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Preparing prospective mathematics teachers to become teachers who recognize and respond to students' mathematical needs is challenging. In this study, we use the construct of critical incident as a tool to support prospective mathematics teachers' reflection on their authentic fieldwork activities, notice students' thinking, and link it to the complexity of mathematics teaching. Particularly, we aim to explore the characteristics and evolution of prospective mathematics teachers' noticing of students' mathematical thinking when critical incidents trigger reflective discussions. Critical incidents are moments in which students' mathematical thinking becomes apparent and can provide teachers with opportunities to delve more deeply into the mathematics discussed in the lesson. In the study, twenty-two prospective mathematics teachers participated in fieldwork activities that included observing and teaching secondary school classrooms. The prospective teachers identified critical incidents from their observations and teaching, which were the foci for reflective discussion in university sessions. By characterizing the prospective teachers' reflective talk in these discussions, we demonstrate the discussion's evolution. In it, participants questioned learning and teaching mathematics and suggested alternate explanations. This characterization also shows that using critical incidents in the university discussions enabled the prospective teachers to link students' thinking with the teacher's teaching practices while supporting their reflection using classroom evidence. We emphasize the importance of descriptive talk in the discussion, which allows for deepening the prospective teachers' reflections. Further, we explore the teacher educator's contributions in those discussions, showing that the teacher educator mainly maintained the reflective talk by contextualizing the critical incidents and pressing the participants to explain further issues they raised in the discussions. Implications for mathematics teacher education are discussed. [ABSTRACT FROM AUTHOR]

In this paper, we draw on our recent research to inspect again some of the theoretical perspectives we have been using to analyze data and to characterize teaching-learning in university settings. We focus particularly, within a sociocultural perspective on Activity Theory and the construct the "Teaching Triad," seeking to embed the Triad within an Activity Theory perspective. To achieve this, we relate the Teaching Triad with aspects of the sociocultural setting both in and beyond direct interactions in face-to-face teaching. While this is mainly a theoretical paper, an example is taken from observations of teaching in university lectures in a Greek university to show how these theoretical perspectives have provided insights to the institutional and cultural complexities involved and in what ways the Triad construct has evolved. [ABSTRACT FROM AUTHOR]

The paper reports on updating and integrating two reformed curricula designed in an era of national crises in Greece. In particular, the curriculum design activity of the Leading Coordinators (LCs) is studied under the lenses of Activity Theory and Positioning Theory. Sixteen transcribed videotaped meetings over a period of ten months are analysed to examine LCs' positionings enacted within the design activity as well as the interaction of the latter with other activity systems in situ. LCs' discourse brought to the fore convergent and divergent positionings mainly related to their research identity, occasionally revealing tensions. Research was the dominant activity system that interacted with that of the curriculum design activity, giving rise to contradictions. These were intensified by socio-political implications of the crises and were mainly related to the formation of a shared object for the design activity, its course of development and the activity of other stakeholders. [ABSTRACT FROM AUTHOR]

Our study focuses on the interplay of two secondary school mathematics teachers' emotions and decision-making in pivotal teaching moments. We highlight the interaction between teacher emotions and in-the-moment decision-making, and resources that coordinate this interplay, suggesting a theoretical and methodological way of addressing it. Our analyzed data comes from three teachers' lessons and four semi-structured interviews. When analyzing pivotal moments where the teacher handles students' errors, it appeared that teacher emotions and resources are interrelated elements of the decision-making. Findings show that: (1) teachers' emotions, while handling students' errors, are mostly negative, but differ in their kind and source; (2) the formation of teacher emotions and actions often seems to draw on the same resources (social-spatial, anticipatory-temporal, moral-ideological dimensions, and teacher's responsibility about mathematics; (3)teachers' actions, while handling students' errors, differ in relation to the resources and the emotions that coordinate their formation. • Teachers' emotions interplay with their decision-making process during teaching. • This interplay can be captured by studying emotions' action tendency and appraisal components in relation to teachers' decision-making resources. • Emotions' appraisal component is directly related to the dimensions of the in-the-moment decision-making's resources. • Teachers' in-the-moment decisions are shaped both by their emotions and by other resources (or combination of resources). [ABSTRACT FROM AUTHOR]

