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## Improving Preservice Secondary Mathematics Teachers' Capability With Generic Example Proofs

Preservice mathematics teachers are entrusted with developing their future students' interest in and ability to do mathematics effectively. Various policy documents place an importance on being able to reason about and prove mathematical claims. However, it is not enough for these preservice teachers, and their future students, to have a narrow focus on only one type of proof (demonstration proof), as opposed to other forms of proof, such as generic example proofs or pictorial proofs. This article examines the effectiveness of a course on reasoning and proving on preservice teachers' awareness of and abilities to recognize and construct generic example proofs. The findings support assertions that such a course can and does change preservice teachers' capability with generic example proofs.

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## For ELLs: Vocabulary beyond the Definitions

A classroom teacher discusses ambiguities in mathematics vocabulary and strategies for ELL students in building understanding.

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## Counterexamples as Starting Points for Reasoning and Sense Making

Asked to “fix” a false conjecture, students combine their reasoning and observations about absolute value inequalities, signed numbers, and distance to write true mathematical statements.

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## Connecting Research to Teaching: Classroom Collaboration: Moving beyond Helping

Do you use group work in your mathematics class? What does it look like? What do you expect your students to do when they work together? Have you ever wondered what your students think they are supposed to do?

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## From the Editorial Panel: Fostering Flexible Mathematical Thinking

Understanding and making connections is at the heart of flexible mathematical thinking. Flexible thinking is generally observed in students' ability and willingness to work with multiple representations, but it can also be seen in their facility in adapting problem-solving strategies when faced with novel situations, reversing thought processes, viewing notation as a process as well as an object, interpreting others' ideas, and posing problems. Although teachers may characterize flexible mathematical thinking differently according to context, we can probably all agree that providing students with opportunities to engage in flexible mathematical thinking is vital for fostering the kind of mathematical understanding that we want them to have

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## Delving Deeper: Approximating π Using Similar Triangles

This article presents a method for approximating π using similar triangles that was inspired by the author's work with middle school teachers. The method relies on a repeated application of a geometric construction that allows us to inscribe regular polygons inside a unit circle with arbitrarily large number of sides.

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## Sound Off!: The Mathematics Teacher's Bookshelf

Books are an excellent source of information and enjoyment and should have a role in every mathematics teacher's professional development plan. This article encourages everyone to have a bookshelf, literal or virtual, and lists several of the author's favorites, noting what each has to offer the reader.

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## Sketching Up the Digital Duck

Two preservice teachers describe how using Google SketchUp, Terrapin Logo, and The Geometer's Sketchpad fosters a deeper understanding of measurement concepts.

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## Connecting Research to Teaching: Making Student Thinking Public

Teachers often have students publicly share their mathematical thinking as part of classroom instruction. Before reading further, we invite you to stop and think about this practice by writing down your responses to the following two questions:

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## Reflections on a Technology–Rich Mathematics Classroom

Integrating technology into the mathematics classroom means more than just new teaching tools—it is an opportunity to redefine what it means to teach and learn mathematics. Yet deciding when a particular form of technology may be appropriate for a specific mathematics topic can be difficult. Such decisions center on what is commonly being referred to as TPACK (Technological Pedagogical and Content Knowledge), the intersection of technology, pedagogy, and content (Niess 2005). Making decisions about technology use influences not only students' conceptual and procedural understandings of mathematics content but also the ways in which students think about and identify with the subject.