Mathematics discussions are central for both teaching practice and student engagement with mathematics, though modal teaching practice in many countries often does not include discussion-leading. There has been a press for professional development that increases teachers’ use of discussion-leading in mathematics. This paper examines the impact of two key components of an intensive professional development on practicing teachers’ mathematics discussion-leading practice: live observation and practice-focused learning sessions. A tool for examining specific discussion-leading moves was used to measure change in teachers’ practice as observed through submitted video recordings of mathematics discussions. Participants included 44 teachers from three school districts who submitted a total of 255 videos across the study. The findings show that both key components of the professional development positively impacted teaching practice but had differential impacts on participating groups. We explore factors that could contribute to these differential outcomes.
This paper presents a systematic review of literature in mathematics teacher education research, which investigated the (1) terminology and theories used to conceptualize ecological approaches; (2) divergences from principles of ecology; and (3) insights from approaches that converged with ecology. Ecological conceptualizations in mathematics teacher education were highly varied and rarely drew from ecology. Most often, ecological was used as an imprecise descriptor, and conceptualizations contained elements that diverged from principles of ecology. Convergences with ecology included (1) a shift from linear, binary, or mechanistic models to participatory, relational, and dynamic processes, (2) de-centering individuals or isolated objects to focus on a unit of analysis that comprises relationships, interactions, and/or communal concerns, (3) weaving together the complex entanglement of interactions that comprise, reciprocally interact with, and elevate the significance of the environment. A comparison of methodological dimensions between the discipline of ecology, Gibson’s (1979) ecological approach to visual perception, Bronfenbrenner’s (2001) bioecological model, and Godino et al.’s (2007) ecological suitability is presented. Methodological questions and tools are introduced as ways to generate future ecological approaches in mathematics teacher education, and the implications of considering timescales using an ecological understanding is discussed using teacher noticing as an example.
Teachers’ emotions both shape and are shaped by their teaching and interactions with others in their educational setting. Researchers have called for more attention to the emotional dimension of teachers’ work, specifically in the context of professional learning for changing practice. In this study, six Australian secondary mathematics teachers at a large low-SES multi-ethnic government school experimented with non-traditional algebra learning tasks and problem-solving pedagogies to engage their Year 8 students. Drawing on a social–psychological theoretical framing of teacher emotions in the context of educational change, we share findings on the teachers’ emotional experiences through their expressions in collective and individual settings. Initially focussing on situational demands in appraisals about their changes in practice, the teachers increasingly evidenced identity talk that brought to light an emerging collective narrative that their experimenting benefited their students yet cost them emotionally. The openness of experienced and leading teachers in a collective setting, and affirmation by colleagues to sharing emotionally charged experiences, was found to stimulate identity talk. In both collective and individual settings, the teachers were more likely to express positive valence with explicit emotion labels but to imply negative valence with descriptions or ambiguous emotion words. Possible implications for school leaders and facilitators of teacher professional learning are discussed.
This study examined the interaction between defining and classifying processes to inform the instructional design for teaching the triangle concept in teacher education programs. To address a gap in the literature regarding the effect of task sequencing on conceptual learning, this study employed independent sequential tasks focusing on the triangle concept. The study, conducted with 209 pre-service teachers, systematically analyzed the effects of defining on subsequent classifying and of classifying on subsequent defining. The findings indicate that pre-service teachers were generally successful in accurately classifying triangle examples and non-triangular non-examples, but that misclassifications were particularly concentrated in triangular non-examples. Furthermore, they frequently provided inappropriate statements when defining the triangle. It was also determined that there was no association between pre-service teachers’ defining performance and classifying performance. Whether the classifying task was administered before or after the defining task did not affect participants’ ability to construct a triangle definition containing both necessary and sufficient conditions. Performance in classifying triangle examples and non-triangular non-examples was unaffected by task sequencing, whereas performance in classifying triangular non-examples was influenced by task sequencing. These findings have implications for the instructional design for teaching the triangle concept in teacher education. Most notably, the findings suggest that reflecting on the formal definition of the triangle before the classifying task may provide a modest advantage in identifying non-intuitive non-examples of triangles.
