@article {MATHEDUC.02366941,
author = {Durand-Guerrier, Viviane and Arsac, Gilbert},
title = {An epistemological and didactic study of a specific calculus reasoning rule.},
year = {2005},
journal = {Educational Studies in Mathematics},
volume = {60},
number = {2},
issn = {0013-1954},
pages = {149-172},
publisher = {Springer Netherlands, Dordrecht},
doi = {10.1007/s10649-005-5614-y},
abstract = {It is widely attested that university students face considerable difficulties with reasoning in analysis, especially when dealing with statements involving two different quantifiers. We focus in this paper on a specific mistake which appears in proofs where one applies twice or more a statement of the kind "for all X, there exists Y such that R(X, Y)", and forgets that in that case, a priori, "Y depends on X". We analyse this mistake from both a logical and mathematical point of view, and study it through two inquiries, an historical one and a didactic one. We show that mathematics teachers emphasise the importance of the dependence rule in order to avoid this kind of mistake, while natural deduction in predicate calculus provides a logical framework to analyse and control the use of quantifiers. We show that the relevance of this dependence rule depends heavily on the context: nearly without interest in geometry, but fundamental in analysis or linear algebra. As a consequence, mathematical knowledge is a key to correct reasoning, so that there is a large distance between beginners' and experts' abilities regarding control of validity, that, to be shortened, probably requires more than a syntactic rule or informal advice. (orig.)},
msc2010 = {E55xx (E35xx I15xx)},
identifier = {2006a.00428},
}