used specially designed Logo computer environments, and one of which used manipulatives and paper and pencil, received eight lessons on geometric.
J. EDUCATIONAL COMPUTING RESEARCH, Vol. 1 l(2) 121-140,1994
EFFECTS OF COMPUTER AND NONCOMPUTER ENVIRONMENTS ON STUDENTS’ CONCEPTUALIZATIONS OF GEOMETRIC MOTIONS* KAY JOHNSON-GENTILE State University College at Buffalo DOUGLAS H. CLEMENTS State University of New York at Buffalo MICHAEL T. BATTISTA Kent State University
ABSTRACT
This study investigated the effects of computer and noncomputer environments on leaming of geometric motions. Two treatment groups, one of which used specially designed Logo computer environments, and one of which used manipulatives and paper and pencil, received eight lessons on geometric motions. Interviews revealed that both treatment groups, especially the Logo group, performed at a higher level of geometric thinking than did a control group. Both treatment groups outperformed the control group on immediate and delayed posttests; though the two treatment groups did not significantly differ on the immediate posttest, the Logo group outperformed the nonLogo group on the delayed posttest. Thus, there was support for the effectiveness of the curriculum and for the notion that the Logo-based version enhanced the construction of higher-level conceptualizations of motion geometry.
The major focus of standard elementary school geometry curricula is on recognizing and naming geometric shapes, writing the proper symbolism for simple geometric concepts, and using formulas in geometric measurement. Too often *This material is based in part upon work supported by the National Science Foundation under Grant No. MDR-865 1668. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the National Science Foundation. We thank the directors of Grant No. H023C70500, U.S.Department of Education, Office of Special Education and RehabilitationServices, for assistance in data entry. Verbal Problem Solving for the Mildly Handicapped.
121 Q 1994, Baywood Publishing Co.. Inc.
doi: 10.2190/49EE-8PXL-YY8C-A923 http://baywood.com
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these cumcula consist of a hodgepodge of unrelated concepts with little attention to dynamic, spatial concepts and little systematic progression to higher levels of thought, levels requisite for sophisticated concept development and substantive geometric problem solving [l-31. This study investigated the effects of computer and noncomputer environments designed to systematically facilitate such conceptual growth in the domain of geometric motions. The goal of the Motions strand of the Logo Geometry curriculum [4, 51 is to introduce students to geometric transformations (via their physical counterparts, slides, flips, and turns) and help them construct cognitive “building blocks” (such as mentally rotating shapes) that are important in dealing with geometric and spatial problems. The curriculum emphasizes helping students build relational understanding of the subject matter. Some research indicates that motion geometry tasks are tractable for elementary school students [6] and that such tasks may improve spatial abilities 171, although there is conflicting evidence suggesting that students this age can not effectively study geometric motions [8-lo]. These studies suggest that while rigorous mathematical transformations may be problematic for elementary age students, a less formal approach, including explicit attention to the transition from physical movements to more mathematically sophisticated representations, may be beneficial 11, 111. The Logo Geometry cumculum employs this approach. Finally, the cumculum is designed to aid students in attaining higher levels of thinking in geometry, as defined by the van Hieles [12]. The van Hiele framework has been applied mostly to the properties of twodimensional figures. In this case, children initially perceive geometric shapes, but attend to only a subset of a shape’s visual characteristics. They are unable to identify many common shapes. This constitutes pre-recognition, or level zero, in the modification and mmmbering of the van Hiele hierarchy used here [l, 51. At the first or visual level, children identify and operate on shapes and other geometric configurations according to their appearance. They recognize figures such as squares and triangles as visual gestalts, often refemng to real-life exemplars, saying, for instance, that a given figure is a rectangle because “it looks like a door.” They do not, however, attend to geometric properties characteristic of the class of figures represented. hoperties are implicit in the recognized figures, of course, but are experienced by the children only intuitively. At the second or descriptive/analyticlevel, children can explicitly recognize and characterize a class of shapes by their properties. For instance, they might think of a rhombus as a figure that has four equal sides. In this article, we apply this framework to the domain of geometric motions. The Logo Geometry cumculum makes extensive use of computer as an instructional tool [4,5]. Although isolated studies have demonstrated increased learning upon the introduction of a new medium, it is often possible to attribute the results to a change of cumculum or teaching strategy [l, 131. Therefore, comparison groups should be included to study the benefits of computer environments per se.
