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:

source: Fibonicci - spatial reasoning test

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."

Help Desk: retaking the SATs?

800 - critical reading; 750 - math; 590 - writing

These are the scores obtained by the son of a friend of mine when he took the SATs for the first time this year as a high school junior. I should mention that he's no longer a high school junior: a few weeks ago he decided to drop out of school because he found it too boring. (The school in question is one of the very top public high schools in Philadelphia). He hasn't taken the PSATs, or done any test prep whatsoever. Nor, as of yet, has he taken any SAT IIs.

So here's the question: given this boy's current scores, would it be a good idea for him to retake the SATs?

December SAT Scores (aka, My Buddha)

I'd characterize yesterday as an epically bad day in my 46 years of life, and while the turmoil had nothing to do with the SAT, my December scores did not help.

Yes, I do realize (intellectually) that I should feel happy about my Reading and Writing scores; but honestly, that Math score feels crushing, like a bully.  Today, well, I'm trying to see it as my Buddha.

The worst part was telling my son. I swear to you, he looked at me with these big, wide, honest to god eyes of surprise, and said "really?" --  like he truly couldn't believe his mom didn't do it.  I think I'd actually convinced him that hard work pays off (that's what I thought!).

But he's a sweetie, and he quickly focused on my Reading and Writing scores, telling me how great they are, blah blah blah. In fact I got all sorts of encouraging emails from friends and family:

"I know it's hard to remember at times like these, but these scores are not a judgment. They're just numbers ..... You did your best and gave it your best shot.  That's what's most important -- the process, not the outcome .... Your scores are fantastic – you’re 40 points away from an 800 on CR – do you know how many parents would kill for that score?? The 730 on writing just puts you in your range."

They made me feel better, in a supported sort of way -- but deep inside I couldn't help feeling like a high school senior who just found out they didn't get into their first choice college, and everyone writes on their Facebook wall: "You're too good for them.... It wasn't meant to be..... There's a better school for you..."

And that's all true, but it still feels devastating.  At least it does for me.

At the end of the day yesterday, I received an email that truly did lift my spirits. It came from a high school senior whom I'd never met:

SAT scores came out today! How did you do? I hope you did well. I know you'll get a good score, and congrats on completing the project! What you did was very inspiring, especially for high school seniors. I just thought that I would let you know that you motivated me to study, and I went from a 1630 (520R 600M 510W) (junior year) to a 2300 (700R 800M 800W) (senior year).

I need to print that out and post it at eye level on my bulletin board.

I haven't fully processed how it's possible that I spent dozens and dozens of joyful hours studying SAT math over the course of 10 months, and hardly improved at all from where I started without knowing a thing last January.  My friend Catherine says it's one more piece of evidence that a solid curriculum is essential, and without that, no amount of SAT prep in the world is going to improve your score.

For all intents and purposes, I didn't learn a lick of math after 9th grade (until I began this project).  I'm thinking about taking a math class at my local community college -- and just starting from scratch.

I'm not done.  I have to pause in order to write a book right now, but I'm not done with the math.  I feel incomplete.

If there's anyone else out there feeling disappointed by their SAT scores, here's a quote that I have posted in a few places around my house that seems to help:

If you have the privilege of being with someone at the time of his or her death, you find the questions such a person asks are very simple:

• "Did I love well?"  
• "Did I live fully?"
• "Did I learn to let go?"                                                      

                                 -- Jack Kornfield

llustrations by Jennifer Orkin Lewis

Cross-posted on Perfect Score Project

SAT story & apologies for the disappearing act

Wanted to say quickly that I've got at least 4 meaty emails waiting for me, which I'm not getting to because they're vying for attention with a stack of student papers and so far the student papers are winning.

Hoping to get to emails later today!

I shouldn't be writing posts, either, but I had to pass this along.

C. just came into the kitchen and told me this story.

One of the math kids in his class, a kid so good at math that the calculus teacher (this would be the calculus teacher whose class C. has dropped) told him he could "sleep through every class" and get an A, got a 650 on SAT math.

650.

I say that's a benchmark. A gifted math student who takes the SAT cold gets 650.

