How to become a good theoretical math player?

 Kimahri
 Posts: 2
 Joined: Mon Apr 20, 2015 10:31 am
How to become a good theoretical math player?
I was wondering what ways there are to become a good math player in college quiz bowl. And when I say math, I mean theoretical math rather than computational math. My dilemma is that most college math textbooks take a computational approach rather than a theoretical approach (for example, a real analysis textbook would go all into how to find the Fourier series of a handful of equations while it does not do much with the theory behind a Fourier series that a quiz bowl tossup may ask). I'm also not a math major so I will not be taking a lot of 300/400level/gradlevel math courses at my university (the highest courses I have taken so far are multivariate/vector calculus, ordinary differential equations and linear algebra). Due to my lack of a university mathematics background, I will probably have a really hard time learning from math textbooks that take a computational approach (especially because the majority of them are proofstructured). So I wanted to know if anyone has advice on highlevel math resources that are heavily theorybased and contains the type of clues that come up in college quiz bowl tossups.
So far, what I have been doing is taking notes on Wikipedia articles. However, I was kind of dubious whether that was the best way to study theoretical math or not.
So far, what I have been doing is taking notes on Wikipedia articles. However, I was kind of dubious whether that was the best way to study theoretical math or not.
John Kim
University of Arizona
Southern Arizona Outreach Director, Arizona Quizbowl Association
University of Arizona
Southern Arizona Outreach Director, Arizona Quizbowl Association
Re: How to become a good theoretical math player?
This is interesting for me, because I was unhappy with my Fourier analysis book (Stein/Shakarchi) for not working a ton of examples (I have to do WORK? To LEARN THINGS??). For analysis, I definitely recommend all four volumes of that set (hopefully your library has them!), although I've only used 13 for classes so far.Macromind101 wrote:for example, a real analysis textbook would go all into how to find the Fourier series of a handful of equations while it does not do much with the theory behind a Fourier series that a quiz bowl tossup may ask.
Jacob Reed
Chicago ~'25
Yale '17, '19
East Chapel Hill '13
"...distant bayings from...the musicological mafia"―Denis Stevens
Chicago ~'25
Yale '17, '19
East Chapel Hill '13
"...distant bayings from...the musicological mafia"―Denis Stevens
Re: How to become a good theoretical math player?
Just posting to agree that Stein/Shakarchi is wonderful.vinteuil wrote:This is interesting for me, because I was unhappy with my Fourier analysis book (Stein/Shakarchi) for not working a ton of examples (I have to do WORK? To LEARN THINGS??). For analysis, I definitely recommend all four volumes of that set (hopefully your library has them!), although I've only used 13 for classes so far.Macromind101 wrote:for example, a real analysis textbook would go all into how to find the Fourier series of a handful of equations while it does not do much with the theory behind a Fourier series that a quiz bowl tossup may ask.
I'm not sure why you don't consider taking more upper level math classes. You may not be a math major, but taking a course in number theory and/or abstract algebra will undoubtedly help you claim a share of the (relatively small number of) math points to be had. Also, those subjects are really interesting.
Understand that studying math for quizbowl points will yield a low return on investment. I would only recommend it if you really want to learn the math also, in which case taking classes is the way to go.
Dr. Noah Prince
Normal Community High School (2002)
University of Illinois at UrbanaChampaign (2004, 2007, 2008)
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 Smuttynose Island
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Re: How to become a good theoretical math player?
As others have said, and as you seem to have surmised, taking upper level proof based math courses is the best and even most efficient way to become good at "quizbowl math." Classes are not only great at providing solid quizbowl knowledge of many clues and topics that come up in quizbowl, but they are also fantastic for developing the base of knowledge that you need to really understand what is going on in math questions and the clues that they use.
If you can't take upper level math classes, then simply reading textbooks is a decent way to improve at math questions. Reading textbooks, just like taking classes, help build a foundation of knowledge that helps make studying math specifically for quizbowl much easier. Reading textbooks will also often help guide you to the more important concepts, theorems, structures etc in mathematics; concepts that tend to dominate the quizbowl math canon. Keep in mind that almost all of the decent textbooks are going to be prooforiented. This is because abstract mathematics, which is the mathematics that dominates the quizbowl canon, is proofbased. While this may take some getting used to if you've only taken computationally based math classes, such as most ODE, calculus and elementary linear algebra classes, you shouldn't have too many problems adapting (If you do there are numerous books out there dealing specifically with introducing students to reading and writing proofs which might be helpful).
Most fields have fairly standard introductory textbooks, although their quizbowl utility varies. Munkres' Topology is a very well written textbook that has a fair bit of quizbowl utility, while Dummit and Foote's Abstract Algebra is excellent as well. A lot of people read Rudin in their introductory analysis courses, but it is pretty sparse. I've gotten a lot of use out of Spivak's Calculus and Calculus on Manifolds.
