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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. 

We identify and document 3 cognitive obstacles, 3 cognitive affordances, and 1 type of integer understanding that can function as either an obstacle or affordance for learners while they extend their numeric domains from whole numbers to include negative integers. In particular, we highlight 2 key subsets of integer reasoning: understanding or knowledge that may, initially, interfere with one’s learning integers (which we call cognitive obstacles) and understanding or knowledge that may afford progress in understanding and operating with integers (which we call cognitive affor- dances). We analyzed historical mathematical writings related to integers as well as clinical interviews with children ages 6–10 to identify critical, persistent cognitive obstacles and powerful ways of thinking that may help learners to overcome obstacles. 

Three meanings of the minus sign are shared.