A large load of topsoil forms a conical pile. Because of its size, you cannot directly measure either its diameter or its height. Find a strategy for estimating its volume.

Theory and Practice

            Historically, education in the United States has vacillated between the liberal and the pragmatic, between Robert Maynard Hutchins and John Dewey. Mathematics reflects a similar tension in the delicate balance of theory and practice, of the pure and the applied (Thurston, 1990). Through most of this century, school mathematics has oscillated back and forth between these poles (Kilpatrick, 1997). Indeed, nearly a century ago, the president of the American Mathematical Society lamented the "grievous" separation of pure from applied mathematics and urged schools to provide a more "practical" mathematics education: "With the momentum of such [education], college students would be ready to proceed rapidly and deeply in any direction in which their personal interests might lead them" (Moore, 1903). Today's effort to make mathematics more functional for all students is just the latest chapter in this long saga.

            In recent years, this debate has been expressed in the form of standards, both academic and occupational. Coordinating these standards will involve not only issues of content and pedagogy, but also the balance of school-based vs. work-based learning (Bailey, 1997). Historically, vocational curricula designed to prepare students for work have been burdened by second-class status in comparison with more rigorous academic curricula. Too often, vocational programs became dumping grounds for students who appeared slow or unmotivated--"other people's children." Most programs responded by limiting goals and lowering expectations, thereby offering stunted education to students who were already behind. In contrast, contemporary career-oriented curricula have been designed not primarily as training for low-skill jobs but as motivation for rigorous study, both academic and vocational (Bailey & Merritt, 1997; Hoachlander, 1997). By setting high standards, these programs offer significant responses to the twin challenges of equity and competitiveness.

            Mathematics provides a microcosm of the duality between the academic and the vocational. Widely perceived as the epitome of theory and abstraction, mathematics is also valued as a powerful, practical tool (Odom, 1998). In many occupations, quantitative literacy is as important as verbal literacy (Steen, 1997); however, if mathematics education is to serve the world of work, a different type of experience than that found in typical mathematics courses is required (National Research Council, 1998).

            Between theory and application lies professional practice--the synthesis of thought and action employed by practitioners in all vocations. Many have argued that practice, properly understood, can be a legitimate and unifying goal of education. Practice is functional knowledge, the kind of know-how that allows people to get things done. According to educator Lee Shulman (1997), practice can provide a context in which theory becomes meaningful, memorable, and internalizable. Peter Denning (1997), a computer scientist, believes that practice--not knowledge or literacy--is what constitutes true expertise. Indeed, practice is what people tend to expect of schools, especially of mathematics education. It is at the heart of functional mathematics.

            An infusion of practice into school mathematics can overcome what Shulman (1997) identifies as major deficiencies of theoretical learning: loss of learning ("I forgot it"), illusion of learning ("I thought I understood it"), and uselessness of learning ("I understand it but I can't use it"). Adults who are not professional users of mathematics will recognize these deficiencies from their own experiences. Little of what adults learned in school mathematics is remembered or used, so the accomplishment of "learning" mathematics is often an illusion. In fact, the mathematics many students are force-fed in traditional school environments creates a severe psychological impediment to the practice of mathematics in adult life (Buxton, 1991; Cockroft, 1982). Functional mathematics avoids many of these pitfalls by emphasizing that the goal of mathematics education is not just mathematical theory and word problems, but authentic mathematical practice.