Meeting the Challenge of High School Calculus, VII: Our Responsibilities

*David M.
Bressoud, *September, 2010

It would be wonderful if all high school calculus students entered that class with a solid knowledge of algebra, geometry, and trigonometry as well as experience in proving mathematical theorems and struggling with complex, multi-step problems. It would be wonderful if they then experienced a conceptually rich and demanding course that fired their interest in mathematics. And, it would be wonderful if the last mathematics course they studied in high school dovetailed perfectly with their first course at college. Unfortunately, none of these happen as often as we would wish.

The reality for most colleges and universities is that far too many students enter with significant gaps in their mathematical preparation, an attitude toward calculus that falls somewhere between fear and loathing, and no attractive courses on offer that address their weaknesses while providing challenging and engaging mathematics. The fact that the number of mathematics majors has remained essentially unchanged over the past thirty years [1] is symptomatic of a deep problem in the articulation from high school to college and the reluctance of many departments of mathematics to address their side of the problem.

While there is much to be done at the high school end of the transition, colleges and universities cannot afford to ignore their responsibilities

Placement Tests:No place is there more potential for harm than in a college’s choice of placement test. Both types of errors are significant. The student who is denied access to a course for which she or he is ready risks boredom and frustration and may be discouraged from pursuing mathematics. The student allowed into a course for which he or she is not ready is likely to flounder, encountering obstacles that are so high that further mathematical progress is barred.It is essential that the placement test truly assesses what the student needs to know and be able to do. For this reason, the MAA has just updated its policy statement on

College Placement Testing in Mathematics[2]. In particular, the MAA has stated that the SAT and ACT are not designed and should not be used for placement testing. Neither should the focus of a placement test be exclusively on computational skill. It must also include procedural fluency, conceptual understanding, and strategic reasoning.I greatly favor on-line testing such as the MAA’s own

Maplesoft-MAA Placement Test Suite[3] which enables students to take multiple versions of the same test. I also like the use of adaptive tests such as ALEKS that are able to burrow down to provide a finer assessment of what the student does and does not know.

Appropriate Courses:Knowing where a student belongs is of no use if none of the available courses meet that student’s needs. This can be especially problematic for students entering with one or two semesters’ worth of credit for calculus. A student may well have mastered the material of Calculus I without being ready for the greatly increased pace and expectations for learning outside the classroom that are required for success in college Calculus II. It also happens that students who are fully prepared to succeed in Calculus II still have gaps or weaknesses in their knowledge of Calculus I or even precalculus that may hinder more advanced mathematical progress.The answer is not to make them retake Calculus I, but to engineer a course that is specifically designed for these students who enter with Advanced Placement credit. Many colleges and universities now do this. It should be common practice at every institution with sizeable numbers of students entering with AP credit in calculus.

Flexible Courses:The number of students who have studied calculus in high school but arrive at college with significant gaps in their knowledge of and ability to use precalculus topics—gaps that may impair the chances of successful continuation in mathematics—is now so large that this population requires particular attention. A standard precalculus course is usually not what they need because too many of the topics constitute unnecessary review. Most of these students need means of addressing their particular gaps and weaknesses and refreshing what they may have known but have forgotten, but they also need recognition of what they have accomplished and the opportunity to build on it.This is the right audience for online, self-paced instruction, possibly as a complement to regular instruction. It is very important to challenge and engage these students while bringing them up to speed on what they will need if they are to succeed in future courses.

Alternate Routes:In the 1980s, Tony Ralston [4] and others raised the question of whether calculus should be the course with which we expect students to begin their college-level mathematics. Why not discrete mathematics? Many voices were raised in defense of calculus, mine among them [5]. Calculus does warrant its privileged position. However, this does not imply that calculus should be the sole gatekeeper to higher mathematics. Too many students are turned off by their experience of calculus, either in high school or in their first term at college. And too many students, because they have seen it as the ultimate goal of their K-12 mathematical preparation, view a passing grade on the Advanced Placement exam as certification that they have completed mathematics.Macalester teaches discrete mathematics in a demanding course required of mathematics majors and targeted to students in their first year. Many take it just because it has a reputation for introducing new and interesting mathematics, and many of these are lured into continuing their study of mathematics. Linear algebra, elementary number theory, and statistics all share the potential to attract those students who have sworn off calculus. Some form of Ralston’s vision of the mathematics curriculum of the first two years should be available as an alternate route into mathematics.

