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To make the complex work of teaching doable and learnable for teachers, Grossman and her colleagues (Grossman et al., 2009) have addressed the need to decompose the work into its constituent components. Although the idea of developing technical language for describing the “implicit grammar of teaching” is appealing, a robust definition of decompositions has not established yet. Drawing on the analysis of the work of supporting students to develop mathematical explanation, this paper advances three features of decompositions of teaching: (1) distinguish between core tasks of teaching and resources for carrying out that work; (2) combine dilemma-based and competence-based characterizations; and (3) use underlying theory as a structure and evidentiary basis for the proposed decomposition.