Computational Mathematics

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Computational Mathematics

Post by wte80 » Wed Sep 17, 2014 5:19 am

There have been plenty of arguments made as to why computational math is a poor topic for quiz bowl. That said, being able to do math is essential to a wide range of other studies. For that reason, mathematics are a major chunk of any child's education, and rightly so. To leave such essential (and indisputably academic) abilities completely untested in an academic competition ought raise serious red flags. The purpose of quiz bowl is not to test knowledge of one or two specific subjects; it is to test to the totality of a team's academic knowledge. As such, tossing out such a large chunk of a proper education as "bad quizbowl" raises questions of how closely we as a community actually choose to stick to the premise of testing knowledge across all academic fields. I'd love to consider alternate ways to present computation questions to bring them more closely in line with other sorts of question, but just tossing them out altogether seems like a non-solution.
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Re: Computational Mathematics

Post by Banned Tiny Toon Adventures Episode » Wed Sep 17, 2014 8:50 am

man, why don't we just have reading comprehension questions in quizbowl too
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Re: Computational Mathematics

Post by Ben Dillon » Wed Sep 17, 2014 9:13 am

Check out this article on the QB Wiki: http://www.qbwiki.com/wiki/Computational_math

In short, modern quiz bowl is about rewarding depth of knowledge (as well as breadth). Hence the pyramidal nature of tossups. Now, can computation be written pyramidally?

Chip/Questions Unlimited would argue that computational math is "by its very nature pyramidal", as students with more depth and experience will know shortcuts to solving a problem than those who don't and therefore buzz in sooner. And perhaps you will agree. This means that a question should merely be asked, e.g. "What is the sine of 30 degrees?" and let students go from there. (Which then leads to a discussion of time parameters for such a tossup, whether or not calculators should be allowed, etc.)

NAQT would argue that such questions are best left to bonuses, not tossups, but if you have to have computational tossups, they should "unfold" in the same way that tossups from other disciplines do. So a question might be asked in stages, e.g. "Arthur wants to know the value of the sine of 30 degrees. He knows that this value would correspond to the y-coordinate of a Quadrant I point on the unit circle that subtends a 30-degree reference angle at the origin. He also knows that this value would be the ratio of the length of the shorter leg to the length of the hypotenuse of any 30-60-90 triangle. Using any method, find the sine of 30 degrees?"

Most others would argue that computational math should never be in a tossup, but, again, if you have to have it, it should be a series of clues that are pyramidal, e.g.
"This fraction is used to compute the constant for magic squares. In binary, it is the terminating decimal 0.1. Name this fraction, the sine of pi/6 radians or 30 degrees."

I'm a math teacher, but I'm comfortable with the computation being left out. I've seen so many quiz bowl matches that hum along nicely, only to grind to a halt when students get 60 seconds (that's the preferred time limit in Indiana, btw) to work on a math problem.

Math theory is excellent, though! There are many tossups that can be written, especially if you include the history of mathematics and mathematicians.
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Re: Computational Mathematics

Post by Cody » Wed Sep 17, 2014 10:20 am

wte80 wrote:That said, being able to do math is essential to a wide range of other studies. For that reason, mathematics are a major chunk of any child's education, and rightly so. To leave such essential (and indisputably academic) abilities completely untested in an academic competition ought raise serious red flags. The purpose of quiz bowl is not to test knowledge of one or two specific subjects; it is to test to the totality of a team's academic knowledge. As such, tossing out such a large chunk of a proper education as "bad quizbowl" raises questions of how closely we as a community actually choose to stick to the premise of testing knowledge across all academic fields.
High school quizbowl has moved towards including more conceptual math as time goes on, which is an increase in focus on mathematics. Mathematics is essential (and accessible), and quizbowl agrees with you (else we wouldn't be moving this way).

However, conceptual math is much better at testing whether someone understands math than math calculation ever could be. In general, players who are good at math calculation (due to them understanding the subject material, at least), or compete in math competitions, do fine to very well on conceptual math questions. To contrast, I've seen many people who have a fine understanding of conceptual math that are too slow to perform calculations and lose math calculation questions to someone able to do math faster in their head.

Quizbowl is not intended to exactly mirror your schoolwork. There are no essays, or multiple-choice tests, or short answer questions. Why would we then attempt to shoehorn in math calculation, when a much better alternative is already included?
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Re: Computational Mathematics

Post by wte80 » Wed Sep 17, 2014 8:22 pm

Cody wrote:High school quizbowl has moved towards including more conceptual math as time goes on, which is an increase in focus on mathematics. Mathematics is essential (and accessible), and quizbowl agrees with you (else we wouldn't be moving this way).

However, conceptual math is much better at testing whether someone understands math than math calculation ever could be. In general, players who are good at math calculation (due to them understanding the subject material, at least), or compete in math competitions, do fine to very well on conceptual math questions. To contrast, I've seen many people who have a fine understanding of conceptual math that are too slow to perform calculations and lose math calculation questions to someone able to do math faster in their head.

