computer environments for learning geometry

J. EDUCATIONALCOMPUTING RESEARCH, Vol. lO(2) 173-197,1994

COMPUTER ENVIRONMENTS FOR LEARNING GEOMETRY* DOUGLAS H.CLEMENTS State University of New York at Buffalo MICHAELT. BAlTlSTA Kent State University

ABSTRACT

Given their graphic capabilities, computers may facilitate the construction of geometric concepts. Comparative media research, however, reveals few differences between media; alterations in curricula or teaching strategies might also explain the positive results of many studies that compare computer to noncomputer media. Yet, there remain certain computer functions that noncomputer media may not easily duplicate. This article reviews research to describe such functions of construction-oriented environments and to evaluate their unique contributions to students’ learning of geometry. Implications for the design of geometric computer environments for geometry education are drawn.

Computers, especially with their graphic capabilities, may facilitate the construction of geometric concepts [l].Students instructed in geometry with computers often score significantly higher than those provided noncomputer instruction [2, 31. Computer-based programs are effective even with young children; for

* Based on papers presented at the Fifteenth Annual Conference of the International Group for the Psychology of Mathematics Education, Geometry Working Group, Assisi, Italy, July 1991 at the Annual Meeting of the National Council of Teachers of Mathematics, NCTM RAC/AERA SIG Research Presession, Salt Lake City, Utah, April 1990. Time to prepare this material was partially provided by the National Science Foundation under Grants No. MDR-8651668, MDR-9050210, and MDR-8954664. 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. 173 0 1994, Baywood Publishing Co.,Inc.

doi: 10.2190/8074-298A-KTL2-UQVW http://baywood.com

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example, such programs can be as effective in teaching about shapes as teacher-directed programs [4]. Computer games have been found to be marginally effective at promoting learning of angle estimation skills [5], and significantly more effective than traditional instruction in facilitating achievement in coordinate geometry [3]. Enthusiasm for the use of computers should be moderated, however, by findings of comparative media research. Decades of pre-computer research reveal few differences between media [6]; changes in curricula or teaching strategies can be postulated to explain the positive results of many comparative studies of CAI [7]. There are, however, certain functions computers can perform that other media cannot easily duplicate. Do these functions affect the learning of geometry? In this article, we review research to describe such functions of construction-oriented computer environments and to evaluate their unique contributions to students’ learning of geometry (for a review of other types of geometric programs, such as CAI and computer tutors, see [l]). Following the frequency of their Occurrence in the research corpus, we will emphasize Logo programming and geometric construction programs. Finally, we will draw implications for the creation of quality computer-based geometry educationalmaterials. We begin with a brief presentation of our theoretical foundation [l]. CONSTRUCTING GEOMETRIC CONCEPTS

We have built our theoretical foundation on a synthesis of several constructivist positions. The first is Piaget and Inhelder’s theory that children construct representations of space through the progressive organization of the child’s motor and internalized actions, resulting in operational systems [8]. Therefore, the representation of space is not a perceptual “reading off’ of the spatial environment, but is built up from prior active manipulation of that environment. While empirical research has tended to focus on other themes, such as the topological primacy thesis, most is consistent with the constructivist theme as well [l]. The second is the van Hiele theory of levels of geometric thinking and phases of instruction [9, 101. The van Hieles’ claim that students achieve higher levels not via direct teacher telling, but through work on a suitable choice of tasks (some of which, of course, may involve students posing their own geometric problems). In addition, “children themselves will determine when the moment to go to the higher level has come” p.van Hiele, personal communication,September 27, 1988). Both theories emphasize the role of the student in actively constructing their own knowledge, as well as the non-verbal development of knowledge that is organized into complex systems. For example, van Hiele emphasizes that successful students do not learn isolated facts, names, or rules, but construct networks of relationships that link geometric concepts and processes. Students eventually organize these into networks [111. Thus, students must abstract mathematics from their own systematic pattern of activities. Teachers cannot successfully provide direct help to students who have not yet attained a certain level, only indirect

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assistance. “If you want to know how far children have made progress, do not wait for their imitation of your argumentation, but listen to them for what they have found out themselves” (P. van Hiele, personal communication, September 27, 1988). Thus, both theoreticians strongly disagree with the belief that good teachers merely explain clearly to children to teach them. Piaget and van Hiele also suggest similar mechanism of development-Piaget stresses the role of disequilibrium and resolution of conflicts; van Hiele implores teachers to recognize students’ difficulties, but not avoid “crises of thinking,” because these facilitate the transition to higher levels. Computer environments for learning geometry may make a substantive contribution to such activity-based,conceptual conflict-driven construction of geometric knowledge. LOGO Logo’s turtle graphics environment was one of the earliest to embody the aforementioned characteristics of computer environments. Because children’s initial representations of space are based on action [S], and because the mathematical concept of path can be thought of as a record of movement, the path concept may constitute a particularly good starting point for the study of geometry. Furthermore, Logo activities designed to help children abstract the notion of path should provide a fertile environment for developing their conceptualizations of simple two-dimensional shapes. For instance, with the concept of rectangle, students initially are able only to identify visually presented examples, a level 1 (visual) activity in the van Hiele hierarchy. In Logo, however, students can be asked to construct a sequence of commands (a procedure) to draw a rectangle. This “. . . allows, or obliges, the child to externalize intuitive expectations. When the intuition is translated into a program it becomes more obtrusive and more accessible to reflection” [12, p. 1451. That is, in constructing a rectangle procedure, the students must analyze the visual aspects of the rectangular path and reflect on how its component parts are put together, an activity that encourages level 2, descriptive/analytic, thinking in the van Hiele hierarchy. Furthermore, asking students if a square or a parallelogram can be drawn by a general rectangle procedure if given the proper inputs encourages students to start logically ordering figures, a level 3 activity. Research suggests that these theoretical predictions are valid. Students work in Logo relates closely to their level of geometric thinking [13]. In addition, appropriate use of Logo helps students begin to make the transition from the levels 0 and 1 to level 2 of geometric thought. For example, Logo experience encourages students to view and describe geometric objects in terms of the actions or procedures used to construct them [14]. Also, when asked to describe geometric shapes, children with Logo experience proffer not only more statements overall, but more statements that explicitly mention components and geometric properties of shapes, an indication of level 2 thinking [14-181. The latter effect may come about

