young children’s ideas about shape - university at...

A toddler, after some experimentation,

puts a square peg into a square hole.

What does she know about shapes?

What more will she learn in preschool and ele-

mentary school. What mightshe learn?

Educators have learned a great deal aboutyoung children’s knowledge of shapes. Much of itis quite surprising. In this article, we describeresearch on young children’s thinking about geo-metric shapes and draw implications for teachingand learning.

What Do ChildrenKnow about Shapes?

Children’s levels of understandingAs children develop, they think of shapes differ-ently. At the prerecognition level,children perceiveshapes but are unable to identify and distinguishamong many shapes. They often draw the sameirregular curve when copying circles, squares, or tri-angles (Clements and Battista 1992b). At the next,visual, level, children identify shapes according to

their appearance (Clements and Battista1992b; van Hiele 1959/1985). For example,they might say that a shape “is a rectanglebecause it looks like a door.” At the descrip-tive level, children recognize and can charac-

terize shapes by their properties. For instance, a stu-dent might think of a rectangle as being a figure that

has two pairs of equal sides and all right angles.Because progress in children’s levels of thinkingdepends of their education, children may achievethis level in the intermediate grades . . . or not until college!

Children at different levels think about shapesin different ways, and they construe such words assquare with different meanings. To the pre-recognition thinker, square may mean only a pro-totypical, horizontal square. To the visual thinker,squaresmight mean a variety of shapes that “looklike a perfect box” no matter which way they arerotated. To a descriptive thinker, a square should bea closed figure with four equal sides and four rightangles. But even to this child, the square has norelationship to the class of rectangles, as it does forthinkers at higher levels. These levels can help usunderstand how children think about shapes. Wemight remind ourselves to ask what children seewhen they view a shape. When we say “square,”they might seem to agree with us for many proto-typical cases but still mean something very differ-ent. The levels can also guide teachers in providingappropriate learning opportunities for children.

482 TEACHING CHILDREN MATHEMATICS

Douglas H. Clements and Julie Sarama

Doug Clements, [email protected], teaches at the State University of New York atBuffalo, Buffalo, NY 14260. He conducts research in the areas of computer applications ineducation, early development of mathematical ideas, and the learning and teaching of geom-etry. Julie Sarama, [email protected], teaches at Wayne State University, Detroit, MI48202. Her research interests include the implementation and effects of software environ-ments in mathematics classrooms. With Clements, she is developing innovative preschoolmathematics materials.

Young Children’s Ideas aboutYoung Children’s Ideas about

Two kindergartners create a rhombus.

Pho

togr

aph

by M

arle

ne E

.And

rusz

;all

right

s re

serv

ed

This material may not be copied or distributed electronically or in any other format without written permission from NCTM. Copyright © 2000 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.

Children’s ideas about common shapesYoung children have many ideas about commonshapes. Levels alone, however, do not give teacherssufficient detail. For this reason, we conducted sev-eral studies of young children’s ideas about shapesto help complete the picture (Clements et al. 1999;Hannibal 1999). In all, we interviewed 128 chil-dren, ages 3 to 6, for a total of an hour over severalsessions. Children identified shapes in collectionsof shapes on paper and manipulable shapes cutfrom wood. The shapes were presented in differentsettings. For example, sometimes we had the chil-dren handle the shapes, and other times we askedthem to identify the shapes placed in various orien-tations in either a fixed rectangular frame or around hoop. Our conclusions from these interviewsregarding the basic shapes are described here.

Circles. Children accurately identify circles,although children younger than six years old moreoften choose ellipses as circles. Apart from theseinfrequent exceptions (only 4 percent incorrect onour paper task), early childhood teachers canassume that most children know something aboutcircles.

Squares.Young children are almost as accurate inidentifying squares (87 percent correct on our papertask) as in identifying circles, although preschoolersare more likely to call nonsquare rhombus“squares.” However, they were just as accurate asolder children in naming “tilted” squares.