Quizbowl is not intended to exactly mirror your schoolwork. There are no essays, or multiple-choice tests, or short answer questions. Why would we then attempt to shoehorn in math calculation, when a much better alternative is already included?
Math bears this distinction from most of the other subjects covered in quiz bowl: conceptual knowledge amounts in almost all cases to little more than trivia. I can teach a third-grader the relationship between differentiation and integration. He would then stand a shot at nabbing some conceptual math tossups, but he has no more real mathematical knowledge (as opposed to facts he can do nothing with) after I teach him that interesting fact than he did before. As a less extreme example, I've met, had classes with, etc. plenty of high-schoolers who could regurgitate formulas (and in many cases, what was meant by what variable, as opposed to just the alphabet soup) all day long, but hadn't a clue at all when it came to using those formulas correctly.

Also, I'll say this: people who as you describe know mathematical concepts, but can't apply them correctly in a reasonable amount of time are very closely analogous to people who have completed a math-free physics course. They know that gravity pulls objects toward each other. They might know that the force is (in the non-relativistic/Newtonian case, of course) proportional to the mass of one object, the mass of the other, and the inverse of the square of the distance between the objects. But unless they can also do the related computations, they cannot be described as possessing anything more than a very basic understanding of the subject.
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Re: Computational Mathematics

Post by Steeve Ho You Fat » Wed Sep 17, 2014 9:23 pm

wte80 wrote:Also, I'll say this: people who as you describe know mathematical concepts, but can't apply them correctly in a reasonable amount of time are very closely analogous to people who have completed a math-free physics course. They know that gravity pulls objects toward each other. They might know that the force is (in the non-relativistic/Newtonian case, of course) proportional to the mass of one object, the mass of the other, and the inverse of the square of the distance between the objects. But unless they can also do the related computations, they cannot be described as possessing anything more than a very basic understanding of the subject.
This is actually a very good description of many non-AP high school physics courses. For example, Paul Hewitt's Conceptual Physics (http://www.amazon.com/Conceptual-Physic ... 0321052021) is the definitive high school physics textbook and has the very word conceptual in its title. People who have read that book or taken a class from it tend to do pretty well on high school physics questions in quizbowl, as well.
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Re: Computational Mathematics

Post by heterodyne » Wed Sep 17, 2014 9:27 pm

wte80 wrote:Math bears this distinction from most of the other subjects covered in quiz bowl: conceptual knowledge amounts in almost all cases to little more than trivia.
What.

Find me a third-grader who can explain Cantor's diagonal argument to me, or for that matter why differentiation and integration are significant or useful, and I'll be pretty impressed. Given that conceptual math questions test those things, I doubt you can say that they are trivia. Quite frankly, it's easier in a qb setting to test knowledge of the foundational concepts behind the math than it is to perform the math itself. It seems like you're implying that conceptual physics questions are just trivia as well, which is ridiculous.
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Re: Computational Mathematics

Post by wte80 » Wed Sep 17, 2014 9:52 pm

Conceptual physics questions are, I'll agree, less trivial, as they still test meaningful, if less exact, knowledge of what will happen in a given circumstance. Conceptual mathematics questions, though, are purely trivial in almost all cases. Knowing a concept but having no clue how to apply it is plain useless in math. Conceptual math questions test knowledge that is only of significance when correctly applied. Therefore, it is far more appropriate to ask players to make such correct applications.
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Re: Computational Mathematics

Post by Steeve Ho You Fat » Wed Sep 17, 2014 9:54 pm

Ah yes, it is not helpful at all to know that the derivative represents the rate of change. Nope, can't think of any way in which understanding that concept would help me be better at math.
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Re: Computational Mathematics

Post by wte80 » Wed Sep 17, 2014 9:57 pm

Knowing that the derivative is the rate of change is helpful if, and only if, you can successfully apply that concept. While you can't very well apply a concept you don't know, a mathematical concept which you can't apply correctly is a nice meaningless fact.
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Re: Computational Mathematics

Post by Ras superfamily » Wed Sep 17, 2014 10:37 pm

This seems like a good set of arguments against computational math in quiz bowl if you care to read it
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Re: Computational Mathematics

Post by Cody » Wed Sep 17, 2014 10:47 pm

wte80 wrote:Math bears this distinction from most of the other subjects covered in quiz bowl: conceptual knowledge amounts in almost all cases to little more than trivia. I can teach a third-grader the relationship between differentiation and integration. He would then stand a shot at nabbing some conceptual math tossups, but he has no more real mathematical knowledge (as opposed to facts he can do nothing with) after I teach him that interesting fact than he did before. As a less extreme example, I've met, had classes with, etc. plenty of high-schoolers who could regurgitate formulas (and in many cases, what was meant by what variable, as opposed to just the alphabet soup) all day long, but hadn't a clue at all when it came to using those formulas correctly.