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because, as recommended by van Hiele researchers, Logo activities encourage students to incorporate implicitly the types of properties that level 1 thinkers need to construct explicitly, something that textbooks often fail to do [19]. Evaluative research of our Logo Geometry curriculum [20] provides additional evidence that students analyze the visual aspects of figures, facilitatinga transition from the visual to the descriptive-analyticlevel of geometric thinking [21,22]. A class of first graders was investigating the concept of rectangle. The students had identified rectangles in the classroom and built them out of various materials such as blocks, tape, clay, and geoboards. They then went to the computer lab and were asked to make the turtle draw rectangles. (These students were in our Singlekey environment in which the turtle is given commands by typing single keys-F for forward (10 turtle steps), B for backward, R or L for right and left turns of 30”.)As the activity continued, all children were drawing rectangles in Logo. One of them tried to be different; he attempted to draw a rectangle that was tilted. He instructed the turtle to draw the first side using 5 Fs. He paused for quite some time as he came to the first turn, so one of the PI’S asked him how much he had turned before. He said three R’s and hesitatingly tried three. It worked to his satisfaction and he then drew the second side. He hesitated again, saying out loud, “What turn should I use?” The PI said, “How many turns have you been using?” He quickly issued three right turns, then hesitated again; “How far? . . Oh, it must be the same as its partner!” Effortlessly, he completed his rectangle.

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Even though this child had built several rectangles with sides horizontal and vertical, it was not obvious to him that the same commands would work for a tilted rectangle (or indeed that there was such as thing as a tilted rectangle). He had clearly abstracted that the opposite sides must be the same length, but he had not abstracted the measure of the turns or even that all the turns were of the same measure. Thus, the appropriate task in this Logo environment provided him with the opportunity to analyze and reflect on the properties of a rectangle. Note also that this child had posed a challenge for himself; the environment encouraged exploration. We have observed a similar phenomenon with second graders using the standard Logo commands. The students can easily give a procedure to make the turtle draw a square with its sides vertical and horizontal. But as soon as they are asked to draw an oblique square, they must confront their lack of abstraction of the required turns (like the first graders, they show evidence of having abstracted the notion that all sides are the same length). Such insights are often facilitated by discussion. A different first-grade class was discussing a group of quadrilaterals drawn on the chalkboard, trying to

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identify the rectangles (in preparation for writing a procedure to draw rectangles). They were focusing on a non-rectangularparallelogram: John: This one is slanted. It can’t be a rectangle. Cathy: Slanted doesn’t matter. It has two long sides, here and here, the same length and two short sides, here and here, the same length (motioning to indicate pairs of opposite sides). Eugene: But it doesn’t have square comers. Thus, the students were grappling with the properties of rectangles. These qualitative findings have been supported by quantitative data (involving 1,624 students in Logo and control groups). For example, Logo Geometry students performed significantly better than control students on items assessing the identification of triangles, rectangles, and squares within a large set of distractors [21,22]. There are often limitations, however, in the extent to which students synthesize visual and symbolic information. Hoyles and Noss reported that after working with Logo parallelograms, middle-school-age students demonstrated greater ability to provide precise definitions, more knowledge of properties, and increased tendency to discriminate at the symbolic level, for example, pointing at Logo code [Z].They noted, however, that some students shifted attention toward only the visual or only the symbolic, especially when they used procedures as tools. There was more evidence of synthesis between the visual and symbolic representations at the level of definition of a parallelogram-that is, how the geometric attributes of the parallelogram such as equality of opposite sides and angles were reflected in the Logo code, than there was at the level of geometric relationships inherent within the construction of the parallelogram, such as supplementary angle relationships [Z]. What specific role did Logo play? According to the authors, the Logo language allowed students to make a rough sketch of the problem and then attend to specific parts of the solution that needed work, without being distracted by other relationships involved. Students could form a solution in this way, then test it, and edit the solution based on visual feedback. Students made progress as they synthesized the symbolic and the visual; Logo allowed them to work on and connect these two representations. Consonant with other work, the nature of the task and teacher interventions were crucial [24]. The tasks required functional use of relationships and thus engendered students’ discriminationof those relationships; teacher interventions helped bring relationships between the visual and symbolic to an explicit level of awareness. Similar results have emerged in the area of motion geometry. -For example, working with a Logo unit on motion geometry, students’ movement away from van Hiele level 1 was slow, but there was definite evidence of a beginning awareness of the properties of transformations [13]. Middle school students

