> The Greek 'Polya' said, more or less, that one learns to solve
> problems by solving problems.
Did he really say that? He said otherwise on
many occasions. He was perhaps the #1 exponent
of the idea that problem-solving techniques could
be taught, i.e. passed from one person to another,
as opposed to being rediscovered _ab initio_ by
each individual.
> That 'aha' moment only arrives through struggle.
I disagree. Saying that's the "only" way is a
tremendous overstatement, to say the least.
To borrow a metaphor from Pólya, that would be
analogous to saying that the "only" way to learn
to swim is to jump into the water and splash
around until you become a skillful swimmer. Maybe
some people learn to swim that way, but I took
swimming lessons, and so did most of my friends.
Of course practice also plays an important role,
but it is not the "only" role and certainly not the
starting place. A proverb alleges that "practice
makes perfect" but that's dangerously wrong; the
true watchword is: practice makes permanent. If you
want to improve, you must practice _the right things_.
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Problem solving is difficult. Teaching it is difficult.
There are, alas, some teachers out there whose
problem-solving skills are so weak that they're in no
position to teach the subject.
Problem solving requires initiative and creativity
(among other things). Kids can be taught to be more
creative. Certainly the converse is true: their
innate initiative and creativity can be crushed out
of them, for instance if they are fed a steady diet
of plug-and-chug problems and aren't allowed to
exercise their initiative and creativity.
I don't think there should be a separate course
called "problem solving 101" or anything like that,
and I think that Pólya's approach (writing books
dedicated to problem-solving) isn't optimal; rather,
problem solving should be taught every day in every
course.
There are dozens of things that can be done to
foster problem-solving skills. I don't feel like
enumerating them all right now. Perhaps other
list members can contribute a few ideas. Here
are a couple of examples, by way of down payment:
Don't just find "the" solution to a problem; find
multiple solutions. For example, under the rules
of geometric constructions with straightedge and
compass: Given a line L and a point P not on the
line, construct a new line through P parallel to L.
Off the top of my head I can think of three nice
_inequivalent_ constructions. (If you think this
is math not physics, consider general relativity,
where questions of what's parallel to what become
nontrivial and central to the physics.)
Another example: If you ask your students to estimate
how much water flows out the mouth of the Mississippi,
they will probably say they have no idea. But that's
a cop-out. Each person in the class probably knows
enough facts to be able to figure it out, to an
order-of-magnitude approximation, without looking
anything up. It requires marshalling quite a few
odd bits of information, but it's quite doable. Then,
to make it an even better lesson: set aside that
solution and find another inequivalent solution. (I
know of two good solutions; there may be more.)
Note that the numerical answer is vastly less
interesting than the methods of solution. That's
my point: methods can be discussed. Methods
can be taught.