A course I like

I like teaching “math for poets” classes and I have taught many. There are so many fascinating things students don’t know about. Many years ago, at the Fashion Institute of Technology in Manhattan, the textbook I chose for the course was Rudy Rucker’s Mind Tools: The Five Levels of Mathematical Reality. It is a wonderful book. In contrast with the publisher’s description I quoted in my previous blog, this one reads:

From mathematics and computers to insights into the workings of the human mind, Mind Tools is a reflection of the latest intelligence from the frontiers of mathematical thought. Illuminated by more than 100 drawings, Mind Tools connects mathematics to the world around us. It reveals that the great power of mathematics comes from the fact that it serves as an alternative language for understanding things — from one’s hand to the size of infinity. Exploring such concepts as digital versus analog processes, logic as a computing tool, and communication as information transmission, Rudy Rucker presents the “mind tools” for a postmodern age.

I enjoyed teaching the course, and I do not think I taught it badly, but as the semester progressed it became more and more clear that I was the only person in class enjoying it. The text turned out to be too philosophical and too difficult for students to digest. Despite my best efforts, they were confused about homework assignments and course requirements (they did turn in some very nice Escher-like drawings though). The course was a failure, and at the end I had no one to blame for it but myself. I underestimated by far the difficulty level of the content. Certain topics are just too hard, not from the technical point of view, but conceptually. It is tempting to teach about Hilbert’s Hotel, Banach-Tarski miraculous doubling of an orange, or Koch’s snowflakes, but in reality a much more successful course is the one that shows how the mathematics that students already know can be put to interesting uses. One such course is offered regularly at Bronx Community College under the title Survey of Modern Mathematics I. The syllabus is flexible and it allows the instructor to make choices of the material. My strategy for teaching it rests on the principle not to introduce mathematical tools beyond what is required by the prerequisites for the course, and this is not much—just elementary algebra. At the same time I try to use those topics that actually use elementary algebra, to show students how what they’ve learned is applied. I will give a brief outline of the course.

The course starts with a short discussion of ancient numeration systems, and then moves quickly to the positional Hindu-Arabic system. I avoid mechanical conversions from one base to another, instead I concentrate on just one system—quintary—which I call the one hand arithmetic. I explain how to add and multiply in base five, and sometimes we go over subtraction and division as well. The goal is to explain how and why the familiar decimal algorithms work. After initial resistance, students get used to quintary operations. It really helps that one can use hands: 4 means four fingers; 12 is one hand and two fingers; 23 is two hands and three fingers, and so on. At the end I introduce the binary system and we talk about its applications in computer technology.

It is hard to teach even a very scaled down probability course in just three or four weeks. To do it properly, one has to introduce some set theoretic notation, cover basic counting techniques, introduce notions of sample spaces, relative frequencies and much more. It takes time, and it is hard to expect that students will absorb much. Still something can be done. I begin with  the definition of theoretical probability for events with finitely many equally likely outcomes. I show how the relevant numbers can be computed on simple examples, and then we go right away to actually performing experiments with random outcomes and to comparing theoretical predictions with relative frequencies recorded in class or in homework exercises. We toss coins, roll dice, and play games. It is really amazing to see how nature obeys theoretical predictions. I particularly like one game I took from Ian Stewart’s Concepts of Modern Mathematics. Four dice A, B, C, D are labeled with numbers in such a way that when a pair of dice is rolled one shows a higher number with probability 2/3, moreover the number are chosen so that B beats A, C beats B, D beats C, but A beats D. It is a two person game. The first player chooses a die, the second another die, and then they roll. It is a very unfair game, if the second player knows the probabilities, he or she has enormous advantage, and that player in class is me. Sometimes we spend the whole class period playing the game and recording statistics. Students get very intrigued, and much can be explained. Among other things, we discuss odds of winning the grand prize in the Monty Hall game (the one with goats behind closed doors), and, we use the Monte Carlo method to estimate π.

In the third part of the course I cover linear programming. It may not be everyone’s favorite, but I like it. Even though one can talk about small systems of constraints for two unknowns, still some real applications (such as some diet problems) can be discussed. Even more importantly, one can see how some meaningful practical problems can be easily solved with very basic algebra, and almost impossible without. Another aspect is optimization: not only we can find solutions to practical problems, but we can also show that they are the best solutions available.

At the end of the course we turn to financial mathematics: the compound interest formula, annuities, mortgages, and loans. This is a very traditional part of the course. Formulas are introduced, examples given, homework exercises assigned, but once it is done one can turn to very practical issues. How to read credit card statements? How to plan savings? How to respond to a refinancing offer from your mortgage company? No college student should graduate without some exposure to these topics.

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