## Pyramidality, Math calculation, and the Goldfish Tournament

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### Pyramidality, Math calculation, and the Goldfish Tournament

One of the major issues arising from the usual math calculation discussions is whether it is even possible for math calculation tossups to be pyramidal. The claim that math calculation tossups are by definition non-pyramidal nearly always is made in math calculation discussions. For example:
http://hsapq.com/policy.html wrote: This article outlines the six reasons why HSAPQ feels that math calculation is not an appropriate topic for tossups in quizbowl.

1. It is not possible to write math calculation tossups in the pyramidal style
Some math calculation proponents, myself included, have argued that it is possible for a math calculation question to be pyramidal. Unfortunately, both sides in this debate have been working mostly off of gut feelings, since we never really had any solid definition of what makes a pyramidal tossup pyramidal, nor any data on how tossups were answered.

I do not seek to reopen the debate over math calculation in quiz bowl; this ground has been covered well enough. However, I would like to discuss pyramidality: how do we define it, and how can we analyze it? In doing so, I will attempt to show that regardless of whether math calculation is pyramidal, the results of playing math calculation questions produce similar buzzing patterns to playing pyramidal tossups.

Definitions

The first issue is defining pyramidality. The QBWiki states:
http://doc-ent.com/qbwiki/index.php?title=Pyramidality wrote: Pyramidality is a concept in tossup-writing that states that clues should be arranged in descending order of difficulty, with the hardest information first and the easiest at the end, after the "For 10 points."
How can we analyze if a tossup is pyramidal, though? Imagine if one were to graph the percentage of buzzes occurred on the y axis against the percentage of the question elapsed on the x-axis; I will call this a "cumulative buzz graph". Certainly, (0,0) and (100,100) are points on this curve for all questions; after none of the question has been read, no buzzes occur; after 100% of the question has been read, 100% of the buzzes have occurred (we normalize the y-axis against all "buzzes occurred", so we disregard teams that do not buzz on the tossup). What characteristics will the graph have? If we see sharp, vertical spikes, this is a sign of a difficulty cliff; at a certain point in the question, lots and lots of teams suddenly buzzed.

The question, then, is what a non-pyramidal question will look like, versus what a pyramidal question will look like on this graph. I claim they will look fundamentally different. When we think of non-pyramidal questions, we think of early buzzer races (because of misordered clues) or late buzzer races (because of misdirection, or too hard clues). If no buzzer races occur in the graph, then I will say that the graph has a "pyramidal buzz distribution."

The Goldfish Tournament

The idea of using cumulative buzz graphs is appealing, but until recently, the data needed to actually create them was unavailable. The Goldfish tournament changed that. We now know exactly where each team buzzed on a given question, for the admittedly small sample size of 50 teams on 30 questions. I have attached the cumulative buzz graph for this year's Goldfish tournament.

Each plot is a tossup; for a given value x on the x-axis, the point on the y-axis is the percentage of buzzes that occurred at or before x percent of the tossup had been read.

What do we notice? First, most of the plots seem to have the "pyramidal buzz distribution" discussed above; there don't seem to be too many buzzer races.

The turquoise plot towards the upper left of center contains the point (45,60), meaning after less than half the question had been read, 60% of the teams that would eventually buzz had already buzzed. That question only had 5 buzzes, though, so the sample size is quite small.

The dark blue plot in the lower right, on the other hand, had no buzzes until over halfway through the question; then, at around 78% of the way through, had a huge leap in the number of buzzes occurred. That corresponds with the end of the question text and the start of the 5-second pause, so we can suppose there was a buzzer race on the giveaway, and that the question was fairly hard.

This brings me full circle back to the original discussion point: is math calculation non-pyramidal? There are two math calculation questions in that graph; they are the ones colored white. Neither one is easily distinguishable from the remaining 28 tossups, which were pyramidal. Hence, I claim that the math calculation questions did have pyramidal buzz distributions.

Methodology

You can look at the spreadsheet containing my data. As a brief explanation of my method: I read a sample question to myself at the rate I read IS sets, and found I read about 4 words a second. Hence, I considered each "second" after the question ended to be the equivalent of 4 words. I considered the length of a tossup to be the number of words needed to find all buzzes when the tossup was in progress, then let the effective length be the length plus the number of waiting seconds times 4; for math questions, one waits 10 seconds, for non-math, 5.

The x-axis of the graph is the percent of question time elapsed; letting each second after the tossup is dead be 4 words, this is simply the number of words read over the total number of effective words. The y-axis is the percentage of buzzes occurred; this is simply the number of buzzes before time t over the total number of buzzes on the question.

The math calculation questions are question 1 and 26, as observed through the reported answer times (they are the only ones which have buzzes between 6 and 10 seconds after the tossup was finished).

I welcome any comments on the methodology or conclusions I discussed here. I would rather avoid this turning into the usual discussion of math calculation in quiz bowl, though, and stay focused simply on the pyramidality aspect of calculation.

EDIT: Described wrong plot originally in analysis; fixed incorrect description.
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### Re: Pyramidality, Math calculation, and the Goldfish Tournament

Two issues I take with this analysis. I come at it with an agenda, but also a relatively open mind.

First, is a "pyramidal buzz distribution" equivalent to a pyramidal tossup? I'd say hardly. A pyramidal tossup can produce a PBD, but the reverse implication isn't necessarily there--you're begging it, really. I could probably invent a game of chicken that would result in players buzzing at evenly-spaced intervals, but that wouldn't be a pyramidal tossup.

Second, is Goldfish data necessarily the best? Since teams are only hypothetically facing other teams, I feel like a lot of the urgency is lost. Even though you do have to buzz early, you have no idea how early--I don't trust that buzz distributions will really remain unchanged.
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### Re: Pyramidality, Math calculation, and the Goldfish Tournament

everyday847 wrote:Two issues I take with this analysis. I come at it with an agenda, but also a relatively open mind.

First, is a "pyramidal buzz distribution" equivalent to a pyramidal tossup? I'd say hardly. A pyramidal tossup can produce a PBD, but the reverse implication isn't necessarily there--you're begging it, really. I could probably invent a game of chicken that would result in players buzzing at evenly-spaced intervals, but that wouldn't be a pyramidal tossup.

Second, is Goldfish data necessarily the best? Since teams are only hypothetically facing other teams, I feel like a lot of the urgency is lost. Even though you do have to buzz early, you have no idea how early--I don't trust that buzz distributions will really remain unchanged.
I agree with you that a pyramidal tossup and a PBD are not necessarily equivalent. I think that a tossup that does not possess a PBD cannot possibly be pyramidal, but there may be requirements other than the PBD that we place on a tossup before we call it pyramidal. I don't necessarily know what those other requirements are, though, which is one question I hoped to raise: what exactly are the properties of a pyramidal tossup?

