In preparing fertilizer for a garden, a homeowner poured one quart of concentrated liquid fertilizer into a two- gallon can and filled the can with water. Then she discovered that the proper ratio of fertilizer to water should be 1:3. How much more liquid fertilizer should she add to the current mixture to obtain the desired concentration? |
Making Mathematics Meaningful
Those who discuss mathematics education frequently describe mathematical knowledge in broad categories such as skills and understanding, concepts and facts, procedures and practices, or insights and knowledge. Whole volumes of educational research are devoted to distinguishing among these different aspects of mathematical knowledge. The standards movement has tended to subsume all these distinctions into two categories of knowledge and performance: what students should know and what students shoud be able to do (Ravitch, 1995; Tucker & Codding, 1998).
The two broad cultures of mathematics education argue with each other largely because they differ in the interpretations they give to these different aspects of mathematical knowledge. Those who favor the traditional curriculum centered on algebra, functions, and Euclidean geometry argue that mastery of facts and basic skills are a prerequisite to understanding and performance. Reformers who favor a broader curriculum take a more constructivist view--that understanding and mastery are an outgrowth of active engagement with contextualized mathematics. Regardless of approach or emphasis, both traditional and reform curricula generally cover a similar set of topics designed to move students along the path from arithmetic to calculus.
Functional mathematics follows much the same path, with variations that reflect its grounding in authentic problems. However, by embedding mathematics in practice, functional mathematics can offer students both theory and know-how. Although in some technical areas, practical "of-the-moment" learning offers little that outlasts the next generation of gadgets, the logical structure that unites mathematics guarantees that all understanding, no matter how specific, has the potential to enhance mastery of other areas. What matters for long-term mastery of mathematics is not so much which particular skills are learned as that the process of learning be, in Shulman's words, "meaningful, memorable, and internalizable." Although topics in functional mathematics may be chosen for proximate utility, their study can provide insight and understandings sufficient for lifelong learning.
A curriculum based on functional mathematics requires appropriate content, authentic contexts, engaging tasks, and active instruction. By featuring mathematics in common contexts, a functional curriculum can motivate students to link meaning with mathematics. The best problem settings offer opportunities for exploration from multiple perspectives, including graphical, numerical, symbolic, verbal, and computational. Technology--from graphing calculators and word processors to spreadsheets and symbolic algebra systems--can enhance understanding from each of these perspectives. Effective contexts provide opportunities for horizontal linkages among diverse areas of life and work as well as vertical integration from elementary ideas to advanced topics. Experience with rich contexts helps students recognize that asking questions is often as important as finding answers. Such contexts invite variations that can stimulate mathematical habits of mind and propel students to deep understanding.