CONCEPTUALIZATIONS OF GEOMETRIC MOTIONS / 123
The present study included such a comparison group, which was provided the same cumcular activities, with one change: Noncomputer manipulative experiences were used that duplicated Logo experiences as closely as possible. Given such similarity, what unique benefits would the Logo computer environment be expected to contribute? First, the computer may act as a transition device to more abstract settings, as helps students tie visual to symbolic representations [l, 141. Students’ symbolic representations can emerge from their activity (using Logo commands to transform geometric objects), rather than being imposed by the teacher, and thus Logo mediates between thought and action by providing a linked, external, symbolic representation [15]. Specifically, compared to manipulating objects, specifying motions in Logo demands a higher level of abstraction. There is empirical support for such a hypothesis in the domain of geometric figures [16-201, and, to a weaker extent, geometric motions [5,21,22]. Second, the Logo motions environment is more precise than paper-and-pencil and manipulatives and can provide a concrete justification for the use of precise and formal language [23]. Nevertheless, it maintains a manipulative character that may be more critical than “real-object” status in contributing to the development of mathematical concepts [ 11. Third, because the computer environments are designed for the quick and easy manipulation of geometric ideas, students can test the ideas for themselves on the computer, and receive feedback regarding discrepancies between what they expected and what their actions produced graphically. Fourth, the computer does exactly what it is told, without mistake-whereas with paper and pencil, students may decide correctly what needs to be done and yet perform the operation incorrectly or vice versa. We posit that this type of manipulative-based, empirical approach is consistent with the way fifth and sixth grade students reason. Fifth, there is evidence that working within such a Logo environment may help students become explicitly aware of, and thus progress beyond, their mathematical intuitions, thus facilitating their transition from the visual to the descriptive/analytic geometric thinking in the van Hiele hierarchy [16,17,20,22]. The major goal of the present study was to examine whether the curriculum positively affected students’ conceptualizations of geometric motions, including their levels of thinking in that domain, and whether the effects differed for a computer and noncomputer implementation of the cumculum. Given the possibility of gender differences, especially in a domain connected closely to spatial visualization, effects of gender were also assessed. METHOD
Subjects Subjects were 223 students, 125 boys and ninetyeight girls, from six fifth-grade and three sixth-grade classrooms (for a total of 9 classrooms, each with a different
124 I GENTILE, CLEMENTS AND BAlTlSTA
tenured teacher). Two fifth-grade classes (1 from a suburban school, 1 from a large urban school in a different district) and one sixth-grade class (from the same urban school) were assigned to each of three treatment conditions (for a total of 3 classes in each condition): 1) the Logo group was instructed with the Motions strand of the Logo Geometry curriculum [4] (n = 74; 2) the non-Logo group was instructed with the same activities but with Logo tasks replaced by noncomputer manipulative experiences designed to match the former as closely as possible (n = 73);or 3) a nontreatment control group that received the regular mathematics program, including two days of textbook instruction on symmetry, included to provide a benchmark for the expected gains of the two treatment groups (n = 76).
Procedure All students completed a test of general achievement in geometry (see the following Instruments section) before receiving any instruction in geometry. Teachers for the two treatment conditions were instructed in the motion geometry curriculum in two separate groups. These teachers worked through the activities [4]; the only difference was the use of Logo or paper-and-pencil manipulatives for teachers in the two treatments. The instruction of teachers consisted of three sessions of two to three hours in length in January. The teachers taught the appropriate version of the eight-day motions unit to their students during the spring semester. Each classroom was visited two times by one of the authors to confirm that the activities were being implemented in a valid manner. This was basically confirmed, although the Logo teacher without previous computer experience did show signs of struggle in dealing both with new content and a new medium (though there were no discernable teacher effects). Upon completion of instruction in the motions unit, an immediate posttest on motion geometry was administered to all students by their teachers. Within four days after completion of the unit, a thirty minute interview was conducted with a random sample of thirty-six students, two boys and two girls selected from each classroom via a stratified random sampling procedure. The motions posttest was readministered to all students one month later; no geometry instruction of any kind was given between the two administrations of this posttest by any teacher. Time and other school constraints prevented readministration of the interview.
Treatments The Logo Geometry motions strand consists of eight lessons focusing on symmetry, geometric motions (slides, flips, and turns), and congruence [4]. The lessons, including differences between the two treatment groups, will be briefly described. For all lessons, students in both treatment groups were provided with activity sheets identical except for small matters such as directions for loading and using computer programs or using manipulatives.
CONCEPTUALIZATIONSOF GEOMETRIC MOTIONS / 125
In the first lesson, students were introduced to the concept of symmetry and used Miras to identify lines of symmetry. (A Mira is a piece of red transparent Plexiglas that allows students to view a geometric figure ‘and its reflection simultaneously.) There was no difference between the treatments for this lesson. In the second lesson, students learned to predict mirror images and to construct symmetric figures. Logo students checked predictions and constructed symmetric figures using a Logo MIRROR program that draws a mirror line and then automatically draws the mirror image of any figure constructed. NonLogo treatment students use paper and pencil and Miras for drawing and checking. The third lesson introduced the concepts of congruence and the slide, turn, and flip motions, including flip lines and turn centers. Students were asked to label the slide, flip, and turn images on an activity sheet. There was no difference between the treatments for this lesson. In the fourth lesson students learned methods for finding slide, flip, and turn images and used these methods for solving problems such as landing a rocket on a pad. Logo students worked with the Logo Geometry MOTIONS environment that extends the capability of Logo by adding pseudo-primitives for performing geometric motions [4,5,24]. For example, students can turn a predefined screen figure such as a rocket around any point, through any amount of rotation. NonLogo students performed motions by drawing geometric figures on sheets of acetate and sliding, turning, or flipping the sheets. For turns, for instance, they placed a pencil point at the turn center and used a protractor to determine the amount of turn. The Logo students used the Logo Geometry MOTIONS microworld and the nonLogo students used the acetate sheet method for all subsequent lessons. In lesson five, students predicted slide, flip, or turn images on paper, then checked their predictions by performing the indicated motion. In the sixth lesson, students determined visually whether two drawn figures were congruent, then checked this visual estimate by attempting to find a sequence of motions that moved one onto the other. For some problems they also recorded the corresponding parts of the figures. In the seventh activity, students created symmetric figures using flips and tested figures to see whether they were symmetric by attempting to flip the figures onto themselves. In the eighth lesson, students were asked to discover how slides and turns can be expressed as compositions of flips.
