Mathematics and calculators

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Mathematics and calculators

Post by First Chairman »

In continuing discussions from the Nationals board, I am interested in the construction of calculation questions involving calculators.

I know I have my particular opinions about it, mostly mixed. But I want to know what makes a good quiz bowl/academic competition question that involves a calculator? Most of us who do academic competition are not familiar with this as a question format.

The other thing that I would have to note is that you have to have a calculator policy. We don't want to be encouraging a calculator "arms escalation" as it were (with someone with a $5 calculator competing against someone with a scientific calculator). I just want to know what other policies on calculators in competition state (like in ARML).
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Post by Stained Diviner »

NSML (Chicago area):
"The only materials students may use during the contest are: pens or pencils, erasers, calculators, and extra batteries. A student may use as many calculators as will fit on the top of their desk. Any battery (or solar) powered calculator that does not do symbolic manipulation is allowed. Freshmen, sophomores and juniors may not use an HP48 series calculator. In some cases, symbolic manipulators may be expected. These contests will be designated as CAS topics. Likewise, some contests may be designated as no calculator contests. Contests with special calculator exceptions will be identified as such on the schedule and on the topics list."

ICTM (Illinois state competition):
"Rulers, protractors, compasses and calculators are allowed in all areas of competition except for the Eight-Person Teams, where no calculators are allowed. Each student is allowed to use any number of calculators that are self-contained and battery operated; however, each student may use at most one desk. No computers are allowed in the competitions. Removable or interchangeable parts are permitted. Electrical outlets in the room may not be used."

ARML:
''Students may use any calculator approved by the CEEB for use on the SAT's given immediately prior to that ARML competition."

CEEB:
"Every question on the math section of the SAT can be solved without a calculator; however, using a calculator on some questions may be helpful to students. A scientific calculator or graphing calculator is recommended.

The only Subject Tests for which calculators are allowed are Math Level 1 and Level 2. A scientific or graphing calculator is required for these tests. A graphing calculator may provide an advantage over a scientific calculator on some questions. The tests are developed with the expectation that most students are using graphing calculators. Calculators are not allowed for any other Subject Tests, including Biology E/M, Chemistry, and Physics.

Students should bring a calculator with which they are familiar and comfortable. Their degree of familiarity with the operation of a calculator may affect how well they do on these tests.

For both the SAT and Subject Tests, the following are NOT allowed:
calculators with QWERTY (typewriter-like) keypads
calculators that contain electronic dictionaries
calculators with paper tape or printers
calculators that "talk" or make noise
calculators that require an electrical outlet
cell-phone calculators
pocket organizers or personal digital assistants
hand-held minicomputers, powerbooks, or laptop computers
electronic writing pads or pen-input/stylus-driven devices"
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Post by pakman044 »

One of the problems with the SAT calculator rules is that they already may be outmoded. I believe the TI-89 already supports symbolic integration (the 86 only supports numerical), which already can make it something of an arms race. For that reason, a number of college professors forbid ALL graphing calculators (and of course there's no way to stop people from writing TI BASIC or assembly programs). In a sense, once you allow any kind of programmable calculator, you already have an arms race--a programming arms race.

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Post by Stained Diviner »

It is getting trickier all the time. There used to be a big gap between the TI-81/82/83/85/86 and the TI-89/92. The TI-84 fits in the middle. Off the shelf, it's a TI-83 Plus. Once you download all the programs, it's close to a TI-89.

As a contest operator, you don't want to tell a kid who bought a $140 calculator that he should have bought the $100 calculator.

It is possible, however, to write questions where the gap between a TI-83 and TI-89 is not that big of a gap. The question writer really needs some familiarity with the different machines, however, and that is a lot to ask of quiz bowl writers. However, it is not so much to ask of math contest writers. A student using a TI-89 on the AMC or SAT exams will not have a huge advantage over a student using a TI-83.
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Post by First Chairman »

I'm wondering... would it be possible for the contest director to provide calculators for all participants? Basically, only the "official" calculators purchased by the TD is the allowed calculator.

For OAD, we strictly tell everyone to bring the calculators that each school system has purchased specifically for the competency tests (for graduation). So the team coaches bring the calculators for their kids, and we know that they conform to the standards set by the state department of education for those exams.
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Post by Ben Dillon »

Allowing programmable calculators would be IMHO dangerous. Teams would start obtaining/writing programs that would speed up calculations for specific problems.

For example, we've all heard this nugget asked: "Suzy Q gets 60%, 50%, and 90% on her first three tests. What score does she need to get on her fourth test to have an average of 75%?"

Now, ignore the fact that this is an extremely basic question. It certainly can be made simpler with a prewritten program that cranks out the answer. And a student adept at working the programmable calculator will end up getting the answer slightly faster than someone with paper and pencil.

Now write a more complicated math question. The result might be that someone with just the right program gets the answer significantly faster than someone without it. Thus we end up rewarding the student who can program rather than the student who can calculate.
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Post by NoahMinkCHS »

Ben Dillon wrote:Thus we end up rewarding the student who can program rather than the student who can calculate.
Neither of which should be tested in quiz bowl, as neither of which requires recall of knowledge, which (IMO and that of, I believe, most of this board) is the goal of our activity.

But I agree -- if calculation questions are expected and accepted, then the focus shouldn't be on the machine, especially when most writers (and educators, for that matter) don't know the nuts and bolts and capabilities of everything on the market.
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Post by No Sollositing On Premise »

The issue of having pre-made programs on calculators is resolved by having the event issue out calculators with cleared memory; I think that it isn't a bad idea to have simple, TI-25x level calculators handed out that only handle basic functions, exponents, and trig. However, most math/calculation questions on conventional formats don't present any need for a calculator at all. In short, calculators aren't necessarily a bad thing for quizbowl, but quizbowl just doesn't need calculators.
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Post by Stained Diviner »

It might be possible for the AD to hand out the calculators. TI is sometimes willing to loan calculators for free or cheap depending on the situation, and many math departments have a classroom set or two lying around.

