Take It Away or Walk the Other Way? Finding Positive Solutions for Integer Subtraction

Practicing teachers as well as researchers, mathematicians, and teacher educators have offered opinions and theoretical critiques of the multiple models used to teach integer arithmetic. Few studies, however, have investigated what students learn with models or empirically compared affordances and constraints of integer models. This led me to investigate how 160 fifth- and sixth-grade students who were learning integer arithmetic for the first time could benefit from a particular model. Each integer model encouraged students to conceive of numbers using distinct conceptual metaphors and move in certain ways to represent integer subtraction. Thus, I used embodied cognition to illuminate ways a manipulative-based cancellation model (chip model) and a physically enacted number line model (walk-it-off model) differentially impacted students’ subtraction knowledge. Integer subtraction, particularly the idea that subtracting a negative number could create a positive solution is especially difficult for students regardless of age, so assessment of this construct deserved a special focus in the test design of the larger study. This chapter reports students’ accuracy and reasoning on this difficult subtraction type 5 weeks after instruction with their assigned model. Findings for practice suggest the walk-it-off model was more effective as the first model students used and more research is needed.