May 2001

Many of us were introduced to the concept of square numbers through the following formula:

Now imagine learning the same concept through this visual pattern:

Which approach would better help you understand the concept of square numbers? Would the visual pattern help you make sense of the formula?

According to researchers in EDC's Center for Mathematics Education, traditional mathematics curricula have neglected visual mathematics in favor of verbal and logical approaches that may not work as well for many students. In fact, for some students, "a visual approach may be absolutely essential," according to E. Paul Goldenberg, Albert A. Cuoco, and June Mark of the Center:

In the most common curricula, both in and out of the United States, geometry represents the only visually oriented mathematics that students are offered. Curricula tend to present an otherwise visually impoverished, nearly totally linguistically mediated mathematics, a mathematics that does not use, train, or even appeal to the "metaphorical right-brain" . . . There is a huge cost in this state of affairs: Some students who would like a visually rich mathematics never find out that there is one because they've already dropped out before they've had the chance to encounter any of the more visual elements. We lose not only potential geometers and topologists in this way, but all students who might enter mathematics through the visually richer domains and then discover other worlds, not as intrinsically visual, to which they can apply their visual abilities and inclinations. For some students, a visual approach may be absolutely essential.

This passage comes from the opening chapter of Designing Learning Environments for Developing Understanding of Geometry and Space (Lawrence Erlbaum Associates, 1998), co-edited by former EDC researcher Daniel Chazan with contributions by several EDC researchers.

In a recent review in the American Journal of Psychology, Professor Chris Donlan of University College London, calls the book "a pace setter for change in mathematics education," observing that it raises important questions about how the mind "thinks mathematically." Donlan, for example, praises the opening chapter by Goldenberg, Cuoco, and Mark for presenting an "enticing range of examples of the ways in which visual process operate," including perceptual illusions and the use of spatial models of multiplication.

In a later chapter, Goldenberg and Cuoco discuss the power of dynamic geometry—computer software that allows users to transform geometric shapes by "clicking and dragging." That ability to perform real-time transformations on shapes can be a powerful tool for exploring geometric concepts. For a sample, try some of these Java applications from EDC's Connected Geometry Web site.

In another chapter on technology applications in the mathematics classroom, EDC's Daniel Lynn Watt describes a computer-aided design (CAD) program developed as part of the Math and More elementary curriculum (a collaboration of EDC and IBM/EduQuest). Students used a CAD program to design a classroom and create a set of scale drawings and maps to illustrate their concept.

For a broader discussion of the relationship between mathematical content and cognitive thinking, read an article about mathematical habits of mind.

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