On student outlines

Oliver Knill, put online May 12, 2013
For intro lectures in the first two years, I kept a small booklet with definitions and theorems. Scanned in 2010, it is now a PDF (269 page). The actual booklet had the size of a modern smart phone and helped me to learn and know the material. For some courses like real analysis, I had to know every theorem, every lemma by heart (with detailed proofs). For exam preparations, I prepared an other outline series PDF (337 pages). These were not lecture notes. The lecture notes themselves were more detailed. Unfortunately, I do not have anymore all freshmen and sophomore outlines.

I can not resist to add a remark about current math education. It concerns a dogma which only exists in math education: "You shall not memorize". Learning formulas or algorithms is considered evil, "understanding rather than memorizing" is the mantra. It was a reaction of the habits earlier in the 20th century, where math teachers would use primarily drill to teach things. Today, we have the other extreme: it is even considered uncool to teach the multiplication table. Why should we learn such a thing if a computer can do it so much better?

But it misses an important point. Understanding can only occur when knowing some stuff already. We need some flagpoles in our memory to learn new things. The multiplication table is a template for many other things. Knowing it will help to understand the concept of a group. When learning about Cayley tables, one can refer to an entrenched entity in the memory, the multiplication table. One is only good friends with something if one knows it well and has learned the multiplication table by heart early on. Understanding will need much more than that. Which digits can appear? How come we assume A*B=B*A? How would it look like in an other base? Are there other number systems? etc etc. Such questions can only be explored if one knows some concept already. I was lucky enough to have teachers who used additional tools to make the first steps in mathematics. One was the Cuisenaire material, wooden sticks, which I still own today. Numbers for me still now have a color attached to them.

Back to practicing basic operations. Nobody would doubt that a pianist, violinist, actor or athlete must have great memory skills and that routine repetition and drill are part of the game. The best rock climbers who win competitions memorize every move before going into the wall. Like a ballet. A pianist works best when playing a piece by heart and when basic moves are internalized (I myself was ordered to play at least half an hour of a 3 hour practice per day on basic études.) Also language is best learned by repetition, by acquiring a solid vocabulary, obtained by memorization and repetition. Mathematics is a language too, the language of all science. Still, memorization is discouraged here. Dramatic failures in math education in the last decades [ December 3, confirmed by Pisa] and apparent gaping holes in mastering even basic arithmetic and algebra still keep mainstream math education daemonize memorization? How come mathematics is the only discipline in which "knowing things" is bad? I don't know. One speculation is because in mathematics, in principle everything can be re-derived. Yes, one can avoid learning about the double angle formulas in trigonometry, because they can be derived quickly. The problem is that not knowing these formulas slows you down in many integration problems, and adds to frustration. Of course, it is not necessary to learn trig-identities for sin(3x) etc like it is not particularly efficient to learn 2 digit multiplication tables.

Even mentioning to students to use "flashcards" to memorize some integrals is for many considered the equivalent to suggesting the use of drugs. Its funny because in the math profession later, especially on a research level, memorized knowledge and technical skill is adored, much more than new ideas or creativity. The modern seminar room is an arena, where those with solid technical expertise and deep knowledge of the current literature do well. An other analogy is programming. I myself program often template based in that I collect examples which I can reuse. Still, it is almost impossible to do any creative work in a programming language, if every detail has to be looked up. Knowing the basic functions and options and pitfalls and shortcuts speeds things up and make programming more rewarding and productive.

Of course one has to have the balance. But as in sports or music, humanities or other sciences, a good amount of drill is needed to learn mathematics. It is an optimization problem like in computer science: to find the right balance of memory or computation. Elegant code derives and computes everything from scratch, efficient code relies on memory and hashed content too. And then there is the synergy: knowing things will help understanding and understanding will help to know. Still, the dirty secret of artificial intelligence is that memory and brute force attempts often beat creativity. Creativity thrives with a rich knowledge. It is a fact that "fast code" is often long and case based, cutting corners wherever possible, treating special cases separately. Some thoughts about that when we built our own "Wolfram alpha" in 2003/2004. Even in chess, computer programs do well, if they have huge opening- and end game libraries as well as can use massive brute force searches over the game tree. Truly powerful AI programs have built in intuition as well as vast and solid knowledge trees.

A side remark: a good vocabulary and a knowledge of the English language is considered a watermark "good education". Those who do well in "spelling bee" contests are considered "smart" even so it is simply a matter of memorizing a dictionary. The rules are currently changed so that contestants also have to know definitions of words. Still, it ramains just memorization.

My experience both as a student and teacher is that students with solid learning skills who can acquire and keep material in the memory do better in math exams. Yes, blind memorization might affect and dampen creativity temporarily, but even the most creative student becomes discouraged when doing not well in exams. Furthermore, the knowledge foundation will be a hotbed for creativity later on. In my first semester of studying math, I essentially had put myself on a "inquiry based learning" (IBL) mode, refuse to look up theorems and tried to derive everything myself. I also spent enormous time on programming. This is the ideal way to really get acquainted with the material, but it was slow and I did not do well in exams simply because I had knowledge gaps. To the point, I almost wanted to give up. After changing strategy I became one of the best (only grade-wise, but I must admit it also reduced my creativity. On the other hand, it allowed me to move forward and - with some luck - do more math later). The above outlines show some of my preparation notes as a student.

So, here is an attempt of an advise from a math teacher (don't tell this to your math teacher!):

When learning math, produce good outlines and learn some basic facts by heart. You will do well. Then use the positive energy from your success to go on and dream up something on your own. If you fail there (and you most likely will, as I and most others did), you at least have solid skills and self confidence that you can excel in something else.