Experience and word problems

Who makes the best teacher?

Our tutoring consultant spends a lot of his time preparing students for the math sections of standardized tests, chiefly the college admissions tests.  He has an extensive stock of material (produced by the company he works for), workbooks full of problems graded from remedial to very challenging.  Most are multiple-choice questions.  The following story might give some impression of its size.  One student had noticed that she had answered “C” five times in a row, and figured this couldn’t happen, so changed one or two answers.  In fact the answer key did indeed have “C” five times in succession.  Our tutor figured the odds, and concluded that, if the answers were randomly distributed among the first four letters, there were enough problems that a run of five Cs was more probable than not sometime.  If one only wanted a run of five of any letter, it was almost certain.  There are a lot of problems.

And yet, after several years of tutoring, our consultant knows many of them by heart.  (Indeed, he has picked up a calculator and been able to identify which problem the previous user had been working on just from the numbers there.)  That means he can set up and work through them quickly, more quickly than any student could.  Even if the material were more extensive, the fact that he has (in effect) been training himself to solve a standard word problem for more years than the students have been adding fractions (this is a slight exaggeration) means he’s become very good at it.  Similarly, he often encounters students who have forgotten much of their Geometry, having had the class two years ago, which is a geological age to an adolescent.  It’s present in our tutor’s mind because he was helping a student with it an hour or two before.

For teaching, this is not necessarily a good situation.  He has caught himself recently becoming impatient with some students who were having trouble with (what seemed to him to be) simple matters.  These were not the beginning or struggling students, but the capable ones, those at the more advanced levels.  It’s important to challenge everyone, rather than spoon-feed answers; but was he really being fair?  And when he outlines the solution to a standard-test word problem in a few steps, is he really doing the student any good?  It could be something similar to a John Henry moment: a demonstration of the skill of the teacher, rather than a help to the student.

It’s possible, then, that a really skilled tutor is not the best teacher.  One who has to think and work out how to approach a problem could be more help than a master-solver.  Everyone knows, though they don’t always recall it, that teaching is a separate skill.