rebted 10 Ihe movements of OUr body and its pans, which constitutes dle intuition of space' (p. 87). 111OS, inruitions consist of sensornnOtor and imeJlec. lOal skills organized into a system of beliefs and expectations thot constitute an implicit theory of space. MOSt important, intuitions thus constructed are enaCtively meaningful: they are sub· jectively self-evident because they e:\-press the direCt behavioral meaningfulness of an idea For e.'(ample, the nOlion of Stro.ight line seems self·evldenL A sophisticated adult is convinced that one may go on e:\1.endillg the line indefinitely, or thm by following the straight line one uses the shonest path to re:lch :l given point. These :tppe:tr uncquiyocal ~faClS,~ propenies of the "obiea~ C:lJled a Straight line. However, the srraight line is an obslraclion. not a perceptual object.. It Is a convention based on lLxioms which could be chonged. It is through e:\1.fapolation from a beh:lVioral mean· ing thar one tends to believe in the absoluteness of the conceptiOn. People know that they can draw :t str:tighl line, recognize a straight line, and run along a s traight IUle 10 reach a goal in minimum distance. The\' imbue the nOlion with Ihe qualities or unequivocal el'idence and credibilily because II is behaviorally meaningful, HOweyer, building intuition based on experience CUts both WilYS. The liOlit:ltions ofhum:1Il f'..)[perience accou m nOl o nl y for the adaptive and organizing functions of intuitions, but also for distorted or erroneous represemations of reality. Thus. space in· tuitions, like other intulfions, do not del'elop inevitablr into increaSing correspondence '>'.1th pure logiC or mathematiCS. as a reading of Piaget may suggest. ImuitiYe represent:ltions of space are non-homogeneous and anisotropic (exbibiting properties with different values when measured :llong a.xes in different directions). For example, people tend to anribute absolutely priVileged direaions to space. such as " up~ and "down. ~ They view space as cemefi.--d (for eX:lmple, at one'S home) and having increasing densl!)' as one approaches the centratioll zones, with the effect th:lt diSl:lnces are increas!nglyampllfied upon approach. Thus, our intuitive represenL'ltion of space is a mixture of possibly contradiCtory properties, all reltl1ed to our terrestrial \lfe and our behavioral adaptive constr:tints (Fischbein, 1987).

The van Hleles; Levels of Geometric Thinking and Phases of Instruction Lewis Of Geometric Thougbt. According to the theory of Pierre and Dina V'..lIl Hiele, students progress through levels of thought in geometry ( Ii:In Hide, 1959: Ii:In Hide, 1986. van Hiele·Geldtlf. 198
of geometric thinking 10 invesllg3l.i"B the continual devel menl of processes such as visual dunking that appear inJll:well deYeloped (Vygotsky, 193411986). Y In ~1. research thai builds on the strengths oralllh ~I'dica.t pe.rspecti~ ~Igtu. tuve potential. For example. ~ agel s schemes, van 1-lIele5 network of relations, and Cognlu science's more explicit declarative netwOrks certainly POISSelie

commonalities In their Y)('WS of knowledge stnIcture and . ~ • ' UIS possible that a symh6iS of these would yield a richer m . • Ore veridical model. Ideally, such a model would have the: eYnI ' ., of ... _ . ..... lea· uon UK: cognitive SCience perspective and the developme I 2SpectS of the Piagetian and van Hiele perspectives. nta

ESTABLISHING TRUTH IN GEOMETRY How do mathetrultlcians establish truth? They use Proof, 1 • ic:al, deduCtive re150nlng based on axioms. How do they fi~~ truth? Most r~quentJy by methods Intuitlve or empirical I rure ( Eves, 1912), In faCt, the process by which new mathemat. ics 15 eStablished 15 belied by the: deductive fonNI in whi h It is recorded (Lakatos, 1976), In creating mathematics :. ......~ . .._ ,pro..,.. Iems are r - ", con~re.s rna""" coumerexamples oITe eel -and conf«rures revised; a theorem resul~ when this r~ , ~t of ide:J5 is ludged to have an5\\'etl!d a Significant questlneRell (1916) dislinguishes three funaions for proof in m ~ mafia: l.Wffiauion, which is con«med ~ith esablishin a truth ora proposition; il/llmmaIlOn, ..-him is concernedgw~ conveying Insighl 1010 why a proposition is UlJe; and ~'$I' ttl lisa/OJ, whkh is ocganization of propositions into a dc!du:;a'

"VII!

