Math Computation, Round 23748234

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Re: Math Computation, Round 23748234

Post by Deviant Insider »

Sir Thopas wrote:
Shcool wrote:However, some learning took place on that question.
I gotta say, I preferred having our math teacher advisor teach us about Lebesgue integrals at HFT than I would if he had taught us a parlor trick for adding consecutive numbers quickly. One is actually useful in the real world; the other, kinda neat for a few minutes.
That problem is a Lebesque Integral.
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Re: Math Computation, Round 23748234

Post by jonah »

Sir Thopas wrote:
jonah wrote:
Sir Thopas wrote:
Shcool wrote:However, some learning took place on that question.
I gotta say, I preferred having our math teacher advisor teach us about Lebesgue integrals at HFT than I would if he had taught us a parlor trick for adding consecutive numbers quickly. One is actually useful in the real world; the other, kinda neat for a few minutes.
Wait, which are you saying is useful and which is neat for a few minutes?
With respect to Gauss, he did plenty more important things than figure out how to add numbers quickly.
You didn't answer my question.

I'm going to assume you mean that Lebesgue integration is useful in the real world while adding numbers quickly is neat for a few minutes, because it seems to support whatever your post's point was. Going forth with that, I think that statement is indefensible. Lebesgue integration is more interesting to me apparently you and probably lots of other people, but the comparative utility is almost beyond question: I can state with near-perfect confidence that whether in terms of the number of people who use each in their daily lives, the gross number of times each is applied, or any other sensible measure, addition (including fast addition) is much more useful (i.e, it is a more valuable skill).

Of course, quizbowl's goals do not necessarily include emphasizing practically-applicable knowledge, though this is not universally accepted. But as to the question of utility in real life, I think you're dead wrong.
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Re: Math Computation, Round 23748234

Post by Sir Thopas »

jonah wrote:Of course, quizbowl's goals do not necessarily include emphasizing practically-applicable knowledge, though this is not universally accepted. But as to the question of utility in real life, I think you're dead wrong.
By "real life", amusingly, I meant "the study of mathematics". Lumps taken accordingly, although I have never needed to use Gauss's method outside of contest math, and can't foresee such a situation.
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Re: Math Computation, Round 23748234

Post by Tower Monarch »

jonah wrote:
Sir Thopas wrote:stuff
You didn't answer my question.

I'm going to assume you mean that Lebesgue integration is useful in the real world while adding numbers quickly is neat for a few minutes, because it seems to support whatever your post's point was. Going forth with that, I think that statement is indefensible. Lebesgue integration is more interesting to me apparently you and probably lots of other people, but the comparative utility is almost beyond question: I can state with near-perfect confidence that whether in terms of the number of people who use each in their daily lives, the gross number of times each is applied, or any other sensible measure, addition (including fast addition) is much more useful (i.e, it is a more valuable skill).

Of course, quizbowl's goals do not necessarily include emphasizing practically-applicable knowledge, though this is not universally accepted. But as to the question of utility in real life, I think you're dead wrong.
The still valid point is that everybody has a cell phone calculator that can add numbers just as quickly, whereas there are only a few computer applications offering Lebesgue integration. By that metric, the adding is actually a less useful skill.
Andy called for an example of an academic studying faster computation methods: guess what? they do study that in computer science to make it so that the average person never has to learn quick math.
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Re: Math Computation, Round 23748234

Post by Mechanical Beasts »

kldaace wrote:To Guy, I'd argue that a lot of those tricks are important in mathematics. The adding of a sequence is a central tenant of good math. In fact, many people argue that you should learn such tricks well before calculus if you really want to understand/master math. A lot of these tricks have much wider applications for the general population anyways.

I believe that such things should be tested more so than things like Cauchy's Residue Theorem. After all, it's more accessible and has wider immediate applications. I'm not the only one who thinks that one should master such arts before learning calculus. The entire (or at least majority) of the Art of Problem Solving community seems to agree.

Viva discrete math!
There are bunches of people who are quite interested in "good math" here on these boards who are very interested in summing sequences but who are terribly disinterested in adding the numbers one through eleven very quickly. I doubt that many of those people buy into the false dilemma that you pose: that either one asks math calculation questions on discrete math (deep knowledge of which, you and I agree is more important for high school; after all, next to no calculus theory is taught in the supermajority of high school classes) that rely on tricks or one asks theory questions about calculus (and a theorem that I think I'll be covering next term in complex analysis, specifically).

