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Recent publications emphasize the need to take greater account of differences in teaching quality between subjects. The empirical analysis of this topic requires a comparison of teaching quality in different subjects to distinguish generic aspects of teaching quality from subject-specific ones. In this paper, we compare teaching quality in mathematics and German lessons using observational data from primary schools in Switzerland (N_Math = 319; N_German = 237). Data were collected using an observation instrument reflecting the teaching dimensions of the MAIN-TEACH model, which was developed based on a synthesis of established observation frameworks. The dimensions of classroom management, motivational-emotional support, selection and implementation of content, cognitive activation, support for consolidation, assessment and feedback, and adaptation were tested for subject-related measurement invariance. With a two-fold measurement invariance approach, differences between the subjects were investigated at both a global and an indicator level. When applying alpha accumulation correction, no significant subject-related differences in factor loadings or intercepts were found. The factorial structure of our data was basically identical for the two subjects. The comparison of latent factor means revealed no mean differences between the subjects. We discuss the implications for both the MAIN-TEACH model and research into subject-related differences of teaching in general.

Research into mathematics education at university level includes a wide range of theoretical approaches. This poses considerable challenges to researchers in terms of understanding and harmonizing the compatibility and commensurability of those approaches. The research community has already problematised and studied these challenges using networking theories. The networking theories framework is taken as a starting point in this study to contrast different approaches and to broaden the comparison of different frameworks. In particular, three case studies framed in the Action, Process, Object, Schema Theory, in the Problem-Solving approach, and in the Anthropological Theory of the Didactic are analysed. The differences and possible similarities between the three with regard to the research questions addressed, their objects of study, their empirical bases, as well as their research ends are considered. The analysis offers an insight into the potential for collaboration and the networking of theories in the field of university mathematics education.

The selection of tasks based on the evaluation of task features can be considered a core practice of teaching and a relevant component of teaching quality. This is typically part of teachers’ preparation for their classroom teaching, which prompts the following question: What are the characteristics of the tasks that teachers use when selecting tasks for differentiated teaching? To answer this question, we analyzed systematic differences in the focus of 78 in-service high school and lower secondary school teachers during the evaluation of task features. The teachers had to select eight tasks about the practice of fractions with respect to their differentiation potential—operationalizing their adaptive teaching competence from a mathematics educational perspective. To analyze the differences, we performed a cluster analysis of the task features that the teachers drew upon. Three groups of teachers could be identified with variations in their focus on directly or indirectly relevant, domain-specific or domain-general task features. Taking into account such variations may explain differences in teaching quality and student outcomes and may be relevant when designing teacher professional development programs.

In this paper, we present a qualitative study on what values students perceive in their abstract algebra course. We interviewed six undergraduates early in their abstract algebra course and then again after their course was completed about what motivated them to learn abstract algebra and what value they saw in the subject. The key finding from the analysis was that participants found intrinsic value (i.e., their enjoyment of the subject) to be essential to learning abstract algebra. While participants desired utility value in the form of mathematical applications, they ultimately did not find this necessary to learn abstract algebra. Finally, some participants had different motivations for learning abstract algebra than for learning other branches of advanced mathematics, such as real analysis, suggesting that motivation research in mathematics education should not treat mathematics as a unitary construct. We offer analysis about how the nature of advanced theoretical proof-oriented mathematics may have contributed to these findings.

While proof has been studied from different perspectives in the mathematics education literature for decades, students continue to struggle to build proof comprehension. Complicating this, the manner in which proof comprehension is assessed largely remains to be the definition-theorem-proof format in which students are asked to reproduce proofs or similar proofs that are presented in class. This approach can encourage students to memorize proofs rather than develop tools for syllogistic reasoning. This paper reports on a seven-year collaboration among research mathematicians and mathematics educators. Following a model for proof comprehension, they implemented a cycle of planning assessments, implementing them and evaluating student responses in several courses each semester for three years at two universities. The model-based assessments were designed with probing questions about proofs or subproofs to point student attention to the relationships among definitions, statements and their relationships, as well as the logic used to help students to both develop proof knowledge and demonstrate their mathematical thinking. Discrepancies between the intention of assessments and student responses led to refinements as the team reviewed the results to inform its practice. Collaboration, experience and empirical data informed the development of the new Promoting Reasoning in Undergraduate Mathematics (PRIUM) Qualitative Framework for Proof Comprehension. The paper discusses three main results: the Framework and how to use it, the Framework’s utility at the individual, course and program levels of departmental evaluation, and a collaborative research process that may be utilized between research mathematics educators and mathematicians.

