Reversible reasoning in multiplicative situations
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While previous research studies have focused primarily on the additive structure a+/-x=b , relatively fewer attempts have been made to explore the multiplicative structure ax=b from a reversibility perspective. The aim of this exploratory study was to investigate, at a fine-grained level of detail, the strategies and constraints that middle-school students encounter in reasoning reversibly in the multiplicative domains of fraction and ratio. In the first phase of the study a mathematical analysis of reversibility situations was conducted. In the second empirical phase, three pairs of above-average students at the grades 6, 7 and 8 were interviewed in a rural middle-school in the United States. Three selected data sets were analyzed using Vergnaud’s theory of the multiplicative conceptual field, the concept of units, and notion of quantitative reasoning. The findings put into perspective the importance of the theorem-in-action ax=b as a key element for reasoning reversibly in multiplicative situations. Further, the results show that reversibility is context-sensitive, with the numeric feature of problem parameters being a major factor. Relatively prime numbers and fractional quantities acted as inhibitors preventing the cueing of the invariant ‘division as the inverse of multiplication’, thereby constraining students from reasoning reversibly. Moreover, the form of reversible reasoning was found to be dependent on the type of multiplicative structures. Among others, two key resources were identified as being essential for reasoning reversibly in fractional contexts: interpreting fractions in terms of units, which enabled the students to access their whole number knowledge and secondly, the coordination of 3 levels of units. Similarly, interpreting a ratio as a quantitative structure together with the coordination of 3 levels of units was found to be essential for reasoning reversibly in ratio situations. This study also shows that students can articulate mathematical relationships operationally but may not necessarily be able to represent them algebraically. The constraints that the students encountered in the multiplicative comparison of two quantities substantiate previous research that multiplicative reasoning is not naturally occurring. Failure to conceptualize multiplicative relations in reverse constrained the students to use more primitive fallback mechanisms, like the building-up strategy and guess-and-check strategy, leading them to solve problems non-deterministically and at higher computational cost.