C. said, "He doesn't care about the SAT. He's going to X University no matter what."

His folks have free tuition at X U.

update: The last two lines above are ironic.

math professor's experience coaching the SAT

The Effectiveness of SAT Coaching on Math SAT Scores
(pdf file) by Jack Kaplan
Chance Magazine | Volume 18, Number 2, 2005

Comment: Jack Kaplan's "A New Study of SAT Coaching"" (pdf file)
By Derek C. Briggs

Here's the Fairtest write-up of Kaplan's coaching results:

Kaplan's course includes 20 hours of classroom instruction and seven to 12 hours of individual tutoring. Of the 34 previously uncoached students he worked with over six summers, 21 increased their Math scores by more than 80 points from a baseline SAT administration with 11 going up by 100 points or more. Average score increases were higher for students with lower initial scores. In a control group Kaplan studied, average scores increased by only 15 points during the same period. Thus, he concludes that "the estimated coaching effect was 63 points" on the Math component of the SAT.

Kaplan was coaching the SAT math test being given prior to the 2006 changes.

the writing test and the math test

Chemprof (and others, I'm sure) pointed out in Comments that math/science professors value the SAT math test for the same reason I value the SAT writing test: both exams test standard mistakes that college students make.

Btw -- this is something I haven't gotten around to putting inside a post -- when I mention "the main errors student writers make," I'm referring to the Connor and Lunsford list of errors compiled in 1988, which is pretty close to the SAT list.

The Connor-Lunsford list is close to the SAT list except for the fact that Connor and Lunsford did not see ginormous numbers of parallel structure problems in the student papers they read, apparently. I find that hard to fathom. I personally do see ginormous numbers of non-parallel structures in the student writing that comes my way.

Faulty comparison, tested on the SAT, does not make the Connor-Lunsford list, either. (I'm not surprised by that.)

In any event, while musing about chemprof's observation (which I agree with, btw), an essential difference between the two tests, one that I hadn't focused on, suddenly leapt out at me: where SAT Math tests content and procedures students have been seeing in school for years,^* SAT Writing tests content students have never seen or even heard tell of unless their Spanish teacher happened to explain what a gerund is in Spanish class.

(I use that example because I asked C. this week whether he knew what a gerund was, and he said he did because he'd learned it in Spanish. I myself had no idea what a gerund was until this semester. Public schools don't teach formal grammar today and haven't taught formal grammar in decades.)

So....when you think about it....isn't the Writing Test a bit of an odd concept?

Students have never been taught grammar, and now they're being tested on grammar?

And why would I be in favor of testing students on content the schools don't teach?

Now I'm thinking: well, maybe I'm not!

Mulling over chemprof's comment, I realize that what I value about the writing test is almost exclusively the test prep kids do for the writing test. The fact of the writing test, the fact that that the writing test exists and students have to take it, gives parents an excuse to insist their kids learn some formal grammar before they graduate high school.

And that's pretty much it; that's what I value about the test.

So, since high scores on writing come entirely from test prep (at least in my experience), what does a high score on the writing test actually mean? Does a high score on the writing section tell us anything about the student's writing?

I don't know the answer to that, and I don't have a good guess.

Basically, I think it's a good thing for a student to recognize a comma splice in an SAT sentence regardless of whether he recognizes a comma splice in his own writing, and effective SAT prep can make that happen. This is a statement of value: I value knowledge of comma splices, and I want my kid to possess it.

^* Most of the content anyway. That's a subject for another post: these days the SAT now features counting problems, and students taking traditional algebra classes don't seem to have counting "units" in their courses (although chapters on counting  units are included in traditional texts). Ditto for the algebra 2 material on the SAT if a student has not taken algebra 2.

SAT math is tricky at all score levels

re: the a in f(x) = a - x² (question below), which appears in the College Board's Online SAT Test 5 and is designated "medium difficulty"

I should have made clear in the post I wrote about the post Debbie wrote^* that this question is not "tricky" for me. This question is easy for me, and the fact that this question is easy for me tells you nothing about whether I have mastery of quadratic equations and their graphical representations. At this point, I do not.

