Letters to the editor

School Science and Mathematics, Feb 1997

Calculus Reform: An Opinion

I am an open-minded mathematics traditionalist who has a Doctor of Philosophy degree in College Teaching with a specialty in Mathematics from the University of North Texas. I have taught mathematics at the college and university level for 30 years. In that time span I have observed changes or reforms in mathematics ranging from the "new math" at the elementary level to the "lean and lively" calculus. I have seen the pendulum of reform swing across various innovations in both arithmetic and mathematics. Eventually, the pendulum revisits the "back to basics" in arithmetic and the traditional, rigorous techniques and methods of mathematics.

At the risk of being classed as a conservative professor, I would like to comment on the current new reform movement: The Harvard Calculus and similarly titled calculus reform movements in higher education. I can say at the outset that the "reform"movement is a new attempt to find a "solution" to the problem of helping a student of calculus learn calculus. This is a never ending problem with no perfect solution. Learning calculus, or any other intellectual subject, will take what it has always taken - time and effort. If enough time and effort are spent on calculus topics whether those topics are presented by "The Harvard Calculus" approach or the traditional "rigorous" scientific-engineering approach, the student will master those topics. Of course, the always-present assumption is that the calculus student has the appropriate background in precalculus mathematics topics and has learned those topics through rigorous use and practice (often erroneously called memorization).

The Harvard Calculus embraces two basic principles: The Rule of Three, which indicates that every topic should be presented geometrically, numerically, and algebraically; and The Way of Archimedes, which is the belief that formal definitions and procedures evolve from the investigation of practical problems. These are admirable principles which all good calculus professors should believe in and practice in their teaching. The administration of those principles to obtain a successful calculus student is the gist of the calculus reform movement. It is my experience in the teaching of mathematics that influences my personal opinion that the current Harvard Calculus reform textbook loses rigor with the geometrical and numerical treatment of the calculus topics because the algebraic treatment is slighted. The introduction of basic calculus topics using numerical and graphical presentations is not in question-that approach is very meaningful, but the development of those topics through rigorous problem solving situations is not as strong in the Harvard Calculus textbook as in most traditional calculus textbooks (Larson, Anton, Salas, Steward, Leithold, Thomas, etc.).

After recent examination in a beginning Harvard Calculus class, I was inspired to write some opinion notes as I evaluated solutions to the calculus exam questions:

1. Unfortunately, many students who could excel in calculus are displaying work consistent with below-average students.

2. Students are being encouraged not to write mathematics and obtain a general solution but to put numbers in answer spaces where a logical symbolic communication is meant to occur. The general solution could be used to obtain the specific results. The student of Harvard Calculus usually begins and ends with specific numerical or graphical information.

3. The student must have a graphing calculator to think. It has become a crutch instead of a supporting companion.

4. The reform approach removes much of the algebraic rigor in favor of arithmetic and graphs. We should be teaching calculus so that pure algebraic results are only initiated and supplemented by numerical and graphical methods.

5. Currently we begin and usually end a problem solution with a graphing calculator by giving the graphical results and written statements that explain those numbers and pictures. The latter is usually lacking a great deal, especially when a student cannot translate the results into mathematical symbols or at least a rigorous, symbolic description of the problem solution. The algebraic problem solution is no longer the "thing" to get in a problem; the graphing calculator gives approximate numerical results that are "good enough" for our real world solutions, and this thinking tends to remove the beauty and importance of obtaining precise results.

We, the Mathematics Academe, need to continue our quest to inform students about mathematics, but not at the expense of the last part of the Rule of Three: the algebraic treatment in the solution of calculus problems. The algebraic process in the solution of calculus problems needs to remain as the principal process with the numerical and graphical approaches used to gain some insight into the complete solution. This should be true of any mathematics problem to be solved. In particular, problems that "only" have approximate graphing calculator solutions need some rigor in the solution process, or we would never know they do not have exact solutions.