Higher math proficiency test: An extended-problem-solving skill evaluation tool(mathvault.ca)
mathvault.ca
Higher math proficiency test: An extended-problem-solving skill evaluation tool
https://mathvault.ca/math-test/
41 comments
Yeah, the wording is quite poor.
The second question about the distribution of random answers to the test is literally incoherent. They probably meant to ask a question about the expected number of random answers on a randomly completed test, but they worded it such that they are asking about the distribution of answers on a single test.
The second question about the distribution of random answers to the test is literally incoherent. They probably meant to ask a question about the expected number of random answers on a randomly completed test, but they worded it such that they are asking about the distribution of answers on a single test.
>Assuming that every question in this 20-question test has only 1 right answer and you choose to answer each question randomly, what is the chance that you're so bad that you only get 4 questions right or less?
They probably mean for you to assume that 1) each question has 5 answer choices, and 2) you select one of them with equal probability. But given that this is question #2, assumption (1) is kind of a guess, and (2) should be explicitly stated... (the answer btw has to do with the binomial distribution - and you definitely need some heavy arithmetic, so are they testing your gut instinct?)
They probably mean for you to assume that 1) each question has 5 answer choices, and 2) you select one of them with equal probability. But given that this is question #2, assumption (1) is kind of a guess, and (2) should be explicitly stated... (the answer btw has to do with the binomial distribution - and you definitely need some heavy arithmetic, so are they testing your gut instinct?)
The messed up question I was referring to, and I think bloaf was referring to, is a later one, the second one involving random answers:
> Assuming that every question in this 20-question test has only 1 right answer and you choose to guess the answer of each question randomly, then in terms of the number of correctly-guessed answers, approximately 68% of them will fall between:
> A. 2.2 and 5.8
> B. 0.8 and 7.2
> C. 6.8 and 12.5
> D. 7 and 13
> E. None of the above
Another messed up question was the one early on asking whether the dot product operator is "associative/commutative on addition" or not. That's not a thing that actually means something.
> Assuming that every question in this 20-question test has only 1 right answer and you choose to guess the answer of each question randomly, then in terms of the number of correctly-guessed answers, approximately 68% of them will fall between:
> A. 2.2 and 5.8
> B. 0.8 and 7.2
> C. 6.8 and 12.5
> D. 7 and 13
> E. None of the above
Another messed up question was the one early on asking whether the dot product operator is "associative/commutative on addition" or not. That's not a thing that actually means something.
I answered none of the above precisely because associativity doesn't apply to an operator that maps 2 vectors to a scalar. But it still felt wrong.
Oh, I was parsing the responses like "(Non-associative, commutative and distributive) on addition". And no, the dot product is not commutative on addition. But it's meant to be "Non-associative, commutative and (distributive on addition)".
I think the following is poorly worded and possibly incorrect:
>What is the area enclosed by the function f(x) = sin²x - x²(sin³x)(cos²x) and the x-axis — between -π/4 and π/4?
Well, the second part is an odd function so we know its areas are 0 over symmetric regions, assuming "area" means "signed area". So we're left with sin²x which has a trick: the integral of sin²x and cos²x are equal over any of the quarter cycles of the graph -- same shape and everything. So we have that over our domain, pi/2 = Int(1) = Int(cos²x+sin²x) = Int(sin²x+sin²x) = 2Int(sin²x)... so the answer would be pi/4 (which doesn't appear).
I also suspect that some of the answers must be wrong, possibly in the counting or probability questions where it is extremely easy to make a logical error.
In my opinion the author should have a third party scrutinize his/her answers. I'm quite certain I didn't get 40% of the questions wrong. (I have a math PhD, taught calculus for 4 years, scored 170 on the quant section of the GRE, and went through this whole "test" very carefully).
>What is the area enclosed by the function f(x) = sin²x - x²(sin³x)(cos²x) and the x-axis — between -π/4 and π/4?
