Michael T. Battista
Douglas H. Clements
Geometry is an essential part of mathematics. Unfortunately, according to evaluations of mathematics learning, such as the National Assessment of Educational Progress (NAEP), students are failing to understand basic geometric concepts and are failing to develop adequate geometric problem-solving skills (Carpenter et al. 1980; Fey et al. 1984; Kouba et al. 1988). This poor performance is due, in part, to the current elementary school geometry curriculum, which focuses on recognizing and naming geometric shapes and learning to write the proper symbolism for simple geometric concepts (cf. Carpenter et al. ; Flanders ). In contrast, we believe that elementary geometry should be the study of objects, motions, and relationships in a spatial environment (Clements and Battista ; cf. Trafton and LeBlanc ). First experiences with geometry should emphasize informal study of physical shapes and their properties and have as their primary goal the development of students' intuition and knowledge about their spatial environment. Subsequent experiences should involve analyzing and abstracting geometric concepts and relationships in increasingly formal settings.
Deficiencies in Current Curricula
Because current elementary geometry curricula focus only on identification of figures and the use of geometric terms (Kouba et al. 1988), little opportunity arises for geometric problem solving. Students have little chance to develop their spatial thinking, a skill that should have primary importance in the geometry curriculum. Students encounter little opportunity to analyze and reconceptualize substantive geometric ideas. Furthermore, the little geometry that is included in textbooks is often slighted (NACOME 1975). It is no wonder that, after experiencing such an impoverished geometry curriculum in elementary school, many high school students do not have the necessary geometric intuition and background for a formal deductive geometry course (Hoffer 1981; Shaughnessy and Burger 1985). They are also not adequately prepared for the later study of important ideas such as vectors, coordinates, transformations, and trigonometry (Fey et al. 1984).
Indeed, in elementary school geometry -- as in almost all of mathematics -- we have, in the words of Skemp, "two effectively different subjects being taught under the same name 'mathematics'" (1976, 22). One consists of learning a limited number of "rules without reasons" -- knowing only what to do. The other, relational mathematics, is "knowing both what to do and why"; it involves constructing conceptual structures from which a learner can produce an unlimited number of rules to fit an unlimited set of situations. As Skemp continues, "What constitutes mathematics is not the subject matter, but a particular kind of knowledge about it" (1976, 26). From this perspective, the current elementary geometry curriculum is deficient because it underemphasizes relational understanding. Too much emphasis is placed on formal symbolism and naming and not enough on spatial explorations, analysis, synthesis, and problem solving.
Another deficiency in the current curriculum is that it does not always emphasize conceptualizations of topics that are most useful in the later learning of mathematics. For instance, the concept of angle normally encountered in elementary school is that of a union of two rays with a common endpoint, the same formal definition used in high school geometry. However, in trigonometry and calculus, an angle is thought of as a rotation. Existing elementary school geometry curricula do not address this second aspect of the angle concept, even though the latter aspect seems more closely related to navigation, "one of the most widespread representations of the idea of angle in the lives of contemporary Americans" (Papert 1980, 68). In fact, interviews that we have conducted with students suggest that many students' notions of angle are connected with navigation in terms of headings or the "tilt" of a line segment (Clements and Battista, in press).
Results of the NAEP indicate a final curricular deficiency in that students at all levels "perceive their role in the mathematics classroom to be primarily passive... They feel they have little opportunity to interact with their classmates about the mathematics being studied, to work on exploratory activities, or to work with manipulatives" (Carpenter et al. 1980, 36). Furthermore, 80 percent of the seventh graders feel that mathematics is rule based, and 50 percent think it is mostly memorizing (Kouba et al. 1988). These results offer empirical support for the National Council of Teachers of Mathematic's recommendation that teachers of mathematics present opportunities for their students to be actively involved in learning, experimenting with, exploring, and communicating about mathematics as part of an environment that encourages problem solving (NCTM 1980). This recommendation is consonant with the belief that "mathematics learning is primarily a person-centered, constructive process: students build and modify their knowledge...[they] must experience opportunities and develop feelings of responsibility for revising, refining, and extending their ideas as the ideas are being constructed" (Hatfield 1979, 53).
