Absolute truths

Some questions are very difficult to answer, even if you know the answer.

Our tutoring consultant has noticed a question coming up more often: “Is this always true?” or “Does it always work out this way?”  Sometimes it’s easy to answer.  Yes, when written out this way the factors of a polynomial multiply out to the constant term; no, the electric currents through each branch of the circuit aren’t always the same. 

Often some qualification is called for.  Yes, the greatest distance happens when you shoot at an angle of 45 degrees, if you neglect air resistance.  No, the roots of your equation won’t be evenly spaced, unless the function has a certain symmetry.  Unfortunately, to make problems in secondary school mathematics tractable they are often made very symmetric, so students come to expect things that aren’t often the case in real situations.

In addition, students in this age group are working out how the world functions.  They are searching for rules to organize the apparent chaos of the impending adult world as well as learning the many subjects they face in school.  There must be simple and immutable laws; the alternative is (at best) distasteful, and may look a lot like madness.  They’re searching for the answers.

So our tutor is becoming more careful about how he responds to this kind of question.  He wants to avoid giving them something that has to be unlearned later on, or misleading ways of judging whether an answer is likely wrong.  At the same time, he has to avoid complications that will only confuse them in their present classes.  Yes, for now, you cannot take the logarithm of a negative number; later on (if at all) you’ll learn how to venture into that aspect of complex numbers.

At least there is a high level of certainty in mathematics, and even in science (if simplifications are allowed for).  His colleagues teaching literature have to get across that there is no absolutely “correct” way of interpreting a certain novel or play.  There are many valid ways of looking at it, a lack of certainty that some students find unsettling.

The hard situation comes when the end of the problem arrives, and the student finds a feature or procedure that seems to work well.  Then our tutor somehow has to foresee all the possible situations in which the student might try to use it in the future.  Often he’s faced with saying, “I’m not sure it works all the time, but I can’t think of a counterexample now.”  Which is unsatisfying.

Sometimes he longs for the days of nineteenth-century prose, when a sentence could go on for a full page before all the qualifying clauses and caveats were done.  But he has to deal with twenty-first century students.