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Running head: A PRESCHOOLER’S UNDERSTANDING OF “TRIANGLE”
A Preschooler’s Understanding of “Triangle:”
A Case Study
Mary Elaine Spitler
University at Buffalo
This material is based in part upon work supported by the National Science Foundation Research
Grant ESI-9730804, “Building Blocks—Foundations for Mathematical Thinking, PreKindergarten to Grade 2: Research-based Materials Development.” Any opinions, findings, and
conclusions or recommendations expressed in this publication are those of the authors and do not
necessarily reflect the views of the National Science Foundation.
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A Preschooler’s Understanding of “Triangle:” A Case Study
For several decades, children and adolescents in the United States have encountered
difficulty in understanding geometrical concepts {Clements & Battista, 1992; Clements,
Swaminathan, Hannibal, & Sarama, 1999; Senk, 1989; Shaughnessy & Burger, 1985; Teppo,
1991; Usiskin, 1987; Yusuf, 1994). “According to extensive evaluations of mathematics
learning, elementary students in the United States are failing to learn basic geometric concepts
and geometric problem solving. They are underprepared for the study of more sophisticated
geometric concepts, especially compared to students from other nations” (Clements & Battista,
1992, p. 66). When compared to other countries, the United States has ranked low in terms of
geometry achievement (Stigler & Perry, 1988). Students enter high school with “low levels of
geometric concept development” (Teppo, 1991, p. 217). Researchers found student
misconceptions and errors, and decreases instead of in geometry performance (Swafford, Jones,
& Thornton, 1997).
In the Netherlands in the 1950s, two Dutch teachers, Pierre van Hiele and Dina van HieleGeldof , were also concerned with the difficulties they were encountering in teaching geometry
to high school students. The van Hieles found that the high school students with whom they
were working did not understand the high school geometry curriculum, no matter what approach
was used. “They believed that secondary school geometry involves thinking at a relatively high
‘level’ and students have not had sufficient experiences in thinking at prerequisite lower ‘levels’”
(Fuys, Geddes, & Tischler, 1988, p. 4). As a result of their studies of Piaget’s work with
children and geometry (Piaget & Inhelder, 1967; Piaget, Inhelder & Szeminska, 1960), and from
their own classroom observations, the van Hieles developed a model of five geometric levels of
thought: visual, descriptive/analytical, informal deductive, formal deductive, and rigor (Crowley,
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1987; Shaughnessy & Burger, 1985). According to the van Hieles, these levels were not
dependent on maturation, but rather on instruction. They contended that high school students
encountered difficulty with proofs and formal geometric thought because they did not have
experiences during elementary school in the descriptive/analytical levels that come before the
level of formal thought required in high school. Children enter elementary school at the first
level of geometric thought, the visual level, and then do not progress; rather, they remain at this
visual level throughout their elementary years due to lack of exposure to descriptive/analytic
experiences (Fuys et al., 1988). Researchers working in the United States have noted that the
curricula and the teachers do not cover the analytic/descriptive level, nor the informal deductive
level. According to the van Hieles, children must proceed through the levels of geometric
thought in order, although the ages vary, depending upon experience with geometry. The van
Hieles contended that “many children have difficulty attaining this formal level of geometric
thought because they lack experience at the preceding levels, especially with exploration,
discovery, and description of a variety of geometric properties” (Fuys, Geddes, & Tischler, 1979,
p. 106).
The van Hieles’ ideas spread to the Soviet Union, and to other countries. These ideas were
introduced in the United States in the 1970’s; subsequently several research projects were funded
to test the model. Eventually, the van Hiele ideas influenced NCTM’s Curriculum and
Evaluation Standards (1989), and the subsequent NCTM Principles and Standards of School
Mathematics (2000), which recommended the incorporation of the descriptive/analytical levels
of geometric thought in elementary school, including at the Pre-K – 2nd grade levels. More
recently, the National Council of Teachers of Mathematics and the National Association for the
Education of Young Children issued a joint position statement on mathematics in the early years
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(NAEYC/NCTM, 2002), in which geometry is featured as one of the five major content areas for
the teaching of mathematics.
Against the backdrop of this budding interest in geometry in preschools in the United
States, and the increase in the number of publicly funded prekindergartens, this ten-month field
observation/case study was conducted, as part of a larger study, in an effort to further understand
the mathematical thinking, including the geometrical thinking, of a four-year-old. The need for
such research into early childhood mathematical thinking is emphasized by Hershkowitz, who
noted that there is
almost no research on early childhood…It would seem very natural to start observations as
early as possible. There is still a need to invest effort in research on the evolution of
geometrical concepts, geometrical thinking, and the development of visual abilities. (1990, p.
93)
The purpose of this case study is to attempt to understand the geometric thinking of a 4-yearold child as she constructs her understanding of the concept “triangle.” Her understanding is
considered in relation to a curriculum that exposes her to consideration of the attributes of
triangles, and provides opportunities for discrimination among prototypical and non-prototypical
triangles and distractors. The argument is that this four-year-old developed some explicit
understandings of the attributes of triangles, and at the same time she entered a period of
disequilibrium in relation to her prototypical image of a triangle. This disequilibrium, brought
about by exposure to nonprototypical triangles and distractors, combined with the child’s
growing explicit awareness of some of the components of triangles, did not result in a single
“triangle” scheme. Instead, two parallel schemes, a “visual triangle” and a “three-sided figure”
co-existed, the threads of which were not interwoven, but rather, loosely connected, during the
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prekindergarten year. These loose connections began to strengthen as the child broadened her
image of “triangle” to sometimes include nonprototypical triangles, and as she became explicit
about certain attributes of triangles. These two schemes grew in parallel; she did not consistently
use her understanding of the attributes of a triangle as a basis to identify triangles. Rather, she
held onto her visual scheme and reasoned visually.
Literature Review
The research corpus on the geometric thinking of preschoolers is dominated by Jean Piaget,
who conducted many clinical interviews of children as he investigated children’s conceptions of
space and shape. Piaget’s work in this area led him to constructivist theories on geometric
concept development in three areas: topological, Euclidean space, and projective space. This
stimulated other research studies that sought to replicate, confirm, or contradict Piaget’s findings,
including the work of the van Hieles. Soviet researchers tested the van Hiele model and
subsequently adopted it in the schools of the Soviet Union. It was introduced in the United
States at an annual meeting of the National Council of Teachers of Mathematics (NCTM) in the
1970’s (Hoffer, 1983). Subsequently, the National Science Foundation sponsored several
research projects in order to test the model. Results of these investigations confirmed, for the
most part, the sequence of the van Hiele levels (Burger & Shaughnessy, 1986; Geddes, Fuys, &
Tischler, 1985; Mayberry, 1981; Usiskin, 1987), although there were mixed results on the
discreteness of the levels (Burger & Shaughnessy, 1986; Fuys, Geddes, & Tischler, 1985).
Fuson and Murray (1978) studied the haptic-visual perception of geometric shapes in 2- to 5year-olds, and found that 60% of 3-year-olds could name a triangle. Vurpillot (1976) reported on
Piaget’s findings concerning young children’s preference for the “good form” – closed and
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symmetrical (p. 55). Burger and Shaughnessy (1986) reported on the responses of six children
(3rd graders to 10th graders) who were asked to draw and sort triangles. They found that young
children often focused on irrelevant attributes of triangles, and ignored relevant attributes. Senk
(1989) looked at the high school geometry students in terms of van Hiele levels. Despite
spending approximately nine to ten years in school prior to taking high school geometry, many
were still at the first level of geometric thought (visual); they had not progressed substantially
since entering elementary school. With progression through levels dependent upon instruction,
and lacking this instruction, they were not yet at the descriptive/analytic level; they did not
explicitly differentiate between shapes based on parts and properties.
Some contemporary researchers have looked at children’s geometrical thinking. Some have
examined young children’s abilities to classify quadrilaterals hierarchically (De Villiers, 1994;
Kay, 1986). A study in 1998 by Lehrer, Jenkins, & Osana shed some light on young children’s
geometric reasoning. Children (ages 6, 7, and 8) looked at triads of polygons, determined which
two were the most alike, and stated their rationale for the pairing. Lehrer et al. found that most
children rationalized pairing based on visual appearance (although nine distinctions were made
between visual descriptions); morphing (“mentally animating the action of pulling or pushing on
a vertex or side [face] of a two- [or three-] dimensional form”) (p. 145) was common. Lehrer et
al. also (1998) found evidence suggesting that “experiences in school and in the world did little
to change children’s conceptions of shape throughout the course of the elementary grades” (p.
145). Clements, Swaminathan, Hannibal, & Sarama (1999) examined children’s reasoning
when distinguishing between classes of shapes. They found that children based their decisions
on visual features, but that children were also capable of recognizing parts and simple properties
of shapes. Hannibal and Clements (2000) examined the processes young children (3 to 6 years of
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age) used to classify new figures. For triangles, they found a high rate of rejection of variants
and distractors that did not match the triangle prototype. Monaghan (2000) studied children to
better understand their concept formation as they differentiated between quadrilaterals. Results
from this study showed students’ over-reliance on standard, prototypical representations of
shapes.
