Some of our colleagues insist that all college students should be required to take a calculus class. It is a wonderful thought and not as outrageous as it may sound. My last year of high school in Poland in the 1970’s included a rigorous introduction to differential calculus (with some proofs). It was possible, because in the three previous years we had very thorough preparation in algebra, geometry and trigonometry. Elements of calculus were required on the final comprehensive exit exam for all students, regardless of their career plans. It was difficult and technical, but we understood from the way mathematics was taught that the purpose was not to learn skills that one would use later, but rather we were studying ideas, and those ideas were somewhat important. Given the level of preparation of the vast majority of students entering CUNY, teaching everyone calculus is not something we will be able to plan any time soon. If mathematics remains a component of general education (as it should), what should we teach students whose majors do not require technical applications of mathematics? In the description of the 7th edition of Peter Tannenbaum’s popular textbook Excursions in Modern Mathematics, one reads:
“Excursions in Modern Mathematics, Seventh Edition, shows readers that math is a lively, interesting, useful, and surprisingly rich subject. With a new chapter on financial math and an improved supplements package, this book helps students appreciate that math is more than just a set of classroom theories: math can enrich the life of any one who appreciates and knows how to use it.”
I have used previous editions of the book and I like it, but the publisher’s description represents all that I find objectionable in recent trends. For example, why does one need to say that the book will show that mathematics is a “useful and surprisingly rich subject.” Aren’t students supposed to know already that mathematics is useful? Why is it surprising that it is a rich subject? It is very sad, and says much about the state of pre-college education, that one has to wait for college to learn that mathematics is rich and useful. And what does it mean that “math is more than just a set of classroom theories”? It is funny, that the blurb promoting a mathematics text is so full of prejudices. One gets the impression that the publisher tells the student: Look, we know you’ve suffered all those years in school learning useless and boring “classroom theories,” but now we will show the true beauty of the subject. In the table of contents one finds a list of interesting subjects: voting schemes, fair-division games, applications of graph theory, elements of statistics, financial mathematics, Fibonacci numbers and the golden ratio, geometry of symmetries, and the mandatory fractals. All these topics are attractive and it would be hard to argue that there is anything wrong with trying to make sure that a college educated person is aware of them. At the same time, one feels that in the attempt to present the discipline through its practical applications, much of its true spirit is lost. Putting too much emphasis on the applied side of mathematics can be dangerous. First of all, how many students really get inspired by the rather mundane applications? If enriching the life of someone who “knows mathematics” means teaching them to make decisions concerning loans or mortgages based on calculations, how is it different from just teaching them to use software that does all those calculations?
So what do we do? It is a really difficult problem and a major source of this difficulty is the enormous gap between what we expect entering students to know and what they actually know.
In the next blog I will describe a mathematics course for Liberal Arts students that I often teach and like.