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Abstract
Poor performance of U.S. middle grades mathematics teachers and their students in international comparison studies and their persistent difficulties with reasoning about fraction arithmetic when fractions are situated in problem situations indicate an urgent need to improve teacher education in the U.S. Therefore, the focus of this dissertation research was to develop methods to study fine-grained components of reasoning about fraction arithmetic and to study the utility of those methods for tracking growth and change in reasoning in teacher preparation programs. In particular, this dissertation research consists of two studies. In the first study, I analyzed responses of a nationwide sample of 990 in-service middle grades mathematics teachers to a novel survey of fraction arithmetic using a contemporary psychometric model called the mixture Rasch model. The results of the first study revealed the existence of three distinct latent classes, and I interpreted those classes in terms of characteristic strengths and weaknesses based on reasoning about fraction arithmetic. In the second study, I administered the same survey to a cohort of 18 future middle grades mathematics teachers in one teacher preparation program before and after 20 weeks of instruction on multiplication, division, and fractions. I used the item parameters from the first study to select a subsample of 8 future teachers from those 18 future teachers. I then studied these 8 future teachers reasoning by interviewing them before and after the instruction to determine if their reasoning was, in fact, consistent with my interpretation of the latent classes in the first study. This study provides evidence for the validity of my interpretation of the three latent classes and demonstrates new ways in which psychometric models can be harnessed both to survey teachers reasoning at scale and to document growth and change in reasoning during teacher education coursework. Such steps are promising in that they may reveal systematic differences in reasoning about fraction arithmetic, help future teachers develop the reasoning needed to meet the demands of recent curriculum standards, and extend existing studies with small samples of teachers to larger samples for obtaining generalizable results and for improving teacher education at scale.