Background
This case study is part of a larger project, Building Blocks, (Clements, Sarama, &
DiBiase, in press), a PreK – Grade 2 mathematics curriculum project, sponsored by the National
Science Foundation. This research project has four purposes: (1) to gain a greater understanding
of the mathematical thinking of children from preschool to second grade; (2) to examine the
research to date on young children’s mathematical thinking and to synthesize that research into
hypothesized developmental learning trajectories in number and geometry; (3) to design and
integrate a curriculum and a mathematical software program, based on the hypothesized
trajectories; and (4) to field test children’s reactions to the curriculum and software (Sarama &
Clements, 2002). This larger study resulted in the production of research-based, hypothesized
learning trajectories for number and geometry (Clements & Sarama, 2001a, unpublished). These
trajectories, each in a specific domain, were constructed and ordered by careful examination of
the literature corpus, and are meant to illuminate processes of learning and provide explicit
cognitive models of early mathematical thinking. The first curriculum produced by this project
is a software integrated preschool curriculum that is a component of the DLM Early Childhood
Express (Schiller, Clements, Sarama, & Lara-Alecio, 2003). It is driven by the Principles and
Standards for School Mathematics, (National Council of Teachers of Mathematics, 2000), and
based on the hypothesized trajectories. The curriculum is committed to “finding the mathematics
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in, and developing mathematics from, children’s activity. The materials are designed to help
children extend and mathematize their everyday activities…” (Sarama & Clements, 2002). This
preschool curriculum was field tested in several sites in western New York, including the urban
public school in which this case study took place.
Two field-testing teams studied the implementation of the curriculum/software in New York
State. The team in this particular setting consisted of seven members: two principal coinvestigators, a videotaper/whole class observer/researcher, a researcher/teacher who taught the
curriculum, the classroom teacher (who acted as an assistant teacher while we were in the
classroom) and two case study researchers.
Four of us visited the classroom three days a week (Monday, Wednesday and Friday) for one
hour each day, during the time regularly devoted to mathematics. The two principal coinvestigators visited the classroom intermittently. During our visits, one of the members of the
research team assumed the role of teacher, and taught the mathematics lesson every Monday,
Wednesday, and Friday. On the days when we were not present (Tuesdays and Thursdays), the
classroom teacher was to reinforce the mathematics concepts we had introduced on Monday and
Wednesday. Our research team worked in this classroom from September to June. We
instructed the children in mathematics from October 10 to May 31, a total of 66 times.
Purpose
The case study within this larger project was an attempt to examine a four-year-old
preschooler’s developing mathematical understandings in relation to the Building Blocks
curriculum and hypothesized trajectories. This particular paper narrows the study, and examines
the evidence of the development of a preschooler’s concept of shape, specifically her concept of
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“triangle.” Seven questions guided this inquiry: What was the child’s understanding of
“triangle” at the beginning and at the end of the school year? What did she understand about the
attributes of a triangle? How did she relate her knowledge of these attributes to her perception of
the visual whole? How did her growing knowledge of some attributes of triangles affect her
acceptance of non-prototypical triangles as triangles? How did she use her knowledge of the
attributes of triangles to help her distinguish triangles from other non-triangular shapes? What
was the effect of the curriculum and software on her understanding of the concept of “triangle?”
How did her developing concept of “triangle” align with the hypothesized trajectory in
geometry/shape?
Participant
The informant is a four-year-old African-American girl, Tania [pseudonym], who
participated, with her classmates, in daily mathematical activities at an urban public preschool in
the northeastern part of the United States during the 2001-2002 school year. Tania was chosen as
the case study because she was an average student, not at the highest or lowest level of academic
performance, in a class of 18 children. The class consisted of 11 girls, six of whom were
African-American and five of whom were European-American, and seven boys, four of whom
were African-American and three of whom were European-American.
Description of Physical Setting/Classroom
Tania’s classroom was one of four state-funded prekindergarten classrooms in an urban
public school early childhood center. The classes in the building ranged from prekindergarten to
third grade. Approximately 18 preschool children were assigned to each of these classrooms,
with one teacher and one assistant in each room. All the teachers and assistants in the
prekindergartens were female. The room was arranged in learning centers, including a
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manipulatives area that served as a focal spot for small group mathematical activities. Three
computers were available at a computer table adjacent to this mathematics center.
During the first 15 minutes of the mathematics activity time, routinely, the children
gathered for a teacher-directed whole group activity or demonstration. Following this teacherdirected activity, the children broke up into three groups of six, and then rotated to three different
learning areas, each set up with mathematics activities that extended and further developed the
concepts the children were working on. The children spent approximately 10 to 15 minutes
engaged in each of these activities. Usually the classroom teacher guided the children through
one of these activities, and the researcher/teacher guided them through another. The third group
of children usually worked independently in an area where materials were set up ahead of time.
The groups sometimes worked at tables, and sometimes in other areas, such as on the floor, in
the dramatic play area, in the block area, and, occasionally, in the hallway).
Data Collection Methods and Analysis
To gain an understanding of a child’s mathematical thinking in the area of number and
geometry, a case study approach was used. I took field notes on the child’s actions, her words,
her interactions with people and materials, and, as much as possible, her thinking, backed up by
videotape and audiotape. I observed and recorded notes as she took part in all the mathematics
activities in her classroom, including large group sessions, small group settings with a teacher
and peers, and independent small group settings with peers. At times I observed Tania as she
worked alone, independent of peers and teachers. I also had some rare opportunities to interview
her on a one-to-one basis. Artifacts of her work were collected, and some photographs of Tania
engaging in mathematical activities were taken. Field notes were expanded as closely after they
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were taken as possible; transcribed notes were e-mailed to the principal co-investigators on an
on-going basis.
Another researcher conducted a case study of a four-year-old boy, keeping detailed field
notes, while another researcher/videotaper set up and maintained videotape of the mathematics
activities, and took whole-class notes on the children’s reactions to the curriculum and the
software, providing context for the case studies. The researcher/teacher also kept whole class
notes, as did the classroom teacher. Notes and records kept by the classroom teacher were
collected and examined; notes were also kept on interviews with the classroom teacher. All of
these notes and records provided triangulation for this study, as did Tania’s school-wide
screening test scores and pre- and posttest mathematics assessment scores. Further triangulation
was provided by the computer-generated database of records on Tania’s use of the software
program, as she progressed through the number and geometry activities. For approximately 15 to
20 minutes after each field site visit, our on-site research team of four met to reflect on the day’s
work. We discussed the children’s thinking, and evaluated the effect of the curriculum and
software on children’s understandings. Records were kept on curriculum implementation, and
suggestions for print curriculum and software improvements were noted. At times, the classroom
teacher was able to participate in these reflective meetings. Her curriculum suggestions were
also discussed and recorded. Minutes of these sessions were submitted via e-mail to the coinvestigators, and became a reflective record of our study.
In addition to these thrice-weekly meetings, our larger research team, including the
principal investigators and those working at other sites, met once a week. The purpose of these
meetings was to discuss the progress of the research, to study our observations and notes, and to
begin identifying bits of data for coding. Notes were also recorded on these weekly sessions.
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We coded field notes individually before coming to the meetings, assigning a code to every
meaningful bit that we encountered in our field notes. At the weekly meetings, we shared codes,
added codes, eliminated duplicate codes, agreed on code names, and eventually collapsed codes.
We also tested for interrater reliability on coding.
We coded our notes on both inductive and deductive levels. On the inductive level, we
coded the bits of meaning as they arose from the pages and came to us individually as we read,
re-read, and lived with our notes. These codes eventually formed the basis of our agreed-upon,
evolving, open code-list. These open codes developed into categories, and categories developed
into themes.
On the deductive level, we coded our notes based on the hypothesized number and
geometry learning trajectories. We assigned codes to hypothesized, incremental developmental
levels in the following areas of number: counting, comparison (both cardinal and ordinal),
estimation, composition and decomposition, embedded units, adding and subtracting. We also
assigned codes to hypothesized developmental levels in the following areas of geometry: shape,
composition/decomposition, embedded/disembedded units, length, area, pattern, representation,
symmetry, congruence, imagery, and transformations. This data analysis included the
construction of a matrix of data.
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Themes
This paper is organized around two themes: the child’s developing understanding of
attributes (in this case, three straight closed sides) of triangles, and her developing understanding
of triangle as a visual whole, viewed under the lens of connectedness of attributes to the whole.
The child’s understanding of triangle (parts and whole) is examined as her imagistic prototype is
challenged and destabilized by curriculum and software activities involving non-prototypical
triangles and distractors, and exposure to attributes.
Findings
This case study of a four-year-old preschooler’s understanding of the concept of triangle
revealed the parallel growth of two separate schemes for triangle. One scheme was the child’s
“visual triangle,” and the other was a “three-sided figure,” a scheme of a figure with certain
observable attributes. These two separate schemes grew in parallel, the threads of which did not
become interwoven, but did become somewhat connected, gradually. During the course of the
preschool year, while participating in a curriculum that exposed her to a wide variety of
nonprototypical triangles and distractors, the child expanded her visual image of a triangle from
a prototypical closed, three-sided figure with approximately equal sides and a horizontal base to
an visual image that included, inconsistently, nonprototypical triangles. At the same time, the
child’s use of the label “triangle” destabilized to sometimes include non-triangles, in response to
the disequilibrium caused by introduction to these nonprototypical triangles, and introduction to
some attributes of triangles. In terms of the attributes, the child progressed from a lack of
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explicit analytical awareness of the components of a triangle (specifically, in this case, three
straight, closed sides), to an explicit awareness of these attributes. The child used her
understanding of these attributes to sort three-sided figures from figures with varying numbers of
sides, but she did not consistently use these attributes to define a triangle. She gradually began to
make connections between these two separate schemes, but the connections were unstable. This
case study provides a better understanding of one child’s geometric thinking. Evidence is
provided, in a chronology of events from fall to winter to spring, by theme (attributes and visual
whole) to shed light on this child’s developing understanding of “triangle” as she experiences the
software-integrated curriculum.
