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This article shares a paper presented as the keynote for the Terry L. Wood Memorial Conference that was held virtually on May 20, 2022. This paper starts with a reference to the project, The Coordination of a Psychological and Sociological Perspective, that was funded by the Spencer Foundation. The results of the research in this project are published in the book, The Emergence of Mathematical Meaning, edited by Cobb and Bauersfeld (1995). [ABSTRACT FROM AUTHOR]

In this paper, illustrative excerpts from mathematics lessons are analysed to examine mathematical and multilingual moves between representations within Sepedi and English medium classrooms. Duval's theory of representational registers and the literature on multilingual practices help foreground similarities within and differences between the instructional practices of four teachers—two teaching in the medium of Sepedi and two in English—as they move between mathematical and multilingual representations in the teaching of numbers. The findings show patterned differences on the basis of the medium of instruction, with the Sepedi medium instruction indicating, primarily, 'restatement' moves between the oral and the symbolic modes of representation, whilst the English medium instruction incorporated a higher incidence of mathematical moves between oral, concrete, iconic and symbolic number-based modes of representation. The evidence of broader use of mathematical representational moves as observed in the English medium lessons offers insights into how and, perhaps, why learners' knowledge development and understanding of early number in English medium classrooms may emerge in ways that are different from the knowledge developed in the Sepedi medium classes. These patterned differences are important to understand in a South African context where language of instruction continues to be associated with differences in mathematical outcomes in ways that, as previously documented, relate to socio-economic disadvantage. [ABSTRACT FROM AUTHOR]

This paper presents the Mathematics Teacher's Specialised Knowledge (MTSK) model. It acknowledges earlier contributions to understanding and structuring teachers' knowledge, in particular, the special debt owed to Shulman's notion of pedagogical content knowledge and to Ball and collaborators' Mathematical Knowledge for Teaching (MKT), influential for the specialised nature of one of its sub-domains. The authors' research with teachers has led them to explore the characteristics of MKT and to refine the descriptors relating to its sub-domains, a task which has underlined the difficulty involved in unambiguously delimiting the boundaries which separate these. As a result, and taking into consideration a broader view of the specialised nature of the teacher's mathematical knowledge, the authors propose a framework which, whilst respecting the major domains of Content Knowledge and Pedagogical Content Knowledge, regards the specialisation in respect of mathematical knowledge as a property which is inherent to the model and extends across all sub-domains. [ABSTRACT FROM AUTHOR]

Mathematics teachers across the Western world are faced with an expectation that they make significant change to their teaching, but repeated attempts have shown little embedded success. This paper draws on a longitudinal study of two apparently well-placed English mathematics departments attempting to make change aligned with both policy and internationally-valued ‘good practice’. It suggests deep teacher change draws on a wide range of both social and affective characteristics, as well as sophisticated professional skills and knowledge. The study supports a construct of ‘(mathematics) teacher capacity for change’ at both individual and group levels within teachers’ ‘personal domains’, synthesising the range of characteristics apparently needed by teachers in times of change. In particular, it argues for the development of dispositions for collaborative learning and of other learning-supportive affects. Such an approach has the potential to place teachers in a better position to respond to demanding aspirations. [ABSTRACT FROM PUBLISHER]

The paper describes a personal experience with using counterexamples as a pedagogical strategy in the teaching and learning of an introductory calculus course at a university of technology. [ABSTRACT FROM AUTHOR]

Attending to student sense-making and enacting asset-based approaches to mathematics teaching are becoming a more central focus of mathematics teacher education. Less attention, however, has been given to supporting early career mathematics teachers with the everyday challenges of attempting to bring this vision into the classroom while teaching within deficit-oriented systems of schooling. This article builds on the concept of educational dignity offered by Espinoza and Vossoughi to investigate how to support beginning teachers to counter dominant deficit discourse by honoring the dignity of young people as mathematical sense-makers. We analyze a co-designed teacher learning community in which a group of first-year teachers and a teacher educator used their own experiences with mathematics as a resource to connect with young people in the human experience of learning mathematics. We show how connecting with young people as mathematical learners fostered generative new understandings of mathematical content and deep engagement with children's dignity as mathematical learners. This research has implications for the design and study of approaches to supporting the disciplinary work of teaching in ways that more fully respect the humanity and potential of young people. [ABSTRACT FROM AUTHOR]

