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The goal of this paper is to share an analytic framework for understanding Students? Ways of Thinking (SWoT) in STEM-rich learning environments. Before revealing our refined coding framework, we detail the nature of our collaborations and the various analytic decisions that led to its formation. These collaborations supported our collective ability to make sense of SWoT and produce a more coherent perspective that can be operationalized in STEM contexts. Our analytic framework foregrounds student claim-making and the related evidence and reasoning used in support. Specific commentary about the development and application of each coding category is provided, including examples of student data and rationale for related coding decisions. Our analytic framework, and discussion of its formation, can help educators, curriculum makers, and policymakers make use of SWoT in the development of meaningful and effective STEM education.

A growing body of evidence suggests developing the attention of preservice, secondary mathematics (PSMTs) teachers towards professional noticing of student thinking should feature in teacher education programs. There were two aims for this qualitative study: first, to explore the extent to which an Interview Module (IM) supported the development of PSMTs' ability to notice and make pedagogical decisions based on student thinking evidenced in video and paper-based work samples. A secondary aim was to establish the viability of the IM in an Australian context. Overall, PSMTs regarded their involvement in the IM as beneficial to their development as teachers. Specifically, participants outlined that the IM helped to shift their beliefs about teaching and learning, and helped promote productive teacher dispositions. Furthermore, these shifts were enabled through opportunities to engage with authentic student work, and the access they were given to new forms of responding.

A critical aspect of Mathematics Teacher Leaders' (MTLs) multi-faceted work is facilitating collective disciplinary discussions relevant and accessible for teachers. While research has identified goals for teacher learning, how MTLs learn to support these goals in content-specific ways is still under investigation. In this study, we lift the classroom design and analytic framework of Productive Disciplinary Engagement to examine the work of 73 MTLs and two expert facilitators in a professional development setting. Our use of the framework as an analytic lens to examine leader professional development revealed critical MTL and facilitator practices that supported productive mathematical engagement and practices that hindered engagement. We contend that productive disciplinary engagement provides a fruitful lens as an analytic framework for research. We also argue that it may serve as a valuable design framework for MTL and teacher professional development in future research and development efforts.

Despite the recognized importance of mathematical proof in secondary education, there is a limited but growing body of literature indicating how preservice secondary mathematics teachers (PSMTs) view proof and the teaching of proof. The purpose of this survey research was to explore PSMTs' knowledge and beliefs about proof and proof teaching in the context of secondary mathematics teaching and learning. Six cases (PSMTs' survey responses) were purposively selected from an entire data set comprising participants from the United States and Australia. Using a situative learning perspective as the theoretical framework, the authors developed a sensible belief system [Leatham, K. R. (2006). Viewing mathematics teachers' beliefs as sensible systems. "Journal of Mathematics Teacher Education," 9(1), 91-102. https://doi.org/10.1007/s10857-006-9006-8] for sampled PSMTs and inductively conceptualized how their knowledge of and orientations toward proof acted as a function of their teaching and learning contexts. Furthermore, an analysis of data revealed that PSMTs' knowledge and beliefs about proof are multifaceted, dynamic and evolving as they engage with proof in their various contexts.

STEM education should be grounded in our knowledge of how students think in STEM-focused learning environments. However, little is currently known regarding specific types of thinking in these contexts. We explore the theoretical notion of STEM Ways of Thinking (SWoT) in interdisciplinary learning situations. Our primary focus is not on the curricular content of STEM education, but rather on the nature of the cognitive activity at play during STEM-focused activity. We give particular focus to students' claim making and reasoning during interdisciplinary STEM activity. Our framework focuses on the disciplines of science, engineering, and mathematics. While these disciplines have vastly different practices and epistemologies, we argue that claim making and reasoning provide an avenue for considering student thinking at their intersections. A brief literature review of SWoT is provided that illustrates trends and limitations amongst current theories and studies. After a detailed discussion of our own theoretical perspective of SWoT, we provide an illustrative learning segment to explore the explanatory and analytic power of our theoretical perspective. Implications of our work on curriculum and instruction are provided.

This study examined proposed teacher responses to students' work to investigate how they respond, what characteristics of a good response are more difficult than others to achieve, and whether particular student error types are more difficult to respond to appropriately. Sixteen preservice secondary mathematics teachers' proposed responses to five students' work to solve linear equations were analysed based on four characteristics of a good response: work toward student learning objective, draw on presented student thinking, draw on research on students' mathematical development, and leave space for students' future thinking. The preservice teachers' responses consistently met the last characteristic, but their skill at meeting the other characteristics differed markedly based on the type of student error in the work sample. An implication is the need to help preservice teachers learn how to address conceptual issues in their responses rather than solely focusing on procedural errors that are often irrelevant to meeting the learning objective.