Also, I'll say this: people who as you describe know mathematical concepts, but can't apply them correctly in a reasonable amount of time are very closely analogous to people who have completed a math-free physics course. They know that gravity pulls objects toward each other. They might know that the force is (in the non-relativistic/Newtonian case, of course) proportional to the mass of one object, the mass of the other, and the inverse of the square of the distance between the objects. But unless they can also do the related computations, they cannot be described as possessing anything more than a very basic understanding of the subject.
Applying a concept correctly "in a reasonable amount of time" is not a standard by which points can be awarded; rather, it is whoever is quickest. As you must well know, the time it takes people to do math calculation varies quite widely, and being able to do calculations more quickly than someone often does not speak to greater knowledge of the material (on a macro level).

Note that I speak here as the person responsible for answering all the math calculation when I was in high school, which was easy enough at the local level because the things you can ask about in math calculation are, by nature, very limited due to time and accessibility constraints. As a result, math calculation can in no way come close to testing the variety of math a person learns, even in high school, which is a real shame because math is a very expansive subject. Additionally - due to the limited nature of math calculation - it becomes very easy to study for and get without ever really understanding it, becoming a game of "regurgitating formulas all day long". One need not understand a formula to apply it correctly, contra your above assertion.

The fundamental precept of quizbowl is that knowledge wins out above all. Math calculation can never meet this. Fortunately for us, conceptual math meets this standard quite easily and can explore all the limits of math it wishes, making it a far superior option to math calculation.
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Re: Computational Mathematics

Post by AKKOLADE » Wed Sep 17, 2014 11:15 pm

I am in pharmacy school. Pharmacists do a lot of math. Doing a lot of math quickly is what calculators do. Understanding how the math works and why things are calculated the way they are is the pharmacist's job.
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Re: Computational Mathematics

Post by wte80 » Wed Sep 17, 2014 11:38 pm

Math computation questions do often test who can solve a given problem the quickest. However, the assertion that ability to perform math efficiently and quickly somehow does not reflect mathematical knowledge seems a bit absurd. If (deeper (as regards the specific topic(s) covered by the problem)) mathematical knowledge is NOT the basis of (greater) computational ability, what is?

As for the claim that computation is for the calculators, I can just as well say that historical knowledge is best left to the internet, or that SparkNotes renders reading a given novel unnecessary to a true understanding of that novel. I can apply the "leave it to technology" argument to just about any academic subject. That we have better technology than ever before is not an excuse to be less learned.
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Re: Computational Mathematics

Post by kayli » Wed Sep 17, 2014 11:42 pm

Computation is completely absent in academic mathematics unless we're talking about numerical or computational methods for solving various problems which it appears we're not. To be able to calculate something is completely meaningless in a mathematical sense unless you can elucidate the underlying theory behind your calculations which quizbowl does much better in its conceptual math tossups than can be done in whatever ersatz computational problem that you could feasibly express in a quizbowl question.
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Re: Computational Mathematics

Post by heterodyne » Wed Sep 17, 2014 11:44 pm

wte80 wrote:If (deeper (as regards the specific topic(s) covered by the problem)) mathematical knowledge is NOT the basis of (greater) computational ability, what is?
Yeah, this is faulty. Computational ability does not require a high level of understanding! If I know how to differentiate using the limit-based definition of derivative, I've got a greater understanding of differentiation than someone who only knows that the derivative of x^n is (n)x^(n-1). That doesn't translate in any way to my ability to, say, apply the power rule.
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Re: Computational Mathematics

Post by wte80 » Wed Sep 17, 2014 11:47 pm

Turner Island wrote:
wte80 wrote:If (deeper (as regards the specific topic(s) covered by the problem)) mathematical knowledge is NOT the basis of (greater) computational ability, what is?
Yeah, this is faulty. Computational ability does not require a high level of understanding! If I know how to differentiate using the limit-based definition of derivative, I've got a greater understanding of differentiation than someone who only knows that the derivative of x^n is (n)x^(n-1). That doesn't translate in any way to my ability to, say, apply the power rule.
However, if you have only knowledge of the definition but not how to apply it, you know a lot about little squiggly things we (humanity) completely made up. If you know how to apply that definition, you might be able to apply it to something more concrete than the little squiggles.
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Re: Computational Mathematics

Post by kayli » Wed Sep 17, 2014 11:47 pm

The real bedrock of mathematics and mathematical understanding isn't computation it's the theorems. It's infinitely more important for a budding mathematician to be able to connect concepts and prove theorems than it is to be able to quickly compute integrals, derivatives, etc..
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Re: Computational Mathematics

Post by wte80 » Wed Sep 17, 2014 11:51 pm

Touko Kettunen wrote:The real bedrock of mathematics and mathematical understanding isn't computation it's the theorems. It's infinitely more important for a budding mathematician to be able to connect concepts and prove theorems than it is to be able to quickly compute integrals, derivatives, etc..
The only reason maths of any sort exist is to perform computations. Newton didn't invent calculus for the sake of fiddling around with squiggles on paper. He sought to create a model of our reality.
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Re: Computational Mathematics

Post by kayli » Wed Sep 17, 2014 11:54 pm

Yeah well, we're not in the 17th century anymore.
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Re: Computational Mathematics

