# Mathematics Learning

## Geometry

Geometry originated in the ancient practice of earth measurement used in agriculture, the building of pyramids, and the observation of the patterns in the movement of the stars applied in navigation. In spite of the very practical origin of geometry in the investigation of the world, geometry is also the subject area where the development of abstract reasoning began, culminating in the first systematic organization of mathematical knowledge by Euclid around 300 B. C. E. Euclid's deductive system, built on definitions, postulates, theorems, and proofs, has served as the blueprint for representing mathematical knowledge since its inception.

Tension between the experiential, empirical origins of geometry and its abstract deductive representation characterizes contemporary instructional practice and research. Critics of the traditional Euclidean approach to the teaching and learning of geometry argue that the severance of geometric knowledge from its foundation in an inherently geometric world is a pedagogical error.

Educators in the United States have been reluctant to introduce geometry in the primary grades. The traditional view of geometry as an exemplification of abstract reasoning and a fear of exposing students prematurely to formal thinking may be among the reasons for this reluctance. When primary grade teachers choose to spend a short instructional period on geometry, it is usually limited to having students recognize and recall the names of prototypical two-dimensional shapes like triangles, squares, and rectangles. This practice fails to take advantage of the host of informal geometric knowledge children bring to school.

Even before entering school, children develop intuitions about geometric shapes and their characteristics during their early explorations with their environment. For example, in exploring the objects around them, children experience that surfaces can be bumpy or smooth. Building with blocks or stacking other objects, children learn about differences in forms and sizes. Using boxes and other containers, they form intuitive ideas of space-filling or volume. As children walk around in their neighborhood they develop informal notions of spatial arrangements, distance, and directionality. The learning of geometry can be built on this naturally acquired spatial sense. Guiding children to reflect on the characteristics and regularities of their spatial experience can easily lead to the development of the basic concepts (abstractions) of geometry, such as straight and curved lines, points as intersections, planes, and planar and three-dimensional shapes. Uncultivated or ignored, however, children's natural spatial sense fades away, and it is difficult to retrieve it for use when students enroll in their first official geometry course in high school.

A programmatic document, the 1989 Curriculum and Evaluation Standards for School Mathematics, produced by the National Council of Teachers of Mathematics (NCTM) to guide reform in mathematics education, recommends that geometric topics be introduced and applied to real-world situations whenever possible. However, this does not imply that immersing children in real-world situations automatically leads to mathematical or geometrical understanding. Hands-on activities are a popular way to establish a connection between instruction and real life, but as instructional means they are only as good as the meanings derived from them. The challenge of geometry instruction is to elevate children's experience with real-world objects to the level of mathematics.

This happens in well-designed instructional tasks that promote reflection on the geometric features of real-life situations, leading to the development of geometric concepts and spatial reasoning. Children learn to generate geometric arguments by participating in carefully orchestrated conversations where they articulate, share, and discuss their ideas regarding spatial problems. Children develop skills of modeling spatial situations when they are invited to publicly display and discuss their visualizations in drawings. These drawings can then be turned into mathematical representations during revision cycles, in the course of which the geometrical features are accentuated while the mathematically irrelevant features (e.g., material, color, and other decorative elements) gradually fade away.

At the secondary level, the traditional Euclidean geometry curriculum that revolves around deductive proof procedures has been criticized because it separates geometry from its empirical, inductive foundation. Critics refer to the typical lack of student appreciation for the subject–often accompanied by low achievement. The deductive organization of the geometry course has been seen as a viable model to help introduce students to mathematical reasoning. However, this is a misrepresentation of the actual reasoning that goes on among expert mathematicians. The deductive logic applied in proofs constitutes only a subset of the rules, and it seldom accounts for the actual thought processes that contributed to the discovery communicated in a proof. Actual discovery usually follows an inductive line of reasoning that begins with empirical investigation and the observation of regularities. It continues with making conjectures based on the observed regularities, and then testing them on multiple examples. Attempts at explaining and generalizing the observed relationship with the help of proof come only after the long process of empirical exploration.

Alternatives to a traditional Euclidean secondary geometry curriculum have been offered based on this more grounded view of mathematical reasoning that incorporates exploration and induction. In the process of exploration, students learn to deconstruct geometric objects into their constitutive elements, and to rely on properties–such as the number and relative size of the sides of the objects, the measure of angles, and their relationships–rather than a prototypical or customary presentation of an image when they identify shapes. Ideally, students will learn to go beyond the appearance of an actual drawing of a shape and argue about generalized concepts of shapes as defined by their properties (for example, a rectangle is a quadrilateral with four right angles and with opposite sides equal and parallel). These skills serve as the foundation of geometrical understanding and need to be acquired–ideally in the primary grades–before students are exposed to proofs.

Some of the new secondary geometry curricula have been organized around technology tools, including geometry construction programs such as the Geometer's Sketchpad and the Geometric Super-Supposer. These programs provide an electronic environment for geometric explorations and allow the learner to generate multiple solutions of geometric construction problems, thus facilitating the generation and testing of hypotheses. Proofs gain a different meaning in this context, becoming the means of explaining why the conjectures developed by the students themselves hold beyond the examples created by the program. This has a motivating effect on the learner. Without such an inductive foundation, students see proofs as an unnecessary procedure to arrive at a simple truth that they already know and accept.

## BIBLIOGRAPHY

CHAZAN, DANIEL, and YERUSHALMY, MICHAL. 1998. "Charting a Course for Secondary Geometry." In Designing Learning Environments for Developing Understanding of Geometry and Space, ed. Richard Lehrer and Daniel Chazan. Mahwah, NJ: Erlbaum.

LAKATOS, IMRE. 1976. Proofs and Refutations: The Logic of Mathematical Discovery. Cambridge, Eng.: Cambridge University Press.

LEHRER, RICHARD, and CHAZAN, DANIEL, eds. 1998. Designing Learning Environments for Developing Understanding of Geometry and Space. Mahwah, NJ: Erlbaum.

LEHRER, RICHARD; JACOBSON, CATHY; KEMENY, VERA; and STROM, DOLORES. 1999. "Building on Children's Intuitions to Develop Mathematical Understanding of Space." In Mathematics Classrooms that Promote Understanding, ed. Elizabeth Fennema and Thomas A. Romberg. Mahwah, NJ: Erlbaum.

NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS. 1989. Curriculum and Evaluation Standards for School Mathematics. Reston, VA: National Council of Teachers of Mathematics.

NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS. 2000. Principles and Standards for School Mathematics. Reston, VA: National Council of Teachers of Mathematics.

SCHWARTZ, JUDAH L.; YERUSHALMY, MICHAL; and WILSON, BETH, eds. 1993. The Geometric Supposer: What Is It a Case Of? Hillsdale, NJ: Erlbaum.

SERRA, MICHAEL. 1989. Discovering Geometry: An Inductive Approach. Berkeley, CA: Key Curriculum Press.

VERA KEMENY