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achieved a working understanding of transformations and used visual feedback to correct overgeneralizationswhen working in a Logo microworld [25]. These results too are buttressed by research on the Logo Geometry curriculum. Intermediate-grade students were engaged in symmetry and motion geometry activities using either Logo or paper and pencil [21,22,26]. Interviews conducted with a subsample revealed that both treatment groups performed at a higher level of geometric thinking than did the control group; Logo students performed at a higher level than the noncomputer students on four of the six interview tasks, noncomputer students performed at a higher level on one. Intriguingly, the two exceptions allowed thinking only at levels 0 and 1. Thus, Logo may not facilitate the growth of abilities at the visual level more so than other media. Instead, Logo may facilitate the transition from the visual to the descriptive/analytic level, which is consistent with the evidence from the other four interview tasks and the posttests. Both Logo and nonLogo groups outperformed the control group on immediate and delayed posttests; in addition, 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 notion that the Logo-based version developed more meaningful ideas that could be more easily reconstructed. Such studies indicate that compared to students using paper and pencil, students using Logo worked with more precision and exactness [21, 22, 26, 271. The need for complete, precise, and abstract explication may account for students’ creation of richer concepts of motions. 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 environments, teachers made attempts 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, the computer environments promoted more discussion of the motions as objects themselves. This was especially true in the error detection and “debugging” phases of problem solving. Past research has shown strongest effects on the metacognitive ability of monitoring [28, 291; again, this may foster high-level metacognitive awareness that supports memory-especially memory as a reconstructive process that rebuilds, rather than recalls, ideas [30]. 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 [31]. Thus, there is evidence in support of the hypothesis that Logo experiences can help children become cognizant of their mathematical intuitions and facilitate the transition from visual to descriptive/analyticgeometric thinking in the domains of shapes, symmetry, and motions. In addition, this work-especially in the areas of symmetry and motion geometry-tends to be more precise and exact than similar work done without Logo.

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Research also supports the hypothesis that work with Logo helps students construct more viable knowledge-that is, knowledge schemes in which concrete experiences are connected to abstractions, which are internally consistent, and which can be flexibly applied to a variety of problem situations. For example, several projects have investigated the effects of Logo experience on students’ conceptualizations of angle, angle measure, and rotation. In one study, responses of control students were more likely to reflect little knowledge of angle or common language usage, whereas the responses of the Logo children indicated more generalized and mathematically-oriented conceptualizations (including angle as rotation and as a union of two lines/segments/rays) [14]. Other researchers studied how Logo might provide experiences at the second and third van Hiele levels for ninth-grade students [32]. Logo students gained more than the control students on interviews that operationalized the van Hiele levels for the concept of angle. Several other researchers have similarly reported a positive effect of Logo on students angle concepts [32, 331, although in some situations, benefits do not emerge until more than a year of Logo experience [34]. In addition, Logo experience generally appears to facilitate understanding of angle measure [14, 16,35371 with students showing progression from van Hiele level 0 to level 2 in the span of the treatment [15]. Benefits may not emerge, however, in unstructured situations [38]. Even if students are competent at producing effects on the screen, they may be confused about turns and angles. Students can change the heading of the turtle, often appropriately. But when the turtle leaves a path, a conflict can occur. “Other angles become candidates for the numbers which the child has entered into the computer and these angles are more obvious than the angle turned by the computer” [38, pp. 381-3821. That is, students assign the number signifying the angle through which the turtle turns to the measure of the angle represented by the line segments drawn by the turtle. Logo experiences may foster some misconceptions of angle measure, included consideringthe amount of rotation along the path (e.g., the exterior angle in a polygon) or the degree of rotation from the vertical [14]. In addition, such experiences do not replace previous misconceptions of angle measure. For example, students’ misconceptions about angle measure and difficulties coordinating the relationships between the turtle’s rotation and the constructed angle have persisted for years, especially if not properly guided by their teachers [31,33,36,39,40]. If, on the other hand, Logo experiences emphasize the difference between the angle of rotation and the angle formed as the turtle traced a path, misconceptions regarding the measure of rotation and the measure of the angle may be avoided [14, 361. Further, such experiences, properly planned, go beyond developing conceptions. Compared to the traditional curriculum, Logo Geometry had moderate positive effects on students’ ability to estimate angle size as measured by paper-and-pencil tests. However, it had large facilitative effects on the application of knowledge of angle measure in problem-solving situations [22].

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One group of sixth-graders used pre-written procedures that took the angles of an isosceles triangle for input, instead of the turn along a path [16]. This was also successful; at the end of the year, almost all could work successfully with tasks involving various isosceles-triangle properties and relations in complex configurations. Kieran and Hillel hypothesize that students had to make sense of what it was that was being controlled by the inputs to the procedures. They had to construct notions about characteristics of the figures that admitted to quantification as they worked with the numbers specifying those quantities. Receiving feedback from their explorations over several tasks, they developed an awareness of these characteristics and quantities. Building robust ideas about rotation and angle measure, and connecting these two ideas, is often a long and difficult process, even when activities are specifically designed for the learning of these ideas [22]. There is some evidence that Logo experiences affect measurement competencies beyond the measure of rotation and angle, because it permits the child to manipulate units and to explore transformations of unit size and number of units without the distracting dexterity demands associated with measuring instruments and physical quantity. For example, Logo children were more accurate than control children in linear measurement tasks 1411. The control children were more likely to: underestimate distances, particularly the longest distances; have difficulty compensation for the halved unit size; and underestimate the inverse relationship between unit size and unit numeracy . Students can learn more sophisticated ideas about measurement, as well as about headings, points, and coordinates, if they are provided with pseudo-primitives designed to make such ideas more accessible and salient [22, 421. Logo’s turtle facilitated learning these ideas as well, even though they are not the usual turtle, or intrinsic, geometry. For example, students adopted and used the notion of the turtle “putting its nose to look” at some direction rather than “turning this much” [42, p. 1061. Several studies, then, indicate that enriching the primitives and tools available to students facilitates their construction of geometric notions and increases analytical, rather than visual, approaches [22,42]. Further, some research has not had positive outcomes. One problem is that students may not always think mathematically, even when the Logo environment invites such thinking. For example, some students rely excessively on visual, or figurative, cues and eschew analytical work such as looking for exact mathematical and programming relations within the geometry of the figure [43]. The visual approach is not related to visualization abilities but to the role of visual perceptionsof a geometric figure in determining students’ Logo constructions;for example, estimating an input to a forward or right command based on what appears on the screen. The visual approach to solving Logo problems involves reasoning at van Hiele level 1 or between levels 1 and 2. Although important in beginning phases of learning, its continued use inhibits children from arriving at mathematical generalizations related to their Logo activity. There may be little

