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Today is SAT Math Day. I used to teach test prep for the SAT and the GRE. I did the best I could, but you could tell certain students had a way with numbers while others didn't.

There's one particular example I remember. I would teach a section on averaging, and later give a quiz. Part of the quiz featured questions with Column A and Column B, and you were given four choices--was A greater than B, B greater than A, were they equal, or was there not enough information to tell? (The joke was the last choice is correct--as far as I'm concerned, there's not enough information to tell.)

The question referred to a room filled with men and women. Every man was 6' 0" and every woman 5' 6". And 60% of the people in the room were women.

Column A was the number of inches for the average height in the room. Column B was the number of inches of someone who's 5' 9".

So the students would take the quiz and they pretty much always got this one right. The issue was how did they get to the correct answer.

Just having received a lesson on averaging, most would take the average of the people in the room. With an unknown number of people, they'd fill in the amount themselves. They'd been taught to choose the simplest numbers, so most would have three women and two men. Maybe some would have six women and four men. I don't think anyone ever had sixty women and forty men.

They'd break down the heights to inches, and figure out 3(66) + 2(72) = 342. Divide that by 5 and the average height is 68.4 inches, or 5' 8.4". That is less than 5' 9" so Column B was greater than Column A.

I'm sure those of you who have a sense of numbers already see the problem. Yes, the answer is correct, but this is a laborious way to solve the problem. If you approach every problem like this you won't be able to finish all the questions on the exam (and are more likely to make simple arithmetical mistakes).

What you need to see is the average of every man and women combined is 5' 9" so if there are more women than men in the room, the average height is below 5' 9".

But how do you teach that sort of insight? I don't know. Something to contemplate on SAT Math Day.

## 7 Comments:

I didn't know you used to teach SAT prep! I did that too, briefly.

There were three times that the SAT or PSAT included math problems that were wrong -- the test authors themselves wrote bad questions (or in one case, wrote a good question but identified the wrong answer as correct). Eventually, disgruntled students convinced the board they were wrong, and so they adjusted the students' scores.

I would try to post them here but they'd require symbols that I can't type in the comment box.

In any event, I agree with your general assessment of teaching test-prep. It might or might not be possible to teach a five-year-old to deepen their mathematical insight. But by the time a kid is sixteen, it's very unlikely that their fundamental interest in math and the way they think about math problems will change.

There are still many useful things you can teach in test prep! Knowing what will be on the test, spending lots of time doing sample tests, knowing what sort of things the SAT or the ACT usually asks (for example, most students take a year of geometry and then forget it, but there are only half-a-dozen geometry topics that ever appear on the test, and it doesn't take long to re-teach these to the students) -- all of these can raise a student's score significantly, no matter where they start. But it's virtually impossible to raise someone from a 600 to an 800 on the math SAT. To get an 800, you have to be someone who has thought about math problems a lot and developed a good intuition for math.

I tried to teach LSAT prep. I thought I had a good chance as I had done well on the LSAT-a perfect score in fact. But just when I was about to get the job, they (Stanley Effing Kaplan) asked about my experience taking the prep course and I said I hadn't taken one so they bounced me.

Maybe that was smart on their part because I don't know if I could have taught them my method. Ignore everyone else for a few weeks and keep the sample tests over and over again (I think I had 8 or them with answer keys that explained the right answer to each question). I think I took 2 a day and ultimately took each test 3 times until I could basically get a perfect score on each one and started to think like an LSAT test executive. I recall there being problem solving but not higher math on the LSAT (And it was all multiple choice). The fact that all my friends were away and I found LSAT studying more enjoyable than hanging out with my folks probably helped too.

I also taught the LSAT. It was fun. The LSAT is the most enjoyable test because you don't need to bring in any outside knowledge. In fact, attempting to use facts not mentioned in the test can only harm you. It's all about logic.

One section--the least favorite, according to my students--was called logic games. I always told them they're looking at it the wrong way. What other test allows you to play games?

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