kitchen table math, the sequel: rat psych - what to do about SAT math (part 2)

Thursday, October 20, 2011

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

17 comments:

Allison said...

--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...By the time you've reached your 17th birthday, "a means quadratic equation" has been deeply imprinted into your brain"

Then you have no mastery of school math at all. You have only superficial and procedural knowledge of school math, and have failed to grasp the most core ideas about symbols or functions.

And your SAT score reflects that.

(btw, no senior has spent 4 years looking at the "ax^2 + bx + c = 0" form at all. They haven't spent a year looking at it unless they flunked a course a few times. At best, they spent a few weeks seeing it spaced out over a couple years. But in TEST PREP, they might have seen it and taught themselves the wrong thing to know about it!)

Amanda said...

I am rather surprised by this series of posts. This particular question does conform with the conventions of teaching quadratics and other polynomials in that 'x' is a variable and 'a' is a constant.

Surely only very weak algebra teaching could lead someone to think that the letter 'a' was always and exclusively the coefficient of x^2 in a quadratic and equation? Or was I just lucky?

SteveH said...

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.

rocky said...

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.

That's why the old grey rats are always mumbling to themselves, "The coefficient of x². The coefficient of x. The constant term..." People think they're mad, but they've just been shocked too much.

SteveH said...

The 'a' problem doesn't bother me much, but it's clear that they are trying to trick students. They try to trick students in most questions. The r^2 problem doesn't bother me too much either. 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.

Allison said...

Catherine, on another thread you said:
"...SAT writing is testing REAL STUFF. SAT writing tests the EXACT errors college students make in their writing. "

You were arguing that the writing test isn't "tricky".

I can't evaluate the truth of that sentence, but I believe you. I believe you because you teach writing to college students, and Ed reads college students' writing all of the time.

But this is EXACTLY what I would say about the SAT math--it's evaluating REAL stuff. The kind of errors that students make on it are exactly the errors seen by college students in their math classes--errors demonstrating misunderstandings and an inability to recognize those misunderstandings. These are errors that undermine their ability to learn other science, eng or math material.

lgm said...

>>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.


That would not be the case in NY. The senior would have seen the Pythagorean Theorem a^2 + b^2 = c^2 numerous times. Hopefully he would understand by senior year what 'coefficient' and 'constant' and 'variable' mean.

I do agree though, that more abstract material should be included in the Regent's math courses, especially in districts like mine that do not offer honours level classes. My sons are in Alg2/Trig this year and they have yet to hear "Let x be..." or "Let's take the case that..." in math class nor have they had to define any variable in their solutions. Algebra for all is not good for all.

cranberry said...

If the student is confused by a, she doesn't understand the math. It's a perfectly fair question. I'd even say it's not tricky at all.

I think part of SAT math is reading the question carefully. A student taking the SAT math section should be able to manipulate simple functions.

ChemProf said...

I do think that too often math is taught in the inflexible way that Catherine is talking about -- I see it in students who can solve for x but who can't do the same math to solve for P (pressure) or T (temperature), which is exactly what they need to do science.

But as Allison says, this is exactly what you need to be able to do in college-level science or math classes, so I don't think it is just "tricks." These are the skills you'll need for STEM fields, where there aren't enough variables to go around.

Even the bad drawings test skills you need. In a physics problem for example, you may draw a picture that turns out not to be as clear an illustration of the final problem as you think.

Catherine Johnson said...

Allison (& all) - I'm going to backtrack a bit from my statement about the writing test.

It tests the specific errors college students are most likely to make in their writing (subject verb agreement; pronoun antecedent agreement; etc.)

However, it does have 'tricky' questions in the sense that the writing is often quite bad, and most students pick up grammar problems by ear. (True of me, too: in fact, I think a person who uses grammar well in writing probably should be picking up grammatical errors by ear.)

A writing question that is badly written that does **not** have an error sounds like it has an error because in fact it does have an error: it is badly written.

I don't see the value of such questions.

Catherine Johnson said...

Here's the thought experiment: what would happen if the question used a different letter in place of a?

Would more students get the problem right?

I suspect more students would get the question right -- and I **strongly** suspect that is what the College Board believes.

Associative interference is a real phenomenon that is different from math and math knowledge. Remember Dehaene's statistics on the number of errors adults make in the times tables?

I don't think many here would say that adults who misremember multiplication facts don't understand multiplication (or am I wrong about that - ?)

There are two separate issues here: one is math, and the other is cognition. The brain is built to function in certain ways, and the SAT systematically exploits phenomenon like associative interference to elicit wrong answers.

Catherine Johnson said...

Even the bad drawings test skills you need.

