Speaker
Description
In the first part of the talk, I will briefly explain the APOS (Action—Process—Object—Schema) theory and how it is applied to the teaching of mathematical concepts as well as to research on learning. In this approach, the mental construction of a mathematical concept is modeled by means of structures that give the theory its name, and mental mechanisms that allow the passage to new structures. The framework helps interpret sources of student difficulties and intuitive ideas in terms of the elements of the theory. It also offers pedagogical suggestions to overcome the challenges.
In our research group, special importance is given to the design of mathematical tasks that promote progress in the learning spiral described by APOS theory. These activities also have the purpose of identifying and differentiating between different mental structures. I will give examples of such questions involving the concept of linear transformation together with empirical data from students’ productions while engaging with the problems. Characteristics of Action, Process and Object conceptions will be discussed both from the point of view of design principles as well as actual student work.
A phenomenon that often happens in linear algebra classes, which can also be observed in textbooks, is that students are expected to deal with Processes and Objects too soon in the instructional development, while they are still constructing Actions. With suitable pedagogical strategies this situation can be dealt with so that students have the opportunity to develop the necessary conceptions for constructing linear algebra concepts.