Even if kids do bring calculators, it is possible to write problems that minimize different types and programs. Students can bring programmed calculators to the AMC, SAT, AP, and ARML, and it has not resulted in a major arms race.
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Post by Romero »

As PhD student in math education, this topic is particularly interesting to me. The following is supported formally in the literature though I will leave out the citations. Inquire if you really want to know.
pakman044 wrote:A number of college professors forbid ALL graphing calculators
Unfortunately mathematics knowledge is widely viewed as the ability to regurgitate algorithms, rather than understanding certain concepts. College professors like those Patrick alludes to, are stuck in some world of the past. Such policies go against EVERYTHING that modern research tells us about learning. The reason you hear about horrible math scores is that it is not taught correctly. Learning does not occur in a vacuum (i.e. without social context or available calculators) and hence no assessment should be completed in a vacuum. That dichotomy is the source of the lack of "transfer" that has plagued cognitive theorists all the way back to Dewey and Thorndike. Most professors of mathematics proper and teachers trained under antiquated theories choose to teach passively because it has "worked" in the past. It actually didn't work (hence all the reports that the US sucks in math), they (just like unsophisticated question sources) just don't know better.
Ben Dillon wrote:Allowing programmable calculators would be IMHO dangerous. Teams would start obtaining/writing programs that would speed up calculations for specific problems.
Ben Dillon wrote:For example, we've all heard this nugget asked: "Suzy Q gets 60%, 50%, and 90% on her first three tests. What score does she need to get on her fourth test to have an average of 75%?"
What Ben is saying here is that we need to insulate incompetent question writers. Calculators (of any type) should be allowed on computation questions. Questions can be written such that the effect of programming is minimized. With respect to an arms-race, yes there are "super-calculators" with on-board computer algebra capability. However such calculators only make a competition unfair if the questions are in and of themselves unfair. Calculators would open a multitude of possibilities for unique and creative question topics.

IMHO, NAQT is particularly guilty of an overreliance on computation questions. There should properly be no more than 1 (instead of 2-3) per match. I would suggest replacing those computation questions with questions on topics such as the square (Leo wrote a horrible question for PACE on that, though I applaud the notion of such a question).

Matt Weiner is on record, I believe, as supporting less didactic emphasis in the social sciences (questions on more germane topics rather than outdated theorists and books). This argument is parallel to the math argument that I make.

In the cognitive scheme of mathematical learning, the ability to compute an integral is trivial; it is almost not worth learning. However the understanding of what an integral represents in the context of area, probability distributions, etc. is fundamental to a conceptual understanding of mathematics.

To sum this up...All calculators should be allowed on computation questions and question writers need to write better math questions.

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Post by First Chairman »

The Duke Academic Festival usually puts in a computation question but only as a handout question. This reduces any translational errors coming from the reader (who may not be mathematically adept), and it allows for having one math specialist do the problem with a bit more time than the hurried 5 second bonus part. That's my philosophy on this.

But as for writing better questions that require calculators for open-ended questions (that is to say, not multiple choice), I would be interested in seeing what would make such questions "better." What is the best way to test particular concepts in mathematics for which calculators are allowed? I am certainly interested in the opinions of the math educators on the board on this.
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Post by Captain Sinico »

The problem with your points, Romero, valid as they may be in the sterile and abstruse vacuum of curriculum theory, is that the objective of quizbowl isn't to teach, but rather to test. Specifically, quizbowl is not meant to instill an understanding of mathematics on whatever level, but to determine something about the competitors: usually who knows more faster, but, in the case of computation questions, who can compute the fastest by whatever means. All of your contentions and, especially, value judgments are made under and depend to whatever extent upon the inherently false pretense that the primary and, indeed, only purpose of quizbowl is education; these same would not stand up with an accurate accounting of the purpose of the game.
So that my position is not misrepresented, I must say that, while learning of whatever kind is a secondary benefit that sometimes accrues to quizbowl players (and, as an aside, is certainly the most healthy reason I can think of for wanting to play to begin with), the purpose of the game itself is and must be to determine who's better at the game. This is so almost by definition. Questions should be written and policies made with this in mind.
Furthermore, I don't see how your contentions, if true, can allow any computational questions of any sort. If the ability to compute an integral is only just better than worthless, then what computation skill (or, especially, computation skill possessed by an actual high school player, with or without a CAS) can be considered valuable? Since the answer admitted by your contentions is clearly none, then the players possess no computation skills of value, so how can one, in good faith, test them on those skills? It is pointless to argue for or against use of calculators using principles that would not validly allow calculation questions to begin with.
I would also especially like to challenge your statement about integrals. Not only is computation of integrals not, in general, trivial, but also the ability to compute them can allow its possessor to realize, in a deep and meaningful way, the meaning of the integral in whatever context. Therefore, the ability to compute integrals, by whatever means, must be considered valuable even in a strictly didactic context, laying to one side the fact that the would-be mathematician unable to evaluate integrals is practically (in the proper sense) useless. For example, I can understand all day that the velocity moment of my favorite distribution function is the particle density, but that won’t help get my plasma discharge running any better until I compute what that is, exactly. This same idea generalizes to all areas in which integrals have concrete interpretations.
To the actual point, I see nothing wrong, in principle, with allowing the use of whatever kind of calculators in quizbowl, so long as cheating can be practically excluded (rules being invalid if unenforceable.) The admission of calculators simply causes computational math questions to test something fundamentally different than they test if calculators are excluded, and the choice of what is valuable and ought to be tested in quizbowl is as close to an unquestionable postulate as you’ll find. It depends entirely on things external to the game itself.

