# Students' understandings of logical implication

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## Abstract

We report results from an analysis of responses to a written question in which high-attaining students in English schools, who formed part of a longitudinal nation-wide survey on proof conceptions, were asked to assess the equivalence of two statements about elementary number theory, one a logical implication and the other its converse, to evaluate the truth of the statements and to justify their conclusions. We present an overview of responses at the end of Year 8 (age 13 years) and an analysis of the approaches taken, and follow this with an analysis of the data collected from students who answered the question again in Year 9 (age 14 years) in order to distinguish learning trajectories. From these analyses, we distinguished three strategies, empirical, focussed-empirical and focussed-deductive, that represent shifts in attention from an inductive to a deductive approach. We noted some progress from Year 8 to Year 9 in the use of the focussed strategies but this was modest at best. The most marked progress was in recognition of the logical necessity of a conclusion of an implication when the antecedent was assumed to be true. Finally we present some theoretical categories to capture different types of meanings students assign to logical implication and the rationale underpinning these meanings. The categories distinguish responses where a statement of logical implication is (or is not)interpreted as equivalent to its converse, where the antecedent and consequent are (or are not) seen as interchangeable, and where conclusions are (or are not)influenced by specific data.

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