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Norms of mathematical definitions: imposing constraints, permitting choice, or both?

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Abstract 

Definitions play an important role in mathematics by stipulating objects of interest to mathematicians in order to facilitate theory building. Nevertheless, limited research has examined how mathematicians approach writing definitions or the values of the mathematical community that are upheld through norms related to definition use and writing. Based on interviews with nine algebraists/category theorists, we characterize two mathematical values upheld through definitions: clarity in and for communication and freedom of choice in the use and writing of definitions. Further results highlight the norms and values related to defining that participants do and do not claim to discuss through their instruction, including a clear emphasis on precision and rigor in mathematics but limited attempts to show that definitions are created by people to serve their needs. Implications include a need for providing instructional opportunities for students to engage with more mathematical values in order to better understand what algebraists do.

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Funding

This project was funded by the Northern Illinois University Division of Research and Innovation Partnerships through a Research and Artistry grant, grant number RA20-130.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Northern Illinois University, DeKalb, IL, USA

    Rachel Rupnow

  2. Department of Mathematics and Computer Science, Augustana College, Rock Island, IL, USA

    Brooke Randazzo

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  1. Rachel Rupnow
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  2. Brooke Randazzo
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The study conception, funding acquisition, and data collection were performed by Rachel Rupnow. The analysis was performed by Rachel Rupnow and Brooke Randazzo. The first draft of the manuscript was written by both authors. Both authors contributed to reviewing and editing the manuscript and read and approved the final manuscript.

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Correspondence to Rachel Rupnow.

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Rupnow, R., Randazzo, B. Norms of mathematical definitions: imposing constraints, permitting choice, or both?. Educ Stud Math 114, 297–314 (2023). https://doi.org/10.1007/s10649-023-10227-y

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