## Metaphors for learning and doing mathematics in advanced mathematics lectures

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Metaphors for learning and doing mathematics in advanced mathematics lectures Joe Olsen 1

2

& Kristen Lew & Keith Weber

1

# Springer Nature B.V. 2020

Abstract

The metaphors that students form and encounter have been shown to exert a powerful influence on how they think about mathematics. In this paper, we explore the linguistic metaphors about learning and doing mathematics that were prevalent in 11 advanced mathematics lectures. We present four metaphor clusters that were common in the corpus that we analyzed: Learning Mathematics is a Journey, Doing Mathematics is Work, Mathematics is Discovery, and Presenting Mathematics is a Story. We conclude the paper by discussing the implications of the metaphors that mathematicians use and students hear in their mathematics courses. Keywords Metaphor . Advanced mathematics courses . Lecture

Perhaps I can best describe my experience of doing mathematics in terms of a journey through a dark unexplored mansion. You enter the first room of the mansion and it's completely dark. You stumble around bumping into the furniture, but gradually you learn where each piece of furniture is. Finally, after six months or so, you find the light switch, you turn it on, and suddenly it's all illuminated. You can see exactly where you were. Then you move into the next room and spend another six months in the dark. So each of these breakthroughs, while sometimes they are momentary, sometimes over a period of a day or two, they are the culmination of—and could not exist without—the many months of stumbling around in the dark that proceed them. – Andrew Wiles (Lynch, 1997, 0:35)

* Joe Olsen [email protected]

1

Graduate School of Education, Rutgers University, 10 Seminary Place, New Brunswick, NJ 08901, USA

2

Department of Mathematics, Texas State University, San Marcos, TX 78666, USA

Olsen J. et al.

In our previous work on lectures in advanced mathematics, we analyzed the ways and the extent to which mathematicians conveyed formal content (definitions, theorems, proofs) and informal content (heuristics and informal representations) in their lectures (Fukawa-Connelly, Weber, & Mejia-Ramos, 2017; Lew, Fukawa-Connelly, Mejia-Ramos, & Weber, 2016). However, one topic we did not examine was the implicit messages professors convey about the nature of learning and doing mathematics. Using a corpus of 11 lectures in advanced mathematics (i.e., proof-oriented university mathematics courses), we explore how mathematicians described mathematical activity during their lectures. In our data corpus, explicit comments about the nature of mathematical activity were rare. However, we found that we could interpret many comments about the nature of mathematical activity as metaphors. For instance, consider the Andrew Wiles quotation above. To make his abstract activity accessible to a layman audience, Wiles compared doing mathematics in terms of wandering around in a dark mansion on the assumption that his audience has had experience wandering around in the dark or at least could imagine

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