Meeting the Challenge of High School Calculus, III: Foundations

*David M.
Bressoud, *May, 2010

Growing ConcernsBut there also was a dark cloud emerging. By the mid 1980s, the mathematical community was becoming concerned about what it saw as the weakening of students’ precalculus preparation in the rush to get them into calculus while still in high school. This alarm was sounded in the 1986 joint statement of the Mathematical Association of America and the National Council of Teachers of Mathematics, which can be found in [7]. It is worth quoting in its entirety because I believe that it is as relevant today as it was a quarter century ago, only more so.

“A single variable calculus course is now well established in the 12th grade at many secondary schools, and the number of students enrolling is increasing substantially each year. In this letter, we would like to discuss two problems that have emerged.

“The first problem concerns the relationship between the calculus course offered in high school and the succeeding calculus courses in college. The Mathematical Association of America (MAA) and the National Council of Teachers of Mathematics (NCTM) recommend that the calculus course offered in the 12th grade should be treated as a college-level course. The expectation should be that a substantial majority of the students taking the course will master the material and will not then repeat the subject upon entrance to college. Too many students now view their 12th grade calculus course as an introduction to calculus with the expectation of repeating the material in college. This causes an undesirable attitude on the part of the student both in secondary school and in college. In secondary school all too often a student may feel ‘I don't have to study this subject too seriously, because I have already seen most of the ideas.’

“Such students typically have considerable difficulty later on as they proceed further into the subject matter.

“MAA and NCTM recommend that all students taking calculus in secondary school who are performing satisfactorily in the course should expect to place out of the comparable college calculus course. Therefore, to verify appropriate placement upon entrance to college, students should either take one of the Advanced Placement (AP) Calculus Examinations of the College Board, or take a locally-administered college placement examination in calculus. Satisfactory performance on an AP examination carries with it college credit at most universities.

“The second problem concerns preparation for the calculus course. MAA and NCTM recommend that students who enroll in a calculus course in secondary school should have demonstrated mastery of algebra, geometry, trigonometry, and coordinate geometry. This means that students should have at least four full years of mathematical preparation beginning with the first course in algebra. The advanced topics in algebra, trigonometry, analytic geometry, complex numbers, and elementary functions studied in depth during the fourth year of preparation are critically important for students' latter courses in mathematics.

“It is important to note that at present many well prepared students take calculus in the 12th grade, place out of the comparable course in college, and do well in succeeding college courses. Currently, the two most common methods for preparing students for a college-level calculus course in the 12th grade are to begin the first algebra course in the 8th grade or to require students to take second year algebra and geometry concurrently. Students beginning with algebra in the 9th grade, who take only one mathematics course each year in secondary school, should not expect to take calculus in the 12th grade. Instead, they should use the 12th grade to prepare themselves fully for calculus as freshmen in college.

“We offer these recommendations in an attempt to strengthen the calculus program in secondary schools. They are not meant to discourage the teaching of college-level calculus in the 12th grade to strongly prepared students.

“LYNN ARTHUR STEEN, President, Mathematical Association of America

“JOHN A. DOSSEY, President, National Council of Teachers of Mathematics”

[7] reprinted in Appendix E in Learning and Understanding: Improving Advanced Study of Mathematics and Science in U.S. High Schools: Report of the Content Panel for Mathematics, 2002, National Academy Press, Washington, DC

I wish to thank Tom Tucker and John Kenelly for their information about and reflections on the AP Calculus Program during the period 1970–90.

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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.

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