Five friends meet for dinner in a restaurant. Some have drinks and others do not; some have dessert and others do not; some order inexpensive entrees, others choose fancier options. When the bill comes they need to decide whether to just add a tip and split it five ways, or whether some perhaps should pay more than others. What is the quickest way to decide how much each should pay?

Teaching Functional Mathematics

            Although the public thinks of standards primarily in terms of performance expectations for students, both the mathematics standards
(NCTM, 1989) and the science standards (National Research Council, 1996) place equal emphasis on expectations for teaching, specifically that it be active, student-centered, and contextual:

• Active instructionencourages students to explore a variety of strategies; to make hands-on use of concrete materials; to identify missing information needed to solve problems; and to investigate available data.
  
  
• Student-centered instructionfocuses on problems that students see as relevant and interesting; that help students learn to work with others; and that strengthen students' technical communication skills.
  
  
• Contextual instructionasks students to engage problems first in context, then with mathematical formality; suggests resources that might provide additional information; requires that students verify the reasonableness of solutions in the context of the original problem; and encourages students to see connections of mathematics to work and life.

            These expectations for effective teaching are implicitly reinforced in recently published occupational skill standards (National Skill Standards Board, 1998) that outline what entry-level employees are expected to know and be able to do in a variety of trades. Although these standards frequently display performance expectations for basic mathematics as lists of topics, the examples they provide of what workers need to be able to do are always situated in specific contexts and most often require action outcomes (Forman & Steen, in press).

            Most students learn mathematics by solving problems. In traditional mathematics courses, exercises came in two flavors: explicit mathematical tasks (e.g., solve, find, calculate) and dreaded "word problems" in which the mathematics is hidden as if in a secret code. Indeed, many students, abetted by their teachers, learn to unlock the secret code by searching for key words (e.g., lessfor minus, totalfor plus) rather than by thinking about the meaning of the problem (which may be a good thing, because so many traditional word problems defy common sense).

            In a curriculum focused on functional mathematics, tasks are more likely to resemble those found in everyday life or in the workplace than those found in school textbooks. Students need to think about each problem afresh, without the clues provided by a specific textbook chapter. Rather than just being asked to solve an equation or calculate an answer, students are asked to design, plan, evaluate, recommend, review, define, critique, and explain--all things they will need to do in their future jobs (as well as in college courses). In the process, they will formulate conjectures, model processes, transform data, draw conclusions, check results, and evaluate findings. The challenges students face in a functional curriculum are often nonroutine and open-ended, with solutions taking from minutes to days, and requiring diverse forms of presentation (oral, written, video, or computer). As in real job situations, some work is done alone, and some in teams.