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reason for students to abandon visual approaches unless they are presented with problems whose resolution requires an analytical approach. Both selection of appropriate tasks and dialogue-between students and between teacher and students-are essential for encouraging analytic and higher-level reasoning (see [44], which provides similar arguments and data). For example, students’ work on another activity from Logo Geometry, “Rectangles: What Can You Draw?” demonstrates progression both 1. in van Hiele levels; and 2. from non-analytical to empirical to logical thought.

Students are shown a variety of quadrilaterals and asked to determine if each could or could not be drawn with a Logo rectangle procedure with inputs and to explain their findings (see Figure 1). This ostensibly simple activity has yielded many riches. Many students (even in grades five and six) often will not try the square, saying “It’s a square, not a rectangle.” However, soon they find that some of their classmates have succeeded. This leads to involved and passionate discussions regarding the relationship of

9

I Figure 1.

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these two shapes. One strength of this activity is that it has forced students to confront the conflict between the relationships as embodied within the Logo procedures and their conceptions of the figures. Otherwise, students may very well have just refused to even consider the possibility of an inclusive relationship between the square and rectangle. After getting the tilted rectangle by first turning the turtle, students often say that the parallelogram can also be drawn. They are quite shocked when it does not work, and many often reflect on the difficulty, concluding that the nonrectangular parallelogram cannot be drawn because it does not have 90”turns. But these students would not have confronted the misconception and they would not have produced the resulting productive reflection without the computer activity. A pair of sixth grade girls who were working in this activity illustrate another valuable aspect of the computer environment. Janet: I don’t think that you can do it. (She was then ready to go on to the next problem.) Sheny: Yes, you can. Janet: You can’t because-[pause] Sheny: Let’s try it. The students try it on the computer and observe the result. Janet: You can’t do it because the turns are not 90”. Interestingly, the first girl did not feel that it was necessary to try this example; she possessed a good reason for her belief, but, in the face of conflict, decided not to share it. She seemed to need to have her theory validated on the computer before she was willing to publicly argue it. Many students at this age level, like Janet, are just beginning to develop the ability to use logical reasoning in the realm of mathematics. They need affirmation that the conclusions that they reach are correct to build confidence in this newly forming mode of thought. Computer experimentation provides them with the opportunity to gain confidence. Also, as previously observed, this episode shows students discovering what is true or not for themselves, with their own reasoning-not using authority to resolve conflicts. This further develops thought that is autonomous and confident.A fifth grade case study student had a similar experience. In response to the parallelogram, Jonathan said, “Maybe” (it could be drawn with the rectangle procedure). He estimated the initial turn, then the side lengths. After typing in his commands, he got a rectangle. He held the sheet up right next to the screen. “No.” Teacher: Could you use different inputs, or is it just impossible? Jonathan: Maybe if you used different inputs.

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(Note in the next section evidence of a transition moment, building up level 2 awareness.) Jonathan: (Types in the initial turn. He stares at the picture of the parallelogram. Pauses.) No, you can’t. (Pause.) Jonathan: Because the lines are slanted, instead of a rectangle going like that. (Traces.) Teacher: Yes, but this one’s slanted (indicating oblique rectangle that Jonathan had successfully drawn). Jonathan: Yeah, but the lines are slanted. This one’s still in the size (shape) of a rectangle. This one (parallelogram) the things slanted. This thing ain’t slanted. It looks slanted, but if you put it back it wouldn’t be slanted. Anyway you move this, it wouldn’t be a rectangle. Jonathan: (Shaking his head) “So, there’s no way.” Note that the Logo environment was important to him in figuring this out. So was the activity. He didn’t even complete his second attempt. Rather, after making the initial turn and trying to choose the inputs, he recognized somehow that the relationship between adjacent sides was not consonant with the implicit definition of a rectangle in the Logo procedure. The first attempt with Logo and his “running through the procedure in his head” contributed to his emerging sense of certainty. Thus, we see that appropriate tasks and dialogue are important features of the educational environment. This should not be taken as a diminution of Logo’s contribution, as the previous example illustrates. Indeed, computational environments may be unique in providing scaffolding that allows students to build on their initial intuitive visual approaches and construct more analytic approaches. In this way, early unsophisticated non-standard concepts and strategies may be precursors of more sophisticated mathematics. In the realm of turtle geometry, research supports Papert’s [12] contention that ideas of turtle geometry are based on personal, intuitive knowledge [22,42]. As another example, students will often declare multiple variables in their Logo procedures, one for each parameter that varies, ignoring necessary relationships among them. It is possible, however, for them to use these procedures. In so doing, they often use values that satisfy the relationships, suggesting that they recognize the relationships, but only implicitly. The initial declaration can therefore serve as a scaffolding for later analysis [45]. One boy, for example, created a procedure to draw a parallelogram, specifying inputs for each item that was varying. He probably related each input to a physical attribute of the figure. However, he gradually transformed the variables to represent values rather than the measure of a specific part of the figure, such as a single side or turn. His revised procedure showed symmetry and explicsit recognition of geometric procedures, such as the equality of the opposite sides. Only later still, did he finally make the supplementary relationship between adjacent turns explicit. This occurred when he synthesized the meanings of his symbolic