When it comes to learning about specific answers and clues, simply looking up the answer on google and reading papers and guides on university websites is a fantastic way to improve. A lot of the search results are documents written for classes. This means that you will typically find a wide variety of sources written for people of differing levels of mathematical maturity. This makes it easy to gain a base understanding of a topic or find excellent hard and middle clues for various answerlines. Additionally, Wolfram Mathworld will often have write ups on important ideas, functions, and techniques. These writeups are excellent for learning about specific topics and even let you explore further by linking to related topics. Over the course of this year, reading the Wolfram Alpha write ups on various topics has been almost as useful as my classes in netting me points.
Finally, simply reading questions is a great way to improve provided that you do so in conjunction with reading online sources, textbooks and taking classes. Math, in a sense, has its own language, which makes it very easy to get lost and confuse concepts if you only learn from reading old questions. Despite this, reading questions is the fastest way to gain a sense of what mathematics topics quizbowlers like to ask about and therefore serves as a great guide for your studying.
Having said all of this, I agree with Dr. Prince. Specifically studying math for quizbowl is not an effective way to win games or learn about mathematics. If you are interested in the subject, it behooves you to take any upper level class that you can. Doing so will have the wonderful secondary benefit of making you better at one of the least asked about areas of the quizbowl distribution.
If you can't take upper level math classes, then simply reading textbooks is a decent way to improve at math questions. Reading textbooks, just like taking classes, help build a foundation of knowledge that helps make studying math specifically for quizbowl much easier. Reading textbooks will also often help guide you to the more important concepts, theorems, structures etc in mathematics; concepts that tend to dominate the quizbowl math canon. Keep in mind that almost all of the decent textbooks are going to be prooforiented. This is because abstract mathematics, which is the mathematics that dominates the quizbowl canon, is proofbased. While this may take some getting used to if you've only taken computationally based math classes, such as most ODE, calculus and elementary linear algebra classes, you shouldn't have too many problems adapting (If you do there are numerous books out there dealing specifically with introducing students to reading and writing proofs which might be helpful).
Most fields have fairly standard introductory textbooks, although their quizbowl utility varies. Munkres' Topology is a very well written textbook that has a fair bit of quizbowl utility, while Dummit and Foote's Abstract Algebra is excellent as well. A lot of people read Rudin in their introductory analysis courses, but it is pretty sparse. I've gotten a lot of use out of Spivak's Calculus and Calculus on Manifolds.
When it comes to learning about specific answers and clues, simply looking up the answer on google and reading papers and guides on university websites is a fantastic way to improve. A lot of the search results are documents written for classes. This means that you will typically find a wide variety of sources written for people of differing levels of mathematical maturity. This makes it easy to gain a base understanding of a topic or find excellent hard and middle clues for various answerlines. Additionally, Wolfram Mathworld will often have write ups on important ideas, functions, and techniques. These writeups are excellent for learning about specific topics and even let you explore further by linking to related topics. Over the course of this year, reading the Wolfram Alpha write ups on various topics has been almost as useful as my classes in netting me points.
Finally, simply reading questions is a great way to improve provided that you do so in conjunction with reading online sources, textbooks and taking classes. Math, in a sense, has its own language, which makes it very easy to get lost and confuse concepts if you only learn from reading old questions. Despite this, reading questions is the fastest way to gain a sense of what mathematics topics quizbowlers like to ask about and therefore serves as a great guide for your studying.
Having said all of this, I agree with Dr. Prince. Specifically studying math for quizbowl is not an effective way to win games or learn about mathematics. If you are interested in the subject, it behooves you to take any upper level class that you can. Doing so will have the wonderful secondary benefit of making you better at one of the least asked about areas of the quizbowl distribution.
Daniel Hothem
TJHSST '11  UVA '15  Oregon '??
"You are the stuff of legends"  Chris Manners
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TJHSST '11  UVA '15  Oregon '??
"You are the stuff of legends"  Chris Manners
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 The Ununtiable Twine
 Yuna
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Re: How to become a good theoretical math player?
Math is basically learned by doing problems. If you don't apply theorems and ideas to other problems the stuff tends not to stick. With that being said if I were a novice again I would take a few of the classes suggested already but to be quite honest it takes years for theoretical math to become a great player of math questions. I would definitely suggest spending time studying other things unless you love math or you are masochistic.
Jake Sundberg
Louisiana '04'10, '14'16, '18'xx
Alabama '1014
President, University of Louisiana at Lafayette Club for Academic Competition
Louisiana '04'10, '14'16, '18'xx
Alabama '1014
President, University of Louisiana at Lafayette Club for Academic Competition
 Lagotto Romagnolo
 Tidus
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Re: How to become a good theoretical math player?
I don't want to turn this thread away too much from the "theoretical" designation, but inasmuch as it seems to define "theoretical" as "not quizbowl computational math," here's another idea.