Fall
Attributes
Tania brought some prior knowledge to the classroom in terms of attributes (in this case,
three straight closed sides) of triangles. She seemed to have an intuitive inner sense of closure, a
sense of the “good form” as described by Vurpillot (1976). Tania applied this sense of the need
for closure early in the school year, as she intuitively closed a shape she was constructing with
sticks, without instruction. She seemed to feel the need to close the triangle, but she did not show
any evidence of understanding that closure is a necessary condition for triangularity.
Tania also exhibited some awareness of straight lines as prior knowledge. She seemed to
have a picture in her mind of what straight lines looked like. She demonstrated “straight” with
her gestures, and she used the word “straight” after teacher modeling.
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October 17
Tania gestured with her hand to indicate that she wanted her teacher to add lines to the
teacher’s drawing of the sun. Mrs. B.: “Tania wants us to add lines to make it a big sun.”
She then asked Tania, “Would those be straight or curved lines?” Tania answered:
“Straight.”
On ten non-consecutive days during the fall months, the curriculum included opportunities for
the children to distinguish between straight lines and curves through tactile, aural and visual
experiences, including experiences with straight lines in relation to 2-D and 3-D shapes. At one
point, Tania responded to a question concerning straight lines or curves by nonverbally moving
her finger down one side of a cut-out paper triangle. On another occasion Tania’s teacher, Mrs.
B., asked the children to close their eyes. She then handed each child a top to a cylinder. Tania
was asked if the shape had straight lines or curves. In response, she put her finger on the edge of
the circle and touched the edge as she turned the circle around in her hand. She was actively
involved, although she did not respond verbally. She appeared to be thinking. The curriculum
provided opportunities and time for these reflective moments. Through her tactile experiences,
and her chance to reflect, she had the opportunity to think about the distinction between straight
lines and curves. The curriculum alternated between number and geometry, returning again and
again to straight lines and curves as it progressed, giving Tania multiple experiences and a
chance to make connections. One activity in the fall involved examining a figure consisting of
three straight lines and one curvy line. Sometimes the discussion involved straight sides;
sometimes it involved straight lines. In terms of shapes, Tania had to make connections between
the words “sides” and “lines,” and sometimes “edges.” At this point in the curriculum, Tania
distinguished between straight lines and curves, but she did not seem to make connections
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between straight sides and triangles, and she did not display any evidence of explicit awareness
of straight lines as attributes of triangles
The attribute of threeness as a condition of a triangle may have been subconsciously part
of Tania’s concept of triangle when she came to prekindergarten, in that she was able to identify
a prototypical triangle on an early screening test given by her classroom teacher. In the fall,
Tania participated in several activities involving three sides as components of a triangle.
Because the curriculum alternated between number and geometry, Tania had opportunities to
apply number to geometry, and geometry to number. Tania worked hard to produce three, and
then applied this knowledge when constructing triangles from parts (sticks, pipe cleaners, coffee
stirrers, playdough, body parts, etc.). In October, Mrs. B. drew a right triangle on the easel during
a whole group session. One of Tania’s peers identified the shape as a triangle. Mrs. B. asked,
“How many lines does this triangle have?” Tania held up three fingers. She counted three sides,
either by subitizing or by nonverbal counting. She represented three separate sides, and
answered the cardinal question “how many” by presenting (producing?) three fingers. In doing
this, she had to identify the sides as separate objects. Her attention was drawn to the idea that a
triangle consists of lines, or sides, and that they can be counted, or at least represented, as
separate objects. Here the curriculum provided Tania with opportunities to make several
connections with other parts of the curriculum.
October 24
Tania made four triangles out of Popsicle sticks, all in a row. Her classmate looked at
them and said that they were triangles. I pointed to the first triangle in the row (on the
left) and asked: “How many sides does that shape have? Tania looked at the triangle,
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briefly. She did not count the sides out loud, and she did not point to them. She looked
at her right hand, which was on her lap. She used her left hand (she is left-handed) to
touch three fingers on her right hand, counting out loud as she did, “One, two three.” She
held up three fingers.
Again, Tania’s attention was drawn to the three sides of the triangle. She used her fingers as an
external representation of her internal concept of the “threeness” of this three-sided figure.
December 17
Tania’s researcher/teacher, Mrs. B., was working with a group of four children, including
Tania, on the large rug where the children gather for their large group activities. Mrs. B.
explained that there were cutout paper shapes on the floor. Mrs. B. instructed the
children to “put a toe on a triangle.” There were no prototypical triangles presented.
Tania listened to Mrs. B., looked around, and then walked over to a chevron. Two
children were already standing on the chevron. Tania left the chevron and went to a
scalene triangle with two very long sides and one very short side. She looked at the
triangle, and then looked down at her hands. She held her fingers together to make a
triangular shape by touching her two index fingers together and her two thumbs together.
She then headed back for the chevron, where there was a conversation going on between
Mrs. B. and some of the children. Mrs. B. invited the children to count the sides of the
chevron. Tania counted to three, out loud, while touching and counting the sides of the
chevron. She counted the two sides of the concave angle as one side. Mrs. B. helped her
to correctly count the four sides of the chevron. “Can it be a triangle if it has four sides?”
Tania: “No.” Mrs. B.: “Okay, find a shape with three sides.”
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Here, Tania’s teacher introduced the strategy of counting sides when discriminating between
shapes. Tania revealed her thinking by walking to and pausing at the scalene triangle as she
looked for a triangle. If she had been thinking about three sides as she looked, this would have
presented her with a dilemma in that the shape had three sides, but did not fit her image of a
triangle. At this point in the school year, Tania’s image of a triangle appears to be a prototypical
image of an “upright” triangle, as described by Hershkowitz et al. (1990, p. 85). She rejected the
scalene triangle and continued to look. She used her hands and fingers as a heuristic to help her
determine if the chevron was a triangle. This led to confusion, in that a “triangle” formed by
joining thumb to thumb, index finger to index finger, looks very much like a four-sided chevron.
It may be that the strategy of counting the number of sides did not occur to Tania at all as she
attempted to respond to her teacher’s request to find a triangle. This episode is presented as an
example of Tania’s exposure to the idea of counting sides to distinguish between shapes, and her
growing awareness of the significance of the sides, as she experiences the act of counting sides.
Visual Whole
Tania brought her prior knowledge and understandings of geometric shapes into the
classroom with her.
October 15
Tania’s teacher handed her a wooden equilateral triangle (part of a wooden puzzle)
while Tania’s eyes were closed. When the teacher asked Tania to open her eyes,
Tania looked at the wooden shape in her hand and said, “Ooo—I have a triangle.”
Tania identified this wooden shape without hearing the word “triangle” from the teacher. This
happened on the first day the class worked with geometric shapes. The word “triangle” had not
been mentioned in the curriculum prior to this occurrence. (Perhaps Tania’s classroom teacher
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had discussed shapes with the children prior to this date.) Possibly Tania had had some other,
outside school experience with triangle shapes. This wooden triangle had the shape that can be
described as the “good” shape. This equilateral or isosceles “upright” triangle represented
Tania’s concept of triangle. Tania held this prototypical “good” triangle shape strongly in her
mind. As she was exposed to, and gained experience with scalene and obtuse triangles, her
concept of “triangle” became conflicted and destabilized. She became less sure of her personal
“definition” of triangle, as she struggled to accept non-equilateral and non-isosceles triangles.
The triangle variants presented by the curriculum caused cognitive conflict between her view of
a triangle, and the exemplars of nonprototypical triangles.
October 24
Tania’s teacher showed the children a large right triangle unit block. She held up a
smaller unit block, also a right triangle. She drew a right triangle on the easel. She
talked about the shape, mentioning three lines. “What shape did I make?” Tania
answered: “Rectangle.”
The right triangle Mrs. B. displayed did not match Tania’s image of a triangle. Tania knew
another word that sounded like triangle, and tried it out as a possible label for this shape.
December 19
Mrs. H. suggested that the children establish families of shapes. She asked the children
to give their rectangles to Martin [pseudonym], so that he could make a “family” of
rectangles. She then told the children to give all their triangles to Tania, and all their
circles to Dexter [pseudonym]. The teacher picked up a scalene triangle. “Is this a
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triangle?” Three children answered, “no,” and one answered “yes.” Tania did not
respond. Mrs. H. “How do we know if it’s a triangle?” In response to this question,
Tania touched the sides of an isosceles triangle and counted the sides as she did, “One,
two, three.” Mrs. H.: “Okay.” She held up an equilateral triangle and counted to three.
Then she held up the scalene triangle and counted to three. She gave the scalene triangle
to Tania to add to her triangle family. Tania held the triangle, looked at it, and said,
“This ain’t no triangle!” Mrs. H. asked Tania, “How many sides does it have?” Tania
counted the sides. “One two, three, but…” She touched one of the vertices. “That don’t
go to me.” She tossed it across the table. She held up the isosceles triangle. “It gotta go
like this.”
Tania rejected a triangle from her “family” of triangles because it did not fit her concept of
“triangle.” She did this despite the fact that she had counted the sides, and there were three. Her
own participation in an tactile attribute “test” did not matter to Tania. Visually, it did not look
like a triangle to her, therefore, to her; it was not a triangle.
December 19
Each child was given a set of nine geometry cards, about the size of playing cards. On
each of the nine cards was a shape, with text identifying the shape appearing under the
shapes. Mrs. B.: “Lay all these cards out in front of you.” Tania lined up her yellow
shape cards, arranging seven of the nine so that the text (the printed name of the shape)
was at the bottom of the card. The eighth card was a triangle with the horizontal base at
the top, and a vertex at the bottom. She rotated the card so that the horizontal base was at
the bottom of the card. The text was upside down.
