Learning Geometry Intuitively with the Aid of a New Computer Tool: The Geometer’s Sketchpad

The Curriculum and Evaluation Standards for School Mathematics (NCTM, 1989) recommends that students in elementary and middle school grades should be studying geometric properties and relationships through practical experiences with shapes and patterns, integrating mathematical experiences with their art work, and through the use of geometric computer programs. The Standards emphasize that “Insights and intuitions about twoand threedimensional shapes and their characteristics, 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 (van Hiele, 1986; Fuys, Geddes & Tischler, 1988; Scally, 1990) 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 while 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 less than a third of 7th grade students, and only a quarter of the 11th grade 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 percent correct) (Lindquist, 1989). It is no wonder then that less than one third of our high school students make it through high school geometry (National Commission on Excellence in Education, 1983). During the past twelve months, faculty and students in the Department of Mathematics Education at the University of Georgia have been involved with the prepublication (beta) testing of a new software environment for exploring geometry called The Geometer’s Sketchpad (Jackiw, 1991). This computer software environment provides teachers and students with an opportunity to explore geometric properties and relations both intuitively and inductively. Its dynamic nature makes it both exciting and accessible for even elementary students.