- Mathematics Teacher Education, Curriculum and Instruction, Teachers Practices, Classroom Discourse Analysis, Mathematics Education, Teachers' professional development, and 6 moreTeaching and Learning, Curriculum Design, Educational Equity and Justice, Classroom Discourse, Teacher knowledge, and Educationedit
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Teacher education in the United States is contextualized by efforts among various actors—for instance, state policy makers, teacher educators, and researchers—to wrest control over the terms, conditions, and consequences of accountability... more
Teacher education in the United States is contextualized by efforts among various actors—for instance, state policy makers, teacher educators, and researchers—to wrest control over the terms, conditions, and consequences of accountability in the field. A key mechanism in this campaign is teacher performance assessments (TPAs), which have evolved with the accountability movement to serve the roles of gatekeeping teachers’ entry into the profession and evaluating the outcomes of teacher education. Tensions over TPAs as policy levers first emerged in California in the early 2000s, when—in response to a law mandating that teaching candidates pass a state-approved performance assessment for licensure—universities within the Performance Assessment of California Teachers (PACT) consortium developed an alternative to the existing test that aimed to emphasize subject-specific student learning, position the TPA as a formative assessment tool, and preserve flexibility in teacher education programming.
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The literature in mathematics education has extensively documented the ways that discourse practices affect student learning and dispositions. However, there has been little discussion about how discourse practices affect what teachers... more
The literature in mathematics education has extensively documented the ways that discourse practices affect student learning and dispositions. However, there has been little discussion about how discourse practices affect what teachers learn in classrooms. This chapter builds from Davis’s (1997) description of listening practices to explore the related propositions that what teachers learn in the context of their teaching is tied to the ways they listen, and their opportunities for listening are tied to the language patterns they enact in their classrooms. I provide examples of classroom discussions to illustrate various types of listening/language patterns and their implications for teacher learning. The episodes of classroom discourse demonstrate a continuum of teachers’ listening practices and associated language patterns, from evaluative listening and teacher-centric language patterns to more interpretive listening and interactive discourse practices. In these episodes, evaluative listening involved less ambiguity than interpretive forms of listening, but revealed little about the deep ways students made sense of the mathematics. Interpretive and hermeneutic listening, by contrast, involved higher ambiguity but yielded deeper insights into student thinking than evaluative forms of listening. These findings highlight the importance of connecting teacher practices not only to student learning but also to what teachers know about how students learn mathematics. They also demonstrate the value of discourse analysis to understanding mathematics classroom practices of teaching and learning.
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In 2013, New York and Washington became the first two states to require that candidates for initial state teacher certification take and pass the edTPA – a performance assessment of teaching structured around artifacts and commentaries... more
In 2013, New York and Washington became the first two states to require that candidates for initial state teacher certification take and pass the edTPA – a performance assessment of teaching structured around artifacts and commentaries related to candidates’ planning, instruction, and assessment practices. With this movement of the edTPA into state policy contexts, we began a study in 2014 to document preservice teachers’ perceptions of and experiences with the assessment as a high-stakes licensure examination. In the summer of 2015, during the second year of our study, we surveyed 193 preservice teachers at 12 colleges and universities across both states, asking questions about: (1) the edTPA’s roles in their teacher education programs and student teaching placements; (2) candidates’ test completion efforts and the supports they received throughout the process; and (3) the extent to which participants found the edTPA to be a fair, credible, and impactful assessment of their teaching.
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We present the results of our analysis regarding the discursive tendencies of four mathematics coaches during planning and debriefing conversations within online coaching cycles. Guided by the Content-Focused model of coaching (West &... more
We present the results of our analysis regarding the discursive tendencies of four mathematics coaches during planning and debriefing conversations within online coaching cycles. Guided by the Content-Focused model of coaching (West & Cameron, 2013), the coaching cycles are a single component of a larger online professional development model for middle school mathematics teachers in rural areas (Choppin, Amador, & Callard, 2015). This paper explores the different ways coaches talk with teachers during coaching conversations. Building on prior studies from literacy coaching (Ippolito, 2010), we found five different discursive moves for how coaches talk with teachers: invitation, suggestion, explanation, description, and evaluation. We use these moves to identify similarities and differences in the discursive tendencies of coaches. The implications of these discursive tendencies are provided.
