kitchen table math, the sequel: SAT math is puzzle math

## Sunday, September 16, 2012

### SAT math is puzzle math

Ed and I had an amazing conversation last night with an SAT math tutor. He told us that for him SAT math is super-easy, so easy that the first time he saw the test he wondered whether it was a joke. He estimates that in his years of tutoring he has encountered 20,000 SAT math items: for each item, he instantly knew the answer.

At the same time, he's not adept at other aspects of math, particularly anything to do with spatial reasoning, which predicts success in math, science, and engineering. He has trouble doing problems like this one:

His brain, he said, works exactly like the brains of the people who write the SAT, and nothing like the brains of other people who are good at math. He himself is good at math, spent most of his life writing software for Wall Street and briefly taught math. As to the latter, he told us US math teaching would be much better if schools cut the curriculum by 2/3 and had students learn the remaining 1/3 really well. He'd never read Schmidt and had never heard of mile-wide-inch-deep.* It just seemed obvious to him that a good 2/3 of US curriculum should go.

Naturally I was keen to know what kind of brain he and the SAT people have, and the answer was: a puzzle brain. He loves, loves, loves puzzles; puzzles are his thing.

That's SAT math, only the puzzles are too easy for him.

A funny moment: he said his ex-wife told him she was reading a book that explained his brain: "You have no right brain at all," she told him, or words to that effect. When he read the list of right-brain characteristics, he agreed.

I wish I'd asked him how he fares on find-the-hidden-right-triangle items specifically.

* Schmidt: "[A]t eighth-grade [we're] telling teachers to teach 35 topics. Other countries are telling their teachers to teach 10 to 15."

#### 30 comments:

SteveH said...

"learn the remaining 1/3 really well"

It depends on which 1/3 he is talking about. I would push for ensuring mastery no matter how much material you cover.

"His brain, he said, works exactly like the brains of the people who write the SAT, ..."

I wouldn't put it that way. I see the issue as a limited domain of material to test and the desire to look for aptitude as much as math skills. If you can't test for trig or calculus, you need to do something else to create a proper distribution curve. Thus, you have unusual questions that often cover special cases. Some of the questions are good checks for flexible math knowledge, but some try to test for whether you "see" the shortcut. Due to the importance of the test, however, many students spend lots of time and money substituting preparation for vague ideas of aptitude. This forces the SAT writers to work even harder to create new questions that maintain the distribution. How meaningful is it to drop from 800 to 700 with just 3 or 4 mistakes?

The parabola question is a borderline case. There are better ones that show how it's not just about flexible math knowledge. Now that I'm deep into analyzing all of the questions, it's simply a matter of preparation. The problem is that above 700, the scoring curve is extraordinarily non-linear.

Catherine Johnson said...

Well, there's also the 'legacy' IQ issue ... the fact that the SAT was originally conceived as an aptitude test.

I wonder whether that's where some of the focus on 'puzzle math' comes from --- ?

Steve, do you see 'puzzle math' as being different in any way from 'math math'?

Agree ***completely*** about the limited domain issue.

If you have to keep your curve, and you have to test ONLY algebra & geometry, then you've got to produce tricky questions and/or increase the number of questions (& steps) to be answered in the allotted time.

Anonymous said...

I'm not Steve, but I find your question "do you see 'puzzle math' as being different in any way from 'math math'?" mystifying.

That's like asking do you see 'number theory' as being different from 'math math' or do you see 'geometry' as being different from 'math math'?

The notion of 'math math' is not at all clear here.

Puzzle math is one branch of mathematics, generally drawing heavily on number theory, combinatorics, and geometry, but potentially from other fields as well.

As for the SAT, it is not really puzzle math, or if it is, they are very simple puzzles that can be solved by straightforward application of algorithms.

I agree that that SAT's insistence on not being an achievement test, and so not covering anything past high-school freshman or sophomore year limits its utility for predicting college performance. There has been some evidence that the SAT Subject tests (which are achievement tests and cover more advanced material) are better predictors, but I don't remember where I saw that so I can't give a citation.

Cal said...

The Subject tests were better only if you ignored the Spanish Language test, and then only because the population studied was UC admitted students, which is the top half of the California tested population. And even then, the only two that were reliable were the Writing test (now incorporated into the SAT) and the Math Ic, whose topics were also incorporated into the SAt.