Our study explores boundaries and boundary crossing between communities of mathematics teaching (MT) and visual art teaching (AT). We focus on teacher collaboration, draw on communities of practice and examine the existing boundaries and the way collaborating members handle them. We present data from 17 group meetings of secondary school art and mathematics teachers who try to develop ways of linking ΜΤ and ΑΤ. Results indicate the emerging boundaries by means of discontinuities regarding mathematical practices and tools used in MT and AT, and the disciplines' teaching and curriculums. Via analysis of the central boundary – the analytical or visual ways of thinking – we present members' boundary handling that indicates the development and developmental process of integrated practice. The study incorporates boundary crossing processes and learning mechanisms in which the two separate ways of thinking are gradually transformed into an integrated one and reveals possible ways of integration in teaching and learning. • Interdisciplinary collaborations interweave with handling of existing boundaries. • Interdisciplinary collaborations need facilitation and support. • Math-art teaching boundaries concern math practices, tools and curriculums. • Boundary objects, brokers, facilitation and coordination are key to boundary handling. • Boundary crossing between analytical and visual thinking leads to their integration. [ABSTRACT FROM AUTHOR]

The reported study is situated within the process of developing a reform-oriented national mathematics curriculum for compulsory education in Greece by a design team that involved teachers, academic researchers, and policy-makers. From an activity theory perspective, we identify the activity systems of mathematics teaching, research in mathematics education, and educational policy interacting in the design process. We focus on the contradictions between the three activity systems and how these were dealt with. We based our analysis on email exchanges during the curriculum design, field notes from whole-team sessions, and interviews with key persons. Our results highlight that the emerging contradictions primarily concerned the teaching and research activity systems. Members of the team who acted as brokers between the different activity systems and facilitated their interaction played an important role in overcoming the contradictions. [ABSTRACT FROM AUTHOR]

The study reported in this paper concerns the tensions and conflicts that teachers experience while they enact a new set of reform-oriented curricular materials into their classrooms. Our focus is οn the interactions developed in two groups of teachers in two schools for a period of a school year. We use Activity Theory to study emerging contradictions and we elaborate on the construct of dialectical opposition to understand the nature of these contradictions and their potential for teacher learning. We provide evidence that discussions about contradictions and their dialectical character in the two groups support teachers to engage differently in mathematics teaching and learning and carry potentials for shifts in the practices that evolve in their classrooms. Our study addresses empirically in the context of mathematics teaching the philosophical claim about the role of contradictions as a driving force for any dynamic system. [ABSTRACT FROM AUTHOR]

In this paper, we studied the transferring process of five apprentice engineering students concerning the place value concept in a telecommunication workplace setting. In this setting, a physical material, a telecommunication closet (TC), appearing as a concrete representation of the ten-numeration system, supported the process of generating composite units in a sequential way. The data were generated by conducting an interview with each student based on two tasks. In the first task, students were asked to locate the place of a particular wire pair on the telecommunication closet representation (TC representation). In the second task, students were asked to determine the range of values that generate this composite unit. Our theoretical and analytical framework is guided by the three levels of students' generalization activity by Radford () and the work on developmental transfer by Engeström (). By identifying shifts in the students' activity (from factual to contextual and from contextual to symbolic), we recognized three transferring cases: the immediate transfer case, the developmental transfer case, and the nondevelopmental transfer case. By comparing and contrasting the different cases, we identified four factors that seem to facilitate or constrain the transfer development. These are the use of metaphors, the development of problem solving strategies, the context dependency, and the students' motivation. In conclusion, we discuss the educational implications of our findings. [ABSTRACT FROM AUTHOR]

This study investigates early childhood prospective teachers' attention to geometrical tasks while designing and using them in the classroom. This is explored in the context of the teaching practice of 11 prospective teachers who taught geometry in early childhood classrooms during the last semester of their university studies. The teaching practice was organized into four stages: design of a lesson plan; classroom implementation; discussion of the lesson with the school practice instructor; and self-assessment report and revision of the lesson. Analysis of data using the Teaching Triad framework (Jaworski, ) shows that although the prospective teachers attended to issues of mathematical challenge, sensitivity to students, and management of learning in their planning, in their actual teaching and after class reflection, their attention was focused mainly on management issues. The findings also show that prospective teachers' attention on geometrical tasks can be developed through a process of reflection on their teaching. [ABSTRACT FROM AUTHOR]