There has been increased attention to instructional practices that teachers can use to support students in constructing an understanding of mathematics through discourse-rich lessons. Teachers need to support the class in synthesizing ideas at the end of such lessons so that students know what to take away from the lesson. We focus on the instructional practice of modeling, one strategy for supporting such synthesis. A decomposition of the practice of modeling content was developed and used to design a prototype formative assessment task that gathers information about prospective elementary teachers’ capabilities with modeling content. In order to establish the potential of the task to elicit variation in performance at a level of granularity sufficient for formative purposes, we examined the following questions: What patterns of PTs’ capabilities with modeling can be revealed through the assessment? Is the level of detail conducive to the provision of feedback and to design subsequent instruction? Ten first-year teachers who graduated from various teacher education programs participated in the study. Findings reveal that our prototype formative assessment works to gather information about prospective teachers’ capabilities with modeling and is capable of eliciting variation in performances. Further, the level of specificity of the associated tools has the potential to support both feedback to teachers and planning for supporting teachers’ learning. This suggests that such a task and associated tools could support the formative assessment of the developing capabilities of prospective teachers.
Many teachers seek to foster students’ conceptual understanding, e.g., for meanings of multiplication, but are only partially prepared to go beyond surface translations between multiple representations. This concerns out-of-field teachers as well as in-field teachers (i.e., with mathematics teaching certificate). The paper presents a professional development program in which teachers engage in PD inquiry activities (a) to specify what understanding multiplication entails and how deeper connections of representations are articulated, (b) to monitor students’ understanding and uncover surface translations, and (c) to enhance students’ understanding by engaging them in explicating unit structures in graphical and symbolic representations. The randomized controlled trial with n = 94 teachers investigates the extent to which out-of-field and in-field teachers’ approximated practices can change in a PD session, and whether their professional learning gains depend on an explicit summary in the PD systematization phase. The comparison of pre-vignettes and post-vignettes reveals significant professional learning gains for all three practices, slightly more strongly for out-of-field teachers. For out-of-field teachers, the explicit summary in the systematization phase has significant additional effects for specifying and enhancing, less for monitoring practices. The findings show that out-of-field teachers can undergo very promising changes, especially when including an explicit systematization phase on specifying the mathematical learning content.
Mathematical tasks play a central role in shaping students’ opportunities to engage in complex cognitive processes such as mathematical problem solving. While previous research has highlighted the importance of teachers’ professional competence for instructional quality, less is known about how different facets of teachers’ dispositions jointly relate to the selection of tasks with high problem-solving potential. This study investigates how combinations of teachers’ mathematical pedagogical content knowledge and beliefs about mathematics and mathematics teaching are associated with the use of problem-solving tasks in lower secondary classrooms. Drawing on data from a German follow-up study of the international Teacher Education and Development Study, the analysis is based on a subsample of 31 in-service teachers whose classroom tasks were systematically collected and analyzed with respect to their problem-solving potential. Teachers’ knowledge and beliefs were assessed using established instruments. To account for the assumed complexity and interdependence of teachers’ dispositions, a Qualitative Comparative Analysis was conducted. The results show that no single facet of teachers’ dispositions is necessary or sufficient for the use of tasks with high problem-solving potential. Instead, configurations combining high levels of mathematical pedagogical content knowledge with either dynamic views of mathematics or constructivist beliefs about teaching and learning emerge as sufficient conditions. These findings underscore the importance of considering the interplay of cognitive and affective-motivational facets of teacher competence when studying task selection in mathematics education.
While much extant work on teacher noticing documents what teachers notice individually or in groups, teacher educators and researchers would benefit from understanding how teachers’ independent noticings may inform collaborative conversations to extend their noticing skills. In this paper, we describe an episode of teachers’ professional learning in which grade-level teams of elementary teachers engaged in a noticing task and were given opportunities to write independently about their noticings while watching a video of a mathematics lesson, then discuss their noticings collaboratively with colleagues. We first examined their independent written responses and then considered how the topic of their noticings and stances toward what they noticed in the video broadened through subsequent collaborative discussions. Findings indicate that when teachers engage in collaborative noticing, they take on more application-based stances than when they notice independently in writing. Additionally, findings suggest that collaborative noticing offers rich opportunities for engaging in pedagogical conversations with colleagues, which are likely to support teachers' professional development.