Recent decades have seen extensive efforts across education policy and research to support mathematics teachers to learn and enact more student thinking-centered approaches in classrooms. While sociocultural theory conceives of learning as involving shifts in practice and identity, studies of teacher learning typically focus on changes in teaching practice alone. Conversely, studies of teacher identity often focus on the narratives teachers construct about their professional lives rather than how they behave in the classroom. In this study, I take up recent calls to investigate mathematics teacher identity as a performance of self in practice, arguing that this enacted identity is visible and open to analysis as teachers participate with students in the classroom and that changes in enacted identity over time are a form of teacher learning. Drawing on qualitative analysis of three elementary mathematics teachers' classroom interactions before and after classroom-embedded coaching, I examine two shifts in teachers' enacted identities – the enactment of control and the enactment of patience – which illustrate how such shifts are intertwined with practice and represent teacher learning. This conceptualization of teacher identity has implications for how researchers study teacher learning and teacher educators design professional development experiences. [ABSTRACT FROM AUTHOR]

For Terry Wood, the personal and professional were seamlessly connected. In her "House with the Blue Door" graduate students and visiting scholars lived; people gathered for conversations and music; the community reflected and found connections between important ideas. This chapter sets the stage for the special issue by contextualizing Terry's personal and professional life with anecdotes shared during the May 2021 Terry Wood Memorial Event. We see how she conceptualized knowledge development as a collective enterprise, putting into practice this worldview as a basis for life and for advancing the field of mathematics education. This reflexive relationship between theory and practice informed a body of powerful scholarship that continues to shape the field. Less recognized—and perhaps more importantly—these stories, and the subsequent chapters, reveal how Terry's investment in teachers, graduate students and colleagues resulted in the growth and development of scholars and teachers across multiple countries and continents. [ABSTRACT FROM AUTHOR]

Research on culturally informed pedagogies (e.g., culturally relevant pedagogy, culturally sustaining teaching, reality pedagogy) has been ongoing for decades, yet very few studies follow teachers into their classrooms after they are introduced to the topic. We conducted an intrinsic case study of three practicing maths teachers after they completed a graduate course specifically on culturally responsive teaching. Our analysis of the data revealed mixed findings. The participants demonstrated sociopolitical consciousness and affirming views of students. However, other aspects of culturally responsive teaching were absent or partial. By interviewing and observing our participants in their classrooms, we gained access to specific and authentic examples from their curriculum and instructional practices, which extended and enhanced their learning about culturally responsive teaching and prompted us to reconsider our course content and pedagogy. We conclude by advocating for continued research that investigates and exemplifies best practices in culturally responsive teacher education. [ABSTRACT FROM AUTHOR]

A current research problem in mathematics education is the characterization of the role of teachers in the processes of technology integration in mathematics classrooms. This article shows how two secondary mathematics teachers taught the concept of derivative of a function at a point and the concept of derivative function, one of them using digital technology and the other one without using it. Their teaching was characterized by describing their hypothetical learning trajectories (learning goals, learning activities and the hypothetical learning processes). APOS theory (which stands for Action, Process, Object and Schema) was used to describe the hypothetical learning processes. The results showed that the use of digital technology in class may promote reflection among students without excessive computations, thus helping them to construct the concept of derivative. [ABSTRACT FROM AUTHOR]

This article provides a concrete illustration of how teachers in a primary school in Singapore discuss students' learning in a lesson study cycle and grew professionally as a community. Specifically, we examined how collaboratively analysing students' work serves as a useful practice for teachers to learn to work with diverse learners. The findings suggested that open discussions around students' work helped teachers to reflect upon their unwarranted perceptions of their students and their teaching. The study provided insights into how teachers' understandings of their students' diverse backgrounds, as well as teachers' understanding of subject content and pedagogy, developed as they participated in lesson study activities that were focused on analysing students' work. Our findings found that lesson study provided the following affordances to foster such changes: (1) eliciting hypotheses in dialogue; (2) creating space for alternative perspectives; (3) collaboratively scrutinizing student learning evidence for follow-up teaching; and (4) identifying problems for further discussion. While the illustration of this case is uniquely Singaporean, implications include concerns about teacher professional learning and teaching for equity common to many other educational contexts. [ABSTRACT FROM AUTHOR]

Mixed-attainment mathematics teaching is not a common practice in England, despite evidence that ability grouping is not an effective strategy for improving educational outcomes. This study compares mathematics in School M (mixed-ability groupings) and School S (sets) in relation to student beliefs, and teacher beliefs and practices. Questionnaire data from 286 students and 12 teachers were triangulated with lesson observations and interviews. This article suggests grouping practices could indeed influence students' mindsets, teachers' mindsets and teachers' beliefs and practices. An above average proportion of students in both schools reported growth orientations, although these beliefs were held more strongly by students in the mixed-attainment grouping. School M teachers also held stronger growth-mindsets than School S teachers. Mathematics teachers in both schools reported connectionist beliefs, but the students' experiences differed. Most students in School M perceive typical mathematics lessons as involving a substantial problem or challenge worked on collaboratively in pairs or small groups, and having several entry points. Students in both schools valued learning from mistakes, but School M students were more likely to both believe this would help them, and have access to this type of learning opportunity. Students taught in sets experienced mathematics as procedures delivered by teachers and reproduced by students. This has implications for further research as mixed-attainment groupings may be a factor in determining the way in which students experience learning mathematics. [ABSTRACT FROM AUTHOR]

