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James Russo

Postscript items are designed as rich grab-and-go resources that any teacher can quickly incorporate into his or her classroom repertoire with little effort and maximum impact. Start the school year right: Build on students' love for doubling by introducing exponential growth with this strategic bingo game.

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Marilyn Howard

The less you know about the patterns in Pascal's triangle, the more fun you will have discovering the triangle's many secrets. I am amazed at how few students and even teachers (especially at the middle school level) have ever explored Pascal's triangle. Although this famous triangle bears the name of Blaise Pascal (1623-1662), who saw many of the patterns when he was only thirteen years old, it had been around for centuries before he was born. See the ancient diagram in figure 1, which appeared at the front of a Chinese book in 1303 (Vakil 2008). Evidence suggests that the properties of the elements of Pascal's triangle were known before the common era. Students and teachers alike can enjoy exploring patterns through problem solving with Pascal's triangle.

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Chepina Rumsey and Cynthia W. Langrall

These evidence-based instructional strategies can lead to deeper mathematical conversations in upper elementary school classrooms.

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Walter Stark

Students must actively engage in exploring math. That is why I am always looking for tasks that will allow my students to to explore problems using the Common Core's (CCSSI 2010) eight Standards for Mathematical Practice (SMP). These standards are vital for developing a deepening understanding of math. They allow students to cultivate skills and thought processes that aid in wiring their brains into being deep thinkers and problem solvers. These skills transcend the classroom and are needed to be successful in the world. I also want my students to revisit ideas that we have already touched on and continue to examine.

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Sandra Davis Trowell

Edited by Denise Taunton Reid

The Teaching and Learning principle in Principles to Actions: Ensuring Mathematical Success for All (NCTM 2014) states,

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Zoe Donoahue

Observe how first graders can learn to routinely theorize and explain their thinking about odd and even numbers, factors, multiplication, and skip counting.

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Debra Rawlins, Natasha Hernandez, and William Miller

Second graders move from counting by ones to counting equal groups to structuring arrays.

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James A. Russo

This game-based activity prompts students to explore the structure of multiplication, experiment with the distributive property, and begin investigating prime numbers.

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Shiv Karunakaran, Ben Freeburn, Nursen Konuk, and Fran Arbaugh

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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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.