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procedure and its visual result. In this case, there was no teacher intervention; his Logo procedure provided an explicit expression of his thought about the shape, one that was linked to a visual representation. As he reflected on this expression and what it meant visually, he was able to see new relationships. Similar facilitative effects of Logo environments have been observed on tasks not tied specifically to properties of geometric figures. Hoyles and Noss found that on a geometric proportion task, students used additive strategies on paper-and-pencil tasks, but none adopted such strategies on the related Logo tasks [46]. The authors trace this catalytic effect of Logo to the interaction between the students’ formalization of the proportional relationships algebraically in the form of a Logo program and feedback regarding their mathematical intuitions in the form of geometrical effects on the screen. On pencil-and-paper, the first of these, while present, is less salient; the second is absent. As an example of the catalytic influence, a strategy of addition with adjusted increments was transformed to adding a fraction of a variable to that variable when students moved to the computer. Hoyles and Noss further posit that students abandoned additive thinking because the computer provided a way to think about the general within the specific. The paper-and-pencil mode activated a fixed answer to a fixed question. The computer allowed exploration as an antidote to mental “blocks” and activated a dynamic answer. Posing the task of writing a superprocedure that would handle all cases promoted additional development. Thus, again we see the encouragement of more generalized and abstract views of mathematical objects within structured Logo environments. It is difficult to emphasize the importance of personal activity and construction of mathematical ideas, as the following vignette indicates. It took place in a fifth-grade classroom participating in the Logo Geometry research program [47]. The class was attempting to write a procedure to teach the turtle to maneuver through a maze to each of several points on the screen, returning home after each. The path home was to be the same as the path to the destination, a restaurant. Teacher: How are we going to get the turtle back? Jonathan: Everything that was a forward is a backward, and everything that is a right is a left, and everything that is a left is a right, probably. Teacher: So you’re telling me what? Sally: Reverse the stuff. Jonathan: You reverse, go from the bottom (of the procedure) to the top. Teacher: What comes first? The students start giving the commands to “undo” the original path as the teacher types them into the computer. Jonathan: We’ve got a problem here, because here’s the restaurant, it goes out of the restaurant BK 10, then right would be up here so it’s going to get screwed up.

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Robin: Now is’s going backwards.

As the students discuss Jonathan’s misgiving, Andy figures out his mistake by standing up and acting out the commands. They reach a consensus, enter the commands entered, and confirm their success. Here, students are discussing mathematical ideas, and making and evaluating arguments. Although the teacher entered the students’ commands into the computer, she did not judge the correctness of the students’ ideas. The studentsjudged their ideas, often by examining the computer screen. At this point, teachers sent students to computers in pairs to continue the activity. Emily and Ryan write a procedure to get the turtle to the second destination, then say “Let’s get him back.” Emily believes that the first command in getting the turtle back to its starting point is the reverse of the first command in the procedure, but Ryan says that it must be the reverse of the last command. After discussion, Emily explains her new reasoning. Emily: We started from here (turtle’s starting point) and ended here (draws in the air a curved path that is not a copy of the screen path), so we want to start here this time (indicates destination) so we have to start with the last one (points to the last command in the original procedure). Even though the idea of undoing a path had been discussed the previous day in class, Emily and Ryan needed to apply and reflect on its themselves to make the idea personally meaningful. This illustrates the power of individual work on computers and the relationship between this work and class discussions. The class discussions offer blueprints for the students’ constructions. The actual construction does not take place until the students themselves manipulate the ideas. Logo allowed the students to personally manipulate the ideas, determining their validity and form [47]. In some cases, such Logo activity may affect the conceptualizations of even young children, even on non-computer tasks. In the final interview, students were to identify all the rectangles in a collection of figures. At the pre-treatment interview most students, from kindergarten to sixth grade, confused non-rectangular parallelograms and rectangles at oblique orientations. First grader Adam was no exception: he included several parallelograms as rectangles on the pretest. After experiencing the Logo-based curriculum, however, Adam correctly identified a rectangle at an oblique orientation and correctly dismissed a parallelogram, saying: “That’s not a rectangle; that’s slanty.” Interviewer: “Well, isn’t this one (indicating oblique rectangle) slanty, too?” Adam: “It’s OK. It’s just turned. Look. If this pencil was a turtle, you could turn it like this (turns pencil from oblique to horizontal orientation). Then it would be a regular rectangle.” Interviewer: “But what if I did that to this one (indicates the parallelogram)?”

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Adam: “No, even if you turned it, it wouldn’t be a rectangle. It would still have slanty sides (indicating non-right angle) slanty to each other!”

