Upper secondary students’ ways of operationalizing the ‘for all’ statement in examining differentiability of a function using CAS

Abstract

Mathematical statements involving logical quantifiers are essential in calculus and for mathematical thinking. At upper secondary level mathematics, students are confronted with the universal quantifier in the definition of differentiability of a function, going from pointwise to global (or piecewise) considerations. Based on empirical cases of students in Danish upper secondary school working with tasks on differentiability, the paper addresses students’ ways of operationalizing the ‘for all’ statement of differentiability when using Computer Algebra Systems (CAS). The analyses of the cases illustrate three different operationalizations, which build on some of the same interpretations of the universal quantifier but turned into actions in different ways. Applying instrumental genesis together with Vergnaud’s notion of scheme, the analyses illustrate how students’ anticipations of dealing with ‘for all’ differ from their operationalizations when transitioning to instrumented techniques. This is important to take into consideration when teaching ‘for all’, designing tasks and selecting if, for what and how CAS should be applied in these settings.