Learning of Geometric Concepts in a Logo Environment Author(s): Douglas H. Clements and Michael T. Battista Source: Journal for Research in Mathematics Education, Vol. 20, No. 5 (Nov., 1989), pp. 450-467 Published by: National Council of Teachers of Mathematics Stable URL: http://www.jstor.org/stable/749420 Accessed: 15-03-2016 19:40 UTC REFERENCES Linked references are available on JSTOR for this article: http://www.jstor.org/stable/749420?seq=1&cid=pdf-reference#references_tab_contents You may need to log in to JSTOR to access the linked references.
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Journal for Research in Mathematics Education
1989, Vol. 20, No. 5,450-467
LEARNING OF GEOMETRIC CONCEPTS IN A
LOGO ENVIRONMENT
DOUGLAS H. CLEMENTS, State University of New York At Buffalo
MICHAEL T. BATTISTA, Kent State University
To investigate the effects of computer programming in Logo on specific geometric
conceptualizations of primary grade children, 48 third graders were randomly assigned to either
a Logo or a control group. The Logo group was given 26 weeks of instruction in a Logo
environment. The children were then interviewed to ascertain their conceptualizations of
angles, shapes, and motions. In both groups children's notions of angle and angle measure were
multifaceted and included a number of misconceptions, although performance was uniformly
higher for the Logo group. The Logo children were more aware than the control children of the
components of geometric shapes and were more likely to conceptualize geometric objects in
terms of the actions or procedures used to construct them.
The Logo programming language was developed to serve as a conceptual frame-
work for learning mathematics. Much of the literature on Logo, however, has
presumed that straightforward exposure to number, estimation, and geometry
concepts within the context of Logo programming increases mathematics achieve-
ment. Research evidence about this presumption is inconclusive. Classroom
observations have demonstrated that children use certain mathematical concepts
in Logo programming (Hillel, 1984; Kull, 1986), but it is not certain that this use
leads directly to increased achievement (Battista & Clements, 1986; Seidman,
1981).
In contrast, using Logo programming as a conceptual framework is not a method
of directly teaching mathematical ideas. Instead, its effects on mathematical
knowledge may result from children's construction and elaboration of schemata
that form a structure upon which future learning and problem solving can be based.
In particular, Logo may permit children to manipulate embodiments of certain geo-
metric ideas. Serving as a transitional device between concrete experiences and
abstract mathematics, it may facilitate children's elaboration of the schemata for
those ideas.
For example, children initially can identify only a visually presented rec-
tangle - a visual activity, Level 1 in the van Hiele hierarchy (Battista & Clements,
1987; van Hiele, 1986). To create a Logo procedure that draws a rectangle, chil-
dren must analyze the visual aspects of the rectangle and reflect on how its com-
ponent parts are put together, an activity that requires descriptive/analytic (Level
2) thinking. The use of Logo commands that correspond to the motions necessary
This project was supported in part by a research appointment granted to the first author
by Kent State University. Thanks are due to the educators and children of Hudson Elemen-
tary School for their enthusiastic participation and to the editor and reviewers for their help-
ful suggestions.
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451
for constructing the rectangle (e.g., FORWARD, RIGHT) increases the salience of
the critical components of the shape. For instance, because turns play a central role
in forming shapes such as rectangles and other polygons, forming such shapes in a
Logo environment may focus the children's attention on, and thereby enhance their
understanding of, the notions of rotation and angle. Indeed, there is empirical
evidence that in Logo, children construct elaborate or interconnected schemata
(rather than simple rules and terms) for specific mathematical topics (Clements,
1987; Findlayson, 1984; Kieran, 1986; Noss, 1987).
It has been demonstrated that certain Logo environments positively affect meta-
cognitive problem-solving skills (Battista & Clements, 1986; Clements, 1986).
Previous research also has indicated, however, that most young Logo programmers
do not engage in high-level construction and abstraction of mathematical ideas
independently (Kieran, 1986; Littlefield, Delclos, Bransford, Clayton, & Franks,
1986). In fact, focusing on Logo as a programming language may even distract
children from the mathematics that is encountered. Thus, the lack of consistent
positive effects on measures of mathematics achievement has led to debate con-
cerning how, and how much, work with Logo should be structured and connected
with the standard mathematics curriculum.
This study is part of a larger project investigating the effects of a specific Logo
environment on certain metacognitive problem-solving skills and geometric con-
cepts. We posit that - if the environment includes sufficient teacher guidance and
peer interaction - mathematically correct concepts can be developed within Logo
without subordinating the goal of providing children with a tool for independent
problem solving. The purpose of this study was to investigate the effects of a Logo
environment that emphasized projects and problem solving, with intensive teacher
guidance and peer interaction, on primary-grade children's geometric conceptuali-
zations. Specific research hypotheses were the following:
1. Compared with those in a control group, children who experience a Logo
environment have conceptualizations of angle and angle measure that are more
consistent with formal mathematics.