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As part of a larger study, we report findings on teachers’ use of the Common Core State Standards for Mathematics (CCSSM) and teacher resources (TR) that were included with teachers’ published curriculum programs. We analyzed 147 lesson... more
As part of a larger study, we report findings on teachers’ use of the Common Core State Standards for Mathematics (CCSSM) and teacher resources (TR) that were included with teachers’ published curriculum programs. We analyzed 147 lesson planning interviews with 20 middle school teachers to understand how teachers interpreted and enacted the CCSSM while working with their curriculum materials. We investigated teachers’ noticing of CCSSM and features of TR in planning lessons. Regardless of curriculum, teachers perceived that the lessons were designed to address the CCSSM. Findings for patterns among curriculum type, teacher orientation, and teachers’ noticing are presented. Implications for curricular policy and design are discussed.
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We share results of a study on the analytic stances of coaches’ and teachers’ as they annotated key moments from classroom video of the teacher’s lessons. In the analysis, emphasis was on the analytic stances of the coaches and how their... more
We share results of a study on the analytic stances of coaches’ and teachers’ as they annotated key moments from classroom video of the teacher’s lessons. In the analysis, emphasis was on the analytic stances of the coaches and how their annotations related to trends in teachers’ annotations. Findings indicate differences in how coaches and teachers noticed across the coaching cycles, suggesting the annotations were influenced by the interactions between the coaches and teachers and the teachers’ perceptions of coaching process. As a result of our analysis, we characterized one coach as having a high ratio of questions to suggestions, another as having annotations coded as interpretation, another as having more evaluations and suggestions, and the fourth as having asked more questions. Some teachers mirrored the analytic stance of their coach over time and other teachers shifted their analytic stance in ways that suggest they were responsive to their coach’s analytic stance.
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In this afterword, I briefly summarize the documentational approach to didactics (DAD) since a number of authors have already comprehensively done so earlier in this volume. The 2018 Re(s)sources International Conference demonstrated the... more
In this afterword, I briefly summarize the documentational approach to didactics (DAD) since a number of authors have already comprehensively done so earlier in this volume. The 2018 Re(s)sources International Conference demonstrated the breadth, promise, and growth of DAD and how it allows the field to productively problematize the interactions between curriculum resources and those who use them. As a friendly outsider, I have explored DAD and grown increasingly familiar with it. Below, I explore key contributions of DAD and the strengths of its theoretical underpinnings. I then provide challenges and limitations of DAD before connecting it to my recent work in curriculum ergonomics.
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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... more
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 annotat...
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We implemented an innovative online mathematics professional learning model designed to create high-impact growth opportunities for teachers in rural school districts. As a part of this model, classroom teachers were partnered with... more
We implemented an innovative online mathematics professional learning model designed to create high-impact growth opportunities for teachers in rural school districts. As a part of this model, classroom teachers were partnered with coaches with whom they worked on content-focused coaching cycles. Within a coaching cycle, teachers and coaches met to plan a mathematics lesson, the teacher video recorded the lesson, the coach and teacher both annotated the lesson video, and finally the two met to debrief the lesson verbally. We report on the process of video annotations. Data analysis focused on identifying the type of response within an annotation as well as the extent to which the teacher or coach noticed students’ mathematical thinking. We contend that the annotation process afforded opportunities for direct suggestions to teachers that may not have otherwise occurred. Findings indicate annotations were predominantly focused on mathematics content goals and classroom discourse pract...
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We describe two metaphors that we hope can be used to better understand the contemporary mathematics curriculum context in U.S. middle schools, to see how this new context is both similar to and different from prior curriculum contexts.... more
We describe two metaphors that we hope can be used to better understand the contemporary mathematics curriculum context in U.S. middle schools, to see how this new context is both similar to and different from prior curriculum contexts. We explain the role and positioning of middle school mathematics curriculum materials over the last century or more and build from learning theory to develop the metaphors. The first metaphor, curriculum as delivery mechanism, builds from technical rational or scientific discourses and encompasses perspectives that are so pervasive they are often unstated and unquestioned. The second metaphor, curriculum as epistemic device, posits that role of curriculum is to provoke interactions that generate understanding. In this metaphor, the role of tasks in curriculum materials is to provoke and progressively refine student thinking, individually and collectively.