In the era of standards-based reforms, informal teacher leadership is a critical factor in realizing instructional improvement. In this paper, we report on data from a one-year study of four early career mathematics teachers engaging in professional development around Common Core mathematical practices and leadership. Our findings highlight how the professional development structure supported the development of early career teachers' leader identity. Through iterative opportunities to participate in two communities of practice (within the professional development setting and in school-based professional learning communities) early career teachers were able to engage in collegial conversations and imagine themselves taking on new roles and responsibilities in order to support the learning of the teachers with whom they worked.

In this paper, we describe the interworking of a group of middle school mathematics teachers who engaged in Mathematics Studio, an adaptation of lesson study, across three years. We use this case to illuminate specific structures, protocols, and norms of interaction within Studio that create conditions for teacher learning and incremental changes in teachers' instruction. Our analysis revealed several discourse practices, including the adoption of a research lens, a shared language of affordances and constraints, and an orientation toward student learning that supported a culture of inquiry as teachers investigated genuine questions they had about instruction. In this paper, we elaborate on these practices and share examples of pedagogically productive talk. We claim Mathematics Studio has the potential to circumvent previously identified challenges to support generative learning and the ongoing growth in teachers' instructional practice. [ABSTRACT FROM AUTHOR]

Classroom observations have become an integral part of research related to mathematics education. In this qualitative study, we describe the current state of the mathematics education field with regard to the use of classroom observation. The research question was: "How is classroom observation being used to measure instructional quality in mathematics education research?" In all, 114 peer-reviewed manuscripts published between 2000 and 2015 that involved classroom observation as part of an empirical study were examined using a cross-comparative methodology. Seventy (61%) did not use a formalized classroom observation protocol (COP), 21 (18%) developed their own COP, and 23 (20%) used a previously developed COP. Of the implemented COPs, 44% have published validity evidence in a peer-reviewed journal. We perceive the great variety of research approaches for classroom observation as necessary and potentially challenging in moving mathematics education forward with respect to research on instructional contexts.

This paper highlights the second phase of a multi-university research project focusing on improving secondary prospective mathematics teachers' abilities to respond to student thinking. This work is based on an Interview Module, created by the authors, which showed gains in prospective teachers' (PSTs) abilities to attend to and interpret student thinking but did not show the same gains in their abilities to respond to student thinking. The Interview Module originally included a pre-post-video, readings, discussion, and one-on-one interview with a secondary student (Lesseig et al. in Math Teach Educ 5(1):29-46, 2016). This second phase adds a take-home assignment based on solutions from the secondary students who were interviewed in Phase 1. This take-home assignment was designed to specifically improve PSTs' abilities to respond to student thinking. Results indicate that gains were made in their ability to respond, as was intended, but, unexpectedly, greater gains were also found in their abilities to attend to and interpret student thinking compared with Phase 1.

Research documenting teachers' fragile understanding of proof and how it is advanced suggests that enhancing the role of proof in school mathematics will demand substantial teacher learning. To date, there is little research detailing what mathematical knowledge might support the teaching of proof or how professional development (PD) might afford such learning. This paper advances a framework for Mathematical Knowledge for Teaching Proof (MKT for Proof) that specifies proof knowledge across subject matter and pedagogical domains. To explore the utility of the framework, this paper examines data from four different PD settings in which teachers and leaders worked on the same proof-related task. Analysis of discussions across these four settings revealed similarities and differences in knowledge of proof made available in each setting. These findings provide a means to detail the MKT for Proof framework and demonstrate its usefulness as a tool for designing, analyzing and evaluating proof experiences in PD.

Calls to advance students' ability to engage in mathematical reasoning practices including conjecturing, generalising and justifying (CGJ) place significant new demands on teachers. This case study examines how Mathematics Studio provided opportunities for a team of U.S. middle school teachers to learn about these practices and ways to promote them in the classroom. Findings demonstrate how CGJ readings and focused discussions, coupled with repeated cycles of collaborative lesson planning, observation and debrief, supported the development of teacher knowledge, professional community, and teaching resources. In addition, this paper explores the role school leadership played in facilitating Math Studio to ensure these learning opportunities were realised. Documenting how Math Studio features and participants contributed to teachers' ability to implement CGJ focused lessons not only provides insights into the difficulties teachers have shifting instruction, but also adds to our understanding of school-embedded professional development more generally.