So why weren't those bad drawings on the test 10 years ago. (Those tests do have the not drawn to scale drawings, but they're different....)

The not drawn to scale problems are huge, huge, huge working memory drains --- massive.

Haven't gotten to that yet: SAT math is a test of working memory.

I'm sure strong working memory is related to math achievement, but it's related to every other kind of achievement, too.

Catherine Johnson said...

rocky!!!!!!!!!!!!!!!

That's why the old grey rats are always mumbling to themselves, "The coefficient of x². The coefficient of x. The constant term..." People think they're mad, but they've just been shocked too much.

That's me!

Allison said...

Catherine,

If you've mastered school math, *there is no need for such HUGE USE OF WORKING MEMORY*.

That's what I'm saying, 1000 different ways.

Yes, this test IS testing YOUR working memory, and it's a big drain--because DUM DUM DUM --MASTERY means you've chunked this stuff!

So the test is trying to overload a student who is relying too heavily on working memory--that student is the WEAK MATH STUDENT.

Allison said...

--I don't think many here would say that adults who misremember multiplication facts don't understand multiplication (or am I wrong about that - ?)

ABSOLUTELY! An adult who misremembers multiplication facts DOES NOT UNDERSTAND MULTIPLICATION.

Why? because there were a hundred ways to mentally compute the same result, all of which should have fired neurons to support solving the problem, and to actually misremember means NONE of those parallel ways noticed--so there's no parallelism happening somehow.

6*7 is 42 is 35 + 7 (that is, 5*7 plus another 7) is 36 + 6 (that is, 6*6 plus another 6) is 48 - 6 (that is, 6*8 - another 6) is 49 -7 (that is, 7*7 minus another 7) is 21 doubled (that is, 2* (3*7)) etc. etc. Understanding multiplication means you have internalized all of the above are true and related and saying the same thing. How could someone possibly misremember that interconnected web of truths without a red flag going up in the brain saying "CONTRADICTION!!!! Something's WRONG!"

Allison said...

--The brain is built to function in certain ways, and the SAT systematically exploits phenomenon like associative interference to elicit wrong answers.

I AGREE with you here, Catherine.

What I'm disagreeing with is
that this makes the test "tricky", that this makes the test somehow invalid/unfair/not a good test of math mastery.

The opposite of associative interference is cueing--I don't know the current cogsci term for it, but it seems like you are mad that the test problems aren't cueing your or prompting your brain to get the right answer. You're mad that you're being made to miscue.

But the only test takers who fall for the miscue, the ones who experience the associative interference, are the ones who don't GROK the problem. They are the ones whose cueing is not just on the wrong item, BUT ON THE WRONG LEVEL.

whether the symbol was an "a" or a "b" or whatever, those of us who understood that picture KNEW IMMEDIATELY what the equation form was. the *x* could have been replaced by *a* and we still would have understood.

Glen said...

An adult who misremembers multiplication facts DOES NOT UNDERSTAND MULTIPLICATION.

Why? because there were a hundred ways to mentally compute the same result, all of which should have fired neurons to support solving the problem, and to actually misremember means NONE of those parallel ways noticed--so there's no parallelism happening somehow.


That there is, in general, no parallelism happening is what Bajic and Rickard's study of this question concluded.

When we face a single-digit by single-digit multiplication, we quickly decide to either retrieve it from memory or to calculate it. We don't do both in parallel and choose the winner of the race; we do one or the other. We could follow retrieval with a double-check calculation, but that's sequential, not parallel, even if quick. The difference is that if parallel, both would happen. If sequential, the second step could be (and I believe it is) optional.

Studies like this and my own experience strongly suggest that I might confidently retrieve the answer from memory, make an association error that looks right because, for example, it is a familiar answer to a similar question, and move on without getting a "second opinion" by calculating it.

Anything slightly more difficult is likely to trigger (in me) a brief sequence of alternative methods just to double- or triple-check, but if I confidently retrieve it from memory, it looks right, and I'm under time pressure, I may very well move on without redoing it, letting the error stand.

I don't believe that these errors (retrieving the wrong item from a set of familiar, memorized facts) is necessarily evidence of lack of understanding. I sometimes call my kids by their siblings' names and don't realize it until they point it out. What am I misunderstanding?

But, as I said, I agree with Allison about this SAT problem. Like many of us, I've seen these downward accelerating curves so many times written so many ways that I don't attach much meaning to what specific letters (or names in computer code) they choose to use for the coefficients, constants, and variables. I knew in advance what the general form of the equation would be, so I recognized which components meant what. Variability of experience (part of what I think Allison is referring to as "maturity") tends to convert incidental associations to meaningful, underlying abstractions.