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We argue that large-scale, standards-based improvements in the teaching and learning of mathematics necessitate advances in our theories regarding how teaching affects student learning and progress in how we measure instruction. Our theory—an embodiment of the interaction of high and low levels of two constructs that past research has shown to influence students’ development of conceptual understanding (explicit attention to concepts and students’ opportunity to struggle)—guided the development of survey-, video-, and artifact-based measures of teaching. Here, we develop a validity argument for the inferences that can be drawn about teaching from these measures by identifying claims and empirical evidence about the extent to which those claims are born out in practice. Results suggest our theory is capturing four patterns of teaching and that it can successfully predict different types of student learning: skills efficiency measured on the state standardized test and conceptual understanding as measured through open-ended task sets

Using Theory and Measurement to Sharpen Conceptualizations of Mathematics Teaching in the Common Core Era