Computer Algebra Systems (CAS) are becoming
more and more important in high school, cegeps and university maths
courses. Whether they involve graphing calculators or software such as
Maple or Derive, their use is often confined to teaching situations
where functions are at stake. In such instances, we discern three main
pedagogical motivations :
- the idea that through the use of CAS, one can give students access
to sophisticated situations and modellings otherwise too complex, the
needed computations lying beyond the scope of students’ techniques ;
- the related and more general idea that by relieving students of
the (tedious) traineeship of computational techniques, more time can be
devoted to conceptual apprenticeships ;
- the idea that these complex functional situations can then be
studied through semiotic representations in various registers (Duval,
1993), giving rise to work involving conversion/coordination between
these registers — researchers in math education being more and more
convinced that this type of work is the basis for solid
conceptualization.
At the secondary level, this third idea prevails, and graphing
calculators in math classes are used mainly in algebra courses, as a
tool to go back and forth between algebraic expressions of a given
function, its table of values and its graph. The aim of the APTE team is
to extend this usage by employing CAS calculators in order to help
students in their construction of meaning and conceptualization, within
the more literal-symbolic segment of secondary level algebra, apart from
any functional consideration. The team has thus designed classroom
activities (i. e. consistent and connected sequences of tasks) targeting
an apprenticeship of techniques in algebra (at the 3rd and 4th
secondary level), such as factorization and expansion, equation solving,
substitution, while fostering the conceptual (theoretical) thinking for
such notions as :
- equivalence of expressions;
- domain of validity for an expression or for an equivalence;
- solution set of an equation or system of equations;
- the distinction equation-identity, etc.
In this communication, we will present one of these activities, with
the relevant work and productions from the students with which it has
been experimented. We will discuss some aspects of its design which we
evaluate as important, in particular regarding the co-emergence of
technique and theory and their mutual interactions (Kieran &
Drijvers, 2006): going back and forth from paper-and-pencil work to CAS
work, comparison between standard algebraic syntax and CAS syntax,
triggering use of the unexpected/startling CAS-output, conjectures,
justifications of the conjectured formulae, large group discussions,
role of the teacher...
References
Duval, R. (1993). Registres de représentation sémiotique et fonctionnement cognitif de la pensée. Annales de Didactique et de Sciences Cognitives, n°5, pp. 37-65. IREM de Strasbourg.
Kieran, C. & Drijvers, P., in coll. with A. Boileau, F. Hitt, D.
Tanguay, L. Saldanha, J. Guzmán (2006). Learning about equivalence,
equality, and equation in a CAS environment: The interaction of machine
techniques, paper-and-pencil techniques, and theorizing. Proceedings of the 17th ICMI Study ‘Technology Revisited’. C. Hoyles & J.-B. Lagrange, eds. Program Committee, Hanoï, Viet-Nam. |