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This theoretical article explores the affordances and challenges of Euler diagrams as tools for supporting undergraduate introduction-to-proof students to make sense of, and reason about, logical implications. To theoretically frame students’ meaning making with Euler diagrams, we introduce the notion of logico-spatial linked structuring (or LSLS). We argue that students’ use of Euler diagrams as representations of logical statements entails a conceptual linking between spatial and non-spatial representations, and the LSLS framework provides a tool for modeling this conceptual linking. Moreover, from our Piagetian epistemological perspective, reasoning with Euler diagrams entails engaging in spatial mental operations and making a logical conclusion from the result. We illustrate the utility of the LSLS framework through examples with two undergraduate students as they reasoned about the truth of the converse and contrapositive of a given logical implication, and we identify specific spatial operations that they used and coordinated in their problem solving. 

Understanding how students reason with logical implication is essential for supporting students’ construction of increasingly powerful ways of reasoning in proofs-based mathematics courses. We report on the results of an NSF-funded case study with a mathematics major enrolled in an introductory proofs course. We investigate the epistemological obstacles that she experienced and how they might relate to her treatment of logical implications as actions. Evidence shows that an action conception may pose challenges when students transform or quantify implications and may contribute to erroneous assumptions of biconditionality. Our report on available ways of operating with logical implications as actions is a first step in designing instructional tasks that leverage students’ existing reasoning skills to support their continued development.