2. Logo children are more likely than control group children to attend to com-
ponents when describing and sorting geometric figures, and they are more likely
to describe these figures in terms of their components and the notion of movement
along a path.
3. Logo children are more successful than control group children at giving di-
rections on a map task.
4. Children, with and without Logo experience, have a variety of conceptions
and misconceptions' regarding angle, angle measure, and shape; however, Logo
children have different misconceptions and fewer of them.
'Terms such as "misconception," "error," and "incorrect" are used as communicative devices to refer to responses
inconsistent with the standard curriculum and are not intended to deny alternative, and possibly valid,
conceptualizations.
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452 Learning of Geometric Concepts in Logo
METHOD
Subjects
The subjects for the study were 48 children from a middle class midwestern
school system. From a pool of children who returned a parental permission form,
20 boys and 28 girls (mean age = 8 years 9 months) were randomly selected from
the third-grade classrooms of seven teachers. The children were randomly as-
signed to either a Logo computer programming group or a control group.
Instruments
Pretests. The total mathematics score of the California Achievement Test, Level
13 (CAT), was used to assess the children's pretreatment level of mathematics
achievement. This test consists of two subtests - computation (40 items) and
concepts and applications (45 items). The published Kuder-Richardson-20 relia-
bility coefficient for the total test is .95 (California Test Bureau, 1979).
Interview. A structured interview containing 24 items in three parts was devel-
oped for this study. An overview of these parts will be presented in this section.
Individual items will be described in the Results section; details on scoring can be
obtained by writing the authors. The first part had a total of 15 items dealing with
concepts of angle and angle measurement. Using Cronbach's alpha, reliability for
this subtest was estimated as .83. The second part consisted of 8 items dealing with
the concepts of shape and motion. Reliability for this subtest was .71. Interrater
agreement was computed for the five items that involved categorization or judg-
ment of responses (Items 1, 13, 16, 18, 22), yielding an average of 96%. The third
part consisted of a single item (Item 24) employed in previous research (Clements,
1986). Test-retest reliability was computed as r = .91.
Treatments
Experimental. The Logo children met for three 45-55 min sessions per week
for a total of 78 sessions. The children's absences ranged from 0 to 15, with a mean
of 5.1 sessions. Two sessions were conducted daily, each for half the children in a
treatment group. During each session, six pairs of children worked on six Apple
microcomputers under the guidance of one or two teachers (the first author or an
experienced graduate assistant). Both teachers were present for about two thirds
of the lessons; one or the other was present for the remainder.
It was not expected that Logo programming alone would produce changes in
children's geometric schemata. Logo was viewed as a tool that facilitated both
geometric explorations and instruction offered by the teachers in the study. The
treatment was designed to promote elaboration and transfer; problems were solved
in several settings (real world, paper and pencil, Logo), and translation between
settings was required. Explicit awareness of the effectiveness of transferring
problem-solving and mathematical skills from one setting (and one problem) to
another was encouraged.
Although new techniques were introduced at a consistent pace, the Logo chil-
dren were encouraged to use those mathematical and programming skills with
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Douglas H. Clements and Michael T. Battista 453
which they were comfortable on projects of their own choosing. This permitted
them to practice applying these skills to the point of mastery. The relatively long
duration of the treatment also contributed to this mastery. Finally, the emphasis
was on solving geometric problems; no attempt was made to introduce all children
to programming concepts such as recursion.
Most sessions consisted of three phases. The first phase was an introduction,
including a review and discussion of the previous day's work. The second phase
was a teacher-centered, whole-group lesson presenting new information, such as a
new Logo command or a problem to solve. In the third phase children worked
independently on either teacher-assigned problems (about 25% of the time) or self-
selected projects. On some of these projects the teachers introduced themes, but
the children were responsible for selecting the specific problem. During this time,
the teachers encouraged children to solve problems by themselves, to predict
(Where will the turtle be then?), and to reflect on their use of geometric knowledge
and strategies (What turn did you use for this shape? How does it compare to this
one? What are you trying to get the turtle to do? What did you tell it to do? What
did it do? Can you find the bug?).
In the initial sessions, the FORWARD and BACK and rotation (RIGHT, LEFT)
commands were introduced and acted out by the children assuming the role of the
turtle. Children were given challenges such as determining the exact length and
width of the screen in turtle steps, creating as many ways as they could to get to a
location, and determining what input to LEFT made the turtle turn halfway around.
Children discussed all new ideas and solutions to problems and were asked to
name other situations, in and out of school, in which such ideas and solutions
would be useful. For instance, it was noted that a 900 turn in a path created a right
angle, so that the two sides were perpendicular. Children were led to identify this
angle inside and outside the room. They also discussed what each command told
the turtle to do and what effect this had on the screen display.
