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This study explored an innovative coaching model termed "video-based online video coaching." The innovation builds from affordances of robot-enabled videorecording of lessons, accompanied by built-in uploading and annotation features. While in-person coaching has proven effective for providing sustained support for teachers to take up challenging instructional practices, there are constraints. Both logistical and human capacity constraints make in-person coaching difficult to implement, particularly in rural contexts. As part of an NSF-funded project, we studied nine mathematics coaches over four years as they engaged in video-based coaching with teachers from geographically distant, rural contexts. We adapted a content-focused coaching model that involved a collaborative plan-teach-reflection cycle with synchronous and asynchronous components. The planning and debriefing sessions were done synchronously via Zoom, while the teaching and initial video reflection on teaching via annotations were done asynchronously. We focused on the coaches' practices in each part of the coaching cycle by analyzing interviews, surveys, annotations of the video, and transcripts of the planning and debriefing sessions. We found that: features of the online environment enabled the coach-teacher pairs to collaboratively discuss the mathematics and how students engaged with the mathematics; the coach used video and annotations to help teachers reflect on specific aspects of their practice; and the coach-teacher pairs formed trusting and productive relationships despite not having met in-person during the duration of their work together. Our findings showed that the online platform is not only an effective implementation for coaching, but also affords new opportunities for teacher reflection and evidence-based discussions.

We studied how the distal policy mechanisms of curricular aims and objectives articulated in official curriculum documents influenced classroom instruction, and the factors that were associated with the enactment of those curricular aims and objectives. The study was set in the U.S. context, where there is an ambitious effort to transform curriculum and instruction via the Common Core State Standards for Mathematics (CCSSM). The CCSSM represented the curricular aims and objectives in most of the U.S. at the time of the study. We analyzed enactments of this official curriculum in terms of the rigor of mathematical activity in 47 middle school mathematics lessons from multiple state and curriculum contexts. The enactment of the CCSSM was not uniform across contexts, and the lack of uniformity was associated in part with the type of instructional materials used by teachers. The use of instructional materials classified as "delivery mechanism" was associated with activity we characterized as routine procedural rigor. In lessons involving instructional materials classified as "thinking device," we found greater variation and more occurrences of non-routine forms of rigor. These differences between types of instructional materials occurred despite the finding that teachers across the sample held similar views of the CCSSM. We conclude that the teachers responded more to features in the instructional materials than to the curriculum aims and objectives articulated in the CCSSM while planning and enacting lessons, which has implications for policy makers who aim to influence instruction through national standards and for school districts as they select materials.

The study explored the verb clauses and thematic development evident in curriculum materials and in transcripts of teachers planning lessons using the materials. A central argument is that though teacher characteristics influence the ways they plan lessons with curriculum materials, the materials themselves influence teachers' planned lessons via the ways mathematics is construed in the materials. We used verb clause and thematic analysis to analyze the features of curriculum materials and teachers' lesson planning using those materials. We found differences between the curriculum materials in terms of how they construed mathematical activity and found that teachers' lesson plans roughly aligned with the features in the materials; this speaks to the impact of features of curriculum materials on teachers' lesson planning. We also found small but perceptible differences between teachers from different curriculum backgrounds, suggesting an enculturating effect from the materials.

An important component of the Common Core State Standards for Mathematics (CCSSM), used by the majority of states in the U.S., has the eight standards for mathematical practice (SMPs). While surveys have investigated teachers' perceptions of the CCSSM few have investigated middle school mathematics teachers' (MSMTs') (grades 6-8) perceptions of the SMPs. Similar to other countries, teachers in the United States frequently use mathematics textbooks. Two different types of textbooks have been in use in the United States, conventional and standards-based. The latter were designed on the basis of earlier standards documents including versions of the SMPs. As a result of these changes to the educational system in the U.S., we sought to characterize MSMTs' perceptions of the SMPs and investigate if these perceptions differed by the type of textbook teachers used. We found that MSMTs struggled in naming the SMPs, misinterpreted the modeling SMP, and conflated components of reform-oriented instruction with the SMPs. MSMTs using a standards-based textbook were more likely to view their textbooks as aligned with the SMPs, x[superscript 2](3) = 7.708, p = 0.026 and to view the SMPs as an instructional philosophy for the CCSSM, Fisher's Exact Test = 3.881, p = 0.05. The implications of these results are discussed.

