Why are the contents of contest maths so different from contents of degree-level maths?

Question

I wonder why topics examined in high school math contests are so different from the maths learned by those who are seriously studying a math major at a university. Firstly, contests like IMO, ARML, AMC and most of the others seem to focus on a very small area of math (graph theory, combinatorics, elementary number theory, planar geometry, etc). Topics in analysis (which include several courses like complex analysis and functional analysis), topics in mathematical physics (quantum mechanics, electromagnetism), and more advanced algebra (rings, modules) are almost completely ignored. I cannot see any obvious reasons why those important topics should be omitted -- it is true that some mathematicians specialise in small fields like combinatorics, but they are definitely not the majority. Most mathematicians are specialists in more "modern" math, like the ones I listed above (analysis, physics, algebraic and analytic number theory, etc).

Also, contests tend to dig unnecessarily deep into something "obsolete". Take the 6th question of this year's IMO as an example:

You can see six or seven extra lines that must be added before one can answer the question. Such questions involve a lot of small tricks that one hardly ever needs in math research. In fact, nobody needs to know how to do this question unless they are a contest math teacher. Yes, one can really dig very deep into planar geometry, and there is beauty in it. But contest-style geometry is very "ancient", and is no longer the focus of research nowadays. Today, computers can already do such geometric proofs in a much more rigorous way than people in the contest. It doesn't seem to be a good idea to play with ancient things too much -- this IMO question is just like making a horse run faster than an airplane.

Another "ancient" technique commonly seen in contests is to construct inequalities without using calculus. It used to be part of the math course at university 100 years ago, but now it is no longer a topic that must be studied -- students simply learn those inequalities when necessary. However, such inequalities are still a large part of the contest.

So, that leads to the question: what it the reason why there are almost no high school math contests that are even close to the style of math at universities?

This might be too opinion based, but note that I am asking "why", not "what should we do", so I believe one can write a very objective answer to this question.

Also, don't tell me that is because degree level maths is too hard -- clearly that IMO question above is much harder than math at any universities. In fact, many PhD students of leading math departments cannot do such questions.

PS: I know that students learn a little bit of degree level math while training for IMO or other competitions, but their knowledge about those more advanced math is likely to be fragmented and incomplete -- they are likely to end up with false impressions about math at universities.

Answer 1

Mathematics constests are a kind of game or puzzle, like chess, poker, sudoku, etc. Not all mathematics adapts well to the context of a competition in a limited amount of time. While it's true that research mathematicians can be very competitive (in Yau's recent autobiography, he several times describes mathematics as a competitive activity - note though that Yau has advanced a model of mathematics competition very different from contest math), the time scale of their competition, when it occurs, is months or years (even if a paper is written in weeks, the preparation required to write occurs on the time scales of months or years), and it is in any case an implicit competition, not occuring in a regimented and programmed context, such as contest mathematics.

The sort of mathematics amenable to contests necessarily cannot require deep, novel, creative thinking, as such just is not possible on the relevant time scales. Contests require mathematics that has been well assimilated into fairly general teaching (to give the illusion of accessibility) and that can be broken into small parts stated in terms that require relatively little formalism. This favors combinatorics, plane and space geometry of the most classical sort, graph theory, finite group theory, elementary number theory, and disfavors that which requires much analysis, any kind of modern algebra or representation theory, or physically related mathematics, just to name a few areas. On the other hand, often problem writers try expressly to avoid the mathematics that is taught in standard courses, so that the contest is not simply testing who is best trained, or something like that. See this essay by Bruce Reznick about writing problems for the Putnam exam.

The most fundamental difference between mathematics contests and doing mathematics is that in a contest one knows there is an answer that can be obtained in a few hours at most. In research sometimes a big part of the problem is deciding what an answer would look like.

That there are people who are successful both at contest math and at doing deep research is in no way an argument against what I just wrote, but it is not the case that ability at either necessarily signals ability at the other (although it is surely an indicator) - for every deep thinking Tao or Perelman who was successful at contest math there is an equally deep thinking mathematician who was not so good at contest math or simply did not like it (I'm tempted to suggest Thurston and Grothendieck as examples, although I have no idea what either felt about or how either fared in mathematics contests).