If you want to learn upperlevel math but, like me, can't stand long purely theoretical books with no applications to wrap your head around, you may find it more interesting to approach the subject from an applied perspective. I find that asking myself "What math do I need to understand physics/chemistry/engineering field X?" lets me learn math in what is, for me, a more engaging manner. Many books on field X have appendices or even full sections that focus on the math required without getting bogged down in ancillary proofs. Of course it's a waste of money to just buy a universitylevel textbook for the appendices, but they're certainly worth checking out from the library. As for the math books that actually are worth reading through (I list only the ones I've used; would love to hear better suggestions) :
Differential Equations: Boyce/DiPrima is the classic, but supposedly Simmons is much betterwritten and much more entertaining.
Statistics: Wackerly/Mendenhall/Scheaffer covers probability distributions and fundamentals well in the early chapters, but doesn't do a great job of explaining hypothesis testing or experimental design (although this stuff is much rarer in quizbowl).
And there are also survey books out there. Mathematical Methods for Physicists by Arfken/Weber covers a lot of material but sacrifices depth, so it provides little detail. Mathematical Methods for Physics and Engineering by Riley/Hobson/Bence is supposed to be good. So, if you're concerned that studying pure math will have too low a return on investment, maybe applied math will give you more knowledge across the whole science distribution.
If you want to learn upperlevel math but, like me, can't stand long purely theoretical books with no applications to wrap your head around, you may find it more interesting to approach the subject from an applied perspective. I find that asking myself "What math do I need to understand physics/chemistry/engineering field X?" lets me learn math in what is, for me, a more engaging manner. Many books on field X have appendices or even full sections that focus on the math required without getting bogged down in ancillary proofs. Of course it's a waste of money to just buy a universitylevel textbook for the appendices, but they're certainly worth checking out from the library. As for the math books that actually are worth reading through (I list only the ones I've used; would love to hear better suggestions) :
Differential Equations: Boyce/DiPrima is the classic, but supposedly Simmons is much betterwritten and much more entertaining.
Statistics: Wackerly/Mendenhall/Scheaffer covers probability distributions and fundamentals well in the early chapters, but doesn't do a great job of explaining hypothesis testing or experimental design (although this stuff is much rarer in quizbowl).
And there are also survey books out there. Mathematical Methods for Physicists by Arfken/Weber covers a lot of material but sacrifices depth, so it provides little detail. Mathematical Methods for Physics and Engineering by Riley/Hobson/Bence is supposed to be good. So, if you're concerned that studying pure math will have too low a return on investment, maybe applied math will give you more knowledge across the whole science distribution.
Aaron Rosenberg
Langley HS '07 / Brown '11 / Illinois '14
PACE
Langley HS '07 / Brown '11 / Illinois '14
PACE
Re: How to become a good theoretical math player?
Go prove more things.
Jonah Greenthal
National Academic Quiz Tournaments
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 The Ununtiable Twine
 Yuna
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Re: How to become a good theoretical math player?
I have a few math text suggestions  if you want to learn math for quizbowl purposes and you want it to stick you will have to understand the context of the material, so you will want to have textbooks which feature nice examples as well. Here, I give a reasonable way to get to the point where you can 20 most math bonuses on regular difficulty:
I used Boyce/DiPrima as a freshman (in an epoch not known to your own eyes): ODE gives a decent return on math points. Any treatment of linear algebra in this book is purely enough to get you to the point where you can solve the ODEs. The chapter on Laplace transforms (Ch 6 in my copy) may be a little useful. Chapter 5 (outside of maybe 5.1, which covers power series) tends to not come up in any quizbowl context, with the exception of an application of the power series method that comes up in an important problem in intro quantum mechanics. Anything in Chapters 811 I would advise you to just know what they are, since they come up as bonus parts a bit. Same with chapters 14. If you know what RungeKutta and SturmLiouville are, you'll be fine. There's a chapter on chaos in this book but I would strongly suggest reading from a classical mechanics textbook. Goldstein is a lot more wordy and presents the material in a way that is much, much easier to remember.
Friedberg/Insel/Spence Linear Algebra is an excellent tool for all of those linear algebra questions. For regular difficulty, Chapters 13 (basic definitions) as well as 5 (eigenvalues, diagonalization). Lots of clues inside of power and lots of 20s and 30s to be had just by reading those chapters. Of course, if it is your life's goal to 30 all of the linear algebra bonuses, chapters 6 (inner product spaces) and 7 (canonical forms) are for you!
As far as probability and stats goes, DeGroot/Schervish is pretty well written, FULL of examples, and has anything you'll need to conquer all of those questions on distributions, the CLT, pdfs, fun Poisson clues, and all sorts of other things. An older edition of this book goes for really cheap and is an invaluable tool if you want to conquer the math canon (which, again, I don't recommend, but hey!). Other than chapter 5 (the one on distributions), I would suggest just browsing through this textbook and Knowing What Things Are. Most of these things don't get tossed up but a lot come up as bonus parts.
