Learning of Geometric Concepts in a Logo Environment Author(s ...

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.


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.


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


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-



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-



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.



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).



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



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



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


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.



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



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?"



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


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.



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



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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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
