I examined the development of three Prospective Secondary Mathematics Teachers' (PSMTs) understandings of connections between concepts in Abstract Algebra and high school Algebra, as well as their use of this understanding while engaging in the teaching practice of noticing students' mathematical thinking. I drew on the theory, Knowledge of Nonlocal Mathematics for Teaching, which suggests that teachers' knowledge of advanced mathematics can become useful for teaching when it first helps reshape their understanding of the content they teach. I examined this reshaping process by investigating how PSMTs extended, deepened, unified, and strengthened their understanding of inverses, identities, and binary operations over time. I investigated how the PSMTs' engagement in a Mathematics for Secondary Teachers course, which covered connections between inverse functions and equation solving and the abstract algebraic structures of groups and rings, supported the reshaping of their understandings. I then explored how the PSMTs used their mathematical knowledge as they engaged in the teaching practice of noticing hypothetical students' mathematical thinking. I investigated the extent to which the PSMTs' noticing skills of attending, interpreting, and deciding how to respond to student thinking developed as their mathematical understandings were reshaped. There were key similarities in how the PSMTs reshaped their knowledge of inverse, identity, and binary operation. The PSMTs all unified the additive identity, multiplicative identity, and identity function as instantiations of the same overarching identity concept. They each deepened their understanding of inverse functions. They all unified additive, multiplicative, and function inverses under the overarching inverse concept. They also strengthened connections between inverse functions, the identity function, and function composition. They all extended the contexts in which their understandings of inverses were situated to include trigonometric functions. These changes were observed across all the cases, but one change in understanding was not observed in each case: one PSMT deepened his understanding of the identity function, whereas the other two had not yet conceptualized the identity function as a function in its own right; rather, they perceived it as x, the output of the composition of inverse functions. The PSMTs had opportunities to develop these understandings in their Mathematics for Secondary Teachers course, in which the instructor led the students to reason about the inverse and identity group axioms and reflect on the structure of additive, multiplicative, and compositional inverses and identities. The course also covered the use of inverses, identities, and binary operations used while performing cancellation in the context of equation solving. The PSMTs' noticing skills improved as their mathematical knowledge was reshaped. The PSMTs' reshaped understandings supported them paying more attention to the properties and strategies evident in a hypothetical student's work and know which details were relevant to attend to. The PSMTs' reshaped understandings helped them more accurately interpret a hypothetical student's understanding of the properties, structures, and operations used in equation solving and problems about inverse functions. Their reshaped understandings also helped them give more accurate and appropriate suggestions for responding to a hypothetical student in ways that would build on and improve the student's understanding. / Doctor of Philosophy / Once future mathematics teachers learn about how advanced mathematics content is related to high school algebra content, they can better understand the algebra content they may teach. The future teachers in this study took a Mathematics for Secondary Teachers course during their senior year of college. This course gave them opportunities to make connections between advanced mathematics and high school mathematics. After this course, they better understood the mathematical properties that people use while equation solving, and they improved their teaching practice of making sense of high school students' mathematical thinking about inverses and equation solving. Overall, making connections between the advanced mathematics content they learned during college and the algebra content related to inverses and equation solving that they teach in high school helped them improve their teaching practice.

This action-based, qualitative research project involving 7 early childhood educators working in a well-established preschool child care program examined the influences of job-embedded professional development on children’s experiences of mathematics. Data was collected through observations, journals, conversations, interviews, and surveys, and then analyzed using a grounded theory model. A number of themes emerged, the strongest being those related to teachers’ increased awareness, interpretation, and support of children’s explorations in mathematics during play. This project provides an example of a successful model of teacher development for early childhood educators, and contributes to the growing field of research in mathematics education related to teacher noticing, but at the preschool level. Further, with the introduction of full day kindergarten and the emphasis on play based learning this project provides many rich examples of the mathematics present in children's every day play that can be used in future teacher development.

This action-based, qualitative research project involving 7 early childhood educators working in a well-established preschool child care program examined the influences of job-embedded professional development on children’s experiences of mathematics. Data was collected through observations, journals, conversations, interviews, and surveys, and then analyzed using a grounded theory model. A number of themes emerged, the strongest being those related to teachers’ increased awareness, interpretation, and support of children’s explorations in mathematics during play. This project provides an example of a successful model of teacher development for early childhood educators, and contributes to the growing field of research in mathematics education related to teacher noticing, but at the preschool level. Further, with the introduction of full day kindergarten and the emphasis on play based learning this project provides many rich examples of the mathematics present in children's every day play that can be used in future teacher development.

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