Developing Conceptions of Algebraic Reasoning in the Primary Grades

摘要

There is a substantial body of research describing the development of children's informal mathematical thinking and how this knowledge can serve as a basis for developing understanding of basic concepts and procedures of arithmetic. The goal of the research described below is to investigate how to design instruction to help children to take the next step to generalize and formalize their knowledge into powerful abstract systems for representing and operating on mathematical ideas (i.e. to move from arithmetic thinking to algebraic thinking). The studies build on existing research on Cognitively Guided Instruction (CGI). Children in CGI classrooms construct a variety of increasingly sophisticated and abstract strategies for solving a range of problems involving the basic operations of arithmetic, and they use these strategies flexibly and creatively in solving problems. The strategies emerge out of and are related to the problem situations that give them meaning. Our goal in the studies reported in this paper is to begin to understand how to provide support for children to reflect on their procedures in order to form generalizations from them and construct notations for representing their procedures and generalizations abstractly.