Integer comparisons across the grades: Students' justifications and ways of reasoning

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 order-based reasoning more often than did the younger students, and it more often led to correct answers; however, the college-track 11th graders, who responded correctly to almost every problem, used a more balanced distribution of order- and magnitude-based reasoning. We present a framework for students' ways of reasoning about integer comparisons, report performance trends, rank integer-comparison tasks by relative difficulty, and discuss implications for integer instruction.