As far as other stuff goes, to be honest I would pick up a copy of a nice discrete math text. I used Fletcher/Patty as a tyke but that particular book is rather expensive. This sort of book will give you very basic details on group theory, graphs, combinatorics, methods of proof, set theory, and the like. I would recommend starting with a book like this if you are going to learn higher mathematics. Math majors are required to take a general course like this and it prepares them for the horror that lies ahead.
If you use those books to study, along with the fancilycoveredtooexpensive calculus textbook of your choice, you should be able to 20 quite a few bonuses on regular difficulty. Those are the primary maths I took my freshman and sophomore years (along with number theory and vector analysis in addition to a couple of other things which are useful in their own right) and it was good enough to get fairly high ppb. It's probably going to be hard for you to ever get past 20 ppb unless you are a mathematician or can store definitions in shortterm memory. As for how to 30 math bonuses, well...math is learned by doing, and so 30ing math bonuses comes with doing enough problems. Eventually you get to the point where you know so many things that you can remember results that are loosely affiliated with any work you've done, but it's hard to remember those unless you understand the context of the material. For instance, maybe in learning the proof of Hard Part Theorem A you used fact B, lemma C.4, and a cutesy algebra trick. Well if you thought that proof was cool chances are it'll stick in your mind. If you didn't follow anything that was going on, it won't stick. Even some of the best science players who aren't mathematicians have trouble 30ing at regular difficulty and above unless they have immersed themselves in the elegance of higher level mathematics, constantly. I don't blame them, really. I spent a decade studying math and it doesn't give you much return on your (quizbowl) investment.
I used Boyce/DiPrima as a freshman (in an epoch not known to your own eyes): ODE gives a decent return on math points. Any treatment of linear algebra in this book is purely enough to get you to the point where you can solve the ODEs. The chapter on Laplace transforms (Ch 6 in my copy) may be a little useful. Chapter 5 (outside of maybe 5.1, which covers power series) tends to not come up in any quizbowl context, with the exception of an application of the power series method that comes up in an important problem in intro quantum mechanics. Anything in Chapters 811 I would advise you to just know what they are, since they come up as bonus parts a bit. Same with chapters 14. If you know what RungeKutta and SturmLiouville are, you'll be fine. There's a chapter on chaos in this book but I would strongly suggest reading from a classical mechanics textbook. Goldstein is a lot more wordy and presents the material in a way that is much, much easier to remember.
Friedberg/Insel/Spence Linear Algebra is an excellent tool for all of those linear algebra questions. For regular difficulty, Chapters 13 (basic definitions) as well as 5 (eigenvalues, diagonalization). Lots of clues inside of power and lots of 20s and 30s to be had just by reading those chapters. Of course, if it is your life's goal to 30 all of the linear algebra bonuses, chapters 6 (inner product spaces) and 7 (canonical forms) are for you!
As far as probability and stats goes, DeGroot/Schervish is pretty well written, FULL of examples, and has anything you'll need to conquer all of those questions on distributions, the CLT, pdfs, fun Poisson clues, and all sorts of other things. An older edition of this book goes for really cheap and is an invaluable tool if you want to conquer the math canon (which, again, I don't recommend, but hey!). Other than chapter 5 (the one on distributions), I would suggest just browsing through this textbook and Knowing What Things Are. Most of these things don't get tossed up but a lot come up as bonus parts.
As far as other stuff goes, to be honest I would pick up a copy of a nice discrete math text. I used Fletcher/Patty as a tyke but that particular book is rather expensive. This sort of book will give you very basic details on group theory, graphs, combinatorics, methods of proof, set theory, and the like. I would recommend starting with a book like this if you are going to learn higher mathematics. Math majors are required to take a general course like this and it prepares them for the horror that lies ahead.
If you use those books to study, along with the fancilycoveredtooexpensive calculus textbook of your choice, you should be able to 20 quite a few bonuses on regular difficulty. Those are the primary maths I took my freshman and sophomore years (along with number theory and vector analysis in addition to a couple of other things which are useful in their own right) and it was good enough to get fairly high ppb. It's probably going to be hard for you to ever get past 20 ppb unless you are a mathematician or can store definitions in shortterm memory. As for how to 30 math bonuses, well...math is learned by doing, and so 30ing math bonuses comes with doing enough problems. Eventually you get to the point where you know so many things that you can remember results that are loosely affiliated with any work you've done, but it's hard to remember those unless you understand the context of the material. For instance, maybe in learning the proof of Hard Part Theorem A you used fact B, lemma C.4, and a cutesy algebra trick. Well if you thought that proof was cool chances are it'll stick in your mind. If you didn't follow anything that was going on, it won't stick. Even some of the best science players who aren't mathematicians have trouble 30ing at regular difficulty and above unless they have immersed themselves in the elegance of higher level mathematics, constantly. I don't blame them, really. I spent a decade studying math and it doesn't give you much return on your (quizbowl) investment.