Procedure writing was introduced, first through discussions of children's expe-
riences with new routines - in familiar contexts such as gymnastics or the class-
room, but in new ideas and words - and then through the notion of teaching Logo
new commands. Children used a support program, TEACH, that allowed them to
define a procedure and simultaneously watch it execute, editing whenever neces-
sary. Various procedural solutions to problems were compared, and discussions
focused on differences in solutions (e.g., What was altered in the procedure to
create this geometric effect?). Given an altered procedure or graphic, children
were asked to predict the graphic that would be produced by the altered procedure
or the procedure that produced the altered graphic. Thus, there was an attempt to
help children construct mappings between components of procedures and the cor-
responding components of the shape formed.
Familiar geometric figures were constructed and manipulated. For example, the
concept of square was discussed, with children offering definitions and exemplars.
The children constructed squares using dramatization, paper and pencil, and Logo.
They constructed several procedures for making squares, each producing a differ-
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454 Learning of Geometric Concepts in Logo
ent side length. Group discussion centered on solution procedures and the ques-
tion "What is a square?" Procedures were compared, and children led to list both
similarities (the pattern of commands, 900 input to the rotation commands) and
differences (the input to the FORWARD commands, right or left turns). Teachers
discussed geometric terms such as line segment, rotation, angle (including the
hidden angle that the turtle turns through in a path and its distinction from the
supplementary angle actually formed), right angle, congruence, and similarity;
however, children were not requested to memorize these terms or their definitions.
The children were finally challenged to plan and construct programs that used their
square procedures as building blocks for constructing such figures as towers, or
squares inside squares.
The concept of procedures with variable inputs was introduced about a third of
the way through the treatment. Comparison of different square-producing proce-
dures was repeated, and children discussed the need for one procedure that would
construct a square of any size. A new square procedure with inputs was written
by the group and dramatized, then incorporated into new projects. Similar gener-
alized procedures were written by the children for rectangles and equilateral tri-
angles. The remaining sessions included challenges, often oriented toward sea-
sonal interests, such as writing valentine-heart procedures; contests, such as writ-
ing the shortest or most elegant program for a stacked-rectangle pyramid, or du-
plicating given shapes and using them as many different ways as possible in the
creation of a picture; explorations of regular polygons and the rule of 3600; and a
final collaborative project to construct a large mural for an end-of-the-year party
for parents.
Control. The control children received computer experience under the same
conditions as the experimental group (six pairs of children working with the same
teachers), with two important differences: First, the content, designed to develop
creativity and literacy, included composition programs (Milliken's Writing
Workshop, which is an integrated package of prewriting programs, a word proces-
sor, and postwriting programs) as well as drawing and music programs. Second,
the group met only once a week for a total of 26 sessions (mean absences = 0.9
days).
Both experimental and control groups. All Logo children and control children
were exposed to Logo for 20 min per day for 2 weeks as part of the regular school
program. During the study, one control child, BJ, also acquired a home computer
equipped with Logo. According to their teachers, all children covered the geome-
try chapter of their mathematics textbook within 2 months preceding the inter-
views. Textbook topics included standard two- and three-dimensional figures,
sides and angles of polygons, line segments, angles and right angles, congruence,
and symmetry.
Procedure
Using the instrument described in the preceding section, the children were pre-
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Douglas H. Clements and Michael T. Battista 455
tested to determine their level of mathematics achievement. The computer activi-
ties were then implemented for 26 full weeks, replacing the last period of the
school day. At the end of these sessions and after a delay of one week, the chil-
dren were interviewed individually by the first author to determine their concep-
tualizations of angle and shape. The interviews lasted from 45 to 75 min.
RESULTS
Pretests
The children in the two groups had nearly identical mean scores on the CAT total
mathematics score (Logo programming, M = 692.46, SD = 38.20; control, M =
692.92, SD = 32.29). The difference was neither practically nor statistically
significant, t(46) = -.04, p > .96.
Angle
Table 1 presents the means and standard deviations of the posttest scores. The
Logo group scored significantly higher on the angle part of the interview than did
the control group, t(46) = 5.88, p < .001. Children's responses to individual items
are, however, more revealing than this global measure.
Table 1
Means and Standard Deviations for Treatment Groups on Posttest Measures
Logo Control
No. Description M SD M SD
2 Draw an angle .94 .17 .77 .33
3 Draw a larger angle .69 .36 .21 .36
4 Circle those figures that are angles .95 .11 .79 .24
5 Number of angles in a triangle .92 .28 .58 .50
6 Number of angles in a rectangle .96 .20 .67 .48
7 Number of angles [3 segments] .92 .28 .50 .51
8 Number of angles ["X"] .57 .35 .29 .41
9 Larger angle [length of segments] .75 .36 .56 .43
10 Larger angle [orientation] .33 .48 .13 .30
11 Larger angle [orientation; same measure] .71 .46 .50 .51
12 Most/least turn on a path .88 .27 .75 .33
13 Sort angles .42 .32 .19 .25
14 Draw an angle twice as large .90 .21 .50 .36
15 Recognition of right angles .76 .16 .54 .17
Angle total 10.70 1.34 6.97 2.84*
17 What are the parts of shapes? .76 .23 .42 .30
18 Tell a Martian how to make a shape .65 .23 .42 .23
19 Transform a template .88 .13 .65 .22
20 Drawing machine .69 .36 .60 .42
21 Which letter is in a circle? .96 .14 .79 .76
22 Sort quadrilaterals .46 .29 .13 .22
23 "Building" problem .65 .38 .40 .39
Shape and motions total 5.04 1.03 3.40 1.17*
24 Map .92 .12 .68 .21*
Note. n = 24 for both groups.
