[Maths-Education] Newsletter on Proof

[David H Kirshner]

Thanks to all who responded for insights and suggestions. 

Alan, for pointing to foundational thinkers, 
Paul, for pointing to one of his FLM papers this year dealing with the
semiotic production of compound texts including text from a social
constructivist perspective--sounds fascinating, I look forward to
reading
Nicolas, for sending me a paper on the limitations of a social
perspective on the didactics of proof, and for continuing off-line
discussion of it
Walter, for suggesting Sfard's discursive approach to understanding
development--which for me doesn't quite fit a sociocultural mold
Barbara, for suggesting Kirsti's thesis, and Kirsti for sending a link
to it--what a fascinating study!
And Stephen, for off-line greetings and encouragement.

Truly a generous and responsive community.
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


I have been trying to offer a semiotic account of mathemtical sign
production and use, writing from what I call a social constructivist
perspective but which has many points in common with a sociocultural
perspective. (My fullest treatment of this perspective is in Social
Constructivism as a Philosophy of Mathematics, SUNY 1998) In my series
of papers in FLM this year I have been looking at semiotic systems in
their social context. One of the key features is a account of the
production of compound texts, and these include sequences of signs that
make up proofs. I think such an approach using semiotics comes the
nearest I have seen so far in answering your question (but I may have
missed something)
Best wishes
Paul Ernest


it is true, in my opinion, that Lakatos has been influencial on the
research and educational community in mathematics. My own work in this
field is largely based on the study of Proof and refutations (which I
translated in French). However, the style chosen by Lakatos in his book
is a bit misleading. The aim is not to explore or value social
interaction but to use it at best to untangle the complexity of the
relationships between proving and conceptualising.  I would put nuances
on the claim that "expertise in proof is primarily a matter of social
development ", and prefer to say that from a learning perspective social
interaction may be a lever and sometimes also an obstacle due to the
intrinsic linguistic, cognitive and social properties of argumentation. 
As for references, I think that you could find many of the Proof
newsletter website : http://www.lettredelapreuve.it/ (including some
discussions on argumentation) 
Best regards Nicolas

I am not aware of such a radical position. But there several projects
with a strong social approach (which could be socio-psychology,
socio-linguistic, social interaction -- although there are relations,
there are also important differences). I would suggest works from Cobb,
Brousseau, Ball, Herbst, Bartolini-Bussi and Boero, and others, I have
not mase a systematic search. I send you an old paper of mine (different
mail), just to give you an idea of the way I understand this issue.
Best
Nicolas

Perhaps Sfard's recent work which I read as locating all of mathematics
(and thinking) in communication, and the cognitive development as the
individualization of the developed shared discourse?
I would not accept that as fully accurate to, say some key Kinesthetic
Reasoning, particularly in early childhood development, and it still
does not quite fit the kinesthetic/visual reasoning (I would say proof)
in undergraduate education through to my mathematics research.
Walter Whiteley

I would like to recommend a Thesis, defended in 2006, by Kirsti Hemmi a
Finnish mathematics education researcher who lives in Sweden. (I'm
copying this to her.) The title is: "Approaching Proof in a Community of
Mathematical Practice".
She interviewed mathematicians and students in a university mathematics
department concerning their perspectives on proof and analysed her data
using a Community of Practice model based in the work of Lave and Wenger
as well as the wider literature on proof.
I hope she will respond to this and tell you how you can access her
thesis.
Regards
Barbara


My thesis "Approaching Proof in a Community of Mathematical Practice" is
published by VDM verlag in 2008 and can be found in many bookstores, for
example Amazon. It is also available as a pdf file at
http://www.diva-portal.org/diva/getDocument?urn_nbn_se_su_diva-1217-2__f
ulltext.pdf
Best wishes
Kirsti Hemmi