Enhancing Geometric Reasoning

Article excerpt

ABSTRACT

Geometry is an important part of the mathematics curriculum. However, students are not demonstrating strong conceptual knowledge of this subject. The research of Van Hiele and Van Hiele-Geldof has focused on the concept of thinking levels in geometry and the role of instruction in raising levels of thinking. This paper describes a field trial of a supplemental geometry unit intended to raise Van Hiele thinking levels in a group of 23 eighth-grade students by having them become more adept at using higher order thinking skills. Sample questions assessing particular Van Hiele thinking levels and attitudes toward geometry, as well as field-tested activities yielding the most positive results, are presented. Educators can benefit from this application of the Van Hiele model of geometric thinking, since the thought processes involved in learning geometry are explained, along with teaching techniques and tools for assessment. By having teachers become more aware of their students' cognitive skills, attitudes, and mi sconceptions, teaching practices and student achievement can be enhanced.

Geometry is a vital part of the mathematics curriculum. However, students have not been demonstrating strong conceptual knowledge of this subject. Senk (1989) reported that many secondary school students in the United States were not prepared for geometry classes. Fuys, Geddes, and Tishler (1988) found that there was too much emphasis placed on formal symbolism and naming in the elementary school geometry curriculum, while relational understanding was underemphasized. Carroll (1998) found that junior high and senior high school students often lacked experience in reasoning about geometric ideas. Carroll argued that middle school students are capable of developing good reasoning about geometric situations when they have had substantial exposure to geometry throughout elementary school. Unfortunately, many students develop misconceptions, and others fail to go beyond simple visualization of geometric figures.

The research of Dutch educators Pierre Van Hiele and Dina Van Hiele-Geldof has focused on levels of thinking in geometry and the role of instruction in helping students to move from one level to the next. Table 1 shows the characteristics of the five Van Hiele thinking levels, which Lawrie and Pegg (1997) described as a developmental model of thought processes through which students progress as they learn geometry. Van Hiele-Geldof (1984) presented phases of learning (see Table 2) that need to be integrated into geometry lessons. Fuys et al. (1988) stated that the teaching techniques advocated by the Van Hieles allow students to learn geometry by means of hands-on activities. The students utilize problem-solving strategies that, when combined with concrete experiences, yield higher order thinking skills.

This paper describes a field trial of a supplemental geometry unit intended to raise Van Hiele thinking levels in a group of 23 eighth-grade students by having them become more adept at using higher order thinking skills. Sample questions assessing particular Van Hiele thinking levels and attitudes toward geometry, along with field-tested activities yielding the most positive results, are presented.

METHOD

In order to assess the Van Hiele thinking levels of eighth-grade students, a pretest consisting of multiple-choice and short-answer questions was administered. The questions were assigned a thinking level according to the guidelines set forth by the Van Hiele model of geometric thinking described by Fuys et al. (1988) and Lawrie and Pegg (1997). Schultz (1989) stated that junior high school students should be able to think at Van Hiele level 2. Therefore, the pretest contained level 0, 1, and 2 questions, involving geometric concepts, shapes, and area. These topics were chosen for their adaptability (they are presentable at Van Hiele thinking levels 0, 1, and 2), their relation to student experience (topics were not taught in detail prior to the supplemental geometry unit), and their adherence to curriculum guidelines. …