What measurements do you need to take in order to tile the floor of a room? How can you use these measurements to determine the number of regular tiles, border tiles, and corner tiles that are needed? What if you decide to lay the main tiles on a 45° angle? |
Functional Mathematics
Functional mathematics comprises content, curriculum, context, and pedagogy. By content we simply mean the mathematics students should know and be able to do after finishing the first three years of high school mathematics (see Appendix B). Because mathematics is mathematics--whether traditional, reform, or functional--most of these elements are unsurprising. Although some topics are uncommon (e.g., index numbers, tolerances, three-dimensional geometry, normal curve, quality control charts, standards of proof, financial mathematics, spreadsheets), most are taught in any high-quality high school mathematics program. To achieve its goal of preparing students for both work and further education, functional mathematics respects the many parts of the traditional curriculum that are broadly useful, even while reshaping the boundaries to reflect its distinctive objectives.
The outline of functional mathematics in Appendix B reflects an inventory of mathematical topics selected for their importance in daily life and modern jobs as well as for their value in providing a strong foundation for further education. This outline is organized in predictable strands that cover what is normally subsumed under the umbrella of mathematics: numbers and data, measurement and space, growth and variation, chance and probability, reasoning and inference, variables and equations, modeling and decisions. Real problems cut across all this mathematics, just as these topics cut across the diverse contexts of authentic mathematical practice.
Clearly, many of the elements of functional mathematics are identical to the mathematics found in both traditional and reformed curricula. The core of school mathematics is more or less the same, even if viewed (or taught) from different perspectives. Percentages and ratios; linear and quadratic equations; areas, angles, and volumes; and exponential growth and trigonometric relations must be included in any strong high school mathematics program. The distinctions among traditional, reformed, and functional curricula lie not so much in core content as in contexts, emphases, and pedagogy.
Nonetheless, prospective employees for the new high-performance workplace need expertise in several aspects of mathematics not now emphasized sufficiently in school. On the one hand, students need greater experience recognizing and using some parts of middle school mathematics such as ratio, percentage, and measurement geometry that, although covered in current programs, are not used sufficiently to be learned well. On the other hand, as prospective employees, they need to understand and be able to use mathematical notions such as data analysis, statistical quality control, and indirect measurement that are hardly ever required in high school (Forman & Steen, 1998).
In addition to shifting the balance of topics, functional mathematics provides much greater emphasis on "systems thinking"--on habits of mind that recognize complexities inherent in situations subject to multiple inputs and diverse constraints. Examples of complex systems abound--from managing a small business to scheduling public transportation, from planning a wedding to reforming social security. At all levels from local to national, citizens, policymakers, employees, and managers need to be able to formulate problems in terms of relevant factors and design strategies to determine the influence of those factors on system performance. Although such systems are often so complex that they obscure the underlying mathematics, the skills required to address realistic problems very often include many that are highly mathematical.
A curriculum built on functional mathematics gives students many opportunities to solve realistic problems and build mathematical understanding. Nevertheless, to make this learning valuable for work and further education--as well as to enhance understanding--such a curriculum must also help students become fluent in the language of mathematics. Individuals need to be able to read, understand, and interpret technical material with embedded charts and diagrams (e.g., property tax bills, stock market reports); they need to be able to speak clearly about mathematical ideas (e.g., as a salesman explaining the interest and payoff on an insurance policy); and they need extensive experience writing reports based on mathematical and technical analysis (e.g., a recommendation to a supervisor summarizing the conclusion of a study).
Functional mathematics channels the much-criticized "mile-wide, inch-deep" curricular river into a narrower but deeper stream of ideas and procedures that reinforce each other as students progress through school and college and on into careers. It provides a rich foundation of experience and examples on which students can build subsequent abstractions and generalizations. Indeed, to fulfill its goals, a functional curriculum must leave students well-prepared not only for work but also for subsequent courses in more abstract mathematics.