*event*Thursday, October 24, 2024

*access_time*10:30am (CDT)

*room*Zoom

*info*Use link above or this Zoom info: Meeting ID: 978 8567 7914 Passcode: RUME2024

**Abstract:** I will discuss my research on how prospective and in-service teachers make connections between abstract algebra and the secondary school mathematics content they teach, as well as how their knowledge of these content connections can influence their teaching. One such content connection is between the algebraic structure of unique factorization domains and the secondary mathematics content of the factorization of integers and polynomials. My colleagues and I performed design research to investigate how graduate students who were prospective and in-service teachers could develop an understanding of this content connection. We conducted a teaching experiment to guide these graduate students to reinvent the definition of UFDs by conjecturing about and generalizing the structure of the factorization of integers and polynomials. We used the instructional design theory, Realistic Mathematics Education, to create a sequence of instructional activities, in which the students used algebra tile manipulatives as a model of factoring integers and quadratics in the polynomial ring, Z[x]. The students used this model for abstracting the shared structure of (ir)reducible elements in Z and Z[x], which the students then used to define reducible and irreducible elements. The students conjectured about the existence and uniqueness of the factorizations of (ir)reducibles using the algebra tiles. The teacher-researcher leveraged these informal, intuitive ways of reasoning to guide the students to formally reinvent the two defining axioms of a UFD. After creating this definition of UFDs, the students were guided to apply their understanding of the properties of UFDs in their teaching practices of noticing and responding to student thinking in approximations of teaching practice. We investigated the studentsâ€™ mathematical reasoning and use of their understanding in these teaching scenarios. We also identified the pedagogical moves used by the teacher-researcher to guide the students into developing these understandings. I will discuss these findings in my talk and provide implications for the mathematical preparation of secondary mathematics teachers.