Students' declining performance in science and mathematics is an issue of great international concern. Recently, educators and researchers have begun to focus on affective factors such as interest to better understand STEM learning and persistence. Therefore, there is a need for effective measures of STEM interest that allow it to be tracked over time and to provide opportunities for early interventions by educators. One such instrument was recently developed to measure youth interest in STEM as a general construct and in four domains associated with STEM: earth and space science, life science, technology and engineering, and mathematics. In this paper, we explore the psychometric properties of the measure scales by administering the instrument to a large sample of youth from both a traditional and STEM-focused school and examining theorized relationships using confirmatory factor analysis as part of a larger structural equation model. Results provided support for a single latent dimension of STEM interest, confirmed the existence of the four individual STEM interest dimensions, and provided evidence of structural and generalizability validity. We conclude that the instrument provides a sufficient means to measure STEM interest for adolescent youth within a variety of populations and educational contexts.

Introductory and remedial courses in college mathematics often serve as a gateway to student success. However, these university courses have seen little innovation despite historically low success rates. We sought to address deficits of the traditional lecture model and enhance student learning in an introductory college algebra course through the implementation of a flipped classroom. In this paper, we fully describe the course design, and discuss the interplay between online and in-class activities. We then present data regarding student self-efficacy and success rates in subsequent quantitative courses from our work across five semesters of implementing and researching the course.

Despite the recognised importance of mathematical proof in secondary education, there is a limited but growing body of literature indicating how preservice secondary mathematics teachers (PSMTs) view proof and the teaching of proof. The purpose of this survey research was to investigate how PSMTs in Australia, the USA and Korea perceive of proof in the context of secondary mathematics teaching and learning. PSMTs were able to outline various mathematical and pedagogical aspects of proof, including purposes, characteristics, reasons for teaching and imposed constraints. In addition, PSMTs attended to differing, though overlapping, features of proof when asked to determine the extent to which proposed arguments constituted proofs or to decide which arguments they might present to students.

Despite the increasing number of inclusive STEM schools, little is known about the cultural dimensions that influence STEM curriculum and instruction within these schools. This paper describes research conducted at three inclusive STEM schools, one each at the elementary, middle, and high school level. We explored similarities and differences in cultural dimensions across the schools with specific attention to how these differentially influence teachers' perceptions and enactment of STEM curriculum and instruction across the elementary, middle, and high school levels. Our cross-case analysis revealed structural aspects (school vision, community partnerships, course scheduling, and testing pressures) as well as professional orientations (i.e., instructional practices, interdisciplinary collaboration, and teacher content knowledge) that appear particularly important to student learning experiences at each grade level. Navigating these factors of STEM school culture requires teachers to not only be knowledgeable but also to draw on a professional orientation that encourages collaboration and risk-taking. We discuss implications for teacher education and STEM school development.

This study describes the development and validation of an instrument to measure graduate teaching assistants' (GTAs) learning about teaching during professional development. In the pilot study, exploratory factor analysis of data from 239 graduate students indicates a single factor structure. The second study, involving 177 science, technology, engineering, and mathematics (STEM) GTAs, confirms the single factor structure of the instrument. The instrument is highly reliable with both populations. The instrument is correlated to the hours STEM GTAs spend in professional development and their self-efficacy in teaching. It is sensitive to departmental differences between GTAs perceptions of their professional development. This instrument has multiple possible users including university faculty involved in GTA professional development as well as educational researchers. University faculty can use it for needs assessment during GTA program development, comparisons among departmental programs, and in improving current GTA programs. It also provides a sensitive measurement of the quality of GTA professional development in multiple program research studies.

Mathematical argumentation is an essential part of the discipline of mathematics and a key indicator of mathematical proficiency. In the process of constructing arguments and critiquing the reasoning of others, students build their understanding of underlying mathematical ideas and engage in critical sense-making activity (Yackel and Hanna 2003). Despite the importance of this practice, teachers--both novice and experienced--have difficulty incorporating argumentation in the classroom. At the high school level, constructing arguments has often been equated to proof writing in geometry. Limiting argumentation to verifying formulas neglects the explanatory role that proof can play in learning mathematics at all levels (Hanna 2000). Adapting or designing activities for students to investigate why something is true and to communicate their reasoning requires intentional work on the part of teachers. In this article, the authors describe strategies for promoting mathematical argumentation developed by a group of teachers as they participated in four cycles of Mathematics Studio focused on argumentation. Mathematics Studio (Teachers Development Group 2010) is a research-based professional development model that is similar to Japanese lesson study in that teachers collaboratively plan, observe/enact, and analyze a classroom lesson. The focus of Mathematics Studio is not on creating a lesson to be used by others but rather to increase teachers' understanding of mathematical argumentation and ways to develop students' abilities related to this practice (Lesseig 2016).

