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Sameness in Mathematics: a Unifying and Dividing Concept

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Abstract

While many aspects of the teaching and learning of specific advanced mathematics courses have been studied, limited work has examined mathematical themes like sameness or its instantiations across disciplines. In this paper, we explore algebraists’ collective example space for mathematical sameness. We used qualitative methods to analyze survey responses from 197 algebraists in order to identify specific mathematical concepts that the algebraists associated with sameness and relevant factors to consider when determining sameness of objects. Using variation theory, we introduce the notion of a community example space for algebraists to highlight specific dimensions of variation in sameness as well as the range of variation for each dimension through specific instantiations given by multiple participants. Dimensions of sameness included contexts, concepts, objects, properties, and qualities. These results suggest potential for building connections across levels and branches of mathematics by highlighting how different choices for each dimension of sameness can produce different instantiations of sameness.

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Acknowledgements

This research was funded by a Northern Illinois University Research and Artistry Grant to Rachel Rupnow, grant number RA20-130.

Funding

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

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Authors and Affiliations

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

    Rachel Rupnow, Brooke Randazzo, Eric Johnson & Peter Sassman

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  1. Rachel Rupnow
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  2. Brooke Randazzo
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  3. Eric Johnson
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  4. Peter Sassman
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Contributions

The study conception, funding acquisition, data collection were performed by Rachel Rupnow. The analysis was performed by Rachel Rupnow, Brooke Randazzo, and Eric Johnson. The first draft of the manuscript was written by all authors. All 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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This study has been approved by the Northern Illinois University Institutional Review Board, HS20-0306.

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Appendix

Appendix

  1. 1.

    What does it mean to be the same in a math context?

  2. 2.

    How do you know two things are the same in abstract algebra?

  3. 3.

    How is “sameness” in abstract algebra similar or different from “sameness” in other branches of math?

  4. 4.

    Which option, in your opinion, best describes the relationship between the positive reals under multiplication and the reals under addition?

    1. (a)

      They are identical structures.

    2. (b)

      They are the same structures in all contexts, but not identical structures.

    3. (c)

      They are algebraically the same structures, but they are different structures in other mathematical settings.

    4. (d)

      They are similar algebraic structures, but they are not the same.

    5. (e)

      They are different mathematical structures.

  5. 5.

    Please explain your response.

  6. 6.

    Which option, in your opinion, best describes the relationship between Z_5 under modular addition and Z/5Z under addition?

    1. (a)

      They are identical structures.

    2. (b)

      They are the same structures in all contexts, but not identical structures.

    3. (c)

      They are algebraically the same structures, but they are different structures in other mathematical settings.

    4. (d)

      They are similar algebraic structures, but they are not the same.

    5. (e)

      They are different mathematical structures.

  7. 7.

    Please explain your response.

  8. 8.

    Which option, in your opinion, best describes the relationship between Z and 2Z under addition?

    1. (a)

      They are identical structures.

    2. (b)

      They are the same structures in all contexts, but not identical structures.

    3. (c)

      They are algebraically the same structures, but they are different structures in other mathematical settings.

    4. (d)

      They are similar algebraic structures, but they are not the same.

    5. (e)

      They are different mathematical structures.

  9. 9.

    Please explain your response.

  10. 10.

    Which option, in your opinion, best describes the relationship between Z_5 and Z_6 under addition?

    1. (a)

      They are identical structures.

    2. (b)

      They are the same structures in all contexts, but not identical structures.

    3. (c)

      They are algebraically the same structures, but they are different structures in other mathematical settings.

    4. (d)

      They are similar algebraic structures, but they are not the same.

    5. (e)

      They are different mathematical structures.

  11. 11.

    Please explain your response.

  12. 12.

    Which option, in your opinion, best describes the relationship between G = Z/2Z x Z/2Z x Z/2Z x … where elements of G must contain a tail of 0's and H = Z/2Z x Z/2Z x Z/2Z x … (unrestricted)?

    1. (a)

      They are identical structures.

    2. (b)

      They are the same structures in all contexts, but not identical structures.

    3. (c)

      They are algebraically the same structures, but they are different structures in other mathematical settings.

    4. (d)

      They are similar algebraic structures, but they are not the same.

    5. (e)

      They are different mathematical structures.

  13. 13.

    Please explain your response.

  14. 14.

    How might sameness be helpful when thinking about isomorphism/isomorphic structures?

  15. 15.

    How might sameness be harmful when thinking about isomorphism/isomorphic structures?

  16. 16.

    How might sameness be helpful when thinking about homomorphism?

  17. 17.

    How might sameness be harmful when thinking about homomorphism?

  18. 18.

    Which abstract algebra topics lend themselves to deepening students' understanding of mathematical sameness?

  19. 19.

    What connections do you see between sameness in abstract algebra and in prior math courses?

  20. 20.

    What sameness connections between abstract algebra and other courses do you (or could you) help students make when teaching?

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Rupnow, R., Randazzo, B., Johnson, E. et al. Sameness in Mathematics: a Unifying and Dividing Concept. Int. J. Res. Undergrad. Math. Ed. 9, 398–425 (2023). https://doi.org/10.1007/s40753-022-00178-9

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