S)'S1en1.

In geometry, as in other areas d mathematics, empirical 2nd deductive methods should Inter2Ct and reinforce each oth For InSI2nce. often when one is stymied in taking 2 ded er. live 2PP~, empiriCLI investigations can generate explora~~ possibilities H~er, for most srudents in geometry. ded tion 2nd empirical merhods are separale domains dlH[and why they -apply. According to van Hieie, the Inrultive foundation of proof "begins with a pupil's statemenr that belief in me truth of some assertton Is connected with belief In the truth of other assertions, The notion of this connection is inruitive. The laws of such a connection can only be learned by analysis" (1986, p, 124), LogIc is created by analyzing and abstraaing these laws, lh3t is, by operating on the network of links between statements. De V!lIiers (1987) concurs that deductive reasoning firsl occurs at LeveJ 3, when the network of logical relations between properties of concepts is established. He continues tha t because slUdents at lb.'els I and 2 do not doubt the validity of their empirical observations, proof is meaningless ( 0 them: they s~ it as justifying the obvious. Van Dormolen (1977) describes three levels of proof performance and relates them to the van Hicle levels. In the first, lustincations are made for single cases; conclusions are restriaed 10 the speCific example for Which the justification is given (for e.xample, a particular reClangle). In me .seeond, justifications and conclusions may be for specific cases, but refer lO collec,ions of similar objeCts (for example, the dass of rectangles): several examples will be considered to illustrate a pattern, with srudcnts capable of generatlng further examples, In the third, srudents justify statements by forming argumeniS that conform (0 Jccepted norms, that is, they are capable of giving formal proofs, Van Dormolen relates his first level to van Hiele's visual level of minking, his second to van Hide's descriptiVe/analytic, and his third to van Hlele's level of formal deduction In whj.:h srudenL~ altend \0 the properties of arguments. It should be observed that although srudents in Van Dormolen's second level h3~,(, made progress. their method is fraught Mth potential for error, For instance, as srudentS reason about a class of shapes by examining specific cases. they often attend 10 properties of the p:miculac insl3llces, thus making mistakes about the class in general. Van Hiele Leve ls and the Ability to Construct Proofs. As can be seen from the above descriptions, a proof-oriented geometry course: requires thinking at least at Level 3' In the VIln Hic\e hierarchy. However, over 70% of srudents begin high

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GEOMETRY AND SPATIAL REASONING

schOOl geometry at Levels 0 o r 1, and only those Students -~ho enter at level '2. (or higher) have a good chance of becommg Illpetcm with proof by the end of the course (Shaughnessy Burger, 1985). II follows, therefore, thai i~S(ruction should help stUdents aila!n higher levels of geometric thought before they begin a proof-orlenced study o~ geo",letry. . Senk (1989) Investigated the relauonslup between van Hle.ie levels. writing geometry proofs, and achievement in nonproor geometry. Students enrolled in full-year geometry classes were