Research on how coaches talk with teachers during coaching cycles is underdeveloped. We analyzed 1,649 discourse moves from 24 mathematics content-focused coaching cycles to determine the extent to which coaches' discursive tendencies vary. We explored variation between coaches, between planning and debriefing conversations, and between cycles for the same coach–teacher pair. Findings indicate there existed significant variability in the coaches' discourse moves during coaching cycles. We also found discursive differences from planning to debriefing meetings, noting that coaches were more directive and less reflective in planning conversations compared with debriefing conversations. Across multiple coaching cycles, we found variation across coaches, with one coach increasing the prevalence of directive moves across four planning conversations and another increasing the prevalence of reflective moves across four debriefing conversations. Although we focus on mathematics coaches, the findings and methodology may be applicable to other disciplines. [ABSTRACT FROM AUTHOR]

The edTPA is a teaching performance assessment (TPA) that the states of New York and Washington implemented as a licensure requirement in 2013. While TPAs are not new modes of assessment, New York and Washington are the first states to use the edTPA specifically as a compulsory, high-stakes policy lever in an effort to strengthen the quality and accountability of teachers and teacher educators. This study examines 24 New York and Washington teaching candidates' experiences with the edTPA during its first year of consequential use for state certification. The data, drawn from qualitative interviews that were part of a larger mixed-methods study, reveal that preservice teachers had to mediate several tensions associated with the edTPA's dual role as a formative assessment tool and a licensure mechanism. In this paper, we identify those tensions, describe candidates' efforts to mediate them, and discuss the extent to which that mediation process may or may not contribute to the improvement of teachers' practices. Given the edTPA's positioning in a policy context--specifically, the potential for the assessment's locus of control, high stakes, and opaque rating process to distort the procedures it is intended to measure--the paper concludes with recommendations for teacher education programs aimed at capitalizing on the edTPA's benefits and mitigating its unproductive tensions.

Coaching is an increasingly popular and promising method of professional learning, but unfortunately, many teachers do not have access to high-quality coaching due to geographic and financial constraints. Technology offers an opportunity to increase access to coaching, especially for educators in isolated rural areas. Research shows video is useful in teacher education and professional learning to focus on moments of practice. Recognizing the potential of technology for coaching in the rural areas where they work, the authors of this article developed an online coaching model in a joint venture between the University of Rochester (New York) and the University of Idaho, with funding from the National Science Foundation. Their goal is to expand their online coaching program to reach more teachers in rural settings, as well as urban and suburban districts.

We studied a case of a school in a high need setting that undertook multiple simultaneous initiatives during a major school reorganization. We focused on the simultaneous implementations of two comprehensive initiatives, one related to ambitious mathematics teaching and one related to the Understanding by Design curriculum writing process. We explored the extent to which educators in a mathematics department saw these initiatives as aligned or in tension. The results show that simultaneous ambitious initiatives may ultimately be mutually reinforcing, especially if grounded in common principles. We also found that initial tensions existed and diminished both initiatives at the outset [ABSTRACT FROM AUTHOR]

Digital content is increasingly present in U.S. K-12 classrooms, with a current push by federal officials to increase the rate at which digital textbooks are adopted. While some teachers' use of electronic resources involves locating activities and lessons from various internet sites, textbook and educational software companies have begun to develop comprehensive programs that can supplement if not fully replace traditional paper textbooks. Digital platforms can be transformative, with possibilities for frequent updating, access to multimedia resources, connection to virtual communities, lower production and distribution costs, and customized instruction. However, there have been no attempts to analyze specific programs in mathematics education with respect to these and other features, a gap we seek to address. In this article, we developed and applied a framework to analyze a representative sample of digital curriculum programs in order to help educators better understand characteristics of these materials. We documented two distinct curriculum types, individualized learning programs and digitized versions of traditional textbooks. While the programs offered some of the features identified as transformative, particularly with respect to assessment systems that rapidly and visually report student performance, there were many features that did not take full advantage of the digital medium.

