[Grassroots-l] [Educators] [IAEP] Concise explanation of Constructionism from the Learning Team

[Bastien]

"Edward Cherlin" <echerlin at gmail.com> writes:

> On Sat, Aug 16, 2008 at 8:32 AM, Bastien <bastienguerry at googlemail.com> wrote:
>> "Bill Kerr" <billkerr at gmail.com> writes:
>>
>>>   • intuition
>>
>> [...]
>>
>>>   • different ways of looking at maths (constructive and intuitive compared
>>>     with rule driven and formal)
>
> It turns out that there is no essential difference between the systems
> constructed under these seemingly quite different programs. Each
> contains a model of the others, in much the same way that one can find
> subspaces of Euclidean space (sphere and pseudosphere) with
> non-Euclidean geometries, and subspaces of non-Euclidean spaces
> (horospheres, Clifford's surfaces) with Euclidean geometry.

The items I quoted were from Bill's list: he listed things a few points
Cynthia makes in her book.

I was interested in better understanding how Cynthia (or Bill's, if it's
his wording and if he knows) relates these "different ways of looking at
maths" to the various learning theories.

The fact that there is no essential difference between these system
doesn't imply that there is no difference in understanding or building
them -- does it?

>>>   • other mathematicians who hold similar views - Poincare, Brouwer, Godel)
>
> In my study of Poincaré, Brouwer, and Gödel, I found little in common
> among their views. What are you talking about?

I was just quoting Bill's items and Bill was summing up Cynthia's book.

So maybe I just should read Cynthia's book and see how she relates these
authors.  I'm not a mathematician, but I thought Poincaré was considered
a precursor of intuitionism, Brouwer being the father, and Gödel being
one of those who made progress in modeling classical maths within
intuitionism. 

So maybe the little in common they have is that they played a key role
in intuitionism history?  Just a thought.

>> I'd be curious on how Cynthia relates mathematical theories (like
>> intuitionism) to pedagogical theories.
>
> Piaget was greatly impressed by Brouwer's Intuitionism, with its
> rejection of excluded middle and other "non-intuitive" ideas, but
> mathematicians are not. It turns out that classical mathematics can be
> completely modeled within Intuitionism.
>
> I found the arguments over mathematical philosophy to be quite arid,
> particularly those about the nature of mathematical objects. Do they
> have independent existence, do they exist only in our minds, or do
> they not exist at all, and only the symbols we work with directly have
> real existence? None of these questions has any bearing on what
> theorems can be proven from what sets of axioms. None of them has any
> bearing on the applications of mathematics. The major practical effect
> that I have seen from these arguments is the refusal of some
> mathematicians to study certain questions, a result that I consider in
> general lamentable. But what can you do? Nobody can study everything
> any more. In some cases, mathematicians have been inspired by
> ontological arguments to take up questions that otherwise would not
> have been studied, which is to the good.

Interesting.

Maybe the fact that metaphysical "intuitions" (or convictions?) lead
mathematicians toward certain kinds of challenges is a good thing on 
the overall.  

> Math teachers need to be aware of some of these views, because
> schoolchildren may well discover them, and other conundrums and
> paradoxes, and may need help at some points to get past the
> difficulties that they can create. It is useful to distinguish
> constructive set theory, as a subset of more general set theories, in
> the same way that it is useful to distinguish problems with
> algorithmic solutions in linear time from those that are more
> difficult (requiring higher-order polynomial time, or even exponential
> time) or are frankly unsolvable (the undecidable, such as the Halting
> Problem for computer programs, or the consistency of arithmetic, or
> membership in any recursively-enumerable but non-recursive set).

Ok.  But I was not interested in maths per se, I was interested in how
maths theories might (or not) relate to learning theories.  

(After all, "mathesis" means (something like) learning, no?  If you
think of Plato, the link is quite obvious.)

-- 
Bastien