Making it harder
We ponder the question of just what makes a math problem difficult.
Our tutoring consultant has at least his share of arrogance. He realized this early on, when he examined a practice problem, checked it again, and pronounced the answer key wrong. Most tutors are reluctant to do something like this, having more doubt about their own ability, or fallibility. Ours, however, has enough experience and enough ways of cross-checking methods and answers that he did not hesitate. He has since become gentler in his pronouncements, because it does the students no good to doubt their materials all the time, and he freely admits he’s made mistakes himself now and then.
But a similar situation arose a week or two ago, when two students asked for help on the same two problems in Calculus. They were assigned to figure out certain limits (if you know what that means, you also know they can be tricky problems at times). Our tutor noticed that, with a small change, each problem could be made into a form that should have been familiar to the students; but as stated, he wasn’t sure how to do them himself.
Well, a little concentrated thought the next day gave him the solutions. But he still thinks the problems were misprinted, because the technique he used was more advanced than most students at this level could be expected to come up with (though they should be able to follow it). More importantly, without the misprint they fit easily into the lesson as taught.
But the episode made him ponder just what areas of math are sensitive to small changes. In a basic algebra problem, for instance 3x + 7 = 4, changing any of the numbers makes no material difference to how difficult it is to solve. He does have students who view with concern answers like y = -11/47, and in general think fractions should be kept away from all right-thinking people, but eventually they can be persuaded to solve problems as given.
As soon as you come to polynomial equations, though, a change can make a big difference in difficulty. x2 + 4x -21 = 0, for instance, factors easily and leads to x =3 and x = -7. Change the -21 to -22, though, and the answers become irrational and must be pried out of the quadratic formula. Change it to +22 and the answers are complex. And as soon as we go to higher powers, the equations that are amenable to factoring become special cases, and the general formulas (they exist for cubics and quartics) are so complicated that they are not generally taught. Indeed, the study of what makes a polynomial equation solvable has led to something called Galois Theory, a very interesting and productive branch of mathematics (though we ourselves are not well-versed in it).
Calculus is at least as bad. While a persevering student can take the derivative of any formula given explicitly, it is depressingly easy to write down an integral that cannot be done. One of our tutor’s students last year miscopied a problem, and had trouble dealing with it. Our tutor eventually turned it into an Elliptical Integral, which has been proven to be impossible to write out in closed form. It doesn’t take much of a mistake.
Of course we must continue to teach the mathematics that we can work with, which is certainly useful and indeed powerful. But sometime each student arrives at the level where the solvable problems are special cases, and must realize that most of the world cannot be factored.
