Finding mistakes

Our tutor ponders techniques he employs unconsciously.

Our tutoring consultant teaches mathematics at all levels through High School, including some glimpses of undergraduate-level material.  That means looking at innumerable problems, easy and hard, short and long, and answering the students’ question: “Did I do this right?”

Often he can tell at a glance, especially if the answer is “no.”  With long practice, and especially after doing scientific research where there is no answer in the back of the book, he’s developed ways of checking what the answer should look like.  The polynomial should have x to the powers of four, three, two, one and zero; the student managed to get x to the sixth.  There should be a cross-term, and there isn’t.  The sine of a small number should also be small.  Most of his techniques, however, he does without thinking, and can seem magical to the students.  He’s trying to catch himself at work, and write down what he’s doing, so he can pass it on.  There’s no point in having another generation learn how to do things the hard way if it can be made easier.

The students do something like this already.  They often say, “This doesn’t look right,” even before they check the answer key.  They’ve been doing math for years, after all, and have had some success.  But sometimes their intuition is wrong.  An answer of 77/13, for instance, “looks wrong,” but only because the student has been given lots of problems with nice integer answers and has come to expect them.  And 182 looks like too big a number, so the student starts looking for where she went wrong.  There’s some unlearning to be done here.  (Eventually, in later science or math classes, one gets to the point that the answers are longer than the questions are, and nice integer answers are a sign that something is wrong.)

A harder challenge is, when we know an answer isn’t right, finding the error.  This can be especially daunting in a long problem with lots of algebra.  When the student has not been very organized in working through the problem and has terrible handwriting, it may be impossible.  It becomes a question of the best use of tutoring session time: tracing through a messy piece of paper, redoing it from the beginning, or just going on.  Unfortunately, there is no way we’ve found to banish mistakes entirely.

We’re not working at the cutting edge of theoretical physics, generating new theories, but we suspect something of this sort happens even there.  Scientsts say things like, “Beauty is a necessary feature of theories that are true,” which may be an expression of something like our tutor’s unconscious answer-checking techniques.  And of course when a theory generates numbers that don’t look right, infinite masses or suchlike, one starts looking for the error.

However, some things may very well be infinite.