Sameness in algebra: views of isomorphism and homomorphism

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.