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Tania preferred prototypical triangles with bases parallel to the table top, in the “upright”
position mentioned by Hershkowitz (1990, p. 85). She was willing to turn the print upside down,
so that she could have the triangle “right side up.”
December19
Teacher: “What are those?” referring to scalene triangles. Tania, “I don’t know.”
December19
A group of six children were sitting at a table with Mrs. H. They were working with
three-dimensional shapes cut from foam board: rectangles, circles, and triangles.
Mrs. H., referring to an obtuse isosceles triangle, asked, “What shape is that, Tania?”
Tania responded, “It’s a house.”
This demonstrates van Hiele’s visual level, when “figures are judged by their appearance….It
looks like one” (van Hiele, 1991).
December 19
Mrs. H. asked Tania to “Point to a triangle.” Tania pointed to an isosceles triangle
without watching to see where anyone else was pointing.
Tania often checked to see what the other children were saying or doing. Tania seemed
sometimes unsure of herself and her answers, and liked to check in with the group before risking
a wrong answer. Occasionally, when she seemed to feel confident, she acted without checking
others’ responses first. In this instance, Tania identified a triangle on her own. Her confidence
was bolstered by the fact that the triangle in question was prototypical.
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Winter
Attributes
During the winter months, Tania continued to have multiple experiences with the
attributes of triangles – three straight closed sides. The curriculum and software offered her
opportunities, through multiple experiences, to explore the attributes of triangles. Tania
continued to apply her own rule of closure to shapes. She consistently rejected unclosed figures
as shapes. She experimented with a different aspect of straight and curved, and she had many
experiences noting sides of shapes. Tania seemed to advance her thinking when she engaged in
reflection, self-talk and individual exploration, using trial and error.
January 23
Mrs. H. gave Tania a large Venn diagram and several wooden unit blocks. She asked
Tania to put the blocks with straight edges in one big circle, and the blocks with curved
lines in the other big circle, and to put blocks with both curved and straight lines in the
overlapping part of the two circles. Mrs. H. then demonstrated a strategy to distinguish
between the blocks. She told her that blocks with curved lines roll, and blocks with
straight lines do not roll. Tania responded by listening, watching, thinking, touching,
testing, trial-and-error, classifying, looking for feedback from peers, private speech, using
her whole body to roll, exploring materials, experimenting (attempting to roll a straightedged block). Tania showed her work, answered pointed questions, rearranged. She
appeared to make connections between her experience with the blocks and prior
experiences with curved and straight. After approximately ten minutes of working alone,
she accurately arranged all the curvilinear blocks on one side of the Venn diagram, and
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23
all the rectilinear blocks on the other. She separated the blocks according to whether or
not they rolled. Tania the summarized her work: she gestured to the blocks that had
curved edges. “This is roll.” She then gestured toward the blocks with straight edges.
“This don’t roll. (She never referred to her two piles of blocks as straight or curved.)
She then celebrated her accomplishment via a dance and exclamations.
Again, the curriculum provided Tania with an opportunity to advance her thinking as she
worked by herself, engaging in self-talk, thinking, and experimenting. Once again, her body
movements and tactile experiences were an integral part of her processing. During this session,
she seemed to teach herself the difference between blocks that roll and blocks that do not roll,
with potential connections to the ideas of straight and curved. Through trial-and-error, practice
and reflection she constructed for herself an understanding of straight and curved, partly based
on the visual, partly based on tactile, and partly based on functional (roll vs. non-roll). Tania’s
periods of reflection sometimes seemed to result in an advancement in her learning, consistent
with Confrey’s premise: “Reflection, as the ‘objectification’ of a construct, functions as the
bootstrap by which the mathematician pulls her/himself up in order to stabilize the current
construction and to obtain the position from which the next construct can be created” (1990, p.
109).
In February, after using the Building Blocks software (Schiller et al., 2003), that
presented scalene triangles as triangles, and after experiences with scalene triangles in the large
and small group activities, Tania began to show evidence of accepting some scalene triangles as
triangles.
Feb 11
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Teacher: “Pick out the biggest triangle in the bag. Reach in.” She held the Feely Bag so
that Tania could reach in and use haptic-perception to discriminate amongst the shapes in
the bag. Tania reached in and then looked in. She pulled out a large scalene triangle.
Teacher: “How many lines?” Tania: “Three.” Teacher: “How does that feel?” Tania ran
her finger along each side of the triangle. Teacher: “Curved or straight?” Tania answers,
“Straight.”
Her teacher guided her in examining the triangle using attributes. Tania used tactile experiences
as well as visual to advance her understanding of “straight.” Again she used the word “straight”
after her teacher modeled it for her.
Feb 22
Exploring on her own, Tania separated some rigid foam shapes into three groups. She
identified the three categories: “One is straight lines, one is curved lines, and one is
pointy. Straight, curved, or pointy corners.”
Rather than two distinctions here, between straight and curved, Tania introduced a third aspect,
“pointy corners.” Tania used her own term to describe the vertices. Tania used her developing
recognition of distinctions between straight and curved (and pointy) to help her to sort geometric
shapes. This time she used the word “straight” on her own, without teacher modeling.
The curriculum provided opportunities for Tania to use her growing ideas about attributes
to distinguish between shapes.
February 22
Tania was playing a game called “Step Shape.” Outlines of shapes were taped to the
floor in her classroom. Her teacher asked the children to step on various shapes. Mrs.
B.: “Find a shape with four sides.” Tania went to a triangle, knelt down, and counted the
Preschooler’s Understanding
25
three sides, out loud, while touching them. “One, two, three.” She got up and walked
away from the triangle and went to a square. She again counted the sides, out loud, while
running her hand along each side. “One, two, three, four. Four!”
Tania distinguished between three-sided shapes and four-sided shapes, at her teacher’s request;
however, she did not use counting sides as a strategy for distinguishing between triangles and
rectangles. She considered the triangle as a four-sided shape, and did not reject it until she
counted the sides. She did not seem to see the “threeness” of the triangle. She did not initiate the
idea of counting sides to distinguish between shapes, but she did successfully count the sides of a
triangle and a rectangle. When counting sides, Tania ran her finger along the entire length of a
side. Her teacher had modeled this earlier, to help children distinguish sides from vertices.
Tania appeared to remember this demonstration, and she put it to use.
February 22
Tania participated in a game called What’s My Rule? Mrs. B. separated blue and red
rigid shapes into two piles, and the children guessed what rule the teacher was using to
separate the shapes. The teacher was using the rule “triangles and non-triangles” to sort
the shapes into two piles. Tania guessed the rule: “Because they’re the same shape.”
Mrs. put a rectangle in the rectangle pile and asked Tania if she agreed with that move.
Tania agreed. Mrs. B. asked for an explanation. Tania: “’Cause it’s got the same things.”
To illustrate her meaning, she reached out and touched the side of the rectangle that her
teacher was holding up.
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Tania appeared to think about the two rectangles in terms of parts that they had in common. Her
comment, “’Cause it’s got the same things” possibly indicated that she was beginning to think in
terms of the attributes of shapes.
March 11
Tania’s teacher spread out several large paper shapes on the floor. She asked six children
in a small group to find a shape with three straight sides. Tania looked over all the
shapes. She moved to a parallelogram. She touched the sides and counted them out loud.
Then she moved to a rectangle and counted all four sides. She then moved to a chevron
and counted the sides. She counted four sides. She then moved to a hexagon. She
counted two sides, touching them as she counted. She stopped. She moved to an
octagon. She started to count the sides and stopped. She went back to the chevron. She
counted sides again and got three. One of Tania’s classmates watched her and said, “Let
me see.” He counted the sides of the chevron. “One, two, three, four.” Tania listened to
her classmate, and then looked at her teacher. Teacher: “We’re going to take a look.”
She told Tania to put the shape down in the area where the other five children were
displaying the shapes they found that had three sides. Tania put her chevron down, and
then covered it up with her hands. “You can’t see mine.” She then picked it up. She put
it down again and looked at it. She watched and listened as the other children in her
group displayed their shapes that had three straight sides. When it was Tania’s turn, she
counted the sides of her chevron out loud and got four. Another child said: “I tried to tell
her.”
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When her teacher asked her to find a shape with three sides, Tania did not go to any of the
triangles. The request to find a three-sided shape did not appear to bring up an image in Tania’s
mind of a triangle. In searching for this three-sided shape, she went to several shapes
(parallelogram, rectangle, chevron, hexagon, and octagon, respectively) and manually counted
the sides, rejecting them as she did so. By this time, the triangles that had been available on the
rug had been picked up by the other children. She returned to the chevron, but she appeared to
be dissatisfied with this choice. This first time she went to the chevron, she counted four sides
and rejected it. When she counted the sides the second time, she made the count conform to
what she was looking for by counting the third and fourth sides as only one side. However, she
seemed to know that there was a problem with it, as evidenced by her covering it up. When her
teacher told Tania to show the group her three-sided shape, Tania was reluctant. She seemed to
be aware that she had chosen a shape that did not have three sides.
During the winter months, the curriculum returned to the topic of straight and curved
sides.
Mar 11
Using shapes cut from rigid foam, Tania and her partner looked for shapes with curved
lines. They found one and handed it to their teacher. Mrs. B. held up the shape they
brought to her. “Is this curved lines?” Tania and her partner simultaneously answered,
“Yes.”
Tania appeared to be able to distinguish between curved lines and straight lines. She does not
indicate that she has made a connection between “straight” and the lines or edges or sides of a
triangle. She does not appear to be aware that straight lines are required as a condition for a
triangle.
