What’s reasonable?

We have to start somewhere.  Sometimes the exact place isn’t important.

In teaching any subject, especially mathematics or science, we have to limit ourselves.  The world is just too large and messy to try to handle everything at once.  As we noted last week, this can lead students to the unwarranted assumption that everything (or at least most school subjects) can be packaged neatly into convenient boxes for solution.  So we’d like to break out of this, if possible.

Our consultants remember several efforts in this direction from their undergraduate teachers, especially in engineering classes.  Problems would be given us, with directions ending, “Make any reasonable assumptions you need, stating them explicitly.”  These were feared.  In part, we were still under the spell of the neatly-solved problems of yore, with specific answers; if we assumed a coefficient of friction of 0.7 when it was actually 0.6, wouldn’t we get it wrong?  But mostly we had no idea what was a reasonable assumption.

Of course we’d been able to reject mathematical answers to problems that required, say, a box to have a negative width or a circle of imaginary radius.  But suppose we were assigned to calculate the mass of gunpowder required to launch the projectile of Jules Verne’s From the Earth to the Moon.  How heavy was the payload?  How long was the gun?  How many joules per kg could we assume of the propellant, and what sort of efficiency?  Even our venerable navigator graduated from the Naval Academy long after any classes in heavy artillery were offered, so none of us had any examples on which to base guesses.  To make assumptions or educated guesses requires experience, sometimes quite a lot of it, and at the time it was one thing we lacked.

But we also lacked an appreciation of the point of the problem.  It didn’t matter, really, what we assumed, within a factor of 100 or so.  To launch an object from the Earth’s surface to orbit (much less as far as the Moon) with a cannon would require more gunpowder than anyone could collect, a barrel longer than anyone could contemplate making, and materials many times stronger than anyone could reasonably imagine.  The technology wouldn’t serve, any more than powering an airplane with a cast-iron, coal-fired steam engine.  And that was the point.

The world is a complex, messy place, but it doesn’t contain everything conceivable, or even possible.  The first step away from the carefully packaged standardized-test problem is often just to show what things are off-limits.