It seems to me a mistake to think of contest mathematics as indicative of what mathematics as a discipline is. There is some part of mathematics contests that is remanent research mathematics, in the same way (although less exaggeratedly so) that solving a sudoku is reminiscent of mathematics (there are even some nontrivial mathematical problems about solvability of generic sudokus, enumeration of sudokus, etc.), but primarily mathematics contests are oriented at giving prizes for solving a certain kind of problem rapidly.

Answer 2

I don’t think it is remotely reasonable to include the areas you mentioned, since (in the U.S., at least) all those that you mentioned are at least 3rd and 4th year undergraduate subjects. Indeed, functional analysis is usually a 2nd year graduate course (often not even required for a Ph.D. in math), and mathematical physics topics such as quantum mechanics and electromagnetism are studied by only a few university mathematics majors (almost certainly fewer than 10%, and probably fewer than 5%, will have taken upper level courses in both of these during their undergraduate years), to say nothing of high school students. Anyway, for these subjects there is the Putnam exam in the U.S. and Canada, the Schweitzer Competition in Hungary, and probably others in other countries. Even these exams avoid mathematical physics (that I am aware of), and the Putnam exam in particular focuses mostly on lower undergraduate level mathematics (with an emphasis on combinatorics and other similar topics). By the way, in the U.S. and Canada, high school students taking one or more college classes are eligible to take the Putnam exam.

Given this, one might wonder if a reasonable argument could be made for including elementary single-variable calculus topics on high school math contests. I think part of the reason for not doing this is historical inertia, part of the reason for not doing this is lack of reasonable access to calculus by many students (especially at the many smaller high schools dotted throughout the U.S., even now few offer the equivalent of a second semester college course in which integration techniques, sequence and series convergence, and the like are covered), and part of the reason for not doing this is in determining what would be fair game for calculus-background problems. Regarding this last reason, would subtle issues such as the fact that a derivative can be discontinuous or derivatives satisfy the intermediate value property (even when discontinuous) be suitable topics? Would the use of topics such as the Leibniz rule for the $n$’th derivative of a product and the more specialized series convergence tests one sees in an advanced calculus course be suitable topics? At least for the examples I’ve mentioned (last two sentences), it seems to me that we primarily would be testing for content knowledge, as someone who isn’t aware of these results would essentially have to discover them while taking the test (an extremely high bar that seems only useful for detecting the next Gauss), and anyone aware of the relevant results will have a MUCH easier task.

I suppose one could counter with the argument that the same is true for the topics that actually appear in contest problems --- the subtle issues and the specialized topics/methods still apply to combinatorics, number theory, Euclidean geometry, etc. However, students taking the tests will have been familiar with these other subjects for a much longer time than with calculus, and thus it is reasonable, I think, to expect students to have explored the more subtle issues and specialized topics/methods in these subjects than is the case with calculus.

Finally, the underlying assumption in all of the above is that these tests are primarily for measuring mathematical ability and potential, not for mathematical achievement/knowledge. We want the questions to be chosen mainly for their cognitive difficulty than for their background-knowledge difficulty. The fact that “many PhD students of leading math departments cannot do such questions” is simply a reflection of the fact that IMO problems are pitched at cognitive difficulty levels beyond that which is needed to obtain a Ph.D. in math. Just to take the U.S., for example, each year about 900 U.S. citizens receive a Ph.D. (in the U.S.) in a mathematical science field (about 940 in 2015-2015 --- see top of p. 354 [= .pdf file page 116] here), and each year 6 U.S. citizens participate in the IMO. Thus, even if every IMO participant eventually gets a Ph.D. in a mathematical science field, then fewer than 1% of those earning a Ph.D. in a mathematical science field will have participated in the IMO. The fact that some IMO participants take the IMO more than one time only reduces the percentage.

Given all this, I certainly sympathize with your view, because when I was in high school I wasn’t interested in learning “contest math topics”. One reason is that there was hardly anything available to me (this was around 1973-1977, in a rural location). Another reason is that I was more interested in learning calculus and other college level math than I was in solidifying my knowledge of advanced/specialized high school type topics. In the years since then, especially since the mid 1990s, I’ve managed to somewhat rectify this (solidifying my knowledge of …), but it was primarily a result of teaching such topics to very strong high school students, and a result of the rise of some internet math discussion groups in the mid 1990s where such topics tend to come up quite a bit more often than, say, the Lebesgue density theorem or the Cantor-Bendixson theorem.