This paper investigates the role of tools in the formation of mathematical practices and the construction of mathematical meanings in the setting of a telecommunication organization through the actions undertaken by a group of technicians in their working activity. The theoretical and analytical framework is guided by the first-generation activity theory model and Leont’ev’s work on the three-tiered explanation of activity. Having conducted a 1-year ethnographic research study, we identified, classified, and correlated the tools that mediated the technicians’ activity, and we studied the mathematical meanings that emerged. A systemic network was generated, presenting the categories of tools such as mathematical (communicative, processes, and concepts) and non-mathematical (physical and written texts). This classification was grounded on data from three central actions of the technicians’ activity, while the constant interrelation and association of these tools during the working process addressed the mathematical practices and supported the construction of mathematical meanings that this group developed from the researchers’ perspective. Technicians’ emerging mathematical meanings referred to place value, spatial, and algebraic relations and were expressed through personal algorithms and metaphorical and metonymic reasoning. Finally, the educational implications of the findings are discussed. [ABSTRACT FROM AUTHOR]

This paper is methodologically based, addressing the study of mathematics teaching by linking micro- and macro-perspectives. Considering teaching as activity, it uses Activity Theory and, in particular, the Expanded Mediational Triangle (EMT) to consider the role of the broader social frame in which classroom teaching is situated. Theoretical and methodological approaches are illustrated through episodes from a study of the mathematics teaching and learning in a Year-10 class in a UK secondary school where students were considered as “lower achievers” in their year group. We show how a number of questions about mathematics teaching and learning emerging from microanalysis were investigated by the use of the EMT. This framework provided a way to address complexity in the activity of teaching and its development based on recognition of central social factors in mathematics teaching–learning. [ABSTRACT FROM AUTHOR]

This study explores the arguments used by 14-year-old students in making decisions about the design of a road in their area. The whole activity was based on an actual problem at the time the research took place. The procedure was a sequence, where students worked first individually, then in groups and finally they had to take a class decision. Then they had to realize one of the final proposals and design the construction of a bridge that their planning had involved. This paper first elaborates the general perspective of such an approach; it then describes the process of argumentation and analyses the nature of the students' arguments, which are discussed on the basis of a specifically constructed network. [ABSTRACT FROM AUTHOR]

Presents a study which explores primary school children's conceptions about volume across a number of different tasks. Six tasks devised to identify commonalities in children's responses; Particular issues addressed by study; Results of study.

• The dynamic triangle of resources, features and learning potentials constituted a promising tool for studying the development of statistics teaching. • The re-sourcing process relates to the links among resources, features and learning potentials. • Software tools seemed to be crucial for statistics teaching and learning as well as for teachers' professional learning. • The Community of Practice context revealed the important role of human resources in developing the statistics teaching. In the context of a community of practice (CoP), eleven secondary mathematics teachers collaborated in exploring various resources that aim to support teaching statistics that fosters learners in developing statistical thinking and reasoning (DSTR). The tracking of a certain material resource, namely the dynamic software tools for statistics instruction (STSI), revealed three phases in the process of the resource's integration in the community's shared repertoire. A qualitative analysis illuminates the characteristics and particularities of each phase in terms of the negotiation of meaning with respect to the practice of the community. We also consider and discuss aspects that seem to frame the transition among the various phases of this process and implications for the re-sourcing process of the DSTR practice. [ABSTRACT FROM AUTHOR]

Abstract: This study investigates teachers’ argumentation aiming to convince students about the invalidity of their mathematical claims in the context of calculus. 18 secondary school mathematics teachers were given three hypothetical scenarios of a student''s proof that included an invalid algebraic claim. The teachers were asked to identify possible mistakes and explain how they would refute the student''s invalid claims. Two of them were also interviewed. The data were analysed in terms of the content and structure of argumentation and the types of counterexamples the teachers generated. The findings show that teachers used two main approaches to refute students’ invalid claims, the use of theory and the use of counterexamples. The role of these approaches in the argumentation process was analysed by Toulmin''s model and three types of reasoning emerged that indicate the structure of argumentation in the case of refutation. Concerning the counterexamples, the study shows that few teachers use them in their argumentation and in general they underestimate their value as a proof method. [Copyright &y& Elsevier]

The article examines ways to implement innovative teaching approaches at scale. It mentions that research and experience reveal that innovative teaching approaches promoted by mathematics education researchers differ significantly from the day-to-day practices of teachers in many countries; and suggests that investigations of scaling up should be broad in scope and attend not only to the practices of teachers and researchers.

Information about several papers discussed at the Seventh Congress of the European Society for Research in Mathematics Education (CERME 7) held February 9-13, 2011 in Poland on mathematical content knowledge needed for teaching (MCKT) is presented. Topics include the lesson study research, the problem solving-oriented approach, and an inquiry-based approach to mathematics. The evolution of teachers' beliefs and practices in collaboration with researchers over the long term is also tackled.