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Conceptualizing the Learning of Algebraic Technique: Role of Tasks and Technology
Location and Schedule
Time slot: 
RL-3 on Friday 11, 11:00 – 12:00
Auditorium: 
S04-Genaro Salinas Quiroga Auditorium – Leyes
Metadata information
Author(s): 
Carolyn Kieran, Université du Québec à Montréal, Département de mathématiques. kieran.carolyn@uqam.ca
Language: 
English
Keywords: 
CAS Technology in high school algebra, Tasks, Technology, Concepts, Techniques, Classroom orchestration
Abstract: 

Research in the 1980s and 90s with Computer Algebra System (CAS) technology has suggested that such tools can be used effectively at the tertiary level. However, more recent studies have been able to provide evidence for the contribution of this technology with much younger students. Despite often-heard arguments that high school algebra is primarily about learning symbol manipulation and that any technology that carries out such manipulation for the student has little place in the high school curriculum, current research is showing that CAS technology can indeed enhance the learning of algebraic technique and even lead to an understanding of algebraic concepts at the secondary level, as long as appropriate tasks are involved in the learning experience. The triad of task-technique-theory is thus considered to be a core component of such technology-supported learning in school algebra.

This presentation is divided into three parts. The first part explores the technical-conceptual interface in algebra and discusses what is meant by conceptual (theoretical) understanding of algebraic technique – in other words, rendering conceptual the technical aspects of algebra. Examples to be touched upon include students’ learning to, for instance, see through symbols and become aware of underlying forms, conceptualize the equivalence of the factored and expanded forms of algebraic expressions, coordinate the nature of equation solutions with the relation between the two expressions that comprise the original equation, take into account the role played by the exponent in factoring families of expressions, and so on.

The ways in which students learn to draw such conceptual aspects from their work with algebraic techniques in technology environments is the focus of the second part of the presentation. Research studies that have been carried out by my research group with a diversity of high school algebra students have found evidence for the kinds of theoretical thinking that can be fostered by specific types of technique-oriented tasks within CAS environments. These tasks, which involve the development of both paper-and-pencil and CAS-based techniques – and the reconciliation of the two – also ask students to make conjectures, predict, and explain their thinking. In the presentation, examples are provided of student work that will also serve to illustrate the nature and characteristics of the tasks, as well as the roles played by the CAS technology in supporting and encouraging students’ technical and conceptual growth in algebra.

The third part of the presentation will shift to the perspective of teaching practice and will discuss some of the issues that, according to this research, are taken into account when planning for the orchestration of such task-technique-theory activity in technological environments. These issues, which go beyond the management of the technical instruments, include consideration of the mathematical content and reasoning processes to be evoked by the tasks, the links between the mathematics and the technological (and other) artifacts, the relation between the scope of the task and cognitive aspects of mathematical learning, and classroom organizational factors and the pedagogical means used to draw out the emergent learning.