The right problem

Why do we only teach the easy problems?

A couple of weeks ago we asserted that being able to program a calculator was, in general, a useless capability.  It was beyond anything needed for the problems you’d use a calculator for, but not enough for something you’d write a real computer program to do.  Hence it went unused.  Now we turn that idea on its head: what would you program a calculator to do?  What sort of problem would you attack?

Well, we need something iterative or repetitive, a task with a lot of steps or something you’d do over and over; but not something that requires the input of many numbers or the manipulation of text.  The fifty-odd examples in the HP-25 Applications Programs booklet are marvelous bits of efficient and ingenious programming, but for almost all we’d only want to do the task once or twice.  So punching in the program and debugging would take more time than just doing the keystrokes.

Half-remembering some things, we went browsing among our library.  We have a number of books that were classics of the age before computers, when any calculation was long and tedious, and we found several example problems that fit our criteria:

Don’t neglect air resistance.  Physics textbooks are full of parabolic trajectories, which result when friction with the air is negligible. This is rare.  Taking one particular rifle, without air resistance it could reach fifty miles; in practice, it’s hard put to attain two.  (Our astronomer, when he taught Physics, resorted to skirmishes on the Moon for more realism.)  Newton’s revolution had little effect on the artillery of his and later eras, which continued to be laid by rules of thumb.  But by solving the (admittedly messy) differential equation that air resistance produces, a calculator program could give a realistic range.  Doing this at a number of angles would produce a proper range table.

Size of a convection cell.  When you heat a liquid from below, at a certain point a steady pattern of upwellings and downwellings appears.  The characteristic size of these can be calculated given certain parameters, but it involves a transcendental equation, connecting the parameters and such functions as circular and hyperbolic sines.  You can’t keypunch the answer, but you can iterate to it.  And then work out how, for instance, raising the temperature of your heater changes things.  (We found this approach doing research on how to improve our crumpets.  Really.)

Designing a lens.  One step in lens design is tracing a ray of light through the several refractions that occur, then tracing another, and eventually building up a picture of the quality the image it would produce.  Before electronic computers, this was extremely tedious; our classic reference devotes several pages (almost an entire chapter) to how to avoid making mistakes using logarithm tables.  With a program, each ray becomes a couple of keystrokes.

These were once cutting-edge research problems, but could now be used in teaching.  In particular the first and third would allow the students to step beyond the oversimplified situations of introductory courses.  But (as far as we’re aware) nothing of this sort has been done.  Of course the first generation of post-calculator textbooks and teachers was unfamiliar with the new capability; but even twenty and forty years later, all the problems we found in Physics textbooks could be solved with slide rules.  We note that calculators are used in mathematics textbooks to do such things as give an idea of what a limit is, and work out the simpler Riemann sums.  We assert the gain here is marginal.

Why not use this extremely powerful resource?  We can think of several explanations.  First, inertia.  This is how we’ve taught, this is what we’ve taught, this is what we have to teach for the follow-on courses.  An air-resistance program doesn’t fit into the format of textbook problems and test questions.  It might become a lab.  But-

These require a significant bit of programming, in no language presented by the Computer Science Department.  But they also require a significant amount of Physics.  So they fall between institutional responsibilities.

The opportunity presented by the programmable calculator, then, has been missed.  What else have we passed up?