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Post by Howard »

Romero wrote:Calculators (of any type) should be allowed on computation questions.
No, no, no. It the question is actually about computation, calculators should not be allowed. That's nothing more than using a machine to do something a human is perfectly capable of doing.

Calculators should be used for computation when the actual concept being tested is not the computation but rather the concepts which determine the calculation which must be made.

This is not to say that calculators in math education is a poor idea. In fact, it's a great idea. Their use, however, should be limited. Modern calculators allow us to estimate solutions to problems for which we may have otherwise never found solutions. They provide to us time-efficient methods of solution which were previously unavailable. It's just this sort of problem solving which should be taught with calculators.

But seriously, they should not be a mechanism for adding, subtracting, multiplying, dividing, nor manipulating variables. Humans can do that and it's essential for everyday life and important for simple problem solving to be able to perform these tasks. Even as an auto mechanic, I can tell you that. If I have a car which blows the 30A fuse every time the rear window defogger is turned on and operating voltage is 15V while the car is running, then I don't need a calculator to tell me that if the resistance of the rear window grid is less than 0.5 Ohms, the car needs a new rear window grid.

Only when the numbers get ugly, does one need a calculator for computation. But the question writer can control the numbers to reflect what's being tested.

The next time I go into a convenience (or any other store) and the bill comes to something like $7.03, and I have a $10 bill in my hand, and see the confusion on the cashiers face as I hand over the $10 bill and dig for the $.03, perhaps I should send them to Romero so he can show them how to solve this problem with an adding machine.
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Post by Romero »

ImmaculateDeception wrote: The purpose of the game itself is and must be to determine who's better at the game.
Amen! Alleluhah! Mike is correct. Quiz Bowl in and of itself is an inadequate testing of information that masquerades itself as "academic" knowledge.

I absolutely agree that there is no currently acceptable computation question that I think is worthy of inclusion; Non-trival computations cannot be done in 10 or 20 seconds mentally. Though I believe that non-trivial computations can be done with calculator in that time. Overall I believe that questions like the square are infinitely more worthy of inclusion.
ImmaculateDeception wrote: All of your contentions and, especially, value judgments are made under and depend to whatever extent upon the inherently false pretense that the primary and, indeed, only purpose of quizbowl is education.
Do the apologists for PACE and NAQT not make such claims. The implication of value-judging NAC as trivial is that somehow that NAQT and PACE are non-trivial. The central issue on which the question of format legitimacy rests is the academic nature of the subject matter. Yet at least in this context, NAQT and PACE are guilty of the same category of offense for which we all are quick to chastise :chip:.

I really have no issue with current NAQT/PACE policy, though I think there are too many computation questions in NAQT. The various responses about calculators specifically Patrick's, Ben's, and Noah's, sparked my desire to speak out against the false pretenses which I believe underpin their arguments. The implication I find in this thread seems to support the notion that somehow calculators undermine the purity of mathematics knowledge demonstration in quizbowl. This is patently untrue. As long as we are willing to concede that computation questions fail to demonstrate any "real" knowledge but rather are pure tests of speed, I have no issue with them. If we aim to test knowledge we must include calculators; if we are willing to test speed then we should disallow them. This is why I really have no issue. For me the case is black and white and any discussion of allowable models or clearing memory is silly.

Back to Mike's points, the ability to conceptually understand integrals is in no way related to the ability to grind out numerical values for them. Modern pedagogies have no need for all of the various substitution tricks that can be done. Any of a number of tools can be used to solve that plasma discharge integral. You don't have to do it. It is a misnomer that somehow the ability to do that integral makes you smarter or better prepared, unless you consider the hypothetical in which you will be held at gunpoint and forced to do that integral by hand. I suspect, however, that your context is from one of the many antiquated pedagogies that are pervasive in academia. In that case your point, though misplaced in a larger scope, is appropriate.

Howard makes a couple of valid points, but the assertion of his affection for number sense baffles me. How is that at all pertinent? The ability to subtract simple change and divide using Ohm's law does not impress. Though I suppose if your intent was satirizing computation questions in general, you did that.
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Post by mreece »

Modern pedagogies have no need for all of the various substitution tricks that can be done. Any of a number of tools can be used to solve that plasma discharge integral. You don't have to do it. It is a misnomer that somehow the ability to do that integral makes you smarter or better prepared, unless you consider the hypothetical in which you will be held at gunpoint and forced to do that integral by hand.
Maybe modern pedagogies don't, but modern scientists certainly do. Certain problems become much more tractable when various "substitutions" are done. Certain approximations become more obvious, for instance. Also, some problems are more numerically tractable when expressed in terms of different variables. And someone had to write the computer programs that do these things, anyhow. It's not as if we can all just "forget" this "antiquated" knowledge. Besides, sometimes if you ask a program like Mathematica for a solution to some integral or differential equation, it will give you an answer that is technically correct but not in the most useful form.

Also, I contend that some fairly nontrivial computations can be done in under 10 seconds. (In Kentucky high school tournaments we had 5 seconds to answer tossups, and it never posed any real problems.)

Maybe I'll respond at more length when I'm not in an airport.
Last edited by mreece on Fri Jun 24, 2005 6:29 pm, edited 1 time in total.
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Post by Stained Diviner »

Dr. Chuck can check out http://www.math.ksu.edu/main/events/hsc ... sample.htm for some sample problems. They are multiple choice, but most of them do not need to be. You would need to allow 30 or 60 seconds for them even with calculators.

