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

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

The UCSMP elementary program is one of the current reform-based curricula attempting to incorporate this research. While a great deal more geometry is done in this program, emphasis is on exploring and building geometric ideas and relationships. For example, in a first-grade UCSMP lesson, students find examples of three-dimensional figures in their homes and bring them to school. The properties of these shapes are discussed, and they are classified into various categories (e.g., cones or rectangular prisms) based on these properties. In one class observed, students discussed whether a truncated cone should be classified as a cylinder or a cone, based on its properties (which geometric features were most relevant). As students progress through the UCSMP curriculum, emphasis shifts from isolated properties to how these properties are related.

Results from this study support the view that elementary school students are capable of developing stronger understandings of geometric relationships. UCSMP students scored significantly higher than comparison students on all measures-mean correct score, van Hiele level, and reasoning score-and maintained or widened this gap across the year. Furthermore, fifthgrade UCSMP students not only outperformed their peers but outperformed comparison sixth graders as well. For example, 20% of the UCSMP fifth graders and 29% of the UCSMP sixth graders scored at the highest van Hiele level measured on the posttest. In contrast, none of the comparison fifth graders and only 7% of the comparison sixth graders scored at this level. While the value of proof-oriented courses might be argued, results suggest that a much higher proportion of UCSMP students were prepared for such a class, even at the end of fifth or sixth grade.

This study also provides further support for the hierarchical nature of geometric learning, as postulated by the van Hiele model, and the usefulness of this model in planning instruction. By including geometry as a regular strand throughout the elementary curriculum, consistent progress can be made. Furthermore, teachers have reported that geometry topics are often of high interest, both to their students and themselves. Rather than teaching geometry simply as vocabulary and formula to be memorized, geometry can be approached from a problem-solving perspective, with a greater emphasis on reasoning, communication, and connections to other topics. The gains in higher van Hiele levels and levels of reasoning are reflective of this approach. While this study looked only at geometry, other related studies have indicated that UCSMP students score much higher than other students on less common topics, such as mental computation and number sense, without losing ground in more traditional areas (Carroll, 1996b; Drueck, Fuson, Carroll, & Bell, 1995).

While the van Hiele model does not postulate a level below Level 0 for geometric reasoning, this study supports previous findings that some students are not even successful at identifying geometric figures. At the end of fifth grade, nearly half of the comparison students had failed to achieve even Level 0 reasoning. Further, 51% of the comparison sixth graders were below Level 0 at the beginning of the year, although all but 25% had achieved Level 0 or higher by the end of the year. These results argue strongly for increasing both the amount and depth of geometry during the primary grades.