Instruments Geometry Achievement Test
Pretreatment geometry achievement was measured by the Geometry Achievement Test developed in the context of the Logo Geometry project [ 5 ] , an assessment administered to all students as part of a larger evaluation project. Items measured student knowledge of a variety of geometric concepts-including angle, angle measure, rotation, two-dimensional figures (identification and properties),
126 / GENTILE, CLEMENTS AND BAlTISTA
congruence, symmetry, and path, as well as students’ levels of thinking and problem-solving ability in this domain. No items were related to Logo or to motion geometry because most students had no previous experience with these topics. Items were scored as correct or incorrect for a total possible score of 116. Using Cronbach’s alpha, reliability was estimated as .75. Criteria for van Hiele Levels of Thinkingin Motion Geometry
The van Hiele theory was used for describing students’ geometric concept development [l, 25-28]. The levels used for motions are as follows [S]. At level 0, prerecognition, students cannot recognize distinct motions, whereas students at level 1, visual, recognize and delineate the motions of slide, flip, and turn. At level 2, descriptivdanalytic, students think of the slide, flip, and turn motions in terms of their components (a slide has a distance and a direction; a turn has a turn center and an amount and direction of turn; a flip has a flip line). Students are also aware of the properties these motions possess; for example, all points on a figure and its preimage are equidistant from the flip line. Students at level 3, abstracthelational, can see and informally justify relationships between motions. Equivalence of certain compositions of motions to other motions is recognized; for example, a rotation can be equated with a sequence of two flips. Minimal sequences of motions to move one object onto another are sought and found. These descriptors were used as criteria for assessing student responses to specific items in terms of van Hiele-based level of thinking. lnterview
A subsample of students was interviewed with the Logo Geometry interview [5]. This structured interview included six items that investigated levels of thinking within the domains of symmetry, motions, and congruence. Two items asked students to group cards with pictures of “horses” (item 1) or “tiles” (item 4)so that the cards that they grouped were “alike in some way” (created in consultation with Richard Lehrer). The tiles were created by crossing two basic figures with the three isometric motions (Figure 1). The horses included a prototype card shown first to the students and several “cousins” that were either noncongruent to the prototype or the result of a single motion performed on it. For item 2, students were shown two congruent triangles on a grid and were asked to explain to someone out of sight how to move one triangle onto the other (Figure 2). Item 3 showed a square grid with three different regions (Figure 3). Students were asked to determine which region covered more space and to tell how they “figured it out.” Item 5 presented half of a symmetric figure and the line of symmetry. Students were asked to complete the figure so that it was symmetric about the line and then to describe what symmetry meant in their own words. On item 6, students were asked to tell what it means to say that two figures are congruent and how they would know for sure that they are congruent.
CONCEPTUALIZATIONS OF GEOMETRIC MOTlONS I 127
4. Place the tiles in the following configuration: axes/slide mountainshurn
mountains/flip axedflip
axes/tum mountains/slide
Say, “Look at these tiles. Put the tiles into groups so that the tiles in each group are alike in some way.” Record the student’s sort on this sheet. Point to one of the student’s groups and ask: Why did you put these together?” Continue asking this question about each group, recording student’s response for each.
Figure 1. Directions for interview item 4.
Interviews were individually administered. Students were asked to think out loud and to explain the reasons for their responses whenever these were not clear. The authors classified each response into response categories and van Hiele level (based on the aforementioned criteria), discussing any disagreements until consensus was reached. Specific descriptions of categories and responses are provided with the results. Interrater agreement, assessed by independent scoring (i.e., before any discussion) of three full interviews by two researchers, was 85 percent.
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2. Say, “You and your friend are playing a game on the telephone. Here is what your friend sees (indicate shaded triangle). You are supposed to tell himher exactly how to move hidher picture so that it fits onto this picture (indicate unshaded triangle) exactly. Remember, you are talking on the telephone, so your friend cannot see you or your picture. You can use this piece to help you”. Give the student the cut out triangle. Record the successive positions in which the student places the triangle.
.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... ........... .. .. .. .. .. .. .. .. .. .. .. ..
.. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. ........... ........... ... ........... ... .
.
I
.................
................. Figure 2. Directions for interview item 2.
