How small is the math canon?

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How small is the math canon?

Post by Dominator » Tue Jul 13, 2010 3:20 pm

I heard something that intrigued me the other day.

My school is writing a tournament for January 2011. Naturally, I am writing the math questions. I told a member of my team that I was about 1/4 done and he said something to the effect that it's hard to write math since the math canon is so small. (Math, unless otherwise stated, is assumed to be noncomputational.)

Is this true? Is the math canon really that small?

I would argue not. Certainly there are reasons to believe the canon is small. The main one, in my opinion, is that the collective answer spaces of math questions in quizbowl is much smaller than in other areas. However, this is also a result of several factors. Math is a very small portion of many distributions, often being demoted to a subfield of science. In addition, in formats in which math plays a larger role (like IHSA, where it makes up 20% of the distro), it is dominated by comp math.

Considering that most students (in my experience) take math every semester of high school and the typical math class covers much more content than a class on novels, for instance, why is it that the math canon appears smaller than the novels canon?

I argue that the math canon is (or ought to be considered) quite large, including:
(1) Mathematicians - If Nobel Prize winners are fair game in science and literature, so too should Fields Medalists be fair game in math.
(2) 20th century math - The development of math did not end when Newton and Leibniz "discovered calculus", but that appears to be the case, with a handful of notable exceptions, in most answer spaces.
(3) History of math
(4) All of computer science
(5) Accessible fields of advanced math, such as graph theory, basic set theory, propositional logic, Boolean algebra, basic abstract algebra, some non-Euclidean geometries and topological surfaces, and basic point-set topology
(6) The entire high school mathematics curriculum

To me, this seems like quite a nice, large canon while still being comparable to the kinds of things we would expect quizbowlers to know in other fields.

Thoughts?
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Re: How small is the math canon?

Post by jonah » Tue Jul 13, 2010 3:42 pm

I think the canon is somewhat bigger than most people—and certainly most Illinoisans—seem to, but probably not quite as big as you do. Possibly your opinion is skewed a bit by having a Ph.D. in math and teaching at a school like IMSA, or maybe you're just more optimistic than anyone involved in Illinois quizbowl should be.

As I pointed out to you in an email a few months ago, one thing to note is that whatever the math canon may be, the number of topics therein that can make for good tossups is significantly smaller. Math (especially easier topics therein) just seems especially prone—not that other sciences and even other disciplines entirely are immune—to topics that are interesting, important, and gettable, but don't have sufficiently many clues that are unique, concise, and suitably graded in difficulty.
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Re: How small is the math canon?

Post by Dominator » Tue Jul 13, 2010 3:57 pm

jonah wrote:Math just seems especially prone to topics that are interesting, important, and gettable, but don't have sufficiently many clues that are unique, concise, and suitably graded in difficulty.
Yes, I see your point, and I have had to throw out some really nice answers for this reason. At the same time, I think that there are times when really good clues do exist, but you really do need to be creative to find them (especially as good examples are hard to find). Having a Ph.D. in math means, among other things, that I do think creatively about math, so yes, I am much more optimistic.
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Re: How small is the math canon?

Post by jonah » Tue Jul 13, 2010 4:00 pm

Dominator wrote:
jonah wrote:Math just seems especially prone to topics that are interesting, important, and gettable, but don't have sufficiently many clues that are unique, concise, and suitably graded in difficulty.
Yes, I see your point, and I have had to throw out some really nice answers for this reason. At the same time, I think that there are times when really good clues do exist, but you really do need to be creative to find them (especially as good examples are hard to find). Having a Ph.D. in math means, among other things, that I do think creatively about math, so yes, I am much more optimistic.
To an extent, yes, but having to think too creatively to come up with such clues often may mean that players would have to think much more creatively to decipher them than they can reasonably be expected to do in the context of a match, especially since by the time they have figured it out, an easier clue may already have been read causing a less knowledgeable player to beat them to it. I'm not saying this is always the case, just that creativity in clue selection can only be taken so far.
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Re: How small is the math canon?

Post by Windmill Tump » Tue Jul 13, 2010 4:51 pm

(I'm only going to address this for high school, since you only mention high school and I wouldn't know anything about the college aspect anyway)

I'm not sure about writing questions on specific things in math, but I'm worried that, with a few exceptions such as Wiles, a lot of the recent mathematicians and math occurrences (20th century math I guess) is too hard to write about, at least for high school. I could see recent math and mathematicians used as harder parts of bonuses at harder tournaments, but writing good tossups of appropriate difficult might be difficult. For example, Perelman, the only recent Fields Medalist I can recognize besides Tao, is obviously most well known for solving the Poincaré Conjecture, so I assume there are high school students that can get Perelman off of that if you were to write a tossup on him. The problem is, what else could you write about Perelman that people would actually get? If this is happening for Perelman, who I'd think is one of the most well known recent Fields Medalists, I doubt that very many tossups could be written on mathematicians who aren't already tossed up. For bonuses, this is different, since you only need to mention one famous detail. The same reasoning applies to 20th century math, where most of the breakthroughs are occurring in fields too advanced for high school.

On the other hand, I'd agree that the math canon is indeed bigger than most people would think, but there's a limit to what you can ask, at least for high school.

EDIT: Replaced tossups with difficulty because I apparently can't type the right thing today.
Last edited by Windmill Tump on Tue Jul 13, 2010 5:50 pm, edited 1 time in total.
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Re: How small is the math canon?