Post by Kouign Amann » Wed Sep 17, 2014 11:55 pm

wte80 wrote:
Touko Kettunen wrote:The real bedrock of mathematics and mathematical understanding isn't computation it's the theorems. It's infinitely more important for a budding mathematician to be able to connect concepts and prove theorems than it is to be able to quickly compute integrals, derivatives, etc..
The only reason maths of any sort exist is to perform computations. Newton didn't invent calculus for the sake of fiddling around with squiggles on paper. He sought to create a model of our reality.
I'm no mathematician, but I'm not sure, let's say, knot theory is good for doing a lot of computations. It's still math.
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Re: Computational Mathematics

Post by kayli » Thu Sep 18, 2014 12:04 am

I'm sympathetic to the notion that computation demonstrates a mathematical understanding outside of what quizbowl usually tests (a position I once erroneously held) and we can get into a protracted discussion of the purpose or role of mathematics (which itself may be a question wrongly asked), but the idea that mathematics exists for modeling physical phenomena has long been since abandoned.
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Re: Computational Mathematics

Post by wte80 » Thu Sep 18, 2014 12:08 am

Touko Kettunen wrote:I'm sympathetic to the notion that computation demonstrates a mathematical understanding outside of what quizbowl usually tests (a position I once erroneously held) and we can get into a protracted discussion of the purpose or role of mathematics (which itself may be a question wrongly asked), but the idea that mathematics exists for modeling physical phenomena has long been since abandoned.
Excuse me as I go motivate some people to invent numbers for the sole purpose of playing with numbers.
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Re: Computational Mathematics

Post by kayli » Thu Sep 18, 2014 12:13 am

Ancient Greece already happened.
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Re: Computational Mathematics

Post by wte80 » Thu Sep 18, 2014 12:19 am

Touko Kettunen wrote:Ancient Greece already happened.
I'm sensing a distinct lack of argument for how math is now somehow divorced from its purposes, just an argument that "oh, so-di-so was discovered/invented a while back."
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Re: Computational Mathematics

Post by Cody » Thu Sep 18, 2014 12:26 am

wte80 wrote:Math computation questions do often test who can solve a given problem the quickest. However, the assertion that ability to perform math efficiently and quickly somehow does not reflect mathematical knowledge seems a bit absurd. If (deeper (as regards the specific topic(s) covered by the problem)) mathematical knowledge is NOT the basis of (greater) computational ability, what is?
Mental tricks and the way a person's brain works.

Do you realize the kind of math calculation we are talking about here? You've thrown out a bit of stuff about derivatives and integrals in your posts, but that can never be the mainstay of math calculation because it's most often a senior-level class. We are talking about stuff like probability, and trigonometry, and algebra, etc. For most problems in such categories, that we can ask about in quizbowl at least, there are only a few levels of understanding possible - ofttimes only one. Even then, a higher level of understanding (i.e. knowing a higher level / more general method of doing the problem) is nowhere near guaranteed to be faster than a lower level of understanding.
wte80 wrote:The only reason maths of any sort exist is to perform computations.
If you actually think this, then you know literally nothing about math.
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Re: Computational Mathematics

Post by kayli » Thu Sep 18, 2014 12:29 am

I'm not going to lecture you on the history or philosophy of mathematics because it's demonstrably not true that mathematics exists to model physical phenomenon. If so the algebraic geometry, algebraic topology, category theory, number theory, and basically all of pure mathematics is, by your implied definition, not math. That's an absurd position to take! For now, I recommend you read Lockhart's Lament (https://www.maa.org/external_archive/de ... Lament.pdf).
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Re: Computational Mathematics

Post by Cold Stone Steve Austin » Thu Sep 18, 2014 12:44 am

Walker, here is another thing you should read. http://en.wikipedia.org/wiki/Philosophy_of_mathematics

I'd like to know exactly which school you adhere to. It seems like you currently have the view of formalism, like David Hilbert. Then again, I am not a philosopher so I could be totally wrong.
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Re: Computational Mathematics

Post by wte80 » Thu Sep 18, 2014 12:52 am

Cody wrote:
wte80 wrote:Math computation questions do often test who can solve a given problem the quickest. However, the assertion that ability to perform math efficiently and quickly somehow does not reflect mathematical knowledge seems a bit absurd. If (deeper (as regards the specific topic(s) covered by the problem)) mathematical knowledge is NOT the basis of (greater) computational ability, what is?
Mental tricks and the way a person's brain works.
wte80 wrote:The only reason maths of any sort exist is to perform computations.
If you actually think this, then you know literally nothing about math.
(A) Said mental tricks are a form of knowledge about the subject; the "way a person's brain works" is not as you seem to think some magical itrinsic characteristic that follows us around from birth. It is a product of what we learn and how thoroughly we seek to understand it.
(B) That I know nothing of mathematics is quite blatantly untrue. You're welcome to ask anyone who knows me whatsoever well whether I "know literally nothing about math."

As for the argument that math is an art form, pure and impractical, I welcome you to divorce yourself further from reality, even when a fifth-grade education is enough to observe that reality.

