The theory and practice of guessing
I recently had an email exchange with Phil Keller, one of my favorite SAT tutors, and the author of the excellent and highly-recommended book The New Math SAT Game Plan. We were discussing guessing on the ACT and SAT.
I’ve excerpted portions of our email conversation with his permission because I thought it would be interesting to see how theory and practice and mathematical probability show up in real-life tutoring situations with hundreds of students.
Phil read this blog post: Strategic guessing leads to a higher SAT score. In it, I advise students to pick A instead of randomly guessing when they don’t know how to answer a given question. I assert that this method of random guessing will result in a much higher test score.
Phil, an actual high school math teacher and author, knows about probability theory and took issue with my assertion. He says:
Let’s consider 3 students who head in to the test with 3 different strategies for when they are guessing. And we are not talking about educated guesses here — just the “I have absolutely no idea” guesses and the “I am out of time because I went slow like Stacey told me” guesses.OK, on to the three kids:
Kid #1I will always put A whenever I guess.
Kid #2I will rotate my guesses: ABCDABCD
Kid #3I will choose my guess at random, not trying to follow any specific pattern but I promise I will choose as quickly as the other kids do.
I do not believe that there is any advantage to any one of these strategies. And certainly not 70 points worth. After the fact, you can analyze tests and find some cases where one or the other of the three plans would have been the luckier choice, but there is no way to know that going in. They are mathematically equivalent.
Probability theory meets teenagers
I replied to Phil that I think there’s a difference between actual probability theory and how teenagers implement their understanding of random guessing.
I can see that how in theory all the choice of the 3 kids is equal, but in practice I think it comes out differently. And I think it’s a combination of several things: speed, mysticism, and control.
Hypothetical kids may choose a random answer quickly but actual kids don’t. Even the smart ones. They pick the “best” random answer choice – taking time and burning brain cells. It’s not a good use of time – as you know you either solve the question, half-solve the question or don’t solve it at all and just choose a letter and move on. Time spent solving or half-solving is a good use of time. Not solving it and trying to magically randomly pick the right answer choice is a waste of time
They usually have various levels of mysticism – “There was a 2 in the problem so I’m going to choose a 2 in the answer; I hate square roots so I’m not going to choose that answer; Hmm..I bet the answer is going to be even so I’m going to choose one of the even numbers.” And somehow the level of mysticism is inversely correlated to their scores. I’ve seen kids focusing on the pattern of the answers, instead of the question. Instead of focusing on solving the problem, kids will wave their hands around and honestly think they are doing “math.”
Kids (especially smart ones) freak out when they encounter SAT problems they don’t know how to solve. They spend enormous amounts of time beating themselves up for not knowing how to solve AND enormous amounts of time trying to solve – often getting more and more mystical. Getting rattled not only hurts you on the current question – it also messes up future questions because you aren’t thinking straight – panic slides right into mysticism, eats up time, and results in lower scores.
I agree that there’s no magic in A or C or ABCD patterns. But there IS magic in having and executing a guessing plan for the SAT and the ACT.
So the end result that I’ve seen over and over is this:
When you give kids a plan to handle ACT/SAT Qs that they don't know how to solve, kids are more confident, efficient, and spend time on the right things (mathematically solving problems that they CAN solve) - leading to higher scores Click To Tweet
There’s no time for mysticism or pattern-hunting
I do agree with this. It was the claim that choosing “A” would lead to a 70 point jump, tied to a specific test answer key, that led to my objections. I have no argument against the “pick a letter and go with it” plan. I have my students do the same. I just think you were over-stating the benefits — or really, not explaining the real benefits (which are harder to quantify). But the points you make in your email are true AFAICT.
For what its worth, I am telling students to go slow, read everything, and take lots of time to play around, giving almost no thought to whether they finish or not. But at the 18 minute mark in the short section and the 40 minute mark in the long section, I have them jump to the grid-ins to make sure they have time to work on them. After all, a kid who cannot get half of the grid-in points was not going to do well at the ends of the multiple choice sections anyway (unless they found “back-door” problems). The last thing I have them do is fill in random letters (or all the same if they prefer) for the multiple choice problems they didn’t have time to address. This keeps them from doing many of the things you are worrying about. There’s no time for the mysticism or pattern hunting.
Mostly, the test continues to be a brutal endurance event and a psychological torment. If you have shaky self esteem, this test will bring that out for sure. The message I am trying to send is: let’s go slow and do the best we can, using the time strategically. What you work on, answer as best you can. What you don’t get to work on, take that random guess.But don’t fall into the trap of bouncing from hard problem to hard problem, giving each a little effort and a little voodoo. That road, tempting as it is, does not lead to happiness.
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