In sum, computer graphics programs such as Logo can help students develop more abstract schema for geometric concepts. Because these environments require students to explicitly specify the construction or manipulation of shapes in mathematical language, they encourage students to elaborate their knowledge. However, as several of the episodes illustrate, we must be careful to not overestimate the amount of abstraction that students achieve. We must more carefully examine the abstractions that students do make and create tasks that encourage even more abstraction. These fiidings are consistent with the results of other researchers (33, 36,37, 39,40,43,48, 491. Within a wider educational environment that includes teacher mediation and appropriate tasks, there is evidence that students develop higher levels of thinking and use Logo as a tool to test out their ideas and thus promote that development. Simultaneously, they develop autonomy in mathematical reasoning. CONSTRUCTION PROGRAMS

Other computer environments, such as the Geometric Supposer software series, have focused specifically on facilitating students making and testing of conjectures. The Supposer programs allow students to choose a primitive shape, such as a triangle or quadrilateral (depending on the specific program), and to perform measurement operations and geometric constructions on it. The programs record the sequence of constructions and measurements and can automatically perform it again on other triangles or quadrilaterals. Thus, students can explore the generality of the consequences of constructions. Anecdotal reports indicate that the Supposers can be used effectively. A study by one of its authors reported that the Supposer, as opposed to traditional instruction, helped students consider nonconventional methods of analysis [50].They posed and solved problems and were claimed to have come to understand the importance of formal proof as a way of establishing mathematical truth (although there is some evidence that this was only in the case of the proof serving Bell’s illumination, as opposed to the verification and systemization, functions). In another evaluation, Supposer students performed as well as or better than their non-Supposer counterparts on geometry exams [51]. In addition, students’ learning went beyond standard geometry content, for example, re-inventing definitions (as they were forced to reconcile images with words), making conjectures, and devising original proofs. Students were able to think about a figure dynamically and see it as a representation of a class. On specially-designed tests, Supposer students produced the same or higher level generalizations than the comparison group and produced more arguments. A concern about this empirical approach encouraged by the

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Supposer was not realized; the Supposer made the need for proof more apparent than a purely deductive approach. Implementing the Supposer’s guided inquiry approach, however, requires effective teaching strategies. Successful teachers encourage inquiry as a way to learn what needs to be known. By basing some lessons on students’ inquiries, they legitimate that approach to knowledge acquisition [52]. In one study, teachers using the Supposers for the first time often promoted a rigid sequence of data collection and analysis, conjecturing, and proof. This led to rote data gathering and obscured the difference between representations of specific instances during data collection phases and more general representations during conjecture and proof phases. Students did not appreciate the different levels of generality these phases represented [51]. Other studies provide confirmation of points already made. For example, Kramer, Hadas, and Hershkowitz developed a microworld in which students performed classical constructions deductively ([48] describes this study). It was impossible to perform a construction without first making a careful analysis; and erroneous analyses led to no product or an incorrect product. The computer monitoring of each move’s legality apparently helped to clarify the “rules of the game.” Present-day media use two-dimensional representations to present three-dimensional information. Research indicates that people find this difficult [53]. Computers offer one solution to this problem because they allow the dynamic manipulation of two-dimensional representations of three-dimensional figures. Osta studied this potential using two commercial programs, one in which operations could be performed on three-dimensional objects represented on the screen and the other a “paint” program in which operations could be performed only on two-dimensional figurative designs ([48] describes this study). The programs have constraints that necessitate the use of geometrical properties rather than just visual information. Osta created problem situations in which students modified figures to move between two- and three-dimensional representations. Solution strategies of students in grades eight and nine were studied. At first, their work was local, dealing with small parts of figures through only perceptual strategies. With experience, students considered more global criteria and replaced inefficient perceptual strategies with strategies based on geometrical properties. Other research confirms that computer programs offering representations of three-dimensional figures mediate students’ learning by allowing them, via images, to validate their actions [54]. Space constraints prevent the discussion of other interesting computer environments, including Cabri-Geometer, the Logo Plane Geometry System, proof checkers and intelligent tutors, and geometry tool kit programs; research on their potential is needed. In summary, it would seem that geometric computer environments can help develop students’ thinking in geometry. The objects on the computer screen become manipulable representations-a “mirror”+f the students thinking. Thus,

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the students can make conjectures, evaluate visual manifestations of those conjectures, and reformulate their thought. This seems to be essential for developing reasoning skills in geometry. CONCLUSIONS AND IMPLICATIONS Unique Benefits of Constructive Computer Programs

There is some support for the educational benefits of the various computer environments. Elaboration

Students’ greater elaboration of geometric ideas within constructive computer environments appear to facilitate their progression to higher levels of geometric thinking. The nature of the interaction in certain environments promotes the connection of formal representations with dynamic visual representations, supporting the construction of mathematical strategies and conceptions out of initial intuitions and visual approaches. In programming, for example, there is a need to make relationships explicit and support for the linkage of symbolic and visual representations. The programming language also permits students to outline and then elaborate and correct their ideas. Objects as Representations of a Class

In a similar vein, certain environments allow the manipulation of specific screen objects in ways that assist students in viewing them as geometric (rather than visual) objects and as representatives of a class of geometric objects. Such activities develop students’ ability to reflect on the properties of the class of objects and to think in a more general and abstract manner. The power of the computer is that students simultaneously consider the specific and grounded with the abstract and generalized (represented by Logo code). Students treat a figure as having characteristics both of a single shape and a class of such figures. Viability

Constructive computer programs help students construct more viable knowledge because students are constantly evaluating a graphical manifestation of their thinking. Linkage of these manifestations to the symbolic helps students reconceptualize (e.g., turn and angle measure) and apply numerical quantification processes to constructs. Some computer environments allow educators to add new primitives that enrich the geometric aspects of the experiences. Further, they facilitate not only concept development, but students’ ability to apply geometric knowledge in problem-solving situations. However, selection of appropriate tasks