EDIT: to respond to Cameron's point, I actually had a TF last term who does fairly similar stuff, and certainly there's an element of formal theory in there; the faster your numerical method for solving an ode converges, for example, the fewer terms you need to add and the faster it goes, et cetera. But as you pointed out, none of these numerical methods are meant to be done by hand, and they certainly aren't giving out degrees to that dude who taught me a fast way to multiply numbers by 49.
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Re: Math Computation, Round 23748234

Post by jonah »

Tower Monarch wrote:
jonah wrote:
Sir Thopas wrote:stuff
You didn't answer my question.

I'm going to assume you mean that Lebesgue integration is useful in the real world while adding numbers quickly is neat for a few minutes, because it seems to support whatever your post's point was. Going forth with that, I think that statement is indefensible. Lebesgue integration is more interesting to me apparently you and probably lots of other people, but the comparative utility is almost beyond question: I can state with near-perfect confidence that whether in terms of the number of people who use each in their daily lives, the gross number of times each is applied, or any other sensible measure, addition (including fast addition) is much more useful (i.e, it is a more valuable skill).

Of course, quizbowl's goals do not necessarily include emphasizing practically-applicable knowledge, though this is not universally accepted. But as to the question of utility in real life, I think you're dead wrong.
The still valid point is that everybody has a cell phone calculator that can add numbers just as quickly, whereas there are only a few computer applications offering Lebesgue integration. By that metric, the adding is actually a less useful skill.
Andy called for an example of an academic studying faster computation methods: guess what? they do study that in computer science to make it so that the average person never has to learn quick math.
The difference I am pointing out is one between academic/theoretical utility (where Lebesgue integration wins) and "real life" utility as Guy mentioned (where I contend addition wins by a landslide). I agree that with the proliferation of computers, calculators, and cell phones that embody both the need for people to perform addition and similarly rudimentary operations is reduced, though it still exists; that's just not my point.
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Re: Math Computation, Round 23748234

Post by kayli »

To reply to Andrew, I agree that summing one to eleven quickly isn't necessarily the best thing to test. Perhaps one to twenty? I believe that there is a good core of mathematical canon that tests useful core mathematical theories that aren't to the level of quizbowl. Sorry if I misunderstood your post.
Tower Monarch wrote:
jonah wrote:
Sir Thopas wrote:stuff
You didn't answer my question.

I'm going to assume you mean that Lebesgue integration is useful in the real world while adding numbers quickly is neat for a few minutes, because it seems to support whatever your post's point was. Going forth with that, I think that statement is indefensible. Lebesgue integration is more interesting to me apparently you and probably lots of other people, but the comparative utility is almost beyond question: I can state with near-perfect confidence that whether in terms of the number of people who use each in their daily lives, the gross number of times each is applied, or any other sensible measure, addition (including fast addition) is much more useful (i.e, it is a more valuable skill).

Of course, quizbowl's goals do not necessarily include emphasizing practically-applicable knowledge, though this is not universally accepted. But as to the question of utility in real life, I think you're dead wrong.
The still valid point is that everybody has a cell phone calculator that can add numbers just as quickly, whereas there are only a few computer applications offering Lebesgue integration. By that metric, the adding is actually a less useful skill.
Andy called for an example of an academic studying faster computation methods: guess what? they do study that in computer science to make it so that the average person never has to learn quick math.
So, multiplication tables are now obsolete. Technology has replaced actual math and thinking. What if I wanted to sum 7+14+21+...+14000? Do I jam them into my calculator? Or do I think? I don't think that series is trivial math. It is the application of one of THE MOST IMPORTANT concepts in mathematics. Monkeys can punch digits into calculators, humans can problem solve and find patterns.
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Re: Math Computation, Round 23748234

Post by Tower Monarch »

kldaace wrote:To Guy, I'd argue that a lot of those tricks are important in mathematics. The adding of a sequence is a central tenant of good math. In fact, many people argue that you should learn such tricks well before calculus if you really want to understand/master math. A lot of these tricks have much wider applications for the general population anyways.

I believe that such things should be tested more so than things like Cauchy's Residue Theorem. After all, it's more accessible and has wider immediate applications. I'm not the only one who thinks that one should master such arts before learning calculus. The entire (or at least majority) of the Art of Problem Solving community seems to agree.