Well, the second part is an odd function so we know its areas are 0 over symmetric regions, assuming "area" means "signed area". So we're left with sin²x which has a trick: the integral of sin²x and cos²x are equal over any of the quarter cycles of the graph -- same shape and everything. So we have that over our domain, pi/2 = Int(1) = Int(cos²x+sin²x) = Int(sin²x+sin²x) = 2Int(sin²x)... so the answer would be pi/4 (which doesn't appear).
I also suspect that some of the answers must be wrong, possibly in the counting or probability questions where it is extremely easy to make a logical error.
In my opinion the author should have a third party scrutinize his/her answers. I'm quite certain I didn't get 40% of the questions wrong. (I have a math PhD, taught calculus for 4 years, scored 170 on the quant section of the GRE, and went through this whole "test" very carefully).
The function is non-negative on [-pi/4, pi/4], so you don't need any assumption about area meaning "signed area."
Your math is wrong on that problem. The integral of sin^2 over [-pi/4, pi/4] is (pi - 2)/4, which was one of the options. The functions sin^2 and cos^2 only have equal integrals over properly aligned quarter-cycles like [0, pi/2].
Your math is wrong on that problem. The integral of sin^2 over [-pi/4, pi/4] is (pi - 2)/4, which was one of the options. The functions sin^2 and cos^2 only have equal integrals over properly aligned quarter-cycles like [0, pi/2].
The second summand due to the sin^3 is not nonnegative (so you do need to use oddness to avoid the integral)
Right on the second part, my bad.
Right on the second part, my bad.
I read that part of your comment as re-interpreting the question, instead of just reinterpreting the prior part of the sentence it's in, never mind.
Regardless -- I think it's a valid reading to say the "area between" say, g(x) = x and the x-axis from -1 to 1 equals 1. Only people who are trained would say it's 0. So I don't think something like that really measures "extended problem solving and deductive thinking skills" which is a pretty lofty goal for a test.
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> Some of the questions are quite poorly worded, and I'm highly suspicious about one question with non-integral boundaries on a probability distribution defined over the integers.
Yup. It sounds like they were assuming that you would use some specific simplifying approximation.
Yup. It sounds like they were assuming that you would use some specific simplifying approximation.
It looks like the "correct" answer was supposed to be "2.2 and 5.8" -- another poster mentioned the server sends answers to the client. But P(X=3 or X=4 or X=5) is .598.
It looks like they used the normal cdf approximation of the Bernoulli distribution, which would give .6857.
It looks like they used the normal cdf approximation of the Bernoulli distribution, which would give .6857.
That is some horrible approximation then!
The closest range sum to 68% is for {3,4,5,6} summing to 70%, but even if available as an answer, the only correct answer remains "none of the above".
Who is this test aimed at? It mentions the math GRE, which would make me think that it targets people applying to graduate programs, but it doesn't really have that vibe and I imagine becoming a test that graduate programs would consider is quite a difficult undertaking.
I pretty much couldn't answer a single question. With a lot of time I could have "bruteforced" a few, maybe. Getting something better than a F? Impossible.
Soon I will have a MSc in Software Engineering... I simply memorized my way through college math classes, just like most of my colleagues.
Math always made me feel dumb. I never had intuition for it. Deeper understanding was always just out of reach. I truly loathe math.
Soon I will have a MSc in Software Engineering... I simply memorized my way through college math classes, just like most of my colleagues.
Math always made me feel dumb. I never had intuition for it. Deeper understanding was always just out of reach. I truly loathe math.
Given what you just said, if I was you, I'd strongly reconsider my career choice (if you ever plan to excel at it, that is).
The test fails to show you which answers (it thinks) you got wrong. Which makes is pretty useless...
The frontend actually receives all the quiz data with the correct answers from the server, here is cleaned up version of it:
https://pastebin.com/MF5TTBGr
Haven't read the article (it's down) but isn't the best test for proficiency in higher maths that you can actually do higher maths? Why do we need these proxies?
