## Studying math theory

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Joshua Rutsky
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### Studying math theory

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.
Joshua Rutsky
Coach, Hoover High School, Hoover, AL

Mnemosyne
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### Re: Studying math theory

A starting point is to look at all the fundamental undergraduate math courses after calculus (linear algebra, abstract algebra, discrete math/graph theory, real analysis, point-set topology, probability/stats, and a few others), and learn (and understand) the basic definitions from each chapter in (at least the first half of) the book. Say you open a linear algebra book: there could be a chapter called "eigenvalues and eigenvectors" or "vector spaces" - you should know what those things are and maybe an example or two of them. Obviously, being able to buzz in early on these tossups requires much more understanding, reading the books, learning theorems, working problems, etc. However, this, along with high school nats/college packet study, is probably the best way to get a grasp of the math canon and convert tossups and bonuses.

A \$15 Mathematics GRE Subject Test book will also cover most of this stuff, if you really want a compact reference source.
Nick Collins
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### Re: Studying math theory

Mnemosyne wrote:A starting point is to look at all the fundamental undergraduate math courses after calculus (linear algebra, abstract algebra, discrete math/graph theory, real analysis, point-set topology, probability/stats, and a few others), and learn (and understand) the basic definitions from each chapter in (at least the first half of) the book. Say you open a linear algebra book: there could be a chapter called "eigenvalues and eigenvectors" or "vector spaces" - you should know what those things are and maybe an example or two of them. Obviously, being able to buzz in early on these tossups requires much more understanding, reading the books, learning theorems, working problems, etc. However, this, along with high school nats/college packet study, is probably the best way to get a grasp of the math canon and convert tossups and bonuses.

A \$15 Mathematics GRE Subject Test book will also cover most of this stuff, if you really want a compact reference source.
On the other hand, I probably wouldn't recommend this unless the kids are really, really inspired. I took everything up through vector calc in high school and even at my best, I don't think I could have really grappled with an abstract algebra or topology book for only quiz bowl purposes even then.

What I would recommend would be some of those softcover popularizations of math books that occasionally pop up. They're technical enough and give enough mathematical meat to keep even a non-math minded person involved and usually drop a lot of historical context/names you should know for tossups.

Some such books on my shelf currently are:

The Golden Ratio - Mario Livio
Number Theory and its History - Oystein Ore (haven't done much more than peruse this one, but it seems good)
Wonders of Numbers - Clifford Pickover (pretty light, but things like "10 strangest mathematicians" and "10 most important unsolved problems" in it started me down the path to learning more about a lot of math and have provided a ton of giveaway-level college math knowledge)
Everything an More: A Compact History of Infinity - David Foster Wallace (yep, it's got footnotes).
Nolan -
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### Re: Studying math theory

The good news is that we're finally at the point where packet study of math is possible. Find some tournaments that have .5/.5 or more of math at a difficulty you want to study, see what answers and clues come up, and learn about them. The answer space and the number of possible clues for each answer is small enough that you can learn a lot from packets, especially if you are willing to look up stuff that looks interesting.
David Reinstein
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### Re: Studying math theory

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.
Jake Sundberg
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### Re: Studying math theory

Habitat_Against_Humanity wrote: Everything an More: A Compact History of Infinity - David Foster Wallace (yep, it's got footnotes).
This is a great book.
Kevin Wang
Amherst College 2019

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### Re: Studying math theory

Your question is a really good one, Coach Rutsky, and people have offered some answers. I'm going to come down heavily on the side of trying to study textbooks, or textbooks and packets together, in order to really learn what's going on. The good thing is, you don't actually have to know calculus to learn a lot of the subjects that traditionally come *after* calculus in the standard undergraduate curriculum. For example. I used very little calculus in my classes on number theory and abstract algebra. In 2003 I went to a summer camp where we were essentially taught a course in abstract algebra in 4 weeks, and I hadn't taken Calc III at that point. Other things, like analysis, may need it, however.

I would also suggest, if you have students who are particularly interested in math, for them to look into...

1. Accelerating their coursework as much as possible, by taking college-level classes online or over the summers. Things like Stanford EPGY are great for this, or at least was in my time.

2. Studying for math contests, as that tends to make you better at learning curricular math as well. Plus, great for college apps.
Eric Mukherjee, MD PhD
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Cheynem
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### Re: Studying math theory

For anyone who wants to discuss the math questions and the amount of math that was in NASAT, I'd encourage you to join the private NASAT discussion forum.
Mike Cheyne
Formerly U of Minnesota

"You killed HSAPQ"--Matt Bollinger

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