What happens when a Swedish mathematics teacher, who has experience of using a specific learning theory—variation theory for learning—as a tool to plan, evaluate, and revise his teaching, is asked to add and apply the Chinese framework of Bianshi (teaching with variation), which also builds on principles of variation? In this paper, we report on a case study where an elementary mathematics teacher in Sweden taught a lesson about fraction multiplication, which was video-recorded, and, in an appurtenant interview, explained the rationale behind his instructional choices. This was followed by a three-month intervention that introduced the teacher to the Bianshi framework of variation. After the intervention, a stimulated recall interview was conducted to obtain information about the way in which the teacher made sense of the Chinese framework when reflecting on his lesson about fraction multiplication. The study contributes to the discussion about whether and how the two frameworks of variation are complementary.
Early childhood teachers and mathematics teacher educators use student-centered mathematics routines like Notice and Wonder and Clothesline Math to promote sensemaking, encourage curiosity, and make mathematics more accessible for students. We used qualitative content analysis to examine 67 digital artifacts of practicing teachers’ implementations of routines they were learning in a mathematics and technology education course. Using the four principles of productive disciplinary engagement (problematizing, authority, accountability, and resources) as an analytic framework, we coded teachers’ photos and written reflections to better understand how enactments of their routines created opportunities for mathematics learning. Forty-nine of the 67 artifacts showed evidence of problematizing, typically featuring explanatory text alongside visuals that connected student mathematical ideas back to posed mathematical challenges. We also created two additional codes (depth of problematizing and depth of accountability) to further describe how the artifacts tied the routine to learning goals and highlighted students’ verbal contributions. Findings suggest that although teachers valued student-centered instruction and mathematics conversations, they may need additional support in connecting classroom activities to curricular learning goals and promoting equitable participation. This research offers implications for mathematics teacher educators seeking to deepen early childhood teachers' understanding of effective student-centered mathematics instruction. The principles of productive disciplinary engagement can scaffold teacher reflection and strengthen opportunities for mathematical reasoning during routines.
The role of explanation in mathematics teaching is central to promoting student understanding, yet research has been limited by the challenges of analysing large volumes of classroom discourse. This study explores the use of AI to identify, segment, and classify explanatory textual sequences in secondary education mathematics classrooms. Drawing on textual linguistics and typologies of explanation in mathematics education, the study analyses 28 h of classroom recordings from six high school mathematics teachers. A two-step methodology was applied: (1) segmentation of explanatory sequences using Whisper and GPT-4, and (2) classification into interpretative, descriptive, and justificative types. The inter-rater agreement between GPT-4 and the coding team reached 92.4
Teachers’ mathematics-related beliefs have a substantial influence on their teaching. Given the variety of opportunities for teaching mathematical modeling supported by digital tools, related beliefs are crucial for classroom implementation. In this paper, we introduce a course that aims to prepare future mathematics teachers for teaching mathematical modeling in school using digital tools (focusing here on MathCityMap as an example). We used linear mixed models to analyze the development of 54 preservice teachers’ beliefs over the course, comparing them with 58 mathematics preservice teachers who did not attend the course. Results show an increase in course participants’ self-concepts about digital tools and mathematical modeling, as well as a tendency toward a more dynamic and less static mathematical worldview. There are hints that different course units contribute differently to the development of beliefs. These results suggest that such a course can provide a foundation for teachers to integrate digitally supported mathematical modeling into the classroom.
Geometry education research has gained momentum in mathematics education because geometry remains a difficult area for learners and teachers. In this paper, we present two key components of an emerging analytical framework, which characterises the learning and relearning of basic geometric ideas for teaching with a specific focus on the lower secondary level. The need for a new framework to map progress in teachers’ emerging understanding at the basic level resulted from the limitations of existing frameworks, which focus mostly on complex geometric problems and figures. We elaborate on two aspects of the emergent framework, namely, geometry tasks and geometric reasoning. The levels within each aspect have the potential to distinguish different levels of complexity, offering a finer grain size to analyse tasks and geometric reasoning. This work emerges from a professional development course for secondary teachers who lacked confidence in their own geometrical knowledge. The research unpacks the constructs needed to understand how to support learning (and relearning) of basic geometry in depth, that is, properties of lines, angles and triangles. We present the rationale, description, and the use of the analytical framework. The proposed elements are useful in designing geometry tasks for pre-service and practicing teachers, and in promoting geometric reasoning among teachers and learners.