Eight middle school mathematics teachers' perceptions and uses of curriculum materials and the Common Core State Standards for Mathematics (CCSSM) were investigated. Adapting a noticing framework and models of dialogic instruction and direct instruction, teachers' noticing practices with curriculum materials and the CCSSM when planning, enacting, and reflecting on lessons were examined. Teachers who were committed to implementing the CCSSM and who were using one of two substantively different curriculum programs were purposefully selected. Data sources included multiple forms of interviews and classroom observations. The teaching evidenced three distinct noticing patterns. These patterns indicated that teachers' curriculum materials were associated with how teachers perceived and enacted the CCSSM. Teaching with a curriculum program that was designed as a thinking device prioritized the Standards for Mathematical Practice of CCSSM evidenced noticing that was consistent with dialogic instruction. Teaching with a curriculum program that was designed as a delivery mechanism prioritized the Content Standards of CCSSM and evidenced noticing consistent with direct instruction. Findings indicated that the designated curriculum and contributed to differing interpretations of CCSSM and served as a lens for noticing. However, a dialogic curriculum program was not sufficient to support dialogic approaches in practice. One pattern showed teachers planning dialogic lessons, but the lesson enactments were not consistent with teachers' plans, with evidence that the teachers were not aware that their practices differed from dialogic approaches. Implications for research and practice are discussed.

This study examines Middle School Mathematics Teachers' (MSMTs') (N = 1,241) perceptions of the Common Core State Standards for Mathematics (CCSSM) and its impact on the instructional environment. A total of eight factors appear in the data. These factors include professional support, teachers' use of district-adopted and non-district-adopted curricular resources, influence of CCSSM assessment and teacher evaluation on the instructional environment, influence of the CCSSM on classroom instruction, teachers' planning practices, and perceived rigor of the CCSSM. The data suggest that teachers' use of digital resources located online was disconnected from their district-adopted resources suggesting a lack of curricular coherence. MSMTs note that the CCSSM had caused them to incorporate more reform-oriented practices such as exploration and productive struggle into their daily instruction. MSMTs also perceive that the CCSSM includes new content that is more rigorous than previous state standards. Sampled MSMTs state that while state assessments will measure CCSSM content, they are less likely to include more complex problems or the standards for mathematical practice. Teachers are more likely to read teacher resources than student textbook activities online and to use digital resources for remediation instead of inquiry activities. Over one-third of MSMTs wanted more CCSSM professional development.

We explore questions around the design, development, and dissemination of digital curriculum materials, the perspectives in these areas, and how these perspectives align with broader discourses in education. We identify and briefly describe four perspectives: (1) designer perspective; (2) policy perspective; (3) private sector perspective (e.g., publishers and philanthropists); and (4) user (teachers and schools) perspective. We discuss how these perspectives converge and diverge by looking at the different features of curriculum materials emphasized by each perspective and the reasons for these emphases. The discussion and findings speak to the promise of digital programs as well as limitations related to the rationales related to the development, dissemination and use of digital curriculum resources. The emergence of a dominant perspective speaks to broader concerns about educational priorities being formulated according to a market-based rationality.

