-- How many school teachers are there in New York     City? How many electricians? How many     morticians? -- How many words are there in all the books in the     school library? How many megabytes of disk     storage would be required to store the entire library     on a computer?

Avoiding Pitfalls

            Those who develop materials and examples for a functional curriculum need to avoid some common pitfalls that plague all attempts at situating mathematics in authentic contexts. On the one hand, there is the temptation to give priority to the mathematics, either by selecting tasks to ensure coverage of mathematical topics without much regard to the tasks' intrinsic importance or by imposing unwarranted structure on a contextually rich problem in the interest of ensuring appropriate mathematical coverage. On the other hand, it is easy to overlook interesting mathematics hidden beneath the surface of many ordinary tasks or to choose problems that fail to help students prepare for advanced study in mathematics. Any curriculum that is to prepare students for subsequent mathematics-dependent courses must recognize the importance of intellectual growth and conceptual continuity in the sequencing of tasks in which mathematical activities are embedded.

            Context-rich mathematics curricula often present tasks in the form of worksheets, outlining a series of short-answer steps that lead to a solution. While ostensibly intended to help students organize their thinking and assist teachers in following students' work, these intellectual scaffolds strip tasks of everything that makes them problematic. Indeed, worksheets reveal a didactic posture of traditional teaching (teacher tells, students mimic) that undermines learning and limits understanding. Students will learn and retain much more from the chaotic process of exploring, defending, and arguing their own approaches.

            Finally, although a functional mathematics curriculum is motivated largely by examples that seem to lie outside the world of mathematics, it is nonetheless very important for students' future study that instructors bring mathematical closure at appropriate points. Students need to recognize and reflect on what they have learned; to be clear about definitions, concepts, vocabulary, methods, and potential generalizations; and to have sufficient opportunity to reflect on the accomplishments and limitations of mathematics as a tool in helping solve authentic problems.