American parents are sending their kids to 'Russian math' (2017)

I liked math, a lot. The SAT Math test said I had a lot of talent in math. I concentrated on math in US grades 9-12, college, and graduate school, did some math research, that later I published, got a Ph.D. in pure/applied math, and am now using some advanced and some original math as advantages in my startup.

So, I struggled through the good and bad but eventually decided that there were a lot of good math books; it was not very difficult to identify the relatively good authors and books; and the keys to learning math well were a stack of blank paper on a clipboard, a sharp pencil, a big, soft eraser, one or a few good math books in the subject being studied, a lot of good exercises, a comfortable chair, a good light, and a quiet room. That's how I learned nearly all the math I did learn; still if I want to learn some math, that is what I use.

This technique of a quiet room worked for me many times, but once was a nice surprise: The college I went to for my freshman year was selected because I could walk to it and it was cheap. The most advanced math course they would let me in was beneath what I'd already done in high school -- the high school was relatively good (MIT came recruiting; 97% of the students went on to college; one year three students went to Princeton). In my class, in 1-2-3 on the SAT Math, I was #2 and #3 went to MIT. So, I didn't want to fall behind in math so got their calculus book and started studying in a quiet room. This effort worried Mom who would find excuses for me to get up and do something else, but I still did well. For my second year of college, I went to a college with an unusually good math department and started on their second year of calculus using the same text Harvard was using. To let me start on that second year, a prof gave me a little impromptu freshman calculus oral exam. So, with the quiet room technique, I never took freshman calculus -- later taught it, applied it, etc. but never took a course in it!

In math written as theorems and proofs, for a big source of good exercises, guess the next theorem. Check your guess. Given the theorem, close the book and prove the theorem. Doing this let me get the solution to a somewhat challenging Ph.D. qualifying exam question -- I did the best in the class on the qualifying exam. This approach to exercises is good, but it is too difficult, that is, too slow, to use for all the math need to learn.

Beyond that quiet room approach to learning, I found that to do well in graduate school, e.g., get respect from the professors, the key was, as soon as possible, do some publishable research. E.g., maybe have been pushed into an advanced course. Okay: Find some places the course and/or texts are not very clear, good, precise, complete, whatever, pick one of those, do some research to improve the situation, and publish the research. Remember: For good results, good initial problem selection can help a lot.

For calculus, yes, work through a good text and then, for a nice advantage, learn measure theory then, in particular, learn probability based on measure theory. Then in, e.g., statistics, you will have a gun while nearly everyone else has at most a knife.

But just calculus from a respected text can be powerful stuff. E.g., at

https://www.youtube.com/watch?v=KZ8G4VKoSpQ

can see Einstein's special relativity done, apparently fully correctly, and where the only math used is ordinary calculus. Some 12 year old students can learn calculus plenty well enough for a lot in applications, including more advanced math.

Here is a special strategy that can work in the US: In US research universities, the math departments typically are in the school of Arts and Sciences. But such universities commonly also have engineering schools! Some of the people who give the big money like engineering more than arts and sciences! And there is the outside world! So, from contact with the outside world, maybe a job, full or part time, pick a problem where a good solution looks promising for some old/new math. Solve the problem, and publish it in a journal with a title like Journal of Theory and Applications in .... Some journals also like to promise candidate readers that they publish not just theory but actual applications!

So, the usual criteria for publication are that the material be new, correct, and significant. Well, easily enough the solution to the new real problem can be new. Since the solution is mostly math, can pass correct. Get significant from the real problem being significant. If the math saves $10 million a month in jet fuel cost for an airline, call that significant!

For the remark, essentially, need to think before writing, I go along with that for research and challenging exercises.

Note: For challenging exercises, I found that some good research is no more difficult than some such exercises -- the exercises are good preparation for research, etc.

Generally in applying some math, will likely find some places where the old math needs some improvement, at least for the application; so, make some such improvements, and get the significance from that of the problem. That is, for picking a research problem, the Riemann hypothesis is not the only option!

For an example, I picked a problem with the Kuhn-Tucker conditions and found a solution and published it. Later I found that the famous paper in mathematical economics by Arrow, Hurwicz, and Uzawa encountered a similar problem and had no solution. My work also solves their problem. I found this research problem just from some careful, quite careful, study of the Kuhn-Tucker conditions.

So that is a way around the bad in programs, teachers, books, exercises, etc.

So I might be comparing the best & brightest of Bulgaria, but in theory at least I'm comparing them against the best & brightest of America (though in practice I suspect it's more like the richest).

It did feel good to see a practical use for the calculus I had learned, much more so than determining how much the surface area changed after adding a layer of paint 0.01" thick to a tank.