Geometry and Op Art

Article excerpt

Each year as I begin to plan a geometry unit for my students, I consider what I am supposed to teach, what I think I ought to teach, and what I currently know how to teach. This year, my planning has taken on a new dimension as a result of a year's coursework in elementary mathematics. The Curriculum and Evaluation Standards for School Mathematics (NCTM 1989), along with a variety of other readings, has expanded my understanding of the importance of providing rich, appropriate opportunities for children to explore all manner of fundamental geometric concepts.

The coursework exposed me to the work of researchers who have conducted investigations into what, when, how, and why various geometry topics should be taught to successfully influence students' learning.

I came away from my studies with a strong sense of responsibility to create a learning environment for my students that will prepare them for the higher-level instruction that they will encounter in middle and high school. The activities that I select, the reasons that I select them, the ways that I present them, and the degree of skill with which I guide my students' understanding lay the foundations for a geometry unit that will build a framework for future learning.

A Framework for Geometry

The focus of my planning is changing from objectives to framework. Just as the larger branches of a tree, its "framework," determine the tree's overall shape, so must I promote a framework for geometric thinking to shape my students' future successes in mathematics.

The curriculum objectives for my grade level include explorations of lines, paths, and polygons. They also include differentiation among horizontal, vertical, parallel, and intersecting lines. While providing a variety of activities for students to investigate these concepts, such as laying wet string on minichalkboards and using geoboards with rubber bands, I also focus attention on developing the specialized geometry language appropriate to these concepts. Standard 9 of the NCTM's

Curriculum and Evaluation Standards for School Mathematics states that "although a facility with the language of geometry is important, it should not be the focus of the geometry program but rather should grow naturally from exploration and experience" (NCTM 1989). The goal, then, is to dovetail the development of concepts and the mathematics language component as seamlessly as possible. The Professional Standards for Teaching Mathematics mentions that the "teacher should monitor students' use of mathematical language to help develop their ability to communicate mathematics" (NCTM 1991). With these points in mind, I try to find ways in which students can meld mathematical concepts and language.

The mathematics activity in this article is actually a lesson in geometry that uses the computer in an artistic exercise to allow students to see the distinct connection between art and mathematics from a personal perspective. The intent is to offer an enjoyable activity that blends concepts and language and avoids the tedium of memorizing definitions (Van de Walle 1994).

Developing a Vocabulary about Lines

In the course of our geometry unit, students are encouraged to share their ideas using language appropriate to the topic. In discussing different types of lines, the terms horizontal, vertical, diagonal, perpendicular, parallel, and intersecting are naturally introduced in the classroom. The expressive language changes from "It has a line on one side and another line just like it on the other side" to the mathematically oriented "It has parallel lines that are vertical." In one paper-and-pencil drawing activity, students eagerly create designs, then describe the figures in geometric terms.

I find that students love to use the language of these special types of lines. The words are big, and the concepts are easy. Remembering which word goes with which particular line, however, proves difficult for the students. …