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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.
19 comments:
Thank you for saying it so PERFECTLY.
Ok, the 'a' I learned about in math class is a constant. What 'a' is this?
I'm with lgm; what's tricky about this question? 'a', 'b', 'c', ... are all just arbitrary constants.
Possibly tricky would be something like:
If this is a graph of f(a) = x - a^2 (and the x-axis were labeled as the a-axis), then what must be true of x?
Catherine,
I know you believe the SAT math is a tricky test, but I just don't see it.
It is a very good test of mastery. The things you see as tricky are perfect examples of why the SAT is a good test of mastery. Mastery is about those who are under pressure vs those who aren't---and for those who have mastered school math, this test isn't asking pressuring questions.
What does school math mastery look like?
It looks like mastering *having a clear mind*.
I mean: to master mathematics, what you master is the ability to know only what it written in front of you, with no additional assumptions.
I know you've got a theory about high working memory and choking, but what I see is that this question, like many you see as tricky, are showing that *you think the wrong things are important* and this test is extremely good at distinguishing those who think the right things are from those who think the wrong things are.
Someone who has mastered school math doesn't HAVE associative interference, because that is EXACTLY what their mastery has taught them: you only know what they gave you.
This question produced no weird associative interference for me or lgm or a few others of us because we know to only read what is in front of us.
In fact, that may be the most important thing to understand: if you have mastered school math, your head is good at being empty! You are clear, and then you only know what you read.
So, no, you aren't going to break set. Those who have really mastered school mathematics aren't breaking set either--they knew immediately that a was a positive constant just by looking at the graph. They didn't know anything about "a" other than what the graph showed them. if they felt nervous, they'd just check a point or two by throwing in a couple values for x and convincing themselves they were right, and then be even more in the flow.
There are "tricky" aspects of SAT math, such as the multiple choice format that offers wrong answers that are designed to look plausible, or asking you to find r^2 when you would normally expect to find r (and offering the value of r as one of the choices to trip you up). Something that is supposed to be testing my math ability that intentionally uses language designed to get me to MISunderstand what the question is asking is what I would call "tricky."
Even so, I would have to agree with Allison here that a lot of what some people think of as trickiness vanishes with familiarity. After thousands of physics problems dealing with ballistic trajectories, I can't even glance at a curve like that without guessing that it's probably an upside-down parabola plus some constant that tells you the height of the high point. It's the same associative phenomenon you describe, but with different associations that are the result of different backgrounds.
You don't need the quadratic formula for this one. What happens when x is 0? y = a. And that little y-intercept is above the x axis, so "a" must be positive. (answer A)
The picture is clear and I also don't see a problem with "a". The one thing they are trying to confuse you with is that students have seen "x^2 - a^2" or something like that a lot, so may expect that in this case a is the square of an integer (E), but there's nothing to show that is the case, so it has to be A (a>0).
That said, your sense is a really good example of why students should be given problems where it isn't all in terms of x, y, and z. It does take practice to see other letters as flexible variables (but that is a critical skill in lots of STEM fields).
I find it fascinating that most of the commenters above don't see what's "tricky" about this problem. I think that reveals a lack of familiarity with how math is taught. Most teachers (and indeed most textbooks) include somewhere in the Alg. 1 chapter on parabolas the following cookie-cutter rule:
If a is positive, the parabola opens upwards; if a is negative, the parabola opens downward.
Now this rule, it should be said, is sort of right. It is a perfectly correct rule if the parabola is specified by an equation written in "standard form", y = ax^2 + bx + c, in which a refers to the coefficient of the quadratic term. Without that crucial bit of context the cookie-cutter rule is not right; it is not even wrong, really, just meaningless.
A student who has learned only the cookie-cutter rule without understanding it will answer the question above (b), because the parabola opens downward, and therefore a must be negative.
I agree completely with Allison that this is not a "tricky" question; it is a question that tests whether you actually understand something or whether you just act on reflex.
Here is a similar question. Draw a right triangle. Next to one leg, write the label a=3. Next to the hypotenuse, write the label b=4. Next to the other leg, write the label c=?. Ask student to solve for c. A very large percentage of students will answer "c = 5", because they have learned the cookie-cutter rule a^2 + b^2 = c^2, and do not understand that what matters is not the letter used to label a quantity, but the role played by the quantity in the figure.
