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This study illuminates claims that teachers' mathematical knowledge plays an important role in their teaching of this subject matter. In particular, we focus on teachers' mathematical knowledge for teaching (MKT), which includes both the mathematical knowledge that is common to individuals working in diverse professions and the mathematical knowledge that is specialized to teaching. We use a series of five case studies and associated quantitative data to detail how MKT is associated with the mathematical quality of instruction. Although there is a significant, strong, and positive association between levels of MKT and the mathematical quality of instruction, we also find that there are a number of important factors that mediate this relationship, either supporting or hindering teachers' use of knowledge in practice. [ABSTRACT FROM AUTHOR]

The article completes the summative stage of the validity argument approach. The interferences and assumptions of the interpretive argument for the Mathematical Knowledge for Teaching (MKT) measures is evaluated. The form and the structure of the interpretive argument for the MKT measures is examined with an eye to generalizations that can be made to other efforts to construct interpretive arguments.