Abstract
Isomorphism and homomorphism are topics central to abstract algebra, but research on mathematicians’ views of these topics, especially with respect to sameness, remains limited. This study examines open response survey data from 197 mathematicians on how sameness could be helpful or harmful when studying isomorphism and homomorphism. Using thematic analysis, we examined whether sameness was viewed as helpful or harmful for isomorphism and homomorphism before examining rationales for those views. Making use of values of the mathematical community, we note that mathematicians saw conceptual and pedagogical benefits to connecting isomorphism and sameness, which connects to leveraging intuition and valuing ways of increasing understanding. Mathematicians’ concerns around using sameness largely revolved around the violation of the mathematical community’s idealized value of expressing a priori truth via a-contextual justifications. However, these concerns can be addressed through only targeted usage of sameness and explicit discussions around the utility and relevance of sameness. Implications include the importance of considering how interventions proposed by mathematics educators align with or provoke tension between values held by the mathematical community in order to mitigate or lean into those tensions and encourage fruitful dialogue between the mathematics and mathematics education communities.
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For reference: “An isomorphism \(\phi\) from a group \(G\) to a group \(\overline{G }\) is a one-to-one mapping (or function) from \(G\) onto \(\overline{G }\) that preserves the group operation. That is, \(\phi \left(ab\right)= \phi (a)\phi (b)\) for all \(a,b\) in \(G\). If there is an isomorphism from \(G\) onto \(\overline{G }\), we say that \(G\) and \(\overline{G }\) are isomorphic and write \(G\approx \overline{G }\)” (Gallian, 2009). A group homomorphism is a group isomorphism without the requirement of bijectivity.
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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. We also thank the editor and reviewers for their helpful comments on earlier drafts of the manuscript.
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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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The study conception, funding acquisition, and data collection were performed by Rachel Rupnow. The analysis was performed by Rachel Rupnow and Peter Sassman. The first draft of the manuscript was written by Rachel Rupnow and Peter Sassman. Both authors contributed to reviewing and editing the manuscript and read and approved the final manuscript.
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Rupnow, R., Sassman, P. Sameness in algebra: views of isomorphism and homomorphism. Educ Stud Math 111, 109–126 (2022). https://doi.org/10.1007/s10649-022-10162-4
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DOI: https://doi.org/10.1007/s10649-022-10162-4