Viva discrete math!
Okay, I would go along with this with some caveats. These things are hard (referring to percentage of teams that understand their basic definitions as there are an infinite number of named tricks) as answers and therefore should really only be clues (in some cases bonus parts).
The way I see it is that the types of named theorems and the like that Olympiad-level math competitors use has a similar place in high school quizbowl as organic reactions have in college quizbowl - they are something that is researched, often outside the classroom, by many people interested in the subject. In college, there is the issue of writing so much on organic chemistry reactions that people come away with (usually rightly) the feeling that most of the clues (or even the answer) were not significant to the subject (chemistry; math in analogy). But just as college quizbowl would not accurately reflect chemists' knowledge without some mention of reactions, high school quizbowl must reserve a place for this field.
Again, I stress that individual bonus parts are more accessible and acceptable here (the early clues to a hypothetical Wilson's Theorem tossup are probably ungettable and definitely not academically significant). Looking through the AoPS Volume 2 text I happen to have on hand, I see plenty of things that could reasonably be clues or bonus answers at various levels of high school play (I'll note that some already do with some frequency):
Cauchy's Inequality
Wilson's Theorem
Pell Equation
Fermat's Theorem (notably the non-proof/history side of it)
Totient Function (hey, this is where I have evidence of the "probably not the best tossup idea")
Brahmagupta's formula
Ceva's Theorem
DeMoivre's Theorem
Euler's formula
Four-Color Problem
Heron's Formula
Pick's Theorem
(Open up the index to this volume and you can quickly quadruple this list, relate these to a common answer like "circle," "triangle," "prime," etc, and you have as many as 20 nonoverlapping theory tossups that are gettable and reward reasonable and respectable knowledge)
I would very much like to see Discrete Math and Number Theory (these are the two fields for which the applies the most, but others do somewhat) come up as they are read about by interested students outside of school (arguably more often than history of art) and are frequently offered in high school (just as often as the history of art).
On a slightly related note, I've been thinking about writing a science subject tournament around NSC level (that could be used to help with NASAT-decision procedures), which would likely be around 15-20% math (at least 8-12%; theory only, no computation, even in bonuses). If coaches and college students feel like sending me hybrid computation-theory questions, or other such math experiments, I would certainly consider them (please don't send anything until I finally decide to do this, likely in the winter).
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Re: Math Computation, Round 23748234

Post by theMoMA »

kldaace wrote:Monkeys can punch digits into calculators, humans can problem solve and find patterns.
And quizbowl players can play quizbowl, which is a game inherently at odds with mathematical pattern-finding.
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Re: Math Computation, Round 23748234

Post by Cheynem »

Clearly we need a contest that will allow computational monkeys an opportunity to play. A league where apes evolved from men, if you will.
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Re: Math Computation, Round 23748234

Post by Mechanical Beasts »

kldaace wrote:To reply to Andrew, I agree that summing one to eleven quickly isn't necessarily the best thing to test. Perhaps one to twenty? I believe that there is a good core of mathematical canon that tests useful core mathematical theories that aren't to the level of quizbowl. Sorry if I misunderstood your post.
Apparently. I don't think a good test of how well you understand discrete math is how fast you can sum 1 to n for any n. I don't think that computation speed has anything to do with understanding.
kldaace wrote:So, multiplication tables are now obsolete. Technology has replaced actual math and thinking. What if I wanted to sum 7+14+21+...+14000? Do I jam them into my calculator? Or do I think? I don't think that series is trivial math. It is the application of one of THE MOST IMPORTANT concepts in mathematics. Monkeys can punch digits into calculators, humans can problem solve and find patterns.
Of course that series is trivial; I multiply seven by the sum of one to two thousand, which is as easy to compute as the sum from one to twenty. How on earth are you saying that this is not trivial math when everyone agrees that the example with the first eleven numbers divisible by eleven is totally trivial? It's the same problem! The other one is, in fact, better quizbowl because it tests knowledge of what the word "divisible" means!

Things can be both trivial and important Napoleon's age when he was defeated at Waterloo may well have been important; maybe he was constipated that morning. It's also pretty trivial. Moreover, I think that while summing series is, indeed, important, summing that series faster than the other seven people in the room is not, and never will be, important.

Here's the issue: any pattern or problem-solving exercise that can be done in a quizbowl timescale is trivial.
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Re: Math Computation, Round 23748234

Post by Rococo A Go Go »

kldaace wrote:Monkeys can punch digits into calculators, humans can problem solve and find patterns.
Quizbowl is not the place to test any one of those skills.
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Re: Math Computation, Round 23748234