Computers and Mathematics Learning
One approach to developing children's mathematics concepts and problem-solving skills is to engage them in active explorations with computer environments. As early as 1972, Hatfield and Kieren found that writing computer programs could be effective for enhancing students' learning of important ideas in mathematics. In the appropriate learning context, a set of computer programming instructions written by a student can be thought of as an active object with which the student can "command the computer to do something that he can observe, study, and modify," and therefore more fully comprehend (Hatfield and Kieren, 101). Used in this way, computer programming is an effective instructional context for "constructive approaches to learning. For example...the student may be called upon to build up a procedure, test it, find the errors or inadequacies, correct or improve it, test it again, and possibly refine or extend it to a more general procedure" (Hatfield 1979, 53). Such approaches are not only problem-solving intensive (i.e., they involve constructing, correcting, and modifying procedures) but also constructive. That is, they encourage students to build networks of conceptual knowledge -- the basis for successful problem-solving.
Although constructive computer approaches may be no more efficient than standard approaches in teaching factual material, they may be more effective in developing mental processes that are important in problem solving and other mathematical endeavors. As the Cockroft report noted, "The idea of investigation is fundamental both to the study of mathematics itself and also to an understanding of the ways in which mathematics can be used to extend knowledge and to solve problems in very many fields.... Mathematical explorations and investigations are of value even when they are not directed specifically to the learning of new concepts" (Committee of Inquiry into the Teaching of Mathematics in Schools 1982).
Logo and Geometry
A natural choice for a programming language to use in geometry activities is Logo. With Logo, children can create geometric figures by typing sets of instructions into the computer that direct the movement of a turtle on the computer screen. These commands can be "relative" and take the perspective of the turtle such as FORWARD and RIGHT; they can be "absolute" by using a set of coordinate axes; and they can include a full range of arithmetic functions. According to Feurzeig and Lukas (1972), Logo "provides an operational universe within which students can define a mathematical process and then see its effects unfold. It is accessible to very young children for simple tasks, yet its operations can be systematically extended to express problems of considerable complexity" (1972, 39).
Logo and Geometric Problem SolvingTo draw a figure in Logo, students devise a set of movement instructions for the turtle. They must determine angle measures and lengths of line segments. They can be asked to analyze the figure and break it into smaller parts that are more easily constructed. Thus, they are constantly involved in geometric problem solving. Such Logo activities encourage students to identify goals and strategies before making overt moves toward a problem solution, create efficient problem representations, make executive decisions, and debug algorithms -- all of which are problem-solving skills too seldom explicitly taught in the schools.
Logo and Geometric Thinking
Taylor (1980) hints at the difference between the traditional geometry curriculum and a Logo geometry curriculum by stating that children who are engaged in Logo activities will invent basic concepts in mathematics, thereby learning "to be mathematicians" versus learning "about mathematics." Furthermore, Logo-based geometry activities can promote substantive rather than factual learning, helping students progress to higher levels of thinking in geometry.
As an example, consider the concept of rectangle. In the usual elementary school geometry curriculum, students are required only to be able to identify a visually presented rectangle -- a level-1, visual activity in the van Hiele hierarchy (see Hoffer ; van Hiele ). In Logo, however, students can be asked to construct a sequence of commands, a procedure, to draw a rectangle. This process forces them to make their concept of rectangle explicit. They must analyze the visual characteristics of the rectangle and establish relations among its component parts. For example, students who think of a rectangle as "a figure with two long sides and two short sides" must be more precise and complete to write a Logo procedure for a rectangle; they must explicitly address properties of rectangles, such as opposite sides being equal in length and adjacent sides being perpendicular. See figure 1.
In the words of Papert, "The computer 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" and can thus be used as material "for the work of remodeling intuitive knowledge" (1980, 145). Therefore, in the context of Logo, the teacher can help students elaborate their intuitions about the concept of rectangle by focusing their attention on its properties and by embellishing those intuitions with verbal labels and descriptions. Such elaboration is essential for progressing toward level 2, the descriptive-analytic level, in the van Hiele hierarchy. Moreover, by designing a rectangle procedure with inputs, students begin to build intuitive knowledge about the concept of defining a rectangle. See figures 2 and 3. They might then be challenged to use this procedure to draw a variety of shapes, such as rectangles, squares, and non-rectangular parallelograms. Note that the last shape cannot be drawn with this rectangle procedure; see Battista and Clements (1988b). These activities, then, serve as an orientation to definitions and to hierarchical classification of geometric shapes and can aid students in the transition to level 3, the abstract-relational level).
Through such a sequence of Logo-based experiences, not only are students progressing into higher levels of geometric thinking in the van Hiele hierarchy, they are building conceptual structures about rectangles that can be useful in other situations, such as drawing quadrilaterals, triangles, or regular polygons. They are thus learning geometry relationally.
