[Maths-Education] Newsletter on Proof
Alan Rogerson
alan at rogerson.pol.pl
Mon Nov 17 17:53:58 GMT 2008
Dear David,
I wish I could agree with you, I am amazed how many people now have
never, or hardly, heard of Lakatos (nor Kuhn) much less Wittgenstein,
and I have never met anyone, in maths education at least, who has read
their major works in entirety, as opposed to reading a few sentences
about them, so I am not sure what "influence" these creative thinkers
are having.
I was reacting mostly to your mention of a "clear and definitive
statement" which was the basis of all of Wittgenstein's early work in
the Tractatus (following to some extent the Vienna School), but despite
its poetic beauty, it was wholly rejected by him in all his later work.
I am not sure if the professional psychologists are any clearer now as
to the proportion of influence of "nature or nurture" as it used to be
called, on human development (it was 40-60 last time I looked) - it is
difficult to say because they continuously co-exist in such an obvious
sense.
When you say something is "primarily a matter of social development" as
opposed to cognitive - that sounds like a very hard question to resolve,
or even to know how it could be resolved in any experimental (as opposed
to anecdotal) sense? Certainly the attempts I have seen have not been in
any way convincing but I hope you have better luck!
The dominant paradigms of the 1960s and 1970s, it is true, are now being
replaced by others, but how far this is due to progress and deep study,
as opposed to fashion, I am not sure.
Best wishes
Alan
PS If anyone on the forum has had the pleasure of absorbing all of the
wondrous works of Wittgenstein and/or Lakatos and would like to discuss
them in a conflict-free way I would really love to hear from them
privately! Last weekend my wife Margaret and I attended the annual MUED
Group meeting in Kassel, Germany, highly recommended if you want to see
the cutting edge of maths education innovation in schools today - and
yes they do follow implicitely Polya, Lakatos, Freire and some other
notables. I mention it however because we had the good fortune to meet a
remarkable man, Mattias, who had taught both philosophy and maths all
his life so it was a real pleasure to discuss our favourite thinkers and
their influence on our lives.
David H Kirshner wrote:
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> Thanks, Alan.
> Both Wittgenstein and Lakatos are important influences on our contemporary conversations about the socially constituted nature of mathematical proof. What I'm looking for are some key papers in the current pedagogical literature suggesting that expertise in proof is primarily a matter of social development, best attended to by nurturing increasingly sophisticated forms of argumentation within the classroom microculture.
> David
>
>
>
> -----Original Message-----
> From: maths-education-bounces at lists.nottingham.ac.uk [mailto:maths-education-bounces at lists.nottingham.ac.uk] On Behalf Of Alan Rogerson
> Sent: Monday, November 17, 2008 9:56 AM
> To: Mathematics Education discussion forum
> Subject: Re: [Maths-Education] Newsletter on Proof
>
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> Dear David,
> I would like to recommend Wittgenstein and especially "On Certainty"
> which I think is one of the deepest and, maybe in some ways, clearest
> exposition of the language usage underpinning why we seem to need...
> clear and definitive answers....and how language functions or behaves in
> trying to meet that need.
> The other seminal work of course would be that of Lakatos, where he has
> some wonderful and trenchant things to say based on his work in the
> history and, we would have to add, the socio-cultural development of
> mathematics.: "The value of a logical proof is .... that it suggests
> doubts", "I respect conscious guessing because it comes from two of the
> best human emotions(? I am quoting from a failing memory, but the sense
> is clear I hope)... courage and humility" .
> The lessons I have learnt from these two have been of immense practical
> assistance in my teaching, over the years, of students from 5 to 55. If
> you are really interested in this line of thinking I would be happy to
> make other suggestions, especially in the fields of sociology and social
> anthropology which also seem to be very useful in "understanding" - ah
> there's the rub - maybe "comprehending", or "apprehending" would be
> better, how we function in a social and cultural complex, in which even
> language of course is but a small element of communication and hence
> of... "meaning"!
> Best wishes
> Alan
>
>
>
>
> David H Kirshner wrote:
>
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>> Maria-Alessandra, and others.
>>
>> I would appreciate your help in finding some sociocultural references to "proof" in the literature.
>>
>> Under the influence of cognitive interpretations of thinking and learning, attaining competence in proof is frequently portrayed as a cognitive accomplishment--achievement of skills and/or concepts related to proof. Thus practice and individual reflection become the vehicles for promoting competencies in proving.
>>
>> >From a sociocultural perspective, one might regard proof as a highly refined form of argumentation practiced within the mathematics community. The associated pedagogical strategy would be nurturing of increasingly sophisticated forms of argumentation within the classroom microculture (e.g., argument from authority --> argument based on material implication --> argument based on abstract logical implication).
>>
>> I know that the sociocultural perspective is reflected in the literature on proof. If anyone can point me to a particularly clear or definitive statement of that position, I'd be most appreciative.
>>
>> David Kirshner
>> Louisiana State University
>>
>>
>>
>> -----Original Message-----
>> From: maths-education-bounces at lists.nottingham.ac.uk [mailto:maths-education-bounces at lists.nottingham.ac.uk] On Behalf Of m.a.mariotti
>> Sent: Tuesday, November 11, 2008 8:57 AM
>> To: Mathematics Education discussion forum
>> Subject: [Maths-Education] Newsletter on Proof
>>
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>> The new issue of the Newsletter on proof is on-line, you can find
>> it at
>>
>> http://www.lettredelapreuve.it
>>
>> Have a good reading!
>> m.alessandra
>>
>> ________________________________________________________
>> Maria Alessandra Mariotti
>> Dipartimento di Scienze Matematiche ed Informatiche
>> Università di Siena
>> Piano dei Mantellini, 44
>> 53100 Siena
>> tel. + 39 0577 233707
>> Fax. + 39 0577 233701
>> e-mail mariotti.ale at unisi.it
>> ____________________________________________________
>> ****http://www.lettredelapreuve.it.****
>>
>>
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