My research agenda addresses the double discontinuity problem posed by Klein (1933): undergraduate students experience a discontinuity in the content they learned in secondary mathematics as they transition to university mathematics, and they again experience this discontinuity when they graduate and go on to teach secondary mathematics. Researchers have documented that secondary mathematics teachers often do not perceive their undergraduate or graduate education in advanced mathematics courses as useful or influential in their teaching. Several students majoring in Mathematics leave their degrees dissatisfied, perceiving their mathematics courses as disparate from their careers. Thus, there is a need for improved continuity among secondary, undergraduate, and graduate mathematics courses, which can guide prospective mathematics teachers to make connections between the mathematical content and mathematical practices used in both secondary and advanced mathematics. In addition to resolving this discontinuity for prospective mathematics teachers, there is a need for research on how to support STEM majors in making connections between advanced mathematics and the content from their respective scientific fields. My research contributes toward meeting these needs by investigating the teaching and learning of the connections that undergraduate students and graduate students make between advanced mathematics and their future career fields in STEM or education.
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My research on students making connections between advanced mathematics content and STEM began when I was a research assistant on Dr. Megan Wawro’s NSF-funded grant. My colleagues and I studied the connections that physics students made between quantum mechanics and the advanced mathematical content of linear algebra. In our paper (Serbin et al., 2020) published in the Journal of Mathematical Behavior (JMB), we focused on physics students’ conceptual and procedural knowledge of deriving and using the characteristic equation. A second paper (Serbin et al., 2021) published in Physical Review Special Topics–Physics Education Research explored discourse about basis and change of basis, which are topics from linear algebra and quantum mechanics. Our most recent paper (Serbin & Wawro, 2022a), published in the International Journal for Research in Undergraduate Mathematics Education (IJRUME), investigated the complexities of undergraduate physics students’ reasoning about linear algebra concepts as they solved quantum mechanics problems. We are currently writing a paper in which we investigate a quantum mechanics class’s development of eigentheory concepts from linear algebra in connection with quantum mechanical systems. This research has culminated in four papers (Serbin et al., 2019; Serbin et al., 2020; Serbin & Wawro, 2022b; Wawro & Serbin, 2023) presented at the national Conference on Research in Undergraduate Mathematics Education (RUME), as well as one paper (Serbin & Wawro, 2020) presented at the international conference of the North American Chapter of the International Group for the Psychology of Mathematics Education.
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My independent research has continued this thread of analysis in exploring undergraduate and graduate students’ understanding of connections between secondary and advanced mathematics, particularly abstract algebra. I investigate how prospective secondary mathematics teachers (PSMTs) can develop meaningful understandings of the content connections and how these connected, coherent understandings are useful in their teaching practices. I began this work in my dissertation by investigating the development of PSMTs’ understanding of connections between Abstract Algebra and secondary Algebra, identifying the changes in PSMTs’ understandings of concepts related to inverses, identities, binary operations, cancellation, and equation solving as they learned about connections between those concepts and the abstract algebraic structures of groups and rings. This solo-authored work has been published in JMB (Serbin, 2023) and has been presented at the national RUME conference (Serbin, 2020; 2022) and the Mathematical Association of America’s (MAA) national MathFest conference in 2022. I am currently writing a solo-authored journal article based on this work that investigates PSMTs’ use of abstract algebra knowledge in their teaching practice of noticing and responding to students’ mathematical thinking. I extended this project in collaboration with Dr. Rosaura Uscanga to further analyze the content connections PSMTs made. This has resulted in a national RUME conference presentations (Serbin & Uscanga, 2023) and an article soon to be submitted for publication.
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I continued this strand of research focusing on PSMTs’ understanding of the content connections by designing, teaching, and researching a graduate-level mathematics for teachers course that connected graduate-level abstract algebra, real analysis, and statistics content to the teaching and learning of secondary and early undergraduate mathematics. My colleague, Dr. Younggon Bae, and I analyzed changes in the graduate students’ perceptions of the relevance of advanced mathematics to their teaching after taking this course and the applications of their advanced mathematical understandings in their teaching. We presented this work at the 2023 RUME conference (Serbin & Bae, 2023), and we are revising an article on it. We wrote an additional manuscript using this dataset, in which we analyzed the pedagogical mathematical practices (Wasserman, 2022) that these graduate students used in the approximation of practice activities (Grossman et al., 2009) that we designed to simulate the teaching practices of noticing, interpreting, and deciding how to respond to student thinking. We conducted this research with my graduate advisee and research assistant, Ashly Olusanya. We also presented this research at the 2024 RUME Conference.
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I am continuing this design research to create innovative graduate-level abstract algebra curricula that emphasize connections from polynomial rings and unique factorization domains to the teaching and learning of concepts in precalculus and secondary algebra, like factorization, polynomial functions, and rational functions. Informed by the theory of Realistic Mathematics Education (Freudenthal, 1991; Gravemeijer, 1999), my colleagues and I recently conducted a teaching experiment and created a local instructional theory on graduate student PSMTs’ guided reinvention of Unique Factorization Domains with connections to teaching secondary mathematics (see the task sequence on my website). In collaboration with Younggon Bae and my graduate advisee, Sthefania Espinosa, we performed analyses of students’ reasoning and the teacher researcher’s (my) pedagogical moves used to elicit and leverage students’ reasoning to guide students’ reinvention. This work was presented at the 2023 MAA MathFest, the 2024 International Congress on Mathematics Education, and the 2024 RUME conference. Dr. Bae and I, along with two other collaborators, Dr. Tenchita Alzaga Elizondo and Dr. Rosaura Uscanga, are pursuing grant funding to extend this local instructional theory and design three others that guide graduate student PSMTs’ reinvention of abstract algebraic concepts in connection to the teaching and learning of secondary algebra and precalculus.
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A related line of research I pursue is investigating how mathematics instructors and teacher educators can guide PSMTs to make such connections between advanced mathematics and secondary mathematics to develop PSMTs’ mathematical knowledge for teaching. In one such project, I, along with Dr. Estrella Johnson and my graduate advisee, Sthefania Espinosa, identified the pedagogical moves that a mathematics teacher educator enacted to evoke PSMTs’ intellectual needs (Harel, 2013) that can be resolved by the PSMTs making content connections between advanced and secondary mathematics. This work was presented at the 2023 RUME conference (Serbin & Espinosa, 2023) and was submitted for publication.
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In summary, my research aims to improve students’ and teachers’ understandings of the connections between advanced mathematics, STEM, and mathematics teaching. In addition to these projects I discussed, I have conducted research in several other areas, such as participatory equity in class discourse, class social norms, factors influencing teachers’ decisions to use inquiry-oriented instruction, mathematical task design, and students’ mathematics identity development. My diverse research experiences well-position me to advise and mentor graduate students’ research projects, theses, and dissertations. My current research focus is on supporting prospective and in-service teachers’ development of understandings of connections between abstract algebra and secondary algebra, as well as mathematics teacher educators’ pedagogical moves that can support PSMTs’ in making such connections. My research has and will continue to have potential for collaboration with other faculty members and student mentees.