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March 18
I asked Tania how she could tell the difference between a triangle and a square. Tania:
“’Cause I seen one line, two lines, three lines.” Using my pen, Tania drew an outline
around the outside of the triangle on my tablet, following the shape of the triangle. Her
outline was rounded on the corners, and a bit wavy on the sides, but basically her lines
along the sides were straight. “And a triangle has four. Watch. One, two, three, four.”
As she said these words out loud, she drew an outline around the outside of the square,
rounding the corners and not quite closing the figure. She did it again, framing the square
twice in all.
In this brief exchange, Tania showed evidence of linking the word “triangle” with three sides.
This time, she responded to the challenge of distinguishing one shape from another by coming up
with the strategy, on her own, of counting sides. Also, she was able to recognize what she had
done, and she was able to explain it explicitly, to “prove” her point. She appeared to be able to
distinguish between shapes based on parts. She was also able to represent her images of triangle
and square on paper by tracing. She continued to make mistakes on the naming, or labeling of
the shapes.
Winter
Visual Whole
During the winter months (January, February, and March) Tania showed evidence of a
beginning acceptance of some non-prototypical triangles as triangles, and more evidence of
instability in her overall concept of triangle.
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29
January 7
Paper shapes wer distributed on the floor. Five children were playing the Shape Sort
game. Teacher: “Now everybody go get a triangle and bring it back to me. The five
children dispersed and each one brings back a triangle to Mrs. B. Tania picked up a right
triangle and brings it back within three seconds. She did not stop to visibly count the
sides. She displayed it on the rug in front of her. The children gathered in a small circle
again to observe all the triangles they had collected. Each of the triangles displayed was
different. Mrs. B. lined up the five triangles in a row. Tania reached out and turned her
right triangle so that the base was horizontal to her own waistline.
In this instance, Tania accepted a right triangle as a triangle, even though it did not match her
original prototypical image.
January 23
Tania’s teacher provided her with pencil and paper, and asked her to draw something she
could make with blocks. Tania drew an isosceles triangle in her math journal, and
described her drawing as a triangle.
February 11
Tania’s teacher pulled a long, skinny scalene triangle out of her bag and said, “Tania,
what is this? Tania did not visibly count sides. She responded, “Triangle.” Teacher:
“How did you know?” Tania: “’Cause I seen it down.” She repeats for me, “’Cause I
seen it down.”
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In this instance, Tania accepted the same rigid foam scalene triangle that she rejected in
December. (“This ain’t no triangle.”) Her rationale for accepting this triangle as a triangle, that
is, “’Cause I seen it down” did not have clear meaning for me. Perhaps she meant the she has
seen a shape like this before. Since December, she continued to have experiences with on- and
off-computer activities involving non-prototypical triangles. She does not show evidence of
basing her acceptance of this shape on its attributes.
February 11
Tania reached in the Feely Bag, looked in the bag, and pulled out a large right triangle.
She then matched the triangle she pulled out with a small equilateral triangle that was
already out of the bag.
Tania matched one triangle to another very different triangle. She seemed to accept as “triangles”
two shapes that did not look alike. This seemed to indicate advancement in Tania’s geometric
thinking.
February 11
Tania identified wooden pattern block shapes as I held them in front of her. She called a
hexagon a hexagon, and a square a square, and a triangle a square.
Tania continues to use other shape words to label triangles.
Feb 20
Tania was playing a geometric shape matching game on the computer. She was
presented with two sets of cards, all face down. Tania turned over two cards; one
depicted a scalene triangle, and one an isosceles triangle. I asked: “What are those
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31
shapes?” Tania pointed to the scalene triangle and said, “Looks like a triangle.” She
pointed to the isosceles triangle and said, “And triangle.” She matched them together.
Twice Tania used the phrase “It looks like a triangle” when referring to a scalene triangle. This
seemed to indicate that she was not completely ready to accept scalene triangles as triangles.
Again, Tania matched two different types of triangles.
Two days after this experience, Tania was playing the What’s My Rule? game. She
matched up two triangles that were not the same size and not the same shape, accepting both as
triangles. One was an equilateral triangle and one was an isosceles triangle. The software and
print curriculum seemed to be having the effect of broadening Tania’s concept of triangle.
Mar 18
While Tania was drawing a picture of her family, I drew a square and a triangle on my
tablet. I asked Tania, “What are these shapes?” She pointed to the triangle and said,
“Triangle.” She pointed to the square and said, “Square.” I asked, “How do you know?”
She replied, “If I turn the triangle upside down, it’s a pizza. If you turn the square upside
down, it’s a diamond. “ I asked, “Upside down?” Tania: “Yeah. No, I mean this way.”
She tilted the tablet.
Orientation of shapes seemed to be important to Tania. Her prototypical image of a triangle was
strong. She wanted triangles to have horizontal bases, and she did not consider a square to be a
square when it was tilted.
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32
Spring
Attributes
By the end of the school year, Tania seemed to be more aware of attributes than at the
beginning of the year. In relation to the attribute “closed,” Tania advanced from an intuitive,
implicit awareness to an explicit awareness, as evidenced by her answer (see May 8 below). In
relation to straight sides, although she distinguished between straight and curved, she still did not
show evidence that she considered straight lines to be an essential part of a triangle. In relation
to thee sides, two vignettes illustrate the advances in Tania’s thinking.
April 15
Mrs. B. held up a wooden isosceles triangle from a set of pattern blocks for Tania to see.
Tania said to her teacher: “Three sides.” Mrs. B. asked the other children, “How did she
know?” In response, Tania held up three fingers.
Early in the school year, Tania referred to a wooden equilateral triangle as a triangle. At
this point, she looked at a triangular shape and described it as “three sides.”
April 17
Tania was working on a paper shape puzzle – a giraffe. “This ain’t hard for me. This
ain’t hard for me. Anything is not hard for me.” She worked on the puzzle. She tried to
put a third trapezoid in a space meant for a triangle, but it wouldn’t fit. She tried to make
it bend. Then she tried a rhombus instead. She took it back out and tried the trapezoid
again. Tania to Mrs. B.: “Can you hand me a green triangle?” Mrs. B. observed that the
trapezoid did not fit, and said to Tania, “Let’s look at it. This one is sliding off the edge.”
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The teacher gestured toward the trapezoid as she spoke. Tania took the trapezoid away.
She counted the sides of the puzzle frame where the trapezoid had been, and then
replaced the trapezoid with a triangle. The teacher said, “How did you know you needed
a triangle?” Tania: “I counted the sides.” Teacher: “How many?” Tania: “Three.”
At this point, Tania seemed to have made some connections. She was initiating on her own a
strategy of counting sides, and she was able to explain what she was thinking. More
significantly, this time she did not just count the sides of the triangle, but rather, she counted the
sides of the frame that partially outlined the shape of a triangle. This advancement was made
while Tania worked with a triangle that matched her prototype – a pattern block triangle. The
use of both prototypical shapes that fit together, such as pattern blocks, and the use of shape sets
consisting of some nonprototypical triangles, seemed to provide opportunities for Tania to
advance in her geometric thinking.
May 8
By the end of the year, Tania exhibited an explicit awareness of closure as a necessary
condition of a triangle . Tania a drawing of a curved, unclosed figure, and a three-sided
unclosed figure. Teacher, referring to a drawing of a curved, unclosed figure, and a three
sided unclosed figure: “Are these shapes?” Tania: “No.” R: “How can you tell?” Tania:
“’Cause this don’t got another part (she points to the circular figure), and this don’t got
another part (she points to the three-sided figure.)
Tania’s statement indicates that she has reached a point where she can be explicit about an
attribute, and she understands that the attribute is conditional to the shape.
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Visual Whole
In relation to Tania’s thinking about triangle as visual wholes, the evidence revealed a
broadening concept, perhaps over-widening, with a great deal of instability. A series of brief
vignettes illustrate this.
April 8
Tania drew a triangular figure on my tablet, a large, tall isosceles triangle with a
horizontal base. She asked me if I knew what shape it was. I responded by asking her if
she knew what shape it was. She answered, “A triangle.”
Tania represented her image of triangle using pen and paper.
April 10
In response to a computer game, Shape Puzzles, which gave the command: “Click on the
triangle,” Tania did.
Tania was successful in participating in the geometry activities on the computer. She was able to
match up different types of triangles.
April 12
Tania was using the geoboard with rubber bands. Researcher: “Can you make a
triangle?” Tania responded by taking all the rubber bands off the geoboard. Then she
put a rubber band in the corner of the geoboard and constructed a small square.
April 12
I constructed a scalene triangle on a geoboard, and asked Tania what shape I made.
Tania looked at the scalene triangle and replied, “I don’t know.”
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35
April 17
Tania was using wooden pattern blocks to solve a puzzle shaped as a wagon. Her teacher
pointed out that Tania used trapezoids and triangles to solve the puzzle. Tania attempted
to correct her teacher: “Squares.” Teacher: “Triangles.” Tania: “Triangles.”
April 19
In reading a pattern consisting of repeating sequences of three shapes (hexagon, square,
triangle), Tania was able to identify the triangle. She read from right to left, triangle,
square, (pause), hexagon.
April 19
Tania’s teacher was working with the children in a large group setting. She showed the
children a few large geometric shapes cut from rigid foam board, including a scalene
triangle. Then she took the scalene triangle away. She asked the group of children,
“What shape did I take away?” Tania answered, by herself, “Triangle.”
This time Tania answered without the preface: “It looks like a….” Her thinking seemed to be
moving in the direction of broadening her definition of triangle, and perhaps recognizing that
triangles do not all look the same -- some triangles look very different from other triangles.
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May 8
Tania’s teacher drew a triangle on the easel chart paper, with a rotated base (30°). and
then asked what shape it was. Tania raised her hand, and Mrs. B. called on her. “Um,
one, two, three.”
