

Observation
and design in mathematical proof

Willi Dörfler


Institut für Mathematik, Universität
Klagenfurt, Austria


Empirical and perceptive observation becomes a decisive part of mathematical
reasoning, of devising and understanding proofs and mathematical arguments.
Mathematical reasoning in this view is not so much the handling of abstract
ideas in one's mind but the observation of the effects of one's manipulations
of diagrams. The mathematical ideas rather reside in the invention of
diagrams and of their fruitful manipulations, transformations, compositions.
From diagrammatic reasoning derives also the absolute reliability and
security of mathematics, its socalled logical necessity. This differentiates
observation of diagrams also from empirical observation in the natural
sciences.
Finally, it should be emphasized that diagrammatic reasoning is very
much different from algorithmic calculations. Though it is rule based
it needs creativity and inventiveness like composing music.
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Argumentation and proof
Group
4 to the CERME 3 conference
Vom Hofe R., Knipping C.,
Mariotti M. A., Pedemonte B

During CERME 3 Conference the topics proposed in the Working
Group 4 "Argumentation and Proof", were the following:
1) Forms and uses of logical and mathematical reasoning from a didactical
point of view. This topic was about logical, historical and epistemological
aspects, related to the nature of mathematical argumentation and proof
(Papers concerning this topic: V. Durand Guerrier, B. Pedemonte, D.
A. Reid, O. Yevdokimov)
2) Argumentation and proof in class  comparing different classroom
contexts. This topic concerned different classroom contexts of student
construction of proof and arguments (Papers concerning this topic: N.
Douek, C. Knipping, A. Scimone)
3) Students explanations, proof competence and experiences with proofs.
This topic dealed with cognitive and epistemological aspects, concerning
the processes of production of conjectures and construction of proofs
(Papers concerning this topic: A. Heinze & K. Reiss, D.
Küchemann & C. Hoyles, C. Misailidou & J. Williams, K.
Nordstroem)
4) Empirical thinking and epistemological obstacles in argumentation
and proof. This topic was about mathematical aspects of realityrelated
thinking, empirical reasoning, and epistemological obstacles for student's
arguing and proving (Papers concerning this topic:
J. P. van Bendegem, W. Blum, R. vom Hofe).
The fulltext versions of these papers are given below (PDF):


Blum,
W. : On the role of "grundvorstellungen" for realityrelated
proofs. 

Reid,
. A. : Forms and uses of abduction. 

Douek,
N. : From oral towritten texts in grade I and the approach to argumentation:
the role of social interaction and task context. 

DurandGuerrier,
V. : Logic and mathematical reasoning from a didactical point of
view 

Heinze,
A. & Reiss, K. : Reasoning and proof: methodological knowledge
as a component of proof competence. 

Knipping,
C. : Argumentation structures in classroom proving situations. 

Küchemann,
D. & Hoyles, C. : The quality of students' explanations on
a nonstandard geometry item. 

Misailidou,
C. & Williams J. : Children's arguments in discussion of a
"difficult" ratio problem: the role of a pictorial representation. 

Nordstroem,
k. : Swedish university entrants' experiences about and attitudes
to proof and proving. 

Pedemonte,
B. : What kind of proof can be constructed following an abductive
argumentation? 

Scimone,
A. : An educational experimentation on Goldbach's conjecture. 

van
Bendegem, J. P. : Proofs and arguments  The special case of mathematics. 

vom
Hofe R. : Epistemological problems with the limit concept  a case
study on communication and argumentation within a computerbased learning
environment. 

Yevdokimov,
O. : The place and significance of the problems for proof in learning
mathematics. 
The Teaching of Proof
D. Loewenberg Ball, C. Hoyles,
H. Niels Jahnke (chair), N. MovshovitzHadar
Paper presented at the last International Congress of
Mathematicians in Beijing

This panel draws on research of the teaching of mathematical
proof, conducted in five countries at different levels of schooling.
With a shared view of proof as essential to the teaching and learning
of mathematics, the authors present results of studies that explore
the challenges for teachers in helping students learn to reason in disciplined
ways about mathematical claims.
This paper was presented at the last International Congress
of Mathematicians in Beijing
The reference is the following: LI Tsatsien (ed.), Proceedings of the
International Congress of Mathematicians, Beijing 2002, August 2028,
Vol. III: Invited Lectures, 907  920.
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