Motions Posttest
Students’ knowledge of motions concepts, including symmetry, identifying motions, and determining congruence via motions was measured by the Logo Geometry Motions Unit test, grades four to six [5]. To assess knowledge of symmetry, students were asked to determine whether a given line was a line of symmetry for a figure (6 items), “draw the other half’ to create a symmetric figure (3 items similar to interview item 3,and draw all the lines of symmetry for a given figure (8 items). In the domain of motions, students were given two congruent figures and asked to identify the motion that would move one onto the other, drawing any turn centers and flip lines (5 items, 9 points), and were given a preimage and a motion and asked to draw the resultant figure (4 items). For determining congruence via motions, students were asked to determine whether pairs of figures (in varied orientations) were congruent (6 items), and find among close distractors a polygon that was congruent to a given polygon (2 items).
CONCEPTUALIZATIONSOF GEOMETRIC MOTIONS / 129
2. Say, “Suppose (pretend) this is a floor. (Show student the grid.) These objects have been placed on the floor. Which object covers more space on the floor? Tell me how you figured it out. (Record spatial moves on the diagram.)”
Figure 3. Directions for interview item 3.
Finally, students were asked about the truth of statements such as “If two figures are congruent, there is a sequence of flips that will move one onto the other” (4 items). Items were scored as correct or incorrect for a total possible score of forty-two. Delayed and immediate posttests were identical. Using Cronbach’s alpha, reliability was estimated as .84 and .82 for the immediate and delayed administrations, respectively.
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GENTILE, CLEMENTS AND BAITISTA
RESULTS Means on the pretreatment geometry achievement test were similar-Logo, M=74.14,SD= 11.29;nonLogo,M=73.59,SD=11.90;andcontrol,M=74.35, SD = 12.14. A 3 x 2 (treatment x gender) ANOVA performed on these data revealed no significant differences (F(2,207)= .15, MS,= .141.09,p = .95); thus, there was no evidence that the groups differed in geometry achievement before the treatment was initiated (there were no grade differences in any analyses). Table 1 presents responses of the interview subsample to the two sorting items. Control students were more likely to sort by location, orientation, and other characteristics irrelevant to motions, or to state that both shapes “looked like” some object. Some sorted by slides, but not one sorted by turns or flips. Thus, all responses for control students were coded at level 0. On the first sorting item, nonLogo students sorted more often by turns and flips than did students in the Logo group. There was no such difference on the second item. This pattern is reflected in the categorization by levels; on the first item, about twice as many nonLogo as Logo students were classified as responding at level 1 (9 versus 4), whereas about the same number were so classified on the second item (8 versus 9). On item 2 (move one triangle onto another), the majority of the control students’ responses were at level 1 or below (Table 2). Most nonLogo students’ responses fell between levels 1 and 2; in comparison, most Logo students’ responses were at level 2. For example, one Logo student said, “turn the figure right ninety degrees from the turn center here (indicating), now flip the figure over this flip line (indicating). It fits!” Table 3 illustrates a similar pattern for symmetry (item 5 ) although with somewhat less striking differences between groups. On item 3, dealing with area, most of the responses were classified at level 2, with the Logo students having the greatest number of these responses, followed by nonLogo and then control (Table 4). This item is the only one for which there was any indication of a gender difference. Table 4 shows the single difference in strategies used: The subtractive strategy was used by a total of nine girls and two boys. The counting half-unit triangles strategy was used by three girls and ten boys. On item 6, no control students explicitly mentioned motions when asked what they would do to “know for sure” that any two shapes were congruent. Five said they would measure but did not describe what would be measured, two mentioned “putting one on top of the other,” and five said they didn’t know or gave inconsistent or uninterpretable responses; for example, “If one was [sic] smaller and one was bigger, they would be congruent” (all level 0 responses for the domain of motion geometry). Five nonLogo students explicitly mentioned motions and their relationship to congruence (a level 2 response); for example, “. . .by flips, slides, and turns you could move one figure onto another to see if they fit. It’s important to check by putting one onto the other.” Seven said they would prove congruence “by taking a piece of tracing paper, tracing one shape, and then putting it on the
CONCEPTUALIZATIONSOF GEOMETRICMOTIONS I 131
Table 1. Frequencies for Treatment Conditions: Response Categories and Levels of Thinking for Sorting Tasks Item 4: Tiles
Item 1: Horses Logo
NonLogo
Control Logo
NonLogo
Control
1 1
4 2 5
~~~
Response Categories
. Both look like Location Heading/orientation (e.g., heading in the same direction) Common characteristic(s) (e.g., "tails are the same") Otherfirrelevant Same shape (only) Same size and shape Slide images Turn images Flip images
1 8
4
1 2
8 11
1
1
1 2
8
5 8 4 4
10 9 9
9
1 3
9 8
11 10 10
3
3
8
9
8
Levels of Thinking 0 Does not sort by motions; for example, They just match," "these are facing this way" (one representing a flip and the other a 180" turn) or "I can't do it." 0 -> 1 1 Sorts consistently, exhaustively, and explicitly by slides, flips, and turns. 2 Sorts as for level 1 and explains the motions used in term of their properties (e.g., "These are turns to the right of 900").
7
2
1 4
1 9
12
Note: Data from interview subsample; for each group, n = 12.