Answer 3

I teach a lot of contest math and I have very mixed feelings. Contests reward repertoire, perspicacity and speed. These are all certainly useful at undergrad, but the emphasis given to them feels wrong. On the other hand, there is a decent overlap between students who can assail these problems and students who will develop the other qualities and attitudes to be good mathematicians.

We cannot expand the syllabus because there is not time enough to teach students at this age about all of those things in any non-superficial way whilst providing a well-rounded education and preparing them for university entrance exams. It's also not worth it because they'd just do it again at undergrad anyway, and I get the sense universities don't want their undergrads turning up with inchoate conceptions of vector spaces and lebesgue integration - they would rather have a voracious and powerful tabula rasa. Also, for some exams, the syllabus is traditional and pretty well fixed.

Given that, I think the answer to your question is quite optimistic: the questions become that complicated and arcane because students are that good. The questions cannot go outside of the syllabus, but they need to really challenge the top students, so examples like the one you gave are the result. Students really are amazing in what they can achieve.

In my opinion, hard problems which get the balance between expounding interesting theory and results, conveying the spirit of university mathematics and providing a fair amount of mechanical difficulty can be found in Cambridge's STEP Papers. I prefer to use these problems over competition problems to challenge my students because they give me the same feeling I had when I was an undergraduate solving problems and putting things together, instead of picking through tedious geometry puzzles.

Answer 4

To be succinct: contest problems are meant to be accessible to a wide audience, so cannot overtly involve somewhat-more-sophisticated mathematical ideas ... that many people would not yet have heard about.

Naturally, since the general degree of sophistication has a bound, the questions will become at least a little byzantine, since standard questions at this introductory level would be answered by everyone, and there'd be no game. :)

Unsurprisingly, although it's good to be able to succeed in such games, it is not at all the same as making a genuine contribution to further understanding of mathematics. There should be no surprise here!

Still, having the enthusiasm, and chops, to do well in such contests is a positive. In my own arc, doing something positive on the Putnam Exam (in the U.S.) helped with my credibility...

But/and I've seen several people who enjoyed contests, and excelled at them, but did not so much enjoy the professional math business. For various reasons, I infer.

So: contests are meant to be accessible.

Accessible-math is not necessarily what any of many sorts of professional mathematicians do, day-to-day. This should not be surprising. :)

Answer 5

I figured that I can expand my comment into an answer, even though it is still more a comment than an answer.

You say that math contest questions "involve a lot of small tricks that one hardly ever needs in math research", and because of this you find contest questions useless for future career as a mathematician. But I do not think that math contests are meant to be a preparatory step for university math, I think they are meant to teach how to attack problems with what you are calling "tricks".

The contest problems are rarely about breakthroughs in mathematics, but rather about smart combination of known techniques. They foster the way of thinking more attributable to an engineer than to a "pure" mathematician: take existing building blocks — "tricks" — and use them to achieve your goal, not much different from chess, where good players know a lot of existing combinations and apply them accordingly. Invention strategies like TRIZ and AIDA rely on this approach.

Contest problems do not require lots of base knowledge, hence are more approachable, and usually can be solved without "tricks", but usage of a particular "trick" can produce a fast and simple solution, while not using it would result in a wall of calculations.

Here is a simple problem that can be solved by a middle-schooler:

Every day an engineer arrives to a station at 8 a.m. by train. Exactly at the same time a car, sent from a factory, drives up to the station, picks up the engineer and takes him to the factory. One day the engineer arrived at 7 a.m., decided not to wait for the car, and started walking towards the car. When the car met the engineer, it picked him up, turned back and arrived to the factory 20 minutes earlier than usual. For how long did the engineer walk? Consider the speeds of the engineer and the car constant.

It can be solved using a rather elaborate approach that produces a system of two linear equations. Alternatively, using a "trick" it can be solved with mental elementary-school math in less than a minute.

In a way, such problems are on the opposite end of the spectrum compared to "pure" math. Whether this sort of mental gymnastics is valuable is a matter of opinion. But I think it is more democratic, and more applicable in real life than "pure" university-level math.