To the question of what school of mathematical philosophy I adhere to: I don't much care for questions asking my to characterize my views by comparison with those of others, but that's a completely separate argument, so I'll give it a go, I do suppose. I'll get back to you on that shortly.
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Re: Computational Mathematics

Post by kayli » Thu Sep 18, 2014 1:14 am

I dunno what more I can tell you other than the fact that all of professional academic mathematics disagrees vehemently with your philosophy, worldview, w.r.t. math.
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Re: Computational Mathematics

Post by wte80 » Thu Sep 18, 2014 1:53 am

Hidehiro Anto wrote:Walker, here is another thing you should read. http://en.wikipedia.org/wiki/Philosophy_of_mathematics

I'd like to know exactly which school you adhere to. It seems like you currently have the view of formalism, like David Hilbert. Then again, I am not a philosopher so I could be totally wrong.
Actually, I'll get back to you on that sometime in the morning/afternoon. Pondering the existence of absolute reality is giving me a major headache, especially as I'm (unsurprisingly) attempting to consider it via highly circular methods, and I "absolutely" (pun/thing/joke/whatever semi-intended) have to wake up early in the morning.
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Re: Computational Mathematics

Post by Cody » Thu Sep 18, 2014 1:55 am

wte80 wrote:(A) Said mental tricks are a form of knowledge about the subject; ...
Well...no. For example, being able to multiply two numbers very fast can be accomplished with a mental trick that requires no understanding aside from how to apply an algorithm. This is true of many mental tricks, which often require no knowledge about the subject to successfully execute them.
wte80 wrote:... the "way a person's brain works" is not as you seem to think some magical itrinsic characteristic that follows us around from birth. It is a product of what we learn and how thoroughly we seek to understand it.
I am aware. Anyone can learn to do math calculation - the idea that you either have the ability to do math or don't is an obvious math. What isn't a myth is that some people are better at quickly doing math calculation, regardless of how "thoroughly [they] seek to understand it".
wte80 wrote:(B) That I know nothing of mathematics is quite blatantly untrue. You're welcome to ask anyone who knows me whatsoever well whether I "know literally nothing about math."
I don't know what to tell you other than that the two views are essentially incompatible.
wte80 wrote:As for the argument that math is an art form, pure and impractical, I welcome you to divorce yourself further from reality, even when a fifth-grade education is enough to observe that reality.
The funny thing about education is that you learn more as time progresses. While a fifth-grader may only know math as calculating things, by the time someone engaged with this kind of material gets to high school or college, they know that math exists for reasons other than performing calculations - if only from popular coverage of the subject! As well, no one in this thread, or in academia, thinks math is impractical - far from it.
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Re: Computational Mathematics

Post by Dominator » Thu Sep 18, 2014 7:49 am

wte80 wrote:As for the argument that math is an art form, pure and impractical, I welcome you to divorce yourself further from reality, even when a fifth-grade education is enough to observe that reality.
I find this argument interesting. To paraphrase: "People with far less education than you hold this belief, and so you should also." This is kind of, you know, antithetical to the whole idea of education at all. You will not win many arguments about what is or is not academic when you pepper them with gems like this.
Touko Kettunen wrote:I dunno what more I can tell you other than the fact that all of professional academic mathematics disagrees vehemently with your philosophy, worldview, w.r.t. math.
As a guy who actually holds a Ph.D. in mathematics: wte80, you are wrong. If I thought you were in an appropriate zone of proximal development, I would take the time to explain it to you. Judging from everything else you've written, I don't think you're there. Instead, I think you are a high school physics student with a high school physics worldview. If you continue to study physics, you will undoubtedly come across things that mathematicians developed for their own interests expecting them to be completely divorced from reality, like complex numbers, which only found non-mathematical applications decades or centuries later.
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Re: Computational Mathematics

Post by jonpin » Thu Sep 18, 2014 9:44 am

Speaking as another person who earned my keep on my high school and college teams with computational questions: They are the Sturgeon's Law of quizbowl; 90% of them are crap (and that's probably an understatement). Those that have some merit are basically restricted to parts of a bonus, where one can test understanding and application together; and legitimately pyramidal tossups that are in practice not actually computational; something like "This is the third symmetric Ramsey number. Pi squared divided by this represents the probability that any two integers are relatively prime, as well as the sum of the reciprocal squares of the integers. It is the first perfect number, since 3 is the first Mersenne prime. For 10 points, name this third triangular number." [clues are out of order since I threw this together in 5 minutes, but the point is gotten across].

In any event, the notion that there is nothing worthwhile to math besides calculation is insulting to a great many people who study theoretical mathematics for their livelihoods. And to say that the best way to incorporate math in quiz bowl is by calculation seems analogous to saying that a tossup on Hamlet should require one to buzz in and give a full soliloquy as, if you don't know that, you clearly don't know Hamlet as well as the other team.
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Re: Computational Mathematics

Post by vinteuil » Thu Sep 18, 2014 11:15 am

Walker, if you're still convinced that computational math can make good tossups, why don't you post some examples (you can even write them yourself if you want) of what you think is "non-trivial" and tests "deeper knowledge" than a conceptual question would?
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Re: Computational Mathematics