1
12
132 I GENTILE, CLEMENTS AND BAlTlSTA
Table 2. Levels of Thinking for Item 2, Move the Triangle ~~
Level 0 Move the triangle without matching corresponding parts; also, state 'I don't know." 0 -> 1 1 Recognize the motions but without necessarily using correct terminology (slides, flips, turns) and display little effort to be precise. For example, "Move it to the right, then down. Now turn it over" (actually a flip). Others may use a term such as "turn," but not specify tum centers or direction and amount of tum. 1 ->2 2 Conceptualize motions in terms of properties; for example, Turn the triangle right 90 degrees from the tum center, which is the vertex of the right angle of the triangle."
Logo
NonLogo
Control
2
2 2
4
3
7
7
1
8
Note: Data from interview subsample.
other" (a level 1 response). Eight Logo students explicitly mentioned motions and the motions-congruence relationship but did not mention computers or specific computer procedures; for example, "You could use a series of motions-flips, slides, and turns-to try to move one figure onto the other. If they match exactly, they're congruent." Three said they would put one figure on top of the other and one said he would measure the sides and angles. Thus, the nonLogo group tended to be more closely tied to a procedural description involving the manipulatives used during the treatment. Table 5 presents the means and standard deviations of the posttest scores of all the students in each of the three conditions, Logo, nonLogo, and control. A 3 x 2 x 2 (treatment x gender x time) ANOVA computed on these scores revealed a significant treatment effect, F(2.191) = 125.32,MS,= 16.32.p c .001. There was no significant effect for time, but a significant interaction between treatment and time was identified, F(2.191) = 6.69, MS,= 5 . 2 4 , = ~ .002.NeumanKeuls analyses revealed that both the Logo and nonLogo scores were significantly higher than the control scores for both the immediate and delayed posttests (p c .01). In addition, the Logo delayed posttest score was significantly higher than both the nonLogo delayed posttest score (p c .01) and the Logo immediate
CONCEPTUALIZATIONSOF GEOMETRIC MOTIONS I 133
Table 3. Levels of Thinking for Item 5, Symmetry Level
0 Does not recognize what symmetry is; for example “Idon’t know“ or guess. 0 -> 1 1 Use informal language, describing symmetry according to its appearance (the way it looks); for example, “It’s when something is the same on both sides.” 1 ->2 2 Conceptualize symmetry in terms of properties, such that to be symmetrical a figure must have a line of symmetry which divides the figure into two congruent parts. For example, “A figure is symmetrical when it has a line of symmetry that divides it exactly in half. If you flip one half onto the other, they would be the same size and shape, or congruent.” Note:
Logo
NonLogo
Control 2
3
1
3
4
5
2 6
5
1 1
3
Data from interview subsample.
posttest score @ < .05). The n o n h g o group’s delayed posttest score was significantly lower than their immediate posttest score @ c .05). There were no significant differences between boys and girls (the main effect for gender missed significance, (F(1,191) = 3.38, MS,= 1 6 . 3 2 ,= ~ .068), nor was there an interaction between treatment and gender. No other interaction was significant. To attempt to further illuminate differences between the two administrations, average scores for individual items were examined. There were few consistent, substantive changes from immediate to delayed posttests for those items dealing with drawing the figure resulting from a given motion, congruence, or “truth of statements.” Patterns did emerge in the other types of items, assessing symmetry and motions identification. For the “draw the other half’ symmetry items, control students scores tended to increase (average of +.06);Logo students remained at the same level (.OO) and nonLogo students’ scores decreased (-.06). On the other two types of symmetry items a different pattern emerged. When asked to draw all the lines of symmetry for figures with one line of symmetry or for figures for which it was visually evident that there was no line of symmetry, there was
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Table 4. Levels of Thinking and Strategies for Item 3, Floor Space Problem
NonLogo
Logo
Level of Thinkindstrategy 0-> 1 Guesses; visually-based but incorrect; e.g., "If you slide them [the two parts of B] together they're bigger than A and C."
G
B
1
Makes visual estimate; e.g., 'It just looks bigger."
1
Counts the squares and then estimates -> 2 Uses a subtractive strategy; that is, counts half unit triangles outside the figures but inside a given rectangle; however, uses it incorrectly and switches the unit.
Control
G
B
G
B
1
1
1
1
1
1
2
1
1
1
2
Counts half-unit triangles that compose the figures.
Uses subtractive strategy as above, but accurately. ~~~
5
5
2
3
1
2
1
1
2
1
~~~
Note: Data from interview subsample. G = girls, B = boys.
little difference between the groups (i.e., the differences between immediate and delayed posttest scores of the groups differed by less than .05 for each item). However, on each of the four figures that either had more than one line of symmetry or had no line of symmetry but appeared that it might (e.g., a parallelogram), the difference between the two scores for the Logo students (average +.lo) was substantially greater than that for the other two groups (nonLogo, 11; control, +.03). Similarly, on the items that asked students to determine whether a given line was a line of symmetry, there was only one item on which there was a substantive difference; Logo students scores on a nonsymmetric parallelogram increased substantially )+.15), whereas those of the control group did not change
-.
CONCEPTUALIZATIONS OF GEOMETRIC MOTIONS / 135
Table 5. Means and Standard Deviations for Motions Posttest Scores
Non-Logo
Logo Motions Posttest Immediate Delayed
Control
M
SD
M
SD
M
SD
36.59 37.50
3.17 2.91
36.88 35.75
2.69 3.54
29.38 29.90
4.03 3.21
Note: Data from all students.