Post by Mewto55555 » Tue Jul 13, 2010 5:11 pm

Dominator wrote: I argue that the math canon is (or ought to be considered) quite large, including:
(1) Mathematicians - If Nobel Prize winners are fair game in science and literature, so too should Fields Medalists be fair game in math.
Despite having a borderline-unhealthy obsession with mathematics, like the above I can only name 2 Fields Medalists off the top of my head, Tao and Perelman. Skimming the list of the rest of them, I can't see any on that list which would have too high of a conversion. Perelman might be tossupable, but Tao is pretty hard (I only know about him due to my experience with math competitions).
(6) The entire high school mathematics curriculum
A lot of this seems like it would be hard to ask and keep it non-computational. With algebra, there's not a lot you can really do. Geometry has a lot of nice theorems with cool names, but most of those are not covered in class, meaning you'd be stuck asking about stuff like SAS similarity :sad:


However, there still is lots of cool stuff to ask about in math while keeping it at reasonable difficulty.
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Re: How small is the math canon?

Post by Deviant Insider » Tue Jul 13, 2010 5:46 pm

Dominator wrote:(1) Mathematicians - If Nobel Prize winners are fair game in science and literature, so too should Fields Medalists be fair game in math.
Actually, many Nobel Prize winners are not fair game in science and literature. If you are talking about the ones that are fair game for tossups in high school matches, then only a small percentage of them qualify.
Dominator wrote:(2) 20th century math - The development of math did not end when Newton and Leibniz "discovered calculus", but that appears to be the case, with a handful of notable exceptions, in most answer spaces.
(3) History of math
The problem here is that very few high school students understand much 20th century mathematics. An occasional question on fractals or topology is OK, but we're talking about a small number of questions. If you want to count non-Euclidean geometry as 20th century math, that increases the number a little bit. Also the trend in writing science questions is to avoid a focus on history so that the focus can be on currently relevant ideas, though it is OK to write about people who are still relevant, especially if their name is attached to a particular concept. As already said by somebody else, you can't write tossups about Wiles and Perelman if the only thing anybody knows about them is only one thing.
Dominator wrote:(4) All of computer science
Computer Science is often placed under science, though I don't think it's a big problem if a tournament is going to have a significant amount of math and reclassify computer science as a subcategory of math. In any case, experience shows that computer science questions generally have low conversion rates, which suggests that the useful canon is fairly small.
Dominator wrote:(5) Accessible fields of advanced math, such as graph theory, basic set theory, propositional logic, Boolean algebra, basic abstract algebra, some non-Euclidean geometries and topological surfaces, and basic point-set topology
(6) The entire high school mathematics curriculum
These two areas should be the focus of quizbowl math questions. The difficulty, as already discussed upthread, is coming up with enough clues that are uniquely identifying and nontransparent and that range enough in difficulty to write a pyramidal tossup, or coming up with three related answers that range enough in difficulty to write a good bonus. It's not impossible, but it's a challenge--more challenging than writing questions in other subjects.

Let's say you wanted to write a tossup on Cramer's Rule, for example. What would you say about it? There's not really any such thing as deep knowledge of it, unless you want to get into its history, which isn't that important, or its derivation, which for a given size isn't that interesting beyond the fact that it works out and which for size n is difficult to describe without giving away the fact that you're using determinants to solve a system. About the only thing interesting about it is that it uses the division of determinants to solve systems. That's OK for a bonus part, but it's not enough clues for a tossup. It seems like this type of problem comes up a lot with topics from the high school mathematics curriculum.
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Re: How small is the math canon?

Post by kayli » Tue Jul 13, 2010 8:03 pm

Currently the math canon is small due to years of computational math. Comp math led to rehashing what are essentially the same problems and severely limited the expansion of the canon. However with the emergence of theory over computation, the canon will have to expand to prevent reusing the same questions and clues. Naturally, there will be growing pains experienced by quizbowlers caused by this sudden expansion. Questions will go dead; bonus parts will go unanswered; and people will be puzzled. But, I think that as long as the math canon doesn't expand at too fast of a rate people will adapt and start learning about things they've never heard before in a process similar to what happens when other categories were expanded. However, one thing I see that could limit that adaptation is the tendency for questions to draw higher up in the math curriculum. There seem to be a lot of lead-ins and bonuses that necessitate understanding in pretty difficult college-level math classes (or blind memorization). I find this detrimental to the math canon because then people won't actually understand the concept much less how to apply it due to the lack of prerequisite knowledge, and this goes against what quizbowl's all about.

In general, I think math tossups should come from an expanded high school math curriculum. What I mean is that instead of stretching up towards higher college level math, we stretch out to include more obscure things from the high school curriculum (geometry, algebra, calculus, number theory, etc). This means that instead of writing a bonus part on analytic functions or partial diffrential equations, we start writing on stuff on Sophie Germain primes and Euler lines. Things like Sophie Germain primes and Euler lines are perfectly good hard math concepts that don't require too much prerequisite knowledge to learn and understand. In contrast, both analytic functions and PDE require very high levels of understanding to get (or plain rote memorization).


By the way, I'm only thinking in context of high school math questions. College and beyond is a completely different story.
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Re: How small is the math canon?

Post by Windmill Tump » Tue Jul 13, 2010 9:18 pm

Arsonists Get All the Girls wrote: In general, I think math tossups should come from an expanded high school math curriculum. What I mean is that instead of stretching up towards higher college level math, we stretch out to include more obscure things from the high school curriculum (geometry, algebra, calculus, number theory, etc). This means that instead of writing a bonus part on analytic functions or partial diffrential equations, we start writing on stuff on Sophie Germain primes and Euler lines.
As much I'd love to see tossups written on the Euler line, I'm thinking that a lot of these obscure things might be too hard or might not have enough good clues to write a good question about. The simplest clue about the Euler line is probably the fact that it passes through the orthocenter, the circumcenter, the centroid, and the nine-point circle. Before that, you could probably mention some rations of distances. At that point, I'm wondering what else you could write in a tossup that would be convertible by a decent amount of teams. Not only that, but how many players who don't participate in math competitions would even get the tossup at the giveaway? Learning these things may also pose a problem, because if you don't actually use the Euler Line and the like in solving problems, would you just randomly decide to look up all these things about triangles? I doubt that I'd know what Menelaus's Theorem was if I hadn't seen it used before in contest problems.

