ABSTRACT
The Curriculum and Evaluation Standards for School Mathematics (National Council of Teachers of Mathematics [NCTM], 1989) recommends that students in elementary and middle-school grades study geometric properties and relationships through practical experiences with shapes and patterns, integration of mathematical experiences with their art work, and the use of geometric computer programs. The Standards emphasize that “insights and intuitions about two- and three-dimensional shapes and their charac-teristics, the Interrelationships of shapes, and the effects of changes to shapes are important aspects of spatial sense” (p. 48). These geometric experiences are critical for our young students. Research based on the van Hiele levels of geometric thinking (Fuys, Geddes, & Tischler, 1988; Scally, 1990; van Hiele, 1986) has indicated that students entering high school have very little knowledge or experience of geometric properties and relationships; most are operating at the visual level of geometric thought in terms of the van Hiele model. These students can do little more than recognize different geometric shapes. They do not realize that a square is also a rectangle and a rhombus, or that all three are also parallelograms. Many do not realize that a square must have four right angles or that all four sides are congruent. Most have never heard of a “line of symmetry,” let alone understand why a square has four lines of symmetry whereas a non-square rectangle has only two. It is not surprising, then, that results of the Fourth National Assessment of Educational Progress (NAEP) indicate that fewer than a third of Grade 7 students (and only a quarter of the Grade 11 students not taking geometry) could identify perpendicular lines, and that for the same set of students, responses to items involving properties of simple figures (such as triangles and circles) were very poor (ranging from 8 to 42% correct; Lindquist, 1989). It is no wonder then that a decade ago fewer than one third of our high-school students made it through high-school geometry (National Commission on Excellence in Education, 1983).