Goldfish data is not ideal because of the exact issues you mention, but it's all we've got right now. If I had this type of data for HSNCT, for example, I'd gladly use that in its place. I suspect the conclusion that math calculation questions possess PBD's is still valid; if math calculation distributions were to differ dramatically from the PBD, I would expect a relatively small 50-team goldfish analysis would show it.
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### Re: Pyramidality, Math calculation, and the Goldfish Tournament

everyday847 wrote: Second, is Goldfish data necessarily the best? Since teams are only hypothetically facing other teams, I feel like a lot of the urgency is lost. Even though you do have to buzz early, you have no idea how early--I don't trust that buzz distributions will really remain unchanged.
Wouldn't the change in buzz distribution tend to amplify evidence of non-pyramidal tossups, since teams will wait to buzz late in the question rather than neg against the goldfish?
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### Re: Pyramidality, Math calculation, and the Goldfish Tournament

bt_green_warbler wrote:
everyday847 wrote: Second, is Goldfish data necessarily the best? Since teams are only hypothetically facing other teams, I feel like a lot of the urgency is lost. Even though you do have to buzz early, you have no idea how early--I don't trust that buzz distributions will really remain unchanged.
Wouldn't the change in buzz distribution tend to amplify evidence of non-pyramidal tossups, since teams will wait to buzz late in the question rather than neg against the goldfish?
Do you mean that it would obscure evidence of non-pyramidal tossups? As in, more things would appear non-pyramidal even if they weren't because of extra hesitancy => clustered buzzing? Then yeah. It would. But we don't see that, precisely because kids know that the buzz time--not just their raw score--matters in computing their matchup against all the other teams. So instead I'm just saying that it makes the data less valid because of whatever unquantifiable and unqualifiable effects a semi-match can produce.
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### Re: Pyramidality, Math calculation, and the Goldfish Tournament

Also, I don't really think you can equate time with words like that. Ignoring math, I think that there's a legitimate difference between clue-filler-clue and clue-clue-clue, even if you get the same amount of buzzes at the same times. What I'm trying to say is that... if people remember/figure out the answer to clue A while clue C is being read and buzz on that, I think that's different than people buzzing while clue C is being read because they know clue B or people buzzing on clue C.

The thing about pyramidal tossups, as opposed to math, is that if you read them in super-slow-motion (like, one word per 2-3 seconds,) the buzz distribution would stay pyramidal.
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### Re: Pyramidality, Math calculation, and the Goldfish Tournament

Having seen NAQT's comp math this year, I'm almost 100% sure that the staggering of buzzes can be attributed to the staggering of arithmetic speeds based on people getting their basis from different clues. Which is exactly what we want to avoid.
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### Re: Pyramidality, Math calculation, and the Goldfish Tournament

Sir Thopas wrote:Having seen NAQT's comp math this year, I'm almost 100% sure that the staggering of buzzes can be attributed to the staggering of arithmetic speeds based on people getting their basis from different clues. Which is exactly what we want to avoid.
This would pretty much be my conclusion as well.
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### Re: Pyramidality, Math calculation, and the Goldfish Tournament

Interesting analysis; I applaud you for doing the legwork. I have to disagree with you, though, because I don't think the buzz distribution is a unique function of question type (which is the tacit keystone to your argument here.) The following point gets at the reason why:
cornfused wrote:The thing about pyramidal tossups, as opposed to math, is that if you read them in super-slow-motion (like, one word per 2-3 seconds,) the buzz distribution would stay pyramidal.
In my experience, the vast majority of calculation question outcomes are dominated by calculation speed, invariably in the form of arithmetic speed, because anyone who cares to knows how to set-up the problem from the lead-in, even if there are multiple clues. Consequently, I'd expect the time distribution of buzzes on such questions to display the features you ascribe to pyramidal questions even in the case of straight-up one-clue calculation questions (which, if we accept my first argument there, is effectively what most calculation questions are anyway), just by the natural variability of calculation speed.

MaS
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### Re: Pyramidality, Math calculation, and the Goldfish Tournament

cornfused wrote:Also, I don't really think you can equate time with words like that. Ignoring math, I think that there's a legitimate difference between clue-filler-clue and clue-clue-clue, even if you get the same amount of buzzes at the same times. What I'm trying to say is that... if people remember/figure out the answer to clue A while clue C is being read and buzz on that, I think that's different than people buzzing while clue C is being read because they know clue B or people buzzing on clue C.
Since the data for Goldfish was half in "words read" and half in "time after completion", I felt I needed some way to convert between the two to display the full data on one scale. I'd certainly be open to suggestions on better ways to do that conversion.
Captain Scipio wrote:Consequently, I'd expect the time distribution of buzzes on such questions to display the features you ascribe to pyramidal questions even in the case of straight-up one-clue calculation questions (which, if we accept my first argument there, is effectively what most calculation questions are anyway), just by the natural variability of calculation speed.
This is an interesting point; a one-line math calculation question yielding a PBD is definitely conceivable. Personally, I don't think we'd get as close to the PBD with one-clue calculation as we do with current math calculation tossups, but I don't have any data whatsoever to back that up, so that is just gut instinct.

I'm assuming you believe one-line math calculation tossups don't make good quizbowl. Hence, we either need to place additional requirements on tossups beyond having a PBD for them to be considered pyramidal, or we need to place additional requirements on tossups beyond being pyramidal for them to be considered good quiz bowl. I think such requirements probably do exist (possibly even in both categories), but I don't know specifically what they are.
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### Re: Pyramidality, Math calculation, and the Goldfish Tournament

dschafer wrote:
cornfused wrote:Also, I don't really think you can equate time with words like that. Ignoring math, I think that there's a legitimate difference between clue-filler-clue and clue-clue-clue, even if you get the same amount of buzzes at the same times. What I'm trying to say is that... if people remember/figure out the answer to clue A while clue C is being read and buzz on that, I think that's different than people buzzing while clue C is being read because they know clue B or people buzzing on clue C.
Since the data for Goldfish was half in "words read" and half in "time after completion", I felt I needed some way to convert between the two to display the full data on one scale. I'd certainly be open to suggestions on better ways to do that conversion.
I'm not sure you get the point. If I hear Clue A, try to solve it using Clue A, and buzz during or after Clue B, there is no way for someone other than myself to know what clue I buzzed on unless you ask me directly.

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### Re: Pyramidality, Math calculation, and the Goldfish Tournament

dschafer wrote:Hence, we either need to place additional requirements on tossups beyond having a PBD for them to be considered pyramidal, or we need to place additional requirements on tossups beyond being pyramidal for them to be considered good quiz bowl.
Multiple clues of decreasing difficulty, such that each clue applies uniquely to the same single correct answer, and each clue can be (but of course, need not be; see the Yaphe method, and so on) buzzed on independently of any other or any external process outside of listening to the clues.
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### Re: Pyramidality, Math calculation, and the Goldfish Tournament

Sir Thopas wrote:
dschafer wrote:Hence, we either need to place additional requirements on tossups beyond having a PBD for them to be considered pyramidal, or we need to place additional requirements on tossups beyond being pyramidal for them to be considered good quiz bowl.
Multiple clues of decreasing difficulty, such that each clue applies uniquely to the same single correct answer, and each clue can be (but of course, need not be; see the Yaphe method, and so on) buzzed on independently of any other or any external process outside of listening to the clues.
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### Re: Pyramidality, Math calculation, and the Goldfish Tournament

