Introducing negative numbers with variation theory – a systematic study of the relation between teaching and learning

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Understanding the rationale behind operations with negative numbers is crucial for students’ development in mathematics. However, there have been few systematic investigations of how teaching affects student learning. This article presents such an approach. Using a learning study-based research design involving 129 Grade 6 students across five teaching cycles, a two-lesson teaching unit on addition and subtraction with negative numbers is designed, implemented, analyzed, and revised in an iterative process. The teaching design is based on variation theory and findings from previous studies concerned with students’ understanding of negative numbers. The results include a complete description of the fifth and final teaching design. An analysis of teaching-learning relationship suggests that a central rationale for addition with negative numbers can be made visible through the sum of opposite numbers. Correspondingly, considering the distance between numbers is powerful to make the result of subtractions with negative numbers visible to students.