I think that the unique thing about math in quiz bowl is that, in contrast to other subjects, what we cover in school doesn't give you much to write about. In school, most students probably learn a few definitions and formulas, and then just blindly use them in problems. This poses a problem to writing about more obscure things, because of the fact that most of those obscure things are never mentioned at all in class, and so you'd probably have to participate in math competitions or do contest problems to even know about those things. On the other hand, at least in what I've experienced in my first year of high school, other subjects like biology and history only need you to learn those definitions, names, etc., so you can learn a lot of stuff just through classes. Additionally, we almost always had questions asked in my classes that diverted into more obscure details, but this isn't very likely to happen in a math class, since the teacher has to stick to the material.

Because of how math is taught in high schools, I'm not actually sure how to expand the math canon. As explained above, I think we'd have to stick to the things that are actually talked about in high school classes. At that point, we need to find good, nontransparent clues of appropriate difficulty to use, which as mentioned, is quite challenging. I think we can slowly expand the math canon, but, unless high school math classes drastically change their manner of teaching, there's going to be a limit on what we can ask.
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Re: How small is the math canon?

Post by New York Undercover » Tue Jul 13, 2010 9:49 pm

Mewto55555 wrote:
Dominator wrote: (6) The entire high school mathematics curriculum
A lot of this seems like it would be hard to ask and keep it non-computational. With algebra, there's not a lot you can really do. Geometry has a lot of nice theorems with cool names, but most of those are not covered in class, meaning you'd be stuck asking about stuff like SAS similarity :sad:
This is an important point, I think. While in many other fields (history, literature, even science) there are definitely questions you can ask on material learned in middle school or possibly even middle school (american revolution, charles dickens, atom, for some terrible off my head answer lines), I'd say there is really almost nothing covered before geometry that can be converted to a good answer line. Examples I can think of that could work but seem terribly difficult are multiplication, fractions... The fact of the matter is that much of what is learned in math, in school, to any degree of depth (which is usually what is wanted for quizbowl) is computation. Certainly we don't want that, and we can say that the canon doesn't need to stick to the curriculum... but don't expect conversion to be high, and I'd say that what people learn will just be what they hear in quizbowl.

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Re: How small is the math canon?

Post by Dominator » Tue Jul 13, 2010 10:42 pm

I think some people are misinterpreting. Certainly I don't think I could tossup Stephen Smale just because he won a Fields medal. What I am saying, though, is that for the math canon to expand, the class of Fields medalists is a nice place to start as they are the analogy to Nobel winners in science and literature. Perhaps they could start appearing more in bonuses. Eventually we may even toss them up. And to the point of only a small positive fraction of Nobelists being in the canon, I would counter that a small positive fraction of Fields medalists should be. Right now that fraction is 0.

From the science questions I have seen, for example, a lot of the clues will never be touched in courses people take in school. The topics are covered, sure, but many of the specific clues will not be in any particular course. Similarly, for literature, any literature course will not cover anywhere close to the hundreds of novels in the NAQT canon. Playing in an orchestra will help you with the music canon, but you will not play nearly enough of the pieces to cover the entire canon. Success in quizbowl demands that students will study above and beyond their coursework. The same principles should apply in all subjects, math included.

I think I align with Kay on this one. There were a lot of bad habits in math question writing from years of comp math. As the canon expands, some questions will go dead. But if the questions represent good canonical knowledge, then it's still good quizbowl imo. Still, this transition should be gradual so that people still enjoy quizbowling.

[NOTE: I also think people should be careful not to fall into the "I can't think of more than two interesting clues on this topic off the top of my head so it's not tossupable" trap. For several of the topics upthread, I think you'd be surprised how many good clues can be written by someone who knows the subject better than you.]
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Re: How small is the math canon?

Post by kayli » Tue Jul 13, 2010 11:21 pm

xpmath wrote:
Arsonists Get All the Girls wrote: In general, I think math tossups should come from an expanded high school math curriculum. What I mean is that instead of stretching up towards higher college level math, we stretch out to include more obscure things from the high school curriculum (geometry, algebra, calculus, number theory, etc). This means that instead of writing a bonus part on analytic functions or partial diffrential equations, we start writing on stuff on Sophie Germain primes and Euler lines.
As much I'd love to see tossups written on the Euler line, I'm thinking that a lot of these obscure things might be too hard or might not have enough good clues to write a good question about. The simplest clue about the Euler line is probably the fact that it passes through the orthocenter, the circumcenter, the centroid, and the nine-point circle. Before that, you could probably mention some rations of distances. At that point, I'm wondering what else you could write in a tossup that would be convertible by a decent amount of teams. Not only that, but how many players who don't participate in math competitions would even get the tossup at the giveaway? Learning these things may also pose a problem, because if you don't actually use the Euler Line and the like in solving problems, would you just randomly decide to look up all these things about triangles? I doubt that I'd know what Menelaus's Theorem was if I hadn't seen it used before in contest problems.

I think that the unique thing about math in quiz bowl is that, in contrast to other subjects, what we cover in school doesn't give you much to write about. In school, most students probably learn a few definitions and formulas, and then just blindly use them in problems. This poses a problem to writing about more obscure things, because of the fact that most of those obscure things are never mentioned at all in class, and so you'd probably have to participate in math competitions or do contest problems to even know about those things. On the other hand, at least in what I've experienced in my first year of high school, other subjects like biology and history only need you to learn those definitions, names, etc., so you can learn a lot of stuff just through classes. Additionally, we almost always had questions asked in my classes that diverted into more obscure details, but this isn't very likely to happen in a math class, since the teacher has to stick to the material.