*p < .001.
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456 Learning of Geometric Concepts in Logo
When asked "What is an angle?" (Item 1), more control children than Logo
children gave definitions that fell into the first three categories in Table 2; these
categories indicate either little knowledge of angle or the influence of common-
language usage. Logo experience appeared to help children develop the concept
of angle as rotation: nine definitions by Logo children, compared with none by
control children, reflected mainly that conceptualization.
To the extent that the "corner" and "union of two rays" categories are thought to
be similar, there seems to be little difference between the responses of the two
groups; when these categories are combined, the Logo and control frequencies are
approximately equal. However, the control children's responses fell more often
into the corner category, a conceptualization that may represent a more constrained
interpretation and that is based more on concrete experience. That is, responses
including the word "corner" usually indicated that a corner must be part of a fig-
Table 2
Frequencies for Treatment Groups: Categories of Angle Definition
Category Example(s) Logo Control
No response Like ... I don't know. (C) 0 1
Line It's like a line. Any line. (C) 0 1
Diagonal line Like a line that goes on a diagonal or a corner, 1 6
either one.
A diagonal line. (C)
It's like a slanted line. (C)
It goes down, it's kind of slanted like that
Indian thing [teepee]. (C)
Orientation/direction A way... going this way. 1 2
Like if you were in the classroom here (to the
side) and the chalkboard were here, then
that would be a bad angle to see. (C)
The way the turtle points in Logo. (C)
Rotation Something that turns. Different ways to turn. 9 0
An angle is a turn. Sort of a line from a turn.
When you turn some degrees.
Corner It's like a corner but it's not . .. it doesn't 3 9
have to be perfect corner, but it can't be
round.
Sort of like a corner of something. (C)
Union or intersection It's when two lines meet each other and they 9 5
of two lines/ come from two different ways.
segments Two segments that come together at a point.
It's sort of a place where two lines come
together.
A right angle or any angle? A line and another
line, slanted, but they don't have to be
slanted, they can be straight. One side, then
another.
Like two lines that they're both straight and
they come together and it's half of a square.
(C)
Anytime two lines meet and form like a corner.
(C)
Line with turn A straight line that makes a sharp turn. 1 0
Note. C = Control group (i.e., those example responses followed by (C) were made by control children;
others were made by Logo children).
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Douglas H. Clements and Michael T. Battista 457
ure or object, often implying perpendicularity. Logo children appeared more likely
to generalize the concept of angle from the more limited context of a corner to that
of the union of two segments (and were more likely to proffer the definition pro-
vided in their regular classes). One Logo child explicitly distinguished between
several conceptualizations: "Do you mean like an angle in math or like the other
kind, like at an angle?" Several other Logo children reflected a process orienta-
tion, for example, "It's like a straight line, and then another line that goes off like
this after a turn" (accompanied by gestures).
Logo children performed well on an angle identification task (Item 4; see Fig-
ure 1); performance for the control children was moderately high. Individual re-
sponses from both groups, however, revealed a variety of misconceptions and in-
terpretations of the angle concept, and the responses of individual children often
varied with the task. For example, BF (a member of the Logo group), who had
consistently replied with an "orientation" conceptualization of angle, correctly
identified all the angles on this task. Conversely, other children were consistent
across items, although their responses were not consonant with mathematical defi-
nitions. For example, two Logo children, SF and KM, selected only the right
angles, and their responses to all other items reflected this conceptualization. In
addition to the misconceptions displayed by all children, Logo children indicated
that angles included (a) "curved angles" (arcs) and (b) lines "slanted from straight
up" (i.e., diagonal and horizontal lines-note that drawing these in Logo would
require a rotation, given the turtle in its home state). Misconceptions held by the
control-group children were more numerous and included choosing all lines,
slanted lines only, angles and slanted lines, nonobtuse angles only, and right angles
only.
Figure 1. Item 4. Children were told,
"Draw a line around all those figures that are angles."
Finally, for both groups, the children's own responses to a previous task influ-
enced their responses to the task in which they were presently engaged. MC was
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458 Learning of Geometric Concepts in Logo
asked why she did not select the two nonintersecting segments. She explained,
"Because you can't fit a square piece of paper in it." She then looked at the other
nonright angles that she had already circled and said, "Oh, no, it's because they
don't close." LH, whose previous definition of an angle was "when two lines come
together," similarly indicated that the curved lines were an angle only after an
apparent internal debate. In justification, she said, "It looks like a semicircle
almost, but there are two lines coming together." Thus, her own definition may
have been in conflict with her intuition and finally was more influential.