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Across the United States, K-12 schools and postsecondary institutions are rushing to address a need for individuals trained in seience, technology, engineering, and mathematics (STEM). Moreover, instead of considering these as four... more
Across the United States, K-12 schools and postsecondary institutions are rushing to address a need for individuals trained in seience, technology, engineering, and mathematics (STEM). Moreover, instead of considering these as four separate content areas, an inte- grated view of STEM (e.g., Capraro, Capraro, & Morgan, 2013) considers these content areas as a whole that is indeed greater than the sum of its parts. The student goals associated with integrated STEM include the following: STEM literacy, 21st century competencies, STEM workforce readiness, interest and engagement, and ability to make connections among STEM disciplines (Honey, Pearson, & Schweingruber, 2014).The National Research Council (2011) recently identified five key educational system elements that cut across content areas to help promote STEM education. These key elements are as follows: "a coherent set of standards and curriculum," "teachers with high capacity to teach in their discipline," &q...
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We analyzed 52 middle school mathematics lessons from multiple states and curriculum contexts to understand how teachers were enacting the CCSSM. The teachers stated that all of the lessons were CCSSM-aligned. We categorized curriculum... more
We analyzed 52 middle school mathematics lessons from multiple states and curriculum contexts to understand how teachers were enacting the CCSSM. The teachers stated that all of the lessons were CCSSM-aligned. We categorized curriculum materials according to two approaches, with one approach associated with curriculum programs funded by NSF and the other representing curriculum programs commercially produced, typically from a large publisher. We analyzed the nature of mathematical activity and level of interactions in the lessons. We found significant differences across curriculum approaches in the mathematical activity categories related to cognitive demand and in the level of interaction. The implications are that curriculum programs strongly mediated the enactment of the CCSSM.
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Coaching has become a common practice to support teachers (Coburn & Russell, 2008; Foltos, 2014; Knight, 2007). While much of the coaching research has focused on the roles and responsibilities of coaches (Gibbons & Cobb, 2017; Mudzimiri... more
Coaching has become a common practice to support teachers (Coburn & Russell, 2008; Foltos, 2014; Knight, 2007). While much of the coaching research has focused on the roles and responsibilities of coaches (Gibbons & Cobb, 2017; Mudzimiri et al., 2014) there is a lack of empirical research on what occurs in the interactions between a coach and teacher and the mechanisms by which these interactions support teacher learning. This study aims to understand both how and why coaches engaged in specific coaching practices. We analyzed interviews with four mathematics coaches in which they reflected on the purposes, goals, and practices they perceived as critical for supporting teachers in a fully online coaching model. For three years, these coaches had used an online content-focused coaching model with rural middle school mathematics teachers. The online coaching model was an adaptation of West and Staub’s (2003) content-focused coaching, which prioritizes focusing on mathematical content ...
This study of six teachers focuses on the ways they organized the classroom discourse, attended to student thinking, and adapted complex tasks from a Standard-based middle school curriculum. The study explores Cohen’s (2011) premise that... more
This study of six teachers focuses on the ways they organized the classroom discourse, attended to student thinking, and adapted complex tasks from a Standard-based middle school curriculum. The study explores Cohen’s (2011) premise that the knowledge teachers develop is related to their attentiveness to student teaching. This study explores the relationship between the extent to which teachers were successfully able to elicit and organize instruction around student strategies and their ability to productively adapt tasks in terms of being responsive and maintaining cognitive demand. The results show that teachers with the most student-centered discourse practices were also able to provide the most detailed justifications for task adaptations and to productively adapt tasks from the Connected Mathematics Program (CMP) curriculum.
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In this chapter, the authors present the design rationale for and empirical results from a predominantly synchronous three-part online model for the professional development of mathematics teachers in rural contexts. They describe how the... more
In this chapter, the authors present the design rationale for and empirical results from a predominantly synchronous three-part online model for the professional development of mathematics teachers in rural contexts. They describe how the design of the components are complementary and are intended to support teachers to develop challenging instructional practices, even when the teachers are geographically remote and dispersed. The three parts include an online course, online video coaching, and online demonstration lessons. They describe how they used conjecture mapping to enhance collaboration within the project team and to inform iterations of the model. They then present empirical results related to each of the components of the model and draw conclusions based upon what they have learned.