dschafer wrote:
Captain Scipio wrote:Consequently, I'd expect the time distribution of buzzes on such questions to display the features you ascribe to pyramidal questions even in the case of straight-up one-clue calculation questions (which, if we accept my first argument there, is effectively what most calculation questions are anyway), just by the natural variability of calculation speed.
This is an interesting point; a one-line math calculation question yielding a PBD is definitely conceivable. Personally, I don't think we'd get as close to the PBD with one-clue calculation as we do with current math calculation tossups, but I don't have any data whatsoever to back that up, so that is just gut instinct.
Okay. Well, I've seen a lot of matches with a lot of outcomes on single-clue computation questions (IHSA-style questions) and I've also seen a lot of matches with a lot of outcomes on (at least nominally) multi-clue computation tossups (NAQT and especially the first couple iterations of Earlybird.) My observation is directly contrary to your instinct.
Also, do we really have no data? What form were these computation questions? Were they, in fact, one-clue questions that still displayed the buzz distribution that you're claiming characterizes pyramidal questions only?
dschafer wrote:I'm assuming you believe one-line math calculation tossups don't make good quizbowl. Hence, we either need to place additional requirements on tossups beyond having a PBD for them to be considered pyramidal, or we need to place additional requirements on tossups beyond being pyramidal for them to be considered good quiz bowl. I think such requirements probably do exist (possibly even in both categories), but I don't know specifically what they are.
Right; I think Guy does a good job of defining pyramidal here. I also think that most people (rightly) add the stricture that outcomes of questions should be dominated by knowledge (measured through recall, obviously) and that's not the case for computation questions (outcomes dominated by arithmetic speed). It's equally not the case for, say, questions with huge difficulty cliffs (outcomes dominated by buzzer speed alone). Properly understood, that's perhaps not distinct from pyramidal, but it does seem to be distinct from the notion of pyramidal you're using.

MaS

PS: I'd also like to re-present my counter-argument. I contend that shoehorning computation into tossups actually does a disservice to computation and math generally because nothing that can be done by 80% of teams in 10 (or even 30) seconds is going to be substantive, novel, thought-provoking math. Rather, it's going to be straightforward number crunching of one of a few isomorphs. In other words, I argue that it's not just that math computation (probably necessarily) fails to be good quizbowl; it's also that quizbowl (almost certainly necessarily) fails to be a good math computation contest.
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### Re: Pyramidality, Math calculation, and the Goldfish Tournament

dtaylor4 wrote:I'm not sure you get the point. If I hear Clue A, try to solve it using Clue A, and buzz during or after Clue B, there is no way for someone other than myself to know what clue I buzzed on unless you ask me directly.
Ah, I think I follow. Let me craft a potential new definition of pyramidality, and let me know if this accurately represents your view (MaS, I think this definition also covers your discussion of one-line math calculation)

On any given question, when someone buzzes, there is some clue that they heard that finally led them to buzz. It might be the most recently read clue, though this is not always true (a player recognizes something at the start of the tossup, but take a few seconds to finally determine what he or she recognizes it from, and another clue that they didn't know is said in the meantime). Let's call the final clue read that affects one's decision to buzz the "last-used-clue", which I will abbreviate LUC.

If we were to examine the LUC distribution as we did the buzz distribution, you would argue the graph would not change significantly for non-math questions (since the pause between "LUC heard" and "buzzing" is small), but would change significantly for math calculation (since the pause between "LUC heard" and "solved problem and am buzzing" is large). For non-math, then, the LUC distribution would be nearly identical to the buzz distribution. For math calculation, on the other hand, the LUC distribution would probably show lots of spikes, since there are only a few spots in the question where "useful" information is given; when the full problem is given, and then maybe when the trick is given, or when some of the solutions to subproblems are given. For one-line tossups in general, the LUC distribution would jump from y=0 to y=100 as soon as that clue was said, since that is the only useful clue, and is by definition the LUC.

An ideal pyramidal LUC distribution, then, must avoid large jumps (since those are indicative of difficulty cliffs). I suggested above that we require pyramidal questions have pyramidal buzz distributions. You claim it is not the buzz distribution that is significant, but the LUC distribution, and that math calculation deviates considerably from the pyramidal LUC distribution. Hence, math calculation is not pyramidal.

Is this a somewhat accurate summary?
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### Re: Pyramidality, Math calculation, and the Goldfish Tournament

dschafer wrote:a lot of stuff
This still doesn't address the problem I implied with math calculation which, handily enough, also covers why questions encouraging (or even engendering, really) lateral thinking tend to be poor ones. I don't think anyone is arguing that point.
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### Re: Pyramidality, Math calculation, and the Goldfish Tournament

argh you people are not addressing my problems regarding computational math.

Problem 1:
Suppose that either I am the only one buzzing on a question, I am the player that wins the buzzer race, or the question is a bonus question directed at my team. If I recall the concept in question incorrectly, I will always get either -5 or 0 points whether or not the question requires an additional computational step. If I recall that concept correctly on a non-terrible quizbowl question, I will always get 10 points, provided that I give a reasonable pronunciation. If I recall it correctly on a computational math question, I am not guaranteed 10 points even if I give a reasonable pronunciation. Since I will always get 10 points on a non-terrible quizbowl question if I recall the concept correctly and pronounce my answer correctly, it stands to reason that computational math questions are either terrible quizbowl questions or not quizbowl questions.

Prove to me that this line of reasoning is incorrect because either:
(1) Given that I have buzzed on a non-terrible question (or it's a bonus question for my team), I have indeed recalled the concept in question correctly, and I am not screwing up the pronunciation, I will not always get 10 points (and therefore one of my premises does not hold);
(2) Given that I have buzzed on a computational math question (or it's a bonus question for my team), have indeed recalled the concept in question correctly, and have not screwed up the pronunciation of my answer, I will always get 10 points (and therefore one of my premises does not hold);
or
(3) My conclusion does not follow from my premises

Problem 2:
Given that I accept the premise that "Can you do this math?" is academic, prove that "Can you do this math faster than other people?" is also academic and is not trivia.

You will note that neither of these problems have any mention of "pyramidality" and that, if no one solves these two problems, I can conclusively state that computational math tossups are something other than good, academic quizbowl regardless of any real or artificial definition of pyramidality used to justify computational math tossups.

But, to answer your question, I do not believe that the "pyramidal buzz distribution" can accurately reflect a pyramidal question, because people who have knowledge will often sit on questions and this will shift the BD to the right. I also think that for a pyramidal question, the cumulative LUC curve (% have buzzed off this or previous clue vs time) should always be monotonically increasing and concave up, like the Lorenz curve. Any such LUC curve that does not satisfy these two requirements has pyramidality problems. I do not believe that the cumulative LUC curve for "standard" computational math satisfies these requirements. The LUC curve for Reinstein computational math may fit such requirements, but I have yet to see defenders of Reinstein math address the above two problems.
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### Re: Pyramidality, Math calculation, and the Goldfish Tournament

Yeah, I'd say that's a pretty good representation of what I'm saying.

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### Re: Pyramidality, Math calculation, and the Goldfish Tournament

cvdwightw wrote:...it stands to reason that computational math questions are either terrible quizbowl questions or not quizbowl questions.

Or a different type of quizbowl questions. You are correct that computational math questions follow different rules, formal and informal, but that does not prove that they cannot fit into a quizbowl match or that they are terrible. Different does not always equal terrible.

Two integers are chosen at random so that each is greater than or equal to negative one and less than or equal to positive two. It is possible for them to be equal. Find the probability that they add up to zero.

