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John P. Smith III

Take the mathematics that you find back to school. With a bit of adaptation, mathematical work can generate real–life problems and projects for your students.

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John P. Smith III

Some years ago, Gila Hanna offered the very insightful and useful distinction between mathematical proofs that prove and those that also explain (Hanna, 1989). Proofs that explain not only state the deductive logic that justifies their mathematical claims, they lay bare the mechanisms and structures that underlie that logic, making content and logic more properly align. In so doing, they more effectively teach mathematics than do proofs that only prove.

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John P. Smith III

This paper analyzes the tension between the traditional foundation of efficacy in teaching mathematics and current reform efforts in mathematics education. Drawing substantially on their experiences in learning mathematics, many teachers are disposed to teach mathematics by “telling”: by stating facts and demonstrating procedures to their students. Clear and accurate telling provides a foundation for teachers' sense of efficacy—the belief that they can affect student learning—because the direct demonstration of mathematics is taken to be necessary for student learning. A strong sense of efficacy supports teachers' efforts to face difficult challenges and persist in the face of adversity. But current reforms that de-emphasize telling and focus on enabling students' mathematical activity undermine this basis of efficacy. For the current reform to generate deep and lasting changes, teachers must find new foundations for building durable efficacy beliefs that are consistent with reform-based teaching practices. Although productive new “moorings” for efficacy exist, research has not examined how practicing teachers' sense of efficacy shifts as they attempt to align their practice with reform principles. Suggestions for research to chart the development of, and change in, mathematics teachers' sense of efficacy are presented.

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John P. Smith III and Jon R. Star

Research on the impact of Standards-based, K–12 mathematics programs (i.e., written curricula and associated teaching practices) and of reform calculus programs has focused primarily on student achievement and secondarily, and rather ineffectively, on student attitudes. This research has shown that reform programs have competed well with traditional programs in terms of student achievement. Results for attitude change have been much less conclusive because of conceptual and methodological problems. We critically review this literature to argue for broader conceptions of impact that target new dimensions of program effect and examine interactions between dimensions. We also briefly present the conceptualization, design, and broad results of one study, the Mathematical Transitions Project (MTP), which expanded the range of impact along those lines. The MTP results reveal substantial diversity in students' experience within and between research sites, different patterns of experience between high school and university students, and surprising relationships between achievement and attitude for some students.

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Leslie C. Dietiker, Funda Gonulates and John P. Smith III

Push your instruction beyond procedures: Enhance student tasks and offer better opportunities to develop conceptual understanding.

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John P. Smith III and Elizabeth A. Phillips

NO PART OF THE K–12 MATHEMATICS curriculum is more fluid and controversial than introductory algebra. Content and assessment issues lie at the core of this debate: What algebra skills and understandings are important? What kind of evidence suggests that students possess these skills? Neither question can be answered in simple terms; in fact, no single “right” answer may exist for either one.

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Jon R. Star, John P. Smith III and Amanda Jansen

Research on the impact of Standards-based mathematics and reform calculus curricula has largely focused on changes in achievement and attitudes, generally ignoring how students experience these new programs. This study was designed to address that deficit. As part of a larger effort to characterize students' transitions into and out of reform programs, we analyzed how 93 high school and college students perceived Standards-based and reform calculus programs as different from traditional ones. Results show considerable diversity across and even within sites. Nearly all students reported differences, but high-impact differences, like Content, were not always related to curriculum type (reform or traditional). Students' perceptions aligned moderately well with those of reform curriculum authors, e.g., concerning Typical Problems. These results show that students' responses to reform programs can be quite diverse and only partially aligned with adults' views.

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Jon R. Star, Beth A. Herbel-Eisenmann and John P. Smith III

New mathematics curricula serve middle grades students well when they provide students with richer and more accessible introductions to a wide range of mathematical content. New curricula also serve teachers well when they lead us to examine and reflect on what and how we teach. When these curricula enter our working lives and conversations, we are often forced to question exactly what is “new” about them and how this “newness” may affect our students' learning. To address this issue and, we hope, to support further reflection and discussion, we take a closer and more careful look at what is new in one middle school curriculum's approach to algebra. The curriculum we examine is the Connected Mathematics Project (CMP) (Lappan et al. 1998), particularly the eighth-grade units, but the issue of what is new in algebra is relevant to many other innovative middle school curricula, as well.