In terms of whether or not to have computation questions, I believe that if you are going to have a significant amout of math, then you need to have a significant amount of calculation. If you are going to have fewer than 10 math questions in your tournament, then you can ask about the four color problem, Gauss, and a few other things that are mildly interesting, but somebody who understands math can do math. Therefore, they should be tested on their ability to do math.

I also would not make the assumption that anybody who does not teach with calculators is a bad teacher. It is possible to have people think without their calculators. I do teach with calculators and am glad that I do so and am well aware of their benefits. However, no calculators=bad teaching is too simplistic.

NAQT writes good computation questions for the most part. Students who have conceptual understanding and are aware of shortcuts and know how to use them will get 15 or 10 points. Students who need everything explained to them will get 10 points if everybody else in the room is slow. This is similar to the way all of their questions are. There will still be some times that somebody is beaten to the buzzer by a tenth of a second, but that can happen on any question.

Computation questions do require longer time limits than recall questions. In Illinois, we allow 30 seconds. NAQT allows 15 seconds, but it is usually possible to get well into the problem before the reader is finished, so it works out about the same. Many of the questions are simple by the end, but many of the recall questions are simple by the end as well.
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Post by Romero »

Matt is correct...and I glossed those points on purpose. Matt and Mike are both physicists whose experiences with mathematics though not unique are not prevalent. In general pedagogies don't consider folks like Matt and Mike, because no matter what teachers do they will learn anyway. It is the other end of the spectrum we have to worry. In higher mathematics (that which is done at and above the ugrad level) I appreciate the need for robustness and intracacy. But the fact remains that in high school (and lower) there is an overreliance on rote memorization of mundane procedure.
ReinsteinD wrote:I also would not make the assumption that anybody who does not teach with calculators is a bad teacher. It is possible to have people think without their calculators. I do teach with calculators and am glad that I do so and am well aware of their benefits. However, no calculators=bad teaching is too simplistic.
It is too simplistic and no they are not necessarily bad teachers. But, there is a better way. It is possible to think (beyond algorithms that is) without a calculator, but the majority of students never get to think because all their time is spent trying to ape their teacher's work.

For what its worth questions about Gauss and the 4-color problem are not math questions, they are history questions. They require no conceptual knowledge. Yes someone who understands math can do math and SHOULD be tested on their math knowledge. They should not be tested on their speed and memory of "tricks." I learned how to do a derivative as a 9th grader and got lots of points for it. By ReinsteinD's argument, the freshmen, who learns the trick, is more worthy of those points that the slower calculus student cause he can do "math." By the conventional logic of qb pyramidality however the freshmen should not get those points.
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Post by Tegan »

I think I've already been on record for this, but I'm not a huge fan of calculators, unless you are going to bring some <real,> tough math in, and give the appropriate time to do it (in this regard, the PAC handles it well, though I think it can really slow down the tournament, not that all of their math is calculator-worthy).

I think I had this argument and got rebutted by the ID before, but if you get rid of math, then who exactly decides what stays? Is math less important than literature? Is literature more important than art? I think it does open up a slippery slope that, IMO, Illinois addressed a long time ago, and has struck a neat balance that seemd to work. One argument is "if you're interested in math, there is mathletes"...but there are also competitions on art, science, the humanities, foregin language, etc. Should all of these be dropped as well? At this rate, you could argue against practically any subject. The big math difference is that it can rely more on skill than fast recall.

On the other hand, as one of my players was registerng a complaint about the PAC format this weekend, my reply was "It's their tournament, and they can run it any way they want, as long as they inform us up front how they are going to run it." PAC has a lot of math, everyone knows that, and so they can't complain once the tournament starts. NAQT and PACE (and a lot of states) tend to de-emphasize math (did someone say that NAQT has too much math???). That's fine, because it is their choice to do so. I love the quality of NAQT and PACE questions, and I like how they reward more obscure knowledge, quicker recall, and in some cases even a bit of synthesis.

Teaching physics, I see that a lot of students (I'm not talking necessarily about the average quiz bowler here) who have seemingly lost the ability to do even basic math unless they have a calculator in front of them. I think they are important tools, and that students should be taught to use them properly, but I see that a great deal of basic understanding regarding mathematical operations is lost when the calculator is used too early as a crutch. For the more advanced students (and I am talking about a lot of your quizbowl types here), the use of a caluclator could be a neat tool to bring in to a competition, if you are going to ask questions worthy of using a calculator.
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Post by Captain Sinico »