This question is easy for me because I have some basic understanding of shifts, because I have memorized the rules about shifts listed in all of the SAT test prep books (and most notably in Phillip Keller's book), and because I see the intended trick of the question the minute I look at it.

The fact that this question is easy for me probably does tell you I am pushing 700 on SAT math, which I am. I scored 680 on the real test, putting me at the 90th percentile for all test-takers; on sample sections my range is well into the 700s. I don't know the exact percentage of test-takers in the 90th percentile who get this question right and find it easy, but it's going to be very high. The top 10% of test takers is the group for whom this question is not tricky.

This question is tricky for test-takers scoring in the 500s, and the College Board knows it. They tell us they know it; they're not keeping it a secret. When the College Board assigns a "Medium" level of difficulty to an item, they are telling us that x number of kids scoring in the 500s will reliably get the question wrong, and x number of kids scoring in the 500s will reliably get the question right. That is the meaning of the words "medium level difficulty."

Kids scoring in the 500s are the ones you can depend upon to see a in f(x) = a - x² and think, in a certain percentage of cases, a-the-coefficient-of-x^2-in-ax²+bx+c=0. Those are the kids getting tripped up by this question.

Test-takers scoring in the mid-500s do not have mastery of quadratic equations and their graphical representations, and neither they nor anyone else is claiming they do. So when a designated percentage of kids scoring in the 500s get this question wrong, we learn nothing about them we did not already know. By the same token, when a designated percentage of kids scoring in the 500s get this question right, we also learn nothing about them. As a general rule, 500-scoring kids do not have mastery of quadratic equations and their graphical representations. I think that is a safe assumption to make.

Ditto for me. When I easily get this question right -- and, again, I stress the word easily -- no one knows anything about my level of mastery of quadratic equations and their graphical representations. I do not have mastery at present, yet for me this question is so easy it's a gimme. In Steve H's words,  I have (near)-mastery of the test, not the math.

So why is this question -- or, rather, questions just like it^**-- on the test?

Questions like this are on the test because they have appeared in the experimental sections of previously-administered SAT tests and have been found to reliably produce a certain level of error. The College Board needs test items that produce reliable levels of test-taker error in order to keep raw scores stable from test to test. They can't have everyone getting all the questions right; they can't have everyone getting all the questions wrong; and, in the big, mooshy middle, they need items that reliably produce a certain percentage of right and wrong answers.

Hence the a-question.

^* this is the mouse that lived in the house...
^** I don't know whether items in Online Course have been tested in Experimental sections. As I understand it, all items in the tests actually administered to real test-takers have indeed been tested in experimental sections taken by other test-takers.

milestone

C. is applying to college today.

Assuming our electricity doesn't go out, that is. A small electrical fire from a fallen wire has been burning in the road next to our side yard since last night.

He wrote his what-I-did-with-my-summer-vacation answer on doing SAT math with his mom every day.

Susan S on her son's experience with competition math

re: the question of whether middle school competition math prepares students for ACT math:

I only have an anecdotal story concerning math contests and the ACT. When my son was taking the ACT in middle school at the beginning of the year, I hired a genius kid to tutor him on some of the things he had never experienced, like algebra 2 and some trig stuff. The tutor thought he would break 30, but he actually was in the middle 20s. Still not bad for a 13 yr. old.

As the year went on he made the Mathcounts team and had to practice the problems every week for a few months. He usually only finished half of them since we couldn't really help him. I signed him up for the actual Midwest Talent Search at the end of the year, but I didn't prep him this time or hire a tutor. His math score jumped to a 29 which placed him in the 99th percentile of the MTS, or Midwest Talent Search, kids and earned him a high scorer medal from Northwestern.

I have no idea if that means anything, but I thought it was interesting. He was only in accelerated algebra 1 at the time. Competition math just may help out in some way. It was the only thing I could think of at the time to explain the jump in scores.

If I had it to do over again, I would have had C. doing middle school competition math problems from the get-go -- and not just because SAT math is so closely related to middle school competition math.

