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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. [ABSTRACT FROM AUTHOR]

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 FROM AUTHOR]