(.OO) and those of the n o n h g o group decreased (-.12). The final group of items asked students to identify the motion that would move one figure onto another, drawing any turn centers and flip lines. There were no substantive changes on the slide scores. On six of the seven scores dealing with flips, flip lines, turns, and turn centers, differences between the Logo group’s scores were greater than those of the other two groups (average difference, +.04),which were near zero or slightly negative (-.04for the control, -.03 for nonlogo; for one flip line score, the nonLogo group showed the greatest increase). DISCUSSION Results supported the efficacy of the curriculum across computer and noncomputer implementations. Groups did not differ significantly on pretreatment geometry achievement, but both treatment groups significantly outperformed the control group on immediate and delayed geometric motions posttests. The Logo and nonLogo interview subsamples substantially outperformed the control subsample on tasks measuring levels of geometric thinking. This lends support to the hypothesis that instruction on geometric motions (rather than mathematical transformations per se) that provides students with a language and an effective mechanism for accomplishing such motions can be effective for intermediate grade students. Given the inconsistent results from previous studies teaching transformations or motions to elementary students, this result provides valuable guidance for teachers and curriculum designers. Gender differences appeared on only one interview item, and this difference was in strategy used, not in level of thinking evinced or correctness. Over three times as many boys as girls used the strategies of counting half-unit triangles that compose the figures and thus “building up” the area measure. In contrast, over four times as many girls as boys used a subtractive strategy, first enclosing the figures with congruent rectangles and then counting the half-unit triangles outside the figures. While based on only one item, such differences are striking enough to recall Erikson’s [29] research, in which children of the same age as those in the present study were asked to build with blocks and other toys. Almost invariably,
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girls first constructed structures that enclosed, such as houses with walls, and then partially filled the resulting spaces. Boys built up designs such as towers, piece by piece. In contrast, Tartre reported that high school males more than females utilized what she called a “complement” strategy (i.e., subtracting from the whole) in solving problems involving representations of three-dimensional objects [30]. Whether such differences in strategies can be replicated and whether they change with development are questions for future research. Across all other measures in the present study, there were no significant gender differences; however, a main effect for gender on the posttests should be tested through replication, considering the significance level reported here @ = .068). Differences between the two treatment groups, while not simple, reveal a pattern suggesting that Logo aids students in the construction of higher-level conceptual structures; that is, Logo students were more likely to forgo visual strategies and use level two, descriptivehnalytic strategies for solving geometric problems. On the paper-and-pencil posttest involving all students, the Logo group’s scores significantly increased from immediate to delayed posttest, whereas those of the nonLogo students significantly decreased, resulting in a significant difference in favor of the Logo group on the delayed posttest. Investigation of individual items suggested specific effects on symmetry and motions concepts. Logo group scores increased, and nonLogo group scores decreased, on tasks involving figures that had more than one line of symmetry or on parallelograms with no lines of symmetry. Logo students appeared more likely to identify horizontal and oblique lines of symmetry and less likely to be misled by the rotational symmetry of the parallelogram. This suggests that differences between the groups were on those items that required students to resist applying intuitive visual thinking and apply analytical thinking in a comprehensive manner. In a similar vein, scores of the Logo group increased on items requiring the identification of motions and their properties (i.e., turn centers). While caution must be used in interpreting these post hoc analyses, they do suggest that significant differences on the delayed test can be attributed to Logo students’ construction of higher-level conceptual structures; that is, Logo students analyzed components and properties of geometric motions (van Hiele level 2). Other evidence supports this connection. For example, on interview item 6, responses of the nonLogo students were closely tied to a procedural description involving the manipulatives used during the treatment; this was not true for the Logo students, whose descriptions were at a more general and abstract level. Further, on four of the six items, the Logo group responded at higher levels of thinking than did the nonLogo group (making more level 2 responses). How might one then understand the ostensibly inconsistent results from the sorting tasks? On the first of the two items that involved sorting, the nonLogo students sorted more often by turn and flip images (i.e., made more level 1 responses); however, this difference disappeared on the second sorting item. If learning in Logo, compared to learning with manipulatives, particularly emphasizes analytic rather than visual
CONCEPTUALIZATIONS OF GEOMETRIC MOTIONS I 137
