This paper provides a summary of the authors' attempts to uncover links between their measures, classroom mathematics instruction, and student learning. This paper also provides evidence regarding one central critique of their measures: that multiple-choice assessments cannot validly represent the knowledge, skills, and judgment involved in actual teaching practice. To the extent that there is a relationship between the test domain and practice domain, it strengthens their claim that they can measure knowledge for teaching in this format. In this article the authors describe the aims, methods, and preliminary findings from a large-scale statistical study of teacher knowledge and student achievement and a videotape study of teachers' classroom mathematics instruction. If a teacher knowledge/student achievement relationship is found in the former, the latter will help identify the mechanisms through which this relationship is realized. Finally, as a means for informing the debate about why test validation is often better conceptualized than practiced, they consider the cost and obstacles to this type of work. (Contains 2 tables, 1 figure and 1 footnote.)