Four items (5-8) asked children to give the number of angles in particular fig-
ures. Performance on these items was strong for the Logo group and fair for the
control group, with the following errors recorded for both groups: For a triangle
(not pictured), children from both groups indicated and counted slanted lines only,
right angles only, "base angles" only, the "vertex angle" only, and sides instead of
angles (L, 2; C, 10). On the rectangle item (also not pictured), errors included
counting slanted sides or only nonright angles (and thus responding "zero") and
counting sides (L, 2; C, 8). For a simple open path consisting of three line seg-
ments (Item 7), errors included indicating and counting slanted sides, the right
angle only, the acute angle only, and the sides (L, 4; C, 12).
Performance on Item 8 (two segments forming an "X") was weaker for both
groups. Errors included indicating no angle in the figure, the point of intersection
only, and a single pair of vertical angles (L, 15; C, 20). The responses of two Logo
children answering "zero" indicated that the figure had no angles but would have
two "if you had turned." For example, MS said, "They're all straight lines. None
of them turn." Tracing an angle, the interviewer asked, "What about if you made
them by turning this way?" MS responded, "Two." One control child had a dif-
ferent notion: If it were "two halves of two triangles, it would be two angles. If it
was an X, it wouldn't be any angles." Thus, it appeared that for these children,
whether or not two line segments constituted an angle depended either on how they
were drawn or on their being a part of a shape.
Angle Measure
Items 3, 12, 13, 14, and 15 examined children's ideas regarding angle measure.
When asked to draw a "bigger" angle (Item 3), 13 Logo and 2 control children
correctly drew an angle with a larger measure. Asked why the angle she drew was
bigger, one Logo girl replied, "The turn at the point is larger." Another Logo stu-
dent stated, "Because of the way the lines come together, it's bigger in between
here," whereas one control student stated, "Because the thing is open wider."
Other Logo children who received full credit were nonetheless struggling with
several possible interpretations. After drawing an angle with the same measure but
longer sides, one boy said spontaneously, "Oh, a bigger angle. This isn't a bigger
angle; the lines are just bigger." Another boy read the question and immediately
asked, "But the same one?" The interviewer replied, "Whatever you think is a
bigger angle." The boy queried again, "But the same turn?" The interviewer said,
"Draw what you believe is a bigger angle." He then drew an angle with segments
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Douglas H. Clements and Michael T. Battista 459
of approximately the same length but with twice the number of degrees. Two Logo
children drew angles with greater rotation along the path (i.e., with greater exte-
rior angle measures) but then took care to justify their responses by saying, for
example, "This is the angle that's bigger."
One Logo child received partial credit for drawing a line segment with a greater
rotation from the vertical than the "slanted line" he had drawn for the previous item
(and justifying this as a "bigger turn"). Six other Logo children received partial
credit because they first drew intersecting segments of greater length but corrected
their responses after the scripted prompt "Is there any other way one could be
bigger?" One control girl asked, "Same or different?" When the interviewer re-
plied, "Just bigger," she drew an angle of similar measure with longer segments.
She was then asked to draw a "different" angle, whereupon she drew a larger angle.
After prompting, five additional control children drew larger angles. One ex-
plained that it was "enlarged," the Logo-experienced boy (BJ) said that "the turtle
turns farther," and the rest explained that it was "wider open." The boy who origi-
nally described angle as perspective maintained that "you can't have a bigger
angle," just a different one. All the other control children drew longer line seg-
ments.
Logo children slightly outperformed the control children on the items in which
they were to select the larger of two pictured angles (Items 9-11). Somewhat
surprisingly, no substantial difference was found on Item 12, designed to measure
children's judgment of the amount of rotation along a path.
Only three children, all from the Logo group, received full credit on an angle-
sorting task (Item 13) by constructing more than one consistent, exhaustive sort,
free of irrelevant attributes. For example, NQ sorted "all the 90 degrees in one
stack, the rest in another." When asked for another sort, he named three catego-
ries, "the 30s (actually the obtuse angles), "the 90s," and "the rest." Last, he asked
if he could sort these categories into "groups by exact degrees." Most children
from both groups who received partial credit constructed a single sort using the
expected three categories - acute, obtuse, and right.
For Item 15, children were given several figures consisting of two intersecting
line segments and were asked to draw a pictured square region "into all corners
where it fits exactly." Using this task, Wallrabenstein (1973) concluded that right
angles are often not recognized. In the present study, scores for this item indicated
better performance by the Logo group. Neither group, however, approached
mastery.
