We describe a case study in which we investigate the effectiveness of a lecture in advanced mathematics. We first videorecorded a lecture delivered by an experienced professor who had a reputation for being an outstanding instructor. Using video recall, we then interviewed the professor to determine the ideas that he intended to convey and how he tried to convey these ideas in this lecture. We also interviewed 6 students to see what they understood from this lecture. The students did not comprehend the ideas that the professor cited as central to his lecture. Based on our analyses, we propose 2 factors to account for why students did not understand these ideas.

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Researchers have increasingly focused on how gestures in mathematics aid in thinking and communication. This paper builds on Arzarello's (2006) idea of a semiotic bundle and several frameworks for describing individual gestures and applies these ideas to a case study of an instructor's gestures in an undergraduate abstract algebra class. We describe the role that the semiotic bundle plays in shaping the potential meanings of gestures; the ways gestural sets create complex relationships between gestures; and the role played by polysemy and abstraction. These results highlight the complex ways in which mathematical meaningsboth specific and general-are expressed in gesture, and to highlight the integrated nature of elements of the semiotic bundle.