## Exploring "Multiple Mathematics"

by W. Norton Grubb, University of California, Berkeley

In the analysis of literacy, the notion of "multiple literacies" has become common. People use reading and writing in many different forms--at work, at home, in trying to get information from various print and electronic media, for amusement and pleasure as well as for more utilitarian goals. The sophistication and tone of different literacies vary; assumptions about form and about what an individual can infer differ as well, and so some understanding of literacy in its different forms is valuable. In fact, the multiple literacies that people employ are often quite different from conventional "school literacy," which usually involves the reading of well-known literature; standard exercises in detailing plot, character, and theme; and familiar drills with synonyms, homonyms, sentence completion, and grammar. For all those who were turned off to Shakespeare in high school, the pain associated with "school literacy" is easily remembered.

The idea of literacy in many different contexts can just as easily be applied to mathematics. Mathematical thinking and calculation (both formal and informal) arise in different ways in various settings. These "multiple mathematics" can be quite different from "school mathematics" with its rigid progression through arithmetic, algebra I, geometry, algebra II, and calculus. In this standard curriculum, which is best suited to the preparation of college mathematics majors, the use of mathematics is ripped from any context and divorced from the various ways people do use mathematics (or could use it if they didn't find it so forbidding). And so--like the concept of multiple literacies--articulating the different worldly manifestations of mathematics might help us appreciate better both the sterility of the standard curriculum and possibilities for alternatives.

When I--an economist, but no expert in mathematics--think about the multiple ways ordinary people use mathematics, the following come to mind:

• Mathematics at work is crucial, as employers have been telling us; it also varies enormously, and provides many examples of application, of "usefulness," and "relevance." Mathematics at work often involves a complex series of applications of relatively low-level mathematics or application to ill-defined problems. The complexity of the application is more important than the sophistication of the "school" mathematics. In other cases some relatively primitive mathematical understanding would help workers interpret better what they do. For example, few people--and not very many doctors--understand the variability inherent in medical tests. As a result their inferences are often wrong.

• People employ mathematical thinking to extract information from graphs, charts, maps, newspaper articles, and other visual devices that display information. (Some of these competencies have been incorporated into the concept of "document literacy" developed by the Educational Testing Service.) Without such facility, individuals may not be able to understand what they need for civic purposes, or parental responsibilities, or simply for daily life.

• In many social and natural sciences, algebraic expressions and geometric displays are used to model complex phenomena. But the notion of modeling in general, and mathematical modeling in particular, is difficult. My economics students have a hard time moving from the reality of economic phenomena to simplified models and back again, and traditional "school math" in no way prepares them to do this.

• Many aspects of "common sense" and "judgment"--competencies in scarce supply in work as well as other settings--require aspects of mathematical thinking even though formal calculations may be irrelevant: strategic thinking in the face of uncertainty, rough calculation of expected outcomes, probabilistic estimates, and rudimentary benefit-cost comparisons.

• "Street mathematics" can be seen in many contexts: merchants who mentally calculate prices and negotiate discounts; children who strategize about sports competitions; even school drop-outs who thrive in a drug economy. Many of these individuals have failed "school mathematics," yet clearly demonstrate a kind of mathematical power in contexts that are meaningful to them.

• Just as there is escapist reading and both reading and writing for aesthetic purposes, so there is escapist mathematics (puzzles and games) and a kind of "mathematics for art"--golden rectangles, classic proportions, tessellations, symmetry, Escher. In recent years, chaos theory and fractals have stimulated the public imagination. Yet these pleasurable uses of mathematics are unavailable to most people because "school math" requires that they progress through a lot of boring stuff before they can understand the fun stuff.

There are surely many other forms of mathematics. As the advocates of multiple literacies stress, the ways in which a subject is encountered are varied, sometimes hidden, often subtle and unsuspected, often providing opportunities for instruction that are otherwise lost. Indeed, by simply articulating a notion of "multiple mathematics," we may stimulate a search for the forms it might take in many walks of life. This is an urgent project for mathematicians, mathematics educators, and all those who mourn the sorry state of mathematics in this country.

Now, I would never argue that "multiple mathematics" should displace conventional school mathematics. Some advocates of whole language and literacy "in context" have gotten into trouble with parents and legislators for saying (or appearing to say) that drill should never be used, or that grammar and spelling are unimportant, or that standard literature is "irrelevant." So too in mathematics: it would be silly and extremist to argue that drills on formal operations or facility with algebraic and geometric representations are unimportant.

The trick is to devise curricula that use different approaches to support one another-- that introduce modeling as a way into the power of algebraic representation; or that examine gambling and the vagaries of the weather to begin the study of probability and stochastic thinking; or that examine the mathematics used at work to demonstrate its relevance and provide facility with application. In this way the notion of "multiple mathematics" could help inspire curricula with greater range, power, and motivation without abandoning the school mathematics that has left so many behind.

W. Norton Grubb, Site Director of the National Center for Research in Vocational Education (NCRVE), is a member of the faculty of the School of Education at the University of California, Berkeley 94720; He can be reached by e-mail at norton_grubb@maillink.berkeley.edu; or by fax at 510-642-3488.

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Last Update: 02/28/96