Back to the Future in Mathematics Education
Recent reports show that little has changed in mathematics instruction.
Just over 20 years ago, A Nation at Risk proclaimed the "imperative of educational reform," arguing (in 1983) that in the two decades since Sputnik, the United States had committed "an act of unthinking unilateral educational disarmament." In the subsequent two decades, K-12 education has been in a constant state of reform, often bordering on upheaval. Nonetheless, both higher education and high-performance industries still complain about "mediocre educational performance," the very issue that animated the authors of A Nation at Risk.
Recent reports dealing with the mathematical expectations of higher education and the world of work show that little has changed in the last 20 years. One report, from the American Association of Universities, focuses on skills required for undergraduate success in research universities; the other, from the American Diploma Project, emphasizes broader skills required for success in high-performance jobs or postsecondary education. Despite their differences—which are important for readers to recognize—the very need for these reports reveals that we are still very much a nation at risk.
Mathematics is central to K-12 education and has played a leading role in all education reform efforts of the last half- century. A Nation at Risk was clear and specific about the importance of mathematics, along with closely related subjects such as science, economics, and computing: The authors’ primary recommendation was that "at a minimum, all students seeking a diploma be required [to take], during their four years of high school, four years of English, three years of mathematics, three years of science, three years of social studies, and one-half year of computer science."
When these words were written, only one in three U.S. high school graduates completed three substantive years (Carnegie units) of high school mathematics. Today, two decades later, two out of three graduates do so. However, according to a recent study by the Urban Institute, only two out of three students graduate from high school. This leaves the percentage of 18-year-olds who have met this two-decades-old standard for mathematics at below 50 percent.
Writing in the aftermath of the controversial "New Math" reforms, the authors of A Nation at Risk knew better than to ask just for more seat time. They were very specific about the kind of mathematics required to prepare students for the world of the 1980s—a world in which the personal computer was still largely a toy for hobbyists. Here is what they said:
The teaching of mathematics in high school should equip graduates to (a) understand geometric and algebraic concepts; (b) understand elementary probability and statistics; (c) apply mathematics in everyday situations; and (d) estimate, approximate, measure, and test the accuracy of their calculations. In addition to the traditional sequence of studies available for college-bound students, new and equally demanding mathematics curricula need to be developed for those who do not plan to continue their formal education immediately.
Mathematics educators responded to this challenge by developing the nation’s first "standards" report, very much along the lines recommended. It framed an approach to mathematics that would both reflect its expanding role in society and appeal to a broader range of students. Statistics, measurement, and computing joined algebra and geometry in the canon of school mathematics; inquiry-based learning joined direct instruction in the pedagogical repertoire of mathematics teachers. "Mathematics for all" became the slogan of the standards movement.
Industry leaders quickly climbed on board. High-performance workplaces require reliable quantitative skills and "six sigma" quality control. Students need "equally demanding" mathematics curricula to prepare for internationally competitive careers. As most such careers began to require some degree of postsecondary education, business leaders argued that the best preparation for work is no different from the best preparation for college.
The purpose of the two recent studies is to clarify just what constitutes high-quality preparation—both for higher education and for the world of work. The first, Standards for Success, is a 2003 project of the Association of American Universities that claims to describe the "content knowledge and habits of mind that are valued by leading research universities" and that are necessary "to succeed in entry-level university courses."
As befits its wide-ranging role in society, mathematics appears in Standards for Success both in its own chapter and in adjacent chapters on science and social science. The mathematics chapter emphasizes primarily just one of the four objectives named in A Nation at Risk, namely, the algebraic and geometric skills required to succeed in university mathematics courses. Other content goals of mathematics suggested in Risk (probability, applications, estimation, for example), as well as uses of high school mathematics in subjects other than college mathematics, are left to these other subjects.
Standards for Success moves beyond Risk in two noticeable ways. First, unremarkably, it acknowledges that the study of calculus requires specialized mathematical preparation that is not a universal requirement for success in college. Examples of such topics (irrational and complex numbers, polynomial arithmetic, trigonometric identities, for example) are marked with an asterisk.
Second, and well worth remarking on, is a strong emphasis on what Success calls "habits of mind." Some are of special importance in mathematics:
- Make and use estimates in appropriate contexts;
- Understand the roles of proof and counterexample;
- Calculate fluently;
- Use mathematical terminology correctly;
- Employ multistep methods;
- Translate words into equations; and
- Use calculators accurately and appropriately.
Others, while important for mathematics, apply to all subjects:
- Think conceptually, not just procedurally;
- Demonstrate flexibility and adaptability;
- Reason logically and use common sense;
- Demonstrate inquisitiveness and think experimentally;
- Revise unproductive strategies;
- Take risks, accept failure, and then try again;
- Question results, explain and defend answers;
- Write clearly (an "important indication" of understanding); and
- Work effectively both in groups and alone.
The second recent study, released in early 2004, is "Ready or Not: Creating a High School Diploma That Counts," from the American Diploma Project. This study seeks to document "must have" competencies needed for high school graduates "to succeed in postsecondary education or in high-performance, high- growth jobs." ("States Must Beef Up Diploma Demands, Study Maintains," Feb. 11, 2004.) The contrast between "Ready or Not" and Standards for Success is instructive.
Both outline required programs in high school mathematics that for average students would require three years of study. Both also signal the importance of certain advanced topics that are required for the study of calculus, but that lie outside the boundary of "must have" competencies for all students. This "asterisked" emphasis on preparation for calculus highlights the importance (some argue, stranglehold) of calculus in the transition from school mathematics to quantitatively intensive college majors.
A major difference between these two reports is their outlook on the school subject called "mathematics." Standards for Success identifies school mathematics primarily with topics in algebra and geometry needed for college mathematics. "Ready or Not" adds aspects of data analysis, statistics, and other applications that are vitally important for other subjects as well as for employment in today’s data-rich economy. These additional topics (data representation, scatter plots, normal distribution, correlation, sampling bias, conditional probability) were documented by the American Diploma Project as essential for high-performance work and higher education.
As Standards for Success moves beyond A Nation at Risk, so "Ready or Not" moves beyond Standards for Success. Yet neither report adds much new to what the authors of A Nation at Risk said so forcefully 20 years ago: All districts should require three years of high school mathematics including algebra, geometry, and statistics; all mathematics courses should be equally demanding; and all students should learn the habits of mind and data-oriented skills that are widely used in society.
Is it any wonder that many see educational reform as just swings of a pendulum?
Lynn Arthur Steen is a professor of mathematics at St. Olaf College in Northfield, Minn., and served as a consultant on the American Diploma Project.
Vol. 23, Issue 30, Pages 34, 36Published in Print: April 7, 2004, as Back to the Future in Mathematics Education