Because of how math is taught in high schools, I'm not actually sure how to expand the math canon. As explained above, I think we'd have to stick to the things that are actually talked about in high school classes. At that point, we need to find good, nontransparent clues of appropriate difficulty to use, which as mentioned, is quite challenging. I think we can slowly expand the math canon, but, unless high school math classes drastically change their manner of teaching, there's going to be a limit on what we can ask.
Stuff like Euler lines and such don't have to be the answer spaces to tossups. In fact, that specific example would be better used as a hard part of a bonus or as a lead-in. Basically, what I tried to say is that there's stuff out there besides what we learned in high school classes that could be totally viable questions or clues.

Dominator wrote:I think some people are misinterpreting. Certainly I don't think I could tossup Stephen Smale just because he won a Fields medal. What I am saying, though, is that for the math canon to expand, the class of Fields medalists is a nice place to start as they are the analogy to Nobel winners in science and literature. Perhaps they could start appearing more in bonuses. Eventually we may even toss them up. And to the point of only a small positive fraction of Nobelists being in the canon, I would counter that a small positive fraction of Fields medalists should be. Right now that fraction is 0.
I think a problem with the Fields Medalists is that a lot of their work requires a high level of prerequisite knowledge to even grasp; and if you don't understand what they did, you just start associating words and titles of papers with names which isn't good. Even a Fields Medalist common link tossup might be a little tough. I think math is unlike science or literature because it's so much more theoretical and often hard to grasp. With literature, it's pretty easy to understand someone's body of work. It's harder with science but still very possible (fiber optics, quarks, jumping genes... we can all sort of understand these things). However, the works of Field medalists are often really theoretical and technical and have crazy concepts no one in high school would have heard of much less understand. There might be some exceptions, but the only one I can really think of is Terrance Tao and maybe other number theorists because their work is somewhat understandable. All this isn't to say that it can't possibly work. I just think that there's a lot of difficulty in writing accessible questions on Fields Medalist winners. Maybe that will change though.

From the science questions I have seen, for example, a lot of the clues will never be touched in courses people take in school. The topics are covered, sure, but many of the specific clues will not be in any particular course. Similarly, for literature, any literature course will not cover anywhere close to the hundreds of novels in the NAQT canon. Playing in an orchestra will help you with the music canon, but you will not play nearly enough of the pieces to cover the entire canon. Success in quizbowl demands that students will study above and beyond their coursework. The same principles should apply in all subjects, math included.
I completely agree with you on this.
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Re: How small is the math canon?

Post by Mechanical Beasts » Tue Jul 13, 2010 11:32 pm

Dominator wrote:[NOTE: I also think people should be careful not to fall into the "I can't think of more than two interesting clues on this topic off the top of my head so it's not tossupable" trap. For several of the topics upthread, I think you'd be surprised how many good clues can be written by someone who knows the subject better than you.]
Well, "tossupable" means many things. I've had an okay education in college math. If I can't think of more than two interesting clues about a math tossup answer, then I sure hope that I'm good enough at math that fewer than 50% of high schoolers--by that logic--would be buzzing before the second-to-last clue. That's never acceptable for a pyramidal tossup.
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Re: How small is the math canon?

Post by Dominator » Tue Jul 13, 2010 11:40 pm

Crazy Andy Watkins wrote:If I can't think of more than two interesting clues about a math tossup answer, then...
I disagree with your logic. The difference is that people recognize clues they would not be able to think of, and recognition is what quizbowlers do. It's like P v. NP: easier to check an answer than come up with it.
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Re: How small is the math canon?

Post by Mechanical Beasts » Tue Jul 13, 2010 11:50 pm

Dominator wrote:
Crazy Andy Watkins wrote:If I can't think of more than two interesting clues about a math tossup answer, then...
I disagree with your logic. The difference is that people recognize clues they would not be able to think of, and recognition is what quizbowlers do. It's like P v. NP: easier to check an answer than come up with it.
Okay, that's fair; that's actually shorthand for more complicated actual thoughts like "even a tossup on Andrew Wiles would be 100% better if written on FLT instead, because at best the top 0.1% of quizbowlers know anything about anything Andrew Wiles has done save FLT; a tossup on someone like Smale should never happen because if I can only right now think of the phrase 'sphere eversion' I bet half of high schoolers won't recognize five clues, which is what you'd want for that sort of thing to be tossupable."
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Re: How small is the math canon?

Post by Mewto55555 » Wed Jul 14, 2010 2:48 pm

xpmath wrote:
Arsonists Get All the Girls wrote: In general, I think math tossups should come from an expanded high school math curriculum. What I mean is that instead of stretching up towards higher college level math, we stretch out to include more obscure things from the high school curriculum (geometry, algebra, calculus, number theory, etc). This means that instead of writing a bonus part on analytic functions or partial diffrential equations, we start writing on stuff on Sophie Germain primes and Euler lines.
As much I'd love to see tossups written on the Euler line, I'm thinking that a lot of these obscure things might be too hard or might not have enough good clues to write a good question about. The simplest clue about the Euler line is probably the fact that it passes through the orthocenter, the circumcenter, the centroid, and the nine-point circle. Before that, you could probably mention some rations of distances. At that point, I'm wondering what else you could write in a tossup that would be convertible by a decent amount of teams.
There's a lot more about the Euler line than just that; you could mention some of the other, slightly more obscure, points which lie on it, you can mention how due to the fact that the orthocenter, circumcenter, and centroid conincide in an equilateral triangle, it can't be determined, you can talk about how the incenter lies on it in isoceles triangles, etc.

Also, a bunch of teams aren't going to be able to convert tossups on INSERT BOOK HERE depending on the level of competition. Obviously no one is going to be asking about Nagel points or Lagrange Multipliers in a novice set.
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Re: How small is the math canon?