Jake Sundberg
Louisiana '04'10, '14'16, '18'xx
Alabama '1014
President, University of Louisiana at Lafayette Club for Academic Competition
Louisiana '04'10, '14'16, '18'xx
Alabama '1014
President, University of Louisiana at Lafayette Club for Academic Competition
 Habitat_Against_Humanity
 Rikku
 Posts: 450
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Re: How to become a good theoretical math player?
I'll note I've learned a fair amount of quizbowl math from reading cheap Dover books. As much as I love Boyce and DiPrima (and its little historical sidebars), I still had to pay an arm and a leg for it. A lot of Dover books are under $10 and if you're Kindleoriented, under $5. Also, at the risk of rebuke, Wikipedia has been pretty good for getting me math points.
Addendum: Dover also has some "History of Mathematics" books that might be what you're looking for if you (understandably) don't necessarily want to delve into a full, theoretical textbook.
Addendum: Dover also has some "History of Mathematics" books that might be what you're looking for if you (understandably) don't necessarily want to delve into a full, theoretical textbook.
Nolan 
UChicago 09
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 Yuna
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Re: How to become a good theoretical math player?
I'll second this  most of their articles are written by folks with reasonable expertise in the field. They tend to be very thorough, so reading some of these articles without a certain level of technical expertise can be a bit difficult. For the most part, anyone can gain an understanding of what a certain thing actually is.Habitat_Against_Humanity wrote:Also, at the risk of rebuke, Wikipedia has been pretty good for getting me math points.
What Wikipedia will NOT help you with (unless you have a certain level of experience) is writing a math question well. Easy, medium, hard, very hard, and impossible clues are often mixed together in these articles. Of course, people can improve by writing questions as well. This method is not nearly as good as working through problems, but quizbowlers suggest writing questions to improve, so it's a thing that I'll address briefly. The majority of what college math students and graduate students learn is pretty canonical, so I would definitely stick to using textbooks intended for undergrads and beginning graduate students for question writing. For those inexperienced math writers writing questions at higher difficulty levels (such as ACF Nationals, CO, etc) I wouldn't recommend using clues from arXiv papers unless you're familiar with the direction research is headed, and even then, use them sparingly as leadins. The majority of that stuff is beyond the scope of a quizbowl tournament and only rarely will the clues be buzzable to more than a couple of players in the country. Sticking to textbooks is definitely your best bet.
Jake Sundberg
Louisiana '04'10, '14'16, '18'xx
Alabama '1014
President, University of Louisiana at Lafayette Club for Academic Competition
Louisiana '04'10, '14'16, '18'xx
Alabama '1014
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 Guile Island
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Re: How to become a good theoretical math player?
Do the people who actually study math in here have a book recommendation(s) for discrete math? I used Kenneth Rosen's for a survey level "discrete math for computer scientists" course and thought it was pretty good, but I'm interested in hearing about alternatives, as well as individual books for graph theory, number theory, etc.
Dylan Minarik
PACE (Former Director of Communications, 201819 season)
Northwestern '17
Belvidere North High School '13
JRPG Champion, BACK TO BACK Robot Slayer
PACE (Former Director of Communications, 201819 season)
Northwestern '17
Belvidere North High School '13
JRPG Champion, BACK TO BACK Robot Slayer
Re: How to become a good theoretical math player?
Doug West's book is pretty much the standard for graph theory.Craise Finton Kirk Royal Academy of Arts wrote:Do the people who actually study math in here have a book recommendation(s) for discrete math? I used Kenneth Rosen's for a survey level "discrete math for computer scientists" course and thought it was pretty good, but I'm interested in hearing about alternatives, as well as individual books for graph theory, number theory, etc.
Jonah Greenthal
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 Yuna
 Posts: 999
 Joined: Fri Feb 02, 2007 11:09 pm
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Re: How to become a good theoretical math player?
I've never actually taken a formal course in graph theory  most of the stuff I learned from algorithm design and simple graph theoretic stuff taught in the other maths has gotten me by pretty well in quizbowl. Of course Jonah will see this and make all of the math for SCT and ICT graph theory.Craise Finton Kirk Royal Academy of Arts wrote:Do the people who actually study math in here have a book recommendation(s) for discrete math? I used Kenneth Rosen's for a survey level "discrete math for computer scientists" course and thought it was pretty good, but I'm interested in hearing about alternatives, as well as individual books for graph theory, number theory, etc.
As far as number theory goes, I am a fan of LeVeque. It's a very cheap Dover book that's pretty comprehensive as far as elementary number theory goes. It also features various biographical snippets of mathematicians that studied number theory. This is the first theoretical book I ever used that was helpful in 30ing lots of bonuses at regular difficulty.
