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As a core practice, teacher noticing of students' mathematical thinking is foundational to other teaching practices. Yet, this practice is difficult for preservice teachers (PSTs), particularly the component of interpreting students' thinking (e.g., Teuscher et al., 2017). We report on a study of our design of a specific approximation of teacher noticing task with the overarching goal of conceptualizing how to design approximations of practice that support PSTs' learning to notice student thinking in technology-mediated environments with a specific focus on interpreting students' mathematical thinking. Drawing on Grossman et al.'s (2009) Framework for Teaching Practice (i.e., pedagogies of practice), we provided decomposed opportunities for PSTs to engage with the practice of teacher noticing. We analyzed how our design choices led to different evidence of the PSTs' interpretations through professional development design study methods. Findings indicate that the PSTs frequently interpret what students understood. Yet, they were more challenged by interpreting what students did not yet understand. Furthermore, we found that providing lesson goals and asking the PSTs to respond to a prompt of deciding how to respond had the potential to elicit PSTs' interpretations of what the students did not yet understand. The study highlights the interplay between task design, prompt wording, and PSTs' interpretations, which emphasizes the complexity of developing teacher noticing 

This study examines secondary mathematics teachers’ anticipations of student responses related to a series of cognitively demanding mathematics tasks from multiple mathematical domains presented in the context of voluntary and asynchronous online professional development modules. We analyze 283 anticipations made by 127 teachers to 17 mathematics tasks and present four distinct foci of teachers’ anticipations. Teachers focused on actions students might take, ways they might think about the task, how they might react emotionally, and what actions they might take in advance or in response to their anticipations. We conclude with a discussion of ways our results can inform efforts to support improvements in mathematics teachers’ practice of anticipating. 

Abstract Middle school students (n = 144) worked with an applet specially designed to introduce the concept of function without using algebraic representations. The purpose of the study was to examine whether the applet would help students understand function as a relationship between a set of inputs and a set of outputs and to begin to develop a definition of function based on that relationship. Results indicate that, by focusing on consistency of the outputs, the students, at a rate of approximately 80%, are able to distinguish functions from nonfunctions. Also, students showed some promise in recognizing constant functions as functions, a known area of common misconceptions. Students' main conceptual difficulty, likely caused by the context, was accepting nonintuitive outputs even if those outputs were consistent. 

Abstract The practice of teacher noticing students' mathematical thinking often includes three interrelated components: attending to students' strategies, interpreting students' understandings, and deciding how to respond on the basis of students' understanding. This practice gains complexity in technology‐mediated environments (i.e., using technology‐enhanced math tasks) because it requires attending to and interpreting students' engagement with technology. Current frameworks implicitly assume the practice includes noticing the ways students use tools (including technology tools) in their work, but do not explicitly highlight the role of the tool. While research has shown that using these frameworks supports preservice secondary mathematics teachers (PSTs) developing noticing practices, it has also shown that PSTs largely overlook students' technology engagement when they are working on technology‐enhanced tasks (Journal for Research in Mathematics Education, 2010; 41(2):169–202). In this article, we describe our adaptation of Jacobs et al.'s framework for teacher noticing student mathematical thinking to include a focus on making students' technology‐tool engagement explicit when noticing in technology‐mediated environments, the Noticing in Technology‐Mediated Environments (NITE) framework. We describe the theoretical foundations of the framework, provide a video case example, and then illustrate how the framework can be used by mathematics teacher educators to support PSTs' noticing when students are working in technology‐mediated environments. 

The authors discuss digital equity from the perspective of using math action technologies to position all students as mathematics explorers. 

A framework to guide teacher noticing when students are working in technology-mediated learning environments. 

Although geometric transformations are functions, few studies have examined students’ reasoning about these two important concepts. The purpose of this study was to examine the various ways students reasoned about functions in the context of preconstructed, dynamic sketches of geometric transformations. We found that, regardless of prior experience, all students were able to reason about important aspects of the notion of function through dragging. Specifically, by using the idea of the semiotic potential of the artifact (the dragging tool), we were able to examine ways in which students with different backgrounds reasoned about functions and how the use of the dragging tool and semiotic mediation contributed to their descriptions of geometric transformations and functions. 

Three different technological activities to explore parameters of quadratic functions each has its own pros and cons.