The End of Algebra?

*David M.
Bressoud, *March, 2011

For his February

President’s Cornerof the NCTM Newsletter,Summing Up, Michael Shaughnessy, wrote about “Endless Algebra—the Deadly Pathway from High School Mathematics to College Mathematics.’’ [1] In this column he asks, “Is the `layer cake’ of algebra-dominated mathematics that pervades our U.S. secondary schools still relevant?’’ and suggests four alternatives to “frequent and repetitive overdoses of algebra”:“One path emphasizes quantifying uncertainty and analyzing numerical trends. Its mathematical foci include data analysis, combinatorics, probability, and the use of data collection devices, interactive statistical software, and spreadsheet analyses of numerical trends.

“A second transition path concentrates entirely on the development of students’ statistical thinking, beginning in high school and continuing into the first year of college. Statistical thinking involves understanding the need for data, the importance of data production, the omnipresence of variability, and decision making under uncertainty. This path differs both in purpose and approach from an AP statistics course.

“A third path recommends building a transition grounded in linear algebra. Linear algebra integrates algebra and geometry through powerful vector methods. It offers an arena in which students can work with important multivariable problems and provides students with general-purpose matrix methods that will serve them well in many fields, including mathematics, science, engineering, computer science, and economics.

“Finally, a fourth transition path incorporates a suggestion that an alternative to calculus can be found in calculus itself—but a vastly different calculus from the traditional calculus I. This path concentrates on multivariate applications of both calculus and statistics, because today’s application problems rarely involve single-variable calculus or univariate statistics.”

Since there are echoes of things I have said in Shaughnessy 's article, I feel that I need to express my strong disagreement with him.

One point of agreement between us is that a singular view of high school mathematics as preparation for calculus has created serious problems. This is particularly damaging when combined with the belief that, if at all possible, students must get through calculus while still in high school, and if they cannot get through it, then, at the least, they have to learn its tricks before they get to college. I believe that one of the most important contributors to student difficulty at the transition to college mathematics is that many, perhaps even most, of the students who study a course called calculus while in high school are simply not ready for college-level work in calculus. They would be much better served if they spent their time broadening and deepening their knowledge of mathematics, especially in algebra and analytic geometry, but also in trigonometry and geometry. We should be very concerned that 17% of the high school class of 2004 who completed a high school course in calculus then took remedial mathematics when they got college. [2]

We have another point of agreement on the other side of the transition, which is that not all students are heading for calculus in college and that it is a serious mistake to direct them into a College Algebra-Precalculus-Calculus track unless this clearly fits their needs and aspirations. Michael Shaughnessy is frustrated, as am I, with the wasted time and resources implicit in the many students who complete Algebra II or Precalculus in high school, yet require one or more semesters of remedial mathematics at university, simply so that they can take a college algebra course that repeats what they have seen before—only faster—and leaves them with nothing more than a checked box for their transcript. There is clearly failure on both sides of the transition: High school has not adequately prepared these students for college mathematics, and our colleges are not giving these students what they need at this point in their education. One of the points I made in my retiring presidential address at the Joint Math Meetings in New Orleans is that we need more good mathematical options for students in their first year at university.

Several of the paths described in the NCTM article reflect thought-provoking presentations from the panel discussion on

Transition from High School to College: Should There be an Alternate to Calculus?at the New Orleans Math Meetings, but the programs I endorse are not alternatives to algebra in high school. I applaud the intention of the Carnegie Foundation’s Statway program for two-year colleges that prepares students for a college-credit-bearing course in statistics through a program that embeds the needed algebraic skills within a course that is focused on statistics, data analysis and quantitative reasoning. [3] I have seen firsthand here at Macalester how effective our Applied Calculus course—with its emphasis on calculus as a tool for mathematical modeling—can be as a terminal course for those who would only take a single semester of calculus and as a window into the power and importance of mathematical analysis. I also can speak highly of Al Cuoco's high school course in linear algebra. It does not attempt to be a college linear algebra course. Rather, it is about engaging students with important mathematical ideas while building facility with the basic tools of multi-dimensional mathematics. Last spring, over 73,000 high school students took the AP Calculus exam before their senior year. There is a real need for courses like this that broaden understanding of mathematics rather than simply accelerating students.But none of this obviates the need for a strong preparation in algebra among high school students. I worry that Michael Shaughnessy’s prescription would lead to a variety of soft options in high school that preclude the possibility of pursuing not just calculus but even statistics and certainly linear algebra once students get to college. We need to focus our attention on improving the development of algebraic knowledge and skills among our high school graduates rather than onfinding alternatives to algebra.

[1] J. Michael Shaughnessy. Endless Algebra—the Deadly Pathway from High School Mathematics to College Mathematics.

Summing Up. February 2, 2011. National Council of Teachers of Mathematics. Reston, VA. www.nctm.org/about/content.aspx?id=182[2] National Science Board. Science and Engineering Indicators: 2010. National Science Foundation. Arlington, VA. Appendix Table 1-20. www.nsf.gov/statistics/seind10/appendix.htm

[3] The Statistics Pathway (Statway) is a program of The Carnegie Foundation for the Advancement of Teaching. www.carnegiefoundation.org/statway

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David Bressoud is DeWitt Wallace Professor of Mathematics at Macalester College in St. Paul, Minnesota, and Past-President of the MAA. You can reach him at bressoud@macalester.edu. This column does not reflect an official position of the MAA.