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The decisions that teachers make in transforming the curriculum into specific lesson plans determine the real enactment or otherwise of curricular ideals. These decisions are shaped by the resources available and by each teacher’s goals and orientations. This exploratory study employs Schoenfeld’s decision-making model to examine how resources, goals and orientations influence lesson planning for mathematics problem solving, for different profiles of primary teachers in Chile. To this purpose, a survey was conducted among 40 teachers of varying degrees of ability and experience: some were beginning teachers, others were experienced but had no further training in teaching problem solving and a third group was composed of experienced teachers with specific training in this question. Interviews with two teachers from each profile revealed important differences between the three groups. Beginning teachers relied more heavily on official resources such as the official curriculum and standard textbooks, aligning themselves with school requirements. Experienced teachers with problem solving training demonstrated a strong inclination towards teaching through a problem solving approach. While beginning teachers acknowledged the importance of promoting problem solving strategies, they did not usually adapt problems to the mathematical content or to the age/competence of their students. Interestingly, all three groups under-utilised sections of curricular resources that emphasise the present curricular focus on problem solving. Finally, the study found that experience alone is not enough to develop a problem solving approach and that focused professional development programmes are needed to equip teachers with the necessary skills. In addition, a diagnostic teaching approach should be incorporated into initial teacher training.

Visualization and argumentation are central processes in the learning of geometry, giving rise to specific lines of research in mathematics education. However, it is worth asking whether they should be developed independently or if it is possible and in what way an articulation between them can be promoted. The objective of this paper is to search for this articulation within the most recent advances in empirical research on the teaching and learning of geometry in Spain. The analysis is approached through the tasks used in research employing two theoretical frameworks which have been developed in Spain, the Onto-Semiotic Approach and Configurational Reasoning. Two main types of task were found in the selected studies: visual tasks and conjecture-and-proof tasks. The results show that in visual tasks the articulation between visualization and argumentation is only necessary in some tasks, in which argumentation aims to justify or to describe the external representation provided as a solution. However, in conjecture-and-proof tasks the articulation between visualization and argumentation must always be present to resolve the task. Analyzing this articulation helps to detect and gain a better understanding of the difficulties that students encounter in solving these types of task. In visual tasks, argumentation helps to understand difficulties associated with the use of visualization. In conjecture-and-proof tasks, visualization determines the argumentation followed, in such a way that many of the difficulties originate more from the visualization than from the process of argumentation.

Existing research has highlighted the key role of observation and reflection in teacher learning during lesson study interventions, and how challenging it can be for pre-service teachers. This study focuses on the post-lesson discussion phase and on the observation of the research lesson of a lesson study process. It is carried out with early childhood education pre-service teachers who studied the Theory of Didactical Situations. The aim of our research is two-fold: finding evidence of the activation of pre-service teacher knowledge during the post-lesson discussion, and identifying mechanisms that may explain said activation as well as the potential development of their professional knowledge. To model pre-service teacher knowledge, the notion of teachers’ praxeological equipment from the Anthropological Theory of the Didactic, specified in the Theory of Didactical Situations, is used. It is hypothesised that potential teacher learning is mainly connected to the identification of particularly relevant episodes during the observation phase, henceforth referred to as critical events. The analysis of a group discussion on three critical events shows the activation and development of mathematical and didactic components of pre-service teachers’ praxeological equipment. However, it is found that it depends on the degree of dissonance these events can provoke, and on the quality and depth of the discussion that takes place around them. It is therefore concluded that the identification of critical events followed by productive discussion is a mechanism that explains teacher learning through lesson study, which can be enhanced by the intervention of teacher educators as mediators.

Although complex analysis is part of the study programs of many mathematics undergraduates, little research has been done on how individuals interpret basic concepts from complex analysis. To address this gap, this paper investigates how experts individually think about complex path integrals. For this purpose, the commognitive framework is used to conceptualize experts’ interpretations of mathematical concepts discursively, namely in terms of so-called intuitive mathematical discourses. A total of nine interpretations of complex path integrals, so-called discursive images, as well as eight sets of rules governing their construction, so-called discursive frames, are derived from expert interviews. These interpretations range from a rejection of intrinsic meaning to connections with real and vector analysis, mean values, and individual formulations of theorems. The paper also raises questions for the inclusion of the results into teaching and addresses further research.

This contribution explores the teaching of graph theory in the context of engineering education. We examine how teachers use collectively designed resources in terms of their backgrounds. After a literature review, we justify the choice to develop an analytical framework for studying the materials in terms of connections to be established in the resources. We suggest developments around the theoretical concept of connectivity, which allows us to analyze the potential connectivity, characterized by the connections enabled by the resources for teaching; and the effective connectivity that develops in teachers’ practices and during the implementation of resources. We study the effective connectivity in terms of each teacher’s personal relationship with graph theory. Graph theory evolves at the interface of mathematics and computer science research. We consider the case of two teachers, a computer scientist and a mathematician in a French Engineering School in France. The data collected consists of resources for teaching graph theory in engineering courses and interviews with teachers. As for results, we characterize how forms of connectivity developed in the use of resources for teaching graph theory are largely determined by the teachers’ backgrounds.