Let's say that you knew you were going to a tournament that was going to ask tossups like this one and you wanted to prepare for it by going through old questions. You would solve a problem like this one and ask yourself how it could be done quickly. You probably would quickly recognize that it is similar to the two dice problem that is covered in many Algebra II classes but that it has some differences that don't allow you to use the exact formulas you would use for those problems. You would think about different ways that the two dice problem could be generalized so that you would be ready for similar problems in the future. In other words, you would think conceptually and learn stuff.
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### Re: Pyramidality, Math calculation, and the Goldfish Tournament

Shcool wrote:Or a different type of quizbowl questions. You are correct that computational math questions follow different rules, formal and informal, but that does not prove that they cannot fit into a quizbowl match or that they are terrible. Different does not always equal terrible.
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### Re: Pyramidality, Math calculation, and the Goldfish Tournament

Shcool wrote:Or a different type of quizbowl questions. You are correct that computational math questions follow different rules, formal and informal, but that does not prove that they cannot fit into a quizbowl match or that they are terrible. Different does not always equal terrible.
If I have understood the concept in question, and pronounce my answer correctly, but because of hurried handwriting I mistake the numeral 3 for the numeral 5, or I'm rushing to beat the other team's player and accidentally forget to carry the 1 when adding 687 and 944, then I do not get 10 points. Computational math is the only category that anyone is arguing is acceptable in which I can be denied 10 points through an error that does not involve failure to correctly recall the concept in question (e.g., remembering "It's that guy who wrote Lolita" but not coming up with his name, or misidentifying him as Pasternak) or failure to correctly pronounce the concept in question (e.g. remembering "It's that guy who wrote Lolita" and saying "Nakobov"). If I can do the two things that all quizbowl questions ask me to do to earn 10 points (recall a person/place/thing/concept/etc. and pronounce my answer such that the moderator can understand that what I am saying is equivalent to the answer on the page), and yet not earn 10 points, the question in question must necessarily be a terrible quizbowl question.
Two integers are chosen at random so that each is greater than or equal to negative one and less than or equal to positive two. It is possible for them to be equal. Find the probability that they add up to zero.

Let's say that you knew you were going to a tournament that was going to ask tossups like this one and you wanted to prepare for it by going through old questions. You would solve a problem like this one and ask yourself how it could be done quickly. You probably would quickly recognize that it is similar to the two dice problem that is covered in many Algebra II classes but that it has some differences that don't allow you to use the exact formulas you would use for those problems. You would think about different ways that the two dice problem could be generalized so that you would be ready for similar problems in the future. In other words, you would think conceptually and learn stuff.
Again, this seems to be in line with my premise that "can you do this problem" is academic. I will even go as far as to say that it is possible that "can you do this problem in the allotted time" is academic, because math tests have actual time limits. I still cannot see how this justifies that "can you multiply two numbers faster than the guy on Wheaton North" is academic.

To expand on my position, "Can you identify TOPIC X faster than the other guy" is academic because it requires having deeper academic knowledge of that topic. At worst, there will be a buzzer race when you and your opponent recognize the same clue, demonstrating equivalent academic knowledge (but only one player can come away with 10 points). With math, either I know the trick, or I don't, or no trick exists. I submit that learning these "tricks" is "trivia" (e.g. knowing the Pythagorean formula is "academic" but memorizing Pythagorean triples is "trivia"), and therefore whether I earn 10 points on this tossup is entirely determined by (1) my ability or inability to recognize a "trivia" clue faster than my opponent (because seriously, most of these "tricks" have no academic value), and (2) my ability or inability to crunch numbers faster than my opponent (which serves no inherent academic purpose, whereas "crunching numbers" or "crunching numbers in a given time period" might).
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### Re: Pyramidality, Math calculation, and the Goldfish Tournament

Questions that involve adding 687 and 944 are bad. We agree on this point. I showed you a computational question that did not involve anything like that or memorizing Pythagorean Triples, yet I am told that computational questions are bad because they involve adding three digit numbers and memorizing things. I have written approximately 500 computational tossups in my quizbowl writing career, and I'm pretty sure that not a single one involved such things.

I don't argue that biology tossups are bad because somebody once wrote "Which organ has nostrils?" Don't argue that math tossups are bad because somebody once wrote, "What do you get when you add 687 and 944?"
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### Re: Pyramidality, Math calculation, and the Goldfish Tournament

cvdwightw wrote:But, to answer your question, I do not believe that the "pyramidal buzz distribution" can accurately reflect a pyramidal question, because people who have knowledge will often sit on questions and this will shift the BD to the right. I also think that for a pyramidal question, the cumulative LUC curve (% have buzzed off this or previous clue vs time) should always be monotonically increasing and concave up, like the Lorenz curve. Any such LUC curve that does not satisfy these two requirements has pyramidality problems. I do not believe that the cumulative LUC curve for "standard" computational math satisfies these requirements. The LUC curve for Reinstein computational math may fit such requirements, but I have yet to see defenders of Reinstein math address the above two problems.
Well, it is non-decreasing by definition. I would agree it should be concave up; if more people buzz based off the first sentence than people buzz based off the second sentence, this is a sign of non-pyramidality.

Is a pyramidal LUC curve the only demand we place on a question to be pyramidal, then? Is a tossup yielding a pyramidal LUC curve necessary and sufficient for that tossup to be pyramidal?
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### Re: Pyramidality, Math calculation, and the Goldfish Tournament

Shcool wrote:Questions that involve adding 687 and 944 are bad. We agree on this point. I showed you a computational question that did not involve anything like that or memorizing Pythagorean Triples, yet I am told that computational questions are bad because they involve adding three digit numbers and memorizing things. I have written approximately 500 computational tossups in my quizbowl writing career, and I'm pretty sure that not a single one involved such things.

I don't argue that biology tossups are bad because somebody once wrote "Which organ has nostrils?" Don't argue that math tossups are bad because somebody once wrote, "What do you get when you add 687 and 944?"
You posted your questions right here. viewtopic.php?f=20&t=4779 Plenty of them involve nontrivial calculations and will reduce to calculation speed. Some of them don't even attempt the "restate the problem and give tricks" faux-pyramidality; they are simply one problem that people will solve faster or slower. Your repeated insistence that math calculation tossups are a test of something other than calculation speed does not make it so.
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### Re: Pyramidality, Math calculation, and the Goldfish Tournament

If the time required to translate the words of the question into a bunch of numbers to be calculated takes significantly longer than the actual calculations, then calculation tossups are a test of something other than calculation speed.
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### Re: Pyramidality, Math calculation, and the Goldfish Tournament

Shcool wrote:If the time required to translate the words of the question into a bunch of numbers to be calculated takes significantly longer than the actual calculations, then calculation tossups are a test of something other than calculation speed.
And until the canon of askable math calculation questions becomes large enough, your premise remains flawed.
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### Re: Pyramidality, Math calculation, and the Goldfish Tournament

everyday847 wrote:
Shcool wrote:If the time required to translate the words of the question into a bunch of numbers to be calculated takes significantly longer than the actual calculations, then calculation tossups are a test of something other than calculation speed.
And until the canon of askable math calculation questions becomes large enough, your premise remains flawed.
Joseph Heller wrote:There was only one catch and that was Catch-22, which specified that a concern for one's own safety in the face of dangers that were real and immediate was the process of a rational mind. Orr was crazy and could be grounded. All he had to do was ask; and as soon as he did, he would no longer be crazy and would have to fly more missions. Orr would be crazy to fly more missions and sane if he didn't, but if he was sane he had to fly them. If he flew them he was crazy and didn't have to; but if he didn't want to he was sane and had to. Yossarian was moved very deeply by the absolute simplicity of this clause of Catch-22 and let out a respectful whistle.
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### Re: Pyramidality, Math calculation, and the Goldfish Tournament