Post by kayli »

everyday847 wrote:
kldaace wrote:To reply to Andrew, I agree that summing one to eleven quickly isn't necessarily the best thing to test. Perhaps one to twenty? I believe that there is a good core of mathematical canon that tests useful core mathematical theories that aren't to the level of quizbowl. Sorry if I misunderstood your post.
Apparently. I don't think a good test of how well you understand discrete math is how fast you can sum 1 to n for any n. I don't think that computation speed has anything to do with understanding.
kldaace wrote:So, multiplication tables are now obsolete. Technology has replaced actual math and thinking. What if I wanted to sum 7+14+21+...+14000? Do I jam them into my calculator? Or do I think? I don't think that series is trivial math. It is the application of one of THE MOST IMPORTANT concepts in mathematics. Monkeys can punch digits into calculators, humans can problem solve and find patterns.
Of course that series is trivial; I multiply seven by the sum of one to two thousand, which is as easy to compute as the sum from one to twenty. How on earth are you saying that this is not trivial math when everyone agrees that the example with the first eleven numbers divisible by eleven is totally trivial? It's the same problem! The other one is, in fact, better quizbowl because it tests knowledge of what the word "divisible" means!

Things can be both trivial and important Napoleon's age when he was defeated at Waterloo may well have been important; maybe he was constipated that morning. It's also pretty trivial. Moreover, I think that while summing series is, indeed, important, summing that series faster than the other seven people in the room is not, and never will be, important.

Here's the issue: any pattern or problem-solving exercise that can be done in a quizbowl timescale is trivial.
Okay, well, I think I was under the impression that the reason it wasn't important was that calculators had overridden the need for finding the sum of a sequence. My bad. From what it seemed, some people were saying that learning how to sum consecutive numbers was unimportant--that one to eleven can be easily punched into a calculator and hence knowing how to find the sum of consecutive numbers is kitsch at best. I think I may have also misunderstood the fact that pretty much every student knows the formula for such a sum also. Again, my bad.
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Re: Math Computation, Round 23748234

Post by theMoMA »

Also, people aren't arguing that computational math skills aren't important in an abstract sense. They are just arguing that the information tested for in computational math questions is inherently different from all other quizbowl questions, and most importantly, that it cannot fit within the parameters of the game. There are plenty of important academic things that just don't lend themselves well to quizbowl, which isn't to say that they're unimportant.
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Re: Math Computation, Round 23748234

Post by mithokie »

kldaace wrote:I think I may have also misunderstood the fact that pretty much every student knows the formula for such a sum also. Again, my bad.
I think this is actually where one of the big mis-understandings occurs. As a math teacher at a small rural public school, I believe that the majority of students do NOT know that formula. On the other hand, I do believe that the majority of quizbowl playing students do know the formula.

I do not think such a question makes a good toss-up, but I believe that would be fine as a bonus part along the lines of Find the Sum - Gauss - Triangular Numbers (Would this be a bad idea as a bonus? I am not a question writer.)
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Re: Math Computation, Round 23748234

Post by cvdwightw »

AHS Eagles wrote:I do not think such a question makes a good toss-up, but I believe that would be fine as a bonus part along the lines of Find the Sum - Gauss - Triangular Numbers (Would this be a bad idea as a bonus? I am not a question writer.)
This would probably be fine as a bonus, though you'd have to do a real difficulty check. Personally, I find nothing wrong with (numerical application of law/theorem)/(name of law/theorem)/(extension of law/theorem) bonuses, as long as they're used sparingly and the computation is not the overriding factor in the bonus (that is, people who know how to apply the law can do the math quickly and accurately).
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Re: Math Computation, Round 23748234

Post by The Toad to Wigan Pier »

I recently read an article which has many parallels with this discussion.
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Re: Math Computation, Round 23748234

Post by Sir Thopas »

The Toad to Wigan Pier wrote:I recently read an article which has many parallels with this discussion.
I love that article!
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Re: Math Computation, Round 23748234

Post by jonpin »

Anti-Climacus wrote:
"This was the length in years of a 17th century European conflict"
I would note that the clue is ambiguous(and therfore not terribly useful) unless you give a specific clue about said war("This was the length in years of a 17th century European conflict that saw the second denfenstration of Prague").
Yeah I figured there'd have to be another phrase in that clue (and wasn't sure "the namesake length in years" would be sufficient either), but I don't know history, so... yeah.