Cost of testing. You can get a good idea of someone's ability to do higher math in an hour with an IQ test. If instead you give them a test of higher math, you'd need to train them to do it first, which would take much, much longer. The added accuracy of that test won't make up for the several-orders-of-magnitude increase in cost.
Fair enough. But there needs to be evidence that this stuff actually works. I'm reminded of the fact that Richard Feynman didn't have a high enough IQ to get into Mensa.
Or rather he did not score well enough on their test to get to enter.
Sure, but he had an IQ high enough to do physics.
> But there needs to be evidence that this stuff actually works.
It is not in short supply.
It is not in short supply.
I'm skeptical. What evidence is there?
Asking questions that imply you've never bothered to read anything about the topic isn't a good way to try to get other people to engage with you.
But if you're looking for a brief overview, you could read https://www.amazon.com/dp/1444791877/
But if you're looking for a brief overview, you could read https://www.amazon.com/dp/1444791877/
Evidence that's built on "statistical correlation" is very weak. A correlation coefficient of 0.6 or less is practically non-predictive. The people in this area need to use better mathematics.
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My guess is at that high cutoff a small number of errors make a big difference to the final score.
> But there needs to be evidence that this stuff actually works.
https://www.gwern.net/docs/www/www1.udel.edu/63a495ef6b79bb4...
Why g Matters: The Complexity of Everyday Life
Personnel selection research provides much evidence that intelligence (g) is an important predictor of performance in training and on the job, especially in higher level work. This article provides evidence that g has pervasive utility in work settings because it is essen- tially the ability to deal with cognitive complexity, in particular, with complex information processing. The more complex a work task, the greater the advantages that higher g confers in performing it well. Everyday tasks, like job duties, also differ in their level of complexity. The importance of intelligence therefore differs systematically across differ- ent arenas of social life as well as economic endeavor. Data from the National Adult Literacy Survey are used to show how higher levels of cognitive ability systematically improve individuals’ odds of dealing successfully with the ordinary demands of modem life (such as banking, using maps and transportation schedules, reading and understanding forms, interpreting news articles). These and other data are summarized to illustrate how the advantages of higher g, even when they are small, cumulate to affect the overall life chances of individuals at different ranges of the IQ bell curve. The article concludes by suggesting ways to reduce the risks for low-IQ individuals of being left behind by an increasingly complex postindustrial economy.
https://www.gwern.net/docs/www/www1.udel.edu/63a495ef6b79bb4...
Why g Matters: The Complexity of Everyday Life
Personnel selection research provides much evidence that intelligence (g) is an important predictor of performance in training and on the job, especially in higher level work. This article provides evidence that g has pervasive utility in work settings because it is essen- tially the ability to deal with cognitive complexity, in particular, with complex information processing. The more complex a work task, the greater the advantages that higher g confers in performing it well. Everyday tasks, like job duties, also differ in their level of complexity. The importance of intelligence therefore differs systematically across differ- ent arenas of social life as well as economic endeavor. Data from the National Adult Literacy Survey are used to show how higher levels of cognitive ability systematically improve individuals’ odds of dealing successfully with the ordinary demands of modem life (such as banking, using maps and transportation schedules, reading and understanding forms, interpreting news articles). These and other data are summarized to illustrate how the advantages of higher g, even when they are small, cumulate to affect the overall life chances of individuals at different ranges of the IQ bell curve. The article concludes by suggesting ways to reduce the risks for low-IQ individuals of being left behind by an increasingly complex postindustrial economy.
The "not much more than g" era in research has ended. TLDR: cognitive ability is not just a single scalar value.
> For several decades, the question of whether measures of specific cognitive ability contributed anything meaningful to the prediction of performance on the job or performance in training once measures of general mental ability were taken into account appeared to be settled, and a consensus developed that there was little value in using specific ability measures in contexts where more general measures were available. It now appears that this consensus was premature, and that measures of specific abilities can make important contributions even if general measures are taken into account.