Post by Windmill Tump » Wed Jul 14, 2010 6:21 pm

Well, if you just want to make clues on something, there are lots of random bits of info that you could potentially turn into a clue; I was trying to think of basic clues that people would get by simply knowing about the Euler Line, instead of it being necessary to dive too deep into it. Again, if you just want to write clues, you can definitely write a bunch of clues. I think the difficulty in writing those tossups would be finding GOOD clues that people, outside of the group that participates in math competitions, would be able to get, unless you want to just let these math tossups be essentially a competition between those people.

As for the difficulty comment, the math in the novice sets wouldn't really be stuff that's not common the canon anyway - at that level, you'd have to ask about the usual high school class things, which are already in the canon. I was examining the math involved at the higher levels, where you'd still have the difficulty of finding clues good enough to write a good, pyramidal tossup. Bonuses would probably be easier to write, as you wouldn't have to come up with as many of those clues, though.

I guess I'd agree with some opinions expressed earlier in the thread, such as the one where math comp covered up a lot of the math canon, fooling people into believing that the canon is smaller than it truly is. I think we all agree that more math theory tossups can be written, but are probably more challenging to write than others, due to the creativity required in thinking of the clues. In any case, I'd really like to see the canon expand and I'd like to see more suggestions as to how to. Maybe someone could try to write a question on some interesting stuff that isn't really mentioned in high school, and we can see how it'd turn out?
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Re: How small is the math canon?

Post by Gautam » Wed Jul 14, 2010 8:46 pm

Dominator wrote:What I am saying, though, is that for the math canon to expand, the class of Fields medalists is a nice place to start as they are the analogy to Nobel winners in science and literature. Perhaps they could start appearing more in bonuses. Eventually we may even toss them up. And to the point of only a small positive fraction of Nobelists being in the canon, I would counter that a small positive fraction of Fields medalists should be. Right now that fraction is 0.
I want to point out that this seems to me a misconception of the how 'the canon' and canon expansion works. I invite you to read Andrew Hart's primer on the topic, especially the caveat in statement 2 G.

"External" canon expansion (which is what you seem to suggest) especially at the high school level, has been a pretty gradual process and I don't suspect that it will continue at an accelerating rate. A lot of the canon expansion you see is internal - i.e. things which once came up as giveaways or late middle clues are now being tossed up, and so on. I would be wary of tossing up these fields medalists even the upper difficulty limits of the collegiate levels, since those questions are either going to be unanswered 90% of the time or they're going to lead to unfortunate buzzer races.

Let's take the example of Stephen Smale: My archive indicates that Stephen Smale has been mentioned 4 times: once in a poorly conceived submission on the h-cobordism theorem (that most likely never made it to the set,) once in a Stephen Smale theme bonus I wrote* for ACF Nationals 2009, once as an answer line in FIST 2009, and once as an answer line in Science Monstrosity 2005. These are about the most difficult tournaments you can find out there. Regardless of how many people end up writing questions on Smale in the upcoming year, I don't think he'll ever be an easy part of a bonus or a tossup answer, unlike the evolutionary path you suggest.

*The only reason why I ever wrote the Stephen Smale bonus was because I had seen the absurd submission on the h-cobordism theorem and I thought it might be something interesting to explore. I remember seeing Stanford play that bonus and they at most 20d it despite having a couple of people who're fluent in math.
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Re: How small is the math canon?

Post by Dominator » Thu Jul 15, 2010 10:10 am

I'm not sure where people got the idea that I was advocating a tossup like "...FTP name this American mathematician who won a Fields Medal in 1928" A: Douglas. What I meant by "the canon" was answer spaces for tossups AND bonuses AS WELL AS clue spaces for both. I never said anything about tossing Fields Medalists up. A lot of what I suggested above would make for good new clues even to more traditional answers.
Arsonists Get All the Girls wrote: I think a problem with the Fields Medalists is that a lot of their work requires a high level of prerequisite knowledge to even grasp; and if you don't understand what they did, you just start associating words and titles of papers with names which isn't good.
I agree with this to some extent. I certainly don't think that every Fields Medalist is appropriate to ask about. However, think of Nobel laureates. Are 10% of Nobel laureates in the science and literature canon? From my experience, I think that is the case. I do think 10% of Fields Medalists could have their work understood, to a reasonable high-school level, by quizbowlers. Keep in mind that there have only been 46 Fields medalists so far, so 10% is only a handful. A great example of how Fields Medalists could fit into the canon is Terrence Tao. He is, by far, the best mathematician in the world. He has won every major award and he is very visible in the public eye (for a mathematician). Quizbowlers can understand the statement of the Green-Tao theorem. They will never understand the 50-page proof using ergodic theory, or any of the other really high-level work he does in analytic number theory, computer science, Fourier analysis, combinatorics, and any number of other fields. There are other examples of Fields Medalists (and non-Fields Medalists, for that matter) whose work high schoolers could understand the results of without needing to know the very technical details.
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Re: How small is the math canon?

Post by Mechanical Beasts » Thu Jul 15, 2010 10:23 am

Well, I think there could be plenty of tossups on primes that use the Green-Tao theorem as a clue. That's perfectly fair to say.
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Re: How small is the math canon?

Post by Windmill Tump » Thu Jul 15, 2010 10:25 am

Dominator wrote:A lot of what I suggested above would make for good new clues even to more traditional answers.
This I can kinda agree with. I think it's still very possible to take a common math tossup answer and find some new clues. It's just pretty hard to write a whole tossup on something that's not so common.
Dominator wrote: Quizbowlers can understand the statement of the Green-Tao theorem. They will never understand the 50-page proof using ergodic theory, or any of the other really high-level work he does in analytic number theory, computer science, Fourier analysis, combinatorics, and any number of other fields
And this is why I think it's a lot easier to write new bonus parts for math than tossups. It's not nearly as difficult to find one or two things that you can mention in a bonus which are comprehensible as it is for a tossup. In fact, I'm pretty sure I've seen the Green-Tao theorem multiple times in bonuses.