Again, Tania seemed to be substituting the words “One, two, three” for the word “triangle.” She
has begun to construct the knowledge that the number of sides have something to do with shapes,
and that triangles and “three” are connected.
May 8
I showed Tania a drawing of a scalene triangle. “Is this a triangle?” Tania, “No.”
During the course of the academic year, Tania rejected scalene triangles as triangles, and then
accepted some scalene triangles as triangles. At the end of the year, she again rejected this
particular scalene triangle, again revealing instability in her geometric thinking about various
kinds of triangles.
May 13
The teacher showed the children some shapes cut from rigid foam. “The shapes are the
same except by size.” She gave a foam triangle to Tania, and told her to reach in the
Feely Box and try to touch, feel, and pull out of the box a shape that was the same shape
as the triangle, but a different size. Tania reached in and pulled out a triangle. Teacher:
“What’s this?” Tania: “A square.”
Again, these three words, triangle, rectangle and square, seemed somewhat interchangeable for
Tania.
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May 17
The teacher demonstrated the construction of an isosceles triangle with straws and pipe
cleaners. “Now what does it look like?” Tania: “A triangle.”
May 20
There were several shapes and some unclosed figures formed with yellow tape on the rug
in the classroom. Teacher: “Tiptoe on a triangle.” Tania walked over to a triangle and
put her toe on it.
Tania is inconsistent and unstable in her use of the word triangle to describe triangles.
May 31
The children played a game, Musical Shapes, similar to Musical Chairs. The children
had to stand on the outline of a shape taped to the floor when the music stopped. Tania
stood inside the outline of a triangle. Teacher: “Is anyone in a triangle?” Tania: “Me.”
Her teacher pointed to a figure consisting of three lines, unclosed. “Is this a shape?”
Tania walked over, knelt down, traced the lines with her finger and answered, “No.”
May 31
Tania was playing a game, sorting rigid foam shapes into shapes with three sides and four
sides. Teacher: “What do you have left in the bag?” Tania looked at an object in the
bag; she did not pull the object out. She says, “Square.” She looked in the bag again,
corrected herself and said, “Triangle.” She shook the shape out of the bag. “It’s a
triangle.”
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38
June 4
In June, Tania was given a posttest, which was an exact version of the instrument used for
pre-testing in September. This was a paper and pencil (actually crayon) assessment administered
on an individual basis, over two sessions (one for geometry, and one for number). This
assessment included an 8 _ inch by 11 inch sheet of paper, on which 17 non-prototypical triangle
exemplars and distractors appeared. There were nine triangles on the page; none were
prototypical with a horizontal base. The instructions were to “Draw a mark on each shape that is
a triangle.” In September, Tania marked only one figure on the page, a figure with curved sides
and a straight base, rotated approximately 30°. All three sides, although not straight, appeared to
be about the same length, and the rotated base was fairly close to a horizontal base, rotated 0°
(cf. Clements, 2001). Tania did not mark a right triangle with a horizontal base, nor an
equilateral triangle with a vertex at the base. This is consistent with her notion of a triangle as a
figure with approximately equal sides and a horizontal base.
In June, Tania marked six figures, four of which were triangles. The two distractors she
marked had straight, horizontal bases, and concave sides. All three sides for both distractors
appeared to be approximately equal in length. She did not apply the constraint of straight lines to
her definition of triangle, despite her ability to discriminate between straight lines and curves. All
the figures she marked were closed. She was not distracted by unclosed figures, although no
unclosed figure came close to the prototypical equilateral, horizontally based figure Tania seems
to hold as a prototypical triangle. She left five triangles unmarked. She did not observably apply
the strategy of counting straight sides to discriminate between triangles and non-triangles. This
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assessment did not provide an opportunity to assess Tania’s ability to discriminate between
triangles and quadrilaterals and other polygons by applying the strategy of counting three sides.
Discussion
Tania’s understandings of “triangle” can be analyzed using four frameworks: Piaget’s
stages of cognitive development; Vygotsky’s work on concept formation; the van Hiele levels of
geometric thinking; and the hypothesized geometrical shape learning trajectory developed by
Clements and Sarama (unpublished).
Piaget gives us a framework from which to begin to analyze Tania’s geometric thinking.
Piaget places Tania’s cognitive thinking on the preoperational level. Fuys et al. (1979)
summarize this level as one in which:
[C]hildren develop their ability to represent or symbolize things that they have physically
experienced, using objects, pictures, and words. Children also exhibit first signs of
reasoning, sometimes drawing conclusions from what has occurred in the past…This
reasoning from one particular situation to another like it is typically based on the
children’s past experience. During this preoperational period, children also begin to
develop intuitive thinking that is based on their perception: the interpretation they attach
to their physical experiences of touching, moving, feeling, seeing, and hearing. Often a
child’s perception of something will be quite different from an adult’s. (p. 33)
Piaget and Inhelder (1967) help us to understand Tania’s process of understanding of the
concept of triangle by elaborating on stages in the recognition of shapes. Their description of the
tactile exploration during the later phases of Stage II seems to describe Tania. This tactile
exploration leads to ”progressive differentiation of shapes according to their angles and even
their dimensions (circle and ellipse or square and rectangles)” (p. 21). Tania often used her arms,
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40
her hands, her fingers, her torso, and her whole body to explore ideas, demonstrate ideas, and to
test her understandings. She actively and repeatedly manipulated triangles. Her tactile sense
enhanced her exploration, and enabled her to construct her own idea of a triangle through the
coordination and organization, in her mind, of her actions. Three straight sides of triangles were
physical experiences for her, as were vertices. As the mathematics dialogic in the classroom
centered around straight sides, Tania was able to use her sense of touch to experience these
triangle attributes, using a variety of three-dimensional objects. This ability to touch the shapes
added to Tania’s visual experiences with triangles.
Vygotsky (1962) sheds some light on concept formation. He discusses the processes
leading to concept formation and describes two major phases: the unorganized “heap” (p. 59),
and the second phase, “thinking in complexes… In a complex, individual objects are united in
the child’s mind not only by his subjective impressions but also by bonds actually existing
between these objects” (p. 61). The last level of the complexes stage is a complex called the
pseudo-concept.
Pseudo-concepts predominate over all other complexes in the preschool child’s thinking for
the simple reason that in real life complexes corresponding to word meanings are not
spontaneously developed by the child: The lines along which a complex develops are
predetermined by the meaning a given word already has in the language of adults. (p. 67)
Along the lines of Vygotsky’s pseudo-concepts, Hershkowitz’s writes of a “concept image”
or prototype example of concepts. “The prototype’s irrelevant attributes usually have strong
visual characteristics, and therefore they are attained first and then act as distracters” (1990, p.
83).
Preschooler’s Understanding
41
Tania’s understanding of triangle can be better understood in light of the construct of these
pseudo-concepts, or concept images. The word “triangle” does have meaning in the language of
adults; the meaning is constructed by the child somewhat within the constraints of adult
meaning. The child develops an early prototype based on the cultural meaning, with all its
irrelevant attributes, and then these early attributes sometimes must be unlearned through a
process of destabilization of the pseudo-concept.
In terms of the van Hiele levels, the first level, the visual level, described the state Tania was
in at the beginning of the year. She judged a triangle by its appearance. If the exemplar “looked
like” Tania’s imagistic prototype, her pseudo-concept, then it was a triangle. She recognized it
with “nonverbal thinking” (van Hiele, 1991), and described the shape as a visual whole (it
looked like an “A,” it looked like a “house,” it looked like a “hat,” it looked like a “pizza”).
The next level, according to the van Hiele (1991), is the descriptive, analytic level,
characterized by a recognition of the parts, or integral attributes of shapes, isolated rather than in
relationships. “[T]hrough observation and experimentation students begin to discern the
characteristics of figures. These emerging properties are then used to conceptualize classes of
shapes. Thus figures are recognized as having parts and are recognized by their
parts….Relationships between properties, however, cannot yet be explained by students at this
level, interrelationships between figures are still not seen, and definitions are not yet
understood.” (Crowley, 1987, p.2).
This second level is further described by van Hiele (1991):
A figure is no longer judged because ‘it looks like one’ but rather because it has certain
properties. For example, an equilateral triangle has such properties as three sides; all sides
equal; three equal angles; and symmetry, both about a line and rotational. (p. 311)
Preschooler’s Understanding
42
This description also begins to fit Tania in that she was able to determine, during the course of
her prekindergarten year, that a shape has a certain number of sides, and that, specifically, a
triangles has three sides. Tania demonstrated an understanding of sides as parts at the same time
that she was grappling with the visual whole. Her visual, nonverbal thinking became disrupted
during the school year as her prototypical visual image was challenged by the introduction of
nonprototypical triangles. Tania’s understanding of “triangle” seemed to operate on more than
one level of geometric thought. Sometimes she saw the triangle as a visual gestalt, and
sometimes she seemed to recognize a triangle by some of its parts. However, she seemed unable
to combine these two developing schemes.