Logo: The path from concrete to abstract
This is not to say, however, that students' geometric experiences should be limited to Logo. According to Piaget, actionis of paramount importance in the development of geometric conceptualizations. The child "can only 'abstract'. . . the idea of a straight line from the action of following by hand or eye without changing direction, and the idea of an angle from two intersecting movements" (Piaget and Inhelder 1967, 43). Indeed, physical actions on concrete objects are necessary. But students must internalize such physical actions and abstract the corresponding geometric notions. Logo can facilitate this process, thus promoting a transition from concrete experiences with geometric ideas to abstract reasoning. For example, by first having children form paths by walking, then using Logo, children can learn to think of the turtle's actions as ones that they themselves could perform. We see evidence of this phenomenon in our studies when, in trying to figure out what path a given set of directions will make the turtle draw, children turn their bodies in an attempt to figure out a turn. That is, they seem to project themselves into the place of the turtle. In so doing, they are performing a mental action -- an internalized version of their own physical movements.
Because children understand beginning spatial notions in terms of action and because the mathematical concept of path can be thought of as a record of movement, it seems natural to emphasize this concept in the beginning study of geometry. For example, having students visually scan the side of a building, run their hands along the edge of a rectangular table, or walk a straight path will help them develop an intuitive concept of straightness. But in Logo, the essence of this abstract concept can be brought to a more explicit level of awareness as a "turtle path that has no turning." Because the concept is explicit and reformulated in a more formal and precise language, it can be internalized in a more abstract form. Thus, we believe that the concept of path should be taught explicitly, that the concept of path can be used to organize beginning geometric notions, and that appropriately connected physical and Logo activities offer an ideal environment for studying paths and related geometric notions.
Pertinent Research on Logo and Mathematics
Many Logo projects have attempted to explore the benefits of Logo programming for mathematics learning. In most, the instructional focus has been on Logo as a programming language and environment for exploration, as a curriculum in itself. Evaluation of these projects has indicated that this approach to increasing mathematics achievement is generally ineffective (Akdag 1985; Blumenthal 1986). However, it is possible that the students in these projects learned concepts that were not part of the standard curriculum and thus were not assessed, or that their teachers did not lead them to see the connections between the Logo-based concepts and other mathematical tasks.
For example, an interview from a longitudinal study conducted by one of the coauthors (Clements 1987) illustrates how one third grader's (let's call him Sam) geometric conceptualizations were significantly affected by work with Logo yet remained unconnected with other aspects of his mathematical knowledge. Sam had previously responded incorrectly to an item from a commercial mathematics achievement test: "How many angles are in a triangle?" When the same question was posed in the follow-up interview, Sam asked, "What do you mean 'angle'. . . corners?" Later in the interview, Sam was asked how he drew a triangle with the turtle when he was in first grade.
"We started, and then I put left and I made it turn so it would be like that [rotating hand]. Then I made it go forward so it would go like that, and then I made it turn on an angle. An angle [shouting and laughing]! A turn! A turn. . . the same thing."
Thus, in his experience with Logo, Sam had developed the useful notion of an angle as a turn. Probably because his Logo experience was not so designed, Sam failed to see connections between the Logo concept of "turn" and the notion of angle he was encountering in his classroom mathematics work. His performance suggests that the standard treatment of the angle concept that he was receiving in his classroom was inadequate in two respects. First, using classroom-based knowledge of angle, he was unable to answer the achievement test item and felt the need to ask for a definition during the interview. Second, it can be hypothesized that Sam's classroom experiences with angle were so infrequent and static that not until the interview did he make the connection. In agreement with this finding, more recent studies that attempted to make connections between students' work with Logo and textbook mathematics have found significant increases on tests of geometric achievement (Howe, O'Shea, and Plane 1980; Lehrer and Smith 1986).
Furthermore, evidence is emerging that supports our prediction that proper Logo environments can help students make the transition from the visual to the descriptive level of thought in the van Hiele hierarchy. In fact, after working with the Logo activities, students attempting geometric tasks were less likely to conceptualize shapes on the basis of their visual appearance, and more likely to conceptualize them in terms of their properties (Battista and Clements 1988a).