I think the “trickiness” of SAT math lies in the fact that the student needs to determine which mathematical approach to use for each problem. For most school math the student knows what tool to use, because they know what topic is currently being covered, they may even be told directly in the instructions. When learning to use any tool, one has to learn not just how to use the tool, but when to use the tool. The when might be more important than the how. Math contests are another place where students need to determine the solution approach to pursue, that's one reason why they might be good SAT math preparation.
This is related to Allison's post regarding the lack of definitions too. Sometimes (often?) definitions and conditions are given, but not given enough attention. Or perhaps there is some other reason that rules stick in students' heads and constraints fall out. (I've heard this complaint regarding college courses too, & not just math.) My daughter's Algebra 2 text has the rule that Michael presents above, but it includes the definition of the function and the constraint that a be non-zero. I suspect my daughter is one of the kids who will forget the constraints, unfortunately. She'd be better off just looking at the problem and reasoning (IMHO).
Good call, Michael Weiss. Teachers need to expose (shock) their students with many of these kinds of examples.
They also need to present the rules as "the coefficient of the second degree term", or "the square on the hypotenuse", even if it sounds high falutin'.
The things you see as tricky are perfect examples of why the SAT is a good test of mastery.
Hey Allison -
I'm trying to find an exchange on one of these posts re: whether or not this is a tricky question -- should have written it down.
I'll find it.
For the moment, though (and without having read all the comments here), I just don't see how a question like the one above tests mastery - unless you're talking specifically about mastery of the concept that the same letter variable can stand for different values in different problems.
This problem is not tricky for me (apparently that's not clear in my post). It was tricky for DS, who never took anything resembling algebra 2 in high school and who is at an earlier stage of learning algebra.
For a person who does not have mastery, this question is tricky. Period. It uses the phenomenon of associative interference to trip up a test-taker who would otherwise have a good chance of answering correctly (and not by guessing).
In other words, this question is tricky for a student who is scoring in the 500s.
What purpose is served by writing questions intended to "trick" or confuse students scoring in the 500 range?
These students aren't claiming to have mastery of quadratic equations and their graphs, and they are perfectly capable of revealing the fact that they don't have mastery without the confusion over 'a.'
I am philosophically opposed to writing a test with the goal of producing maximum error by any means.
I realize others disagree, but that is my position.
But beyond my philosophical position, I question whether writing a test so as to maximize error gives you more information than writing a test so as to minimize error due to associative interference, working memory overload, working memory collapse due to stress, and mental fatigue.
Basically, the test is built to produce the maximum number of "careless errors" possible.
I no longer think careless errors are due to careless people, but if I did I would feel the same way.
Michael Weiss wrote:
I find it fascinating that most of the commenters above don't see what's "tricky" about this problem. I think that reveals a lack of familiarity with how math is taught. Most teachers (and indeed most textbooks) include somewhere in the Alg. 1 chapter on parabolas the following cookie-cutter rule:
If a is positive, the parabola opens upwards; if a is negative, the parabola opens downward.
Now this rule, it should be said, is sort of right.
Yes! Yes! Yes! Yes! Yes!
Students have it drilled into their heads that "a" means xxxxx.....and the College Board knows it.
That's why "a" is there; it's there because the College Board problem writers know it will knock some mid-500 scorers down another 10 points.
We're not talking about mastery.
We're talking about the College Board maintaining a perfect curve.
I should look at the curves. At the top of the curve, on all of the tests, you always have the same number of items wrong for each score. (I think the number varies by no more than 1 or 2 on any given test -- but please correct me if I'm wrong.)
Each and every test item has been test-run in experimental sections, and they choose the items that will give them exactly the number missed they're looking for.
It's not about mastery, not when you're producing an error in a mid-500 scoring student.
What does school math mastery look like?
It looks like mastering *having a clear mind*.
I don't see this at all, and I don't believe it's true.
Math mastery means having a clear mind for math.
Having a clear mind for math doesn't mean having a clear mind for history or social science or other subjects.
That is a core finding of cognitive science.
btw, I see this all the time because I'm constantly digging into disparate fields. Smart people who reason well in their own fields make major errors of logic and assumption outside their fields.
Here is a similar question. Draw a right triangle. Next to one leg, write the label a=3. Next to the hypotenuse, write the label b=4. Next to the other leg, write the label c=?. Ask student to solve for c. A very large percentage of students will answer "c = 5", because they have learned the cookie-cutter rule a^2 + b^2 = c^2, and do not understand that what matters is not the letter used to label a quantity, but the role played by the quantity in the figure.
