Study on the Introductory Process of Algebra from the Holistic Perspective : On the Emergence of Algebraic Ideas

Abstract

In the atomistic perspective which teaching materials are divided in detail teaching takes many small steps. Students may acquire a certain amount of knowledge and skills, however they may be in danger of learning without knowing what they are doing as a whole and why. So this study aims to rethink mathematical learning from the holistic perspective, especially the 7th grade's algebraic contents. In order to design classroom lessons from the holistic perspective, we need to know beforehand what sort of problems encourage students' mathematical ideas to emerge and how they develop their ideas through solving the problems. The main purpose of this article is to make it clear how 7th grade students who have not learned the contents of letters and algebraic expressions can emerge and develop their algebraic ideas through solving problems in the context of the linear equation. We conducted several case studies and analyzed students' problem solving activities. The results are as follows; 1. Students could create naive algebraic ideas through solving problems presented by Bednarz and Janvier (1997). 2. The interplay between using trial-and-error method and setting up the expression could help them to understand local relationships and to create algebraic ideas. 3. The idea of uniting several terms emerged in two situations. One was the situation of simplifying the numerical calculation and the other was that of understanding the structure of the parts in the expression that they set up. 4. It was effective for the emergence of the idea of distributive law that they conceived the problem situation and the numerical computations structurally. The idea became explicit when the explanation was sifted from numerical calculation to algebraic expression using letters and when they reinterpreted the expression as the number of pieces of one letter. 5. Students could create a feeling of necessity and appreciation of using letters through the problem solving in the context of linear equation before they learn algebraic expressions.