Well Thank Christ for That

Uh, should any reader of Crooked Timber get less than 9 or 10 (allowing for carelessness or a second glass of wine or beer) correct?

Of course, it should also be said: this is about as hard as I remember the GRE general exam math portion to be, on which I stunned myself by scoring an 800.

The fact that I got 10/10 on this quiz is proof positive that it cannot be testing math at anything higher than the 4th grade level!

I didn’t submit the form, but isn’t -7 both an integer and a prime number?

bmm6o–definition of prime is that it is a positive number divisible only by itself and one.

Well, I’ll go the contrarian route and say if the test were timed (10 minutes) and that the scores would cause a moderate level of anxiety ( say, you lose your job if don’t get at least 9 correct answers) then no more 5% of the total US population would get all correct and that if there were partial points deducted for wrong answers the modal score would be close to 0.

Questions 2 and 6 both seem definitional and likely to trip up a significant share of people that are otherwise algebra competent.

The ^ symbol for exponentiation and the definition of squared would take another batch, the use of symbols for unknowns and the use of fractions would be enough to make the mode 0.

I think you would need to count none of the above on Q7 as correct.

Here is a question for you testing geeks out there. Which question do you think would be the most highly correlated with doing well or poorly on the test. In othe words which question would be done right by almost everyone that scores well yet ansered wrong by almost everyone that scored poorly?

I would go with question 3.

Question 6 refers to the “set of numbers’ {2, 2, 3, 4, 5}. On any standard conception of set, this is equivalent to {2, 3, 4, 5}. But none of the answers are true of 2 on such a conception. The “correct” answer requires us to view {2, 2, 3, 4, 5} as not equivalent to {2, 3, 4, 5}. But that means we can’t really be talking about a set of numbers (as opposed to, eg, a set of inscriptions).

Maybe you shouldn’t worry so much if you didn’t pass “eigth grade math.” The people who set it don’t understand basic mathematical concepts.

The people who set it don’t understand basic mathematical concepts.

Ummm… set theory may be a shade higher level than grade 8 math. I for one don’t recall properly covering that until undergrad.

Andrew Edwards: the basic concepts of set theory can’t be “a shade higher level than grade 8 math.” They use the expression “set of numbers” right there in the question. Oughtn’t the people who set the test know the meanings of the mathematical expressions they use?

j.doyle is wrong — or at least he’s demanding a precision of word usage which is not general to ordaniry maths teaching practice: a set of data doesn’t automatically collapse repeat elements into a single representative, and it’s perfectly reasonable, certainly not unmathematical, to refer to data that’s in number form as “numbers”

What lalala said. 8th Grade algebra was by invitation only, and I had to promise that I’d keep up before they gave me permission to attend. But there was no mention of statistics, and I never came across ‘mode’ in that or subsequent courses. Had to Google that one in order to get it right…

But that means we can’t really be talking about a set of numbers (as opposed to, eg, a set of inscriptions).

Ah, but this is really philosophy! And I can be sure that 4th graders wouldn’t get this point…

Look, to suggest that the question is wrong requires a willfull misreading of the question. The notions of “mean”, “mode” and so forth blatently apply to data where repetition is allowed. The notion of “mode” is formally undefined for a set of integers in the sense you mean “set”.

Stommeling Jones got thirteen out of ten, as can be seen here.

I have never seen it built into the definition of a prime that it must be positive, and I don’t think you need to jury-rig the definition like this. Why not just stick with the standard def, sc. a number divisible only by itself and one, and then point out that -7 is divisible by itself, one, and *negative* one?

On modes: data points are implicitly ordered pairs. The values we were given (i.e. “2,2,3,4,5”) are the second members of a set of ordered pairs which in full reads “(susy, 2), (tommy, 2), (billy, 3), (cindy, 4), (casey, 5)”, e.g. if this was data on the number of siblings of various students. Read that way, the two ‘2’s are indeed distinct members, since they belong to distinct ordered pairs.

Sure, that’s not spelled out. But if we’re talking means and modes and all, we’re almost certainly talking data-points.