Shapes and Their Components
The Logo group significantly outperformed the control group on the shape-and-
motions part of the interview, t(46) = 5.17, p < .001. As with the angle-definition
item, the question "What are shapes?" (Item 16) elicited divergent responses from
children in both groups (see Table 3). Overall, however, differences between the
Logo and control groups are less palpable than for the angle-definition item. There
was some tendency for control children to respond that they did not know or to
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Table 3
Frequencies for Treatment Groups: Categories of Shape Definition
Category Example(s) Logo Control
No response I don't know. (C) 0 2
Objects They're objects. 1 1
Objects, big or small. (C)
Instances Like circles. 1 4
Like a triangle, circle, squares. (C)
Dimensions They're 2-dimensioned things. 2 2
Two- or three-dimension objects. (C)
Closed path/region Things that are enclosed. 1 5
If it was an angle, it wouldn't be a shape. It
wouldn't be unless the whole thing connects.
(C)
Things with space in between them and they
close. (C)
How do I explain it? A shape can never end.
There's no beginning or end. (C)
Things that have their own area . . . space and
size. Squares only has so much room and a
triangle only has so much room, too. (C)
Form Objects that have a definite form. Their shape 8 3
cannot be changed.
Different kinds of forms. Every kind of shape
in different ways.
Objects that are shaped . . with directions.
Shapes are objects-even people-every
person has a shape.
Shapes are like things. And without shapes,
there wouldn't be anything. Everything has a
shape. (C)
That frame is a shape, this (bookcase) is a
shape, a window is a shape. Liquid changes
shape. (C)
Part/whole Like things you make things out of. Everything 3 1
relationship is made out of a shape. Like the towel
(components of dispenser is a square and my shoe is an oval.
objects) Things that you can put together that can make
other things.
Like things that are just one piece of
something. (C)
Combination of lines A group of lines that form a shape ... uh, 4 5
that forms a picture.
Something with more than two lines. It could
be a circle.
They're lines made up into things. (C)
Shapes are lines together to form a shape. (C)
Combination of Things that have angles and sides. Some have 4 1
straight lines, curved the same sides, like squares.
lines, and/or angles They're like things that have different angles
and are different from each other. They have
different kinds of sides.
They're like angles, curves, lines, and they
make shapes.
Different pieces of curvy and straight lines
connected to each other with points.
Note. Those example responses followed by (C) were made by control children; others were made by Logo
children.
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Douglas H. Clements and Michael T. Battista 461
name instances of shapes. In addition, they were more highly represented in the
"closed path/region" category, possibly reflecting the topological emphasis in their
knowledge that Piaget described (cf. Piaget & Inhelder, 1948/1967). The Logo
group was more likely to proffer definitions that included the term "form." In
addition, although equal numbers of Logo and control children defined shapes as
a combination of lines, several more Logo children than control children included
"curved lines" and angles in their definitions, possibly indicating a greater
awareness of these components of shapes.
The latter possibility is supported by responses to Item 17, "What are the parts
of shapes?" Logo children named more parts of shapes (usually angles and arcs)
than did control children. It should be noted that all Logo children had used and
examined Logo procedures that directed the turtle to draw screen representations
of circles and arcs. Interestingly, even though these procedures actually consisted
of forward movements and turns, most Logo children included arcs or circles as
components of shapes, distinct from line segments and rotations.
The Logo group scored higher on Item 18, which asked children to describe over
the telephone how to make a triangle to an extraterrestrial who had never seen this
shape. This item was designed to measure whether the Logo experience did in fact
alter children's ideas regarding the construction and description of plane figures.
Curiously, only 14 Logo children (along with 1 control child, BJ) used Logo ter-
minology. Whether this was the result of the assumption that Logo was not acces-
sible to extraterrestrials is not known. The descriptions given by 12 Logo children,
compared with 3 by control children, consisted of sequences of movements and
turns as opposed to compilations of segments, which suggests a path-oriented or
procedural conceptualization of the task. In addition, Logo, or the use of variables
within Logo, may have influenced several children to suggest a surprisingly ab-
stract solution for the task, as in the following example: "Turn your pen to any
angle. Then go forward any amount, then turn again. Go forward any amount, turn
to where you started, and draw a line."
Previous research with Item 21 (Figure 2) indicated that under the CAT scoring
system (with "none of the above" considered correct), Logo children scored lower
than control children (Clements, 1987). That research suggested that the Logo
experience created "circle schemata," in which sections of circles were accepted
as bona fide circles in that many of the children's instantiations of these procedures
split the circle on the screen. As hypothesized in the present study, Logo children
performed no worse than control children under a revised scoring system, in which
"j" was also considered a correct response. Children's comments illustrated their
thinking on this item. All but one of the Logo children indicated that "j" was part
of a circle and thus was the best response; only 13 control children so indicated.
Seven control children selected "none of the above," stating simply that no figure
was a circle.