But your larger point is correct, gas station. The SAT tutor and the blogger don't seem to have a good grasp on what, exactly, "math" is.

It's true that many people (myself included) are good at some aspects of math whilst having more of a struggle with the spatial aspects. To call these people "bad at math" is amazingly ignorant.

Glen said...

The SAT tutor and the blogger don't seem to have a good grasp on what, exactly, "math" is.

Nor does anyone else. What is math--exactly?

Any facile one liner ("the study of patterns") simply begs the question. Okay, fine, then what exactly are patterns?
My own definition is that it is the study of abstract properties, but that is only for guidance in questions of what math is useful for, not for defining what is and isn't math.

Math means different things to different people and---unless I missed the memo about the definition of math finally having been "proved"---all of these definitions are matters of opinion and convention. It's no wonder that some people's questions or statements about math seem "mystifying" or "ignorant" to others. They're operating with different subjective definitions of "math" in a world where the objective definition doesn't exist.

SteveH said...

"Steve, do you see 'puzzle math' as being different in any way from 'math math'?"

I wouldn't use the word puzzle. If you take out the time constraint, then few questions need the shortcut. If you prepare more, then you probably won't see them as tricks. It might even look more like real math. I would call it contest math. It's mostly about preparation. Aptitude might make preparation easier, but preparation will hide aptitude. The ability to prepare is a useful skill to test for, but not over such a limited domain of material. Ironically, the SAT (aptitude) test makers make lots of money selling Blue Books with 10 real practice tests.

Instead of talking with my son about things like physics and calculus, I have to ensure that he can solve odd/even problems immediately.

If 2x + 10y = 53, is there a solution where x and y are integers?

Yes, this is math and not a puzzle. Is it a test for aptitude? No, it's a test of preparation.

Crimson Wife said...

SAT math is definitely trickier than ACT math. ACT math is much more straightforward even if it does touch upon more advanced topics than SAT math.

I agree with the PP who compared it to contest math- the same people who are really good at solving math contest problems tend to score super-high on the SAT-M, even if they're not necessarily the ones who earn the highest grades in high school math courses.

One of my best friends in high school was the only girl on our school's math team to make it to the AIME. I had a higher average in our honors high school math classes, but she beat me by quite a bit on the SAT-M.

allison said...

I've commented long and often on my opinions of the general SAT math portion, (see here http://kitchentablemath.blogspot.com/2011/10/rat-psych-why-sat-math-is-tricky-redux.html, and here: http://kitchentablemath.blogspot.com/2011/10/rat-psych-what-to-do-about-sat-math.html, if you care) but I take exception to this:

--His brain, he said, works exactly like the brains of the people who write the SAT, and nothing like the brains of other people who are good at math.

I don't know who to attribute the "nothing like brains of other people who are good at math" to, but that is a gross overgeneralization.

I am "good at math", for a large assortment of definitions of "math". I was a successful scientist and engineer. I was a successful teacher of college students in a variety of math related subjects. My spatial rotation skill, though, is limited, and required explicit instruction to improve. And lo, explicit instruction in spatial reasoning improves spatial reasoning.

Watching half a dozen mathy folks solve a puzzle is to watch at least 3 solutions. Math people don't all think the same way, even when taught the same courses and techniques. Some are naturally more spatial--on average, men are better than women at this, for instance--but there is no monolith of how math people think at this level of problem.

Brains are finite. You can't be good at everything. My brain gave up naturally being good at spatial reasoning out of the box and compensated elsewhere. To imply all other math folks are naturally gifted spatially is false.

SATVerbalTutor. said...

Just for clarification, the math tutor in question is my partner. He grew up with a bunch of true math geniuses (one of his best childhood friends is a math prof at Harvard), and as a result, he's developed somewhat warped sense of what it means to be "good" at certain areas of math (he can also be very self-deprecating). When he says he's "not good" at certain areas of math, what he actually means is that he's not at the very, very top in those areas -- he just doesn't see it that way because he's known so many people better than him. Yes, he's naturally great at SAT math, but he's plenty good at "real" math too, by pretty much anyone's standards other than his.

Anonymous said...

The SAT isn't really a math test. It's an intelligence test. You can get an excellent score on the math section without taking much math if you are smart enough.

SteveH said...

