Theorizing and Measuring Mathematics Teaching at Scale

• In Event: Measuring and Supporting the Improvement of Common Core State Standards–Aligned Mathematics Teaching at Scale

Abstract

The purpose of this paper is to provide early evidence of the viability of a theory of mathematics teaching and how it relates to student learning. In past standards eras, researchers often relied on “reform” or “student-centered” instruction as a way of theorizing the kind of teaching needed to achieve desired student outcomes. This approach ultimately proved limiting because it “lumped together features of teaching in ill-defined ways” (Hiebert & Grouws, 2007). Scholars of teaching (e.g., Stigler & Hiebert, 1999; Schoenfeld, 1998; Simon et al, 2004) now agree that instruction is a system of interacting factors. The impact of any one feature of teaching (e.g., higher order questioning) is dependent on its interaction with other features (i.e., the cognitive demand of the task on which students are working) and the overall system within which it is embedded (developed norms for student discussion).
We have placed our bets on two complex (yet measurable) dimensions of teaching that early evidence suggests support students’ development of CCSS-aligned proficiencies: explicit attention to concepts and opportunities for students to struggle (Hiebert & Grouws, 2007). The interaction of high and low levels of each of these dimensions form four “quadrants.”

In Quadrant 1 (high concept/high struggle), students are provided with open-ended tasks for which there is not a predictable, well-rehearsed approach or pathway to solve the task; students have to exert considerable cognitive effort to solve the task. In the process of grappling with the task—and/or as a wrap-up to the task—ties between the student-generated ideas/methods and the concepts embedded in the task are explicitly addressed. Quadrant 2’s (high concept/low struggle) distinguishing feature is explicit attention to concepts. Robust understanding of mathematical concepts can be built in a variety of ways (e.g., developing definitions across multiple examples/non examples; relating concept to procedures). However, students don’t have to struggle to figure out how to solve an open, challenging task on their own. Quadrant 3 (high struggle/low concept) refers to classrooms in which students are provided with open-ended tasks for which they do not have an immediate strategy for solving and they struggle, but not in a productive way. Lacking teacher guidance toward the mathematical ideas embedded in the task and/or strategic scaffolding, students do not engage with the mathematical concepts. Finally, Quadrant 4 (low concept/low struggle) represents a type of instruction in which there is both low tolerance for student struggle and limited attention to concepts (e.g., worksheet-driven instruction in which the teacher demonstrates the procedure to use and then students do similar problems using that exact procedure).
After describing the theory, this paper will provide evidence that teachers’ survey responses to beliefs and frequency-of-teaching-behaviors items (n = 453 teachers) can be used to assign teachers into each of these quadrants and that this assignment aligns with quadrant placement based on videotapes (n = 49 teachers). In addition, we will provide evidence of consistency in how teachers reported their instructional practices in different parts of the survey.

Authors

• Mary Kay Stein, University of Pittsburgh

• Richard James Correnti, University of Pittsburgh

• Jennifer L. Russell, University of Pittsburgh

• Katelynn Kelly