C. and I both learned a tremendous amount prepping for SAT math. We learned a tremendous amount about algebra 1, geometry, and arithmetic, and we learned a tremendous amount about what we didn't know we didn't know.

puzzle math - from the comments

in an earlier Comments thread about SAT Math, gasstationwithoutpumps writes:

My son did no prep for the SAT when he took it in 6th grade and got a 720 on the math part. I'm quite sure that he'll do better on it when he takes it as high schooler next year without doing any SAT prep. The math questions just aren't all that hard.

to which SteveH responds:  

He had all of the material on the SAT by 6th grade? You are using this as normal?

gswp's reply:

No, he had not had all the SAT math. He'd had a very feeble algebra series in school (The Key to Algebra) and had self-taught geometry from a good book (he'd gotten through about half of Art of Problem Solving Geometry). The SAT supposedly tests much more than he had had at that point, but the questions are more like simple puzzles than testing content knowledge.

Gswp is exactly right! Setting aside the issue of the SAT using core features of cognitive function to produce error, which it does, SAT math is middle school puzzle math.

an SAT math tutor on seeing the 'trick'

A friend forwarded me an email from a parent whose high school age child rapidly sussed out SAT math and scored somewhere in the neighborhood of 800 with minimal test prep. The dad said his child has what he sees as a kind of "insight." After reading through some SAT math tests, the child could instantly see the 'trick' involved in the questions (trick used in the nonpejorative sense of the word). The dad said, too, that he believes this form of insight isn't necessarily present in students earning physics degrees from Columbia.

An SAT math tutor I know agreed and, when I asked why, wrote this response and gave me permission to post:

I've interviewed LOTS of math/physics/computer science majors from NYU and Columbia for potential teaching gigs who couldn't make sense of relatively simple SAT problems. Almost universally, the issue is that they WAY overcomplicate things.

Steve H mentioned Richard Feynman's hobby practicing "divergent thinking" problems. I think that's probably what this dad (and this tutor) are talking about.

Steve H on SAT math and math prep

from the comments:

SAT math tries to trick students. You could say that the tricks relate directly to whether or not they really understand math. However, when you add in the time constraints, it really relates to preparation. Is preparation the same as mastery? Yes. Mastery of the test. Is this equivalent to mastery of math or whether you will do well in college math? Not necessarily. There are better ways of determining that than with the limited material included on the SAT. Why not just require students to take the Achievement Test? Look at the AP Calculus grade.

What is it about SAT-Math that is so important? They are trying to test something other than just math knowledge. They think that these tricky questions reflect on how well you think on your feet, but what it really does is test preparation and whether you have seen these questions before. The questions don't reflect on whether you have a wide body of knowledge and skills in math.

They create problems where you have to "see" the shortcut. You get problems with hidden 3-4-5 triangles. Add a time constraint and then what do you call those problems? It's not just about math knowledge and skills. The problem has to do with trying to determine the difference between aptitude and preparation. The tricks may have some basis in meaningful math, but that's not what they are trying to test.

It reminds me of questions companies like to ask at job interviews, like "Why is a manhole cover round", and "How many golf courses are there in the US.?" Preparation can make you look like you have a great aptitude. Preparation is directly related to math knowledge, and that is important, but identifying aptitude is an arms race for something like the SAT. That's causing the tricky problems, not any desire to test a breadth and depth of math knowledge.

In Dick Feynman's books, he talks about how he spent a lot of time in high school learning about all sorts of trick, lateral thinking problems. He would challenge people to ask him questions. There is nothing like preparation to make you look like a genius, although he really didn't need help with that. It really annoyed some of his colleagues.

My son will get to calculus in his junior year and he always gets A's. He still has to prepare for SAT-Math. He can't let others, with specific SAT-Math preparation, seem like they have a better aptitude than him.

and:

They try to trick students in most questions....What bothers me the most are the shortcut problems where using standard math techniques cause you to take too much time. This is supposed to identify aptitude, but it really tests preparation for the test.

