Reversible reasoning in multiplicative situations

Understanding children's thinking is a challenging endeavor that necessitates both robust methodological tools as well as a thorough theoretical and analytical set-up. This study illustrates how researchers in mathematics education can access children's mathematics, at a fine-grained level of detail, through videotaped analyses. Effective mathematical thinking involves both doing and undoing processes within the range of structures that constitute the body of mathematics. In a given problem situation, the path from the result to the source is often more challenging than from the source to the result. This study focuses on reversible reasoning in the multiplicative domains of fractions and ratio.The systematic analysis of qualitative data is presented through the articulation of three analytical tools, namely Vergnaud's concept of theorem-in-action, the notion of coordination of units, and quantitative reasoning. This study is useful not only for researchers in Mathematics Education but also for teachers, curriculum developers, and test writers. The theoretical analysis contains a wide range of problem situations that are pedagogically useful in fostering multiplicative reasoning.