The centrality of student-generated representation in investigating generalizations about the operations

Abstract

This article addresses the nature of student-generated representations that support students’ early algebraic reasoning in the realm of generalized arithmetic. We analyzed representations created by students for the following qualities: representations that distinguish the behavior of one operation from another, that support an explanation of a specific case of a generalization, and that support justification of a generalization. One key finding is that representations in the form of pictures, diagrams, arrangements of manipulatives, or story contexts that embody the meaning of the operation(s) allow students to distinguish between operations. Such representations can be used by young students to illustrate relationships conveyed in specific instances of a general claim. Further, extending these representations to class of numbers is a mechanism for proving a general claim.