The Logo group outperformed the control group on a quadrilateral-sorting task
(Item 22), although overall the scores were relatively low. It is revealing to exam-
ine the types of categories constructed by the children. Two kinds of sorts occurred
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462 Learning of Geometric Concepts in Logo
most frequently: (a) the separation of all seven types of quadrilaterals into the clas-
sical mathematical categories (4 L, 1 C) and (b) similar sorts (14 L, 13 C) with the
exception that two of these types were collapsed into one category; for example,
parallelograms and rhombuses were combined. Only 3 children, all from the Logo
group, re-sorted dichotomously, using the following categories: with and without
parallel lines, with and without 90 angles, and convex and nonconvex. All other
children considered irrelevant variables, separated figures of the same type, or did
both (3 L, 10 C).
Applying Arithmetic
On Item 23 (Figure 3), 11 Logo and 5 control children correctly used computa-
tion; 9 Logo and 9 control children estimated correctly, basing their estimates on
the lengths provided; and 4 Logo and 10 control children responded incorrectly,
using no apparent estimation strategy. Thus, the Logo experience appears to have
facilitated children's recognition of the relevance of arithmetic processes in the
solution of geometric problems.
Which letter is in a circle in this figure?
Of Og
Oh Oj i
O None of the above
Figure 2. Item 21. (From the California Achievement Tests, Form C, Level 13. By permission of
the publisher, CTB/McGraw-Hill, 2500 Garden Road, Monterey, CA 93940. Copyright 1977 by
McGraw-Hill, Inc. All rights reserved. Printed in the U.S.A.)
100
30 30
Figure 3. Item 23. Children were told, "Here is a simple building. The builders
want to build a sign here [indicate]. How wide must the sign be to fit exactly?
How do you know for sure?"
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Douglas H. Clements and Michael T. Battista 463
Motions
The Logo group tended to score higher on Items 19 and 20 (Figures 4 and 5) than
did the control group. It was hypothesized that the Logo experience might posi-
tively influence children's ability to use the turn operation, possibly to the exclu-
sion of their use of the flip operation. That is, because the flip operation was not
available in Logo, it might have become relatively less accessible as a mental
operation. This hypothesis was not supported. In fact, of those children who made
errors, 31% of the Logo group and 43% of the control group evinced errors only
on the flip operation. All errors on Item 20 involved left-right reversals.
Map
In agreement with previous research (Clements, 1986), Logo children outper-
formed control children on Item 24, a task that asked them to describe a path
through a map, t(46) = 4.94, p < .001. Virtually all of this difference was accounted
for by the Logo children's greater accuracy in discriminating right from left turns.
DISCUSSION
Angle
Several conclusions may be drawn about the ideas of children from both groups
regarding the nature of angle. First, their conceptual scheme for angle is complex
and multifaceted and appears to include subschemata from the domains of com-
mon language, normal school experiences, and Logo (with the verbal textbook-
definition subscheme possibly the weakest). Their angle conceptualizations are
not well discriminated from related concepts, such as orientation, corner, rotation,
and right, and often include other nonstandard notions. In addition, for some chil-
dren, even the way a figure is formed determines if it contains an angle. Children
seem to lack the control structures necessary to select the appropriate subscheme
for mathematical tasks.
Second, the children's responses often indicate that they struggle with the idea
that a geometric object-say, a side of a polygon-is simultaneously part of more
than one object, such as two adjacent angles. For example, when asked to circle
the angles in a polygon, JH said, "The circle shouldn't be too big. ... You can't
circle the whole thing [indicating the segment], because there's nothing left for this
one [the adjacent angle]." JH believed that part of each line segment must be
included, but not so much that the whole segment (and thus the side) was sub-
sumed. Children had a similar difficulty on the vertical-angles task (Item 8).
Similar responses have been observed on other class-inclusion tasks (Inhelder &
Piaget, 1969).
Third, the conceptualizations guiding children's responses apparently vary with
the task. For example, several children defined or drew angles that reflected ori-
entation conceptualizations, but selected only standard angles when asked to iden-
tify angles.
Fourth, comparison of the groups indicates that although the Logo experience
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19. This is a template. I use it to draw shapes like this.
Could I use it to make this? [Indicate the first figure.]
How would I have to move it to do that?
[Repeat for each figure.]
Figure 4. Item 19.
20. I have a drawing machine. [Indicate picture at left.]
I put it down and it draws this.
[Indicate picture at right.]
If I put it here, what do you think it would draw?
[Indicate picture at left. Repeat for picture at right.]
Figure 5. Item 20.
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Douglas H. Clements and Michael T. Battista 465
by no means eliminated errors, it appears to have significantly affected children's
ideas about the nature of angles. The responses of the control children were more
likely to reflect little knowledge of angle or common-language usage. The re-
sponses of the Logo children indicated more generalized and mathematically ori-
ented conceptualizations (including angle as rotation and as a union of two lines/
segments/rays). The combination of such conceptualizations that some Logo chil-
dren displayed across items intimates that their schemata for angle might consist
of two structures-the union of rays or segments, and rotation. The degree to
which these structures are integrated (or possibly in conflict), as well as instruc-
tional methods that will facilitate proper integration, should be ascertained in fu-
ture research.