There are also the problems where using a brute force or direct counting technique works better than any applied math technique. In some cases, there is no math to apply. One question on a sample PSAT test asked for the number of positive integers less than 1000 which don't have a '7' as one of the digits. (notice - "don't have" and positive integer) This simply checks how well you work under time pressure. Nobody expects you to apply any fancy math to this problem. One of the answers was the "have" solution. This tests preparation and practice, not aptitude or math ability. There may be a correlation between the test and aptitude or math ability, but not to the resolution colleges use it to select students. At the top levels, it correlates to preparation. That's not necessarily a bad thing, but there are better ways of figuring that out.

rat psych - what to do about SAT math (part 3)

Your typical high school student, I presume, has spent several years setting up equations and solving for x. At least, let's hope so. I certainly did.

The SAT uses this fact to elicit many wrong answers from test-takers who have worked a problem correctly. The student gets the solution right but the answer wrong because the answer isn't x. The answer is 3x, say, or xy. I seem to recall a problem or two where the answer was -x, for god's sake, but I might be making that up.

Other times the test will give you a value for x + y, say, and you're supposed to see that you should simply insert that value some place else in the problem, et voilà: the answer they're looking for pops up.

Here's a typical problem, medium difficulty (according to the College Board):

If 4(x + y)(x - y) = 40 and (x - y) = 20, what is the value of x + y?

A kid who's had no test prep at all will likely miss this question -- either miss it outright or take too much time spotting the solution, thus leaving him too little time to finish the test and increasing the likelihood he'll make "careless errors" on the questions he does get to because now he's working too fast trying to make up for the time he lost on the x + y problem.

For what it's worth, I think using x + y as the value, instead of x or y alone, is an interesting and instructive way to write a problem. (I'm curious what math people think). It seems to me that writing problems in which x + y is the salient unit may be a way of teaching what Ron Aharoni calls the fifth fundamental operation of arithmetic:

In addition to the four classical operations, there is a fifth one that is even more fundamental and important. That is, forming a unit, taking a part of the world and declaring it to be the “whole.” This operation is at the base of much of the mathematics of elementary school. First of all, in counting, when you have another such unit you say you have “two,” and so on. The operation of multiplication is based on taking a set, declaring that this is the unit, and repeating it. The concept of a fraction starts from having a whole, from which parts are taken. The decimal system is based on gathering tens of objects into one unit called a “10,” then recursively repeating it.

The forming of a unit, and the assigning of a name to it, is something that has to be learned and stressed explicitly. I met children who, in fifth grade, knew how to find a quarter of a class of 20, but had difficulty understanding how to find “three-quarters” of the class, having missed the stage of the corresponding process of repeating a unit in multiplication. What I Learned in Elementary School by Ron Aharoni

Maybe I'm wrong, but it seems to me that the x+y questions test math as opposed to obedience under pressure, which is what the Find xy questions test.

Still, there is no doubt in my mind that these questions elicit wrong answers from test takers who know the math involved, can do the math involved, and have a reasonable understanding of the math involved. Students who have spent years of their lives solving for x aren't going to break the Solve for x habit for the first time ever when they're working at breakneck speed and their eyes are bleeding from the Ella Baker passage.

Which brings me back to extinction learning. Test prep for SAT math involves spending a fair amount of time building new habits that conflict with ingrained old habits. You've been conditioned to solve for x; now you have to condition yourself not to solve for x. Also, you have to build as much speed as possible at not solving for x because you are never going to forget solve-for-x. The two impulses are inside your head, competing with each other, and the competition takes time (and probably eats up some precious working memory resources to boot).

Funny thing: during the time we spent doing SAT math prep around here, I overlearned don't solve for x to the degree that a couple of weeks before taking the real test I came across a practice problem that did ask the test-taker to solve for x. I was so surprised that I wasted several seconds reading and re-reading and re-reading again to make sure I hadn't misunderstood. You can't win.

For parents: your child needs to spend enough time not solving for x that he or she gets to be really, really fast at not solving for x.

Then he should be on the lookout for problems that say Solve for -x.


I'm a 10
rat psych: what to do about SAT math (part 1)
rat psych: what to do about SAT math (part 2)
rat psych: what to do about SAT math (part 3)
rat psych: careless reading errors on the SAT

rat psych - what to do about SAT math (part 2)

What makes SAT math “tricky” (part 1): here and here.

Picking up where I left off a little while ago, the answer to SAT math trickery, part of the answer, is extinction learning.

