id: 02370437
dt: j
an: 2007a.00234
au: Otte, Michael
ti: Proof-analysis and the development of geometrical thought. (Análise de
prova e o desenvolvimento do pensamento geométrico.)
so: Educ. Mat. Pesqui. 5, No. 1, 13-55 (2003).
py: 2003
pu: Pontifícia Universidade Católica de São Paulo (PUC-SP), São Paulo
la: PT
cc: E20 E50
ut: proof; structuralism; representation; complementarity; cognitive
development; mathematics and philosphy
ci:
li:
ab: Author’s abstract: Piaget characterizes the historical development of
geometry as a succession of three periods of thought: intrafigural,
interfigural, and finally, transfigural or structural. We discuss an
example to illustrate Piaget’s conception of geometrical development
and to provide a particular interpretation of it. The example concerns
Euler’s theorem, according to which the concurrent points of the
perpendicular bisectors, the medians and the altitudes of any triangle
are collinear. What we want to show is that by conceiving mathematical
activity as essentially constructing proofs one might better understand
Piaget’s conception. In this context, Rotman’s criticism of Piaget
is presented and discussed. Rotman argued that Piaget’s
characterization of mathematics and its creation is limited by his
misunderstanding of ’the nature and status of proof’ (Rotman).
Rotman, who concentrates on the semiotic and social aspects of
mathematics, continued, ’The central error of Piaget’s
structuralism is the belief that it is possible to explain the origin
and nature of mathematics independently of the non-structural
justificatory questions of how mathematical assertions are validated’
(Rotman). Rotman completely misses the point that proof and
justification always depend on structural contexts as represented by
signs and language, and that the objective meanings of mathematical
signs are nothing but structural determinations. No isolated sign can
be intrinsically a sign. Thus, our aim is to show that it may be
worthwhile trying to combine the approaches of Piaget and Rotman.
rv: