A round chimney 8" in diameter protrudes from a roof that has a pitch of 3:1. Draw a pattern for an aluminum skirt that can be cut out of sheet metal and bent into a cone to seal the chimney against rain.

Mathematics in Context

            Students' achievements in school mathematics depend not only on the content of the curriculum and the instructional strategies employed by the teacher but also on the context in which the mathematics is embedded. Traditionally, mathematics has served as its own context: as climbers scale mountains because they are there, so students are expected to solve equations simply because it is in the nature of equations to be solved. From this perspective, mathematics is considered separate from and prior to its applications. Once the mathematics is learned, it supposedly can then be applied to various problems, either artificial or real.

            Many of the new curricula developed in response to the NCTM standards or state frameworks give increased priority to applications and mathematical models. In some of these programs, applications are at the center, providing a context for the mathematical tools prescribed by the standards. In others, applications serve more to motivate topics specified in the standards. In virtually all cases, the applications found in current curricula are selected, invented, or simplified to serve the purpose of teaching particular mathematical skills or concepts. In contrast, the mathematical topics in a functional curriculum are determined by the importance of the contexts in which they arise.

            For most students, interesting contexts make rigorous learning possible. Realistic problems harbor hidden mathematics that good teachers can illuminate with probing questions. Most authentic mathematical problems require multistep procedures and employ realistic data _--which are often incomplete or inconsistent. Problems emerging from authentic contexts stimulate complex thinking, expand students' understanding, and reveal the interconnected logic that unites mathematics.