Still, I grant that if the context were asking you to think about *sets*, rather than about data points, one could legitimately test for whether the student understands the point Mr. Doyle makes.

tad #27

Thanks for backing me up. I was posting after several beers, so didn’t want to take too strong a stance. I don’t remember dealing with negative primes by 8th grade, but they are legitimate. The undergrad definition of “prime” (or rather, “irreducible”) refers to division by “units”, which in the integers are {-1, 1}.

re: set vs. data point:

The statement was imprecise, sure. But if you look at that question and say something like “sets of cardinality not equal to 1 don’t have modes”, is this test going to tell you something you didn’t already know?

Tad: By your definition, there are no primes. For example, -1 divides 2: (-2)(-1) = 2.

you just can’t catch a break around here. I come up with a truly epochal mathematical discovery–a proof that there are no prime numbers–and people start dissing me.

Okay, yeah, now I see the need for a restriction to positive numbers, and I see why it isn’t ad hoc.

I stand corrected re: primes being positive. I guess it’s required for unique factorization. That -1 divides all numbers doesn’t bother me.

7/10, which is still passing and not too shabby for an art School Grad whose last math class was an astronomy class for Architects, eight years ago.

I’ve found that algebra doesn’t give me nearly the trouble it used to in high school. I suppose my reasoning skills are more developed, or something.

Depending on your definition of “whole number,” “-7” is a whole number as well as an integer. Wikipedia says that whole number is defined both as the positive integers, and as all integers. As a math major, it troubles me to have gotten one wrong. But I can argue that it was a question with two correct answers.

Prime numbers have to be positive because prime factorizations are unique.

Um, isn’t “none of the above” the correct answer there?

The prompt in no way refers to percentages. Percentage is a different concept from decimal number. If I asked you what .3 minus .1 was, you’d be wrong to say “20%” Yet I think the test presumes “45%” is somehow the same as “.45”

John Quiggin:

I don’t know, I had to add 3/4 and 2/3 yesterday when baking a loaf of bread.

S/he whipped out the 1/12th-of-a-cup spoon, of course.

Where do you cook?

I got a perfect score and felt awesome until the same site told me that my “outrageous name” is “Harry Box” and that I’m an “alcoholic”

If there are several different definitions of a term (such as “prime”), and some of them yield several correct answers, and some yield one correct answer, then obviously you should use the definition that yields one correct answer. This is elementary test-taking. It is also useful in real life: when voting, or choosing a job, or buying a house, or marrying, it isn’t possible to choose either “none of the above” or “several of the above.”

I don’t have much use for math in my personal or work life, but one thing which I often use is proportions of the a:b=c:x type. Being able to estimate numerical products closely enough not to get factor-of-ten errors is also a good skill, and requires some actual training even though it’s a rough commonsense skill.

But the quadratic equation never comes up any more, and I never have to figure out how long it takes to fill a tank when the intake pipe is 2″ in diameter and the drain is 3/4″ in diameter. (Yeah, I learned nonmetric math back in the stone age).

Michael–

Not sure about that. I, like you, took my GRE’s in the mid-80s, and I, like you, have been struck when doing grad admissions by the increasing discrepancy between the absolute and percentile score in the math GRE. (I.e., the fact that a very high absolute score is now a relatively unimpressive percentile score).

Could be they have watered the test down. But it could also be that the GRE is taken by a much larger number of people now, including a very large contingent of foreign applicants to engineering, math, science, etc. It’s a big world out there, and there’s a lot of mathematical talent that would like a US graduate education. So it may be that the test has stayed just as hard, but there are a lot more people scoring 800’s on it.

This would also explain why the same inflation has not occurred in the Verbal portion, i.e. that it is still providing anglophones with the Mandarin advantage.