Triangles.Young children are less accurate inidentifying triangles (60 percent correct). They arelikely to accept triangular forms with curved sidesand reject triangles that are too “long,” “bent over,”or “point not at the top.” Some three-year-oldsaccept any shape with a “point” as being a triangle.

Rectangles.Again, children’s average accuracyis low (54 percent). Children tend to accept “long”parallelograms or right trapezoids (see fig. 1,shapes 3, 6, 10, and 14) as rectangles. So children’s

image of a rectangle seems to be a four-sided fig-ure with two long parallel sides and “close to”square corners. Only a small number of three- andfour-year-olds seem not to have any ideas orimages of rectangles or triangles.

Children’s OftenSurprising Thoughtsabout ShapesWe were impressed with how much preschoolersknew about shapes yet were more surprised byother findings.

483APRIL 2000

FIG

UR

E 1 A set of quadrilaterals

GeometricShapes


Not that much changesIt was surprising how little children learn aboutshapes from preschool to middle school. Forexample, our preschoolers identified about 60 per-cent of the triangles correctly. In a large study ofelementary students doing the same task, scoresranged from 64 percent for kindergartners to 81percent for sixth graders (Clements and Battista1992a). Similarly, the preschoolers’ scores were 54percent on rectangles, and elementary school stu-dents’ scores ranged from 63 percent to 68 percent.Although approximately eight years separate thesegroups, the scores on the same tasksincreased onlyminimally. Perhaps we could be doing a better jobteaching geometry in every year.

Individual differences can be dramaticOne of the six-year-olds in our study chose rectan-gles randomly and described them as “pointy.” Incomparison, one of the three-year-olds outscoredall the six-year-olds on the rectangle task. Goodopportunities to learn are more important thandevelopmental level when it comes to children’slearning about shapes.

Given good opportunities,children pose and solveinteresting problemsA teacher in our study challenged her kindergartnersto make various shapes with their bodies. Two boystried to make a rhombus. They sat down facing eachother and stretched their legs apart. With their feettouching each other, they made a very good rhombus.One of the children in the circle suggested that “if weput another child in the middle, we would make twotriangles!” Immediately, the children called into ser-vice a boy named Ray, because he was the smallest,and they asked him to scrunch in the middle. Theboys posed their own problems about decomposingshapes and devised their own solutions.

Computers help push back the boundaries ofwhat children can do with geometric shapes. Forexample, preschooler Tammy was working with acomputer program that presented a fixed set ofshapes. She overlaid two overlapping triangles onone square and colored selected parts of this figure.This process created a third triangle that had notexisted in the program! Computers can also bepowerful and flexible manipulatives (Clements andMcMillen 1996). For example, Mitchell wanted tomake hexagons using the pattern-block triangle.He started without the computer and used a trial-and-error approach, counting the sides and check-ing after adding each triangle. However, usingShapes, a computer program, he beganby planning(Sarama, Clements, and Vukelic 1996). He first

placed two triangles, dragging them and turningthem with the turn tool. Then he counted with hisfinger around the center of the incomplete hexa-gon, visualizing the other triangles. “Whoa!” heannounced. “Four more!” After placing the nextone, he said, “Three more!” Whereas without thecomputer Mitchell had to check each placementagainst a physical hexagon, his intentional anddeliberate actions on the computer led him to formmental images. That is, he broke up the hexagon inhis mind’s eye and predicted each placement.

What Is Trickyfor Children?

Different settings, differentdecisionsSpecific settings affect children’s decisions. In ourfirst setting, we included obvious nonexamples,such as circles, in with a group of triangularshapes. Children accepted more types of trianglesand more “pointy” forms that were not triangles.Similarly, children accepted more ovals as circleswhen shapes were drawn inside each other. Thus,what children choose is strongly influenced by theshapes they are comparing.

