Geometric knowledge of middle school students in a reform-based mathematics curriculum

School Science and Mathematics, Apr 1998 by Carroll, William M

National and international studies have found U.S. elementary students to be weak in their understandings and applications of geometric concepts. The University of Chicago School Mathematics Project's (UCMSP) Everyday Mathematics Program is one of the current reform-based elementary curricula incorporating geometry throughout the K-6 curriculum, with an emphasis on hands-on and problemsolving activities. In this study, the geometric knowledge of fifth and sixth graders using the UCSMP curriculum is compared to the knowledge of students using more traditional curricula. Because UCSMP students had been in the program since kindergarten, this research attempts to measure the longitudinal effects of such an approach. Along with an overall score, a subset of test items was used to assign each student a van Hiele level for geometric thinking, as well as a reasoning score. On all measures, UCSMP students substantially outperformed their counterparts, and nearly all differences were significant. Aspects of the UCSMP curriculum and the van Hiele model for learning geometry are discussed relative to these results.

Although U.S. students begin school recognizing basic geometric shapes, little progress is made in furthering their understandings of or their ability to apply geometric concepts. For example, while 90% of seventh graders who took part in the Fourth National Assessment of Mathematical Progress identified simple geometric figures, fewer than one third could solve missing angle measurements or identify properties of angles and triangles (Lindquist & Kouba,1989). Results of international studies suggest that the cause of this deficit lies in the U.S. curriculum, not in students' developmental capabilities (McKnight, et al., 1989; Stigler, Lee, & Stevenson, 1990). In Stigler, Lee, and Stevenson's study (1990), Japanese and Taiwanese fifth graders consistently outperformed U.S. fifth graders on all questions requiring analysis of geometric properties and relationships. Only on questions of simple recognition, such as identifying a square or parallel line, did U.S. students' scores approach those of their Asian peers. While other factors may also be important, international analyses of curriculum and instruction indicate that U.S. elementary students have less opportunity to learn geometry beyond basic vocabulary and recognition of simple figures (McKnight et al., 1989).

The van Hiele model for geometric understanding provides a framework for understanding the development of geometric understanding (Clements & Battista, 1992; National Council of Teachers of Mathematics [NCTM], 1989). This model, developed in classroom studies by Dina van Hiele-Geldof and Pierre van Hiele (van Hiele-Geldof, 1984), postulates five levels of geometric reasoning-from basic recognition through mathematical rigor (see Appendix A). Progress through these levels is sequential and is based on regular investigations of geometry relevant to the student's current level of reasoning. That is, in order to progress from Level 0 (recognition) to Level 1 (description and analysis), students need to participate in activities allowing them to investigate the geometric properties of figures. Hands-on activities, such as using geoboards or dot paper to construct and explore properties of various polygons, are especially relevant to this development. In these activities, students move beyond simple recognition of the figures ("It's a rectangle") to analysis of the properties ("All interior angles are right angles"). Memorization of definitions, perhaps meaningful at a higher level, does little to help students' progress at this level (Fuys, Geddes, & Tishchler, 1988). Similarly, progress from Level 1 to Level 2 (order and informal proof) is contingent upon activities requiring students to go beyond single propertiesactively examining and explaining how these properties are related in figures and families of figures.

A number of studies have provided support for the van Hiele model, as well as further evidence for the poor performance of U.S. students in geometry. In a large-scale study of secondary school students enrolled in geometry, Usiskin (1982) found that nearly three fourths of the students entering high school geometry scored at or below van Hiele Level 0, and few scored at a level of understanding necessary for success in a proof-oriented geometry class (Senk, 1989). Few of the students showed evidence that they were capable of analyzing or applying geometric concepts. Other studies also support the sequential nature of geometry learning and provide evidence that reasonable progress can be made when students are provided with wellplanned activities relevant to their current level of thinking (Fuys, Geddes, & Tishchler, 1988; Mayberry, 1983). Because the levels of thinking are sequential according to the van Hiele model, students cannot "jump" levels, moving from recognition to formal proof in any meaningful way. For this reason, development of geometric reasoning requires that geometry be an integral part of the mathematics curriculum from kindergarten onward, especially if students are to be ready for understanding of proof concepts in secondary school. Throughout the elementary school curriculum, activities should be appropriate to the students' current level, while leading toward more sophisticated levels of reasoning