The question for today is how and when to teach people
to handle ill-posed problems in general, and paradoxes
I like to say that physics isn't hard. We do problems
because they are important, not because they are hard.
Sometimes we do things that /would have been hard/ if
you didn't know the tricks ... which is why it is worth
learning the tricks.
Also: Different games are played by different rules.
You don't play baseball by football rules or vice versa.
In teaching, the introduction to a subject is different
from mastery of the subject.
In an introductory situation, it is best to emphasize
correct ideas. Later on it becomes appropriate to delve
into ill-posed questions and paradoxes.
Usually on this list we look at topics that arise in
introductory situations, topics that arise on the first
turn around the spiral. That's fine as a starting point,
but it's not enough. Where you start out is not where
you want to end up.
At some point, people need to learn to handle ill-posed
problems. In the real world, most decent jobs involve
lots of problem-solving. Unless your job consists of
asking "would you like fries with that", most of the
problems are ill-posed. Otherwise they would have been
solved long ago.
Problem-solving skills can be used for lots of things,
not just physics. However, all too often, the physics
teacher gets stuck teaching remedial problem-solving,
along with remedial math and everything else.
Suggestion: Start with Fermi problems that do not
require very much physics domain knowledge, e.g.
-- how many hairs on your head?
-- how much water in the Atlantic ocean?
Another suggestion: Start with underspecified problems
(such as the examples above). Later move on to wrongly
specified problems (as we now discuss).
If you want to see what it looks like when things go
horribly wrong, look at the Wikipedia article on the
"Faraday paradox" https://en.wikipedia.org/wiki/Faraday_paradox
The only part of the article that makes sense to me
is the banner at the top that points out that the
rest of the article is incomprehensible and self-
contradictory. I find it encouraging that somebody
noticed the problem. OTOH I find it discouraging that
AFAICT nobody bothered to write a clear, correct, modern
I put the Faraday rotor squarely in the category of
problems that /would have been hard/ if you didn't
know the tricks. Michael Faraday couldn't figure it
out. James Clerk Maxwell couldn't figure it out.
In general, a frontal assault on this problem using
19th-century techniques is going to be a disaster.
So don't do that.
On the other hand, this does /not/ mean this is a
hard problem. It is a would-be-hard problem. It
would be hard if you tried to do it without the
MF and JCM died long before there was any such thing
as vectors, long before there was any such thing as
spacetime, and long before there was a Drude model.
This gets back to the pedagogical question: How should
we teach people to deal with problems like this? I don't
have all the answers, but here are some ideas, to get the
discussion started: IMHO it does not make sense to assign
this problem to naïve students. We can talk about guided
inquiry, with the emphasis on guidance, but if you start
with this problem, AFAICT there is not much guidance you
can give, short of giving away the answer, which defeats
the purpose. Therefore it seems to me that good guidance
requires teaching people to recognize when they are in
over their head. They should put problems like this on
the back burner. Do not attempt a frontal attack on this
problem using 19th-century techniques. Wait until you
have a deeper understanding of the physics, whereupon
this problem becomes a gazillion times easier.
Also, good guidance means starting out with a succession
of easier problems, rather than tossing non-swimmers into
the deep end of the pool.
At some point people need to learn to survive with less
guidance, and indeed to survive a certain amount of
wrong guidance. Sometimes it is important to /not/ follow
directions. If the question is whether the magnetic field
lines rotate along with the magnet, the only reasonable
way to proceed starts by pointing out that magnetic field
lines don't actually exist.
This is the skill that a lot of otherwise-smart people
seem to lack. Even though they nominally have all the
tools needed to solve this problem, if it is presented
to them as a paradox they take the bait and start thinking
about it in all the wrong ways. They get wrapped around
the axle. They have a hard time disengaging from the
wrong approach. They misapply some tools, and leave
other tools completely unapplied.
I guess this gets back to the connectionist theory of
learning. It doesn't matter whether you nominally
"learned" something. What matters is whether you can
recall it when needed. This requires hard work in
advance. By the time you need to recall it, it's
(mostly) too late. It doesn't matter how much you
currently "want" to recall it. What matters is how
hard you thought about it in previous weeks and months
and years, figuring out what it's good for, figuring
out what it's connected to. This is not a new theory.
William James was big on this, back in the 1890s.
More generally, there are lots of people running around
trying to figure out how to teach students to handle
FCI problems. That's fine as far as it goes, but it
doesn't go nearly far enough. At some point we need
to figure out how to teach people to handle problems
that are orders of magnitude harder than FCI problems.
Surely the folks on this list have thought about this.
Any suggestions? Anecdotes?