Clements and Battista (1992) suggested a syncretic level as a replacement for van Hiele’s
first level, visualization. This syncretic level combines two levels of geometric thinking that
seem to co-exist – visual and the analytic. They expand on this idea in a later publication, based
on further work with young children and their geometric thinking:
[T]he results support a reconceptualization of van Hiele Level 1. The high proportion of
visual responses was in line with theoretical predictions. However, among these young
children there is also evidence of recognition of components and properties of shapes,
although these features may not be clearly defined (e.g., sides and corners). Some
children appear to use both matching to a visual prototype (via feature analysis) and
reasoning about components and properties to solve these selection tasks. Thus…we
provide evidence that Level 1 geometric thinking as proposed by the van Hieles is more
syncretic than visual….That is, this level is a synthesis of verbal declarative and imagistic
knowledge, each interacting with and enhancing the other. Thus, we suggest the term
syncretic level, instead of visual level, signifying a global combination without analysis
Preschooler’s Understanding
43
(e.g., analysis of the specific components and properties of figures). At the syncretic
level, children more easily use declarative knowledge to explain why a particular figure is
not a member of a class because the contrast between the figure and the visual prototype
provokes descriptions of differences (Gibson, 1985). Children making the transition to
the next level sometimes experiences conflict between the two parts of the combination
(prototype matching vs. component and property analysis), leading to incorrect and
inconsistent task performance. (Clements, et al., 1999, p. 206)
Tania has not yet reached this syncretic level. She does not easily use declarative
knowledge to explain why a particular figure is or is not a triangle. She does not reason about
shape components. She has not syncretized these two levels. Not only did Tania experience
conflict between the prototype and the attributes, she experienced a separation too. Rather than a
syncretization of the levels, Tania is operating with two separate schemes: she has a visual
scheme for triangle, and a scheme that recognizes some attributes of a triangle; these attributes
are only gradually being used as a basis to define the whole.
Clements et al. (1999) shed further light on Tania’s geometric thinking about triangles as
she progresses toward this syncretic level. This description seems to fit Tania’s over-widening
scheme which accepts non-triangular forms. The connection between more variance and less
consistency for triangles rings true.
“We propose that children are developing stronger imagistic prototypes and gradually
gaining verbal declarative knowledge….[R]ecognition of shapes such as triangles, the
least definable by imagistic prototypes of those we studied, may show complex patterns
of development while the schema widens to accept more forms, over-widens, and then
must be further constrained. Supporting this theory is evidence that the largest internal
Preschooler’s Understanding
44
consistency was for those shapes with less visual and property variance within the class
(circles and squares); the more variance (from rectangles to triangles) the less internal
consistency we found. (p. 207)
Support for this syncretic level can be interpreted from the work of Beilin, (1984) who
writes that researchers who tested Piaget’s theories on geometry found that “the developmental
order reported by Piaget was preserved, but children identified at the formal operation level still
showed many characteristics of concrete operational thinking” (p. 54), and that children’s
“performance tended to distribute over more than one stage level (p. 56).
Lehrer et al. (1998) lend some support to this position, stating: “The children’s thinking
about shape can be characterized as appearance based…However, children distinguished among
many different features of form. And the nature of the distinctions children made varied greatly
with the contrast set involved in the similarity judgment” (p. 145).
Hershkowitz also considers a mixture of the two processes. “There is some evidence that
the construction of the concept image is a mixture of visual and analytical processes. For
example, subjects’ behavior changed from one concept to the other: Students and teachers who
showed analytical behavior in a quadrilateral task failed to identify nonprototypical right
triangles” (1990, p. 84).
Tania’s developing concept of shape was also examined using the Building Blocks Learning
Trajectories (Clements & Sarama, 2001, unpublished). The hypothesized learning trajectory
provides a tool to understand where a child is on his/her understanding of number and geometric
concepts, and then to anticipate the child’s next steps. The trajectory most applicable to this
particular study is the trajectory on Geometry/Shapes, which designates twenty-three levels of
understanding of shape concepts for children between the ages of 2- and 8-years-old. From
Preschooler’s Understanding
45
observations and recordings of Tania’s actions and words, particularly from her answers to
reflective questions posed as part of the implementation of the curriculum, an analysis of Tania’s
thinking in relation to this trajectory was possible. During the course of the year it was evident,
from her actions and speech during on and off computer activities, that she moved through some
stages of “Shape Matcher.” She progressed from being able to match prototypical equilateral
and isosceles triangles with same size and same orientation to matching same-shaped, different
sized triangles, some of which had different orientations (“Shape Matcher-4”). For much of the
year, the “Shape Prototype Recognizer and Identifier” (Recognizes and names prototypical
circle, square, and, to lesser extent, triangle. Accepts many variants and distractors) (Clements
& Sarama, 2001a, unpublished), described Tania’s thinking. Although she tended to preserve and
favor her imagistic prototype, Tania’s repeated and varied experiences with a variety of triangle
exemplars, including scalene triangles, and the dialogue accompanying such exposure, resulted
in Tania’s progression to higher levels of thinking on the shape learning trajectory. During the
course of the year, Tania could sometimes be identified by the designation, “Shape Recognizer Basic 1,” indicating that she “recognizes some nonprototypical …triangles and, may recognize
some rectangles, horizontal, and vertical lines, but not rhombi. Often doesn’t differentiate
sides/corners” (Clements and Sarama, 2001a, unpublished). At times during the school year, as
evidenced during Tania’s multiple experiences touching, counting and dialoguing about the sides
of triangles, Tania’s actions revealed that she was able to separate sides as distinct geometric
objects, identifying her as an emerging “Side Recognizer” (Clements & Sarama, 2001a,
unpublished). She did not reach the higher level of “Parts of Shapes Identifier” (Identifies
shapes in terms of their components; e.g., That’s a triangle because it has 3 sides and 3 angles).
Preschooler’s Understanding
46
The presence of two separate schemes, not yet interwoven, indicates why this designation would
not yet be descriptive of Tania.
Attributes
At the beginning of the year Tania was not consciously aware that triangles had parts. A
statement by the classroom teacher corroborates this. "In the beginning Tania had no
understanding that shapes had properties…” (Classroom teacher, personal communication, June
17, 2002). She viewed a triangle not in terms of it attributes, but as a visual whole (Crowley,
1987, Van Hiele, 1991). During the course of the school year, Tania began to become more
consciously aware of the components, or attributes of a triangle. She could easily count three
sides, and she could look for three sides to determine if the shape was a three-sided figure. But,
if the sides she encountered were longer or shorter than the other lines, and if angles were very
acute, or very obtuse, she only occasionally identified this shape as a triangle, even if it had three
straight closed sides. At first, she did not necessarily see these sides as attributes of triangles.
Rather she approached the sides as something to count. The sides were related to physical
action, often involving Tania’s sense of touch. Only gradually did she seem to begin to
understand that if a figure had three sides, and other conditions were met (e.g., closed, straight
sides), then she could use this information to distinguish between triangles and other polygons.
Slowly she began to understand that counting three sides gave her a tool to identify the large
varieties of shapes that are triangles.
At times she seemed to relate the number of sides to the word triangle – and to the threedimensional and two-dimensional presentations of a triangle. She began to substitute the words
“one, two, three” for the word “triangle.”
Preschooler’s Understanding
47
Although Tania learned to distinguish between straight sides and curved sides, and between
objects that roll and objects that do not roll, she did not develop an understanding that triangles
must have straight sides. To Tania, triangles could have curved sides. She was exposed to the
idea that the three sides of a triangle have to be straight, but she did not explicitly apply this
attribute to her own developing concept of triangle.
In contrast, Tania did seem to understand that the sides of a triangle must be closed, and that
if they were not closed, the shape was not a triangle. For Tania, the idea of closed shapes came
intuitively. For Tania, topological aspects of triangles seemed to came more easily, and earlier,
than Euclidean and projective aspects (Piaget & Inhelder, 1967; Piaget, Inhelder, & Szeminska,
1960). Hershkowitz: “According to Piaget’s theory, the child’s early transformations are those
that conserve topological attributes of objects (e.g., interior and exterior of a set, boundary of a
set, connectedness, and openness and closedness of curves) (Hershkowitz, 1990, p. 72).
Tania started the school year with a prototypical concept of triangle (symmetrical, three-sided,
closed figures with horizontal bases). During the year she developed a growing understanding of
the attributes of triangles; she slowly began to make the connections between the analytical cues
and the shapes, allowing her to make distinctions between triangles and non-triangles, and
prototypical and nonprototypical triangles.
Visual Whole
When Tania came to preschool, she had prior knowledge that there was such a thing as a
“triangle,” and that the word named a shape. She was able to identify a triangle as such when
first handed one early in the year. This is consistent with research indicating that most children
can identify some basic shapes by three-years-old (Fuson & Murray 1978; Clements et al.,
Preschooler’s Understanding
48
1999). Tania seemed to make this identification based on the visual whole. This resonates with
van Hieles’ first level.
Tania seem to attach two meanings to the word “triangle.” The first meaning was that of
an isosceles or equilateral triangle with a horizontal base. Intermittently throughout the year,
when she was faced with a triangle that met her prototypical image of a triangle, she referred to
that shape as triangle. The second meaning was one she attached to any rectilinear shape, after
her original, underdeveloped definition of a triangle became destabilized, setting the stage for
growth. Tania had been set on her visual image of a triangle. When we introduced scalene
triangles to Tania, she did not at first put them in the same class as triangles. She seemed to put
scalene triangles in their own category. In order to make sense to Tania, this category should
have had a name to distinguish it. Because her teachers called this second class of shape a
triangle also, she seemed to make the decision that the shape words (square, triangle, rectangle)
could be somewhat arbitrarily assigned to rectilinear shapes. She then used any shape word that
described rectilinear shapes. Throughout the year, Tania identified a triangle as a triangle or a
rectangle or a square. She even used the word “triangle” to refer to hexagons. She sometimes
used these words interchangeably. The word “triangle” to Tania seemed to be a word that named
a non-curvilinear shape. She did not use the words triangle and circle interchangeably, nor did
she use the words triangle and oval interchangeably. She did seem to distinguish between
rectilinear and curvilinear shapes (Piaget and Inhelder, 1967). Tania and her teachers had
different meanings for the same word, “triangle” (Gravemeijer, 1998).
