“Because You’re Exploring this Huge Abstract Jungle…”: One Student’s Evolving Conceptions of Axiomatic Structure in Mathematics

Kathleen Michelle Clark,Cihan Can

doi:10.29333/iejme/8566

Kathleen Michelle Clark, Cihan Can

Open Access

https://doi.org/10.29333/iejme/8566

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Abstract

For several decades, literature on the history and pedagogy of mathematics has described how history of mathematics is beneficial for the teaching and learning of mathematics. We investigated the influence of a history and philosophy of mathematics (HPhM) course on students’ progress through the lens of various competencies in mathematics (e.g., mathematical thinking and communicating) as a result of studying mathematical ideas from the perspective of their historical and philosophical development. We present outcomes for one student, whom we call Michael, resulting from his learning experiences in an HPhM course at university. We use the framework from the Competencies and Mathematical Learning project (the Danish KOM project) to analyze the evolution of Michael’s competencies related to axiomatic structure in mathematics. We outline three aspects of axiomatic structure to situate our analysis: Truth, Logic, and Structure. Although our analysis revealed that Michael’s views and knowledge of axiomatic structure demonstrate need for his further development, we assert what he experienced during the HPhM course regarding his mathematical thinking and communication about axiomatic structure is promising support for his future mathematical studies. Finally, we argue that a HPhM course has potential to support students’ progress in advanced mathematics at university.

Highlights

For several decades, literature focused on the history and pedagogy of mathematics has described the ways in which the history of mathematics is considered beneficial for the teaching and learning of mathematics
In the remainder of this section, we describe the perspective on axiomatic method that we believe is closest to that used in advanced mathematics at university, together with the aspects of axiomatic structure that we will highlight in the analysis of our data
Http://www.iejme.com instructor in his notes), we identified a strong contribution of the history and philosophy of mathematics (HPhM) course on Michael’s mathematical thinking related to axiomatic structure, in that it provided him with a context in which to discuss his views on this issue

Summary

Introduction

Literature focused on the history and pedagogy of mathematics has described the ways in which the history of mathematics is considered beneficial for the teaching and learning of mathematics. In an early contribution to the re-popularization of using history in teaching and learning mathematics,1 Fauvel (1991) outlined 15 reasons for using history in mathematics education, including that it: helps to increase motivation for learning, gives mathematics a human face, changes pupils’ perceptions of mathematics, provides opportunities for investigations, and provides opportunity for cross-curricular work with other teachers or subjects It can be argued that much of the literature on the benefits of using history for the teaching and learning of mathematics focuses heavily on affective-motivational contributions and often lacks empirical support. To this end, Barnett et al outlined reasons for using primary sources in undergraduate mathematics teaching that draw explicit attention to student learning, including:

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