strategies, this result is not incongruous: the sorting tasks emphasized visual thinking. Indeed, as has been argued previously [ 16, 171, Logo seems to facilitate the transition from the visual to the descriptivdanalytic level. This is consistent with research indicating that van Hiele-based instruction using the Geometric Supposer significantly raised students’ van Hiele level of thought, more so from level 1 to level 2 than for any other levels [31]. The pattern of results on the sorting tasks suggests that administration of assessment in a different medium should be considered. In all assessment, we took the conservative stance of paper-and-pencil testing-a format more familiar to the nonLogo students. It may have been more difficult for Logo students to transfer what they had learned; to “see” motions in the paper representations on the first sort. The need in Logo environments for more complete, precise, and abstract explication may account for students’ creation of conceptually richer concepts for motions. That is, in Logo students have to specify steps to a noninterpretive agent, with thorough specification and detail. The results of these commands can be observed, reflected on, and corrected; the computer serves as an explicative agent. In noncomputer manipulative environments, one can make intuitive movements and corrections without explicit awareness of geometric motions. For example, even young children can move puzzle pieces into place without conscious awareness of the geometric motions that can describe these physical movements. In the noncomputer environment used here, attempts were made to promote such awareness, but descriptions of the motions were generated from, and interpreted by, physical motions of students, who also understood the task and thus interpreted the descriptions in that context. In contrast, when “intuition is translated into a program it becomes more obtrusive and more accessible to reflection” [32, p. 1451. Observations during the sessions indicated that Logo students were more likely to discuss the motions themselves, treating the motions as concrete processes and objects of reflection, as opposed to discussing the figures or manipulatives undergoing the motion. Such facilitation of reflection was especially in the error detection and “debugging” phases of problem solving. Past research has shown strongest effects on cognitive monitoring [33, 341; again, this may foster highlevel metacognitive awareness that supports memory-especially memory as a reconstructive process that rebuilds, rather than recalls, ideas [35]. This interpretation of the results is consonant with previous research that indicated prolonged retention and continuous construction of early Logo-based schema for geometric concepts [36]. In summary, considering the interview and motions posttest data together, there is no evidence that the use of the Logo motions cumculum helped children perform tasks at level 1 of geometric thinking; that is, such use did not appear to facilitate the ability to draw symmetric figures, draw figures given specified motions, or apply motions to problems such as sorting tasks. There is evidence that the use of Logo facilitated students’ transition to level 2 thinking. Logo students were more likely to describe the properties of geometric motions and
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symmetry constructs and more likely to solve problems involving these constructs at a more abstract and analytical level. This adds to the body of evidence in support of the hypothesis that Logo experiences can help students become cognizant of their mathematical intuitions and facilitate the transition from visual to descriptive/analytic geometric thinking, not only in the domain of twodimensional figures, but also in that of motion geometry [l, 16,17,20,22,37,38]. This is significant, given that even in secondary school, students may be at the lowest two van Hiele levels in transformation geometry, showing limited use of precise language and little knowledge of properties [27]. It is important to recall that the main difference between the treatments was the use of the Logo environments and this difference existed for only six of the eight lessons. If replicated, these results have implications for research on the effects of media-there are domains in which computer tasks, embedded thoughtfully in a complete educational environment, can make a significant contribution.
REFERENCES 1. D. H.Clements and M. T. Battista, Geometry and Spatial Reasoning, in Handbook of Research on Mathematics Teaching and Learning, D. A. Grouws (ed.), Macmillan, New York, pp. 420-464, 1992. 2. V.L.Kouba, et al., Results of the Fourth NAEP Assessment of Mathematics: Measurement, Geometry, Data Interpretation,Attitudes, and Other Topics, Arithmetic Teacher, 33, pp. 10-16, 1988. 3. A. Porter, A Cumculum Out of Balance: The Case of Elementary School Mathematics, Educational Researcher, 18, pp. 9-15, 1989. 4. M. T. Battista and D. H. Clements, Logo Geometry, Silver Burdett & Ginn, Momstown, New Jersey, 1991. 5. D. H. Clements and M. T. Battista, The Development of a Logo-based Elementary School Geometry Curriculum (Final Report: NSF Grant No.: MDR-8651668), State University of New York at Buffalo, Buffalo, New YorkKent State University, Kent, Ohio, 1991. 6. K. A. Schultz and J. D. Austin, Directional Effects in TransformationalTasks, Journal for Research in Mathematics Education, 14, pp. 95-101, 1983. 7. J. J. Del Grande, Can Grade Two Children’s Spatial Perception be Improved by Inserting a Transformation Geometry Component into their Mathematics Program?, Dissertation Abstracts International, 47, p. 3689A, 1986. 8. S. A. Shah, Selected Geometric Concepts Taught to Children Ages Seven to Eleven, Arithmetic Teacher, 16, pp. 119-128, 1969. 9. F. R. Kidder, Elementary and Middle School Children’s Comprehension of Euclidean Transformations,Journal for Research in Mathematics Education, 7, pp. 40-52, 1976. 10. H.J. Williford, A Study of Transformational Geometry Instruction in the Primary Grades, Journalfor Research in Mathematics Education, 3, pp. 260-27 1, 1972.
.
