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This paper describes an emerging approach to the design of task sequences and the theory that undergirds it. The approach aims at promoting particular mathematical concepts, understood as the result of reflective abstraction. Central to this approach is the identification of available student activities from which students can abstract the intended ideas. The approach differs from approaches in which learning to solve the problem posed is the intended learning. The paper illustrates the approach through data from a teaching experiment on division of fractions.

This paper is part of a wider study focused on divisibility. Participants were prospective elementary teachers. One of the aims of the research is to characterize the meanings of multiple shown by prospective teachers. In this paper, we present the results concerning this aim. We analyse the productions of 37 prospective elementary teachers collected in a practice session, designed and developed in the context of a teaching experiment. We characterize the meanings through the following elements of the didactic analysis: conceptual structure, representation systems and phenomenology. Prospective teachers showed mostly three meanings of multiple: product, relationship and dividend in an exact division.

The importance of good mastery of specialised mathematics knowledge for mathematics teachers and prospective teachers is now recognised internationally. This paper presents an exploratory study aimed at investigating how mathematics education courses affect the Mathematics Teachers’ Specialised Knowledge (MTSK) of prospective primary teachers. A questionnaire was administered to a sample of 118 future teachers, and an analysis of their answers was carried out with the MTSK model. The research highlights positive feedback on the effectiveness of the mathematics education laboratory that was staged, and provides an outline of the beliefs and the knowledge of the prospective teachers.

This paper describes a study of mathematical argumentation in primary school. The principal aim is to explore the nature of complex argumentation at a structural level. The context of the study was a teaching experiment involving nine tasks that promoted argumentation among fifth graders. We use the framework and method of reconstructing complex argumentation in the classroom proposed by Knipping (2008). The findings show that complex argumentation at a structural level in the context of refuting conclusions is characterized by a source-like structure with the addition of a new refutation argument element.

In this paper, the effect of holding a math training workshop using GeoGebra software has been studied on the changes on teachers' knowledge and beliefs . The selected sample is 40 male and female teachers in Iran. Before and after the intervention were administered a pre and post questionnaire with two components: TPACK knowledge and teachers’ beliefs. Fuzzy logic and Fuzzy TOPSIS methods were used to analyze the data. The results of this method showed a significant difference between the results before and after the workshop.

The paper situates mathematics teaching-learning processes within a multimodal perspective and discusses a semiotic approach apt to seize this dimension, namely the Semiotic Bundle lens. This analytical tool considers the great variety of semiotic resources through which mathematical meanings emerge and evolve in the classroom, ranging from embodied ones such as gestures, to symbolic systems. In particular, the analysis considers them in a systemic and dynamic way. The theoretical account is illustrated by means of an example on children spatial conceptualization, carried out in kindergarten. The data analysis will constitute a background against which the connections with the Theory of Objectification will be highlighted, showing a constructive resonance between the two theories.

This paper presents a study of networking of theories between the theory of registers of semiotic representation (TRSR) and the onto-semiotic approach of mathematical cognition and instruction (OSA). The results obtained show complementarities between these two theoretical perspectives, which might allow more detailed analysis of the students’ performance.

This paper describes the ideas that mathematics teachers (grades 5-9) have regarding mathematical models of real-world phenomena, and explores how teachers’ ideas differ depending on their educational background. Participants were 56 United States in-service mathematics teachers. We analyzed teachers’ written responses to three open-ended questions through content analysis. A varied landscape of ideas was identified. Teachers referred to different entities as constituting models, expressed different ideas about whether data points can be part of models, and whether models convey more information than data. Interesting differences according to educational background were identified, especially between teachers with and without mathematics backgrounds.

This paper’s aim is to discuss the concept of width of a proof put forward by Timothy Gowers. It explains what this concept means and attempts to show how it relates to other concepts discussed in the existing literature on proof and proving. It also explores how the concept of width of a proof might be used productively in the mathematics curriculum and how it might fit with the various perspectives on learning to prove.

In this paper, we focus on topological approaches to space and we argue that experiences with topology allow middle school students to develop a more robust understanding of orientation and dimension. We frame our argument in terms of the phenomenological literature on perception and corporeal space. We discuss findings from a quasi-experimental study engaging 9 grades 5-8 students (10-13 years old) in a 6-week series of school-based workshops focused on knot theory. We discuss video data that shows how students engage with the intrinsic disorientation of mathematical knots through the use of gesture and movement.