C;

tested in the fall for van Hiele level and entry level geometry knowledge. and in the spring for van Hiele level, knowledge'of geomeuy, llnd proof-writing ability. It w~ . found mal achievement in writing geometry proofs was positively correlated with van Hiele level (.50 in (he fall , .60 in the spring) and to achievement on nonproof coment (.70 in the spring), Senk argued chat srudents who stare geometry ar level 0 have little chance of learning to wrlle proofs , students at Levell have less than a one-in--three chance, and s tudems at Level 2 have a 50· 50 chance. level -Z Is lhe critical emry leveL Senk nOled thaI at the end of the school yetr. students at Level 3 or above Significantly outperformed on proof students at ~ev:l 2 or below, but students at levels ;tS and a "gravit:l!lonlll factor" (that is. 3 figure is ·slable" only if II has one horizontal side, wi!h the other side ascending). (The findings of other srudies agree that students limit concepts to studied exemplars and consider inessential bul common features as essential to the concept; Burger & Shaughnessy, 1986: Fisher, 1978. Fuys et al, 1988; Kabanova· Meller, 1970; Zykova, 1969). Components of concepl Images were also Identified; for elGlITlple, srudems' concept Image for a right triangle were most likely to include :1 right triangle with a horizontal and a ven-ical side, less likely ( 0 include a simIlar triangle rotated sllghd y, and least likely to Include a right isosceles triangle with a horizontal hypotenuse. Stud y of .such concept images may provide useful informatio n abolll errors !llal srudents make. For example, students who know a correct verbal descriplion of a concept, but also have a specific vtsual Image or prototypC associated tightly with thai concept, may have difficulty applying the verbal deSCription correaly (Clemenl5 &: BattiSt3, 1989: Hershkowitz et al., 1990. Vinner & Hershkowitz, 1980). Aschbein (1987) relates concept images to inru itlon. SuDjeas anach a particular presentation [0 the concep4 which has

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a strong impaq on !heir cogOflive decisions. Even when the definition is e.xp l icjtl~ mentioned, most subjects are not able to respond correctly. 1l1e manner in which a concept functions in a reasoning process is highly dependent on its paradigmatIc connections. The filCI of knowing explidtly lhe definition does nOi eliminate the constraints Imposed by the tacitly intervening paradigm" (p. 146). This also helps explain StudenlS' resistance to hierarchical relationships among quadrililterals, The images anached to elIdl figure [unction cognidvely, not as panicular cases but as generoll models. Thus, srudents ha\'C to learn the decisive role of e.xpliCitly defining concepts 10 avoid errors In using the terms !hal signify them. They have to conStruct a meaningful syntheSiS of !his definition with II range of e:»em· plars. Employing such a synthesis of an31~tic and verbal pro· cesses 10 consrruCt robust concepts is possible, especially for studentS in grAde .5 and be~'ond ( HershkOWitz et at, 1990). This formulation is highl)' conslstem with Reirs (1987) ~ ideal ~ model for interpreting malhematical concep(s reliably and efficiently. In f.uniJiar situations, this ]1lodel fi rst appiles no nformal, case-based knowlet!ge (see Fischbeln's [1987] p3radigmatlc models), then checks doubtful conclusions wilh e.xplicil fo rmal knowledge. in unfamiliar situations or when· ever Inconslstendes or needs to make general Inferences arise, the Ideal model rums directl y to formal know ledge. Nonfo,... mal knowledge, then, Is still useful in providing checkpoinl5 for more abstract arguments. Reii' found the processes of ef· fective e;;amples, a Level J (visual) activity in the 'lOIn I-Jiele hierarch)'. In Logo, however, Students can be asked 10 construct a sequence of commands (a procedure) to draw a rectangle. This · .. , allows, or obliges, the child to externaJi7.e Inruitive expectatiOns. When the intuition is translated inlo a progr.lm il becomes more obtrusive and more accessible to reflection" (Papen, 1980, p. 14;). nut is, in COll$tructing OJ rectangle procedure, Ihe students must analyze the visual aspectS of the reaangle and reflect on how its component pans are pUl together, an aetivity that encourages Level 2 thinking. Funhennore, If asked to design a recungle procedure that takes the length and width as lnpUl5, srudems must construci 3. form of definition ror a rect:lngle, one that the computer understands. Thus, they begin 10 build intuitive knowledge aboUI the concept of defining a reaangle, knowledge that can later be imegrnlCd and formalized into l11l ab-