"Extinction learning" means learning that what you've previously learned no longer applies.

Say you're a rat in a cage and the sound of a buzzer means you're 2 seconds away from receiving an electric shock. You learn this lesson very, very well.

Then one day, things change.

Now, under the new regime, the sound of a buzzer means a piece of kibble, or perhaps a morsel of cheese.

You -- the rat whose life fortunes have taken such a dramatic turn for the better -- don't just go with the flow. You don't hear the buzzer and say to yourself: Buzzer means kibble---oh, boy!

No. You remember the dark days when buzzer meant shock, and it will take time and many repetitions to learn that buzzer means kibble now. You will learn that buzzer means kibble now, but you will never forget the old days; buzzer means shock will always be with you. There's a saying that once you've scared a person or an animal, you can't unscare him, and that's true for many things, including a. When you've learned something very, very well, you can't unlearn it; you can't not know. You just have to learn the new thing on top of the old.

So say you're a high school senior and you have spent the last 4 years of your life seeing the letter a only in the context of the quadratic equation in its standard form: ax² + bx + c = 0.

You are the rat, a is the buzzer, and buzzer means quadratic equation coefficient of x² in the standard form of the quadratic equation. By the time you've reached your 17th birthday, "a means quadratic equation coefficient of x² in the standard form of the quadratic equation" has been deeply imprinted into your brain; you have learned this so well you’ll remember it when you’re 80. There are probably people with dementia who still remember ax² + bx + c.

Now it's senior year and you're taking the SAT -- a math section -- and time is running out. Your eyes are bleeding from the protracted Ella Baker critical reading passage you’ve just hacked and bashed your way through, your future is being decided and your fate being sealed -- and all of a sudden, here you are, staring at the letter a inside a quadratic function.

And you blow it.

You don't see that this a isn't the a you know.

Which is exactly the effect the question has been written to produce.

Getting late – will finish this up tomorrow.

I'm a 10
rat psych: what to do about SAT math (part 1)
rat psych: what to do about SAT math (part 2)
rat psych: what to do about SAT math (part 3)
rat psych: careless reading errors on the SAT

rat psych - why SAT math is tricky, redux (part 1)

from perfectscoreproject:


SAT math questions use the phenomenon of associative interference against the test taker. That's what makes the questions tricky: each of the problems Debbie has posted on her site is designed to activate the wrong associations inside the student's mind. Why else choose the letter a in question #7? If your goal as a problem writer were to avoid associative interference, you would choose a different letter.

Put College Board math trickery together with the high-stakes, time-pressured, mentally grueling nature of the entire 4-hour ordeal, and you radically increase the odds that students will take the bait, especially students with high working memory. (pdf file)

As for people who breeze through the test racking up correct answers, I would be interested to see how they fare on find-the-missing-figure puzzles. I'm guessing many of them would do well. I have no idea whether aptitude for missing figure tests is associated with aptitude for math. I wouldn't be surprised to learn that it is, but I've never read anything about it one way or the other. The point is: doing SAT Math is about perception as much as anything else. SAT math is about finding the hidden right triangle in the not-drawn-to-scale figure that looks exactly like something else altogether because it is something else altogether, in real life. To do the problem you have to look at the figure, but when you look at the figure you have to not see the figure that's actually there on the page. You have to see the other, not-there figure. Finding a hidden figure that is not present on the page is a couple of quanta more challenging than finding a hidden figure that is present on the page, I think. ^*

Find the implied hidden figure!

Infer the implied hidden figure, and then solve a problem about it!

Apparently, people who breeze through the test racking up correct answers easily break free of the actual figure on the page (or so I gather).

For the rest of us, it is simply not possible to "break set" in the heat of the moment. It is not possible because breaking set in the heat of the moment is precisely what our brains are built not to do. Under pressure, normal human beings become less flexible, not more.

So, for the rest of us, the answer (one part of the answer) is "extinction learning," which is a critical component of SAT math test prep. All parents should know this.

Back in a bit.

