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Endless Algebra—the Deadly Pathway from High School Mathematics to College Mathematics

Shaughnessy_52010by NCTM President J. Michael Shaughnessy
NCTM Summing Up, February 2011 

For some time I have been concerned about the mathematics that we are teaching our K–12 students and whether it will prepare them for the problems and challenges that they will encounter in mathematics, science, technology, and engineering in the 21st century. Recently, my concern has reached the red-alert level, especially my concern about how we prepare secondary students for the transition to college. I constantly ask myself two questions: (1) Are we really offering our secondary students an appropriate mathematics experience? (2) What can we do to provide students with relevant, coherent mathematical options on the pathway throughout high school and as they move into college? Or to put it another way: (1) Is the “layer cake” of algebra-dominated mathematics that pervades our U.S. secondary schools still relevant? (2) Is calculus the be-all and end-all goal for the preparation of students for a successful transition to college? My answer is, I think not.

For more than 20 years national organizations and prominent leaders in mathematics education, including NCTM, have warned that our national rush to calculus is misguided and not even an appropriate path for many students. For secondary school mathematics, NCTM recommends a different approach: “These four years of mathematical study will revolve around a broadened curriculum that includes extensions of the core topics and for which calculus is no longer viewed as the capstone experience.” (Curriculum and Evaluation Standards for School Mathematics. NCTM, 1989). According to Lynn Steen, former president of the Mathematical Association of America (MAA), “It is probably about time that we face facts: Aiming school mathematics for calculus is not an effective strategy to achieve the goal of improving all students’ mathematical competence.” (Mathematics Teacher 100 [February 2006]). More recently, the MAA published the Curriculum Renewal Across the First Two Years (CRAFTY) report, which examined the mathematical needs of many client disciplines, such as biology, chemistry, economics, engineering, physics, and others. CRAFTY advocates secondary mathematics that facilitates students’ transition from high school to college by providing (1) a greater emphasis on modeling; (2) consideration of multivariate topics; (3) an emphasis on computational skills that are useful in other fields; and (4) a strong foundation in units, scaling, and dimensional analysis.

Last month, I participated in a panel presentation, Transition from High School to College: Should There be an Alternate to Calculus? at the Joint Mathematics Meetings conducted by the Mutual Concerns Committee of NCTM and the MAA. This experience caused me to look even more critically at what many of our students are experiencing as they move through high school and on to college. I am not the only one who is concerned. My co-presenters and many of the participants packed into the room echoed similar thoughts. Both high schools and colleges are operating under outdated assumptions. Among these are the assumptions that high school students should take or be prepared to take calculus, and that the path to calculus needs to be paved with frequent and repetitive overdoses of algebra.

Consider, for example, a typical student’s mathematics transition path. In high school, a student takes algebra I, algebra II, and perhaps pre-calculus. In college, this student may be put into Intermediate algebra, followed by college algebra, and perhaps, yet again, pre-calculus. This endless sequence of algebra courses is not an uncommon experience for many students, and the attrition rate along this path is very high. Many students thus mired in algebra discover they don’t need calculus, and they exit math at the level of college algebra, never to return. This is an out-of-date, wasteful, and repetitive transition path for our students. Worse, it does nothing to improve our students’ disposition toward mathematics. (Read NCTM President’s Message, October 2010) When students are confined to this tunnel of repetitive algebra, they never have opportunities to experience the beauty, excitement, power, or usefulness of mathematics as called for in the NCTM Standards (1989, 2000) or suggested by the CRAFTY report from the MAA.

The NCTM/MAA Mutual Concerns panel presented four concrete, relevant, alternative mathematical transition paths for high schools and colleges to consider. 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. We live in a multivariate world. Therefore, students’ mathematics experience in preparation for their transition to college should emphasize multivariate functions, partial derivatives, multivariate data sets, and analyzing covariance.

The Common Core State Standards provide us with an opportunity to rethink the sequence of school mathematics, as well as a challenge to provide exciting new pathways and transitions from high school to college mathematics. We need to offer students alternative pathways as they make their transition from secondary school and into colleges. The mathematics paths that we provide for our students need to prepare them for existing fields that are changing rapidly, as well as for emerging fields—and for fields that don’t yet exist. In my view, the current deadly sequence of ever-repetitive and out-of- touch experiences in algebra—the sequence intended to lead students to a single variable calculus course—will not accomplish this goal. It is time that we replace the eternal algebra transition from high school to college with some viable and exciting 21st century mathematics alternatives. 

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