
02366941
j
2006a.00428
DurandGuerrier, Viviane
Arsac, Gilbert
An epistemological and didactic study of a specific calculus reasoning rule.
Educ. Stud. Math. 60, No. 2, 149172 (2005).
2005
Springer Netherlands, Dordrecht
EN
E55
E35
I15
AE statements
EA statements
proving
calculus
dependence rule
didactic inquiry
historical inquiry
mathemaitcla practice
natural deduction
predicate calculus
semantics
syntax
student errors
tertiary education
doi:10.1007/s106490055614y
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.)