In this conceptual review paper we draw on recent literature with respect to digital curriculum resources (DCR); we briefly outline and explain selected theoretical frames; and we discuss issues related to the design, and the use (by teachers and students) of digital curricula and e-textbooks in mathematics education. The results of our review show the following. Firstly, whilst there are some contrasting tendencies between research on instructional technology and research on DCR, these studies are at the same time predominantly framed by socio-cultural theories. Secondly, whilst there seems to be a continuing demarcation between the design(er) and the use(r), there is at the same time an emerging/increasing understanding that design continues in use, due to the different nature and affordances of DCR (as compared to traditional text curriculum resources). Thirdly, there is an apparent weakening of traditional demarcations between pedagogy and assessment, and between summative and formative assessment techniques, due to the nature and design of the automated learning systems. Fourthly, there is an increasing need for understanding the expanded space of interaction associated with the shift from static print to dynamic/interactive DCR, a shift that has the potential to support different forms of personalised learning and interaction with resources. Hence, we claim that DCR offer opportunities for change: of understandings concerning the design and use of DCR; of their quality; and of the processes related to teacher/student interactions with DCR--they provide indeed the foundations for change.

In this study, U.S. middle school teachers' perceptions of Common Core State Standards for Mathematics (CCSSM), CCSSM-related assessments, teacher evaluation processes, and resources for implementing CCSSM were investigated. Using a mixed methods design, a national sample of 366 teachers was surveyed, and 24 teachers were interviewed. Findings indicated that teachers viewed CCSSM as including new content for their grade level. Teachers also reported using multiple curriculum resources to align with CCSSM and indicated that new assessments would serve as a proxy for CCSSM. Implications for rapidly changing policy, curriculum, assessment, instruction, and professional development related to CCSSM are discussed.

The edTPA, a performance assessment designed to generate reliable and valid measures of teaching practice, increasingly is used as a gatekeeping mechanism for beginning teacher licensure in various states, including New York, Washington State, Wisconsin, and Georgia. One of the edTPA's key components is the demonstration of instructional practice by video recording. This article explores threats to validity associated with using video segments as part of the edTPA. Based on interviews with 24 teaching candidates from New York and Washington State, results show that candidates had difficulty fully addressing the competencies assessed by the edTPA, thoroughly representing their teaching practices, and learning from the process of analyzing their videos, affecting content validity, ecological validity, and consequential validity, respectively. One implication is that the utility of the video records may be limited to corroborating and triangulating claims made in the written commentaries, rather than serving as authoritative approximations of teaching practice.

Productive adaptations at the classroom level are evidence-based curriculum adaptations that are responsive to the demands of a particular classroom context and still consistent with the core design principles and intentions of a curriculum intervention. The model of design-based implementation research (DBIR) offers insights into complexities and challenges of enacting productive curriculum adaptations. We draw from empirical research in mathematics and science classrooms to illustrate criteria for productive adaptations. From these examples, we identify resources needed to encourage and sustain practices to promote productive adaptations in classrooms.

The eight mathematical practices explored in the Common Core Math Standards are the following: (1) Make sense of problems and persevere in solving them; (2) Reason abstractly and quantitatively; (3) Construct viable arguments and critique the reasoning of others; (4) Model with mathematics; (5) Use appropriate tools strategically; (6) Attend to precision; (7) Look for and make use of structure.; and (8) Look for and express regularity in repeated reasoning. If teachers are going to take the Common Core Math Standards seriously, they need to think of them as more than simply a reordering of content. That means focusing on the practices they associate with mathematical understanding. A major implication is that "developing practices" rather than "covering content" requires a focus on task sequences rather than singular lessons; these sequences provide repeated opportunities for students to reason about ideas before they are formalized. Most students can reason mathematically but few get the opportunity to publicly test ideas and conjectures as they are forming. Participation in such practices leads not only to increased understanding but also to the development of mathematical dispositions that are valuable as students move to more advanced mathematics. (Contains 1 figure.)