In our second setting, we placed the woodenshapes inside a hoop. Children were less likely tonotice or care about whether they were “long” or“upside down.” That is, not having a rectangularframe of reference affected what shapes theyaccepted. In the third setting, we asked children tojustify their choices. They often changed theirdecisions, usually to correct ones. For example,one five-year-old increased her score by 26 per-centage points when asked to explain her choices.

Limited understanding ofpropertiesEven though talking about the shapes often helped,children’s knowledge often has limits. Four-year-olds would frequently state that triangles had “threepoints” or “three sides.” Half of them were not, how-ever, sure what a “point” or “side” was (Clements1987). For example, a student was asked to stand ona triangle from among several taped outlines ofshapes on the floor. The student immediately stoodon the triangle and explained that “it has three sidesand three angles.” When asked what she meant bythree angles, she said, “I don’t know.”

What Can TeachersDo to Help?The studies’ results that we have described havemany implications for geometry instruction. The

remainder of this article offers research-basedsuggestions for improving the teaching of basicshapes and for enhancing children’s understand-ing of these shapes.

Reconsider teaching “basicshapes” only through examplesAs we continually contrast squares and rectangles,do we convince children of their separateness?Does teaching shapes mainly by showing examplessometimes build unnecessarily rigid ideas? In onestudy (Kay 1987), a first-grade teacher initiallyintroduced the more general case, quadrilaterals, orshapes with four straight sides. Then she intro-duced rectangles as special quadrilaterals andsquares as special rectangles. That is, she discussedthe attributes of each category of shapes and theirrelationships; for example, a rectangle is a specialkind of quadrilateral. She informally used termsthat reflected this relationship; for example, that“this is a square-rectangle.” At the end of instruc-tion, most of her students identified characteristicsof quadrilaterals, rectangles, and squares, andabout half identified hierarchical relationshipsamong these classes, even though nonehad done sopreviously. This teacher concluded that the typicalapproach of learning by showing examples isappropriate only for shapes that have few suchexemplars, such as circles and possibly squares.For other shapes, especially such hierarchical-based classes as triangles and quadrilaterals, thisapproach alone is inadequate. We might questionhow deep these first graders’ understanding of hier-archical relations was (Clements and Battista1992b). However, we might also question the wis-dom of the traditional, show-examples approach—it may lay groundwork that must be overturned todevelop hierarchical, that is, abstract-relational,geometric thinking.

Give children credit for whatthey knowBy the time they enter kindergarten, most childrenhave many ideas about shapes. Yet teachers oftendo not ask children to extend their ideas. In onestudy (Thomas 1982), about two-thirds of teachers’interactions required only that children parrot whatthey already knew in a repetitious format, such asthe following.

Teacher:Could you tell us what type of shapethat is?

Children:A square. Teacher:OK. It’s a square.

Most questions that teachers asked were closed-ended and dealt only with memory. Few children

asked questions, but when they did, the teacherresponded with silence. Instead, teachers shouldbuild on what children know and generate richdiscussions.

Avoid common misconceptionsIn the Thomas (1982) study, most teachers did notadd new content. When they did, however, thestatements were often incorrect, such as saying,“All diamonds are squares.” Others were, at best,unfortunate, such as telling a child who chooses asquare rectangle, “No, find a rectangle.” Otherunhelpful statements included that “two trianglesput together always make a square” and that “asquare cut in half always gives two triangles.” (Seefig. 2 for counterexamples.)

Expand the limited notions thatare too often “taught”Everywhere, it seems, unfortunate geometricideas surround us! As part of our research, weclosely examined toy stores, teacher-supplystores, and catalogs for materials on “shape.”With few exceptions, these materials introducechildren only to “best examples” of triangles, rec-tangles, and squares. Most triangles are equilat-eral or isosceles triangles with horizontal bases.Most rectangles are horizontal or vertical,between two and three times as long as wide.Most squares have horizontal bases. Curricularmaterials are also often limited. As just one exam-ple, Leah was working on a computer programthat asked her to choose fish of various shapes.When asked to pick square fish, she chose onewhose square body was oriented at 45 degrees.The program announced that she was wrong, say-ing that it was a “diamond fish”!