In philosophy admissions, I take the math GRE very seriously, but it’s far from the only criterion. I feel sorry for people doing graduate admissions to a math program, though, since that’s exactly where you would hope the Math GRE would provide valuable information, and it really cannot. Presumably there is some sort of informal super-GRE that people who care about math refer to, when trying to select among the hundreds of people who maxed out on their math GRE.

tad, michael –

I took the GREs in early 90s. It should be noted, however, that I was an undergraduate teaching assistant in formal logic for 2 1/2 years–so I expected to do well in math and logic sections. What I didn’t expect, and what threw me for a loop, was that I’d get such a low verbal score (670 or something, based on at least two questions I now remember getting wrong on that damned electronic version, which doesn’t let you change your answers) && that, when the 800s came in for math and logic they’d be, as you said, so low percentile-wise (though I remember the logic being a stronger score than the math).

Also–there is a math-specific GRE that’s much more difficult.

That’s right, joel–no professional philosopher would *ever* inflate their own credentials to make themselves look smarter–even in informal recollection. Unflnching honesty and humility about our intellectual limitations–that’s what sets us academic philosophers apart.

For what it’s worth, I worte the GMAT a couple years ago and they’ve set it up as an adaptive test, so that as you kick the crap out of the easy math questions (or verbal questions), the questions get harder, but if you fail at the easy ones, they get easier.

This means that there is still a lot of gradation at the top of the scale. I also think that it’s kinda cool.

That said, my business school has done some regression to determine what impact GMAT scores have on grades. This research has been informally communicated to me. Turns out that the best performers are not people with high scores per se, but people whose verbal and quant scores do not deviate much from each other.

So, all else equal, you’d rather have a person with 85th percentile scores in both math and verbal than a person with 99th percentile math and 75th percentile verbal.

This has always puzzled me, but what is the level of learning that graduate tests presume to test(other than a math-specific GRE)? I’ve only taken the GMAT (in the early ’80s) and I (hazily) recall being a bit surprised that the quantitative portion seemed to top-out about 10th or 11th grade level; i.e., geometry and some Algebra 3-4 stuff, but minimal, if any, college-level calculus or statistics. I had always been a good, but not “advanced,” math student, but nonetheless scored at the 99th percentile on the GMAT about 5 years after college with what today apparently would be minimal “preparation” (i.e, a half-day “prep class” for about $50 through UC Extension). I recall that the Harvard Biz School eliminsted the GMAT from their admissions process around 1990 or so, which struck me as a wise move, though don’t know if it caught on elsewhere.

Hunh. So after my comments above I decided to go look at some other Grade 8 math problems.

These ones are MUCH tougher, and also much tougher than I remember Grade 8 math being:

I should say so! Question 1 requires you to consider the time-value of money!

“The Adams family was going to buy a car for $5800. The car dealer offered the Adams family two options for buying the car. They could pay the full amount in cash, or they could pay $1000.00 down and $230.00 a month for 24 months on the installment plan. How much more would they pay for the car on the installment plan?”

If there’s high inflation, all of a sudden the installment plan looks like it could be a sweet deal!

bmm60–

Alas, contrary to kb’s suspicions, I revealed my scores only because a) they were not brilliant but didn’t suck and b) it was empirically interesting (statistically and as an very obvious instance of the moral failure of self-aggrandizement). I only took the GREs, in fact, as a last ditch effort to salvage some attempt at academic career after doing very poorly early in college (several incompletes turned F, C’s and D’s in Calculus sort of thing) & when my just-below 3.0 average (though much, much higher within my philosophy major) and high GRE scores didn’t cut it for early acceptance to grad schools at the end of my junior year I… dropped out. My two other passions, writing and software, both happened to be fields in which one could succeed regardless of academic credentials and I (rather foolishly, in retrospect–though I don’t wholly regret it) decided it was high time to start working rather than studying if I was going to make something of myself. Having since started and sold both a book and a software company (and, as visiting writer, taught at universities), I can’t say I’ve done too badly–though I hardly hold this as badge of honor–or, at least, if a badge of honor (I confess!), one badly tarnished by more than one act of foolhardy fuck-u-ism and unwarranted bravado.

bmm60–

where did I reveal or offer to reveal my GRE scores? So far as I know, they are still a closely-guarded secret. I’m looking up-thread, and I just don’t find it.

Oh–I did say that I *took* them in the mid-80’s. But I’m pretty sure that my *score* was higher than the mid-80’s. I mean, I think I broke into at least the low triple-digits.