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Elaine M. Purvinis and Joshua B. Fagan

In first- and second-year algebra classrooms, the all-too-familiar whine of “when are we ever going to use this in real life?” challenges mathematics teachers to find new, engaging ways to present mathematical concepts. The introduction of quadratic equations is typically modeled by describing the motion of a moving object with respect to time, and typical lessons include uninspiring textbook practice problems that portray dropping or shooting objects from given distances or at particular time intervals. For a novel approach to exploring quadratics, we chose to step outside the classroom to look at some phenomena in the field of acoustics. Our activity incorporates mathematical modeling to provide a multirepresentational view of the math behind the physics and to provide a conceptual basis for analyzing and understanding a real-world quadratic situation.

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Kathleen Melhuish, Eva Thanheiser and Joshua Fagan

In classrooms, students engage in argumentation through justifying and generalizing. However, these activities can be difficult for teachers to conceptualize and therefore promote in their classrooms. In this article, we present the Student Discourse Observation Tool (SDOT) developed to support teachers in noticing and promoting student justifying and generalizing. The SDOT serves the purpose of (a) focusing teacher noticing on student argumentation during classroom observations, and (b) promoting focused discussion of student discourse in teacher professional learning communities. We provide survey data illustrating that elementary-level teachers who participated in professional development leveraging the SDOT had richer conceptions of justifying and generalizing and greater ability to characterize students' justifying and generalizing when compared with a set of control teachers. We argue that the SDOT provides both an important focusing lens for teachers and a means to concretize the abstract mathematical activities of justifying and generalizing.