Liora Linchevski

The Gains and the Pitfalls of Reification — The Case of Algebra

Algebraic symbols do not speak for themselves. What one actually sees in them depends on the requirements of the problem to which they are applied. Not less important, it depends on what one is able to perceive and prepared to notice. It is this last statement which becomes the leading theme of this article. The main focus is on the versatility and adaptability of student’s algebraic knowledge

Using Intuition From Everyday Life in 'Filling' the gap in Children's Extension of Their Number Concept to Include the Negative Numbers

We report here an instructional method designed to address the cognitive gaps in childrens mathematical development where operational conceptions give rise to structural conceptions (such as when the subtraction process leads to the negative number concept). The method involves the linking of process and object conceptions through semiotic activity with models which first record processes in situations outside mathematics and subsequently mediate activity with the signs of mathematics. We describe two experiments in teaching integers, an interesting case in which previous literature has focused on the dichotomy between the algebraic approach and the modelling approach to instruction. We conceptualise modelling as the transformation of outside-school knowledge into school mathematics, and discuss the opportunities and difficulties involved.

Together-and-APart for Quality and Equity in Mathematics Education

Those who teach mathematics in same-ability groups genuinely strive to foster equity in mathematics education. In this chapter, we first analyze why their approaches do not really achieve equity. We then describe our research-based TAP (Together-and-APart) mixed-ability teaching approach that has been implemented in junior high schools in two projects in very different contexts: TAP in Israel and MALATI in South Africa. We describe TAP’s rationale and the way TAP’s implementation achieves quality and equity by allowing the co-existence of two types of groups within the heterogeneous class—heterogeneous and homogeneous groups—so that each student is simultaneously a member of two groups.

Sharing Teacher Training Methods

This chapter illustrates the value of international collaboration. The research, which originated in Israel, arose from a study aiming to enhance teachers’ awareness of their pupils’ thinking processes by exposing them to cognitive theories and to authentic Problematic Learning Situations (PLS). For that purpose, we detected and analysed PLS and used them as video clips in a yearly course for pre- and in-service junior high school teachers. The universality of these learning situations was then pilot-tested across cultures. We provided some of the material on difficulties in geometry to U.K. students training to be elementary school teachers in an effort to learn how these materials affect student teachers in a different country. We were interested in the value of using these materials, differences in Israeli and British reactions to the materials, and the feasibility of conducting such a collaboration. The findings show that viewing the Israeli video clips triggered a change in the U.K. students’ reactions to teaching episodes: it was the first time they took the pupil’s side rather than the teacher’s. However, the findings also show that the dominant influence on the responses of both groups of student teachers (from Israel and the U.K.) was the content of the course in which the video clips were presented.