Romero wrote:...the ability to conceptually understand integrals is in no way related to the ability to grind out numerical values for them. Modern pedagogies have no need for all of the various substitution tricks that can be done. Any of a number of tools can be used to solve that plasma discharge integral. You don't have to do it. It is a misnomer that somehow the ability to do that integral makes you smarter or better prepared, unless you consider the hypothetical in which you will be held at gunpoint and forced to do that integral by hand. I suspect, however, that your context is from one of the many antiquated pedagogies that are pervasive in academia. In that case your point, though misplaced in a larger scope, is appropriate.
First of all, the values of integrals don't necessary have to be numerical, as you well know. The particular integral that I'm talking about certainly would not be, since it's an implicit function of a large number of other variables.
Secondly, your assertion that the general integral can be trivially computed in a large number of ways is simply untrue. In fact, certain integrals are only tractably computable in one or a small number of ways. Other integrals are not computable at all. This was all only incidental to my main point, however; what I did say (and you didn’t address) was that the ability to analytically compute certain integrals facilitates the understanding of the meaning of the integral in various contexts. For example, if I cannot verify that a probability density function integrates to unity over the range of its variate, I cannot understand it to be a probability density function.
Finally, there's no need to resort to the hyperbole of computational stand-and-deliver to find a situation in which the ability to actual compute an integral by myself renders me better prepared. For example, just now, I am working with colleagues at New Mexico Tech to design a certain device, most of the important parameters of which depend on several integrals. However, I do not at the present time know the dimensions of the device. When I go into the design meeting and the information that I request to conveyed to me via speakerphone, my PI, who pays me, is going to expect me to give my evaluation of the device's feasibility quickly. Because and only because I can evaluate the relevant integrals myself, I will be able to. Conversely, a hypothetical version of me unable to compute these integrals would be at the marked disadvantage of having to find some outside assistance to perform the same evaluation. Whether I'm smarter or not as a result of what I can do is a moot point since it depends entirely on what's on the intelligence test implicitly in question (intelligence being operationalized by the social sciences as "what is tested by an intelligence test") but there can be no doubt whatsoever that in today's situation, which will actually occur in about half an hour, and in a large number of future situations my ability to compute integrals myself does and will in point of actual and measurable fact render me better prepared than I otherwise would be.
Also, for your information, I taught myself differential and integral calculus and have never taken (and shall never take) introductory calculus at any level. I don't know what pedagogy that places my observations of the subject in, but I also think that's a bunch of obfuscatory nonsense, myself.

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Post by iamsam »

I agree with Mr. Sorice on the "numerical values," I remember an question by NAQT that required you to differentiate functions, now, without a calculator, you either know how to do it, or you don't. (To Mr. Romero) Your argument that computation questions are non-academic is fundamentally flawed. Your hypothetical question is not going to occur in solid formats, most "good" math questions I heard are more word problems than they are computation, they test whether or not you will be able to put all the information together and then churn out the right answer. If you claim that knowing how to do this correctly in irrelevant--partially because we can just have calculators do it--then why even bother memorizing facts, why don't we all just bring encyclopediae to matches?

Also, where has it ever been stated that "convention" states that "freshmen" should not get points for things. It may be a de facto situation that freshmen are not as aware of the canons of quiz bowl, but nowhere does any state that if you place a smart "freshmen" in a room with a "slower" senior, the senior should win. Now you are putting words in other's mouths and crying them to your own tune. Memorizing your so-called "tricks" is not useless and inconsequential, while it may not be the "slippery slope" as Tegan described, it certainly is analogous. If computation questions that require fundamentals of math are "mundane," then why even bother having questions on, for example, Charles Dicken's more famous works? Knowing how to do the chain rule for derivatives or impartial fractions for integrals is just as academic as knowing that John Updike wrote the four "Rabbit" novels. As far as dismissing Mr. Sorice's plasma physics as trivial and possibly solved plainly by calculator, I recomend that you read some of the material--I took one look at some of that stuff, and I wanted to pass out.
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Post by Stained Diviner »

Two respectful points:
Romero--If the differential calculus you learned as a freshman consisted of pretty much differentiating polynomials, and the question writers kept asking you to differentiate polynomials, then that is bad question writing. It isn't much different than having several questions at each tournament about who won the Presidential Election of ____ and having some team become significantly better by memorizing each year's election winner. If you learned how to differentiate lots of functions using the chain rule and how to handle exponential, logarithmic, and arctrig functions, and how to find maxes and mins and tangent lines as a freshman, then you deserved some of those points you got.

ID--If you brought a good calculator or computer to the meeting, it could do the integrating for you. A TI-89 easily does double or triple integrals and can give answers that include variables.
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Post by solonqb »

Pardon my ignorance here, but can a TI-89 do indefinite integrals that are evaluated in terms of nonelementary functions (Gamma function, etc.)?
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Post by Stained Diviner »

I'm not 100% sure, but I think the answer is No.
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Post by Howard »

Romero wrote:Howard makes a couple of valid points, but the assertion of his affection for number sense baffles me. How is that at all pertinent? The ability to subtract simple change and divide using Ohm's law does not impress. Though I suppose if your intent was satirizing computation questions in general, you did that.
You're exactly right. It is unimpressive. And that's why we shouldn't allow calculators for the purpose of performing such tasks. Humans have the ability to perform those tasks efficiently and accurately. I hadn't intended any satire, but I agree I've heard many questions that deserved it.

While I'm speaking from an educational standpoint more than a quizbowl standpoint, my contentions are the same. Calculators are important tools which can and should be included in mathematics curricula, but only for the purpose of problem solutions which would be impossible or tremendously laborious without the calculator.

To allow calculators for the purpose of solving problems which can be solved reasonably efficiently by humans will eventually have us living in a society where scientists and mathematicians won't be able to provide an exact solution to a differential equation or a family of equations. There are certain reasoning skills humans possess that computers and calculators will never possess. To dismiss these skills simply because we have a shiny tool at our disposal is to willingly make ourselves ignorant.
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Post by Stained Diviner »

In the hope that it might improve the debate, here are some questions I wrote for quiz bowl matches in which 30 seconds is allowed and no calculators are allowed:

From questions meant for one-on-one play:
* Ignore units. Find the distance between the two vertices of the hyperbola given by the equation: 4x2-y2=36
* There are no parentheses in this problem. Find the maximum value of y if x is real and y=x-2x4
* Points A, B, and C are collinear, with B between A and C. Q is not on line AC. Find the measure of angle AQC if angle QAB is 25 degrees, angle QBA is 110 degrees, and angle BQC is 70 degrees.
* Give your answer as a single number. What do you get when you divide ‘four raised to the tenth power’ by ‘two raised to the seventeenth power’?
* How many handshakes will take place if everybody in a room shakes hands with everybody else once and there are twenty people in the room?
* Find the focal length of a lens if an object and its image are both located ten centimeters from the lens.