485APRIL 2000

FIG

UR

E 2 Triangles and squares

Two triangles put together do not always make a square, and a squarecan be cut in half many ways.


Help children break free of these limitations byusing a wide range of good examples and non-examples. To do so, try these tactics:

• Vary the size, material, and color of examples.For rectangles and triangles, vary their orienta-tion and dimensions, such as the rectangles infigure 1. For triangles, vary their type, includingscalene—no sides equal in length—andobtuse—one angle larger than 90 degrees.

• Compare examples with nonexamples to focuschildren’s attention on the essential attributes(see fig. 3).

• Encourage children to change shapes dynamically.

FIG

UR

E 4 Manipulating a category of shape in

Math Van

Math Van (McGraw-Hill School Division1997) allows children to draw and manipu-late a certain kind, or category, of shape.Here children defined an isosceles triangle.The double-color on the two equal sidesrepresents their equality (a). Children canslide, turn, or flip this triangle. They canalso distort it, but no matter how they do, itremains an isosceles triangle [(b) and (c)].

(a)

(b)

(c)

FIG

UR

E 3 Tricky Triangles

The Tricky Triangles sheet includes triangles (shapes 1–5) and nontri-angles (shapes 6–10) that are paired to encourage contrasts. For exam-ple, shape 1 is a triangle, but shape 6 is not a triangle because it is notclosed. Shape 2 does not have a horizontal base and is not symmetric,but it is a triangle; however, shapes 7 and 8 are not triangles, becausethey have four sides.

1

2

3

4

5

6

7

8

9

10

For example, some computer programs allow chil-dren to manipulate a figure, such as an isoscelestriangle, in many ways, but it always stays anisosceles triangle (see figs. 4a, 4b,and 4c).

Matching activities to children’slevel of thinking about shapesChildren at the prerecognition level might bestwork with shapes in the world. Help these chil-dren understand the relevant and irrelevant—forexample, orientation, size, color—attributes ofshapes with such activities as the following:

• Identifying shapes in the classroom, school,and community

• Sorting shapes and describing why theybelieve that a shape belongs to a group

• Copying and building with shapes using awide range of materials

Children at the visual level also can measure,color, fold, and cut shapes to identify theirattributes. For example, they might engage in thefollowing activities:

• Describing why a figure belongs or does notbelong to a shape category

• Folding a square or rhombus in various waysto determine its symmetries and the equalityof its angles and sides

• Drawing shapes with turtle geometry, whichhelps children approach shapes actively but with precise mathematical commands andmeasurements; or having children give com-mands to a “child-turtle” who walks the path

• Using a “right-angle tester” to find angles inthe classroom equal to, larger than, or smallerthan a right angle

• Using computers to find new ways to manip-ulate (fig. 5a) and make (fig. 5b) shapes(Clements and Meredith 1998)

• Discussing such general categories as quadri-laterals, or “four-sided shapes,” and triangles,counting the sides of various figures tochoose the category in which they belong

ConclusionsAn adult’s ability to instantly “see” shapes in theworld is the result, not the origin, of geometricknowledge. The origin is our early active manipu-lation of our world. Young children learn a lotabout shapes. Geometry instruction needs to beginearly.Our research and that of Gagatsis and Patro-nis (1990) show that young children’s conceptsabout shapes stabilize by six years of age but thatthese concepts are not necessarily accurate. Wecan do a lot as teachers to supplement curriculathat too often are part of the problem rather thanthe solution.

ReferencesClements, Douglas H. “Longitudinal Study of the Effects of

Logo Programming on Cognitive Abilities and Achieve-ment.” Journal of Educational Computing Research3(1987): 73–94.

Clements, Douglas H., and Michael T. Battista. The Develop-ment of a Logo-Based Elementary School Geometry Cur-riculum. Final Report: NSF Grant No. MDR-8651668.Buffalo, N.Y./Kent, Ohio: State University of New York atBuffalo/Kent State University, 1992a.

