While reading many of the solutions to the last AlgPoW of this year (The Professors' Primes), I was struck by the frequency of how many students immediately jumped to the conclusion that they would have to solve it by massive amounts of what we call "guess-and-check". Yet when looked at more calmly, it turns out to be more like a "domino" game we all played in our youth. You know the one: standing several dominoes on their small ends close together, and by flipping the first one, it tilts the second one, which tilts the third, and so on till all are down.
Many students solved the problem like that: one basic idea led to a 2nd one, which led to a 3rd, and before you know it ÃÂ the solution was done! And as for "guess-and-check"? Well, it could be said they hardly used it.
But I digress from my main reason for titling this post as I didÃÂ
When I was a kid (and it's still used today), we were taught to solve some problems by "trial and error". Now what kind of mixed message are we sending to a student with that duality? "Trial" is good [as in "If at first you don't succeed, try, try again."]. But "error"? Who likes to make errors? "Error" is bad. And, gee, if your first trial was lucky, and you got the right answer needed, you didn't make any error in the first place. But I was never lucky.
Later, along came another duality, just about as bad (IMHO): "guess and check". Now everybody agrees that it's good to check your work. That's one of the staples of the Polya scheme: check your work to see if it's correct and reasonable. But guessing is, I feel, a little on the negative side of the "scale of things". When taking a multiple-choice exam, mere guessing from the 4 or 5 choices is not usually a wise strategy for success. If students are taught to reflect on the outcomes of successive guesses, in order to make future guesses better, that would be good. But that takes time and good number sense on the part of the solver.
I think the result here is that we've just replaced a "good-bad" duality with a "bad-good" one. There were times in my teaching that I would have liked to try a radical idea and use a "good-good" duality philosophy, saying "trial and check" instead. I can't help but wonder if that might not have had a better effect in the minds of the learners as they attempted to solve hard and unfamiliar problems. But I guess I'll never know, because I am currently in the process of retiring from classroom teaching. I must leave that question to be answered by other colleagues.
In closing, I'd like to cite the lyrics of a famous hit song popular during the big band days of swing music:
"Accentuate the positive, eliminate the negative, Latch on to the affirmative, and don't mess with Mr. In-between."