So I guess I'm agreeing that bonuses and actual clues could be written. Still not sure about how effective the tossup writing will be.

EDIT: Hmm darn I got beaten to the post. I definitely agree with the above poster; mentioning the Green-Tao theorem in a prime tossup is a good example of what you said about clues for traditional answers.
EDIT2: Why do I always mix other things up with tossup. Replaced bonus with tossup in my first edit.
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Re: How small is the math canon?

Post by Gautam » Thu Jul 15, 2010 4:32 pm

Dominator wrote:I'm not sure where people got the idea that I was advocating a tossup like "...FTP name this American mathematician who won a Fields Medal in 1928" A: Douglas.
We know you aren't advocating anything like this. Nobody is claiming so.

However,
Dominator wrote:I never said anything about tossing Fields Medalists up.
seems to directly contradict
Dominator wrote:...the class of Fields medalists is a nice place to start as they are the analogy to Nobel winners in science and literature. Perhaps they could start appearing more in bonuses. Eventually we may even toss them up.
(bolding mine).

Anyway...
Dominator wrote:A lot of what I suggested above would make for good new clues even to more traditional answers.
Is true and does happen about as frequently as it should be happening: they probably come in a question or 2 per year. It's easy to overlook this fact, given that there are litearally thousands of HS questions being produced, but I'm willing to bet that the Green Tao theorem has been used as a clue before.

Lastly,
Dominator wrote:I agree with this to some extent. I certainly don't think that every Fields Medalist is appropriate to ask about. However, think of Nobel laureates. Are 10% of Nobel laureates in the science and literature canon? From my experience, I think that is the case.
They're in the canon because they come up frequently enough in high school courses!

The work of Van't Hoff, Arrhenius, Haber, Nernst, Pauling, Mulliken, Ramsay, Rutherford (8 out of approx. 100 awards is close enough to the magic 10% number) keeps coming up in tournaments and people keep asking about them simply because the people and their theories are covered (to some extent if not in depth) in regular and AP Chemistry classes. You can't claim that the same is true with Field's medalists!

With physics, you have people like the Curies, Roentgen, van der Waals, Michelson, Thomson, Bohr, Einstein, Pauli, Miliken, Chadwick, Fermi, etc who fall in the same boat. Take a standard HS physics textbook such as Serway and Faughn (or whatever you wish) and you'll find those people and their work discussed in that book.

Here's a not-so-outlandish claim: What the Curies and the Paulings are to physics and chemistry, the Eulers are to math. There are plenty of mathematicians (Leibniz, Fermat, Gauss, Bernoullis, etc.) who enjoy the same rock-star status in the high school math canon that the Nobel laureates in chemistry and physics do, and there's good reason for it: all of those mathematicians worked on stuff that comes up really frequently in high-school math.

In a similar vein my claim can be extended as, what Terence Tao is to math, Robert Woodward is to chemistry and 't Hooft is to physics; people might be able to appreciate their contributions, but to be able to understand their work requires a substantially deeper understanding of the subject matter, and will pretty much be impossible at the HS level.

I apologize if I'm misrepresenting anybody in this conversation. Good conversation to have regardless.

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Re: How small is the math canon?

Post by Mechanical Beasts » Thu Jul 15, 2010 4:38 pm

That's pretty accurate: the most recent contributions of chemists are never ever going to be part of the high school canon because they're so far removed from even an undergraduate college curriculum. Andrew Wiles is an odd exception because of his celebrity status--the Nova episode, etc.
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Re: How small is the math canon?

Post by kayli » Thu Jul 15, 2010 4:51 pm

There's one quirk about the Fields Medalists that doesn't apply to Nobel laureates. Fields Medalists have to be 40 years of age or younger. So, there are a lot of really famous modern mathematicians like Kurt Godel, David Hilbert, etc. that aren't Fields Medalists. I think the works of modern mathematicians would be great clues, and I think some difficulty-appropriate tossups on modern mathematicians would be a good idea too.
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Re: How small is the math canon?

Post by cvdwightw » Thu Jul 15, 2010 5:31 pm

Arsonists Get All the Girls wrote:difficulty-appropriate tossups on modern mathematicians
Has nothing in this thread demonstrated to you that the above phrase is an oxymoron at anything below open post-nationals level?
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Re: How small is the math canon?

Post by kayli » Thu Jul 15, 2010 5:42 pm

cvdwightw wrote:
Arsonists Get All the Girls wrote:difficulty-appropriate tossups on modern mathematicians
Has nothing in this thread demonstrated to you that the above phrase is an oxymoron at anything below open post-nationals level?
What's wrong with a nationals level tossup on Kurt Godel?
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Re: How small is the math canon?

Post by Auks Ran Ova » Thu Jul 15, 2010 6:37 pm

Arsonists Get All the Girls wrote:
cvdwightw wrote:
Arsonists Get All the Girls wrote:difficulty-appropriate tossups on modern mathematicians
Has nothing in this thread demonstrated to you that the above phrase is an oxymoron at anything below open post-nationals level?
What's wrong with a nationals level tossup on Kurt Godel?
Nothing, but he died 32 years ago, and most of his most famous work (correct me if I'm wrong) was done in the 1930s. "Modern" usually implies (and, as far as I can tell, thus far in this thread has been used to mean) "contemporary".
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Re: How small is the math canon?

Post by Mewto55555 » Thu Jul 15, 2010 7:19 pm

What about Erdos?
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Re: How small is the math canon?