487APRIL 2000

FIG

UR

E 5 Manipulating shapes in the Shapes program

The Shapes program (Clements and Meredith 1998) allows children to manipulate shapes in waysnot possible with usual pattern blocks, such as to define new units-of-units, which would turn andduplicate together. They can enlarge or shrink shapes, as Leah did in making her snow person (a).They can hammer shapes to decompose them into parts (b).

(a) (b)

☛


———. “Geometry and Spatial Reasoning.” In Handbook ofResearch on Mathematics Teaching and Learning,edited byDouglas A. Grouws, 420–64. New York: Macmillan, 1992b.

Clements, Douglas H., and Sue McMillen. “Rethinking ‘Con-crete’ Manipulatives.”Teaching Children Mathematics2(January 1996): 270–79.

Clements, Douglas H., and Julie Sarama Meredith. Shapes—Making Shapes. Palo Alto, Calif.: Dale Seymour Publica-tions, 1998.

Clements, Douglas H., Sudha Swaminathan, Mary Anne ZeitlerHannibal, and Julie Sarama. “Young Children’s Concepts ofShape.”Journal for Research in Mathematics Education 30(March 1999): 192–212.

Gagatsis, A., and T. Patronis. “Using Geometrical Models in aProcess of Reflective Thinking in Learning and Teaching Math-ematics.”Educational Studies in Mathematics 21 (1990): 29–54.

Hannibal, Mary Anne. “Young Children’s Developing Under-standing of Geometric Shapes.”Teaching Children Mathe-matics5 (February 1999): 353–57.

Kay, Cynthia S. “Is a Square a Rectangle? The Development ofFirst-Grade Students’ Understanding of Quadrilaterals withImplications for the van Hiele Theory of the Development ofGeometric Thought.”Dissertation Abstracts International47(1987): 2934A. University Microfilms No. DA8626590.

McGraw-Hill School Division. Math Van. New York: McGraw-Hill, 1997. Software.

Sarama, Julie, Douglas H. Clements, and Elaine Bruno Vukelic.“The Role of a Computer Manipulative in Fostering Spe-cific Psychological/Mathematical Processes.” In Proceed-ings of the Eighteenth Annual Meeting of the North AmericaChapter of the International Group for the Psychology ofMathematics Education,edited by E. Jakubowski, D.

Watkins, and H. Biske, 567–72. Columbus, Ohio: ERICClearinghouse for Science, Mathematics, and Environmen-tal Education, 1996.

Thomas, Betsy. “An Abstract of Kindergarten Teaching’s Elici-tation and Utilization of Children’s Prior Knowledge in theTeaching of Shape Concepts.” Unpublished manuscript,School of Education, Health, Nursing, and Arts Professions,New York University, 1982.

van Hiele, Pierre M. “The Child’s Thought and Geometry.” InEnglish Translation of Selected Writings of Dina van Hiele-Geldof and Pierre M. van Hiele,edited by D. Fuys, D. Ged-des, and R. Tischler, 243–52. Brooklyn, N.Y.: BrooklynCollege, School of Education, 1959/1985. ERIC DocumentReproduction Service No. 289 697.

Time to prepare this material was partially providedby two National Science Foundation ResearchGrants: NSF MDR-8954664, “An Investigation of theDevelopment of Elementary Children’s GeometricThinking in Computer and Noncomputer Environ-ments,” and ESI-9730804, “Building Blocks—Foun-dations for Mathematical Thinking, Pre-Kinder-garten to Grade 2: Research-based MaterialsDevelopment.” Any opinions, findings, and conclu-sions or recommendations expressed in this publica-tion are those of the author and do not necessarilyreflect the views of the National Science Foundation.We also extend our appreciation to Julie JacobsHenry for her suggestions regarding the article. ▲

young children’s ideas about shape - university at...

Documents