Two different tools for assessing pedagogical content knowledge (PCK) of mathematics teachers used in the framework of the COACTIV study are systematically compared in this paper, namely the paper-and-pencil test consisting of items on the three facets "knowledge of explaining and representation," "knowledge of student thinking and typical mistakes," and "knowledge of the potential of mathematical tasks," and the video vignettes instrument that examines teachers' "proposed continuations for presented lesson video clips" specific to their subject-related and methodological competence aspects. Initially, both COACTIV PCK assessment tools are systematically contrasted for the first time with respect to their predictive validity for instructional quality (N = 163 German secondary mathematics teachers) as well as student learning gains (N = 3806 PISA students from 169 different classes) by means of path models showing that PCK, when assessed by the paper-and-pencil method, can better predict instructional quality than the video vignettes instrument can. Next, we theoretically propose the "cascade model" as capable of integrating pertinent theories on teacher competence and instructional quality. This model implies five 'columns' that are ordered according to a "sequential causal chain" (teacher disposition [arrow to the right] situation-specific skills [arrow to the right] observable teaching behavior [arrow to the right] student mediation [arrow to the right] learning gains). Finally, we specify four out of the five 'columns' of this cascade model, based empirically on the COACTIV data.