Also, Lockhart's Lament is awesome.
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Re: Math Computation, Round 23748234

Post by Kouign Amann »

The Toad to Wigan Pier wrote:I recently read an article which has many parallels with this discussion.
This is so win.
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Re: Math Computation, Round 23748234

Post by Byko »

The Toad to Wigan Pier wrote:I recently read an article which has many parallels with this discussion.
I'm only a few pages into reading it, and, even as someone with a math degree, this explains so much about what I've been missing. Thank you!
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Re: Math Computation, Round 23748234

Post by evilmonkey »

Byko wrote:
The Toad to Wigan Pier wrote:I recently read an article which has many parallels with this discussion.
I'm only a few pages into reading it, and, even as someone with a math degree, this explains so much about what I've been missing. Thank you!
QFT - well, I mean, I don't have the math degree yet, but you get the picture.
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Re: Math Computation, Round 23748234

Post by Mettius Fufetius »

evilmonkey wrote:
Byko wrote:
The Toad to Wigan Pier wrote:I recently read an article which has many parallels with this discussion.
I'm only a few pages into reading it, and, even as someone with a math degree, this explains so much about what I've been missing. Thank you!
Hey, so when I was a kid, I was really into math. I thought it was the most fascinating thing ever, and spent a great deal of my time reading books like "The Mathematical Universe" by William Dunham. Even though I didn't always understand what was going on, the fact that this was an interesting subject was obvious. Some people told me I was a "math prodigy" and I even met a few times with a graduate student from USC to discuss interesting stuff like topology. But when I hit algebra in 8th grade, it turned out that I was only okay at memorizing formulas and rather bad at grinding out a page and a half of arithmetic operations without making any mistakes. I came to the conclusion that I was actually bad at math, which is what I believed throughout high school, because those things continued to be the measuring stick.

Flash forward to sophomore year of college, where I'm feeling really desperate because I consider myself bad at math and I'm trying to find a course to fill the requirement that I could actually pass. I take something called "Ideas in Mathematics," offered mainly to the liberal arts kids, eschewing the alternative, a word problems class called "Finite Mathematics." I'm excited to learn that it uses another Dunham book, "Journey Into Genius." It basically reexplores a selection of the high school and very early college curriculum, plus a few interesting outliers (we finished by doing RSA), in largely the way Lockhart wants it: by situating everything in its historical context, showing what ideas came before it, and then showing the proofs in all their elegance. There was also a great deal of opportunity to prove stuff we'd never seen in the non-systematic, guided, "throw ideas out there" way that Lockhart advocates. It was the greatest class I'd ever taken.

They've gotten rid of it now. They kept the word problems class, of course. :roll:
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Re: Math Computation, Round 23748234

Post by Mechanical Beasts »

I think "Count Down" is a book that profiles the US's IMO team of a few years back. It has some interesting life stories, for one thing, and I enjoyed looking at the solutions provided to the actual problems (I was only able to solve one of six, sadly, and my solution probably wasn't rigorous enough to get more than a five from them). There's a great chapter that contains the opinions of their coach (I believe--in any event, it's Titu Andreescu, the dude whose signature was on the certificate you got for doing the AMC) on how math curricula in Eastern Europe and the US differ. He supports the problem-based approach, certainly; he says the difference is so stark that when he's in Romania, taxi drivers upon hearing his profession will talk about how good they were at math in high school and how much they loved it.

He could be making it up, but frankly, with how stupid math before college is, just about anything is worth a try!
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Re: Math Computation, Round 23748234

Post by Tower Monarch »

everyday847 wrote:I think "Count Down" is a book that profiles the US's IMO team of a few years back. It has some interesting life stories, for one thing, and I enjoyed looking at the solutions provided to the actual problems (I was only able to solve one of six, sadly, and my solution probably wasn't rigorous enough to get more than a five from them). There's a great chapter that contains the opinions of their coach (I believe--in any event, it's Titu Andreescu, the dude whose signature was on the certificate you got for doing the AMC) on how math curricula in Eastern Europe and the US differ. He supports the problem-based approach, certainly; he says the difference is so stark that when he's in Romania, taxi drivers upon hearing his profession will talk about how good they were at math in high school and how much they loved it.

He could be making it up, but frankly, with how stupid math before college is, just about anything is worth a try!
That is a great book, and I definitely came in line with that opinion (ie, bring more problem solving into lower level math) after reading it.
On another note, that guy should totally try writing on quizbowl: he did a great job with asking the players, coaches, and organizers what the competition is about and why it is worthwhile (he also highlighted distinctions like how many former high-level competitors rarely use more than arithmetic in their daily jobs, but continually apply the reasoning/problem-solving strategies they learned).
Cameron Orth - Freelance Writer/Moderator, PACE member
College: JTCC 2011, Dartmouth College '09-'10, '11-'14
Mathematics, Computer Science and Film/Media Studies
High School: Home Schooled/Cosby High '08-'09, MLWGSGIS A-E '06-'08

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Mechanical Beasts
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Re: Math Computation, Round 23748234

Post by Mechanical Beasts »

He and Christian Flow and Stefan Fatsis (Washington Post, I think? Wrote Word Freak) should fight for it.
Andrew Watkins

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