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Despina A. Stylianou

Representation lies at the heart of doing mathematics. It is the activity of creating and using mathematical symbols, signs, and diagrams. NCTM (2000) advocates that students be fluent users of representations and that instruction should support students in learning to navigate mathematical concepts and problem solving through the use of a variety of representations.

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Rachel Lambert and Despina A. Stylianou

One middle school teacher developed classroom routines to make challenging questions accessible to all learners in her class.

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Despina A. Stylianou and Maria L. Blanton

Proof is central to doing mathematics. Yet proving is challenging for most students. In this article we describe the findings of a yearlong teaching experiment that focused on developing proof. We discuss the role of classroom discourse in developing argumentation practices that lead proof using examples from our work. The teacher's role in managing and promoting this discourse is explicitly discussed.

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Despina A. Stylianou, Patricia Ann Kenney, Edward A. Silver, and Cengiz Alacaci

The task in Figure 1A, which was given to middle school students, asked them to find the number of dots in the twentieth step of the pattern. As shown in figure 1b, the answers that students gave ranged from 20 dots to well over 60 million dots. If your students gave these answers without providing work or explanations, would you be able to tell how they obtained them? Probably not. Without looking closely at the students' work or explanations or talking to them about their solution strategies, it is difficult to understand how they were thinking about the pattern task and how their thinking could have produced such a wide range of answers.

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Kara Louise Imm, Despina A. Stylianou, and Nabin Chae

The NCTM's Standards (2000) suggest that a representation is not only a product (a picture, a graph, a number, or a symbolic expression) but also a process, a vehicle for developing an understanding of a mathematical concept and communicating about mathematics. To serve as a vehicle in learning and communication, however, a representation must be personally relevant and meaningful to a student. Therefore, when choosing a representation to explore with a group of students or when reviewing student work, we ought to consider everything that students bring to the classroom. Even at a young age, students come to school with their own, often culturally influenced, valid representations (Lave 1998). Because those representations have been crafted, interpreted, and modified by the students themselves, they become vital to classroom instruction. To dismiss what students bring naturally to the classroom reduces mathematics to a one-way transaction between teacher as expert and student as novice, confirming the notion that a student's own thinking and all that he or she brings to mathematics is marginal at best. By relocating student-generated representations to the center of the instruction, the nature of how students experience mathematics changes dramatically. It reconsiders mathematics as a vibrant dialogue among different but equally valued thinkers. This deliberate approach to the teaching of mathematics, we believe, becomes vital if we are serious about creating greater equity for our students.