I can not resist to add a remark about current math education. It concerns a

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

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

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. |