Teaching is not always effective. One of the reasons for
this is that teachers are often bored by trivia and skip as
much of it as possible. Why bother, there more interesting
and important things to emphasize.
Let me illustrate this with two examples. One is the
refraction of light on a spherical surface. The well known
formula (Jenkins and White) is
n/s + n'/s' = (n'-n) / r
It is derived for one particular configuration and then
generalized in the form of the "sign convention". The
exercise of deriving the formula for all possible
configurations (using only positive distances) and
of inventing a sign convention which turns many
formulas into one is too trivial. And it takes too much
time. To do this one must deal with concave and
convex surfaces, with n>n'and n<n', with a converging
input beam and diverging input beam, in all possible
combinations. So what do we do instead? We impose
a sign convention and ask students to memorize it.
The second illustration is also from optics. It has to
do with the mathematical description of an ideal wave.
After dealing with harmonic motion of a single point-like
mass we introduce the kinematical wave equation:
y=A*sin(k*x-w*t)
Then we use it in dealing with interference, diffraction, etc.
Just think how much is skipped. Is it really obvious to kids
that the above equation describes waves? So many masses
are in motion. Some points are going down while others are
going up. One equation for all of them? This is something
very different from what was called kinematics up to that
time.
How do we" sell" the above equation to students? We
present it first. Then we ask them to plot it and thus to verify
that it indeed corresponds to waves. What is skipped? Trivial
and boring analysis in which the above equation becomes the
end product, not the starting point to be verified.
That is why less is "more". And that kind of more is better.
No I am not suggesting to replace interesting topics by
trivia. I am saying that what is trivial to us is often not
trivial for students. And I wish there were a White Paper
on "less is more" among those listed as "critical issues"
in the last Announcer (page 8).