"The SAT isn't really a math test."

Yes and no. A low score says a lot about your lack of knowledge of math.

"It's an intelligence test."

I disagree. It's a preparation test. Students at every level of aptitude or IQ will have to prepare or run the risk of looking less smart than their peer group.

"You can get an excellent score on the math section without taking much math if you are smart enough."

It depends on what you call excellent. Very smart students are spending a lot of time preparing and they are not doing it for no good reason. Your score can drop to 720 with just three errors. A 700 might be excellent, but not if your peers are getting 750. In spite of what colleges say about the importance of other factors, nobody wants to start out the admission process at the 25 percent SAT end.

My son is taking AP calculus in his junior year. His math grades have always been 98 or higher, but he has a lot of preparation to do before he takes the SAT. I can see exactly what he is missing and intelligence won't give him the practice and speed he needs for the test. Speed is being used as a proxy for intelligence, but it really reflects preparation.

Anonymous said...

"My son is taking AP calculus in his junior year. His math grades have always been 98 or higher, but he has a lot of preparation to do before he takes the SAT."

So Steve, you think there's calculus on the SAT?

SteveH said...

"So Steve, you think there's calculus on the SAT?"

Do you think I'm talking about calculus preparation? Did you read all of my posts?

Cal said...

"Yes, this is math and not a puzzle. Is it a test for aptitude? No, it's a test of preparation"

No, it's a test of math. Number properties, in fact.

Anonymous said...

Steve, I doubt anybody has read all your posts. I'd wait until you say something insightful to read further.

You talking about your kid's early calculus is just bragging and beside the point. It's not preparing him for the SAT. Don't you worry it may be distracting him, taking time away from reviewing the math he'll actually need to ace the SAT? You seem like a dedicated striver, so this must worry you. He could be three years ahead in math and still not do as well in the Algebra and Geometry tested on the SAT as a kid cruising along at grade level.

There's nothing past Algebra II on the SAT, which any kid at grade level would get by the time they take it. It's not a test of math advancement, just a test of how well students understood the math all takers encountered in school. If you understand the math, there's plenty of time to finish it. You don't even have to memorize anything, as the formulas are given. You just have to be smart enough to apply them.

It's an intelligence test, where money (spent on paid test prep, tutoring, and the like) may substitute in part for intelligence. The prevalence of rich parents pouring cash into prep has changed both the test and its meaning.

SteveH said...

"No, it's a test of math. Number properties, in fact."

As usual, you're being churlish.

SteveH said...

"I'd wait until you say something insightful to read further."

Nothing like starting out with a good ad hominem attack to set the tone.

"Don't you worry it may be distracting him, taking time away from reviewing the math he'll actually need to ace the SAT?"

You really should read the entire thread.

"There's nothing past Algebra II on the SAT,..."

I know that, thanks. Again, read the thread.

"...just a test of how well students understood the math all takers encountered in school."

"just"?

"You just have to be smart enough to apply them."

Is it really that simple; smarts, not preparation?

"It's an intelligence test, where money (spent on paid test prep, tutoring, and the like) may substitute in part for intelligence. The prevalence of rich parents pouring cash into prep has changed both the test and its meaning."

So, it's an intelligence test, but it isn't? How big is that "in part"? Yes, those darn pesky rich parents ruin everything. They must be the driving force. It couldn't be the colleges putting too much emphasis on the scores.

It appears that you agree with me. It's not just an intelligence test, but a preparation test. Either the extra preparation is effective or it isn't. Either it helps your score or it doesn't. It's clear that you think it does.

Some of the preparation is simply review of basic math, but it's much more than that. It's supposed to be an aptitude test.

It's not just rich parents who are driving this change. The College Board promotes this preparation, and they make a lot of money off of it. If you read my posts carefully, you would see that I'm annoyed with this game. I've talked in the past about the arms race between aptitude and preparation. Preparation has gone far past a simple review of basic math.

lgm said...

I agree prep has gone past review, but that is because Algebra for All means that some of the needed number theory has been dropped from the curriculum. Those topics have been pushed into math club where math club still exists (not here) and into prep review. Of course one has to spend more time prepping...the school is not offering as much. In my time, pre-recentered, I was able to score in the 700s without an ounce of outside prep;; no honors math classes either, just what Dolciani, Beckbenbach et all included in their curriculum. I did however have a quiet classroom - no disruptors - which is quite an advantage over today's students.

