Examining Connections among Instruction, Conceptual Metaphors, and Beliefs of Instructors and Students

Algebra Context The context in which I examine beliefs, content knowledge, and instruction is Abstract Algebra. Special attention is paid to the instruction of isomorphism and homomorphism, but instruction throughout a 3000-level Abstract Algebra course is examined. The two classes chosen were taught using different instructional approaches. One teacher used the Inquiry Oriented Abstract Algebra (IOAA) instructional materials, based on students reinventing the course content. The other teacher used both lecture and “lab” sessions in which students worked on problems together.Algebra (IOAA) instructional materials, based on students reinventing the course content. The other teacher used both lecture and “lab” sessions in which students worked on problems together. 2 Figure 1.1. Broad framework of influence of teachers on students through instruction. Abstract algebra is important because it is a class required for graduation for 89% of math departments training secondary pre-service teachers, and either required or generally taken in 95% of programs (Blair, Kirkman, & Maxwell, 2013, p. 54). For math majors in general, Abstract Algebra is required by more than 85% of math departments across types of institutions (Blair, et al., 2013, p. 53). Not only is Abstract Algebra important, but it might also be considered a “best case scenario” for implementing and studying a variety of types of instruction. Many research-based reform materials, including the Inquiry Oriented Abstract Algebra (IOAA)algebra is important because it is a class required for graduation for 89% of math departments training secondary pre-service teachers, and either required or generally taken in 95% of programs (Blair, Kirkman, & Maxwell, 2013, p. 54). For math majors in general, Abstract Algebra is required by more than 85% of math departments across types of institutions (Blair, et al., 2013, p. 53). Not only is Abstract Algebra important, but it might also be considered a “best case scenario” for implementing and studying a variety of types of instruction. Many research-based reform materials, including the Inquiry Oriented Abstract Algebra (IOAA) curriculum (Larsen, Johnson, & Bartlo, 2013), have been developed for Abstract Algebra, providing more materials for active instruction in this course than for many other upper-level math courses (Johnson, Keller, & Fukawa-Connelly, 2017). Additionally, Abstract Algebra is often a terminal course, allowing more flexibility with the course material. Previous research has characterized what a lecturer teaching Abstract Algebra could look like, but instruction using inquiry-oriented materials or blended lecture and lab sessions has not been examined in detail. I seek to give descriptions of what these types of instruction in Abstract Algebra could look like to fill this gap in the literature. Abstract Algebra provides special advantages for studying university teachers’ and students’ beliefs and conceptual knowledge as well. Because Abstract Algebra is often a terminal course, instructors can use their flexibility with the course material to allow students to look back at the courses they have taken and examine their views of mathematics as a whole. This could lead to students reflecting on their beliefs about mathematics. Specific content knowledge that is addressed in Abstract Algebra, such as isomorphism and homomorphism, has not been studied