The limiting factor is the fact that questions need to be stated in about six lines and answered in about ten seconds, not the fact that a very few good formats don't let unique and beautiful math snowflakes fall from the sky. In case you haven't noticed, bad quizbowl is still the norm in Illinois and elsewhere, and there are literally thousands of distinct calculation tossups asked each year. If there were more than 30 types of problems to be asked, surely someone would have found them by now.
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### Re: Pyramidality, Math calculation, and the Goldfish Tournament

cvdwightw wrote:"can you multiply two numbers faster than the guy on Wheaton North?"
Note: the answer here is "no."
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### Re: Pyramidality, Math calculation, and the Goldfish Tournament

Well, Coach Reinstein's right to a point here: certainly not writing the same isoform problems increases the importance of the modeling time, which is a laudable goal since that really gets at mathematical knowledge and understanding, which is what we'd like to test ideally. However, in the end, as a writer one is constrained to ask problems from the set that 80% of the field can find the answer to in some small amount of time (~10 seconds). In practical terms, that has always meant asking the same few (arithmetic-dominated) problems, possibly with different numbers.
I leave it as an exercise for the reader to determine whether Reinstein's integer problem is actually of that kind (i.e. the kind that is dominated by modeling, not application of a memorized formula and arithmetic). My own inclination is to say not - it's just a four-sided dice problem at heart and I think most people able to solve the corresponding dice problem will see that. Certainly, if that form of problem became popular, everyone would see that (their coaches would teach them how to solve such problems!)
Now, irregardless of the character of the integer problem, perhaps it's possible to write a non-trivial number of questions with difficult modeling and trivial arithmetic. However, if that is so, I've never seen a tournament (including those written by Reinstein and other defenders of computation of this kind) that has featured predominately questions of that kind. Therefore, it's disingenuous to say "We don't call for the end of biology because someone once wrote a bad one-liner on 'the nose,'" i.e. to implicitly accuse the anti-computation camp of drawing a hasty generalization. The fact is that, in my experience, computational math has been more or less uniformly of the same (bad, arithmetic-dominated) type for my entire career, which is not at all true of other fields. I can point one to any number of tournaments that had successful biology (or chemistry or literature or even geography) by whatever standard, but I cannot point to any tournament that had truly successful computation, even by Reinstein's standard, much less my own. Since I know a number of clever, well-intentioned people, (including myself) were responsible for the writing of those computation questions I saw, I'm understandably led to the conclusion that writing good computation, even by Reinstein's standard, is not really consistently possible. I'd love for someone to prove me wrong here, though.

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### Re: Pyramidality, Math calculation, and the Goldfish Tournament

30 Types of Computational Math Questions
1. Combining algebraic fractions by +,-,x,/
2. Algebra Word Problems involving days of the week, coins, etc
3. 2- or 3-digit numbers which meet certain criteria due to factors/digits
4. Matrix Manipulation
5. Linear Systems
6. Nonlinear Systems
7. Polynomial Expansion
8. Solving linear, quadratic, polynomial, rational, or irrational equations
9. Graphs of linear, quadratic, polynomial, rational, or irrational functions
10. Complex Numbers, including DeMoivre
11. Exponential growth and decay
12. Logarithmic Properties
13. Quadratic/Polynomial coefficients from graphical information
14. Arithmetic & Geometric Sequences & Series
15. Other Sequences & Series
16. Eccentricities of Conic Sections
17. Focus/Directrix/Vertex/Latus Rectum Problems
18. Finding the number of vertices, edges, or diagonals of 3D figures
19. Areas & Perimeters
20. Volumes & Surface Areas
21. Inscribed and Circumscribed Figures
22. Triangle Angles, including Law of Sines and Law of Cosines
23. Polygon Angles, Apothems, and Areas
24. Circle chords, inscribed angles, tangents, etc
25. Similarity & Dimensions (eg x2 length is x8 volume)
26. Variation
27. Analytical Geometry Areas, Perimeters, and Distances
28. Converting between rectangular, polar, cylindrical, and spherical coordinates
29. Permutations, Combinations, and Factorials (including license plates/codewords)
30. Binomial expansion/Pascal’s Triangle/Flipping coins
31. Rolling dice, picking cards, etc
32. Conditional Probability/Bayes
33. Features of Trig Graphs
34. Evaluating Trig Expressions, including arcfunctions
35. Simplifying Trig Expressions
36. Parametric Graphs
37. Vectors
38. Limits
39. Evaluating integrals
40. Evaluating derivatives
41. Tangent Lines
42. Max & Min
43. Taylor Series
44. Differential Equations
45. Multivariable Calculus, including rotation volumes
46. Statistics
47. Number bases
48. Logarithmic Scales
49. Distance/Rate/Time problems
50. Riemann Sums

Is this what people mean when they state that there are 30 types of math problems? Even if you change 30 to 50, pretty much every one of these categories contains several different types of problems. If writers are asking the same questions over and over again, then that is the fault of the writers.

I think Mike's case against computational math is the most serious. If all computational questions are bad because nobody, despite best efforts, can write questions that reward what we want to reward, then it is time to give them up. This is somewhat related to Dwight's second point, and I think this issue to some extent comes down to opinions. I claim that students with a good mathematical understanding will be able to answer the probability problem above faster than students without a good mathematical understanding, but it's not a claim that I can prove or state with 99.9% certainty.

For the record, I make no claim that quizbowl requires computational math. Most of the people I am arguing with in this thread have written/edited tournaments without computation that I have taken my team to and praised. I will continue to do so. However, I disagree that NAQT, IHSA, or I commit some horrible act by including computational math in quizbowl. I believe it does test understanding, promote learning, and fit in with the flow of a match.
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### Re: Pyramidality, Math calculation, and the Goldfish Tournament

Shcool wrote:For the record, I make no claim that quizbowl requires computational math. Most of the people I am arguing with in this thread have written/edited tournaments without computation that I have taken my team to and praised. I will continue to do so. However, I disagree that NAQT, IHSA, or I commit some horrible act by including computational math in quizbowl. I believe it does test understanding, promote learning, and fit in with the flow of a match.
Did you see the HSNCT finals last year?
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### Re: Pyramidality, Math calculation, and the Goldfish Tournament

Hi Coach,
Okay, well it seems there's some grounds for discussion at least. I'd challenge you to post a problem that I can't rather quickly reduce to one of a few well-known forms of quizbowl math (and, therefore, an arithmetic contest among people clever enough to see the reduction.)
Also, your list is interesting, but I think it suffers a few flaws. First of all, not all these problems are properly distinct: for example, solving linear systems is generally a matter of matrix multiplication, as is any manipulation on a vector space; for further example, Taylor series and Riemann sum problems are generally matters of evaluating several derivatives.
Secondly, I don't see many of these types of problems in actual quizbowl. My experience dictates that the actual number of common isoforms is much fewer and less interesting. When I tried to stretch the edges of askable math at Earlybird the last couple years to include things like integration by parts, higher-order determinants, Taylor series, etc. I wound up with questions that went dead a lot (even more than computation usually does). This feeds back to one of my original points: we systematically wind up with so many boring, straightforward arithmetic contests because that's what people can do in ~10 seconds. As educators, we both know that rewarding problems in math, ones that really get it what someone knows or understands, systematically take much longer than that.
Thirdly, even when such interesting problems occur, they very generally devolve into arithmetic (or, worse, guessing) contests, too. That's because everyone knows we have to ask a problem that a clever person could generally answer in ~10 seconds (unless you're heartless like me.) For example, I wrote a problem like this for Earlybird at some point:

Code: Select all

``````Find the two real values of x that make T singular when T is the linear transform given by the three-by-three matrix with first row 1, 0, 0; second row 6, 2, 5; and third row 3, x-squared, 10. Note that a singular transform corresponds to a singular matrix, i.e. one with determinant zero. Further note that the required three-by-three determinant can expanded across the top row to yield a two-by-two determinant of the matrix with first row 2, 5 and second row x-squared, 10. For ten points, find the two real roots of 20 - 5 x-squared.
I found that this question (or one just like it) went dead an awful lot, even though it wasn't very hard in the end. I also asked one person who got it early how they'd got it. It turns out they'd just guessed that the determinant of the lower-right submatrix had to be zero, so they just picked the value of x that made it so without understanding. I think that kind of heuristic thinking is wildly applied and successful in computation questions, especially when they're on unfamiliar topics. Also, even though the arithmetic in this problem is very simple if one knows how to reduce the problem to an algebraic equation (find roots of 5x**2 - 20 = 0...), there's still arithmetic to be done there and, if two players both know how to perform the reduction, the quicker arithmetician will win in the end.

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### Re: Pyramidality, Math calculation, and the Goldfish Tournament

Shcool wrote:I believe it does test understanding, promote learning, and fit in with the flow of a match.
I do not have problems with the first claim. I do not think that as currently written, these questions promote learning to the same degree that a question on, say, Bismarck or Santa Anna does, but I am willing to admit this is a matter of opinion. I also think that, due to the extra time given for computation, and the expectation that these questions be of similar length to non-computational questions, these questions have to be shoehorned to fit within the flow of the match - whether or not this is acceptable leeway for "flow of the match" is also a matter of opinion that we apparently disagree on.
Shcool wrote:I think Mike's case against computational math is the most serious. If all computational questions are bad because nobody, despite best efforts, can write questions that reward what we want to reward, then it is time to give them up. This is somewhat related to Dwight's second point, and I think this issue to some extent comes down to opinions. I claim that students with a good mathematical understanding will be able to answer the probability problem above faster than students without a good mathematical understanding, but it's not a claim that I can prove or state with 99.9% certainty.
I have no issue with your claim. However, when two teams have good mathematical understanding of probability (as often happens with places like Illinois) then the deciding factor is speed of computation, with accuracy a distant second. Ideally we would like to test understanding and knowledge of the mathematical concepts behind the problem - and I think we can do that, in bonuses, where "if you know how to set up the problem, you can do it quickly." I do not see how we can consistently do that in tossups, where if two people understand the concept (or don't understand the concept) it becomes a computation race(/guessing game) rather than rewarding the player with more knowledge of the concept.
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### Re: Pyramidality, Math calculation, and the Goldfish Tournament

Mike's example is a good one, and I think it helps my case. There's a problem that, by the end, requires only a little mathematical understanding. Finding the roots of a very simple quadratic that works out about as easily as a quadratic can work out is about 100 times easier than solving my probability question by taking advantage of the patterns and similarity to dice problems. Yet teams at Earlybird, which are as a group well above average in a state that does lots of math, had trouble with it.

I think some of the concerns in this thread are along the lines of what happens if you ask a question and a player on each team immediately sees the most efficient way to set it up, and one player is able to do the calculations in 2.4 seconds while the other one takes 2.6 seconds, and that 0.2 second advantage leads to 10 points and the bonus. In such a case, it is true that one team is getting an oversized reward for a negligible, and perhaps meaningless, level of superiority. If this is what actually happened time after time, it would be a good argument against computation. However, what typically happens is that at most one student takes the efficient path toward an answer. Often, no students do, and it eventually goes to a student who gets it with his second line of attack or a student who finds the answer somewhat inefficiently, but more efficiently than anybody else in the room.

Sometimes computation comes down to a split second, but sometimes regular questions come down to a split second, and in those cases the student who was able to buzz on hearing half the key word hasn't really shown some meaningful superiority over the student who waited to hear the whole word.

In a way, it is similar, though different, than what happens with pyramidal tossups, which is demonstrated somewhat by the original posts in this thread by Dan.
Here is the first Goldfish question. (Ignore the multiples of 5.)
Pencil and paper ready. Eric5 has a balance scale and10 six objects that weigh 2,15 [pause] 7, [pause] 8, [pause]
10, [pause] 14, [pause] and20 17 kilograms. He wants to25 put three objects on each30 side of the scale so35 that it
balances; he thinks40 the best way to do45 that is probably to find50 three objects whose total mass55 is exactly
half that of60 all six. (*) For 10 points65—find either of the two70 sets of three weights that75 will balance the scale.
My sense is that the limiting factor on this question is computational speed. You have to add up the masses, divide by 2, and then add different combinations of them until one works. The point that a few people made about students buzzing in after hearing Clue E based on what they heard in Clue A probably happened a fair amount of the time.

However, here is the second question. (Ignore the multiples of 5 again.)
Pencil and paper ready. A is5 the point (9, 7) [“nine comma seven”]. B is10 the point (-4, 2) [“negative four
comma two”]. ABC is15 a triangle with its centroid20 at the origin. The coordinates25 of the centroid of a30
triangle are just the arithmetic35 means of the coordinates of40 its vertices. (*) For 10 points45—use this
information to find50 the coordinates of point C.
In this question, understanding mathematical principles is a big advantage. For one thing, understanding centroids allows one student to start solving the problem a sentence before students who don't understand centroids. Since the calculations are easy, a one sentence advantage is very significant. Second, understanding averages around zero is a big advantage, since a student who immediately recognizes that adding the x-coordinates together and taking their opposite gives the correct answer is going to have a big advantage over a student who takes time to recognize that fact. It may be easy to assume that every team will have somebody who sees that right away, but that's not the case. Remember Mike's example.

I'm not sure I understand Mike's challenge. A good problem is one that somebody with good understanding can quickly reduce to an easy problem. You have good understanding, so you should be able to do most quizbowl math at least somewhat quickly.

I'll agree that my list is flawed. My point, however, was that the argument about there not being a wide enough variety of problems is wrong. There is a wide variety of problem types--I think that the number of problems, if you considered the different combinations of facts that could be presented and the different quantities that could be asked for but ignored just changing the numbers, is wider than the high school fine arts canon. It's of course true that Taylor Series and derivatives are connected, but the action of taking a derivative varies widely depending on the function you are taking a derivative of and whether or not you are given an explicit formula for that function.

I did not stick around for the HSNCT Finals. I remember reading on this board that the last question was computational and that the team with a super computational player got it, which upset some people.
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### Re: Pyramidality, Math calculation, and the Goldfish Tournament

Shcool wrote:I did not stick around for the HSNCT Finals. I remember reading on this board that the last question was computational and that the team with a super computational player got it, which upset some people.
Actually, said super computational player took a lot longer than I expected. The question gave you either the numbers 17 and 15 or 8 and 15, which means you're obviously talking about a 8-15-17 triangle, and the only thing they're going to ask is the area; I'm surprised it took as long as he did to buzz with it. Perhaps he wanted confirmation from the question about what it was asking? That's actually a great illustration: at high levels, you're not going to have teams with players who aren't sure how to speed up a calculation once they're given that the average of some numbers is zero; you'll probably also have teams with at least one player who knows the very basic definition of what a centroid is. So they all hear two ordered pairs, write them down, and wait for the question to actually be asked--then they average each coordinate, add a minus sign, and hit the buzzer. This is exactly how I played every "find the midpoint of this stupid hypotenuse" question in high school, and I won't pretend I'm great shakes at math. But I could strategize around my lack of real math knowledge simply by finding the obvious desired combination of numbers. You can't avoid real math knowledge if you ditch computation.