>>If 2x + 10y = 53, is there a solution where x and y are integers?

This is a test of prep. for those who never considered the properties of odd/even numbers. For those who did (likely sitting in the back of a fully included class forced to review material mastered years before) it isnt even a test of reasoning. It is obvious. Really, the best thing to come out of full inclusion in elementary and middle school is the time the math inquisitive kid now has to practice mental math and ponder the interesting questions from math club or afterschooling.

Anonymous said...

As an SAT guy, I find a lot in this thread that is interesting.

You know, it does not matter what kind of math you call "SAT math". And you can't define "math" in any way that is productive for the matter at hand.

And it does not matter whether you believe the SAT tests intelligence -- again, assuming you can define that.

It tests what it tests the way that it tests it. And you better believe that it tests DIFFERENTLY than a school math teacher does. So yes, review whatever material you have not seen. And then do a boatload of practice. A great way to solve a puzzle it to have seen it before.

I'm sure that SAT math scores will always correlate with "intelligence " -- measured however you will. I'm not sure that there is ANY talent at all that does not correlate statistically with any other. (Yes, I am exaggerating...but only a little.)

And as for wo to blame for the arms race...lots of blame to go around. But part has to go to schools that lose their will to use grading to make distinctions among their students.

Phil Keller

Glen said...

Anonymouse to SteveH: You talking about your kid's early calculus is just bragging and beside the point.

No it's not; it's an integral part of Steve's position on a problem with the SAT. Steve says, and I agree, that he could do more good for his son by spending his math time on more calculus-level work instead of going back and running laps on the middle school algebra track, trying to shave seconds off his time.

I like the SAT more than Steve does because our sons are different ages. Mine is just coming out of elementary school, and trying to polish SAT-type math to fluency is just what he ought to be working on in his math education right now. Studying for the SAT is both fun and useful to him right now--useful for his math learning, not for scoring well on an SAT. Unfortunately, the time in his math studies when it is most useful to focus on SAT-type math and the time he will have to take the SAT for credit toward college admissions are a few years apart.

When it's time for my son to take the SAT for admissions purposes, he'll have to take some time away from learning higher math to polish the speed of his basic math. If we do a good enough job at this age, I'm hoping that bringing him back up to his best speed won't take too long later. Even so, I see Steve's point. For kids who are good at math like Steve's son, those who are most likely to become STEM majors, math study for college admission is something of a distraction from the best math preparation for your college major.

Glen said...

"...for their college majors," that is. (ahem)

SteveH said...

"...and trying to polish SAT-type math to fluency is just what he ought to be working on in his math education right now."

Some tricky things in SAT are good tests of knowledge. The drumbeat of simplify in regular math classes is countered by the need to anti-simplify for some of the SAT questions. Negative exponents are not illegal. I would say that up to some SAT level (650?), I have few complaints. However, I do have a problem when you get to the nonlinear point of diminishing returns at and above 700. The SAT has to resort to more quirky problems to maintain their distribution. This is the area where students do the most preparation with the least real math benefit. The SAT counters that preparation by making the top few questions trickier. Sometimes it's just the wording of the problem. Other times, it's presenting a special case they've never used before.

Anonymous said...

Although preparation can compensate for lack of general problem-solving skill, the SAT can be done at a fairly high level without prep. My son got a score of 720 on math at the end of 6th grade with no SAT prep, having only self-studied geometry (half a text book) and had a little algebra. Most of SAT math is middle school math, not high-school math.

A student who has continued doing math throughout high school probably won't benefit much from specific SAT preparation, which I believe benefits more those who've just been phoning it in during math classes.

The writing portion of the SAT is the only part he had trouble with at the end of 6th grade, since he had never been taught the 5-paragraph essay (or really any essay writing techniques). I believe that the writing part of the SAT does require specific prep, though most high school English classes seem dedicated to providing that prep (to the exclusion of more interesting content).

Anonymous said...

Glen comes perilously close to making the point that Steve only imagines he's made (I'm guessing Steve is not so much help with the persuasive essay portion).

For the handful of 'tie-breaker' questions that differentiate between 800 and 700, should higher math (trig, calc) be used instead of just trickier puzzle math questions? I hear Glen saying yes, because otherwise test prep takes away from more useful study (and that Steve says so too).

