A Model for Teaching Mathematical Argument at the Elementary Grades

Over years of collaborating with elementary-school teachers to research students' thinking about the "big ideas" of K-6 mathematics, particular attention was given to generalizations about the operations-addition, subtraction, multiplication, and division-and arguments that explain why these generalizations are true. Through this work, we created a model of five phases that separate different points of focus in the complex process of formulating and proving such generalizations: 1) noticing patterns, 2) articulating conjectures, 3) representing with specific examples, 4) creating representation-based arguments, and 5) comparing and contrasting operations. In this paper, we illustrate the phases with classroom examples as students investigate a set of generalizations. We then present assessment results from classrooms of project teachers who engaged their students in this content.