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### Re: Pyramidality, Math calculation, and the Goldfish Tournament

I brought up the HSNCT situation because of this comment:
However, I disagree that NAQT, IHSA, or I commit some horrible act by including computational math in quizbowl. I believe it does test understanding, promote learning, and fit in with the flow of a match.
That question was a great example of the damage that a math contest does to "the flow of a match." Everything came down to tossup 26, at which point the tournament decided the result of a game that knocked the fifth-place team out of the tournament and decided who advanced to the semifinals, based on finding a cube root. It was completely divorced from everything else in that game--and, by the way, it may also have been unpowerable (certainly a nonzero number of the calculation tossups in the HSNCT were), as if we needed even more reasons why math calculation tossups don't fit in NAQT. The tossup, by the way, was followed by a conceptual math bonus, which just goes to show that there's no reason you can't ask about math without having calculation tossups.

NAQT takes itself very seriously, as it should, and its teams take it even more seriously. If we're going to have hundreds of NAQT tournaments per year, leading up to a 200-team HSNCT, and then winnow the field down after we've selected the top 5 teams out of the 3000 or so that played a full year of events, then we need to do it on good quizbowl. That tossup wasn't good quizbowl, it was a sore-thumb-like remnant of NAQT's decision to placate states like Illinois that aren't interested in good quizbowl and wouldn't buy NAQT's product if it didn't stop the quizbowl match every 3 minutes for a math contest.
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### Re: Pyramidality, Math calculation, and the Goldfish Tournament

Shcool wrote:However, what typically happens is that at most one student takes the efficient path toward an answer. Often, no students do, and it eventually goes to a student who gets it with his second line of attack or a student who finds the answer somewhat inefficiently, but more efficiently than anybody else in the room.
Stop saying this! It's simply untrue! I've played enough high school tournaments, and talked to my team enough, to know that these tossups, no matter how initially pyramidal or whatever, consistently come down to either applying a form of a problem that's come up in QB before, or pure computation speed. I find it difficult to begin to accept your point when your premise flies in the face of all my empirical evidence.
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### Re: Pyramidality, Math calculation, and the Goldfish Tournament

Also, another problem I have with the examples of pyramidal comp math that I haven't seen come up yet is that, everything else aside, they do a worse job of differentiating players simply because they have fewer clues. With respect to the concepts, at least, the numbers are literally just filler for these clues of concepts, while math theory tossups can go more into depth into individual clues, and also provide more clues in the same amount of space.
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### Re: Pyramidality, Math calculation, and the Goldfish Tournament

My challenge is that computation systematically rewards arithmetic speed and little else. I further claim that this is perhaps necessarily so due to the stricture that most teams must be able to get each question in only a few seconds. I back this up by the fact that I've never a tournament's worth of computational math of which I wouldn't say that (with the possible exception of the impossible math I wrote for Earlybird 2005 which, as I noted, went dead everywhere because even math-immured Illinois Scholastic Bowlers can't do real math problems in 10 seconds.)
I don't think anyone's problem is that computation is potentially rewarding a negligible level of superiority: that happens all the time, even in the very best quizbowl, when two players know the same clue but one gets in just a moment earlier. My problem is that computation systematically rewards negligible superiority at a skill (arithmetic speed) that is not recall of knowledge/understanding and is consequently fundamentally unlike what other good quizbowl should be. It's also a skill that is already massively over-emphasized by lower-level math curricula, especially given how useless it is for doing real math (a point on which I think we agree.)
I ask: do you disagree with this? Do you think math as written now (by anyone... or really as written ever) doesn't massively reward arithmetic speed primarily?

MaS
Mike Sorice
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### Re: Pyramidality, Math calculation, and the Goldfish Tournament

I invite Mike to conduct an experiment tomorrow. You are going to moderate 14 matches, and each match is going to have three computational tossups. One of the three will be a pyramidal math question (like the ones in the thread previously linked to), and the other two will consist of single math problems (one in Algebra and one in either Geometry or Trigonometry). In a significant minority of rounds, the pyramidal math will be noncomputational, but those rounds will each have a computational physics question instead. By Illinois standards, the time limit on computational questions is 30 seconds.

When students come into your room, tell them you would like them to stick around for two minutes after the match to complete a short survey on computational math questions in quizbowl. You can either get data from an oral interview or written response. I don't have enough time today to design a survey, but I can run it off for you in the morning if you email it to me. I think the key thing we want to find out here is, if a student did not answer the tossup, either because the other student got it first or time ran out, where were they in the problem. However, I'll leave it to Mike to ask what he wants to ask, and of course anybody can make suggestions. You can hold on to the packets used in your room, and you'll probably want to take notes on how much time it takes students to answer questions and whether students buzz in with wrong answers. Of course, all students should be given the right to opt out of this survey. Given the time allotted per match and Mike's ability to read quickly, this will not mess up the tournament as long as the surveys can be done in something like two to three minutes. (I run a tight ship that keeps things moving, but Mike reads faster than most of my moderators.) Mike's not the only one on this board who will be moderating for me tomorrow--other people interested in collecting data should post here or email me in case we come up with a written survey or agree on what questions to ask.

This is not standard quizbowl, since it is a Solo Tournament. This difference is significant, since there are many quizbowlers who during a normal tournament put their pencils down and let their teammates take care of it. However, I think it is close enough to standard that it could give us interesting data. Mike will have some repeat students in his room, but he probably will read for about twenty unique students over the course of the day.
David Reinstein
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### Re: Pyramidality, Math calculation, and the Goldfish Tournament

Please give me copies of the survey when it's made; I'd like to give it to willing players in my room.
Jonah Greenthal

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### Re: Pyramidality, Math calculation, and the Goldfish Tournament

I don't think that the claims I'm making can be falsified or supported by a survey of this type (though other claims against computational math certainly could.) The problem is that I need to know not where in the computation (non-)respondents were in their process, but rather why they were only there and not at the end. To really measure this, I'd need to ask everyone a number of computation questions, allow them to solve the questions (while still demanding that they do so as quickly as possible,) and measure the following things for each student:
0. Did they succeed? Why?
1. Among those who eventually succeeded, what was the last information they used to succeed? I.e. where on each question I could effectively stop reading because, if the student was ever going to know how to do the computation, they know at that point. What did the student need to know to establish the point where it was?
2. How long after I reach point 1 for each student does it take that student to model the problem (say by reducing the problem to an algebraic expression or root of an algebraic equation.) What techniques are used in doing so?
3. How long after the reduction achieved in part 2 does it take the student to compute the answer and what techniques were used in doing so?

For computation to be valid by your criterion, you must contend one or more of the following:
a. Point 1 is significantly earlier in the question for players with a lot of legitimate math knowledge/understanding/insight, so that the resulting time gain dominates steps 2 and 3 and such players tend to win.
b. If point 1 is approximately the same for different players (as I contend,) step 2 dominates step 3 on average and is significantly shorter for players with knowledge/understanding/insight, so that such players tend to win out.
