Many empirical studies documented students’ challenges with operation-choice problems, in particular for multiplication and division with rational numbers. The design principle of problem variation was suggested to overcome these challenges by engaging students in making connections between inverse operation-choice problems of multiplication and division, and between problems with natural numbers and fractions/decimals, but so far, this approach was hardly investigated empirically. In this study, we investigate 17 sixth graders’ modelling pathways through sets of operation-choice problems that are systematically designed according to the variation principle. In the qualitative analysis, we identify five pathways by which students solve the problems and sometimes connect them. While one pathway uses deep relational connections, others only draw superficial and operational connections and others stay with informal strategies without connecting them to formal operations.

This study is supported by an FPU grant FPU19/02965 from Ministerio de Universidades (Spain) to Cristina Zorrilla under the supervision of Ceneida Fernández and Salvador Llinares. The analytic approach and the paper have been developed collectively by the first, second and last author during the first author’s research stay in Dortmund, Germany, with Susanne Prediger and Anna-Katharina Roos. This stay was funded by the Ministerio de Universidades (EST21/00333).