Or not

Sometimes following a procedure is necessary; other times, it’s not an option.

Our tutoring consultant is now in summer mode.  That means going through reams of problems like those on standardized tests.  The most distasteful part of this, though unfortunately necessary, lies in test-taking strategy: how to choose or eliminate multiple-choice answers for instance; mathematical tricks of various kinds that have no application outside the testing room.  A more useful skill lies in sorting all the confusing information into a coherent form and identifying a procedure to solve the problem.  We will get a quadratic equation and the problem asks for the maximum; find the vertex.  In many situations following step-by-step reasoning will bring order (and the right answer) out of confusion.

This doesn’t always work, even in non-test mathematics.  The time-hallowed problem of factoring a quadratic can be done by trying every possibility, but that’s the long way around.  After a number of examples most students pick up tricks that sometimes work and sometimes don’t.  If they do, the problem goes much more quickly.  And with experience (there seems to be no shorter way) a student will develop a feel for which tricks will work.

More advanced students have the task of solving integrals in Calculus.  You can always find a derivative, but there are integrals no one can do.  There are techniques that work sometimes but not others.  There are tricks and combinations of techniques for some of the more recalcitrant forms.  And while there are indications of things that might work, the real skill of integration only comes from doing a lot of integrals.  Like factoring, the student must develop a feel and a kind of creativity.

Hardest to teach are the challenge problems, those exercises in geometry or logic or whatever that do not obviously test a particular mathematical topic or skill, but require insight.  It’s quite possible for a student working on a Math Competition practice set to come up with a problem our tutor can’t see how to approach (at least immediately and in the context of a tutoring session, where he might have been laboriously guiding another student through the task of adding fractions).  Even when the required bit of lateral thinking comes easily to him, he’s sometimes hard put to suggest a way for the student to find it.

As before, there are approaches and tricks, questions to ask and ideas to try; but no guaranteed procedure.  It makes the job of tutor more difficult.  But also much more interesting.