Using CAS in Letter-Symbolic Algebra
at the Secondary Level : a Classroom Activity

Workshop by André Boileau and Denis Tanguay from the Université du Québec à Montréal
15th  International Conference ACA 2009 (Montréal, june 2009)

• 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.

• Presentation used (PDF)
• References (PDF)
• APTE (link to the research team site)