Joshua Rutsky wrote:With the new move towards tossing up theoretical math in the last couple of years, and particularly coming off the NASAT, I was interested in hearing how people are studying or prepping math theory. Are there any good texts or clusters of information hat would help a team get started with this? I have several math team kids, but they are lost on much of this material.
Well I think the answer to this question depends on the difficulty level you're looking to study for. For easier high school tournaments, baseline knowledge of topics that come up in high school tends to be just about enough as far as topics are concerned. As you stretch toward the upper end of the high school canon (NSC and HSNCT levels) some of the questions tend to use introductory college materials in their cluing. It's important to note that a lot of the clues inside of power are geared toward the curious high school student who has either come across things in math competitions or has a natural curiosity about mathematics. It's also important to note that the editors and writers typically understand that most high school students have only rudimentary technical knowledge of mathematics and so questions are typically written so that they can be converted by most high school students with said knowledge. However, earlier clues will often cover simple results in abstract algebra, linear algebra, analysis, and whatnot. With that being said:
Definitions, definitions, definitions.
It's hard to get any context whatsoever in mathematics without first knowing those definitions. If you, as a player, do not understand the basics of what something is, you certainly won't be able to grasp the deeper meaning of more powerful theorems and lemmata. It sounds like a simple thing, but I can tell you as a former undergraduate that just understanding what things are before trying to grasp the importance of results makes a world of difference. Know what things are before studying them in depth. This idea is lost on so many people trying to study quizbowl math.
I find that reviewing a few examples from each studied result can work wonders. Even if a student knows the terminology, results can sometimes be difficult to grasp without looking into how the results can be applied to a problem. Although most of these examples will not manifest themselves within the context of quizbowl questions, they are quite often illustrated in ways that illuminate the purpose of the theorems. Also, mathematics is learned by doing, so actually working a few similar exercises can make the material stick.
Becoming good at mathematics questions takes time and the rewards are very slim. Becoming great at mathematics takes lots of time and, to be honest, is only truly rewarded by the most knowledgeable editors and writers. Why is that? Well it's hard to reward someone's depth and breadth of knowledge in the subject unless you understand how such questions should be structured. It can also be tough to even write a math question in an accessible yet not stale way unless you have experience with exactly how difficult certain topics are in actual practice. (As an aside for another day, it is unreasonable to expect a non-math person to be able to gauge what both math and non-math people know about math with utmost precision. Unfortunately, math writers capable of such precision are rare at this time because it takes years to learn that stuff. You surely can't blame most math editors for not having this knowledge. It's tough to get!) I can say with certainty that almost all math writers and editors have pretty decent knowledge of math that was important to their plans of study in their undergraduate programs. So definitely study those areas. (ODE, PDE, calculus, stats, linear algebra)
When you get to regular college difficulty and beyond, the issue of broadness of topics also manifests itself. In high school, strong knowledge of high school level algebra, geometry, trigonometry, prob and stats, and calculus is good enough to make a player somewhat effective even without any extra knowledge. Once you get into college regular difficulty, other topics show up. Popular ones include linear algebra, introductory stats, differential equations, and anything engineering and science players have to take as prerequisites. I strongly suggest looking at material from those required courses. Of course, math is a rather broad subject, so things that math majors study such as topology, algebra, analysis, number theory, discrete math, and numerical methods will also show up. I never really studied quizbowl math for the sake of studying it, however I think this is where math can become really challenging for non-math people. However, the idea for studying this stuff is actually the same as before. Know your definitions then go and read some theorems and try to understand those with some (hopefully) simple examples. In that way, clues will stick better. Of course, herein lies the difficulty of math. It's not easy to develop a technical understanding of these things without some sort of fundamental math background past, say, AP calculus.
So where do you start? Any player looking to develop into a strong math player should look into reading a fundamentals of mathematics textbook. Not only do such textbooks lay the groundwork for learning how to prove basic theorems, but they also touch on a lot of different topics that one will come across during his or her careers in mathematics. Quite often, when one doesn't understand a result in introductory level undergraduate courses, I find that person is either lacking knowledge of or misapplying their knowledge of a simpler result. This creates confusion and frustration. So, it sounds simple, but learning the fundamentals is key. This takes time and a desire to develop mathematical maturity beyond that which is offered at most high schools.
tl;dr - learning mathematics at more advanced levels can be a bit challenging! Know where to start. Also know what you're getting yourself into and don't expect to be rewarded for your knowledge as often as you would like as math is a broad topic that is represented by such a small fraction of the distribution.