stract definition (a level 3 aCtivity). Asking students if a squ3re or a parallelogram can be drawn by their rectangle procedure if given the proper inputs encour:tges studentS to sran ordering figures logically (another Level 3 activity), Research suggests that these theoretic-..u predictiOns :] [e valid. Grade 7 studen[5' work in logo rel:lles closely to theit level of geometriC thinking (Olson. Kieren, & Ludwig, 1987), tn addition, appropriate use of Logo helps elementary srudems begin to make the transItion from the Levels 0 and I to Level 2 of geometric tho ught. For example, Logo experience has been shown to have a significaml~' positive effect on elementary school children's plane figure concepts (Clements, 1987; Clements 8: B:lttiSta, 1989: Hughes 8: Macleod, 1986). This may be because, as recommended by van Hiele researchers, logo incorporales impliduy the types of propenies which will be developed by level I thinkers e.>;plidtl)\ something that te:>.'books oflen fail to do (8.'miSt:l & Clements, 1987, 1988, 1991; FUrs et aL, 1988). logo experience encourages students 10 view 3nd describe geometriC objects in terms of the actions o r procedures used to conSlruct them (ClemenlS &: Bauista. 1989). When asked to describe geQrnetric shapes, children with Logo experience prolTer not only more statementS overnll, but also more Statements that expliCitly memion components and geometric properties of shapes, an indication of leI'el 2 thinking (Clements & B3ttista, ]989, 1990; Lehrer & Smith, 1986). Similar results have emerged in the area of symmetry and mexion geometrY. Working with a logo unit 00 motion geometry, students' movemem to van Hide levels was slow, but there was definite evidence of a beginning awareness of the propenles of lransformations (Olson et at. i987), Similarly, intermediate grade students in the United StateS were engaged in symmeO)' and mexion geomeuy activities using eiuler Logo Of paper and pendl Oohnson·Gemile, Clements, & Battista, 1990). Interviews conducted with a subsample revealed that both treatment groups performed al a higher level of geometric thinkIng than did the control grouPi logo students performed at a higher level than me noncomputer SlUdents o n four of the six lmerview taSks, noncomputer studenlS performed at a higher level on one. Both logo and non·Logo groups oUlperformed the l.'Ontrol group on immedi:lIe :1nd del:lyed post-tests: in addition, though the twO treaLmenl groups did not significant.ly differ on u,e Immediate poSt-test, the Logo group outperfonned the non-Logo group on the delayed post-test. Thus, there was support for the notion that the logo-based version enhanced conceprual reco nstruction of previously-learned ideas. Compared to Students using paper 3nd pencil, students using Logo worked with more precision 3nd exactness (GalJou·DumieJ, 1989; )ohnson-Gemile el aL, 1990). Thus, there IS evidence in suppon of the hypothesis th:n Logo experiences can help e1emeom ry to middle school sTudeo[5 become cognizant of their mathematical inruitlons and f3cilitate the transldo n from visual to descriptive/analytic geometric thinking in the domains of shapes, symmetry; and motions (Clements & Balllsm,I990). Several research prOjects have Investigated the effectS of logo experience on srudents' conceptualizations of angle, angie measure, and rolation. In one study. responses of Intermediate grade control students were more likely to refleC1 little knowledge of angle or common language usage, whereas the responses of th~ logo srudents indicated more generallzed,