From questions meant for team play:
* What is the current age of Larry? Larry is currently five times older than Michael. In twelve years, Larry will be twice as old as Michael.
* What is the cotangent of the arcsine of 7/8?
* A circuit has two resistors in parallel. If one of the resistors is three-and-one-third ohms and the total resistance is two ohms, what is the resistance of the other resistor?

Here are questions from the 2001 AMC 12 that could be used in a quiz bowl match. Calculators, including TI-89s, are allowed:
+ The sum of two numbers is S. Suppose 3 is added to each number and then each of the resulting numbers is doubled. What is the sum of the final two numbers?
+ Let P(n) and S(n) denote the product and the sum, respectively, of the digits of the integer n. For example, P(23) = 6 and S(23) = 5. Suppose N is a two-digit number such that N = P(N) +S(N). What is the units digit of N?
+ The state income tax where Kristin lives is levied at the rate of p% of the first $28000 of annual income plus (p + 2)% of any amount above $28000. Kristin noticed that the state income tax she paid amounted to (p + 0.25)% of her annual income. What was her annual income?
+ Let f be a function satisfying f(xy) = f(x)/y for all positive real numbers x and y. If f(500) = 3, what is the value of f(600)?
+ How many positive integers not exceeding 2001 are multiples of 3 or 4 but not 5?

Some, but far from all, of my questions would become stupid if students had calculators. Some, but not all, of the AMC problems would require much more than 30 seconds without a calculator. Unfortunately, I cannot make an apples vs. apples comparison because I do not have any question source that makes some questions for calculators and some not for calculators. I think pretty much all of the questions are challenging. That is, if you asked them at a local tournament, many of them would go unanswered. I have little doubt that there are people on this board who could answer all of them within the time limit.

PS Computers can provide an exact solution to a differential equation or a family of equations. While it is still true that there are certain reasoning skills humans possess that computers and calculators will never possess, our ideas regarding what those are decreases significantly every decade.
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Post by jonpin »

ReinsteinD wrote: * There are no parentheses in this problem. Find the maximum value of y if x is real and y=x-2x4
Is something missing in this problem? An exponent?
* Points A, B, and C are collinear, with B between A and C. Q is not on line AC. Find the measure of angle AQC if angle QAB is 25 degrees, angle QBA is 110 degrees, and angle BQC is 70 degrees.
This is a good problem with 30 sec to do it. Read it last month at a practice at my old high school.
* How many handshakes will take place if everybody in a room shakes hands with everybody else once and there are twenty people in the room?
* Find the focal length of a lens if an object and its image are both located ten centimeters from the lens.
[gasp] Applied mathematics?!

Here are questions from the 2001 AMC 12 that could be used in a quiz bowl match. Calculators, including TI-89s, are allowed:
(Wow. I remember everyone of these questions from when I took that.)
+ The sum of two numbers is S. Suppose 3 is added to each number and then each of the resulting numbers is doubled. What is the sum of the final two numbers?
I think it's quite clear that anyone who reaches for a calculator on this is going to get easily beaten to the buzzer.
+ Let P(n) and S(n) denote the product and the sum, respectively, of the digits of the integer n. For example, P(23) = 6 and S(23) = 5. Suppose N is a two-digit number such that N = P(N) +S(N). What is the units digit of N?
A calc is just not going to help you on this, you need to do it algebraically. If calculator questions are included in HSQB, it's my opinion that there should be some questions of this type (where a calc is no use).
+ The state income tax where Kristin lives is levied at the rate of p% of the first $28000 of annual income plus (p + 2)% of any amount above $28000. Kristin noticed that the state income tax she paid amounted to (p + 0.25)% of her annual income. What was her annual income?
Another one where a calculator can provide the answer, slower than algebraic work, which itself is slower than a trick of treating the total tax as a weighted average. While either of the first two methods would do fine on the AMC, where you have an average of 3 minutes per problem and aren't competing head-to-head against anyone else, in a QB environment, someone who can do this problem in their head or on paper is going to buzz first, which is good.
+ How many positive integers not exceeding 2001 are multiples of 3 or 4 but not 5?
On the other hand, this question was one I didn't want to do by hand because of the likelihood of error. Easier, but by no means trivial, with a calculator.
While AMC-type questions are good as bonus parts, it takes a different type of computation to be a solid tossup question, and especially if it's NAQT format. One of my pet peeves over my high school career was the impossible-to-power math tossup. Not hard-to-power, but impossible, because you weren't told what you were looking for until after the asterisk.
On the other hand, I remember other questions that were perfect for QB, in that they were easily doable (in fact the question specifically told you how to do them), but in pyramidal form. In paraphrased wording, I think this was from the 2001 NAQT HSNCT:
You are looking for a number that is the cube root of 17576. Since 17576 ends in a 6, you can use that to find the [*] units digit of the cube root, and since it is between 8000 and 27000, you can find the tens digit. FTP, what is the cube root of 17576?
ANSWER:26[/b]
The one problem is that in a match between two teams competent enough to make HSNCT, this question should never go unanswered, though people might get it wrong. However, someone who knows the tricks named in the question may very well get the question by the power mark. [If memory serves me, the question may have crossed the line and said the units digit was 6, and it was between 20 and 30, but not sure.] Good computation question.

Bad computation questions require units and do not specify what unit early in the question. I believe a question from CBI Regionals this year ended "How many hours a week does Dude spend sleeping?" and had an answer of "49 [hours per week]" I interrupted before the end of the question and, since most of the other times in the question were x per day, I said "7 hours per day" and was given a -5. Bad computation question.
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Post by Stained Diviner »

You're right--the 4 should have been an exponent in the first problem. Also, the lens problem was a physics problem.

