Reversible reasoning in fractional situations: Theorems-in-action and constraints

• The aim of this study was to investigate how students reason reversibly in the domain of fractions.
• Two seventh grade students were interviewed.
• The results show that reversible reasoning is sensitive to numeric features of problem parameters.
• Two key resources were found to be essential for reasoning reversibly in fractional contexts.
• Failure to conceptualize multiplicative relations in reverse led to fallback mechanisms.

The aim of this study was to investigate, at a fine-grained level of detail, the theorems-in-action deployed and the constraints encountered by middle-school students in reasoning reversibly in the multiplicative domain of fraction. A theorem-in-action (Vergnaud, 1988) is a conceptual construct to trace students’ reasoning in a problem solving situation. Two seventh grade students were interviewed in a rural middle-school in the southern part of the United States. The students’ strategies were examined with respect to the numerical features of the problem situations and the ways they viewed and operated on fractional units. The results show that reversible reasoning is sensitive to the numeric feature of problem parameters. Relatively prime numbers and fractional quantities acted as inhibitors preventing the cueing of the multiplication–division invariant, thereby constraining students from reasoning reversibly. Among others, two key resources were identified as being essential for reasoning reversibly in fractional contexts: firstly, interpreting fractions in terms of units, which enabled the students to access their whole number knowledge and secondly, the unit-rate theorem-in-action. Failure to conceptualize multiplicative relations in reverse constrained the students to use more primitive strategies, leading them to solve problems non-deterministically and at higher computational costs.