I'm a 10
rat psych: what to do about SAT math (part 1)
rat psych: what to do about SAT math (part 2)
rat psych: what to do about SAT math (part 3)
rat psych: careless reading errors on the SAT

^*The hidden right triangle problems do not appear on 10 Real SATs. At least, I haven't come across one leafing through the book. For me, those are the hardest problems bar none, a reaction I've heard from numerous others.

not your father's SAT

I've been amazed by the difficulty of SAT math in its current incarnation; I remember SAT math as being pretty easy. Now it's hard.

I'd been wishing I could look at an old test -- and I seem to have misplaced the email Akil sent me that included an old test (must find email...) when I raised this issue before --

Anyway, long story short, I've carried on being mystified over the question of what has or has not happened to SAT math.

Suddenly, the other day, it hit me: 10 Real SATs! The book was published before CollegeBoard changed the test in 2005 (2006?) It has real SAT tests -- 10 of them! -- with real prior-to-2006 SAT math.

So I ordered it.

And ----- wow.

The math on the earlier tests is so much easier. Easier at the level of: I found myself doing the final, hardest problem in a section in my head, in bed, in a state of sleep deprivation, and after drinking a glass of wine.

I'll have to sit down and take a timed section and see what happens.

kp on the MathCounts course and SAT math

kp writes:

I took the AOPS Advanced MathCounts class this summer (I'm a coach and wanted to see what it would be like for my students and what new things I could learn from it.) I appreciated that the class taught the shortcuts but also focused on how the shortcuts worked and how you could adapt them when the problem was given a new twist. (For example, to find the number of factors a number has, first find its prime factorization, then add 1 to each of the exponents, then find the product of these numbers. They explained why this made sense, then assigned different variations of problems on this topic.)

I'm no expert on the SAT (I'm a middle school teacher), but there does seem to be a large overlap between hard middle school math and what is on the SAT. We sometimes use SAT practice problems in our MathCounts practices. Our district's merit scholars often participated in MathCounts in middle school. Perhaps that is because the type of kid who stays after school to do math is the type of kid who is also successful on the SAT, or perhaps it is because MathCounts helps to prepare them for the SAT.

The MathCounts course sounds like a blast.

Teaching the shortcuts is a great idea -- it's the shortcuts that help you see what's actually going on, I think.

I remember years ago reading an article -- it may have been a study -- about smart-works-hard type students versus the 'naturals.' The smart-works-hard types went on wild goose chases trying to solve problems, while the naturals produced short, elegant proofs and solutions. I laughed, reading that, having been on many a wild goose chase myself.

add this problem to the curriculum

re:

r² is a multiple of 24 and 10. What is the smallest value?

Here, from a few weeks ago, is Stanislaus Dehaene on multiplication:

[O]ur intuition of quantity is of very little use when trying to learn multiplication. Approximate addition can implemented by juxtaposition of magnitudes on the internal number line, but no such algorithm seems to be readily available for multiplication. The organization of our mental number line may therefore make it difficult, if not impossible, for us to acquire a systematic intuition of quantities that enter in a multiplicative relation. (This hypothesis is supported by the fact that patients can have severe deficits of multiplication while leaving number sense relatively intact; in particular, patient NAU (Dehaene & Cohen, 1991), who could still understand approximate quantities despite aphasia and acalculia, was totally unable to approximate multiplication problems).

This passage precisely captures my experience learning arithmetic.

Addition and subtraction make intuitive sense to me; multiplication and division do not. It's really that simple. For me, "math" - math as opposed to simple counting - begins with multiplication and division.

This observation brings me to a corollary: people always say kids fall off the math cliff when it's time to learn fractions, but I think the math cliff comes sooner. I think the math cliff is multiplication, only nobody knows it.

Nobody knows it because falling off a math cliff isn't like falling off a real cliff; with a math cliff, you can walk right over the edge and just hang there for awhile, suspended in mid air, like Wile E. Coyote.

The real drama comes after the fall, which is when kids finally get to fractions. Fractions aren't the cliff, and they aren't the fall. Fractions are the crash-landing at the bottom.

At least, that's my guess for the moment.

Setting metaphor aside, though, if it's true that we are not equipped with an intuitive understanding of multiplication and division, and I believe it is true, why don't more people know this?