When analyzing instructional quality in mathematics, observational measures can consider domain-specific instructional approaches (e.g., effective teaching of mathematics) and content-focused practices (e.g., effective teaching of a specific mathematical topic). In this study, we analyze lesson videos from secondary algebra classrooms using two different instruments—one focused on ambitious mathematics teaching practices and one focused on instructional features that support students’ learning opportunities in algebra specifically. We consider the information provided by overlaying these instruments to raise implications for instructional improvement efforts, particularly related to how information from the instruments might inform formative feedback to teachers. We consider how alignment between instrument and instructional emphasis can influence the strengths and areas of instructional improvement identified in a given lesson. We further model a collaborative process for looking across observation instruments to deepen understanding of the information provided.

The potential of tasks to foster mathematical learning and understanding is an important aspect of instruction and their implementation in teaching is thus often viewed to be positively related to the quality of instruction. Both the selection of tasks as well as their implementation in the classroom depend on many factors, with teachers’ knowledge and skills as one of the most important ones. The present study aims to analyze the relations between different aspects of task potential, the quality of instruction, and teachers’ competence in order to investigate whether task potential can be seen as an indicator for teaching quality, for teacher competence, or as an independent construct in models of educational effectiveness. To this end, we draw on data from the TEDS-Validate study, namely tests of mathematics teachers’ competence (n = 31) observations in their classrooms (n = 60), and an in-depth analysis of all tasks used in the respective lessons (n = 2490). Multiple regression analysis suggests that while some facets of task potential are related to either teaching quality or teacher competence, the potential of tasks emerges as an independent construct with some characteristics predicting the teaching quality of the respective lessons. Implications of these results for the role of tasks in educational effectiveness research are discussed.

This paper focuses on in-service teacher education as a crucial agent in driving educational paradigm shifts for the teaching and learning of mathematical modelling. Our paper presents empirical research based on two case studies, examining training courses for in-service secondary school mathematics teachers. The first course integrates different modelling approaches, while the second one is based on the anthropological theory of the didactic and uses the proposal of the study and research paths for teacher education. Both experiences conclude with teachers working on the design of modelling tasks to be implemented in secondary schools. Both courses are analysed through the lenses of how teachers progress on the collective identification of conditions and constraints for modelling, when teachers deal with the design of modelling tasks and analyse their implementations. First, the results show the kind of modelling tasks teachers design and the level of disciplinary specificity at which they plan modelling to be implemented. Second, we analyse how teachers address the “ecological analysis”, that is, the identification of conditions and constraints that facilitate or hinder the integration of modelling in secondary school classrooms after their experiences.

This paper studies the knowledge developed by a group of sixty-two pre-service primary teachers during a training session on the definition of a polygon and the mathematical practice of defining. We explore the knowledge developed by pre-service teachers when they carry out a series of training tasks oriented around the practice of defining with primary pupils, within the theoretical framework of the Mathematics Teachers’ Specialised Knowledge model, and using videos of an authentic lesson as prompt. Data were collected by means of video recordings of the implementation of the training tasks, in conjunction with the pre-service teachers’ own written observations as they watched the recorded lesson. A content analysis was carried out on the collected data using the Mathematics Teachers’ Specialised Knowledge model. The results highlight the way in which knowledge of the mathematical practice of defining is constructed, along with the mathematical objects involved in this practice, and pedagogical aspects of defining with young pupils. They also illustrate how a sequence of tasks oriented around a model of teachers’ knowledge can contribute to the development of this knowledge in initial training programmes, and more especially provide orientation about the training of pre-service teachers in the mathematical practice of defining.

Mathematics instruction is not race or culture neutral. For students who have been historically marginalized in mathematics classrooms, high-quality mathematics instruction, instruction that helps students build conceptual understanding, on its own might not be enough to disrupt inequities. These students might also need instruction that is culturally relevant, with teachers who demonstrate cultural competence, build critical consciousness, and support student learning. Our goal in this study was to understand which components of culturally relevant pedagogy (CRP) are captured and which are missed in a typical U.S. framework of high-quality mathematics instruction. To find the overlaps and gaps, we analyzed the mathematics lessons of three elementary teachers through both the lens of CRP and the Mathematics-Scan, a mathematics observation tool. We found the strongest overlap between the two frameworks in the patterns of strengths and weaknesses in the teachers’ lessons. When the teachers were delivering high-quality instruction, they were also often supporting students’ learning or showing cultural competence. When the teachers were delivering lower quality instruction, they were also often missing opportunities to enact CRP. At the same time, key elements of CRP including linguistic support for students, high expectations, critical consciousness, and nuances within cultural competence, were missed by the high-quality instruction framework. High-quality instruction was the foundation for CRP in the teachers’ classrooms, but CRP was more than just high-quality instruction. We conclude with recommendations for increasing the alignment between the frameworks and implications for international educators also grappling with equity in their own frameworks of mathematics instruction.