CONCEPTUALIZATIONS OF GEOMETRIC MOTIONS I 139
11. J. C. Moyer, The Relationship between the Mathematical Structure of Euclidean
Transformations and the Spontaneously Developed Cognitive Structures of Young Children, Journalfor Research in Mathematics Education, 9, pp. 83-92, 1978. 12. P. M. van Hiele, Structure and Insight, Academic Press, Orlando, Florida, 1986. 13. R. E. Clarke, Reconsidering Research on Learning from Media, Review of Educational Research, 53, pp. 445-459,1983. 14. C. Hoyles and R. Noss. Formalizing Intuitive Descriptions in a Parallelogram Logo Microworld, in Proceedings of the Fifteenth Annual Conference of the International Group for the Psychology of Mathematics Education, International Group for the Psychology of Mathematics Education, Veszprem, Hungary. pp. 417-424, 1988. 15. R. Noss and C. Hoyles, Afterword: Looking Back and Looking Forward, in karning
Mathematics and Logo, C. Hoyles and R. Noss, (eds.), MIT Press, Cambridge, Massachusetts, pp. 427-468, 1992. 16. D. H. Clements and M. T. Battista, Learning of Geometric Concepts in a Logo Environment, Journal for Research in Mathematics Education, 20, pp. 450-467, 1989. 17. D. H. Clements and M. T. Battista. The Effects of Logo on Children’s Conceptualizations of Angle and Polygons, Journal for Research in Mathematics Education, 21, pp. 356-371, 1990. 18. H. M. Findlayson, What do Children Learn Through Using Logo? D.A.I. Research Paper No. 237, British Logo Users Group Conference: Looghborough, U.K.. 1984. 19. R. Noss, Children’s Learning of Geometrical Concepts through Logo, Journal for Research in Mathematics Education, 18, pp. 343-362, 1987. 20. J. Olive, C. A. Lankenau. and S . P. Scally, Teaching and Understanding Geometric Relationships through Logo: Phase 11. Interim Report: The Atlanta-Emory Logo Project, Emory University, Atlanta, Georgia, 1986. 21. L. D. Edwards, Children’s Learning in a Computer Microworld for Transformation Geometry, Journal for Research in Mathematics Education, 22, pp. 122- 137. 1991. 22. A. T. Olson, T. E. Kieran, and S. Ludwig, Linking Logo, Levels. and Language in Mathematics, Educational Studies in Mathematics, 18, pp. 359-370. 1987. 23. W. Feurzeig and G. Lukas, LOGO-A Programming Language for Teaching Mathematics, Educational Technology, 12, pp. 39-46, 1971. 24. M. T. Battista and D. H. Clements, Using Logo Pseudoprimitives for Geometric Investigations, Mathematics Teacher, 81, pp. 166- 174, 1988. 25. W. Burger and J. M. Shaughnessy, Characterizing the van Hiele Levels of Development in Geometry, Journal for Research in Mathematics Education, 17, pp. 31-48, 1986.
26. D. Fuys, D. Geddes, and R. Tischler, The van Hiele Model of Thinking in Geometry among Adolescents, Journal for Research in Mathematics Education Monograph, 3, 1988. 27.
Y.P. Soon and J. L. Flake, The Understanding of Transformation Geometry Concepts of Secondary School Students in Singapore, in Proceedings of the Eleventh Annual Meeting, North American Chapter of the International Group for the Psychology of Mathematics Education. C. A. Maher. G. A. Goldin, and R. B. Davis, (eds.), Rutgers University, New Brunswick, New Jersey, pp. 172-177, 1989.
140 / GENTILE, CLEMENTS AND BATTISTA
28. I. Wirszup, Breakthroughs in the Psychology of Learning and Teaching Geometry, in Space and Geometry, Papersfiom a Research Workshop, J. L. Martin and D. A.
29. 30.
31.
32. 33.
34. 35. 36.
Bradbard, (eds.), University of Georgia, Georgia Center for the Study of Learning and Teaching Mathematics. (ERIC Document Reproduction Service No. ED 132 033). Athens, Georgia, pp. 75-97, 1976. E. H. Erikson, Childhood and Society, Norton, New York, 1%3. L. A. Tartre, Spatial Skills, Gender, and Mathematics, in Mathematics and Gender: Influences on teachers and students, E. Fennema and G. Leder. (eds.), Teachers College Press, New York, pp. 27-59. 1990. J. C. Bobango, P. M. Van Hiele Levels of Geometric Thought and Student Achievement in Standard Content and Proof Writing: The Effect of Phase-based Instruction, Dissertation Abstracts International,p. 2566A. 1988. S. Papert, Mindstonns: ChiMren, Computers, and Powerful Ideas, Basic Books, New York, 1980. M. T. Battista and D. H. Clements, The Effects of Logo and CAI Problem-solving Environments on Problem-solving Abilities and Mathematics Achievement, Computers in Human Behavior, 2, pp. 183-193, 1986. D. H. Clements, Metacomponential Development in a Logo Programming Environment, Journal of Educational Psychology, 82, pp. 141-149, 1990. M. Minsky, The Society ofMind, Simon and Schuster, New York. 1986. D. H. Clements. Longitudinal Study of the Effects of Logo Programming on Cognitive Abilities and Achievement,Journal of Educational Computing Research, 3, pp. 73-94,
1987. 37. E. Gallou-Dumiel, Reflections, Point Symmetry and Logo, in Proceedings of the
Eleventh Annual Meeting, North American Chapter of the International Group for the Psychology of Mathematics Education, C. A. Maher, G.A. Godin, and R. B. Davis, (eds.), Rutgers University, New Brunswick, New Jersey, pp. 149-157. 1989. 38. R. Lehrer and P. C. Smith, Logo Learning: Are Two Heads Better than One?, paper presented at the American Educational Research Association: San Francisco, California, 1986.
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