Background: Despite increasing attention to STEM education worldwide, there is considerable uncertainty as to what constitutes STEM education and what it means in terms of curriculum and student outcomes. The purpose of this study was to investigate the commonalities and variations in educators' conceptualizations of STEM education. Sensemaking theory framed our analysis of ideas that were being selected and retained in relation to professional learning experiences in three contexts: two traditional middle schools, a STEM-focused school, and state-wide STEM professional development. Concept maps and interview transcripts from 34 educators holding different roles were analyzed: STEM and non-STEM teachers, administrators, and STEM professional development providers. Results: Three themes were included on over 70% of the 34 concept maps: interdisciplinary connections; the need for new, ambitious instructional practices in enacting a STEM approach; and the engagement of students in real-world problem solving. Conceptualizations of STEM education were related to educational contexts, which included the STEM education professional development activities in which educators engaged. We also identified differences across educators in different roles (e.g., non-STEM teacher, administrator). Two important attributes of STEM education addressed in the literature appeared infrequently across all contexts and role groups: students' use of technology and the potential of STEM-focused education to provide access and opportunities for all students' successful participation in STEM. Conclusions: Given the variety of institutionalized practices and school contexts within which STEM education is enacted, we are not convinced that a single worldwide definition of STEM education is critical. What we do see as essential is that those working in the same system explore the common elements that are being attributed to STEM education and co-construct a vision that provides opportunities for all their students to attain STEM-related goals. This is especially important in the current reform contexts related to STEM education. We also see that common conceptions of STEM education appear across roles and contexts, and these could provide starting points for these discussions. Explicitly identifying the ideas educators are and are not selecting and retaining can inform professional learning activities at local and larger scales.

In this article, we report on "Researching Mathematics Leader Learning" ("RMLL"), a project designed to support leaders in learning how to facilitate robust opportunities for teachers' mathematical learning. Our two-phase research design allowed us to construct a set of videocase seminars, enact the seminar design with leaders, analyze these data, refine our seminar design, and implement a second set of seminars with a new group of leaders. We drew on the noticing literature to examine leaders' pedagogical reasoning as they discussed videocases of professional development. In this article, we demonstrate how changes in our framework for leader development and the resulting changes in the prompts and tasks shaped leader noticing in three ways: (a) accounting for the mathematical work of the facilitator and teachers in the videocase; (b) linking the mathematical work to goals for teacher learning; and (c) reasoning around the facilitator's work in advancing those learning goals. Analysis indicates that in Phase II, leader discussions were more focused on the mathematical and pedagogical work needed to advance teacher learning. Based on our research and development work with over 70 leaders, we offer a set of design principles for leader professional development.

Flipped instruction is increasing in popularity, however research that moves beyond descriptions of its implementation in mathematics classes is lacking. We sought to better understand how students taking an introductory college algebra course used the resources provided within a flipped instructional model and how students viewed such resources as useful to their learning. Data from course activity records and student usage surveys were used to characterize the frequency of student access to course resources (e.g. video lectures and comprehension assessments) and students' perceptions of the usefulness of these resources. Students naturally fell into one of four categories with respect to video usage and success in the course. Students who employed videos as intended were successful in the class. Students rated watching videos and working on homework in class as the activities that best supported their learning. Students did not perceive group work and in-class presentations as helpful. Our findings have implications for other instructors implementing flipped instruction. We argue that while flipping collegiate mathematics courses has great potential, instructors need to explicitly attend to the design of the in-class components in order to maximize student learning.

Applying the Mathematical Knowledge for Teaching framework, we discuss the components of teacher knowledge that might be useful in supporting mathematical inquiry, and examine ways in which we strive to develop this knowledge within a middle grades mathematics program for undergraduate students who are prospective teachers. Using sample activities from multiple courses in the program, we offer general principles of instruction for supporting mathematical inquiry at all grade levels. We contend that an awareness and application of the multiple facets of mathematical knowledge for teaching can be critical to supporting mathematical inquiry across the K-16 spectrum.