Throughout the school year, Tania showed instability, but gradual progress, in her ability
to discriminate between prototypical and nonprototypical triangles as she was exposed to several
exemplars that caused her to over-widened her concept.
Preschooler’s Understanding
49
Consistent with our suggestion that children possess multiple types of geometric
knowledge, we suggest that the children’s knowledge of geometry might be enhanced in
different ways. First, their imagistic prototypes might be vastly elaborated by the
presentation of a variety of exemplars, through the systematic variation of irrelevant and
relevant attributes… (Clements, 2001, p. 127)
As teachers and peers used the word “triangle” to describe non-canonical triangles that to her
were not triangles, she seemed to lose confidence in her own use of the word. Her concept of
triangle seemed to destabilize as she attempted to accommodate her understanding of “triangle”
to these new shapes. As her confidence in her own meaning ebbed, she began to use other
polygon terms she knew to describe triangles. She began to refer to squares as triangles,
rectangles as triangles, and even hexagons as triangles. As her confidence returned, as a result of
multiple experiences with scalene triangles, she gradually became able to use the term with
confidence.
At first, she accepted non-prototypical shapes as triangles, inconsistently and reluctantly.
Twice she referred to scalene triangles as “Looks like a triangle” contrasted with “triangle.” In
identifying triangles during the school year, she was sometimes able to operate as if orientation
was not an essential property of a triangle (Hannibal & Clements, 2000).
Tania’s ability to distinguish between triangle variants, and her ability to accept these
variants as triangles is dependent, in part, on her ability to detect which features are invariant
(Gibson, 1962). Exposure to these triangle variants, as opposed to exposure strictly to visual
prototypes, may result in more flexible thinking.
Tania encountered differently shaped triangles one at a time, and then worked to identify
them as triangles or not, individually. She was exposed to equilateral, scalene and isosceles
Preschooler’s Understanding
50
triangles with obtuse, right and acute angles. She gradually accepted some scalene triangles, as
she began to develop the broad understanding that there are many different kinds of triangles,
that they are all called “triangle,” and that they all have three straight, closed sides. She did not
test a polygon by applying three rules, or three “testers” – three straight, closed sides. Tania did
indicate her awareness of vertices by touching the “pointy parts” and by referring to “pointy”
parts. She did not look for or count three vertices in triangles. She did not develop an
understanding that more than one condition had to be met at the same time in order for a shape to
be a triangle. She did not apply three conditions (three, straight, closed sides) to a rectilinear
shape to distinguish it from other rectilinear shapes, although at times she did this to distinguish
between the number of sides of polygons. She did not seem to develop an awareness of more
than one set of attributes at a time. Fuys and his colleagues (1988) quote Piaget on this:
Preoperational children use their perceptions to make judgments about shape. However, their
perceptions can fool them and lead them to make incorrect judgements. …This reflects the
preoperational child’s general inability to consider two or more characteristics of a situation
at the same time….[T]his inability affects children’s work with class inclusion relations. (p.
33)
During the winter, the meaning Tania attached to the word triangle did not seem to have to
do with a shape with three sides; rather, it seemed to have to do with a closed shape that was not
curvilinear. She was clear on circles in October, and clear on ellipses without instruction on
ellipses. (She seemed to have some prior knowledge of ellipses.) One meaning Tania attached to
the word triangle had to do with her prototypical image. As her concept destabilized, she applied
a second meaning to her idea of “triangle.” The second meaning did not have to do with the
particular shape (nor its attributes). It seemed to have to do with shapes that did not have curves.
Preschooler’s Understanding
51
More specifically, she seemed to use the label triangle to refer to triangles, squares or rectangles
(and hexagons).
During her preschool year, Tania had an understanding of “circle.” The circle did not
vary in shape. It did not vary in orientation. It varied only in size. Tania was able to distinguish
circles from ovals. For Tania, an oval was different than a circle, and it had a different name.
Tania may understand ovals to be variants of circles, in that they both have curves. If so, these
variants have a different name. She seemed to want this for triangles.
The square did not vary in shape. It varied in size, and in orientation. To Tania, a square
is not a rectangle; it is a separate class. For Tania, a square was a square only when it was in a
certain orientation. When it was presented as a tilted square, Tania indicated that it would be a
square if it was turned to an orientation that matched her image of a square, with the top and
bottom lines horizontal to her waistline.
For Tania, in the fall of her preschool year, a “triangle” was as invariant as a circle and a
square. For Tania, a triangle had no variants. It had a triangular appearance with sides that were
approximately equal in length, and it had a base that was horizontal in relation to Tania’s
waistline. The triangular appearance was enhanced by the shape having three sides, but for
Tania, having three sides did not mean that the shape was a triangle. In the fall, if the three sides
were not approximately equal in length, for Tania, it was not a triangle. It was something else
that needed another name. Also, for Tania, a triangle had closed sides (not necessarily straight).
The angles were acute, not right or obtuse. The orientation of a triangle, a non-attribute, was for
Tania, an important attribute. For Tania, triangle variants (variants from equilateral or isosceles
triangles with approximately equal side lengths), such as scalene triangles, were not triangles.
She admitted that they were “like triangles,” but she reserved the word triangle for her own
Preschooler’s Understanding
52
prototypical invariant triangle image. It seems as if she would have liked to have another word
to apply to these other shapes, just as there is another word to use for shapes with curved lines,
like circles, that are not circles (ovals).
When the teaching staff did not supply another word for non-prototypical triangles, and
indeed, when the teaching staff applied the word “triangle” to non-prototypical triangles – shapes
that were to Tania, non-triangles, her concept of triangle destabilized. She began not to trust her
formerly trustworthy word “triangle.” She began to apply it, somewhat arbitrarily – to other
rectilinear shapes. At times she called squares triangles and triangles squares. She called
rectangles triangles and triangles rectangles. She even called hexagons triangles. She aimed to
please the teacher, and therefore, she tried to go along with the teacher when the teacher wanted
Tania to use the word “triangle” to describe shapes that, to Tania, were not triangles. Tania
slowly began to give up her strong prototypical view of triangle. Counting sides seemed to be a
separate activity that at first did not necessarily distinguish triangles from other shapes. To
Tania, it distinguished 4-sided shapes from 3-sided shapes. Slowly she began to make the
connection that counting sides had to with proving that a figure was a certain shape. Tania’s
developing understanding of the attributes of triangles brought about cognitive conflict for her.
During the course of the school year, her concept of triangle became unstable. Her strong
prototypical idea of a triangle conflicted with new information and understandings about
triangles, creating a disequilibrium that set the stage for deeper, more complex understandings.
These understandings led Tania to a richer, more abstract concept of triangle that allowed for
more variants, and opened the door to analytical as well visual perspectives.
This four-year-old child seemed to be dealing with two separate schemes. She saw the
triangle in its wholeness and likened it to other objects of similar shape. At the same time she
Preschooler’s Understanding
53
was beginning to analyze the attributes of the triangle. She slowly and gradually began to make
connections between these components of a triangle and their relation to the whole.
Implications
This case study has implications for the classroom, in terms of mathematical curricula and
methods. Understanding that young children can develop understandings of attributes, and that
through exposure to a variety of exemplars, children gradually expand their understanding of the
large and varied class of triangles, and eventually apply their knowledge of attributes to these
exemplars, may enhance preschool teaching and learning. A curriculum that offers substantial
geometry, integrated with number, offering repeated, varied opportunities for reflection, a deep
exploration of the concepts, and supporting a wide variety of non-prototypical shape exemplars,
will support the growth and mathematical development of preschoolers whose thinking differs
from that of older children and adults. The combination of on- and off-computer activities
provides opportunities to make connections, and to reinforce and strengthen her ideas. The
finding that two parallel schemes developed simultaneously stimulates interest. The possibility
that scalene triangles may represent a whole different class of shapes to young children adds to
our understanding of the complexity of the development of the concept of “triangle” with all its
variants. Additional understandings of a child’s thinking can help researchers, curriculum
writers and teachers to make thoughtful decisions when working with four-year-olds on the
development of shape concepts.
Preschooler’s Understanding
54
Conclusion
This field study examined the geometrical thinking of a four-year-old girl in an urban
preschool in the northeastern United States. The study focused on the formation of the child’s
concept of “triangle,” organized by the themes of attributes and the visual whole. It documented
how one child progressed in her concept of a triangle from fall, through winter, to spring, as she
experienced a software-integrated mathematics curriculum that included exposure to a wide
variety of non-prototypical triangles, and offered opportunities for making connections. The
child’s thinking was examined within Piagetian and Vygotskian frameworks, as well as the van
Hiele levels of geometric thought, and Clements’ and Sarama’s hypothesized learning trajectory
in the development of shape concepts.
Tania developed two separate schemes for triangle. She viewed triangles as visual
wholes, and yet same time she developed a scheme around the attributes of triangles. Tania
became more aware of the parts of a triangle, but did not always relate these parts to the class of
triangles. Her concept of triangle, which was inaccurate but stable at the beginning of the year,
showed instability as the school year progressed. This instability was brought on by the
cognitive conflict she encountered between her fixed, prototypical idea of triangle that she came
into the preschool with, and interference with this prototype, as non-prototypical triangles were
introduced, and as she gradually applied the attributes to these shapes, and became more attuned
to analytical approach to shapes. Tania’s efforts to understand where non-prototypical triangles
fit into classes of shapes caused conflict, and set the stage for growth. Through this conflict her
fixed prototypical view of triangles began to give way to a broader, more inclusive definition of
the many non-prototypical triangles that the word “triangle” symbolizes.
Preschooler’s Understanding
55
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