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One great challenge of the reform inspired by NCTM's Standards (1989, 2000) has been to transform classroom discourse in ways that actively involve students and use their ideas as sources of mathematical ideas in the classroom... more
One great challenge of the reform inspired by NCTM's Standards (1989, 2000) has been to transform classroom discourse in ways that actively involve students and use their ideas as sources of mathematical ideas in the classroom community. In this article, I present a type of classroom discourse—teacher-orchestrated classroom argument—that serves those functions.
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Asking middle school students to verify the math they do requires them to think about proof. By doing so, students construct arguments in the middle school and are more ready for proof in high school.
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Finding and emphasizing connections between strands of mathematics is Standard 4 of the Curriculum and Evaluation Standards for School Mathematics (NCTM 1989). The activity described in this article is designed to review and connect a... more
Finding and emphasizing connections between strands of mathematics is Standard 4 of the Curriculum and Evaluation Standards for School Mathematics (NCTM 1989). The activity described in this article is designed to review and connect a number of algebra and geometry topics, as well as to involve students in formal calculations, visualization, pattern discovery, series, estimation, and recursion. This activity centers on Baravelle spirals created in regular polygons, as discussed by Newman (1989).
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Abstract We report findings on teachers’ noticing of features in the teacher resources of mathematics curriculum programs. Based on prior analysis, we selected teachers using one of two curriculum types: delivery mechanism or thinking... more
Abstract We report findings on teachers’ noticing of features in the teacher resources of mathematics curriculum programs. Based on prior analysis, we selected teachers using one of two curriculum types: delivery mechanism or thinking device. The participating teachers and the curriculum programs aimed to align with the Common Core Standards for Mathematics, and thus, they ostensibly held a common aim for instruction. We analyzed 147 lesson planning interviews with 20 middle school mathematics teachers. We found that teachers attended to similar features of teacher resources; however, patterns for interpreting and planning decisions varied based on teachers’ orientations and curriculum type.
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Abstract We conceptualize curriculum ergonomics as a field that studies the interactions between users and curriculum materials. We identify five themes that a curriculum ergonomics lens brings into sharper focus: (1) teachers’... more
Abstract We conceptualize curriculum ergonomics as a field that studies the interactions between users and curriculum materials. We identify five themes that a curriculum ergonomics lens brings into sharper focus: (1) teachers’ relationships with and capacity to use curriculum resources; (2) alignment between design intentions and patterns of curriculum use; (3) ways in which curriculum resources influence instruction; (4) ways in which curriculum features are purposefully designed to achieve an educative purpose; and (5) the dissolution of boundaries between design and use. We first summarize the literature in curriculum ergonomics and then show how key themes from that literature inform the discussion of curriculum ergonomics.
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Abstract 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... more
Abstract 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.
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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... more
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.
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Student-generated algorithms, despite being inelegant and cumbersome, can nevertheless highlight a Common Core standard on rational number subtraction to show flexibility and understanding.
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ABSTRACT 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,... more
ABSTRACT 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.
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Introduction An enduring challenge in mathematics education reform is to help teachers develop competency at new and complex practices (cf. Smith, 1996). The mathematics reform represented by the NCTM Standards implicates teaching... more
Introduction An enduring challenge in mathematics education reform is to help teachers develop competency at new and complex practices (cf. Smith, 1996). The mathematics reform represented by the NCTM Standards implicates teaching practices that involve highly developed ...
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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... more
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 improve...
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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... more
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.
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... 1992; Yackel and Cobb 1996) or teachers who are considered experts in mathematics education (eg Ball 1993; Lampert 1992; McClain and ... In the first, Robyn Jorgensen employs a Bourdieuian analy-sis to explore the 'synergy'... more
... 1992; Yackel and Cobb 1996) or teachers who are considered experts in mathematics education (eg Ball 1993; Lampert 1992; McClain and ... In the first, Robyn Jorgensen employs a Bourdieuian analy-sis to explore the 'synergy' between the culture of school mathematics and ...