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5. A Cross-Sectional Investigation of Students’ Reasoning About Integer Addition and Subtraction: Ways of Reasoning, Problem Types, and Flexibility

   Lamb, L.L. ; Bishop, J.P. ; Philipp, R. A. ; Whitacre, I. ; & Schappelle, B. P. ( November 2018 , Journal for research in mathematics education)

   In a cross-sectional study, 160 students in Grades 2, 4, 7, and 11 were interviewed about their reasoning when solving integer addition and subtraction open-number sentence problems. We applied our previously developed framework for 5 Ways of Reasoning (WoRs) to our data set to describe patterns within and across participant groups. Our analysis of the WoRs also led to the identification of 3 problem types: change-positive, all-negatives, and counterintuitive. We found that problem type influenced student performance and tended to evoke a different way of reasoning. We showed that those with more experience with negative numbers use WoRs more flexibly than those with less experience and that flexibility is correlated with accuracy. We provide 3 types of resources for educators: (a) WoRs and problem-types frameworks, (b) characterization of flexibility with integer addition and subtraction, and (c) development of a trajectory of learning about integers.