In terms of the per day vs per week question, it was a bad question. It is not just a math issue though. One of the question companies has as a sample question on their website something to the effect of:
'He sculpted David and painted the Sistine Chapel ceiling. What is the last name of Michelangelo?'

I actually was able to find some calculator-allowed questions I wrote:
* What is the fifteenth term of the Fibonacci sequence? The first two terms both equal one.
* Give the coordinates for the only local minimum of the graph y=x^3-12x.
* Rounded to the nearest thousandth, what is the golden mean?
* A circle is inscribed in a square of side length 15. Rounded to the nearest tenth, what is the area that is inside the square but outside the circle?
* Find the area contained in the enclosed region formed by the graphs: y equals the absolute value of x and y equals ten minus the absolute value of x.
* (Note to moderator: Read the digits individually.) Convert from base two to base three: 1001011. Make sure when giving your answer to read digits individually.

As jonpin said, a very good player can do some of these in his/her head without touching a calculator and will be able to beat out any player who does use a calculator. However, the calculators will be helpful to the more average teams. On some questions, the best players will use their calculators.
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Post by Howard »

ReinsteinD wrote:I actually was able to find some calculator-allowed questions I wrote:
* What is the fifteenth term of the Fibonacci sequence? The first two terms both equal one.
I can't think of an expedient way to do this without using a calculator. (Others feel free to chime in if you know a way, my Fibonacci knowledge is weak). In my opinion, what your testing here is knowledge of the Fibonacci sequence and the ability to program a calculator. Both are useful knowledge and skills. I like this as a calculator question.
ReinsteinD wrote:* Give the coordinates for the only local minimum of the graph y=x^3-12x.
This is trivial without a calculator. Most teams, even at teams well below those referred to as second tier (at least in my area), have at least one student with significant knowledge of calculus. Allowing a calculator on this question serves little purpose except to encourage use of a device which isn't necessary. Any reasonable calculus student who has learned about minima and maxima can quickly differentiate and solve 3x-12=0 to get the coordinates (4,16).
ReinsteinD wrote:* Rounded to the nearest thousandth, what is the golden mean?
I like the fact that this question requires the use of a calculator to get the desired answer unless the player happens to have memorized the golden mean to three places, and in that case, I think he deserves the points anyway. What I don't like is that once you get beyond knowing the golden mean, you're testing the speed of punching buttons. I also have a fundamental problem with placing more emphasis on rounding to three places rather than knowing the exact answer, which embarassingly, I can't remember.
ReinsteinD wrote:* A circle is inscribed in a square of side length 15. Rounded to the nearest tenth, what is the area that is inside the square but outside the circle?
Same as last problem.
ReinsteinD wrote:* Find the area contained in the enclosed region formed by the graphs: y equals the absolute value of x and y equals ten minus the absolute value of x.
A good quizbowl team will know the area is d squared divided by two. Even a moderate quiz team will be able to take the diagonal of ten and go throught the 45-45-90 ratios to get the answer. Once again, I'm not sure what this question gains by allowing the use of a calculator.
ReinsteinD wrote:* (Note to moderator: Read the digits individually.) Convert from base two to base three: 1001011. Make sure when giving your answer to read digits individually.
Again, a good team will be able to do this quickly without a calculator.
ReinsteinD wrote:As jonpin said, a very good player can do some of these in his/her head without touching a calculator and will be able to beat out any player who does use a calculator. However, the calculators will be helpful to the more average teams. On some questions, the best players will use their calculators.
I have to question the value of allowing the use of calculators to help average teams, when we are well aware that better teams can do the same problems without the calculators. In my opinion, this is doing nothing more than teaching teams to use a crutch rather than to learn principles and procedures they can do themselves more efficiently than a machine.
I don't mean to be overly critical of your questions and apologize if you perceived my above remarks that way; I am simply trying to illustrate my points with the questions you've provided.
What I'd really like to see are questions which actually require the programming of calculators to approximate a solution which would otherwise never be obtainable in half a minute or so. I have to admit that my age prevented me from learning most of these tasks in high school and college and that I'm relatively ignorant of the features of modern calculators. Can they solve systems of equations in, say, 20 variables in a matter of seconds? I would think that an application question requiring programming of a number of equations in a number of variables would be a reasonable task. (Okay, 20 is a bit large, but even three variables becomes difficult in short periods of time). This has actual real world application and importance.
Perhaps someone with better knowledge of these types of problems and solutions can chime in and tell me whether I'm off base either with regard to the abilities of the calculators or with the practicality of fitting such questions into quizbowl.

Edited for proper formatting.
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Post by MLafer »

ReinsteinD wrote:* Give the coordinates for the only local minimum of the graph y=x^3-12x.
This is trivial without a calculator. Most teams, even at teams well below those referred to as second tier (at least in my area), have at least one student with significant knowledge of calculus. Allowing a calculator on this question serves little purpose except to encourage use of a device which isn't necessary. Any reasonable calculus student who has learned about minima and maxima can quickly differentiate and solve 3x-12=0 to get the coordinates (4,16).
:w-hat:
Last edited by MLafer on Mon Jun 27, 2005 7:25 pm, edited 2 times in total.
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Post by NoahMinkCHS »

(mostly in response to Howard's post)
I agree that using calculators to help average teams does become something of a crutch and shouldn't be added for that purpose. No doubt, a "one Google-enabled laptop per team" rule would significantly help marginal teams, but this isn't the NFL -- parity isn't the point. If average teams learn the math -- and the information -- needed to succeed at quiz bowl, they deserve to win. If not... they deserve to be average.

