Mind the gap

Sometimes there’s a space between too little and too much.

Last week we mentioned that our tutoring consultant sometimes finds himself too skilled at certain things, having done them much and often, making it hard to help students who are encountering them for the first time.  There are also things he doesn’t work with often enough, mostly through having progressed to a more advanced way of addressing them.  Some examples may help explain this.

He tutors Geometry.  The ins and outs of triangles and parallel lines are familiar enough, and he has learned not to be as afraid of teaching how to do proofs as the students are to try them.  But there are a set of elegant theorems concerning secant lines that appear only at the end of the course.  He has never used them in his own work (though perhaps if our astronomer carries out his threat to study Ptolemy they might show up there).  Some years he receives no questions on them; in any case, he doesn’t remember them, and if pressed would do any required calculations with advanced trigonometry.

In Chemistry, several students are now studying rates-of-reactions.  This is a subject tailor-made for Calculus, which is all about rates of change.  But Chemistry students are not expected to have seen Calculus, and are given ways to arrive at their answers without it; ways that vary from class to class.  So our tutor is sometimes at a loss how to explain what is going on.

Then there is Multivariable Calculus, mostly carried out in three dimensions, sometimes for simplicity restricted to two.  There are calculations and theorems that look very strange to him that way, because of his experience with four-dimensional tensors and differential forms, with which much of the work is simpler and almost tautological.  He doesn’t remember ever having to work with the equations of a line or plane in three dimensions.  This may be because the techniques used to derive them don’t generalize to four-dimensional space-time, his preferred habitat.

The gap is generally less trouble in Physics, mostly because few of his students get to an advanced level in that subject.  However, he does remember one problem in orbital mechanics and another in electrical circuits that he would instinctively have approached through differential equations and vector calculus, tools not available to his students.  He is particularly proud to have found other ways to do them, though not always quickly enough to help.

He is now of two minds on these gap subjects.  On the one hand he feels he should remember and master these theorems and techniques, in order to be of most use to his students.  On the other hand, he rebels at the thought of fixing in his mind a slow and inefficient way of doing something that he already knows how to do better and faster.

In either case, he consoles himself with the thought that by using the more advanced technique he can generally check an answer found through the more laborious method.  Sometimes an answer key is all the student needs.