Everyone knows fractions are hard; why doesn't everyone know multiplication and division are hard?

It's true most people perceive that certain aspects of multiplication and division are hard. Namely: memorizing the times tables is hard (for many children) and learning to do long division is hard (for many children). But I've never seen anyone take these facts to mean that there is something intrinsically challenging about multiplication and division in a way that is not the case with addition and subtraction.

Why?

I don't know, but I have some thoughts.

Which.... will have to wait. It's getting late, and I'm still trying to edit the body of this post into shape, so I'm going to set that aside and skip to the end, and just say that I think children should probably be taught to solve problems like the one above, which appeared on the October SAT. I'm pretty sure problems this can be used to find out whether students are suffering associative interference between addends and factors, which I bet an awful lot of students are no matter how quickly and accurately they can construct factor trees.

I'm also thinking more attention should be paid to teaching young children the terminology of arithmetic: addends, subtrahends, factors, and the like. I think -- I don't know -- that fluency with the terminology might help reduce associative interference. "All math looks alike": the 5 and the 2 in 5+2 look exactly like the 5 and the 2 in 5x2. But the words addend and factor have nothing in common whatsoever.

More later.

what is SAT math?

I'll be interested to hear from all of you if I'm wrong about this, but as far as I can tell, SAT math is middle school competition math.

It's hard math for middle school.

If C. and I had worked our way through Art of Problem Solving's Competition Math for Middle School, neither of us would have gotten stuck on the factoring problem we both got stuck on:

r² is a multiple of 24 and 10. What is the smallest value?

Competition Math for Middle School, a book for mathematically gifted middle schoolers, explicitly teaches the answer to this question:

441, 256, and 576 ... are all perfect squares. If a number is a perfect square, each of the prime factors will have an even exponent in its prime factorization.
J. Batterson, p. 139

Leafing through the two books, I am struck by the the amount of explicit, procedural teaching directed to mathematically gifted students. No one's asking them to figure these things out or to "problem solve." Instead, they're being directly told that all of the prime factors in a perfect square will have even exponents.

Ditto for the how many diagonals in an n-gon, a perennial favorite amongst SAT math writers these days, it seems.

correction: My wording above -- explicit, procedural teaching -- implies that these books teach procedures instead of concepts. That's not at all the case and isn't what I meant to convey.

What I meant to convey was the fact that gifted middle school students aren't being asked to figure out  concepts for themselves; they're being explicitly told how the problems work, and why.

SAT reading vs math

If I had to come up with an artificial number of what constituted truly impressive combined scores on the recentered SAT I, I would say over 1,490 or 1,500. Those are still considered very high scores, ones that are not easy to attain, no matter how bright a student is.

I hesitate to give this example, because, as always, it is not the combined score that matters as much as the breakdown. It is still, as it has always been, more impressive to see high verbal scores than high math scores, since most students at the highly selective colleges will be doing much more writing and reading than math. Verbal ability is still a good indicator of how strong a reader the student is. The ability to read well will ultimately have a bigger impact on most college students than the ability to do SAT I math very well, especially since the level of SAT math is not particularly high. There are many students who do terribly on the SAT I math and yet who manage to get the highest score of 5 on the AP calculus exam. If any math is useful at the college level, it is calculus, not the basic math covered on the SAT I. Therefore, SAT I scores of 750V, 630M would be much more impressive for most highly selective colleges than a 640V, 780M, even though the latter score has a higher combined total by forty points.
A is for Admission by Michele A. Hernandez 1997

From what I can see, this view still holds true (though take my observation with a grain of salt).

I do wonder whether the high-end test prep industry may have actually increased the value of a high Critical Reading score, given that the reading test is the one you can't tutor.

I use the words "can't tutor" because those are the words everyone uses: it is extremely difficult to raise the critical reading score via tutoring, and everyone knows it, including college admissions officers presumably.  Nonetheless, I believe there are tutors who do raise reading scores.^*

But it's not easy.

My guess is that college admissions officers at highly selective colleges assume the parents bought 50 points on math, but the reading score came from the kid. 


^* I should probably add that C. wasn't tutored in reading. I have confidence in Erica because I've met her and I like her book.