SIXTH GRADERS’ CONSTRUCTION OF QUANTITATIVE REASONING AS A FOUNDATION FOR ALGEBRAIC REASONING

Abstract

In a year-long constructivist teaching experiment with four 6th grade students, their quantitative reasoning with fractions was found to form an important basis for their construction of algebraic reasoning. Two of the four students constructed anticipatory schemes for solving problems that could be solved by an equation such as ax = b. In doing so, these students operated on the structure of their schemes. In solving similar problems, the other two students could not foresee the results of their schemes in thought—they had to carry out activity and then check afterwards to determine whether their activity had solved the problems. Operating on the structure of one’s schemes is argued to be fundamentally algebraic, and a hypothesis is proposed that algebraic reasoning can be constructed as a reorganization of quantitative operations students use to construct fractional schemes.