Post by Halved Xenon Stinging » Thu Sep 18, 2014 12:39 pm

Dr. Loki Skylizard, Thoracic Surgeon wrote:I am in pharmacy school. Pharmacists do a lot of math. Doing a lot of math quickly is what calculators do. Understanding how the math works and why things are calculated the way they are is the pharmacist's job.
Comp math tossups aren't simply a calculation either. Having a calculator won't help find the probability of say, me wearing a black shirt and white socks this morning. You need a level of understanding of the problem to be able to use a calculator. This level of understanding isn't shown by simple math theory tossups, anyone can remember facts off of quinterest without understanding the material at all.
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Re: Computational Mathematics

Post by Cody » Thu Sep 18, 2014 2:57 pm

Shangdevin wrote:Comp math tossups aren't simply a calculation either. Having a calculator won't help find the probability of say, me wearing a black shirt and white socks this morning. You need a level of understanding of the problem to be able to use a calculator.
Yes, that would be what Fred said in his post: it is his job to have a conceptual understanding of the problem. As I have noted previously, math calculation in quizbowl - by definition - can only reward the person who does the calculation the fastest, not who has the better conceptual understanding (with a few exceptions - such as only one person in the room have a conceptual understanding of the problem. This only further highlights the flaws of math calculation, however). If you wanted to test who has a better conceptual understanding, you would write a question that could actually take advantage of that: a conceptual math question.
Shangdevin wrote:This level of understanding isn't shown by simple math theory tossups, anyone can remember facts off of quinterest without understanding the material at all.
As I have noted previously in the thread, the level of understanding needed to do math calculation is necessarily quite limited by the time one can give for a problem and accessibility concerns - generally, the bar is purposefully set low (and I speak here as someone who formerly wrote a lot of math calculation questions for VHSL Scholastic Bowl). I am not really sure where this specter of people memorizing things without understanding the material is coming from. It's possible, but necessarily more involved than preparation for math calculation could ever be (see the first sentence in this paragraph). The real problem is that "getting math calculation questions demonstrates conceptual understanding" does not follow from such an objection. For what is possibly the third time in this thread: math calculation is only capable of giving points to the person who performed the calculation the fastest - conceptual understanding only enters into it insofar as you have to do the problem (and a fairly simple one at that - see again my first sentence in this paragraph).

The idea that math calculation could possibly cover more ground or test conceptual understanding better than the ever-varying conceptual math is empirically untrue. Anyone who has read or written math calculation and conceptual math knows this.
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Re: Computational Mathematics

Post by Benin Rebirth Party » Thu Sep 18, 2014 3:08 pm

wte80 wrote:Math bears this distinction from most of the other subjects covered in quiz bowl: conceptual knowledge amounts in almost all cases to little more than trivia. I can teach a third-grader the relationship between differentiation and integration. He would then stand a shot at nabbing some conceptual math tossups, but he has no more real mathematical knowledge (as opposed to facts he can do nothing with) after I teach him that interesting fact than he did before. As a less extreme example, I've met, had classes with, etc. plenty of high-schoolers who could regurgitate formulas (and in many cases, what was meant by what variable, as opposed to just the alphabet soup) all day long, but hadn't a clue at all when it came to using those formulas correctly.
Being able to do math is very important, and that's where math contests come into play, requiring both computational and conceptual knowledge. I did all the math contests I could and I answered most of my teams computational questions like many other people who have posted in this thread, and I have similar sentiments. The way every other tossup in quizbowl works make computation impossible to fit in. Quizbowl is about knowing and not doing. You can write a physics bonus on bridge building dynamic, but it's certainly not practical to actually do that in a tournament.

Also, as a not math person in university, I'd rather be spending my free time I devote to math by watching Numberphile videos instead of learning how to do vector calculus and multivariable limits.
wte80 wrote: Also, I'll say this: people who as you describe know mathematical concepts, but can't apply them correctly in a reasonable amount of time are very closely analogous to people who have completed a math-free physics course. They know that gravity pulls objects toward each other. They might know that the force is (in the non-relativistic/Newtonian case, of course) proportional to the mass of one object, the mass of the other, and the inverse of the square of the distance between the objects. But unless they can also do the related computations, they cannot be described as possessing anything more than a very basic understanding of the subject.
I had a pleasant conversation years ago with my freshman physics prof how I thought the best type of physics exam would be one that was purely conceptual without any super intense computations. His exams were 50% of each and also wanted to push for a 100% conceptual exam, but the department wouldn't be happy with that. For example, it's more important to make sure students understand concepts than be able to solve the actual algebra involved in projectile motion questions. Even in harder examples, you the situation where you set up an uber complicated free-body diagram and a corresponding equation for the variable you desire and then after that, all you really need to do is plug it into wolfram alpha. Or when you're dealing with fields and need to set up some sort of complicated integral. Once you get an equation, you're basically done the problem. A lot of the time, setting up proper equations is not trivial and certainly require more than just a basic understanding of the subject.
Shangdevin wrote:
Comp math tossups aren't simply a calculation either. Having a calculator won't help find the probability of say, me wearing a black shirt and white socks this morning. You need a level of understanding of the problem to be able to use a calculator. This level of understanding isn't shown by simple math theory tossups, anyone can remember facts off of quinterest without understanding the material at all.
The comp math tossup in NAQT's sample IS packet on their website asks you to solve log_10 (3x-2) = 2. The only knowledge tested here is that 10^2 is 100 (e.g. how a basic log works), everything else isn't more than some basic one variable algebra. This tossup would be better as one that conceptually tests what logarithms are, allowing it to go significantly more deep than just log base 10 of 100 is 2.
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Re: Computational Mathematics