A Logo-based Elementary Geometry Curriculum
Thus, as the Sam vignette illustrates, Logo activities can help students develop intuitions about geometric ideas. But teachers must help students elaborate these intuitions and connect them to formal geometric concepts. For example, in a standard textbook approach, the concept of regular polygon is defined, examples and nonexamples are presented, and students are required only to accurately discriminate examples from nonexamples. But a teacher using Logo could supplement this standard approach by asking students to have the turtle draw several regular polygons. This would force students to analyze their conception of regular polygon, searching for a way to reconceptualize it in terms that they can explain to the turtle through the Logo language. See Figure 4. Teachers can use this procedural perspective to enhance students' understanding of regular polygon by helping them connect the corresponding visual characteristics, procedural components, and mathematical properties of regular polygons.
The teacher might then have the students determine the relationship between the amount of turtle turning at each vertex and the number of vertices of the polygons. Finally, to connect their Logo work to the more traditional perspective, students should investigate the relationship between the amount of turn at each vertex and the measure of the vertex angles of the polygons. The whole process helps bring the properties of regular polygons to an explicit level of awareness, helps illustrate how knowledge of these properties can be applied and generalized, and furthers mental development in geometry by encouraging students to function at higher levels in the van Hiele hierarchy.
As another example, most geometry activities in the current middle school curriculum assume the perspective of a person who is observing figures from outside the plane. They use letters to label points or require students to measure line segments and angles. In Logo, however, middle school students are generally restricted to giving relative commands. Logo primitives, such as FORWARD, BACK, RIGHT, and LEFT, for example, give directions from the turtle's perspective within the plane of the screen. To help students connect Logo to the standard geometry perspective, a set of mathematics utility procedures, called MATHSTUFF, has been created (Battista 1987). These procedures permit students in a Logo environment to create and label points with letters and to measure line segments and angles specified by these letters, thus enabling them to use standard geometric ideas to solve problems in the dynamic environment of Logo. For example, students attempting to draw a house with a rectangular, but not square, bottom could use the MATHSTUFF procedures to measure distances and angles so that a triangular roof could be constructed -- not an easy task without the MATHSTUFF procedures. Or, students might construct several triangles by connecting three randomly placed points and then might measure their angles in an attempt to discover the sum of the measures of the vertex angles of triangles. See Figure 5.
Expanding the Geometry Curriculum
A Logo-based elementary geometry curriculum also permits a richer study of oft-neglected topics, as well as investigation of topics heretofore not studied in the elementary school. For example, geometric transformations is an important but frequently neglected topic in elementary school geometry. In Logo, translations, rotations, reflections, and compositions of these transformations can be dynamically illustrated using procedures presented by the teacher. These visual experiences can help students develop the ability to manipulate images mentally -- the essence of spatial visualization. A problem such as "What composition of transformations will move triangle A onto triangle B?" necessitates that students formulate and test hypotheses about sequences of transformations. This "transformation approach makes geometry an appealing, dynamic subject that will develop spatial visualization ability and also the ability to reason" (Fey et al. 1984, 44).
Logo is also "an obvious computer tool for work with vectors" (Fey et al. 1984, 45). For example, predicting the final position of the turtle after issuing a sequence of commands (FORWARD, BACK, RIGHT, LEFT) builds an experiential foundation for later work with vector addition. Properly designed vector microworlds could help students bridge the gap between informal Logo experiences and the later formal study of vectors.
We believe that Logo environments have the potential to transform both the method and content of the elementary geometry curriculum. Properly used, such environments can help alleviate deficiencies in the current curriculum. Students can learn geometry relationally. They can learn to analyze and reconceptualize their geometric ideas, progressing to higher levels in the van Hiele hierarchy. They can engage in the exploration of significant geometric problems. To realize this potential, though, teachers not only must involve students in properly structured Logo explorations, they must help them elaborate these explorations to construct formal mathematics.References
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Campbell, Patricia F. "Microcomputers in the Primary Mathematics Classroom." Arithmetic Teacher 33 (February 1988):22-30.
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Craig, Bill. "Polygons, Stars, Circles, and Logo." Arithmetic Teacher 33 (May 1986):6-11.
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This material is based upon work supported by the National Science Foundation under Grant No. MDR-8651668. 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.
Michael Battista teaches mathematics education, computer education, and mathematics courses at Kent State University, Kent, OH 44242. Douglas Clements teaches early childhood, elementary mathematics, and computer education courses at the State University of New York at Buffalo, Buffalo, NY 14260. Battista and Clements are codirecting an NSF project to develop a Logo-based elementary school geometry curriculum.
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