Convergence of Sequences and Series 2: Interactions Between Nonvisual Reasoning and the Learner’s Beliefs about their own Role

By Lara Alcock, Adrian P. Simpson. 2004

87 citations recorded by the source corpus.

This paper examines the work of students who, when reasoning about real analysis, do so almost exclusively by means of verbal and algebraic reasoning, and tend not to incorporate visual images into their work. It examines the work of students from two parallel courses of introductory real analysis, whose reasoning ranges from those who introduce definitions appropriately and work with them competently, to those who cannot recall definitions and appear to manipulate notation without regard for its reference. It presents a theory that relates the differences to students’ expectations regarding their role as learners of mathematics. Throughout, the argument is illustrated with interview data from which the theory was inductively generated.

Subfields: Education, Statistics and Probability, Theoretical Computer Science

View Paper: https://www.semanticscholar.org/paper/ad979de28e5334cbce94f1b1a2fd84088ab4be9a

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Citations in the corpus

Doctoral students’ use of examples in evaluating and proving conjectures (2008, background): “Both Pinto & Tall (1999, 2002) and Alcock & Simpson (2004, 2005) studied students’ thinking in Real Analysis, and both teams noticed that some students regularly used visual representations to assist in their reasoning, whereas others rarely did this, and instead worked with mathematical concepts…”
An empirical study of students’ understanding of a logical structure in the definition of limit via the ε-strip activity (2010, background): “On the other hand, Alcock and Simpson (2005) contended that, in the case of students who possessed strong links between logical structure and underlying meaning for convergent sequences, they could recall and use the definition of convergence in appropriate ways.”
Proofs and refutations in the undergraduate mathematics classroom (2008, background): “Many mathematics educators have made brief reference to Lakatos’ (1976) historical case studies and his description of processes of mathematical discovery (e.g., Alcock and Simpson 2004; Mariotti and Fischbein 1997; Zaslavsky and Shir 2005).”
Students’ images and their understanding of definitions of the limit of a sequence (2008, background): “On the other hand, Alcock and Simpson (2004, 2005) could not confirm that learning by visualization was helpful in advanced mathematical thinking.”