The Korea Society Of Educational Studies In Mathematics
[ Special ]
Journal of Educational Research in Mathematics - Vol. 30, No. SP1, pp.15-28
ISSN: 2288-7733 (Print) 2288-8357 (Online)
Print publication date 31 Aug 2020

A Model for Teaching Mathematical Argument at the Elementary Grades

Deborah Schifter*, ; Susan Jo Russell**
*Principal Research Scientist, Education Development Center, USA
**Principal Research Scientist, Education Research Collaborative at TERC, USA

Correspondence to: Email:,

Please cite this article as: Schifter, D. & Russell, S. J. 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.


Argument, Generalization, Conjecture, Representation, Operations


Material in this paper was supported in part by the National Science Foundation under grants no. HR-1019482 and ESI-0550176. Any opinions, findings, conclusions, or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the National Science Foundation. We thank our collaborating teachers and their students, especially those whose classroom work appears in this paper: Sarah Bent, Quayisha Clarke, Caitlin Doering, Emmanuel Fairley-Pittman, Natasha Gordon, Isabel Schooler, and Jan Szymaszek.

We want to thank Megan Franke, Professor, Department of Education, University of California, Los Angeles who was the evaluator for the project that produced student assessment results reported in this paper; she collaborated with the authors in the design of the assessments and carried out the analysis of results.


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