Affective and cognitive processes may be jointly researched to better understand mathematics learning, paying special interest to emotions related to knowledge acquisition. However, it remains necessary to explore these processes in studies linked to the education of pre-service mathematics teachers. This study aims to characterize epistemic emotions in different practices linked to the practice of mathematics teaching: problem-solving, anticipating what would happen with the students and reflecting on classroom implementation. It considers the theory of Mathematical Working Spaces to describe the mathematical and cognitive dimensions generated by epistemic emotions, paying special attention to the cognition-affect interaction and the workspace created. The results indicate that the epistemic emotions of the pre-service mathematics teachers associated with the distinct practices were different. Differences are observed in the interaction between emotions and cognitive epistemic actions, depending on whether the pre-service mathematics teachers analyze them within the framework of their own solving or anticipate them in their students. This reveals how personal work relates to what is considered to be suitable for students. Specifically, certain antecedents and consequences have been specified for the emotions of surprise and boredom in relation to the characteristics of the optimization problems and the cognitive activity of the subject when solving them. These results highlight the need to enhance the education of pre-service mathematics teachers through training that helps regulate their epistemic emotions and model effective strategies for regulating their own emotions and those of their students.

This research, following a sequential mixed-methods design, delves into metacognitive control in problem solving among 5- to 6-year-olds, using two floor-robot environments. In an initial qualitative phase, 82 pupils participated in tasks in which they directed a floor robot to one of two targets, with the closer target requiring more cognitive effort due to the turns involved. The results of this phase revealed that younger students often rationalised their decisions based on reasons unrelated to the difficulty of the task, highlighting limitations in children’s language and abstract thinking skills and leading to the need for a second quantitative study. In this subsequent stage, involving 117 students, a computerised floor-robot simulator was used. The simulator executed the students’ planned movements and provided feedback on their validity. Each participant had three attempts per problem, with the option to change their target each time. The simulator stored the information pertaining to the chosen resolution path, design of the plan, and re-evaluation of decision making based on the results and feedback received. This study aims to describe the criteria upon which students base their metacognitive control processes in decision making within problem-solving programming tasks. Additionally, through a comparative analysis focusing on age and gender, this research aims to assess the relationship between metacognitive processes and success in problem-solving programming tasks.

Identifying mathematically gifted students is an important objective in mathematics education. To describe skills typical of these students, researchers pose problems in several mathematical domains whose solutions require using different mathematical capacities, such as visualization, generalization, proof, creativity, etc. This paper presents an analysis of the solutions to two problems by 75 students (aged 11–14), as part of the selection test for a workshop to stimulate mathematical talent. These problems require the use of the capacity for mathematical generalization of solution methods. We define a set of descriptors of such capacity, use them to analyze students’ solutions, and evaluate how well students with high capacity for generalization can be distinguished from average students. The results indicate that the two problems are suitable for identifying potential mathematically gifted students and several descriptors have high discriminatory power to identify students with high or low capacity for generalization.

The development of professional digital competencies is an important condition for a meaningful integration of digital technologies in mathematics education. A widely used and empirically studied approach for the professional development of in-service teachers are teacher design teams, in which lessons are collaboratively planned, implemented and reflected. However, this approach has not yet been considered for the collaboration of pre-service and in-service teachers. Therefore, this article examines in a holistic case study mixed teacher design teams composed of pre-service and in-service teachers. A total of 23 interviews were conducted with pre-service and in-service teachers and analyzed by inductive development of categories using the method of qualitative content analysis. The case study shows among other things, that the different prerequisites of the participants can lead to a relatively clear division of roles within the collaboration, which turns out to be suitable for the professionalization of both pre-service and in-service mathematics teachers.

This study contributes to Action, Process, Object, Schema (APOS) theory research by showing two approaches used by advanced mathematics students to construct relations between higher-order derivatives to solve complex problems. We show evidence of students’ ability to perform Actions on their graphing derivative Schema, that is, of its thematization. It also contributes to the literature on the learning of differential calculus by showing how advanced students use their knowledge to construct relations between concepts when facing complex situations. The work of three graduate students on transforming complex graphs and determining their properties and their relation to the domain structure is analyzed to determine their solution approaches. Their graphing derivative Schema is analyzed in depth in terms of the construction of relations among the Schema structures and assimilation and accommodation mechanisms involved in thematization in APOS theory. These findings are important in informing and developing didactic strategies to foster university students’ understanding of derivatives, which can smoothe the transition to the study of advanced mathematics courses.