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Re: Hand waving in physics teaching?



Savinainen Antti wrote:

In my opinion the real understanding involves well developed
conceptual understanding linked with mathematical apparatus in
physics.

Sure.

... he continued that this "hand waving" is not enough, and of
course students need also to master the topics; by this he meant the
mathematical mastery.

I think that he identifyed conceptual understanding with hand waving.

That's not really the right identification.

There are at least three factors to consider:
-- Qualitative versus quantitative expression.
-- Mathematical versus empirical justification.
-- Correct or not.

These factors can be combined in various ways:

1) It is, alas, possible to have a correct mathematical
formulation without real physical understanding, which
can be a big problem, as described for instance in
_Surely You’re Joking, Mr. Feynman!_
http://www.chemistry.ucok.edu/Jezercak/Feynman.pdf

2) Much more common is to have a more-or-less correct
qualitative picture without much mathematics. In real
life, there are many problems for which a qualitative
answer is required and a quantitative analysis would
be an utter waste of time. For instance, should you
put on your socks before your shoes, or vice versa?

Even in physics sometimes a qualitative understanding
suffices. An elementary-level student may comprehend
the analogy between interference of water-waves and
the interference of light-waves, without being able
to write down (let alone derive) the wave equation for
either one.

3) To my ears, "hand-waving" is a pejorative term that
applies to alleged explanations that are unduly
superficial, misleading, and/or completely incorrect.

As a common yet extreme example, consider the notion
that "a wing produces lift because it is curved on
top and flat on the bottom". I don't object to its
qualitativeness; I object to its wrongness.


Other things to remember:
-- Qualitative is not the same as hand-waving.
-- Empirical is not the same as hand-waving.
-- Qualitative is not the same as inexact.
-- Quantitative is not the same as exact. Most of the
equations of physics are only approximations.
-- Qualitative understanding and quantitative understanding
are not mutually exclusive. In many cases you want both.
Sometimes one proceeds from the other; for example given
the wave equation you can (with a little work) show that
it has qualitatively wavelike solutions.


Sometimes there may not be any good way of explaining a piece of
physics for high school students (e.g. invoking quantum mechanics or
Maxwell's equations is pretty meaningless for them);

Mendeleev did not derive the periodic table by starting from
QM and the Maxwell equations. Instead, he first collected a
ton of raw data. Then he detected empirical patterns in the
data (along with many exceptions to the patterns). Long after
his death there began to emerge a mathematical explanation for
the facts.

What matters is knowing what's true. Mathematical derivation
is by no means the only way of knowing something is true.

People have made empirical checks of the Maxwell equations.
In particular the predicted 1/r law has been checked to high
accuracy. (See Jackson for a discussion.) Obviously such
an exercise does not start by assuming that the Maxwell
equations are axiomatic. High-school geometry takes
exclusively the axiomatic approach; physics doesn't and
shouldn't.

===========

Often there is a choice whether to introduce an idea in
mathematical/axiomatic terms or in empirical terms. This
choice does _not_ count as a dilemma, because you can make
it work either way.

More-serious choices involve deciding what level of
approximation is appropriate. The choice is easy if the
candidates are really good *or* really bad.
*) The notion that "a wing produces lift because it is
curved on top and flat on the bottom" is so useless that
it is easy to dismiss it. Better explanations are readily
available.
*) The notion that F=ma in the lab frame is such a good
approximation that it can be introduced without worry.
The limitations on its validity can be learned later,
without requiring any troublesome unlearning.
*) The real dilemmas come from notions that are gravely
flawed but not entirely useless. Particularly annoying
examples include Lewis-dot diagrams and the associated
doctrine of filled octets on atoms in molecules ... which
remains popular even though it is based on wrong physics,
is grossly inconsistent with observations (paramagnetism
of O2, spectroscopy, reactivity, etc.), and must be
unlearned when it comes time to study molecular-orbital
theory.