Teachers' knowledge of mathematical content and children's mathematical thinking have been identified as critical elements related to teachers' ability to effectively teach mathematics (Fennema & Franke, 1992; Kazemi & Franke, 2001; Ma, 1999; Peterson, Carpenter, & Fennema, 1989). Literature on teachers' knowledge suggests that teachers need not only to hold a deep understanding of the mathematics they teach, but also have detailed knowledge of the common correct and incorrect mathematical conceptions their children and research-based knowledge of the progression of the development of children's mathematical ideas. Research-based learning trajectories and learning progressions of children's mathematical development represent a potentially valuable resource for teachers to help make sense of children's thinking, and research on teachers' use and understand of such materials is much needed. This study investigated teachers' understanding and use of the Cognition Based Assessment (CBA) learning progression materials for multiplication and division concepts. The data sources included structured clinical interviews based around teachers' use of the CBA materials in analyzing written student work episodes, determining learning goals based on student work, and proposing instructional plans to help students progress in their mathematical understanding. Additionally, three case studies of teachers using the CBA materials in live one-on-one teaching and assessment situations are investigated. Analysis of teachers' use and understanding of the CBA materials consisted of grounded theorizing and retrospective analysis, involving iterations of hypothesis/conjecture formation based on data, hypothesis/conjecture testing, and hypothesis/conjecture revision. The study had several key findings. First, as complexity of either the CBA level, or a student's mathematical reasoning within the CBA framework increased, so did the inconsistency and variation in teachers' interpretations of the student thinking. This finding led to the conceptualization of teachers' "consistent," "partially consistent," and "inconsistent" use of CBA materials. A second important finding was that CBA language and research-based descriptions of children's thinking were occasionally quite challenging for teachers to interpret or understand. The terminology and conceptual ideas embedded in the CBA multiplication and division framework occasionally involved mathematical or technical descriptions that led to inconsistent teacher conceptualizations and misinterpretations of the CBA framework. A third important finding was that CBA consistent conceptualizations of student thinking within the CBA framework were frequently related to learning goals and instructional plans that were more informed by children's thinking. This relates to similar findings by the research related to Cognitively Guided Instruction (CGI) (Fennema et al., 1996; Franke, Carpenter, Levi, & Fennema, 200; Kazemi & Franke, 2001; Peterson, Carpenter, & Fennema, 1989). Overall, the data demonstrated that the CBA multiplication and division learning progression materials could become a powerful framework for teachers to analyze their own instruction and assessment of children's mathematical thinking. [The dissertation citations contained here are published with the permission of ProQuest LLC. Further reproduction is prohibited without permission. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Web page: http://www.proquest.com/en-US/products/dissertations/individuals.shtml.]