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and dialogue-between students and between teacher and students-are essential for encouraging analytic and higher-level thinking and for constructing connected, viable concepts. Precision

Computer environments demand and thus facilitate precision and exactness in geometric thinking. In contrast, when working with paper and pencil, there is imprecision and students are distracted by the actual effort of drawing. (This precision does not extend to screen representations for all systems; e.g., the “jaggies” on low-resolution systems. Though what is meant here is mathematical precision and the use of algorithms in geometric construction, higher resolution displays are also palpably desirable.) Explication

The need, for example in Logo environments, for complete and abstract explication may account in part for students’ creation of richer geometric concepts. 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 paper-and-pencil or manipulative environments, descriptions of the geometric objects and processes are generated from physical motions of students, who also understand the task and thus interpret the descriptions in that context. These interpretations can remain tacit and thus conceptualizations-both consistent and inconsistent with traditional mathematics-can remain implicit as well. In contrast, when “intuition is translated into a program it becomes more obtrusive and more accessible to reflection” [12, p. 1451. Personal and Intuitive

Logo’s turtle graphics environment encourages students to build geometric knowledge upon personal, intuitive, experiential knowledge. Thus, there is a firm reason to begin geometry explorations in this realm. Mirroring Thinking

The computer environments mirror students’ geometric thinking. Researchers and teachers consistently report that in such contexts students cannot “hide” what they do not understand. That is, difficulties and misconceptions (or alternate conceptions) that are easily masked by traditional approaches emerge and must be dealt with, leading to some frustration on the part of both teachers andstudents, but also to greater development of mathematics abilities [14,51]. On the positive side, the aforementioned meaningfulness of the visual representation, and the necessity (e.g., in Logo) of the formal symbolic representation, makes knowledge

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and strategies explicit and provides an opportunity to build on and extend students’ intuitions (it should be noted that the lack of connections between visual and symbolic representations is not endemic to computer work; it is merely more salient in this work, which is precisely the point here). For teachers willing to work with and listen to students, such environments provide a fecund setting for learning to take the student’s perspective on analyzing geometric situations and for discovering previously unsuspected abilities for students to construct sophisticated ideas if given the proper tools, time, and teaching. Ways of Thinking

Because students may test the ideas for themselves, computers can aid students in moving from naive to empirical to logical thinking and encourage students to make and test conjectures. In addition, the environments appear conducive not only to posing problems, but to wondering and to playing with ideas. In early phases of problem solving, the environments help students explore possibilities rather than become “stuck” when no solution path presents itself. Overall, research suggests that computer environments such as considered herein, can enable “teaching children to be mathematicians vs. teaching about mathematics” [12, p. 1771 insofar as making and testing conjectures, posing problems, and playful engagement with ideas is considered the role of the mathematician. Autonomy

For similar reasons, computers can facilitate students’ development of autonomy in learning (rather than seeking authority) and positive beliefs about the creation of mathematical ideas, if employed in an ecology designed to support such development. Implications for the Design of Computer Environments

What should the ideal computer environment be? Of course, this depends on many factors, including one’s goals for geometry education and the developmental level of the student. However, we can draw several implications from existing research. Building on Visual Strengths to More Powerhl Geometric Thinking

Computer environments should be designed to allow students to build on their visual strengths, but concomitantly demand complete and precise specifications, symbolic representations, and analytic thinking. More general, exact, symbolic, and abstract thinking should have a “pay off” in increased power and control within the environment. This suggests that “user-friendly” drawing tools will not likely lead to increases in sophistication for students’ geometric (as opposed to visual) knowledge. Computer geometry tools need to constrain students’ actions.

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They must serve as a transition device, connecting the intuitive and the visual to the symbolic and abstract. Connected Representations

Close ties between representations, such as Logo code and the resultant figure, need to be continuously (and simply) maintained. Moving bidirectionally between the visual and symbolic modes may facilitate the establishment and strengthening of these connections. Expanded Primitives

Enriching the primitives and tools available to students can facilitate their construction of geometric notions and increases analytical, rather than visual, approaches. Facilitation of Change and Exploration

Ease of editing and repeating constructions and operations, along with “undoing” and similar functions, should characterize any geometric environment. The rationale for such functions goes beyond simple convenience; the tools should embody the critical Piagetian concept of reversibility. Procedures

The ability to create procedures, alter them, and reflect on them, is powerful because it allows students to treat sequences of actions as cognitive objects that can be altered and reflected upon. Whatever form this takes (Logo procedures, macros, or storage and replay of actions), this power should be available and easy to use. The procedural and algorithmic approach to the construction of geometric figures should highlight procedural-conceptual connections. It should be possible for teachers to create and present these as well. Freedom within Constraints

Computer tools should allow for students and teachers to pose and solve their own problems, encouraging exploration and conjecture. That is, there should be a balance-freedom within constraints. At the same time, the computer environment should allow teachers to present and students to engage in structured tasks. Plumb the Depths of Simple Tasks

Achieving these goals may not require-nor even warrant-increasingly complex tools or programming techniques. It is often surprising what cognitive riches stem from ostensibly simple tasks (e.g., drawing a “tilted” rectangle in Logo). Also, there should be a continuing research focus on how children learn in various

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computer environments, instead of a never-ending chase for the newest technological “bells and whistles.” Theory

