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

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

Reasoning Questions

While other questions clearly required students to reason about geometric properties, two required students to explain their reasoning on each of the tests. The purpose was to further investigate students' reasoning and their ability to communicate their thinking. Results on the questions included on the posttest are discussed here. One reasoning question and the accompanying 5point rubric used on both the pretest and posttest are shown in Appendix B (Sheila's triangle) and Appendix C. A higher score required both the correct answer and a clear explanation. A second reasoning question showed an angle of 135 deg and asked students to estimate the measure and explain how they arrived at this estimate. Students could approach this problem in various ways, such as using benchmark angles (e.g., "This angle is greater than 90 degrees but less than 180 degrees.").

On the two reasoning questions, a total of 7 points were possible. The UCSMP sixth graders scored the highest with 4.4 points, followed by UCSMP fifth graders with 2.9 points, comparison sixth graders with 2.8 points, and comparison fifth graders with 1.6 points. The results on the reasoning question involving a triangle with two right angles, shown in Table 3, indicate a pattern similar to van Hiele levels achieved by UCSMP and comparison students. For example, 80% of the comparison fifth graders and 47% of comparison sixth graders scored at the two lowest levels on this question, compared to 34% and 19% of the UCSMP fifth and sixth graders. Over 50% of the UCSMP sixth graders and nearly one third of the UCSMP fifth graders scored at the two highest levels on this question. A significant correlation was found between the students' reasoning score and van Hiele level on the posttest (Pearson correlation coefficient .61).

Discussion

While other nations introduce and integrate geometry, algebra, and other mathematical topics much earlier in the elementary curriculum (McKnight et al., 1990), the U.S. K-8 curriculum has focused on arithmetic (Flanders, 1987). Algebra and geometry have generally been delayed until secondary school and have often served to "filter" students from further mathematics classes. This view of mathematics education has changed drastically, with a push toward increasing the number of higher mathematics classes taken by all secondary students (National Research Council, 1989; Silver, 1997). One model for improving students' success in high school geometry is to offer more "informal" geometry classes in place of proof-oriented classes. This solution is reasonable for those who have had little experience with geometric skills and concepts in the elementary school. However, a second approach, based on the van Hiele model for geometry and supported by the NCTM Standards ( 1989, 1991), is to change the elementary school program, making geometry an integral part of the curriculum from kindergarten through junior high. Rather than simply identifying geometric figures and memorizing names and formulas, elementary students can explore various aspects of geometry and progress from simple identification through reasoning about mathematical relationships. Because the emphasis is on understanding rather than memorization, these activities are often hands on and involve student explorations, mathematical connections, and discussions.