This study explores the characteristics of four secondary school mathematics teachers’ orientations when noticing their students’ mathematical thinking using real-time learning analytics (LA) on a digital platform. Teachers conducted lessons using tasks that allowed students to generate mathematical examples that were automatically assessed and analyzed. The digital platform provides automated visual reports to support teachers’ interpretations of student responses. Before using LA, teachers tended to adopt a deficit-based view. Interaction with LA encouraged a shift toward strength-based noticing, highlighting students’ mathematical reasoning rather than their mistakes. Visualizations, such as grids and histograms, enable teachers to identify patterns and common mistakes, promoting a more inquiry-based and evidence-driven approach. While normative responses remained central, teachers began to recognize the value of unexpected work. The findings underscore the role of interactive LA tools in shaping teachers’ orientations when they notice their students’ work, thereby supporting more responsive, student-centered, and equity-focused practices.
This study examines how a mathematics teacher, whose noticing skills developed through a 2-year online professional development program, implemented responsive teaching moves in a classroom setting while teaching the concept of the mean. Accordingly, the study was built on the responsive teaching model, which stems from noticing students’ mathematical thinking. After discussing various scenarios involving student strategies for alternative mathematics content with 30 other participating teachers in an online environment, a teacher with eight years of professional experience implemented two tasks related to the concept of the mean in two sixth-grade classrooms. The analysis revealed that the teacher employed responsive teaching moves, albeit in varying orders, with different frequencies, and at varying durations. This led us to conclude that a sustained professional development program focused on developing teachers’ noticing of students’ thinking appeared to improve teachers’ in-the-moment responses regarding the concept of the mean.
Mathematics teachers’ instructional vision, or the image they hold of valued mathematics teaching and learning, is a cornerstone of instructional improvement. Prior studies of instructional vision have focused on accessing teachers’ visions through interviews in which teachers imagine a hypothetical classroom that illustrates their aspirations. How they notice such a hypothetical classroom is connected to their professional vision, or how teachers notice specific artifacts and events. In this study, we examine how teachers’ instructional vision can be accessed and developed in professional development by asking teachers to notice and name strengths in video records from one another’s classrooms. We analyzed the strengths that 23 teachers noticed in 126 video discussions across a 2-year professional development program for early career secondary mathematics teachers. We grouped the 1186 strengths described by participants into 14 strengths categories, which we argue paint a rich portrait of participants’ collective instructional vision. Further, we identified three discursive mechanisms through which teachers developed—defined and refined—their instructional vision: leveraging engagement, leveraging growth, and leveraging agreement. These findings indicate that video-based professional development can both reveal and create opportunities for teachers to refine their instructional vision by discussing strengths. Methodological implications for accessing teachers’ instructional vision are discussed.
The publication Building Thinking Classrooms in Mathematics (BTC) generated remarkable worldwide interest from mathematics teachers, contrasting a marked resistance to instructional change in secondary mathematics classrooms over the past four decades. BTC’s practices align with existing research, yet create opportunities for support and coaching to fill gaps in addressing the needs of all learners. This paper uses narrative inquiry to capture the experiences of a first-year teacher whose read of BTC launched classroom experimentation in a high school with a high percentage of minoritized youth. The narrative highlights key obstacles and underscores the value of additional resources and coaching, grounded in mathematics education research, to sustain the momentum for classroom change that BTC inspired. As the teacher shifted from non-curricular to curricular tasks, new challenges emerged, requiring strategies such as high-leverage practices, conceptual models, and piloted task sequences. Findings situated in this teacher’s case illustrate a developmental progression of instructional shifts that characterize this teacher’s changes in practice towards student thinking-based instruction. This series of instructional shifts provides a testable model for future research. This model holds implications for supporting coaches and teacher educators in strategizing and targeting stimuli for teachers’ adaptive teaching as they learn to enact and fine-tune student thinking-based instruction inspired by BTC.