malhematical1y-oriemed conceptualizations (including angle as rcxatJon :tnd as a union of twO lines/segme nts/rays) (ClemenlS & 8.1111513, 1989). Other researchers stud ied how logo might provide experiences a! the second ilnd third van Hielc levels for ninth-grade students (Olive, 13nken:tu, & Scally, 1986), logo students gained more Ihan the control SfudentS on Interviews that operatlOnalized the 'I:ln Hiele levels for the concept of angle. Sever:tl other researchers ha\'e similar!~' reponed a positive effect of logo on Students' angle concepts (Kieran, 1986a: Olive et aL, 1986), although in some sItuations, benefilS dId not emerge until after more than a year of Logo experience (Kelly, Kelly. & Miller, 1986-1987). 11'ls line of research also Indicates that students hold many dllTereru schemas regarding not only angle concept but :11so angle measure. Third graders frequently relate the size of an angle tQ the length of the line segments that formed its sides, the tUt of the tOp line segment, the area enclosed by the lri:tngular region defined by the dr.lwn sides, the length between the sides (from poInts sometimes, but not always, equidistant from the vertex), the proximity of the twO sides, or the rum at the venex (Clements & BattisL'l, 1989). Intermediate grade students often possess one of twO schemas, In [he ~lIalfc(lllmowledge. Paper presented at the meeting of the Nonh American Chapter of [he internaIIOflal Group for the ~rchoJog}' oi MlIthemalic5 Education. New Brunswick. NJ

Brumfield, C. (l973). Convention.11 Jpproodlt$ using ~)'I11hedc Euclidean geometr}'. !n K. B. Henderson (Ed), GlN>menJ' (1Ilbe II1f11be· I/I(I/;a cllrricllllllll ' 1973 Jetll'book (pp 9S-115). RCSIOl1. VA< N:IIlomo! Counc!! ofTh'~c!lers of M;tdH:matic5 Durger. '1(( . &: Sh:tughnessy,1. M. (1986). Char~clerlzing the \';11'1 Hie1e 1f.'\'C.'ls of devl:lopmcm in I!,o::ometr-,·. jollrl/flfjol' HesoetIrd) III M(lII)(!'mnlfcs £dU((I//oII, 17,31-46. Clm~!!, ? F. (19117) MfI{/£WlllS tits/mIce.. Cbildrell~ I~ oj 1ll1II1be!' (ll1d /tml. Final repon submlueQ [u the Nmiooa! Institute of Ml:maJ He'Jlth Under the A[)M.lHA Sm:lll Gram A~",lrrl Program. Gr:1nt No. MSMA 1 R03 MH423435{l1. UniverSl!r of M:Ir) bnd. College. Plirk. C lrpemer, t P., Corbitt. M. K.. Kepner. Ii. S.. lindquist. M. M., & Re)·s. R. E, (1960). NatlQn:tI assessment. In E. Fennema (Ed.), Mm/)emfllics !!d1lCf/llon f~(//l:JJ.. Implkations fol' lbe 80s (pp. 22-38). Ale"andri,t, VA: Assocl:uion for Supenoislon a/td CurriC1,llum Development. Charle:>. R I (1960). E.~emplific:t(ion and cil:lr:IC(t'riz'lIion mO'o'e; In the classroom le:,chlng of geometr-,' collr:eptS.jollmal for ResMrr:b in ,I/(IIIJemlllicsEdIIQ'llio'1. 11, l(}... 21 _ Olal:ln. 0 (1989). I nstruoion~1 !mplicuions of a research project on ~udentS' under.;t.1ndings ot the differences ber.'t.'t!fl emplriCll veri· fie-Juan and m;ltllCcm:.ltical proof. In D. Hergert (Ed.). PI'OCeedlllSs of I/~ FirsJ 11I/eIlUlIIaIl(l1 Conferrmce 0'/ Ibe His/OF)' and P/;ifOropl~I' of Science III SClel1ce TMdJing T;1)(:dt:lSSt:e. FL: Florida Slme UnlversiC)' Science EdUClUOfl and Philosoph)" OepartmCI'\L Clark, R. E. (1983). Recoflsidering n:sC'~n::h on Ie'Jrning from medi:l. Ret'few of Edllen/ioll(ll Resemrb. 53. cbno/Offl'. pp 7-16. . Clemenl5,~. H. (1987). Longiludill~l Stud) of the effects of'Logo progr:Immmg on cognitIve abilities JI1d xhievement.jOIll'l',al of EdllcaliOl/(I1 ComplIIllIS Re.q?fIl'clJ, 3. 73--94. demefll5, D. H .. & Bm1151..1. M. T. (1989). Lt;:lminl! of geometrIc conceptS in a Logo environmem.jol//"I1f11 jor Resea/'ch ill ,l/mfJem(llics Edlleo/;ol/.20.450-467.

