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## Description

Teachers' mathematical content knowledge is one of the most important constructs considered by researchers studying elementary mathematics education (Fennema & Franke, 1992). One component of mathematical content knowledge that is complicated, ill-defined, and oft-ignored is productive disposition, defined as the "habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one's own efficacy" (NRC, 2001, p. 116). In this dissertation I undertook two separate but related studies about the productive dispositions of K-3 elementary school teachers to better understand the construct. In Study 1, my overarching research question was What differences in evidence for productive disposition can be found through the analysis of teachers' engagement in a mathematical task? To answer this question, I assessed for differences and highlighted what might be taken as evidence for teachers' holding strong productive dispositions among 136 preservice and in-service K-3 teachers who worked on a challenging mathematical task in focus groups. Building from the mathematics education literature and the focus-group data, I created a list of potential productive-disposition indicators. In Study 2, my overarching question was What evidence for productive disposition is self-reported by teachers who have been identified as having strong productive dispositions? I observed 10 of the in-service elementary school teachers who had participated in Study 1 while they engaged in several mathematical tasks. These teachers were also asked to complete a mathematical autobiography and were interviewed individually. My purpose was to verify with the participants my interpretation of the indicators they exhibited when working on the mathematical tasks and to identify others not recognized in the analysis. Seven productive-disposition traits resonated with all 10 of the teachers, and 3 new traits arose from our discussions. I conclude by discussing why mathematics educators and mathematics teachers should care about productive dispositions.