abstract: This dissertation reports three studies about what it means for teachers and students to reason with frames of reference: to conceptualize a reference frame, to coordinate multiple frames of reference, and to combine multiple frames of reference. Each paper expands on the previous one to illustrate and utilize the construct of frame of reference. The first paper is a theory paper that introduces the mental actions involved in reasoning with frames of reference. The concept of frames of reference, though commonly used in mathematics and physics, is not described cognitively in any literature. The paper offers a theoretical model of mental actions involved in conceptualizing a frame of reference. Additionally, it posits mental actions that are necessary for a student to reason with multiple frames of reference. It also extends the theory of quantitative reasoning with the construct of a ‘framed quantity’. The second paper investigates how two introductory calculus students who participated in teaching experiments reasoned about changes (variations). The data was analyzed to see to what extent each student conceptualized the variations within a conceptualized frame of reference as described in the first paper. The study found that the extent to which each student conceptualized, coordinated, and combined reference frames significantly affected his ability to reason productively about variations and to make sense of his own answers. The paper ends by analyzing 123 calculus students’ written responses to one of the tasks to build hypotheses about how calculus students reason about variations within frames of reference. The third paper reports how U.S. and Korean secondary mathematics teachers reason with frame of reference on open-response items. An assessment with five frame of reference tasks was given to 539 teachers in the US and Korea, and the responses were coded with rubrics intended to categorize responses by the extent to which they demonstrated conceptualized and coordinated frames of reference. The results show that the theory in the first study is useful in analyzing teachers’ reasoning with frames of reference, and that the items and rubrics function as useful tools in investigating teachers’ meanings for quantities within a frame of reference. / Dissertation/Thesis / Doctoral Dissertation Mathematics 2019

Lesson study is a form of professional development for teachers adopted in recent years from Japan. Introducing lesson study to U.S. teachers and researchers has been the focus of most of the literature on this subject. Much of the literature outlines how lesson study works and describes its essential features. One of the features of lesson study is anticipating student responses, also known as anticipating student thinking. Anticipating student responses is passingly described in lesson study literature. This research was conducted to understand what it means to anticipate student responses for preservice mathematics teachers in a lesson study group. Lesson study literature indicates that anticipating student responses is to anticipate conceptual development from the students' perspective, and the purpose is to be prepared to have meaningful discussions and questions to enable students to develop the understanding. Anticipating student responses is highly related to the hypothetical learning trajectory described by Simon (1995), the self directed anticipative learning model described by Christensen and Hooker (2000) and the expert blind spot discussed by Nathan and Petrosino (2003). While their work does not stem from lesson study, they add theoretical perspective to the idea of anticipating student responses. Their work indicates that anticipating student responses is difficult, valuable, that one gets better at it through experience, and that it is very useful in refining lessons. Participants were enrolled in the mathematics education methods class of a large private university in the U.S. A characterization of anticipating student responses was developed as the participants met in group meetings to create a lesson. They anticipated student responses in ways that facilitated lesson planning and task design. Participants did not anticipate student responses toward students' conceptual development. This research reports five particular ways that anticipating student responses was used as a tool to define and refine the lesson so that it ran smoothly toward lesson goals. These ways are related to: goals, tasks and materials, procedural mathematical reasoning, successful student efforts, and emotional responses. It is believed that anticipating student responses towards task design is a necessary precursor to anticipating student responses toward students' conceptual development.

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