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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 

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.