Given the complete record of a soccer league that ended in a three-way tie, devise a fair means of determining which team should be crowned champion. |
Need and Urgency
Despite mathematics' reputation as an ancient subject consisting of indisputable facts, mathematics education has recently become the source of passionate public debate. At stake is nothing less than the fundamental nature of school mathematics: its content (what should be taught), pedagogy (how it should be taught), and assessment (what should be expected). At times, these "math wars" have become so heated that Education Secretary Richard Riley (1998b) has issued a public call for a truce.
At the risk of oversimplifying, this debate can be characterized as a clash between "traditionalists" who expect schools to provide the kind of well-focused mathematics curriculum that colleges have historically expected and "reformers" who espouse a broader curriculum that incorporates uses of technology, data analysis, and modern applications of mathematics. The reform approach is championed by the National Council of Teachers of Mathematics, whose standards (NCTM, 1989) advocate a robust eleven-year core curriculum for all students, with supplementary topics for those who are "college-intending." Critics argue, however, that the NCTM standards are diffuse and ambiguous (Cheney, 1997; Raimi & Braden, 1998), that they are based on questionable assumptions about how students learn (Anderson, Reder, & Simon, 1997), and that curricula based on these standards will not provide the kind of rigorous preparation students need to succeed in calculus and other college-level mathematics courses (Wu, 1997).
Largely left out of this debate is a major constituency of mathematics education: employers. In today's information age, economic prosperity--for individuals as well as for the nation--depends on "working smarter, not just working harder." Yet, a majority of America's businesses report deficiencies in the technical and problem-solving skills of their workers and a severe shortage of prospective employees with these requisite skills. The cost of finding skilled employees has become a serious impediment to growth in many sectors of the U.S. economy (Carnevale, 1998).
Also left out are the voices of democracy and citizenship which were so important in the development of public education in the United States. Now, two centuries later, quantitative literacy is every bit as important as verbal literacy for informed participation in civic affairs. Today's news is not only grounded in quantitative issues (e.g., budgets, profits, inflation, global warming, weather probabilities) but is also presented in mathematical language (e.g., graphs, percentages, charts).
Neither traditional college-preparatory mathematics curricula nor the newer standards-inspired curricula were designed specifically to meet either the technical and problem-solving needs of the modern workforce or the modern demands of active citizenship. Although each includes much that is of value for citizenship and employment, neither provides the context, motivation, or balance of mathematical topics necessary for citizens or prospective employees in a data-drenched world.
The common curricular alternative--vocational or consumer mathematics--is significantly worse. Historically, vocational mathematics has provided only a narrow range of skills limited to middle school topics and devoid of conceptual understanding (National Center for Education Statistics [NCES], 1996). Such programs leave students totally unprepared--not only for modern work and postsecondary education, but even for advanced secondary school mathematics. Although some innovative school-to-career programs are seeking to change this pattern of low expectations, the vast majority of secondary schools in the United States offer students no effective option for mathematics education that meets the expectations of today's high-performance workplace.