We describe and analyze a professional development (PD) model that involved a partnership among science, mathematics and education university faculty, science and mathematics coordinators, and middle school administrators, teachers, and students. The overarching project goal involved the implementation of interdisciplinary STEM Design Challenges (DCs). The PD model targeted: (a) increasing teachers' content and pedagogical content knowledge in mathematics and science; (b) helping teachers integrate STEM practices into their lessons; and (c) addressing teachers' beliefs about engaging underperforming students in challenging problems. A unique aspect involved low-achieving students and their teachers learning alongside each other as they co-participated in STEM design challenges for one week in the summer. Our analysis focused on what teachers came to value about STEM DCs, and the challenges in and affordances for implementing DCs. Two significant areas of value for the teachers were students' use of scientific, mathematical, and engineering practices and motivation, engagement, and empowerment by all learners. Challenges associated with pedagogy, curriculum, and the traditional structures of the schools were identified. Finally, there were four key affordances: (a) opportunities to construct a vision of STEM education; (b) motivation to implement DCs; (c) ambitious pedagogical tools; and, (d) ongoing support for planning and implementation. This article features a Research to Practice Companion Article.

The purpose of this study was to detail teachers' proving activity and contribute to a framework of Mathematical Knowledge for Teaching Proof (MKT for Proof). While working to justify claims about sums of consecutive numbers, teachers searched for key ideas and productively used examples to make, test and refine conjectures. Analysis of teachers' mathematical activity revealed knowledge of the proving process that would be useful for and useable in the teaching of proof. This includes knowledge of the interconnections among empirical exploration, conjecturing, generalizing, and justifying as well as an understanding of the characteristics of examples and conjectures that could support the proving process. The central premise of this paper is that delineating aspects of teacher knowledge is a first step to supporting teachers' efforts to engage all students in fundamental mathematical practices of conjecturing, generalizing and justifying.

Background: This study is about teachers' collective activity during the development and initial year of a science, technology, engineering, and mathematics (STEM)-focused school in the USA. The target school of this study was inclusive, as it sought admission of students from varying backgrounds and levels of ability. Drawing from narrative inquiry and case study methodologies, we examine the collective work of the teachers in the target school from 6 months prior to school start-up through the end of the first year. We focus on visioning, collaboration, and curriculum development in our analysis of the teachers' collective work. Results: We analyze the collective sense-making activity of the teaching staff regarding key facets of the start-up process. While the teachers received a variety of supports, including time and resources for collaborating, there was a lack of specific support for the conceptualization and creation of multi-disciplinary, STEM-focused projects. The risk-taking and collaborative actions of the teachers led to three specific instructional approaches that were continuously adjusted to respond to the evolving vision of the STEM-focused school. The teachers also solicited the needs and interests of their students and utilized these in curricular design and instruction, which promoted student buy-in and participation. By the end of the school year, a common vision for STEM-focused, project-based learning was emerging, but not solidified. Conclusions: Our study confirms the power of "doing" and "risk-taking" in teacher development, particularly in the ways in which teacher collaboration advanced curriculum and instruction in this STEM-focused school context. The intellectual supports that teachers require in this context are numerous and must be carefully identified and nurtured, and the subsequent teacher activity must be monitored as contextual shifts occur and sources of pressure (e.g., external learning standards) become relevant. The teachers' role is a complex mixture of learner, risk-taker, inquirer, curriculum designer, negotiator, collaborator, and teacher. Instructional and curricular supports require substantial time to synthesize and eventually enact, and more than a few months prior to school start-up are necessary to fully engage and prepare teachers for the collective task of visioning, collaborating, and planning the curriculum and instruction of an innovative school.

Filling the knowledge gap in the limited research on professional development leaders is an urgent issue if teacher learning is to be improved. This research and development project is studying how leaders learn to cultivate mathematically rich professional development environments. The authors adapted two frameworks from classroom-based research--sociomathematical norms and practices for orchestrating productive discussion--to support leaders' understanding of facilitation of mathematics professional development. In this article, the authors describe the use of these frameworks in their work and argue for a third framework--the mathematical knowledge for teaching. Based on the analysis of their work, they believe that mathematics professional development leaders need to cultivate particular sociomathematical norms for teacher explanation and employ practices for orchestrating discussions to achieve the purposeful development of teachers' specialized knowledge of mathematics for teaching. (Contains 1 table and 8 notes.)