ConclusionThe tidal wave of AP has carried calculus into the high school curriculum. It does no good to complain that it should not have happened, or that there has been too little preparation for its consequences. Our job now is to deal with it. I close this series where I began, with the three urgent tasks that lie before us:

- to better understand the effect of high school calculus on our students and their subsequent study of mathematics,

- to ensure that those who study calculus in high school are prepared and that when it is billed as college-level mathematics it truly meets that standard, and

- to adapt our college and university curricula to meet the needs of the students who now are entering.
Throughout the past two decades of transition, we have stumbled in meeting these challenges, with the result that too few students are getting the mathematical preparation they need to succeed in technical and scientific careers. I am cheered by the many local efforts I see being made. I am hopeful that the mathematical community will now rise to these challenges.

[1] According to CBMS data, the number of majors in Mathematics (excluding majors in mathematics education, statistics, actuarial science, operations research, or joint majors) stood at 11,541 in 1980, increased to 13,303 in 1990, and dropped back to 12,316 in 2005. US Dept of Education data suggest that by 2008 it may have climbed back to around 13,000. From 1980 to 2008, total undergraduate enrollment increased by 58% (NCES fall enrollment data) and the US population increased by 36%.

[2] The following is the policy statement adopted by the MAA Board of Governors on August 4, 2010:

COLLEGE PLACEMENT TESTING IN MATHEMATICSEducational accomplishments in mathematics can be key to successful careers. College-level mathematics study should build on and extend prior experiences. Students entering higher education have diverse preparations for college mathematics due to many factors: academic background, time since high school graduation, age, work experience, etc. As a result, college and university mathematics departments may misplace students in their first college mathematics courses if they use only data such as high school transcripts, class rank, or GPA.

Carefully developed placement tests can be effective parts of a comprehensive placement process. The Mathematical Association of America recommends the following strategies for college placement tests in mathematics:

MEASURE DEVELOPED MATHEMATICAL REASONING SKILLS. College admission tests such as the SAT or ACT measure students' general readiness for college, whereas placement tests seek to measure students' knowledge and skills that are prerequisite for specific entrylevel college mathematics courses. Nationally administered tests such as SAT and ACT measure a broad range of quantitative skills, and this measure is often too general to distinguish between readiness for entry-level mathematics courses such as college algebra, trigonometry, pre-calculus and calculus. Therefore, very often, high school record and admission test scores need to be supplemented to make decisions about placing entering students into their initial mathematics courses.

EMPHASIZE REALISTIC AND CURRENT EXPECTATIONS. Placement tests should not reflect obsolescent expectations in mathematics preparation in the secondary schools. Placement tests must be carefully reviewed as more is learned about what contributes to success in post secondary education and in light of changes in content and effectiveness of pre-collegiate mathematics programs.

AVOID SINGULAR FOCUS ON COMPUTATIONAL SKILLS. Good placement tests assess computational skills in unexpected contexts and a balance of procedural fluency, conceptual understanding, and strategic reasoning.

INCORPORATE APPROPRIATE TECHNOLOGY. Calculators and computers are an integral part of most pre-collegiate mathematics instruction. Even though prerequisite skills for a college mathematics course can be assessed without computers or calculators, students may be more comfortable working on a placement test in the familiar environment that includes use of technology. Therefore, calculators and computers should be considered for use in placement testing programs.

USE APPROPRIATE TESTING METHODS. Great care should be used in the design and administration of placement test programs. Informed consultants and helpful literature should be utilized in the design of placement test programs.

[3] See

Maplesoft T.A. MAA Placement Test Suite 6, www.maplesoft.com/products/placement/[4] Anthony Ralston. Computer Science, Mathematics, and the Undergraduate Curricula in Both.

The American Mathematical Monthly, Vol. 88, No. 7 (Aug. - Sep., 1981), pp. 472–485.[5] David Bressoud. Why do we teach calculus?

The American Mathematical Monthly, Vol. 99, No. 7 (Aug. - Sep., 1992), pp. 615–617.

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