Post by Mechanical Beasts » Thu Jul 15, 2010 7:22 pm

Mewto55555 wrote:What about Erdos?
Far more famous for his ludicrous life story than for his mathematics, much of which (actually, everything I've encountered) is well out of the reach of high schoolers--for the most part, anyway.
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Re: How small is the math canon?

Post by Dominator » Thu Jul 15, 2010 7:25 pm

Ukonvasara wrote: "Modern" usually implies (and, as far as I can tell, thus far in this thread has been used to mean) "contemporary".
Considering that most high school math courses include only work done centuries ago (algebra is all 1000 years old, geometry older, and the calculus is about 200 but tends to give all credit [undeservedly] to Newton and Leibniz who worked about 350 years ago), I would consider the 1930s relatively modern.
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Re: How small is the math canon?

Post by Dominator » Thu Jul 15, 2010 7:28 pm

Crazy Andy Watkins wrote:
Mewto55555 wrote:What about Erdos?
Far more famous for his ludicrous life story than for his mathematics, much of which (actually, everything I've encountered) is well out of the reach of high schoolers--for the most part, anyway.
And this is really a problem. I've studied a lot of the work of Erdos. He did a lot of stuff that is beyond reasonable high school level, but he also did a lot that is within reach. With over 1500 papers, he can do a lot of both. Also, there are (at least) two "popular math" biographies about him. He should be in the canon somehow.
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Re: How small is the math canon?

Post by Geringer » Fri Jul 16, 2010 3:58 am

Dominator wrote:
Crazy Andy Watkins wrote:
Mewto55555 wrote:What about Erdos?
Far more famous for his ludicrous life story than for his mathematics, much of which (actually, everything I've encountered) is well out of the reach of high schoolers--for the most part, anyway.
And this is really a problem. I've studied a lot of the work of Erdos. He did a lot of stuff that is beyond reasonable high school level, but he also did a lot that is within reach. With over 1500 papers, he can do a lot of both. Also, there are (at least) two "popular math" biographies about him. He should be in the canon somehow.
Yeah, I feel like the giveaway for this question would have much more to do with how he was sort of itinerant and published lots of papers and his namesake number. That's dangerously close to the "science biography" that people really hate.

Could a tournament benefit from 0/1.5 or 0/2 math instead of 1/1? I feel like this thread has established that the tossup canon for math isn't nearly as large as other subjects but well-written bonuses could test knowledge and help expand the canon (if my knowledge of canon-expansion is accurate).
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Re: How small is the math canon?

Post by Carambola! » Fri Jul 16, 2010 1:17 pm

Psychology and the Social Sciences could probably use the distro more, given that they cover a wider range of topics and could expand their canons a little easier that way.
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Re: How small is the math canon?

Post by Deviant Insider » Fri Jul 16, 2010 5:25 pm

Despite the fact that writing a lot of math tossups is difficult, it's not like it is impossible to write some. To make math 0.5/1 or something along those lines does not require a huge canon, and if you have a writer on board who is willing and able, 1/1 is certainly possible.

Scobol Solo will have 40 noncomputational math tossups this year. If anybody not in high school wants to see our currently messy work in progress, send me an email.

EDIT: Also, math questions probably are more interesting and encouraging to novice teams than Social Science.
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Re: How small is the math canon?

Post by kayli » Fri Jul 16, 2010 10:25 pm

I think 1/1 is fair. Even though quizbowl isn't driven by the curriculum, I think the fact that everyone has to take ~4 years of math in high school is pretty notable. Also, math is pretty important as a subject. If physics, chemistry, and biology (all courses which are pretty much taken universally less than math) can get 1/1, I don't see why math should be the same.
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Re: How small is the math canon?

Post by DongDonger » Sat Jul 17, 2010 3:04 am

Dominator wrote:
Crazy Andy Watkins wrote:
Mewto55555 wrote:What about Erdos?
Far more famous for his ludicrous life story than for his mathematics, much of which (actually, everything I've encountered) is well out of the reach of high schoolers--for the most part, anyway.
And this is really a problem. I've studied a lot of the work of Erdos. He did a lot of stuff that is beyond reasonable high school level, but he also did a lot that is within reach. With over 1500 papers, he can do a lot of both. Also, there are (at least) two "popular math" biographies about him. He should be in the canon somehow.
I always thought Erdos numbers were quite well known, so there's something you could work with.
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Re: How small is the math canon?

Post by kayli » Sat Jul 17, 2010 3:29 am

DongDonger wrote:
Dominator wrote:
Crazy Andy Watkins wrote:
Mewto55555 wrote:What about Erdos?
Far more famous for his ludicrous life story than for his mathematics, much of which (actually, everything I've encountered) is well out of the reach of high schoolers--for the most part, anyway.
And this is really a problem. I've studied a lot of the work of Erdos. He did a lot of stuff that is beyond reasonable high school level, but he also did a lot that is within reach. With over 1500 papers, he can do a lot of both. Also, there are (at least) two "popular math" biographies about him. He should be in the canon somehow.
I always thought Erdos numbers were quite well known, so there's something you could work with.
The problem is that Erdos numbers aren't of any significance mathematically. It doesn't actually deal with anything Erdos has done... except author a lot of papers with people. It's just a fun, cool thing.
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Re: How small is the math canon?

Post by Deviant Insider » Sat Jul 17, 2010 10:18 am

I think that many good tossup answers are very elementary concepts: planes, slope, prime numbers, etc. It also is possible to write questions with numerical answers that are not computational. I'm not saying that all math tossups should be this way, but I believe that overall this is the best source of tossup answers.

The problem with writing a tossup with the answer Erdős is that few people are going to get it until near the end, and the end probably will focus on stuff that isn't all that mathematical, such as amphetamines, his sayings, the number of papers he wrote, and Erdős numbers. A decent number of teams won't get it even then.