As far as abstract algebra goes, I think the easiest way to go would be Dummit/Foote. I never took a course out of that book but it seems pretty comprehensive and a lot of schools use it. As far as more advanced topics go: Rotman is very good for early clues at regular difficulty and higher, but is a little more technically advanced so it might be tough to read unless you have some expertise. Passman's ring theory is a good book but doesn't have many examples. Hungerford is a pain in the ass. Robinson is beyond the scope of quizbowl. Of the four, Rotman is probably your best bet. With that being said, it's probably best not to touch those books unless you have an idea of what things are and can digest notation pretty well.
Topology? Munkres is great reading  it's pretty comprehensive as far as general topology goes and is pretty standard in the field too. There are several sections on simple ideas in algebraic topology near the back of the textbook. It's probably a good idea not to take many clues from algebraic topology textbooks because they can be rather tough to read, but asking about those topics by branching off from Munkres is a great idea for tournaments whose prescribed difficulty is above regular. (At any rate, neglecting most of algebraic topology is one of the very few shameful things the promoters of this game have done. The number of quizbowl players who quiver at the very thought of a tossup on fundamental groups is quite astonishing!) Munkres will get you many and many a 15 and 30. General topology is not something people really research anymore, so everything you could possibly want is pretty much in this book.
As far as complex analysis goes, I think John B. Conway's book is the best I've encountered as far as quizbowlcentric topics are concerned. Rudin is a better book but the topics are a little more advanced. For a little softer reading, Mathews/Howell Complex Analysis for Mathematics and Engineering presents topics very well. There are also chapters on applications and Fourier transforms that might prove to be useful as well. You can't go wrong with Conway or Mathews/Howell.
For baby real analysis I used Gaughan's Intro to Analysis which I think is used in several schools. It's a small book, but has several few ideas that can help. Current editions are very expensive. For real analysis my prof used Royden, but I'm not very fond of that book. Learning out of it just wasn't very enjoyable. Rudin is better for these things, methinks.
If you can master those books (good luck!) you'll be set to get 25+ ppb on math bonuses for the rest of your career.
Jake Sundberg
Louisiana '04'10, '14'16, '18'xx
Alabama '1014
President, University of Louisiana at Lafayette Club for Academic Competition
Louisiana '04'10, '14'16, '18'xx
Alabama '1014
President, University of Louisiana at Lafayette Club for Academic Competition
 Vainamoinen
 Lulu
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Re: How to become a good theoretical math player?
Just chiming in on topology books I recommend Topology Without Tears as a quick intro for beginners, especially since it's free. http://www.topologywithouttears.net/
Will Overman
MW '16
Caltech '20
MW '16
Caltech '20
Re: How to become a good theoretical math player?
I'll take a slightly different tack from what people above are suggesting. The above advice is fairly solid for learning math for its own sake, and it will translate into getting lots of quizbowl questions. The problem is that this advice is basically asking you to undertake a twoyearplus course load of highlevel mathematics, and you won't see gains in quizbowl until much later. If you want to learn math specifically for quizbowl and do so quickly, then your best bet is to focus less on working through Rudin or Dummit and Foote and more on acquiring a good mathematical vocabulary (i.e. know the definitions of terms and their basic relation to each other) and then skimming through packets and Wikipedia. Like other categories, math in quizbowl is often out of sync with what people actually study, and it becomes helpful to know what clues keep coming up. Studying Wikipedia is useful because a lot of quizbowl questions on math get written from Wikipedia. It might be useful to look at all the weird things that are named after really famous mathematicians like Fermat or Weierstrass or Hilbert since driveby common links on their tertiary accomplishments are fairly common. Wikipedia studying is also helpful because the names of theorems and definitions is not standardized across texts so you may end up needing to recall a name for a theorem or object that is unnamed or differently named in your book.
If you want to study math for its own sake, then I highly recommend the UChicago Undergraduate Math Bibliography (https://www.ocf.berkeley.edu/~abhishek/chicmath.htm) for help on choosing which books to look at. Starting out, I'd encourage you to work through Spivak's Calculus to make sure you're comfortable with basic proofbased mathematics and then familiarize yourself with the big three subjects of analysis, topology, and algebra through Baby Rudin, Munkres, and Dummit and Foote (Jacobson if you're feeling ambitious) respectively. Additionally, while learning algebra and topology, you should learn some basic category theory. Don't get crazy with it (unless you want to) but category theory will really help you organize your thought and see connections between mathematical structures.
If you want to study math for its own sake, then I highly recommend the UChicago Undergraduate Math Bibliography (https://www.ocf.berkeley.edu/~abhishek/chicmath.htm) for help on choosing which books to look at. Starting out, I'd encourage you to work through Spivak's Calculus to make sure you're comfortable with basic proofbased mathematics and then familiarize yourself with the big three subjects of analysis, topology, and algebra through Baby Rudin, Munkres, and Dummit and Foote (Jacobson if you're feeling ambitious) respectively. Additionally, while learning algebra and topology, you should learn some basic category theory. Don't get crazy with it (unless you want to) but category theory will really help you organize your thought and see connections between mathematical structures.
