Search

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.

Given the vast experiences pre-service mathematics teachers (PSMTs) have with functions, one may assume that they have existing personal definitions and concept images. The purpose of this paper is to build an understanding of the variety of images that one might associate with a definition of function that includes 'for each input there is one output' and how these images are coordinated with the definition to determine whether an object is a function or not. Two cases are presented to describe PSMTs who were working from the same personal definition of function, yet used that definition differently within a task designed around the function concept. The task itself was novel in that it is presented in a non-algebraic context, namely a vending machine. As such, PSMTs were required to coordinate their personal definition and images within the novel context. Attending to PSMTs' concept images revealed that though they were working from the same personal definition, images they drew upon to determine whether or not an object is a function were varied.

The teaching practice of noticing students' thinking includes both attention to students' mathematical strategies and interpretation of their mathematical understanding. This practice is further complicated when students are engaged with mathematical action technologies while learning mathematics content because, in these contexts, teachers must also notice how the students' engaged with the technology to develop their mathematical understanding. For this crossinstitutional study, we focus on how preservice secondary mathematics teachers (PSMTs) coordinated their noticing of students' mathematical thinking and engagement in a technology-mediated environment. The PSMTs viewed artefacts of high school students working with a Desmos task focused on rational functions, specifically locating vertical asymptotes, and noticed the students' mathematical thinking. Results showed it was easier for PSMTs to attend to and interpret the students' spoken and written responses than for them to attend to and interpret the students' engagement with the technology. Further, it was even more difficult for the PSMTs to coordinate the students' spoken and written responses with their technology engagement.

When thinking about digital equity, often what comes to mind are issues of access to digital technologies. Although that is a divide that must be conquered, once students have access to digital technologies, the next step is to ensure that they wield these technologies in such ways that they help to address the existing inequities in mathematics classrooms. The focus in this article is to discuss the ways that digital technologies can be positioned to meet this challenge through providing ways for "each and every student" to: (1) enter a mathematical problem; (2) mediate mathematical discussions; and (3) build personal and powerful ways of mathematical thinking. These three characteristics position all students as mathematical explorers when using technology in the classroom.

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 pre-constructed, 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.

The goal of this article is to give examples of high school students exploring parameters of quadratic functions using different tools as a catalyst to discuss the pros and cons and demonstrate the mathematical power of each. The authors illustrate how teachers' learning goals should navigate them toward a productive option.

Eliciting and using student thinking (NCTM 2014) is a foundational pedagogical skill that informs the decisions teachers make during instruction. It informs the questions teachers pose, the focus of small-group and whole-class discussions, and even the small adjustments teachers make during a well-planned lesson. To use student thinking to strengthen the teaching of mathematics, a teacher needs to learn how to professionally notice students' mathematical thinking. This article discusses what it means to professionally notice students' thinking in a technology-mediated learning environment.

The field of mathematics teacher education has been moving collectively toward a common goal of preparing prospective mathematics teachers to teach with technology. In this article, we present findings from a national survey of accredited university secondary mathematics teacher education programs. The purpose of the study is to describe the current state of the ways in which programs are preparing preservice teachers to teach secondary mathematics with technology.

If the goal is to move beyond the shape of the graph to consider the dynamic aspects of functions, students need to explore dynamic representations of functions. To address this, the authors combined the use of a dynamic representation of function with the instructional routine, Which One Doesn't Belong? [WODB]. In this article, they briefly introduce the WODB routine and the dynamic representation of function that they use, dynagraphs (for more information on dynagraphs, see Edwards and Nickell 2014). Then, they share their dynamic WODB task along with examples of ways in which their students engaged with it. The intent is for readers to consider how dynamic representations situated within WODB can extend students' exploration and recognition of the dynamic characteristics of functions.

In this article, we examine the ways in which the creation of a third space can bridge the divide between coursework and practice for preservice secondary mathematics teachers (PSTs) taking a technology, pedagogy, and content course. A university-based instructor partnered with two high school teachers to create a space in which PSTs draw upon and use both academic and practitioner knowledge while creating technology-based tasks for high school students to use. Our results revealed increased focus on pedagogical decisions in areas such as technology-task design and questioning techniques. The data also indicate that the success of this collaboration was connected to fair distribution of work, feeling valued, and personal benefit and challenges centered on maintaining rejection of hierarchy.

In this article, we present a set of design principles to guide the development of instructional materials aimed to support preservice secondary mathematics teachers (PSMTs) examining student practices in technology-mediated environments. To develop design principles, we drew on the literature related to technological pedagogical content knowledge (TPACK; Niess, 2005), video cases as learning objects (Sherin & van Es, 2005), and professional noticing (Jacobs, et al., 2010). After presenting the design principles, we share a task created using these design principles. Finally, we share PSMTs' reflections about changes in their own understanding after examining students' practices. Their responses provide insights into the usefulness of the design principles for deepening PSMTs' mathematical knowledge and knowledge of students' understanding, thinking, and learning with technology.

