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## Responsiveness to Students’ Mathematical Thinking in Middle-Grades Classrooms

. 46). In this spirit, we define responsiveness to studentsmathematical thinking 1 as a characteristic of discourse that reflects the extent to which students’ mathematical ideas are present, valued, attended to, and taken up as the basis for

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## Do You See What I See? Formative Assessment of Preservice Teachers’ Noticing of Students’ Mathematical Thinking

title rather than building on the studentsmathematical thinking to reach the goal of the lesson. Erica, on the other hand, spent the debriefing discussing the instructor’s classroom management moves and the behavior of students in the class . Dr

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## Focusing on Students' Mathematical Thinking

When viewing videotaped examples of his classroom teaching, Anthony, a veteran ninth-grade teacher, was surprised that he focused more on the students' responses than on the students' thinking. For example, he realized that he was not asking questions to understand what or how the students were thinking but rather to test their knowledge.

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## Lenses for Examining Students' Mathematical Thinking

A few years ago, a colleague shared a video from a first-year algebra class that he had observed. The video captured a class discussion about slopes of horizontal and vertical lines. At the beginning of the discussion, the teacher, Ms. Milner, asks, “How can we have a slope of zero?” Students respond in various ways; one student, Peter, explains that a horizontal line would have a slope of zero “because it's never moving up.” Later, Milner asks about the slope of a vertical line, and another student, Alex, replies that because “it went … up and down and didn't move at all, it would be zero.” Milner then asks the class about the slope of a line through the points (0, 0) and (0, 5). Peter says that “the slope is zero, because you subtract the change in x and the change in y. On the top there'd be zero and on the bottom there'd be 5, and any division problem that has zero in it has to be zero.” Finally, Rafael disagrees and suggests that the slope is undefined because “in division there can't be a number over zero.”

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## The “MOST” Productive Student Mathematical Thinking

Identify student thinking that has potential to support significant mathematical discussion and pedagogical opportunity.

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## On Patterns, Conjectures, and Proof: Developing Students' Mathematical Thinking

A useful strategy in solving problems is “look for a pattern.” In using the strategy, we start with simple cases or versions of the problem and from these cases discover a pattern or rule that can be applied to find the general solution.

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## Using Portfolio Assignments to Assess Students' Mathematical Thinking

A model in use at Prospect Hill Academy in Cambridge, Massachusetts, provides a skills-based portfolio assessment for mathematics classes.

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## Conceptualizing Mathematically Significant Pedagogical Opportunities to Build on Student Thinking

The mathematics education community values using student thinking to develop mathematical concepts, but the nuances of this practice are not clearly understood. We conceptualize an important group of instances in classroom lessons that occur at the intersection of student thinking, significant mathematics, and pedagogical opportunities—what we call Mathematically Significant Pedagogical Opportunities to Build on Student Thinking. We analyze dialogue to illustrate a process for determining whether a classroom instance offers such an opportunity and to demonstrate the usefulness of the construct in examining classroom discourse. This construct contributes to research and professional development related to teachers' mathematically productive use of student thinking by providing a lens and generating a common language for recognizing and agreeing on a critical core of student mathematical thinking that researchers can attend to as they study classroom practice and that teachers can aspire to notice and build upon when it occurs in their classrooms.

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## Supporting PSTs in Using Appropriate Tools Strategically: A Learning Sequence for Developing Technology Tasks That Support Students' Mathematical Thinking

This article describes an intervention with preservice mathematics teachers intended to address the use of Interactive Geometry Software (IGS) for mathematics instruction. A unit of instruction was developed to support teachers in developing mathematical tasks that use IGS to support students' high-level thinking (Smith & Stein, 1998). Preservice teachers used the IGS Framework (Sherman & Cayton, 2015) to evaluate 3 tasks, to revise a task, and ultimately to design a task using the framework. Results indicate that a majority of preservice teachers in this study were successful in creating a high-level task where IGS was instrumental to the thinking demands, and that the IGS Framework supported them in doing so. The article concludes with suggestions for use by fellow mathematics teacher educators.

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## Understanding the Impact of Racial Attitudes on Preservice Teachers’ Perceptions of Children’s Mathematical Thinking

., 2017 ; Stockero et al., 2017 ). Across all this work, a robust literature has developed around studentsmathematical thinking, what teachers notice about this thinking, how they evaluate it, and how they use it to make instructional decisions