Existing research indicates inconsistent or at best weak predictive effects of teacher knowledge on student achievement. Data from Germany were used to examine the relation between teachers' content and pedagogical content knowledge, their perception, interpretation, and decision-making skills, the instructional quality implemented in class, and students' learning progression in mathematics. Rather than direct effects of teacher knowledge on students, we hypothesized an effect chain with multiple mediation processes while controlling for school type and student background. Multi-level modeling with 3496 students from 154 classrooms revealed a mediating role of teachers' skills and their instructional quality for the relation between teacher knowledge and students' learning progress. Effect sizes were medium to strong, and the model explained a large amount of variance. No direct effects of teachers' knowledge on student progress were found. We discuss our findings with respect to the teacher-competence-as-a-continuum model and with respect to future research.