As for calculator features -- I can't say for sure, but I'm pretty sure that at least the TI-83+ (the most common graphing calc, at least for a couple more years) can't do a 20-variable equation on its own. Numerous other tasks -- like the Fibonacci question above -- can be automated rather easily. I've done it myself at math team-type events... BUT those events are timed tests, where you have like 30 or 60 mins to do some number of problems -- 50 maybe? Anyway, clearly NOT quiz bowl. The only way to make such an application work in qb format would be to allot at minimum a minute for the question (to allow time to decide how to write the program, then write it, then run it), and even then a lot of questions would go dead unless the particular topic was familiar. (e.g., if there was a "Give the nth term of the Fib. sequence" at every tournament, and you just had to wait to find which one to give -- which would, IMHO, not be fun or worthwhile.) And taking a minute on one question, even in an untimed format, would drag the game and knock its rhythm off course. Such a topic might work at PAC, but not a normal tournament.

I will add that there were many, MANY times when I let a math question that I knew how to do go dead -- because I knew how to do it with a calculator only. Though I learned nearly every skill on paper first because my math teacher felt we should know why and how, we learned the TI-83 method too, because it was much more useful, simple, and expedient. (For example, we learned an easy way to solve systems of equations using matrices; doing the matrix method on paper would've been harder than doing the system the normal way, but doing it on calculator let us get that task out of the way, since (in higher math classes) that wasn't the actual question, just a part of it.)

So the point of that last paragraph was to say, essentially, that I wish I'd remembered which factorials divided to solve a permutation, because it would've given me more quiz bowl points. But for academic purposes, knowing the calculator way was, and, given that it's 2005, should always be, all I needed. If I needed the qb points that badly, I'd've learned it, but it wasn't that crucial to me. And opening the door to calculators -- and calc-dependent problems -- would've been far more of a cost than I'd be willing to pay.
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Post by Stained Diviner »

Don't worry about criticizing my questions--most of what you said was pretty accurate.

One thing that needs to be kept in mind here is how little an above average math student can do in 30 seconds. If you have to think a little bit about what type of problem you are dealing with or what exactly is being asked for, then your 30 seconds is up. It is also easy to make a mistake even when you know what you are doing, as demonstrated above.

If it comes to writing programs on a calculator, which is an inconvenient medium for writing programs, then 60 seconds is not enough time to accomplish anything useful. If you wanted to program Cramer's Rule for a 3x3 system, it would take you a few minutes even if you had Cramer's Rule memorized just because of typing speed. (Fortunately, as was said before, your calculator does not need to be programmed to do this.)

Calculators would be a crutch for some teams on some problems, but continuing to read a question after passing the power mark can also be seen as a crutch. (Excuse me for playing devil's advocate here--why can't teams figure out the answer to a history question before hearing the end of the question?)

As to the power issue that came up earlier--it is not something I have to worry about because powers are foreign to Illinois. NAQT does write good power calculation tossups. They tend to make up about 5-10% of their tossups, and they tend to focus on probability, combinatorics, and number theory. It is difficult to write a lot of good powerable calculation tossups--I don't know if geometry, trigonometry, calculus, and basic algebra topics lend themselves so well.
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Post by Howard »

MLafer wrote:
ReinsteinD wrote:* Give the coordinates for the only local minimum of the graph y=x^3-12x.
This is trivial without a calculator. Most teams, even at teams well below those referred to as second tier (at least in my area), have at least one student with significant knowledge of calculus. Allowing a calculator on this question serves little purpose except to encourage use of a device which isn't necessary. Any reasonable calculus student who has learned about minima and maxima can quickly differentiate and solve 3x-12=0 to get the coordinates (4,16).
:w-hat:
DOH! I'll go back and learn to differentiate now! How 'bout solving 3x^2-12=0 and getting coordinates (2,-16) and (-2,32). Here, (2,-16) would be a minimum (the answer) and
(-2,32) would be a local maximum. This is a little harder than originally thought, but I still don't see the need for a calculator, just the need to pay attention and check work. :smile:

Edited to make coordinates reasonably readable.
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Post by fluffy4102 »

ReinsteinD wrote:Don't worry about criticizing my questions--most of what you said was pretty accurate.

One thing that needs to be kept in mind here is how little an above average math student can do in 30 seconds. If you have to think a little bit about what type of problem you are dealing with or what exactly is being asked for, then your 30 seconds is up. It is also easy to make a mistake even when you know what you are doing, as demonstrated above.

If it comes to writing programs on a calculator, which is an inconvenient medium for writing programs, then 60 seconds is not enough time to accomplish anything useful. If you wanted to program Cramer's Rule for a 3x3 system, it would take you a few minutes even if you had Cramer's Rule memorized just because of typing speed. (Fortunately, as was said before, your calculator does not need to be programmed to do this.)

Calculators would be a crutch for some teams on some problems, but continuing to read a question after passing the power mark can also be seen as a crutch. (Excuse me for playing devil's advocate here--why can't teams figure out the answer to a history question before hearing the end of the question?)

As to the power issue that came up earlier--it is not something I have to worry about because powers are foreign to Illinois. NAQT does write good power calculation tossups. They tend to make up about 5-10% of their tossups, and they tend to focus on probability, combinatorics, and number theory. It is difficult to write a lot of good powerable calculation tossups--I don't know if geometry, trigonometry, calculus, and basic algebra topics
lend themselves so well.
What's the point of Cramer's rule if you have a calculator? Just use RREF on the matrix. Cramer's rule is basically obsolete with the presence calculators. The only way it wouldn't work is if you put in variables instead of numbers in the matrix. A 3x3 matrix takes no time to plug in for the standard TI-83 or TI-84.
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Post by Stained Diviner »

(Fortunately, as was said before, your calculator does not need to be programmed to do this.)
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