This study focused on the use of curriculum materials for three teachers who had enacted instructional sequences from the materials on multiple occasions. The study investigated how the teachers drew on the materials, what they understood about the curriculum resources, and how they connected their use of the materials to their observations of student thinking. There were similarities across the teachers, particularly with respect to their goals and how they read and followed recommendations in the teacher resource materials. There were differences in how their task revisions were in response to what they observed about student thinking. The teacher who most intensively observed student thinking made connections between her interpretations of students' strategies and her use of the curriculum resources, allowing her to design learned adaptations. Learned adaptations required both an understanding of the design rationale and empirically developed knowledge of how that rationale played out in practice. The empirically developed knowledge could not be totally anticipated by the designers, in part because it developed within a particular context by a teacher with particular characteristics. The case of the teacher who developed learned adaptations showed how these complementary forms of knowledge helped her to use the curriculum resources in ways that enhanced students' opportunities for sense making. Furthermore, her adaptations were intended to facilitate success not only at the task level, but also across instructional sequences as well. This study also shows how professional vision is not limited to informing only in-the-moment instructional decisions, but also to the use of curriculum materials.

This study explores the extent to which a teacher elicited students' mathematical reasoning through the use of challenging tasks and the role her knowledge played in doing so. I characterised the teacher's knowledge in terms of a "local theory" of instruction, a form of pedagogical content knowledge that involves an empirically tested set of conjectures situated within a mathematical domain. Video data were collected and analysed and used to stimulate the teacher's reflection on her enactments of an instructional sequence. The teacher, chosen for how she consistently elicited student reasoning, showed evidence of possessing a local theory in that she articulated the ways student thinking developed over time, the processes by which that thinking developed, and the resources that facilitated the development of student thinking. Her knowledge informed how she revised and enacted challenging tasks in ways that elicited and refined student thinking around integer addition and subtraction. Furthermore, her knowledge and practices emphasised the progressive formalisation of students' ideas as a key learning process. A key implication of this study is that teachers are able to develop robust knowledge from enacting challenging tasks, knowledge that organises how they elicit and refine student reasoning from those tasks. (Contains 3 tables and 1 footnote.)

This study investigates how teacher attention to student thinking informs adaptations of challenging tasks. Five teachers who had implemented challenging mathematics curriculum materials for three or more years were videotaped enacting instructional sequences and were subsequently interviewed about those enactments. The results indicate that the two teachers who attended closely to student thinking developed conjectures about how that thinking developed across instructional sequences and used those conjectures to inform their adaptations. These teachers connected their conjectures to the details of student strategies, leading to adaptations that enhanced task complexity and students' opportunity to engage with mathematical concepts. By contrast, the three teachers who evaluated students' thinking primarily as right or wrong regularly adapted tasks in ways that were poorly informed by their observations and that reduced the complexity of the tasks. The results suggest that forming communities of inquiry around the use of challenging curriculum materials is important for providing opportunities for students to learn with understanding. (Contains 2 tables and 2 footnotes.)

The goal of this article is to help teachers both recognize and capitalize on classroom opportunities so that students can be meaningfully engaged in mathematical proof. To do so, the authors discuss students' thoughts about proof and their abilities to construct arguments. They also offer suggestions for teachers that are intended to support their efforts to provide students with meaningful experiences with proof throughout their middle school mathematics education. (Contains 4 figures.)

This study characterizes the teacher learning that stems from successive enactments of innovative curriculum materials. This study conceptualizes and documents the formation of curriculum-context knowledge (CCK) in three experienced users of a Standards-based mathematics curriculum. I define CCK as the knowledge of how a particular set of curriculum materials functions to engage students in a particular context. The notion of CCK provides insight into the development of curricular knowledge and how it relates to other forms of knowledge that are relevant to the practice of teaching, such as content knowledge and pedagogical content knowledge. I used a combination of video-stimulated and semistructured interviews to examine the ways the teachers adapted the task representations in the units over time and what these adaptations signaled in terms of teacher learning. Each teacher made noticeable adaptations over the course of three or four enactments that demonstrated learning. Each of the teachers developed a greater understanding of the resources in the respective units as a result of repeated enactments, although there was some important variation between the teachers. The learning evidenced by the teachers in relation to the units demonstrated their intricate knowledge of the curriculum and the way it engaged their students. Furthermore, this learning informed their instructional practices and was intertwined with their discussion of content and how best to teach it. The results point to the larger need to account for the knowledge necessary to use Standards-based curricula and to relate the development and existence of well-elaborated knowledge components to evaluations of curricula.