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.
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5 comments:
This sounds right to me. I see a lot of engineering students, and not many of them understand math very well. Some have been trained to follow instructions well, so that they can duplicate the examples in the text, but their ability to apply math in new contexts is often quite poor.
Bad programming is also very common and does usually take the form of being far more complicated than it needs to be.
I think that a fair number of the students getting near 800 on the math part of the SAT are doing it without much (or any) prep.
I think what the SAT tutor is saying is that SAT math is not about applying math in new contexts.
Some kids might not need much test prep because they are the ones already involved with math competitions. Competition math IS test prep.
Clearly, extra work outside of regular math classes is required. This is not just because K-12 math is poorly taught on average. Even if you have a good background in math, and even if you have strong honors classes, you still have to prepare. The purpose of SAT is to separate kids. To do this, they have to resort to competition-type math questions. Some might think that doing well on competition math reflects a proper understanding of math, but I think that is a very limited view. The speed factor is critical and the test uses many special cases. These special cases penalize those who take a general problem solving approach to a problem.
"I was just providing a counterexample to the insistence that test prep is essential for getting a good math score."
What is a "good math score"? I claim that the specialized SAT prep starts to matter above a score of 600. Maybe one could make the case for 700. These are good scores, but are they good compared to other students with an equal math background? If you don't prepare specifically for the SAT, then you are put at a disadvantage.
If you have never seen a SAT substitution problem before, you might use a general approach to solve for x & y before solving (x+y)^2. Only because it is a test, and only because you have so little time, you might figure out that a simple substitution is needed. Unless you are ready for this, you've already wasted too much time. We are talking about very little time for this. This is not what some would call a normal speed in math. This requires specific preparation. It requires knowing a lot of special cases.
I always approached problems in trig using sines and cosines. I never bothered remembering special cases like 45-45-90 triangles. Now that I am studying the questions, I know that it is mandatory to know the radical (not decimal) forms for these triangles. You have to know them instantly. This skill is completely unnecessary for a proper understanding of math, or to show any particular level of aptitude. it is strictly competition math. If you get a problem wrong, SAT doesn't know if you don't know the material or if you just aren't fast enough. Speed is used to separate kids, but to get to that level of speed requires special preparation, not just general math knowledge.
I am unimpressed with the speed requirement of competition math. I think there are better things my son can focus on. A while back, my son spent time deriving series formulas. Last night we talked about whether X^4 has a curvature of zero at x=0. He was plotting it using GeoGebra. We discussed derivatives and integrals. However, I really have to get him to learn how to immediately calculate any side of a 30-60-90 triangle in radical form.
The main problem I have is at the top end on an extrordinarily competitive test covering a limited domain of material. A few questions can have a big impact on one's SAT score and one's college opportunities.
I agree that many engineering/CS students have memorized their way through math. I have also noticed that many high tech and financial companies, who need to hire technical people who are truly good at math and computational thinking, will include lots of puzzles and trick questions in their technical interviews. Google of course, is famous for this, but the financial places here in NY all do it too. They are looking for deep understanding and divergent thinking.
My husband had a perfect math score on the SAT, with no prep (he came from a working class town where many people did not go to college at all). He now has a PhD in CS from a program knows for "quants", and works at a hedge fund as a software engineer. He is one of those people who can look at a hard problem and immediately have the insight to the core of the problem. He can write tight, beautiful code without seemingly even thinking about it. His specialty is high performance algorithms and he is very good at it.Interestingly, back when he was in academia, he was at the forefront of the reform calculus movement, because he was appalled at the students coming in from good high schools, having memorized their way through AP calculus.
"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."
My point about that was to show the difficulty of trying to distinguish between aptitude and preparation. One of my colleagues loves to pretend to answer questions as if he is just figuring them out off the top of his head. Many think he is a genius. He also loves to explain things in ways that he knows are way over the listeners' heads. It's so easy to fool some people, especially those in personnel departments who think up those silly questions.
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