Of course, the question goes back to what the A stands for in SAT. It's not intended to be a test of knowledge, but aptitude (i.e. intelligence). Gassy's kid, for example, clearly has the aptitude. If it were a test of knowledge, there's no way he could score that high above the median in sixth grade.

There really should be some sort of SAT Math test that includes more advance study. Something like a Math II Subject test, say. Oh, wait... there is.

Perhaps the energy spent bemoaning the unfairness of the scholastic APTITUDE test attempting (poorly) to measure aptitude instead of advanced knowledge should better be directed towards wishing the math subject test covered more advanced material.

Personally, I always found those tests (PSAT, SAT, GRE) wicked easy. That's why I got my BA at 19 and have a couple 800s on my record (not that employers care much about that). Oh, and let me brag about my kid now. He's eight and enthusiastically studying pre-algebra two hours a day. And did I mention how handsome he is?

SteveH said...

"... probably won't benefit much from specific SAT preparation, which I believe benefits more those who've just been phoning it in during math classes."

"Probably"? Based on just your son?

SteveH said...

"(I'm guessing Steve is not so much help with the persuasive essay portion)."

You can't make an argument without an ad hominem attack? Is that your form of persuasion?

"If it were a test of knowledge, there's no way he could score that high above the median in sixth grade."

Is that the extent of your argument? Equal SAT scores mean equal aptitude?

"Perhaps the energy spent bemoaning the unfairness of the scholastic APTITUDE test attempting (poorly) to measure aptitude instead of advanced knowledge should better be directed towards wishing the math subject test covered more advanced material."

"Poorly?"

Thanks for making my point.

This is not a discussion about where best to apply one's energies. If so, then how is "wishing" energy more effective than "bemoaning" energy?

There is little need to improve AP Calc and SAT II math tests. There is, however a need to reduce the importance of the math SAT, especially if it does its job "poorly". Perhaps we could apply some wishing energy. Perhaps parents can better apply that energy to preparation as a substitute for aptitude.

Anonymous said...

I believe Steve has powers of persuasion and argument which he has not deigned to deploy in this discussion. In other forums, instead of merely sniping at words or phrases, Steve exercises his mastery of complete development of ideas in paragraph form to convince his interlocutors of his perspectives. We wish this forum weren't beneath such a full implementation of his skills.

Crimson Wife said...

"For the handful of 'tie-breaker' questions that differentiate between 800 and 700, should higher math (trig, calc) be used instead of just trickier puzzle math questions? I hear Glen saying yes, because otherwise test prep takes away from more useful study (and that Steve says so too)."

The ACT does this. Is there any data on whether the SAT-M is a better predictor of performance in college STEM classes than the math portion of the ACT?

SteveH said...

It's a math mint. It's an aptitude mint. It's a preparation mint. It's three, three, three mints in one.

A major issue is supply and demand. The more important a metric is, the more money and effort people will throw at it to achieve even marginal gains; the more the metric controllers will fight back. With a limited domain of material to test and a need to show separation, the preparation variable looms large. Colleges may take many things into account, but SAT scores still play a major sorting and cutoff role. Aptitude may get you a high score without much preparation, but there will still be huge competition and a need for preparation at that level. If you get a 700 with little preparation, will you be happy when your friend works her butt off and gets a 750?

I am, however, hearing stories that SAT CR and writing scores are being valued more highly than math scores. Does anyone have any specific evidence of that?

SteveH said...

My niece got a 34 on the ACT, but she is no hot STEM prospect. The ACT gets into trig, but the questions are very simple and do not reflect any real potential for success in a STEM department.

David Coleman, one of the architects of the Common Core Standards, will become president of the College Board on Oct. 15. He said that one of his top priorities is to reshape the SAT to better reflect the new standards. Unfortunately, they are having trouble trying to get colleges to accept CCSS proficiency in place of remediation.

Since the ACT is their real competition, will the SAT evolve into a CCSS version of the ACT? Are they going to give up on aptitude? From what I've read, it seems that way. Even a little more content in math would push the effort more towards math over vague ideas of aptitude and test preparation. Will they add a science component?

The real problem is that a test of limited material cannot separate all students into a meaningful curve at the top end.