The theoretical models that underlie the environments should be used explicitly in the environments’ design and should be explicated for educators who will use the environments with students. Other Educational Implications

Computer environments alone are insufficient-the wider educational environment must be considered. This leads to eight additional implications. Tasks and Teacher Mediation

Thoughtful sequences of computer activities and teacher mediation of students’ work with those activities+specially the promotion of dialogue among students, their peers, and the teacher-are important for encouraging analytic thinking and higher-level reasoning. Teachers must guide students to predict turtle behavior and to reflect on their actions. Curriculum materials and teachers must provide tasks designed to create disequilibrium (often using computer feedback as a catalyst) and thus promote new ways of thinking. Hands-on Experience

Students need to personally manipulate computer representations and reflect on these actions to construct concepts, even after “clear” explanations and demonstrations by the teacher. Adequate Time

Students need time to construct, or reconstruct, ideas in computational environments. Initial strategies and conceptions may be precursors of more sophisticated mathematics, but only if students are given the time and encouragement to revise their ideas and strategies over a considerable developmental period, using computer scaffolding to fully synthesize visual and symbolic representations. Rethinking Assessment

Evaluation of learning in such environments must be reconsidered, as traditional approaches do not assess the full spectrum of what is learned, especially students’ posing and solving ill-defined problems.

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Grouping Students

Two students working cooperatively at a computer seemed ideal in the more exploratory environments of the Geometric Supposer and Logo [14,51]. Whole-ClassDiscussions

On a practical note, a monitor or projector for group discussions is noted by researchers as essential for all types of environments. At least at the high school level, a single location for computer work, discussion, and lecture may alleviate confusion about the overall’structure of the course. At all levels, such a location can increase the already high motivation of cooperative, meaningful computer explorations in geometry. Preparatory Work

Teachers should understand that much may have to be introduced before moving to work within a computer laboratory, including: 1. the software; 2. ways to operate the hardware and software; 3. mathematics content and problem-solving strategies; and so on.

Students’ engagement with the computer environments often precludes the insertion of such information within the laboratory. Teachers should also be encouraged to reflect on students’ conceptualizations and problem solving efforts in the laboratory to know what and how to introduce additional information during the next whole-class session. Teacher Education

Teacher educators must continue to investigate what experiences pre- and in-service teachers need to enable them to construct pedagogical environments that incorporate geometric computer tools. Integration with Research

Development of computer environments for learning geometry should be synergistically integrated with research efforts. Researchers must discover how we can systematically build on the geometric knowledge students learn each year. 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. 420-463,1992.

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21. M. T. Battista and D. H. Clements, Using Spatial Imagery in Geometric Reasoning, Arithmetic Teacher, 393, pp. 18-21,1991. 22. 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/Kent State University, Buffalo, New York/Kent, Ohio, 1992. 23. C. Hoyles and R. Noss, Formalising 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. 24. C. Hoyles and R. Noss, Children Working in a Structured Logo Environment: From Doing to Understanding, Recherches en Didactique des Matht!matiques, 8, pp. 131174,1987. 25. L. D. Edwards, Children’s Learning in a Computer Microworld for Transformation Geometry,Journal for Research in MathematicsEducation, 22, pp. 122-127,1991. 26. K. Johnson-Gentile, D. H. Clements, and M. T. Battista, The Effects of Computer and Noncomputer Environment on Students’ Conceptualizations of Geometric Motions, Journal of Educational Computing Research, in press. 27. E. Gallou-Dumiel, Reflections, Point Symmetry, and Logo, in Proceedings of the EleventhAnnual Meeting, North American Chapter of the International Groupfor the Psychology of Mathematics Education, C. A. Maher, G. A. Goldin, and R. B. Davis (eds.), Rutgers University, New Brunswick, New Jersey, pp. 149-157,1989. 28. 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. 29. D. H. Clements, Metacomponential Development in a Logo Programming Environment, Journal ofEducationa1Psychology, 82, pp. 141-149,1990. 30. M. Minsky, The Society of Mind, Simon and Schuster, New York, 1986. 31. D. H. Clements, Longitudinal Study of the Effects of Logo Programmingon Cognitive Abilities and Achievement,Journalof Educational ComputingResearch, 3, pp. 73-94, 1987. 32. J. Olive, C. A. Lankenau, and S. P. Scally, Teaching and Understanding Geometric Relationships through Logo: Phase II. Interim Report: The Atlanta-Emory Logo Project, Emory University, Atlanta, Georgia, 1986. 33. C. Kieran, Logo and the Notion of Angle among Fourth and Sixth Grade Children, in Proceedings ofPME 10,City University, London, England, pp. 99-104,1986. 34. G. N. Kelly, J. T. Kelly, and R. B. Miller, Working with Logo: Do 5th and 6th Graders Develop a Basic Understanding of Angles and Distances?, Journal of Computers in Mathematics and Science Teaching,6 , pp. 23-27,1986-87. 35. H. M. Findlayson, What Do Children Learn rhrough Using Logo? D A I . Research Paper No. 237, paper presented at the meeting of the British Logo Users Group Conference, Loughborough, United Kingdom, September 1984, ERIC Document Reproduction Service No. ED. 36. C. Kieran, Turns and Angles: What Develops in Logo?, in Proceedings’of the Eighth Annual PME-NA, G. Lappan (ed.), Michigan State University, Lansing, Michigan, 1986.

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Direct reprint requests to: Dr. Douglas H. Clements State University of New York at Buffalo 593 Baldy Hall Buffalo, NY 14260