Peer feedback is increasingly used to train assessment skills in teacher-training programs. However, studies investigating the roles of peer feedback providers' beliefs about peer feedback provision, their perceptions of their peer feedback message and their experienced emotions are still limited. This study aimed to explore how peer feedback accuracy and providers' comprehension of the learning task were related to these characteristics. Analysis of peer feedback provided by 53 preservice mathematics teachers revealed that peer feedback accuracy was predicted by cognitive-verification, cognitive elaboration and self-efficacy types of peer feedback. Providers' perceptions of their peer feedback message and anxiety also predicted peer feedback accuracy. No evidence was found that peer feedback related beliefs and perceptions predicted providers' comprehension of the learning task. [ABSTRACT FROM AUTHOR]

This article argues that mathematical knowledge, and its related pedagogy, is inextricably linked to the tools in which the knowledge is expressed. The focus is on digital tools and the different roles they play in shaping mathematical meanings and in transforming the mathematical practices of learners and teachers. Six categories of digital tool-use that distinguish their differing potential are presented: (1) dynamic and graphical tools; (2) tools that outsource processing power; (3) tools that offer new representational infrastructures for mathematics; (4) tools that help to bridge the gap between school mathematics and the students' world; (5) tools that exploit high-bandwidth connectivity to support mathematics learning; and (6) tools that offer intelligent support for the teacher when their students engage in exploratory learning with digital technologies Following exemplification of each category, the article ends with some reflections on the progress of research in this area and identifies some remaining challenges. [ABSTRACT FROM AUTHOR]

Whilst research underlines the importance of early mathematics in kindergarten, practitioners need effective and innovative approaches to pedagogy. Currently, very different approaches are deployed from an instructional, educator-led approach based on training programmes to a play-based approach. This intervention study examines the effects on the mathematical competency of these two pedagogies. Thirty-five kindergarten educators and 324 six-year-old children were randomly assigned to either a training programme, a play-based approach with card and board games or to the control group. Educators’ views on the interventions were gathered in semi-structured interviews. The results indicate higher learning gains overall for the play-based approach. Differentiated effects were found as tendencies: children with low competencies tend to gain more from training programmes compared to no intervention; children with high competencies gain more from the play-based approach than the training. Educators evaluated the play-based intervention with card and board games as better suited to children’s diverse needs. [ABSTRACT FROM AUTHOR]

As part of recent scrutiny of teacher capacity, the question of teachers’ content knowledge of higher level mathematics emerges as important to the field of mathematics education. Elementary teachers in North America and some other countries tend to be subject generalists, yet it appears that some higher level mathematics background may be appropriate for teachers. An initial examination of a small sample of textbooks for teachers suggested the existence of a wide array of treatments and depth and quality of mathematics coverage. Based on the literature, a new framework was created to assess the mathematical quality of treatments for both specialized knowledge and horizon knowledge in mathematics textbooks for teachers. The framework was tested on a sample topic of the circle area formula derivation, chosen because it draws heavily on both specialized and horizon knowledge. The framework may contribute to similar analyses of other topics in a broader range of resources, in the overall quest to better describe the details of what constitutes appropriate mathematics horizon knowledge for teachers. [ABSTRACT FROM AUTHOR]

The authors examined the implementation of written reflections in a Grade 4 mathematics classroom over the course of 8 weeks. Students in this case study engaged in a workshop modeled after Calkin's Writers' Workshop and within this workshop the use of writing as a reflective tool in mathematics was introduced. The authors explore how students used writing to evaluate their learning and how the teacher used the students' written reflections as a formative assessment for instructional purposes. Students' written reflections were coded and these codes were used to conduct an inductive thematic analysis. Analysis of written reflections via constant-comparison analysis was used for further differentiation. The findings show students' ability to accurately self-evaluate their problem-solving skills and highlighted students' confidence level with certain mathematical concepts. Teachers were able to use students' reflections as a place to begin conferring with a student for further clarification. The written reflections aided in instructional decisions and increased individual instruction when needed. The authors include implications for teacher practice and areas for future research. [ABSTRACT FROM AUTHOR]