Post by cvdwightw » Thu Sep 18, 2014 6:52 pm

I am speaking here as a person who has written an absurd amount of math calculation bonus questions for HSAPQ, and a not-insignificant number of math calculation questions for NAQT, and thus am probably somewhere in the top 5 writers of math calculation by volume in "good quizbowl" over the past five years.

I had a much longer post, but it repeated with a lot of what's said here and I'm pretty sure that I've said the same things more eloquently in mathcalc threads past. This is something new and is possibly the best argument I've found for not including computational math.

Look at the Common Core High School Mathematics Standards. I counted 156 of them. But this argument isn't about just counting them - there are certainly things, like computer science and world literature, that have a defined (if small) place in the distribution without a very large canon at the high school level.

The point that I want to make is that most of these standards are things like "Represent data with plots on the real number line (dot plots, histograms, and box plots)" or "Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials" or "Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures." These are things that can't be easily distilled into a non-trivial math calculation question.

Under Common Core, it's more important to internalize understanding of mathematical concepts than to do the actual calculations. Due to the adoption of Common Core Standards, then, math is becoming increasingly like every other course where the emphasis is on something that we can't test in quizbowl. Accordingly, we should treat math like every other subject is treated: by asking questions that test factual knowledge about important concepts.
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Re: Computational Mathematics

Post by wte80 » Thu Sep 18, 2014 8:11 pm

I can't agree that conceptual knowledge of mathematics is more meaningful than any actual application of those concepts. The applications are facts or models of facts about reality, or at least can be so, although I recognize that computation questions are sometimes posed without any real-world context. By comparison, mathematical concepts are tools by which we can make models of our reality. They are in essence facts about facts, which are always of less worth than facts about, you know, the reality in which we reside. That said, I'm approaching dangerously close to a philosophical argument, and we all know how likely those are to (not) be settled by squabbling on the internet. As such, I might just shut up at this point.
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Re: Computational Mathematics

Post by alexdz » Thu Sep 18, 2014 8:24 pm

wte80 wrote:... facts about facts ... are always of less worth than facts about, you know, the reality in which we reside.
Which statement below is "worth more" to you?

(A) The number 2 is an even number.
(B) Even numbers greater than 2 can be expressed as the sum of two primes. (Goldbach's conjecture)

Statement (A) is a fact about the reality in which we reside. Statement (B) is a fact "about a fact," namely, it is a fact that relates even numbers and prime numbers to one another. I am by no means a mathematician - my closest claim to fame is that I taught 7th grade math test prep once - yet I can clearly see that one of those facts above is "worth more," mathematically speaking, than the other.
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Re: Computational Mathematics

Post by wte80 » Thu Sep 18, 2014 8:26 pm

Statement A is not a fact about reality.
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Re: Computational Mathematics

Post by vinteuil » Thu Sep 18, 2014 8:33 pm

vinteuil wrote:Walker, if you're still convinced that computational math can make good tossups, why don't you post some examples (you can even write them yourself if you want) of what you think is "non-trivial" and tests "deeper knowledge" than a conceptual question would?
In case I wasn't being clear: this was an extremely serious and very relevant question. If it doesn't work as a quizbowl question, it doesn't work.
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Re: Computational Mathematics

Post by Sam » Thu Sep 18, 2014 8:36 pm

People have sort of been referring to this implicitly, but it's worth noting that plenty of high school quiz bowl does not in fact get rid of computational questions, they get rid of computational tossups. A computational tossup will reward someone who can do the calculation half a second faster than the other team, which is a poor indicator of who actually knows more. A computational bonus will give points to those who know how to do the computation and not give points to those who do not, as it should. What one believes constitutes a real understanding of math is mostly irrelevant, unless you believe split-second differences in calculating speeds can actually distinguish mathematical ability.
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Re: Computational Mathematics

Post by Halved Xenon Stinging » Thu Sep 18, 2014 8:46 pm

I'd like to point that in 9 cases out of 10 the person who can do the problem faster is the one who knows more. What i mean by this is that someone who has more experience with problem solving will have a stronger grasp on the technique, thus allowing them to solve the problem quicker. Players who are less familiar with the technique required to solve the tossup will take longer to execute it.
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Re: Computational Mathematics

