Misc comments on categorisation, problem solving, etc (Was Re: Aaron Somon's....)

From: Wolf Kirchmeir (wwolfkir_at_sympatico.ca)
Date: 08/04/04 

Date: Wed, 04 Aug 2004 11:37:51 -0400

Neil W Rickert wrote:

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> Wolf Kirchmeir <wwolfkir@sympatico.ca> writes:
>
>
>>Again, granted that "categorisation" takes place. But what, exactly, is
>>it?
>
>
> That would actually be a good topic, if only people could discuss
> it intelligently.
>
>> I submit it's behaviour.
>
>
> It seems that to a behaviorist, everything is behavior. And that
> makes your statement vacuous.
>
> For me, categorization has to do with inputs, while behavior has to
> do with outputs.

No wonder you fail to get it. The behaviorist looks for the relationship
between "inputs" (ie, the ennvironment) and "outputs" (ie, the actions
of the organsim.) A behaviorist that ignored the environment in studying
behavior wouldn't get very far. NB that for "private behaviors" the
environment _includes_ other private behaviors.

Anyhow, if categories are "inputs," they're data. Where did the
categorising take place? Or are you saying that categorising operates on
inputs? If so, it's behaviour.

>>But mostly, categorisation is language use.
>
>
> I guess that is what you would conclude if you insist on construing
> everything as behavior.

You snipped whole bunch of stuff here. If you must snip, kindly show
where you did it!

>>** When I was first teaching, a question that vexed me from the
>>beginning was, How do I know that a student has understood a tex? The
>>answer is, of course, that the only evidnce we have is his or her
>>language about the text.
>
>
> Maybe we mathematicians have it easier. To a mathematician, a
> student has understood the text if he is capable of solving problems,
> including original problems that are significantly different from any
> in the text. We judge understanding on the basis of behavior, but
> this is not restricted to behavior that is in the form of "language
> about the text".

The subject of literary analysis is the text, hence "language about the
text" is all that's relevant. IMO, "language about the problem" is also
relevant in mathematics. Solutions can be produced by computer programs,
after all, and I do think you expect more from your students than
emulation of a computer program.

I find your comment more than somewhat beside the point, actually. Maybe
you should consider what sort of "language about the text" would
demonstrate "understanding." Or "language about the problem", for that
matter.

> I guess that makes me a behaviorist of sorts. But I am not a
> radically stupid behaviorist. As far as I can tell, the "operant
> conditioning" account cannot adequately explain the acquired ability
> to solve original problems.

That's because
a) you persist in thinking of behaviour as wholly external, and ignore
private behaviors (yet you use signs of private behaviours as criteria
for "understanding"!);
b) you fail to keep in mind that only behaviours that are performed can
be conditioned;
c) you forget the reward system that operates on "thinking." (Recall
Archimedes' "Eureka!")
d) you forget about the experimentation that occurs during "solving an
original problem", and don't seem to see that we pursue those
experiments that are "rewarding" (to use a phrase I've encountered more
than once in books about the pleasures of math.) **

I referred to poetry writing programs in another post: such programs use
a mix of rules and random selection to prduce phrases. Some of these are
  "striking" and "intersting" metaphors. Poets have reported that they
experiment withn variations on their pharses that "come to mind", or
construct such variants according to some rules. Manuscript evdince
supports their reports, by showing how many variants were tried before
the darft was finalised. Eventually, the poet _selecst_ the variants
that "sound right," ie, the ones that provide the reward they are
seeking. If the poets gets poditive feedback from an audience, (s)he'll
repeat that behaviour of "composing a poem." The whole process
illustrates operant conditioning, IMO.

** I personally like thinking about the simpler math problems)I'm not a
trained mathematician.) The pleasure I feel when arriving at some
theorem is reinforcer for this behaviour. The pleasure I feel when I
confirm that my solution is correct is a further reinforcer. You see, my
"private behaviours" are udner the same constraints and shaping
influence's as my public ones.

Or consider a child that's "frustrated" with a math problem. Frustration
is a powerful conditioner - allow the frustration to continue or repeat,
and the child may well be turned off math altogether. So what the good
teacher do? Provide some reinforcemnent for the problem solving
behavior, or reconstruct the problem into smaller chunks, so that the
child will continue to "try" -- until the problem is solved. At which
point the child feels more or less satisfaction, which tends to
condition the child to continue to work on math.

But to get back to "categorising": I can't really follow Solomonoff's
papers, but he claims to have made a mathematical model of a
categorising process that starts with a mess of data (inputs) and uses
"similarity" and statistics to pridduce categories. Have you read his
work? Does it make mathematical sense? Etc.

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