Despite over thirty years of theoretically based research investigating "how" teacher mathematical knowledge and instructional practice relate to student learning, it is still largely unclear how these constructs are related, and policy makers and practitioners are still situated in a context with insufficient data to make decisions. Thus, there is a need for further research that examines the credibility of such theories and understands the mechanisms behind "how" teacher knowledge (MKT) influences the mathematics instructional quality (MIQ) and promotes student achievement, particularly in large samples of teachers and children. In addition, although much work has been dedicated to testing the efficacy of social emotional learning (SEL) interventions for promoting academic growth, there is very little research that integrates SEL and math. This study has two goals. First, this study aims to validate existing theoretical frameworks and question and/or replicate initial findings about how mathematical knowledge for teaching, instructional practices, and student learning are related. Specifically, this study examines the direct and indirect relations of these constructs, as guided by the following four questions: (a) What is the relation between higher MKT and improved student achievement?; (b) What is the relation between higher MKT and higher quality of instruction?; (c) What is the relation between higher quality instruction and improved student learning?; and (d) Is MKT indirectly related to student mathematics achievement through instructional quality? To the authors' knowledge, no large quantitative study to date has examined how teacher knowledge and MIQ collectively contribute to student achievement, or have tested the potential role of mathematics instructional quality as a mediator between teacher knowledge and student achievement. This has largely been due to the complexity of reliably measuring the quality of instruction in the math classroom (Kersting, et al., 2010). This study addresses these gaps by using multiple methods (i.e. teacher-report, direct assessment, and classroom observations), which together can provide important information about the complex relations between these constructs. Second, this study examines the extent to which the Responsive Classroom (RC) approach strengthens the relations between MKT, MIQ, and achievement (compared to teachers delivering "business as usual" instruction). By looking at intervention and control schools simultaneously, this study is able to better understand the processes by which teacher knowledge and SEL interventions can facilitate higher quality instruction and student outcomes. (Contains 1 table and 1 figure.)