Similarly, many different schemata emerge regarding angle measure. Some
children in each group related the size of an angle to the length of the line segments
that formed its sides, the tilt of the top line segment, the area enclosed by the trian-
gular region defined by the drawn sides, the length between the sides (from points
sometimes, but not always, equidistant from the vertex), the proximity of the two
sides, or the turn at the vertex. Misconceptions that may have been fostered by
Logo experiences include considering the angle of rotation along the path (e.g., the
exterior angle in a polygon) or the degree of rotation from the vertical. Some chil-
dren avoided ordering angles by size. That is, some children accepted different
kinds of angles, or different orientations or perspectives, but not the existence of
bigger or smaller angles.
In agreement with previous research (Findlayson, 1984; Kieran, 1986), Logo
appears to have facilitated children's understanding of angle measure, although for
many Logo children, numerous experiences emphasizing angle measure as amount
of rotation did not replace previous misconceptualizations of angle measure
(Davis, 1984). In general, however, Logo children's conceptualizations of larger
angle were more likely to reflect mathematically coherent and abstract ideas.
Because the Logo treatment emphasized the difference between the angle of rota-
tion and the angle formed as the turtle traced a path, it is interesting that, in direct
contrast to Kieran's children, the children of this study performed better than
control children on items assessing the measure of angles, but not noticeably bet-
ter on the item (Number 12) assessing the angle of rotation along a path.
Shapes and Motions
In general, third graders from both Logo and control groups possessed a variety
of unsophisticated conceptions of shape. Their tendency to begin definitions with
"It's like" may indicate that they were initially retrieving and generating instances
of that object. Some named these instances as their definition. Others constructed
a definition from these instances. This use of prototypes rather than properties as
a basis for the formation of definitions seems to reflect a visual, or Level 1, opera-
tion.
Logo experience appears to engender greater awareness of the components of
two-dimensional figures, especially angles and arcs. Logo children were more
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466 Learning of Geometric Concepts in Logo
likely than control children to construct mathematically acceptable categories of
quadrilaterals, to discriminate on the basis of the properties of these figures, and
to sort dichotomously according to a geometric property. That is, the Logo chil-
dren, more than the control children, exhibited signs of transition from van Hiele's
visual level to the descriptive/analytic level of thought.
Although only slightly more than half the Logo children employed Logo termi-
nology in describing how to make a triangle, their responses, compared with those
of the control group, were more likely to emphasize the idea of path and the pro-
cess of constructing paths to create the figure. Additional items also indicated that
the Logo children were more likely to conceptualize geometric objects in terms of
the actions or procedures used to construct them. Thus, the Logo experience
appears to have engendered within the children an explicit awareness of those
action-based conceptualizations that Piaget identified as fundamental to the
development of geometric ideas. Finally, the Logo experience appears to have
facilitated children's use of arithmetic processes in solving geometric problems.
Map Item
The Logo children's greater accuracy in discriminating the direction of turn in
the map task is significant, given the difficulty of applying these concepts inde-
pendently of one's own position (Wallrabenstein, 1973). Of course, since the task
is almost isomorphic to turtle programming, generalization to other domains and
other tasks, such as work with more realistic maps, should not be assumed.
Final Words
Because the children appeared to be constructing and reconstructing their con-
cepts about geometric entities during the interviews, caution should be used in
generalizing responses from single items. However, this reconstructive process is
quite revealing; it appears that these third-grade children had not consolidated their
conceptualizations of the geometric ideas dealt with in this study and that their
geometric conceptualizations were only marginally affected by verbal definitions.
This suggests that these children lacked experiences requisite to the development
of meaningful verbal definitions or that they had not begun the process of elabo-
rating these experiences. One implication is that children's classroom experiences
should involve a variety of explorations and discussions of issues that build on
their intuitions of geometric concepts, gradually discriminating among these intui-
tions to acquire conceptions that are more mathematical. Only then should pre-
cise verbal definitions be introduced. There is strong evidence that the Logo envi-
ronment employed in this study promnted such explorations and discussions and
thus positively affected the children's geometric conceptualizations. Indeed, re-
sponses of Logo children tended to be more mathematically coherent and abstract.
Thus, these results support the use of Logo as a conceptual framework for learn-
ing aspects of geometry, given that the educational environment in which Logo is
embedded is structured to guide children's experiences within this framework.
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Douglas H. Clements and Michael T. Battista 467
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AUTHORS
DOUGLAS H. CLEMENTS, Associate Professor of Early Childhood and Mathematics Education,
Learning and Instruction, Faculty of Educational Studies, State University of New York at Buffalo,
593 Baldy Hall, Amherst, NY 14260
MICHAEL T. BATTISTA, Associate Professor of Mathematics Education, Teacher Development and
Curriculum Studies, College of Education, Kent State University, 404 White Hall, Kent, OH 44242
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