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.

With online learning becoming a more viable option for teachers to develop their expertise, our report shares one such effort focused on improving the teaching of statistics. We share design principles and learning opportunities in an online course developed specifically to serve as a widescale online professional development opportunity for educators, thus deemed as a massive open online course for educators (MOOC-Ed). In this report we focus on a subset of 412 participants who identified themselves as classroom teachers. We use multiple data sources, quantitative and qualitative, to characterize changes in teachers' beliefs and perspectives about statistics and identify triggers in the course that appear to influence teachers' sense making about issues related to teaching statistics. Implications about specific course experiences that served as triggers for critical reflection and change are discussed.

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.

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 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.

Recently adopted state standards for middle grades and high school mathematics content have an increased emphasis on statistical topics. With this change, teacher education programs may need to adapt how they prepare preservice secondary mathematics teachers (PSMTs) to teach statistics and require measures related to statistics teaching to assess the impact of programmatic changes and track teacher growth. Using responses from a sample of 290 PSMTs from 20 institutions across the United States, this study presents validity and reliability evidence for the high school version of the Self-Efficacy to Teach Statistics (SETS-HS), which could be used to assess statistics teaching efficacy. Confirmatory factor analysis results via Rasch modeling support the use of three subscales, which exhibit adequate reliabilities and correspond to the three levels in the "Pre-K-12 Guidelines for Assessment and Instruction in Statistics Education" endorsed by the American Statistical Association. Item and rating scale analyses indicate that the 46 items and the six-category scale employed in the SETS-HS perform as intended.

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.

Amid the implementation of new curriculum standard regarding statistics and new recommendations for preservice secondary mathematics teachers (PSMTs) to teach statistics, there is a need to examine the current state of PSMTs' knowledge of the statistical content they will be expected to teach. This study reports on the statistical knowledge of 217 PSMTs from a purposeful sample of 18 universities across the United States. The results show that PSMTs may not have strong Common Statistical Knowledge that is needed to teach statistics to high school students. PSMTs' strengths include identifying appropriate measures of center, while weaknesses involve issues with variability, sampling distributions, p-values, and confidence intervals.

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 teacher education programs often need to respond to changing expectations and standards for K-12 curriculum and accreditation. New standards for high school mathematics in the United States include a strong emphasis in statistics. This article reports results from a mixed methods cross-institutional study examining the preparedness of preservice secondary mathematics teachers to teach statistics and identifying factors and experiences that influence their preparedness. Our results suggest that the cohort of teachers entering secondary mathematics classrooms in 2015-2016 were not well prepared to teach statistics. Specific suggestions are given for how teacher education programs must rise to the challenge of preparing their graduates to teach statistics.

Repeated sampling approaches to inference that rely on simulations have recently gained prominence in statistics education, and probabilistic concepts are at the core of this approach. In this approach, learners need to develop a mapping among the problem situation, a physical enactment, computer representations, and the underlying randomization and sampling processes. We explicate the role of probability in this approach and draw upon a models and modeling perspective to support the development of teachers' models for using a repeated sampling approach for inference. We explicate the model development task sequence and examine the teachers' representations of their conceptualizations of a repeated sampling approach for inference. We propose key conceptualizations that can guide instruction when using simulations and repeated sampling for drawing inferences.

The Common Core State Standards for mathematics promote ideals about learning mathematics by providing specific standards focused on conceptual understanding and incorporating practices in which students must participate to develop conceptual understanding. Thus, how we define learning is pivotal because our current definition isn't aligned with how we teach and assess. But how do we reconcile the two? What does this look like? And what does this imply about how we assess learning?

To draw conclusions from data, we need to understand the context of the data and the measurement units. In 2007, Franklin and colleagues presented the "Guidelines for Assessment and Instruction in Statistics Education (GAISE) Report" to provide teachers with a framework to assist in developing students' investigation of statistical questions in an investigative cycle. Although the role of context and the use of measurement units for data are important components, they continue to be underemphasized in many curricula (Franklin et al. 2007). Are teachers providing opportunities for students to engage in this type of reasoning? Too often, the answer is "no." The Common Core State Standards for Mathematics (CCSSM) recommends using tasks in which students consider the context and in which measurement units of data align with a sixth-grade standard that encourages "describing the nature of the attribute under an investigation, including how it was measured and its units of measurement" (Common Core State Standards Initiative 2010). Although others (e.g., Groth and Bargagliotti 2012) have provided more recent guidance for teachers on how the GAISE framework can be used to complement the CCSSM's Standards, more attention is still needed concerning the role of context and measurement units that can support students' statistical reasoning. This article discusses how data collected from a personal information survey provides students such an opportunity to reason about data by considering the units of measurement and reasonable values within a context.

The purpose of this study was to investigate a function machine in the form of a Vending Machine applet as a means to motivate preservice teachers to examine their own understanding of the function concept. The applet was designed to purposefully problematize common misconceptions associated with the algebraic nature of typical function machines. Findings indicate that the preservice teachers for which the applet provoked a dilemma elaborated on or transformed their understandings related to the definition of function.

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]

The purpose of this paper is to present a multitechnology-enabled lesson used with secondary preservice mathematics teachers to develop their technological pedagogical statistical knowledge. This lesson engages preservice teachers in a statistics lesson aimed at developing their reasoning about the measurement units of data using TinkerPlots and then engages them in reasoning about students' approaches to the task. A description of the lesson, preservice teachers' approaches, and how they reasoned about sixth graders' strategies are included. The authors further discuss the affordances of the specific technologies used in creating the learning opportunities for these preservice teachers and implications for teacher education.