Understanding factors that contribute to the variation in student mathematics achievement has been a critical issue to researchers, teachers, and policy in national efforts to improve mathematics achievement. Teachers are largely responsible for students' mathematics learning (Hiebert & Grouws, 2007; Nye, Konstantopolous, & Hedges, 2004; Rowan, Correnti, & Miller, 2002). However, there is a need for further research examining how teachers influence student achievement through the quality of the instructional opportunities they provide. Existing work directs attention toward mathematical knowledge for teaching (MKT) and mathematics instructional quality (MIQ) as critical factors predicting student achievement (Fennema & Franke, 1992; Hill et al., 2004; 2008; US Department of Education, 2008). Despite theoretical frameworks suggesting positive links between MKT, MIQ, and achievement, it is still largely unclear how these constructs are related. Contradictory evidence linking MKT to MIQ and achievement encourages us to press more deeply about how we can better support teachers' ability to translate their MKT into high quality classroom practice.

            Social emotional learning (SEL) interventions aimed at helping teachers create optimal classroom learning environments that enhance children's ability to learn effectively provide promise for improving the quality of instruction teachers provide. One such intervention, the Responsive Classroom¨ (RC) Approach (NEFC, 2003) is consistent with many of the NCTM (2000) process standards for creating high quality mathematical learning experiences for children and has early evidence demonstrating improved classroom quality and gains in mathematics achievement (xxx, 2007). However, there is a need to understand whether RC can help teachers translate their MKT into higher quality instruction.             This study aims validate existing theoretical frameworks and question and/or replicate initial findings about how MKT, 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 achievement through MIQ? To our knowledge, no large quantitative study to date has examined how MKT and MIQ collectively contribute to student achievement, or have tested the potential role of MIQ as a mediator between MKT and student achievement. This has largely been due to the complexity of reliably measuring the quality of instruction in the math classroom. This study addresses these gaps by using multiple methods to provide important information about the complex relations between these constructs. Second, this study examines the extent to which the RC approach strengthens the relations between MKT, MIQ, and achievement.

            The present study is part of a larger three-year longitudinal cluster randomized controlled trial examining the impact of the RC approach on classroom quality and student achievement. Twenty-four schools in a mid-Atlantic school district were matched demographically and assigned randomly into intervention (n=13) and control (n=11) schools. Participants in the present study include 88 third grade teachers and their 1,533 students from the 24 schools. Half (n = 44) of the teachers taught in schools receiving the RC intervention. 50.8% of the students were male, 41.5% were Caucasian, 43.1% were English Language Learners, and 32.4% of the students received FRL. Data were collected from three sources: student achievement tests, online questionnaires, and classroom observations. Students' math achievement was assessed in both 2^nd and 3^rd grade, using the Stanford-10 and the state standardized math assessment. Demographic information about the students was also collected at this time by the school administration. Teachers completed the online MKT assessment (Hill et al., 2004) (13 items, µ=0.84). In addition, research assistants videotaped all 88 third grade teachers for three mathematics lessons. Upon the completion, tapes were sent to the laboratory for off-site observational coding using two different measures: Classroom Practices Observational Measure (CPOM): (Abry, et al., 2010); and the Mathematics Scan (M-Scan) (Berry, et al., under review). The M-Scan is a new classroom observational measure of MIQ. Reliability and validity work has been conducted supporting its utility and will be described in more detail in the paper (µ=0.93). A composite score was created for each teacher representing his or her average MIQ.

            Two level multi-group path analysis techniques were used to acknowledge the nesting of children in classrooms, simultaneously test the direct and indirect effects of each variable for predicting student achievement for each group separately, and compare path coefficients (directionality and strength) across the two groups of teachers.

            Two major findings emerged from this study. First, this study provides empirical evidence supporting two of the three direct links, and initial evidence for an indirect link between MKT, MIQ, and student achievement; however, such findings were only evident in the RC group. On average, RC teachers who scored one point higher on the MKT showed a 0.42 point gain on MIQ (roughly 1/3 of a standard deviation). Further, teachers who integrated more RC practices into their teaching provided higher quality mathematics instruction (p<0.05). In addition, students who had RC teachers who scored one point higher on the M-Scan performed, on average, 5.95 points higher on their mathematics assessment (p=0.04). MKT was not directly related to student achievement in the intervention or control groups (p>0.05), but was indirectly related in the RC group. Second, group differences suggest that training in the RC Approach strengthens the relations among MKT, MIQ, and student achievement.
            A current push in policy and teacher education is to improve teachers' content knowledge and effectiveness (U.S Department of Education, 2008; 2010). However, solely focusing on increasing the amount of MKT teachers hold without helping teachers understand how to translate their knowledge and create classroom environments that foster higher quality instructional practices falls short in efforts to improve teacher quality or raise student achievement. SEL interventions (such as RC) may provide teachers with the necessary organizational and pedagogical skills that allow them to effectively translate their knowledge into high quality practices and support student learning. Findings will be discussed in relation to potential methods that enhance the translation of content knowledge to mathematics instructional quality and achievement. Further, the paper describes limitations of the research pertaining to statistical power and sampling.