school organizational structures, school leadership characteristics, the nature and organization of teachers’ work, and the social matrix in which the school is embedded. These matter principally as they permeate instruction—that is, whether and how they enter into the interactions among teachers, students, and content.2 Hence, what goes on in classrooms to promote the development of mathematical proficiency is best understood through an examination of how these elements—teachers, students, content—interact in contexts to produce teaching and learning.
Much debate centers on forms and approaches to teaching: “direct instruction” versus “inquiry,” “teacher centered” versus “student centered,” “traditional” versus “reform.” These labels make rhetorical distinctions that often miss the point regarding the quality of instruction. Our review of the research makes plain that the effectiveness of mathematics teaching and learning does not rest in simple labels. Rather, the quality of instruction is a function of teachers’ knowledge and use of mathematical content, teachers’ attention to and handling of students, and students’ engagement in and use of mathematical tasks. Moreover, effective teaching—teaching that fosters the development of mathematical proficiency over time—can take a variety of forms. To highlight this point, we use excerpts from four classroom lessons and analyze what we see going on in them in light of what we know from research on teaching.
The pedagogical challenge for teachers is to manage instruction in ways that help particular students develop mathematical proficiency. High-quality instruction, in whatever form it comes, focuses on important mathematical content, represented and developed with integrity. It takes sensitive account of students’ current knowledge and ways of thinking as well as ways in which those develop. Such instruction is effective with a range of students and over time develops the knowledge, skills, abilities, and inclinations that we term mathematical proficiency.
The four classroom vignettes we present below offer four distinct images of what mathematics instruction can look like. Each vignette configures differently the mathematical content and the roles and work of teachers and students in contexts; hence, each produces different opportunities for mathematics teaching and learning. Two points are important to interpreting and using these vignettes. First, to provide a close view, each vignette zooms in on an individual lesson. Effective instruction, however, depends on the