Abstract: While learning mathematics is often characterized as a process of abstractions from more concrete prior understandings, students’ reasoning in advanced contexts can be a more dynamic process of reflection across multiple layers of complex cognitive structures. To explore the ways that real analysis students’ understandings of abstract mathematical concepts evolve in the context of multi-faceted proofs, we observed students’ attempts to reason about convergence in functional analysis. We describe the understandings from which students built to engage in more complex reasoning and demonstrate their folding back (Pirie & Kieren, 1994) to facilitate the co-evolution of their foundational and advanced understandings.
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