The purpose of this study is to examine the ways in which PSMTs' professionally notice middle school students' mathematical thinking on a technology enhanced mathematical task. Findings show that every preservice mathematics teacher included a discussion of the middle school students' interaction with the technology in their noticing prompts, demonstrating that PSMTs recognized that the middle school students' mathematical understanding was tied to their interactions with the technology. Additionally, results from PSMTs' justifications for their predictions of middle school students' responses to the task, incorporated the middle school students language and described how the middle school students would interact with the technology.

Noticing students' mathematical thinking is a complex, but important practice for preservice secondary mathematics teachers (PSMTs) to develop. This practice is further complicated when secondary students use technology, as it requires the dual and interconnected attention to students' mathematical thinking and the ways they engage with the technology as they are learning. The purpose of this study was to examine how explicitly sharing a framework for noticing students' mathematical thinking in technology-mediated learning environments and providing opportunities for practice supported PSMTs' noticing. Pre- and post-video-based assessments were used to examine changes in PSMTs' noticing as a result of engaging with the framework. The findings of this study suggest that using this framework to support PSMTs' development of the teaching practice of noticing students' thinking has promise, especially related to coordinating students' written and spoken mathematical thinking with their technology engagement.

Context is what makes mathematical modeling tasks different from more traditional textbook word problems. Math problems are sometimes stripped of context as they are worked on. For modeling problems, however, context is important for making sense of the mathematics. The task should be brought back to its real-world context as often as possible. In this article the authors present a GeoGebra-based trigonometric function modeling task along with examples of student thinking from implementation with second year algebra students. The goal is to draw attention to the importance of context throughout the modeling process.

Mathematics autobiographies have the potential to help teachers reflect on their identities as mathematics learners and to understand their role in the development of their students' mathematics identities. This paper reports on a professional development project for K-2 teachers (n = 41), in which participants were asked to write mathematics autobiographies. Using an adaptation of an existing framework for characterizing teachers' mathematics stories, we describe the consistencies among the participants' experiences as mathematics learners and the events that are identified as being the impetus for a transition from a negative to a positive attitude toward mathematics. Implications for both teachers and teacher educators are presented.

This paper reports on a mixed-methods study of 111 Advanced Placement calculus students' self-reports of their graphing calculator use, comfort, and rationale for trusting a solution produced with or without a graphing calculator when checking written work. It was found that there was no association between gender, teacher-reported mathematical ability, or comfort with the graphing calculator and students' trust in either a graphing calculator-produced solution or a solution produced without a graphing calculator. Furthermore, regardless of solution choice, the same four categories were evident in students' rationale for their solution choice: (a) an awareness of the possibility of careless errors, (b) the importance of checking over work, (c) a recognition of the limitations or affordances of the graphing calculator, and (d) a confidence (or lack thereof) in their own mathematical abilities. These results have implications for mathematics teaching as graphing calculators are used extensively in middle and high school mathematics classrooms and standardized tests in the United States. (Contains 2 tables.)

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]

STEM-based understandings and experiences that prepare learners beyond the classroom are of imminent need, as today's STEM education students are tomorrow's leaders in science, technology, engineering, mathematics, and education (Prabhu, 2009). Integrative STEM education signifies the intentional integration of science and mathematics with the processes, content, and procedure of technology and engineering education (Sanders & Wells, 2010). Intentionally embedding signature approaches of technology/engineering education into STEM-based activity presents a unique method of educational integration that is grounded in the approaches, pedagogies, and strategies that have contributed to student educational engagement and successes over the years. The intent of this article is to provide an example of how the signature approaches of technology/engineering education can be used as the basis for an authentic integrative STEM education experience for middle school students. (Contains 5 figures.)

This article uses data from 65 teachers participating in a K-2 mathematics professional development research project as an example of how to analyze video recordings of teachers' classroom lessons using mixed methods. Through their discussion, the authors demonstrate how using a mixed methods approach to classroom video analysis allows researchers to draw more comprehensive conclusions about participants both collectively and individually. Implications for using a mixed methods approach to analyze classroom video research are provided. (Contains 1 figure, 4 tables, and 2 notes.)

This article reports on a qualitative study of six high school calculus students designed to build an understanding about the affect associated with graphing calculator use in independent situations. DeBellis and Goldin's (2006) framework for affect as a representational system was used as a lens through which to understand the ways in which graphing calculator use impacted students' affective pathways. It was found that using the graphing calculator helped students maintain productive affective pathways for problem solving as long as they were using graphing calculator capabilities for which they had gone through a process of instrumental genesis (Artigue, 2002) with respect to the mathematical task they were working on. Furthermore, graphing calculator use and the affect that is associated with its use may be influenced by the perceived values of others, including parents and teachers (past, present and future).