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Dawn Teuscher and Robert E. Reys

How do we mathematics teachers introduce the concepts of slope, rate of change, and steepness in our classrooms? Do students understand these concepts as interchangeable or regard them as three different ideas? Here we report the results of a study of high school Advanced Placement (AP) Calculus students who displayed misunderstandings about the meaning of these three concepts. We then share an example of a typical rate-of-change problem commonly found in mathematics textbooks that may contribute to the development of these misconceptions. Finally, we provide a classroom task that supports the discussion of the differences among these three concepts, and we ask readers to reflect on ways to help students differentiate among them.

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Dawn Teuscher, Kylie Palsky and Charlie Y. Palfreyman

An “undoing” process can improve students' conceptual understanding.

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Dawn Teuscher, J. Matt Switzer and Tyler Morwood

Researchers have called on teacher educators to break down complex teaching practices to assist preservice teachers in learning these practices. In this article, we unpack the practice of probing student thinking while providing evidence that as sophisticated users of various teaching practices, mathematics teacher educators may be unaware of preservice teachers' varied and naïve images and understandings of teaching practices.

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Dung Tran, Barbara J. Reys, Dawn Teuscher, Shannon Dingman and Lisa Kasmer

This commentary highlights the contribution that careful and systematic analyses of curriculum or content standards can make to questions and issues important in the mathematics education field. We note the increased role that curriculum standards have played as part of a standards-based education reform strategy. We also review different methods used by researchers to compare and analyze the Common Core State Standards for Mathematics, each method designed for a particular purpose. Finally, we call upon mathematics education researchers to engage in careful analysis of curriculum standards and to share their findings in ways that can inform public debate as well as support education professionals in improving student learning opportunities.