This study is an investigation of students' reasoning about integer comparisons-a topic that is often counterintuitive for students because negative numbers of smaller absolute value are considered greater (e.g.,-5 >-6). We posed integer-comparison tasks to 40 students each in Grades 2, 4, and 7, as well as to 11th graders on a successful mathematics track. We coded for correctness and for students' justifications, which we categorized in terms of 3 ways of reasoning: magnitude-based, order-based, and developmental/other. The 7th graders used orderbased reasoning more often than did the younger students, and it more often led to correct answers; however, the college-track 11th graders used a more balanced distribution of order-and magnitude-based reasoning correctly for almost every problem. We present a framework for students' ways of reasoning about integer comparisons, report performance trends, rank integercomparison tasks by relative difficulty, and discuss implications for integer instruction.
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