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6526477/
> For several decades, the question of whether measures of specific cognitive ability contributed anything meaningful to the prediction of performance on the job or performance in training once measures of general mental ability were taken into account appeared to be settled, and a consensus developed that there was little value in using specific ability measures in contexts where more general measures were available. It now appears that this consensus was premature, and that measures of specific abilities can make important contributions even if general measures are taken into account.
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6526477/
You'll note that the paper you cite observes that g consistently predicts performance well, and that assessing "more than g" consistently fails to improve the quality of predictions.
And then it goes on to conclude that "If the goal is to do a good job in prediction... it makes sense to use the most general measures available" and "If predictive efficiency is the goal, it will be hard to consistently beat g."
It's an opinion piece arguing that, because the concept of g provides no mechanistic insight into how people solve problems, studies that seek to explain how problems get solved (as opposed to whether problems will get solved) should try to explain that process in terms of factors at the second-highest level of abstraction, rather than the highest level.
And then it goes on to conclude that "If the goal is to do a good job in prediction... it makes sense to use the most general measures available" and "If predictive efficiency is the goal, it will be hard to consistently beat g."
It's an opinion piece arguing that, because the concept of g provides no mechanistic insight into how people solve problems, studies that seek to explain how problems get solved (as opposed to whether problems will get solved) should try to explain that process in terms of factors at the second-highest level of abstraction, rather than the highest level.
The "g" hypothesis itself was built upon an abuse of factor analysis. An abuse that had a high chance of producing something spurious.
For a much, much longer treatment of the argument ogomad gestures towards, read Cosma Shalizi[1] g, A Statistical Myth. Note that most of the criticism applies as well to temperature or pressure as to g. For a detailed response to Shalizi, see Human Varieties[2].
[1] http://bactra.org/weblog/523.html
[2] https://humanvarieties.org/2013/04/03/is-psychometric-g-a-my...
[1] http://bactra.org/weblog/523.html
[2] https://humanvarieties.org/2013/04/03/is-psychometric-g-a-my...
Not only that they would have either been using it consistently since they learned it or just learned it. I at one point could do differential equations (linear alg + calc). If you put one of the remedial problems in front of me I would not be able to do it. That info is just 'gone' from 25 years ago. But obviously I can do it. I got an A. I know enough right now to know what to go lookup so I could re-learn it if I needed to.
Data point: 70%.
Background: Software Engineer.
I found this to be challenging.
Background: Software Engineer.
I found this to be challenging.
Just so you know, the test sends requests to the server on every answer and sometimes these requests silently fail with 503 error code, you don't get any points in such case, even if you answered correctly.
looks like it's been slashdotted.
(The link will not load for me, as of writing.)
The US SAT I of the late-1990's was far too easy (I only missed one Q with a mis-selected answer, but I'm no genius and didn't study or prep for it one iota (or epsilon, as the case maybe)). Static, multiple-guess Q&A tests aren't able to assess a broad range of orders-of-magnitude of capabilities because of their various vulnerabilities.
Professional Engineer and physics tests that include fewer, open-ended written-response problems that build on each other tend to be more rigorous forms of domain knowledge testing. Some Q&A can be used as a first-pass filter, but it shouldn't be relied-on how the US K-12 under NCLBA leans on excessive multiple-choice standardized testing.
The US SAT I of the late-1990's was far too easy (I only missed one Q with a mis-selected answer, but I'm no genius and didn't study or prep for it one iota (or epsilon, as the case maybe)). Static, multiple-guess Q&A tests aren't able to assess a broad range of orders-of-magnitude of capabilities because of their various vulnerabilities.
Professional Engineer and physics tests that include fewer, open-ended written-response problems that build on each other tend to be more rigorous forms of domain knowledge testing. Some Q&A can be used as a first-pass filter, but it shouldn't be relied-on how the US K-12 under NCLBA leans on excessive multiple-choice standardized testing.
Some of the questions are quite poorly worded, and I'm highly suspicious about one question with non-integral boundaries on a probability distribution defined over the integers. Since I didn't get to see results, I can't say whether they screwed up or not.
It seems like the question level was appropriate for students that took Calc I/II and some course like Discrete Structures.