c. Step 3, when important, is significantly shorter for players with knowledge/understanding/insight because the best techniques are non-trivial.
Otherwise, pretty much everyone's going to know how to do the problem at the same point and the straightforward arithmetical computation steps that comprise part 3 dominate outcomes, so players who can execute those quickly are the ones winning the questions. That is precisely what I'm contending.

Now, I think most of these things could be measured, though not without being a good deal more invasive than a simple after-round survey. If someone sees a way to measure these things with the means at my disposal tomorrow, I'll gladly take that up with vim; otherwise, this will have to wait until we devise a better way to take data.

MaS
Mike Sorice
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### Re: Pyramidality, Math calculation, and the Goldfish Tournament

Send me a copy as well. As a player familiar with computational math, I would be interested to see how well students these days do computation math, and whether it is really math theory or simple number-crunching (which is not real math).

On a side note, why aren't there more math theory questions? I went to the Math Olympiad Summer Program in 2005, and there were very challenging and useful theorems from Algebra (Descartes Rule of Signs, AM-GM-HM inequality, Cauchy-Schwarz), Geometry (Ceva's Theorem, Pascal's Theorem, the nine-point circle, Euler's Line), Combinatorics (Catalan numbers, derangements), and Number Theory (Fermat's Little Theorem, the four-squares theorem, Wilson's Theorem). None of those theorems go beyond calculus. All of these are infinitely more applicable than knowing how to multiply shinty-six by twentington faster than the opponent.

Math is certainly a field where there could be a lot of canon expansion. I think computational questions are a stop-gap until players are good enough to handle advanced math theory, but 10 years from now, I see quizbowl players learning these advanced math skills outside of class, just as literature experts learn from reading plot summaries and works outside of class time.
Greg (Vanderbilt 2012, Wheaton North 2008)

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### Re: Pyramidality, Math calculation, and the Goldfish Tournament

Okay, so I thought I'd look at Reinstein's second example (which he seems to find acceptable) using my proposed inventory.
0. I was able to solve this problem without any difficulty because it's easy (requires, at the ceiling, math I learned in 7th grade.)
1. I don't know how to answer this question until the end; nobody possibly can because the question isn't posed until the end. Now, an astute player might guess what they want somewhere in the middle; that's fine. However, any other question that requires you to do that is a hose and so is this one. I guess I won't know if I'd have guessed what they wanted in the middle somewhere because I saw the whole question at once.
Supposing this question were re-written not to be a hose, I might be able to leverage my knowledge of how to find a triangle's centroid (learned in 7th grade) into knowing what the question wanted in the middle (at "origin.") That's as early as anyone who isn't blatantly guessing can possibly get this question... though I should note that I expect that, if I read this to enough people, at least one will get it before that point because people will guess the centroid's the origin; after all, any sensible player will know that it almost always will be, because that makes the problem easiest!
2. This problem is extremely easy to set up, requiring me to know only the definition of "mean," which I learned in 6th grade. The set-up time is negligible; certainly less than a second and certainly longer than it took me to just write down the expression.
3. This problem is just straight-up arithmetic; add two numbers, negate the result. The useful skills are: second-grade arithmetic. Actually doing the arithmetic took me a little under 10 seconds (which is abominably slow; certainly a good player could do this in under 5 seconds.)
So we see that this was a bad question and, for me, it was a calculation-dominated question requiring only very rudimentary knowledge. I don't see how this question is going to be otherwise for anyone else; if you see a way, let me know.
rjaguar3 wrote:...why aren't there more math theory questions?
See, now that's what I want to hear! I have a feeling you're going to be enjoying college quizbowl math, my mellow.

MaS
Mike Sorice
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### Re: Pyramidality, Math calculation, and the Goldfish Tournament

Captain Scipio wrote:
rjaguar3 wrote:...why aren't there more math theory questions?
See, now that's what I want to hear! I have a feeling you're going to be enjoying college quizbowl math, my mellow.

MaS
I have enjoyed college-level quizbowl math. Unfortunately, in Illinois, it would be unacceptable because it would go dead in every single room.
Greg (Vanderbilt 2012, Wheaton North 2008)

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### Re: Pyramidality, Math calculation, and the Goldfish Tournament

rjaguar3 wrote:Send me a copy as well. As a player familiar with computational math, I would be interested to see how well students these days do computation math, and whether it is really math theory or simple number-crunching (which is not real math).

On a side note, why aren't there more math theory questions? I went to the Math Olympiad Summer Program in 2005, and there were very challenging and useful theorems from Algebra (Descartes Rule of Signs, AM-GM-HM inequality, Cauchy-Schwarz), Geometry (Ceva's Theorem, Pascal's Theorem, the nine-point circle, Euler's Line), Combinatorics (Catalan numbers, derangements), and Number Theory (Fermat's Little Theorem, the four-squares theorem, Wilson's Theorem). None of those theorems go beyond calculus. All of these are infinitely more applicable than knowing how to multiply shinty-six by twentington faster than the opponent.

Math is certainly a field where there could be a lot of canon expansion. I think computational questions are a stop-gap until players are good enough to handle advanced math theory, but 10 years from now, I see quizbowl players learning these advanced math skills outside of class, just as literature experts learn from reading plot summaries and works outside of class time.
I would like nothing more than to see MOP-level knowledge in math theory questions, and the answers certainly are out there to be written about. Unfortunately, we'd have to fire the canon out of a cannon to get from where it is right now to MOP-level. As an example, "derangement" appeared in my archive exactly once in a math context, Ceva's Theorem was rightly in the middle of the triangle tossup at EFT and Fermat's Little Theorem was a middle part in CMU's ACF Fall packet last year. All of the items that you list would make for fine math theory tossups (and hearing a tossup on any one of them would make my day), but I suspect they are all too hard to toss up right now even at the college novice level, let alone in high school.

I think part of the issue is the low representation of math theory at both the high school and college levels; the standard ACF distribution includes no more than 1 math question per packet, with that not even required, and few if any high school competitions go beyond 1/1 math theory. It is necessary for the lit canon (as an example) to evolve and grow with 5/5 a packet, since if it didn't, we'd hear the same things constantly asked about every tournament. A canon of 100 answers easily suffices for the entirety of math questions in an entire calendar year; 100 answers wouldn't get you enough lit for half a tournament.
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### Re: Pyramidality, Math calculation, and the Goldfish Tournament

think part of the issue is the low representation of math theory at both the high school and college levels; the standard ACF distribution includes no more than 1 math question per packet, with that not even required, and few if any high school competitions go beyond 1/1 math theory.
I think part of that is that this is representative of a difference between tournaments/formats which emphasize math and those that don't. The latter, logically, do not see a great need for a large number of math theory questions and the former tend to represent math with computation.
Douglas Graebner, Walt Whitman HS 10, Uchicago 14
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### Re: Pyramidality, Math calculation, and the Goldfish Tournament

I understand Mike's objections, and he is correct that any such survey will give us a very limited amount of knowledge. My offer stands for anybody, however. If there is a set of questions emailed to me or posted here by 7:00 AM tomorrow morning that seems like they might tell us anything, I'll make some copies and find out which moderators are interested in handing them out.
David Reinstein
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### Re: Pyramidality, Math calculation, and the Goldfish Tournament

I don't think your post, Dan, quite gets everything right. I know we write 1/1 math for pretty much every ACF packet ("Other Science" being under our control) and it's not especially uncommon to see that level in the edited product. Also, the MOP topics given seem to cover only Euclidian plane geometry and rudimentary number theory. The askable math topics are much, much broader than that, even at the high school-level (including things like the entirety of calculus and statistics, for example.) So, while some of those topics are beyond what's askable, even in lower-level college tournament, there are whole swathes of other math theory that are askable (and are asked!)

MaS
Mike Sorice
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