The art of the possible

We consider being more realistic in education.

Some weeks ago, as we mentioned, our tutoring consultant made up a set of mathematical word problems for a particular type of student.  There are, of course, difficulties in making sure any problem set does what it’s intended to do, and he watched carefully as the students set to work.  One problem seemed much easier than he anticipated, another a bit harder.  What caught his attention, however, was that each student attempted to calculate a particular curve, a pursuit trajectory.  This is a problem in Calculus, and not a trivial one; it was years beyond anything these students had seen.  They certainly could not do it.  He diverted them and modified the question.

But it got him thinking.  In class, students are only given problems they can solve.  All the equations in the “factoring” lesson can be factored, indeed factored into expressions with integers.  Polynomial equations above degree two are mostly not solvable in closed form; students are given those that can be.  Now and then a question comes up for which the students are forced to write, “no real numbers,” but those are rare and well-prepared in advance.  He has seen a few Calculus questions in which the student is asked which of the following integrals can be done and which can’t, but not many, and the students have a lot of trouble answering.  Nowhere are students systematically taught the limits of their techniques.

Maybe it would be a good thing to address.  In the first place, it would start to dispel the illusion, fostered by standardized tests, that the world is a multiple-choice word problem.  It would be a great step in maturity, we think, for students to understand how sheltered they have actually been in math class.  It could lead to a really illuminating set of lessons on the difference between absolute and approximate mathematics, something not one student in ten (a hundred?) understands.  (That’s even when approximation is the point of the exercise.)

But, on the whole, we’ve decided against it.  First and most cynically, there is a large population of students who need no encouragement to decide there’s something they can’t do.  Among the others, we suspect that more would be discouraged by being told they have to wait to solve a problem, than would be excited by the prospect that someday they’ll have the tools.

And considering the few students who would best benefit from this sort of lesson, we think that they would learn more from attempting the impossible.  After all, that’s the first step toward making it possible.