[From Chris Cherpas (971022.1023 PT)]
Chris Cherpas (971015.1752 PT)--
My interest is in trying to decide whether/how to teach "mathematical
problem solving" as an explicit subject, as opposed to a more implicit
treatment. My understanding is that attempts to teach the methods of
Polya, for example, do not have much impact.
Fred Nickols (102197.1800)--
...which is it: math problems or general problems?
I had assumed that there was nothing written in the PCT literature
about mathematical problem solving, per se, but hoped to find more
about "problem solving" from a PCT perspective and then apply it to
understanding mathematical problem solving.
_My_ actual problem is deciding whether it is worthwhile to put
into a mathematics curriculum a strand devoted to problem solving,
as opposed to just making that material implicit in existing
strands devoted to fractions, equations, probability, etc. I think
of the putative strand on problem solving as "generic" with respect
to mathematics. In fact, there is already a practice strand on word
problems, so what I had in mind was something more general than just
practicing word problems.
I had heard that attempts to teach the method described in "How to
Solve It" by Polya were not found to have much impact in children's
performance when confronted with math problems. I have the book
and have read it, but do not know specifically how the attempts to
teach its methods were implemented. I will have to follow-up on
my sources there.
Anyway, I hope this long story clarifies my intent.
Bill Powers (971016.0711 MDT)--
...solving any problem would seem to benefit from
setting it up as a _control_ problem. There has to be some way to know when
the problem has been solved (a reference condition), and there must be some
way to compare the result of the current solution being tried (a
perception) against the reference condition, to tell whether you're getting
closer to or farther from the solution.
One thing that may be helpful is to develop some ways of reifying the
steps one has taken, as well as the possible moves one could make. On
a computer screen, one represent a state space, showing what operations
are possible and how they would transform the state. This would provide
at least way to reduce the memory load and may even offer an image that
one could learn to internalize and apply to future situations. Also,
given a variety of problems, one could practice perceiving what _kind_
of answer would count as a solution (which relates to Polya's first step).
Bill Powers (971016.0711 MDT)--
Ideally, we would like the errors to decrease or at least never increase.
Whether this actually happens clearly depends on picking the right
operations in the right sequence -- and of course on defining what we mean
by "error" appropriately. In many mathematical derivations and proofs that
I have seen, there is an enormous increase in complexity between the first
and last statements; I wonder whether it might be possible to find either a
measure of error or a sequence of operations that would keep the error from
ever increasing. Complexity is only one measure; maybe it's not the most
useful one. Would it be possible, through studying mathematical derivations
and proofs, to find measures of error that consistently decrease with each
step of the process? Or at least that never increase? If so, this would be
of enormous help in the teaching of mathematics -- as well as to
mathematicians.
There's been a lot of activity in the area of automated theorem-proving that
may suggest something. Non-monotonicity in the perceived distance between the
reference state and the current state with successive steps is possible for
some problems, but not others; yet, it would help if one could establish
sub-goals along the way to test and determine when to turn back to
a previous state, go deeper on the current part of the problem, or just
switch to another part and see if you've got coherence (assuming you've
already decomposed the problem somewhat)...
I work with someone who has done huge numbers of non-trivial proofs and
perhaps I see if he has something to add to what Polya has rendered.
Bill Powers (971016.0711 MDT)--
I don't know if any of this is relevant to your project.
Yes. Thanks for your thoughts and examples.
Best regards,
cc