More ETS Math "Thinking"

Greetings From "Princeton, NJ" 

Another day, another math SAT Problem of the Day from your friendly Educational Testing Service/College Board (always like to mention that while the ETS has a "Princeton, NJ" address - and also one in Berkeley, CA - they are not affiliated with Princeton University or UC-Berkeley. Of course, neither is test prep giant The Princeton Review, but they're small fry compared with the ETS. Even better, the ETS is actually located in nearby but less academically-impressive-sounding Lawrenceville. I suppose it's "poetic license." Or something like that).

So, here's today's problem, answer, and ETS explanation:

Mathematics > Standard Multiple Choice

Read the following SAT test question and then click on a button to select your answer. 

The stopping distance of a car is the number of feet that the car travels after the driver starts applying the brakes. The stopping distance of a certain car is directly proportional to the square of the speed of the car, in miles per hour, at the time the brakes are first applied. If the car’s stopping distance for an initial speed of  miles per hour is  feet, what is its stopping distance for an initial speed of  miles per hour?

• (A)  feet
• (B)  feet
• (C)  feet
• (D)  feet
• (E)  feet

Correct!

Explanation

The stopping distance is directly proportional to the square of the initial speed of the car. If  represents the initial speed of the car, in miles per hour, and  represents the stopping distance, you have that the stopping distance is a function of  and that , where  is a constant. Since the car’s stopping distance is  feet for an initial speed of  miles per hour, you know that . Therefore, , and the car's stopping distance for an initial speed of  miles per hour is  feet.

Okay, so what's my gripe this time? Did they screw the pooch again? No, not insofar as their indicated answer and the explanation thereof are both consistent and correct.


But pity the high school kid prepping with their advice (and I will bet a lot that the test-prep they charge for is JUST like this less-than-incisive example). I looked at this problem and reasoned that if the relationship is based on the square of the speed, and if the second speed was double the first that the stopping distance had to be 2^2 or 4 times the first. 17 x 4 = 68. End of problem. No 0.0425 necessary. You'd have to be blind to SAT math best practice (it IS, after all, a TIMED test, folks) to set up that ridiculous equation and solve it, even with a calculator. While you're writing that down and pushing calculator buttons, I've done no more than 10 seconds of thinking and mental math. And am already on the next problem.

So, I'm REALLY smart, right? No, not at all. What I am is very test-wise, experienced, on to the ETS tricks, aware of the mind-set they WANT you to have, versus the mind-set the best, most efficient test-takers bring to the table. When I tutor, I want students to avoid as much as possible getting bogged down in this sort of ETS swamp-thinking. All that needless formula-writing that has its place in some contexts, but not here. All that US math class donkey arithmetic that eschews THINKING and jumps into writing down computations and carrying them out without any consideration of the real mathematical and/or physical situation under consideration. If nothing else, it should hit students that a 1:2 ratio, squared, becomes a 1:4 ratio. This comes up a lot on the SAT and ACT, often having to do with linear vs. area situations in geometry (also 1:2 versus 1:8 when volume is involved, and to generalize a/b versus (a/b)^2 or (a/b)^3. It also arises, as in the current example, in some physical problems. In these kind of questions, the word "square" is explicit, and it's a huge hint.

Of course, students being clever test takers shouldn't operate mindlessly, either. That is, my suggestion that the test is giving a huge hint as to a more efficient strategy isn't to be used without consideration of its applicability or lack thereof. Sometimes, the test-makers pull a fast one, like on the ACT test that asked for the ratio of one linear measure (diameter) of two circles and another linear measure of those circles (the circumference). That stays the same, and students were offered the trap answer in which the first ratio is squared. Nice! But the cautious test-taker won't miss that the squaring of the ratio isn't appropriate if we're not talking about areas or some other situation that calls for squaring.

The bottom line in doing mathematics is to actually THINK. I know that doesn't fit the mindset of US mathematics education, publishing, or assessment. All that thinking gets in the way of making money. But if we want kids to grow up other than as little robots, we need to push them out of their personal comfort zones and the infantilizing, passive world that we see further evidence of in the "explanation" of this problem from our friends at ETS.

Note: Yeah, I know that there's nothing mathematically WRONG with the explanation, and I anticipate comments from those who did it JUST as the ETS suggests. Such folks will resent my suggestion that it's a mindless approach. So let me be crystal-clear: what's mindless is to do it that way on a timed test where every second spent doing something unnecessarily slowly is a second you don't have to work on more challenging problems. Just sayin'.