You are helping your brother-in-law build a garage on gently sloped land next to his house. After leveling the land, you begin staking out the foundation. To check that corners are square, you measure the diagonals and discover that they differ by 3 inches. Is that because the corners may not be perfectly level, or because they are not perfectly square? How can you determine what needs fixing to make sure that you start with a foundation that is both level and square?

Functional Curricula

            The elements of functional mathematics can be embedded in many different curricula--the paths students follow through their education. Although some parts of mathematics impose a necessary order on the curriculum (e.g., arithmetic before algebra; linear equations before quadratic), large parts of mathematics can be approached from many different directions. Data analysis can be either a motivation for or an application of graphing and algebra, geometry can either precede or follow algebra; and each can reinforce the other. The order in which elements are listed bears no relationship to the order in which they may be taught through a three-year core curriculum.

            The elements of functional mathematics arise from common contexts of life and work--measuring objects, managing money, scheduling time, making choices, and projecting trends. Although it is possible to organize a curriculum around such contexts, without a list of elements such as those in Appendix B to guide instruction, the mathematics itself may remain largely hidden. Alternatively, a functional curriculum can be organized around mathematical themes such as the sections of Appendix B. Indeed, the latter fits better the experience of most mathematics teachers and is more likely to be adaptable to most school settings.

            Any mathematics curriculum designed on functional grounds--whether organized around external contexts or mathematical themes--will emphasize authentic applications from everyday life and work. In such a curriculum, students will gain considerable experience with mathematical tasks that are concrete yet sophisticated, conceptually simple yet cognitively complex (Forman & Steen, 1995). A functional curriculum compels a better balance of statistics (numbers), geometry (space), and algebra (symbols)--the three major branches of the mathematical sciences. By highlighting the rich mathematics embedded in everyday tasks, this approach (in contrast to traditional "vocational math") can dispel both minimalist views about the mathematics required for work and elitist views of academic mathematics as an area with little to learn from work-based problems (Bailey & Merritt, 1997; Forman & Steen, 1998).

            Because of the history of low standards in traditional vocational programs, many teachers and parents believe that a work-focused curriculum will necessarily lack the rigor of a precollege academic track. Contrary to this belief, the "zero-defect" demands of the high-performance workplace for exacting standards and precise tolerances actually impose a much higher standard of rigor than do academic programs that award students a B for work that is only 80-85% accurate. Moreover, the lengthy and subtle reasoning required to resolve many problems that arise in real contexts provides students with experience in critical thinking that is often lacking in academic courses that rush from topic to topic in order to cover a set curriculum.