On the other hand, you can write a tossup with the answer prime numbers, sets, or a few other things that use clues based on the work of Erdős (and of course the list gets much longer if you include other recent mathematicians). That way, you can include some late middle clues and a giveaway that are mathematical and that some people will understand (which should be the two basic requirements for a giveaway in a math tossup).

I think this is analogous to what we do with other subjects. We don't write tossups with the answer Devil's Den, but we use that as a clue for the Battle of Gettysburg. I think for similar reasons, we might want to avoid tossups with the answer Euler line, but Euler line could be used as a clue for Euler (whose work is more relevant to the high school curriculum than Erdős', and thus easier to give away using math), triangle, line, and possibly a few other things.
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Re: How small is the math canon?

Post by Down and out in Quintana Roo » Sat Jul 17, 2010 2:41 pm

DongDonger wrote:
Dominator wrote:
Crazy Andy Watkins wrote:
Mewto55555 wrote:What about Erdos?
Far more famous for his ludicrous life story than for his mathematics, much of which (actually, everything I've encountered) is well out of the reach of high schoolers--for the most part, anyway.
And this is really a problem. I've studied a lot of the work of Erdos. He did a lot of stuff that is beyond reasonable high school level, but he also did a lot that is within reach. With over 1500 papers, he can do a lot of both. Also, there are (at least) two "popular math" biographies about him. He should be in the canon somehow.
I always thought Erdos numbers were quite well known, so there's something you could work with.
I enjoy math, was good at it, and was a math major for a year and a half in college.

And i have never heard of these things.
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Re: How small is the math canon?

Post by Auks Ran Ova » Sat Jul 17, 2010 2:52 pm

Carangoides ciliarius wrote:
DongDonger wrote:
Dominator wrote:
Crazy Andy Watkins wrote:
Mewto55555 wrote:What about Erdos?
Far more famous for his ludicrous life story than for his mathematics, much of which (actually, everything I've encountered) is well out of the reach of high schoolers--for the most part, anyway.
And this is really a problem. I've studied a lot of the work of Erdos. He did a lot of stuff that is beyond reasonable high school level, but he also did a lot that is within reach. With over 1500 papers, he can do a lot of both. Also, there are (at least) two "popular math" biographies about him. He should be in the canon somehow.
I always thought Erdos numbers were quite well known, so there's something you could work with.
I enjoy math, was good at it, and was a math major for a year and a half in college.

And i have never heard of these things.
You assume they have anything to do with actual math!
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Re: How small is the math canon?

Post by Down and out in Quintana Roo » Sat Jul 17, 2010 2:59 pm

Ukonvasara wrote:
Carangoides ciliarius wrote:
DongDonger wrote:
Dominator wrote:
Crazy Andy Watkins wrote:
Mewto55555 wrote:What about Erdos?
Far more famous for his ludicrous life story than for his mathematics, much of which (actually, everything I've encountered) is well out of the reach of high schoolers--for the most part, anyway.
And this is really a problem. I've studied a lot of the work of Erdos. He did a lot of stuff that is beyond reasonable high school level, but he also did a lot that is within reach. With over 1500 papers, he can do a lot of both. Also, there are (at least) two "popular math" biographies about him. He should be in the canon somehow.
I always thought Erdos numbers were quite well known, so there's something you could work with.
I enjoy math, was good at it, and was a math major for a year and a half in college.

And i have never heard of these things.
You assume they have anything to do with actual math!
Ha, then why the hell are we talking about them (and all this other stuff i don't understand/know) in the "math canon"?
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Re: How small is the math canon?

Post by Captain Sinico » Sat Jul 17, 2010 3:09 pm

I think Coach Reinstein has the right idea here. There's plenty of stuff to fill out a year's tournaments without resorting to un-mathematical answers like Erdős numbers (I guess people, perhaps just Auroni, don't know that they're really not mathematical at all) or other biographical junk. Rather, you can include the work of Erdős or your favorite Fields Medal winner or whatever in a tossup on a more general, more accessible topic to which their work pertains.

To take an example from upthread, one could write a question on closed or compact or spheres rather than Grigori Perelman or the inscrutable details of his book-length, opaque proof of of the Poincaré conjecure, or even the conjecture itself; simply use the latter as clues. In an analogous case from my own work, I'd probably have a much better time with questions on matrices or commutation or the trace than I would on Lie algebra representations or the Lie bracket or the Killing form.

M
Mike Sorice
Coach, Centennial High School of Champaign, IL (2014-) & Team Illinois (2016-2018)
Alumnus, Illinois ABT (2000-2002; 2003-2009) & Fenwick Scholastic Bowl (1999-2000)
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Captain Sinico
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Re: How small is the math canon?

Post by Captain Sinico » Sat Jul 17, 2010 3:20 pm

Erdös' work is, to my knowledge, serious math; he was certainly a serious mathematician. It also happens that he led an interesting, curious life that makes him the subject of popular interest, at least among some, c.f. Erdös numbers, which were/are in no way the work of Erdös per se. Consequently, it's tempting to say, as some here have seemingly said, "Well, let's write Erdös semi-biography and call it math because people might know him and he's a mathematician."
I'd say the problem is that idea is that Erdös' work is by and large beyond what people will know or understand such that we'd be better served with questions on the things Erdös was talking about, as questions on Erdös himself will inevitably be very difficult and fail to interface with what people understand, lack proper mathematical character, or both.
I'd suggest that epi-academic material like Erdös biography doesn't necessarily belong entirely outside the game. However, its place is definitely not as math. I'd say that's one class of thing people ought write for "your choice/other" in academic tournaments, rather than trash.

M
Mike Sorice
Coach, Centennial High School of Champaign, IL (2014-) & Team Illinois (2016-2018)
Alumnus, Illinois ABT (2000-2002; 2003-2009) & Fenwick Scholastic Bowl (1999-2000)
ACF
IHSSBCA
PACE

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