Kay, Chicago.
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Re: How to become a good theoretical math player?
One thing I thought of last night, that I don't think has been mentioned anywhere for any subject, but was helpful to me, is to go to talks and colloquia. When I was in undergrad, the Society of Physics Students would mail out lists of talks being given and every now and then (it probably should have been more often than that), I'd go just to break up my otherwise boring schedule of studying. I learned a lot from these talks that I wouldn't otherwise learn in undergrad textbooks, as they usually involved recent research and more advanced material. What's great is that this gradschool+ material was condensed into a PowerPoint slide, so you could get a rough idea of what something was and look it up later. Case in point: I never came across Feshbach resonances in my undergrad studies, but I did encounter them in more than one talk and since have gotten points from it on the ubiquitous BEC questions found in many question sets. I guess this isn't math specific, but I think it applies there (and to other subjects as well).
Nolan 
UChicago 09
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Re: How to become a good theoretical math player?
This has definitely helped me get biology questions too, for what that's worth. I imagine it's the same for every subject, like Nolan says.Habitat_Against_Humanity wrote:One thing I thought of last night, that I don't think has been mentioned anywhere for any subject, but was helpful to me, is to go to talks and colloquia.
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Re: How to become a good theoretical math player?
I think that Shurman's notes is a good introduction to multivariable calculus. It's available for free here.
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Re: How to become a good theoretical math player?
For those unaware, if you want to learn science in general, it's not difficult to find lecture notes for pretty much anything you want to learn about, or to obtain pdfs of textbooks in dubiously legitimate manners.
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Re: How to become a good theoretical math player?
Pinter's A Book of Abstract Algebra is quite good if you're just starting out.Habitat_Against_Humanity wrote:I'll note I've learned a fair amount of quizbowl math from reading cheap Dover books. As much as I love Boyce and DiPrima (and its little historical sidebars), I still had to pay an arm and a leg for it. A lot of Dover books are under $10 and if you're Kindleoriented, under $5. Also, at the risk of rebuke, Wikipedia has been pretty good for getting me math points.
Addendum: Dover also has some "History of Mathematics" books that might be what you're looking for if you (understandably) don't necessarily want to delve into a full, theoretical textbook.
Speaking as someone who studies math and is pretty bad at quizbowl math, this has been my experience.kayli wrote:If you want to learn math specifically for quizbowl and do so quickly, then your best bet is to focus less on working through Rudin or Dummit and Foote and more on acquiring a good mathematical vocabulary (i.e. know the definitions of terms and their basic relation to each other) and then skimming through packets and Wikipedia. Like other categories, math in quizbowl is often out of sync with what people actually study, and it becomes helpful to know what clues keep coming up. Studying Wikipedia is useful because a lot of quizbowl questions on math get written from Wikipedia. It might be useful to look at all the weird things that are named after really famous mathematicians like Fermat or Weierstrass or Hilbert since driveby common links on their tertiary accomplishments are fairly common.
If anybody interested in getting into statistics or probability (my personal specialty), I'd really recommend Blitzstein/Hwang's Introduction to Probability. A lot of it is lowerlevel, but they also get into some pretty approachable linear algebra applications with Markov chains/Monte Carlo methods.
Alex D.
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Re: How to become a good theoretical math player?
I'll toss in my two cents:
Math is one of the trickiest subjects to learn about for the purposes of quizbowl. For one, quizbowl has usually favored theoretical math, but recently the door has been blown wide open by asking about applied math  look at this year's ICT for example and Cody's MUT math (like the inequalities question.) Secondly, math is one of those topics that at higher levels, answers can come out of nowhere: this year's nats had tossups on spacefilling curves and Yevgeny Lifshitz  looking through old packets isn't really going to help you out, you just better have some amount of class knowledge, math curiosity, blah.
All of this is really just to say, mastering the math canon is nighimpossible, moreso than any other subject, I think you just have to accept that at times, you won't get certain questions if you are trying to learn math without devoting your life to math; even if you do have a math degree it probably doesn't mean you're going to be able to answer most questions early  I don't think knowing topology inside out is something most math majors do.
I might make a post about this in the future, but I think there is a very good argument for having .5/.5 math in every packet. I don't like cutting any of the other sciences for that (I'd prefer cutting bio  take that Eric!*). As Seth and I talked about once, the number of times in which we took numerical methods or differential equations or probability and stats for our applied science degrees was just astounding, and those topics could stand to come up much more (please don't use all of the extra math questions to write about Alexander Grothendieck and category theory, you fools)
*Also, it would be good for penance for that medical paper that "derived" the trapezoid rule.