Clements. 0, H" &: nauist.l, M. T. (1990). The efferu of t.ogo on chll· dren's conceptuali~lioos of lingle and polygons. jOllm(l1 for Re5e(II'ch (II ,I/(iriJell1a1K3 EdIlCfl/iOli. 21, 3.56-~71.

Clemenl5, M. A. ( 1979). Sex dIfferences in m:uhematic;1J performance: An hiSlOrical perspectl\l{". EtlllcmiOllal SlIIdles in MmbellinfiC!i 10 30)-322.

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ClementS. M. A.. (!983). l1le quesllon of how spallal ~1li1!t)'ls defined ~nd ItS rdevJ/lce to m:lthem~tics edUt:"Jlion Zemmlblnll fm' Didllk· Ilk df!l' MfIIIJe/1/fllik. 15. &-ZO. Clements, M A.., & Del C:lmpo. G. (1989). LInkIng verb."tI knowledge. ",su:11 Inwgcs. Jnd episodes ror m~lhemallcal leaming. Focus on teaming Problems ill ,!imbem(l/ICl. t 1(1 ).25-33. CObb. P.. (1969)...... amSM ,Clil'isJ ~rsp«III1! 011 infonllallOlI·processlllg lheortes of mmbem(lliw/ n"II·'(I'. Unpublished ffi,1nuscflpt. Pun:lue tJnlversiq; ''('est llIbreue. I!'I. Cousins, ~., & Abr~'"lIlel. E. (971). SOme findings re!evanl to the hr' pothesis tll:ll topoJogiC:11 sp~lial fC:IIUres are dilferentr.lled prIor 10 euclldl::ln fe:nures during gro"'1h. Brirl'w joll/J/(lt o'D...'CiJ%a" 62 -175-479 1I .~) 0'" CtOW!e); M. L. ( 1990). Cl'l.lerion referenced reJtalliliq' indIces associlUed ,,'lth d1e V'JIl Hiele geomClf)' ICS!. 10/1"/(11 for Rese(lrc:b in ,II(lII}f!lII(IIics Educarioll, 21, 238-241 I>~rlIe. I. (1982). A review of reselrch re l:lted 10 the topologit:"J! prima"}' thesis. Edllcmioll(lf SlIId;e:; ill Mmf)fll/Ulria, 13, 119--142. ();)vl5. fl B. (1984) Umnling marbemarics' 71Je cogl1f1./:e SiC/ella! (lp" prptIcb 10 IIIf11!Je1l1mics edl/cmiolJ. NorwooL1, NJ: Ables. O::ivis. R. B. (198G). Conceptual ;md procedur.d knowledge in mmhem:ulcs: Asumm:lIT aNlI}'SiS. InJ. H!ebert (Ed.). Conceplll(ll (mdpnr cMumi knowledge· nJe crue of II/mbelllfllics (pp. 265-300). Hillsd:Jle. NJ: L.:lwrt:rn:t" ErllJaum

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Del GrJnde. J.J. (1986). Can grode ~'O child ren's spatial perceplion be lm~rovcd bj' insening a tr.losfoomulon geometry componr!l1t Imo theIr lnm!lemmlcs progrnm? Dissertmloll Abstracts Il1Iem(ll/olln/ 47, ~689A. ' Denis. L P. (1987). Rel~tJonships between suge of cognitive devl.'lopment and I':m Hlele 1~1 of goomt'tric thought among Puerto RIcan adole:;rents. Di5se11(lIioll !\bslmcls IlIIemmioll(ll, 48, 859A. (UniverSity Mlcrofilrn~ No. DA8715795.)

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