Post by alexdz » Thu Sep 18, 2014 9:06 pm

I learned how to use the power rule to solve derivative problems in about November of my freshman year of high school, expressly for the purpose of scoring points in quiz bowl. I literally knew nothing about derivatives, what they were useful for, or how to do anything except mindlessly apply a rule I read in a textbook. I could do it very quickly. I was quite good at doing it. I often buzzed in the fastest. I was quite familiar with the technique - but I didn't actually KNOW anything. A pyramidal tossup on a derivative will actually determine whether or not someone has actually learned about derivatives *regardless of* their ability to memorize a formula. One could theoretically demonstrate superior knowledge on a computation, but a computation does not by default indicate more knowledge. Any knowledge can be obtained "fraudulently" (like by memorizing lists or formulas) but it is the job of a quizbowl question writer to minimize that from happening. One easy way: don't include math computation tossups.
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Re: Computational Mathematics

Post by Halved Xenon Stinging » Thu Sep 18, 2014 10:01 pm

alexdz wrote:I learned how to use the power rule to solve derivative problems in about November of my freshman year of high school, expressly for the purpose of scoring points in quiz bowl. I literally knew nothing about derivatives, what they were useful for, or how to do anything except mindlessly apply a rule I read in a textbook. I could do it very quickly. I was quite good at doing it. I often buzzed in the fastest. I was quite familiar with the technique - but I didn't actually KNOW anything. A pyramidal tossup on a derivative will actually determine whether or not someone has actually learned about derivatives *regardless of* their ability to memorize a formula. One could theoretically demonstrate superior knowledge on a computation, but a computation does not by default indicate more knowledge. Any knowledge can be obtained "fraudulently" (like by memorizing lists or formulas) but it is the job of a quizbowl question writer to minimize that from happening. One easy way: don't include math computation tossups.
Pyramidal questions don't prove deep understanding either; you don't need to know anything about calculus to remember that the Weierstrass function is "associated" with derivatives. It is extremely difficult for a writer to find unique clues that are not too obscure to be gettable, so inevitably "fraudulent" knowledge will come up from time to time. Though concepts of computational math can be repeated, one must apply an understanding to arrive at the solution in ALL cases.
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Re: Computational Mathematics

Post by cvdwightw » Thu Sep 18, 2014 10:15 pm

Okay, look, I ought to be tired of wading into these messes but I guess I'm a sucker.

The following people who use math all the time generally agree that conceptual math is more important than computational math:
  • People with (or working on) college and graduate degrees in math, who agree that computational math is trivial and uninteresting compared to conceptual math
  • People with (or working on) college and graduate degrees in science/engineering/other quantitative fields, who use math all the time in their courses/real world jobs and agree that constructing the appropriate model is infinitely more important than "plugging in the numbers"
  • Educators who teach high school math, who almost universally agree that performing computations is secondary to understanding the actual concepts that allow those computations to be performed
The following people who use math all the time generally agree that conceptual math is less important than computational math:
The following people generally agree that computational math should be limited or eliminated in quizbowl:
  • People who have written literally thousands of computation questions, have watched as teams regularly fail to answer them, and are at wit's end regarding how to write interesting computation questions that players can actually answer
  • Some other people who espouse various theoretical reasons why computational math questions are fundamentally different from every other type of quizbowl question, and therefore should not be part of quizbowl
The following people generally agree that computational math should be a significant part of quizbowl:
  • Quizbowl coaches who don't like that math is marginalized in quizbowl relative to its importance in a high school education, and recognize that the imbalance cannot be corrected by including more questions on conceptual math
  • High school players who are good at computation relative to other subjects, and thus have a selfish interest in maintaining or increasing the amount of computational math
  • People who do not understand how math works, how quizbowl works, or both
I hate to use the argument from authority, but in this case it's justified. People who know a whole lot more about math than you do right now, and who work with math a whole lot more than you do right now, agree that you're wrong about how math works. People who have been playing quizbowl a whole lot longer than you, and have been writing math calculation questions since before you started playing quizbowl, agree that you're wrong about how math works in quizbowl.

The only evidence that you will ever present to get any of us to change our minds about any of this is a set of non-trivial, grade-appropriate, computational math questions that are converted at a rate similar to other categories. This set has not existed historically and will probably continue to not exist.
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Re: Computational Mathematics

Post by Halved Xenon Stinging » Thu Sep 18, 2014 10:24 pm

The following people agree that pure memorization without understanding is bad in general, and includes quizbowl:
Math teachers/ professors and basically everyone you named

Unfortunately, in my understanding, pure memorization is what many conceptual math tossups are down to now. People don't care to understand the concept, they just memorize the clues that come up often.
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Re: Computational Mathematics

Post by Guile Island » Thu Sep 18, 2014 10:29 pm

Not to mention, despite all of your ridiculous comp math apologetics, you both have failed to address the problem that math computation tossups are, by default, not pyramidal and therefore impossible to write in a manner that all other good tossups are properly written in.
Shangdevin wrote:The following people agree that pure memorization without understanding is bad in general, and includes quizbowl:
Math teachers/ professors and basically everyone you named

Unfortunately, in my understanding, pure memorization is what many conceptual math tossups are down to now. People don't care to understand the concept, they just memorize the clues that come up often.
If this is really your grievance against conceptual math questions, then why the :capybara: are you playing quizbowl in the first place? This argument has been used by apologists for pretty much every bad practice in quizbowl and has been proven many times over to be... not a very good argument.
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