Ike
Math is one of the trickiest subjects to learn about for the purposes of quizbowl. For one, quizbowl has usually favored theoretical math, but recently the door has been blown wide open by asking about applied math  look at this year's ICT for example and Cody's MUT math (like the inequalities question.) Secondly, math is one of those topics that at higher levels, answers can come out of nowhere: this year's nats had tossups on spacefilling curves and Yevgeny Lifshitz  looking through old packets isn't really going to help you out, you just better have some amount of class knowledge, math curiosity, blah.
All of this is really just to say, mastering the math canon is nighimpossible, moreso than any other subject, I think you just have to accept that at times, you won't get certain questions if you are trying to learn math without devoting your life to math; even if you do have a math degree it probably doesn't mean you're going to be able to answer most questions early  I don't think knowing topology inside out is something most math majors do.
I might make a post about this in the future, but I think there is a very good argument for having .5/.5 math in every packet. I don't like cutting any of the other sciences for that (I'd prefer cutting bio  take that Eric!*). As Seth and I talked about once, the number of times in which we took numerical methods or differential equations or probability and stats for our applied science degrees was just astounding, and those topics could stand to come up much more (please don't use all of the extra math questions to write about Alexander Grothendieck and category theory, you fools)
*Also, it would be good for penance for that medical paper that "derived" the trapezoid rule.
Ike
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Re: How to become a good theoretical math player?
This deserves its own thread, but Alexander Grothendieck is the most important mathematician of the past 60 years. His contributions to algebraic geometry should come up much more than spacefilling curves, which in topology are objects of secondary importance.
However, I do agree that applied mathematics should come up more.
However, I do agree that applied mathematics should come up more.
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Re: How to become a good theoretical math player?
And for Economics! This is a great idea that I presume the innumerate hordes will vociferously oppose. Let me further add that applied math should replace the Mythology of Agave Nectar distribution. And I also like the idea that quizbowl distributional space should be allocated so as to compensate for past thought crimes perpetrated by various academic fields, though I fear Economics would be adversely affected by such a regime.Ike wrote:I might make a post about this in the future, but I think there is a very good argument for having .5/.5 math in every packet. I don't like cutting any of the other sciences for that (I'd prefer cutting bio  take that Eric!*). As Seth and I talked about once, the number of times in which we took numerical methods or differential equations or probability and stats for our applied science degrees was just astounding, and those topics could stand to come up much more (please don't use all of the extra math questions to write about Alexander Grothendieck and category theory, you fools)
*Also, it would be good for penance for that medical paper that "derived" the trapezoid rule.
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Re: How to become a good theoretical math player?
I don't agree with your assessment either of the historical importance of these two topics or how much each should come up.kayli wrote:This deserves its own thread, but Alexander Grothendieck is the most important mathematician of the past 60 years. His contributions to algebraic geometry should come up much more than spacefilling curves, which in topology are objects of secondary importance.
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Re: How to become a good theoretical math player?
The point I'm making isn't trying to assess whether space filling curves or Alexander Grothendieck should be tossed up .5 times per a year or 1 time a year  let's face it, you can only toss these things up at Nats or higher difficulty. My point is that almost every engineer, physics, chemistry, mathematics, and apparently economics major will encounter differential equations, probability and statistics, and numerical methods, and other applied math fields over and over in their academic lifetimes, and we can and should be asking a lot more about these topics.
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Re: How to become a good theoretical math player?
Yeah, stats and probability are grossly underrepresented, especially given their wide applicability and use in not only the "hard" sciences, but also in things like Psych, Anthropology, Sociology, and the like.
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Re: How to become a good theoretical math player?
I'd also recommend Ross's First Course in Probability for a lowerlevel probability book.Aaron's Rod wrote:If anybody interested in getting into statistics or probability (my personal specialty), I'd really recommend Blitzstein/Hwang's Introduction to Probability. A lot of it is lowerlevel, but they also get into some pretty approachable linear algebra applications with Markov chains/Monte Carlo methods.
Shan Kothari
Plymouth High School '10
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Re: How to become a good theoretical math player?
I would love to have more stats in Quizbowl. I was ridiculously excited when we read MUT packet 11 in practice.Habitat_Against_Humanity wrote:Yeah, stats and probability are grossly underrepresented, especially given their wide applicability and use in not only the "hard" sciences, but also in things like Psych, Anthropology, Sociology, and the like.
Alex D.
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Re: How to become a good theoretical math player?
As an economics person, I'll echo the sentiment that statistics is an Important Subject that could use more questions.
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Re: How to become a good theoretical math player?
I will just briefly state that college players have many opportunities to make more questions come up on what they're interested in seeing more of, by submitting good questions to packetsub tournaments or (when sufficiently qualified) by writing their own sets and tweaking the distribution or focus of some parts of the set somewhat. The current preferences of quizbowl writers are not fixed in stone for all eternity.
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