What's Your Choice for Twelfth-Grade Mathematics?

Bob Rosenbaum, Chair of PIMMS  7/26/2006

As one method of improving the learning of mathematics, a considerable number of educators recommend a requirement of four years of the subject in high school. Such a suggestion leads to the question, "What should senior math look like?"  Here is my answer to this question:

1. While Advanced Placement calculus is a good 12^th grade course under certain circumstances, it is not the best course for the majority of twelfth graders in contemporary U.S. high schools.  I start with this negative statement because calculus is all too often the course recommended in high schools for students who have had three years of math, including a course named, in whole or in part, "pre-calculus."  The recommendation usually stems from a mistaken sense of "prestige," both for the school and for the students' parents!

A calculus course on a transcript is often thought to enhance the student's chance for admission to college.  For over 30 years the Wesleyan Admissions Office has included a statement to the effect that the University's mathematics department prefers a solid grounding in a traditional curriculum of algebra, geometry, and periodic functions, without calculus, to a less-thorough introduction to the traditional curriculum in order to make room for a (perhaps truncated) introduction to calculus.  I am pleased to see this Wesleyan choice (if a choice has to be made) endorsed by other selective colleges and universities.

Please understand that I am not objecting to calculus as the (principal) content of a high-school course, taught by competent instructors to adequately prepared students.  (Indeed, a few strong secondary school math departments regularly offer a course in multivariable calculus, and one of my current tutees, after earning a 5 on the BC calculus exam last year, solved an integration problem on this year's Putnam Exam.) But if the teacher has only a dim view of the notion of limit, and the student is weak in techniques of algebra (so that, for example, the student can't break up a rational expression into partial fractions), the course won't contribute much to the student's intellectual development, and the following dialogue may epitomize the residue:

Mathematician: "You've had a course in calculus. What did you learn?"

Student: "Well, I had the course last year, so you can't expect me to know much about it now.  But I do remember something like this:  If         or perhaps          and there may be a 'plus C' somewhere."

2. There are several possibilities for a Math course for the average high school senior.  In my opinion the best such course might be titled "Quantitative Thinking, Data Analysis, Statistics, and Probability," with the applicability of the material of such a course to problems in science, technology, engineering, mathematics, and the social sciences being the strongest argument for offering the course to a broad spectrum of students, or even for requiring the course for graduation from high school.

In fact, some elementary material of this course might be introduced at the ninth grade, furnishing examples of the use of algebraic notions, and other material of this course might be introduced in grades 10 and 11, continuing to reinforce algebraic and geometric concepts, so that, by the time the student is in twelfth grade, the course may be fairly advanced.  The CEEB does offer an A.P. course in statistics as an alternative to an A.P course in calculus.)

3.Another possibility for the twelfth grade is a course in Discrete Mathematics (as in the Kemeny, Snell, and Thompson book), involving a selection from the following topics, among others:

a. Linear Transformations and Matrix Algebra

b. Statistics

c. Combinatorics

d. Number Theory

e. Analytic Geometry

f. Vector Geometry

g. Solid Geometry

h. Advanced Algebra

i. Symbolic Logic

There is no priority ordering intended in the foregoing list.  Strengthening students' powers of visualization is an important purpose of a course in Solid Geometry; and, if such strengthening is not accomplished in the tenth grade Geometry course, it should be addressed elsewhere.

I will welcome your comments on these thoughts. You may contact me at rrosenbaum@wesleyan.edu.