That problem is on the SAT. (Not quite: the version I've seen applied 3 & 4 to the angles in a triangle -- something like that.)
Michael could get a job writing problems.
These questions are not separating the 800 scoring kids from the 700 scoring kids.
They are simply producing reliable misses in kids whose scores are in the 500s and below.
That is their function.
I guess I'm not sure why you are surprised that their goal is to produce a curve, or why you think that's a problem. Here's the deal -- schools looking for science students are looking for math SATs at 650+. That means a student who has at least some ability to deal flexibly with variables, which is an important but under-taught skill. Students who don't have it will fail general chemistry because they will use the ideal gas law when Clausius-Clapeyron is needed, because they are just plugging in numbers.
Again, the errors that the SAT is setting up in the math test are the same kinds of errors you see in the writing test. They are exactly the kinds of errors that beginning STEM students make. You can argue the whole thing is set up to blow out your working memory, but the individual questions are absolutely appropriate.
I am surprised that commenters here think the SAT is testing for mastery!
I'm actually kind of stunned, and I guess I'm speaking as a person who writes about research psychology. This test has a gazillion confounding variables -- it's all over the place. It's testing working memory; it's testing working memory under pressure (a different issue & bad for kids with high-IQ); it's testing associative interference; it's testing test prep; and then it's also testing some math.
It's noisy.
I completely agree that 650 works as a cut-off, but that's a different issue.
At this point, I'm not sure you're getting any information from SAT math that you don't already get from the high school transcript -- and I think that's pretty much what the various people who've run the correlations have found, right?
+++++
I also don't think we can assume that kids who miss this question don't know what variables are (and I'm pretty sure a psychometrician would say you can't).
I think the only thing that's safe to assume is that most students who miss this question are novices on the subject of quadratic equations.
You would have to test students on the math they know to see whether they understand that variables vary.
A big issue is the time factor and stress. Do they test to see how increased time affects the error rate? I never liked take-home tests because that is just too much time, but what does speed tell you on a SAT test? It's too little time to tell you something about a broad level of mastery of math. It tells you about over-preparation for a narrow domain of material. SAT problems may catch some who have a rote or superficial understanding of the material, but that isn't true for all students. Many are caught by the time, stress, and stupid mistakes.
How fast do you have to be at any particular question in math to claim mastery? What does it mean to be faster than that?
--"This test has a gazillion confounding variables..It's testing working memory; it's testing working memory under pressure (a different issue & bad for kids with high-IQ); it's testing associative interference; it's testing test prep; and then it's also testing some math. "
You seem to think that there is some *thing* called "mathematical maturity" or that I called "math mastery" that is orthogonal to these other cognitive attributes (like working memory and the like.)
Why would you think that?
I'm saying that these aren't confounding variables, because what you think of as separate cognitive attributes are *part* of this maturity, flexibility, agility in seeing a math problem's answer.
Michael, thanks for clearing up the mystery of what is tricky about this problem. It never would have occurred to me there was anything tricky about it. And I haven't seen a problem like that since I took the SATs and GREs myself back in the eighties.
I guess this problem is intended to differentiate between students who have been taught math and students who have learned it. If students just have a mishmash of partly memorized rules in their heads, the "a" might trigger one of them and lead them to a wrong answer. If students actually understand the math, it won't matter how the question is phrased, because the math is so simple.
I look at this after a thirty year lapse and say to myself, "It's one of those whatchacallit parabola thingies. It starts with a middle point and moves away exponentially. They're calling that middle point "a" and it's above the zero." I don't remember why they call it "f(x)" instead of "y" I don't remember what "quadratic function" means. I don't remember much else. But I can look at this figure and see what it is, just like I did thirty years ago, and it's a dead easy question. I never paid much attention in math class, I never had any math past tenth grade, but really you shouldn't have to spend more than a minute looking at that if you've never seen it before to understand what it is.
Honestly, I feel like the SAT math portion isn't primarily a math test at all but an intelligence test. You can teach rules but you can't teach smart. And it seems to me that being able to understand what makes a figure like this is some kind of floor for smart. So tossing in some chaff like "a" to throw off the pretenders and rule half-memorizers is fair enough.
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