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Typically, curriculum resources including textbooks are seen to reside at the interface between policy and practice (e.g., Valverde et al. 2002), as they translate policy (the intended curriculum) into practice (the enacted curriculum). More recently mathematics teachers increasingly rely on digital resources to prepare their lessons and to design their mathematics curriculum, and students use such resources in class and to complement their courses. These materials are said to become key tools for teachers; as in many countries (e.g., France, the Netherlands, the United Kingdom, the United States), teachers are increasingly encouraged to (re) design the curriculum in planning their instruction.In the next section we define curriculum resources; in particular we distinguish digital curriculum resources from educational technology. In the subsequent section, we discuss the design and “use” of mathematics curriculum resources by teachers (and students). In the last section, we develop...

International audience; This chapter proposes a theoretical frame, the documentational approach to didactics (DAD), as a tool for analyzing the changes brought about by digitalization in the design and uses of mathematics teaching resources. One of the major changes appears to be initiated by the profusion of Open Educational Resources (OER), which provide new opportunities for the design and use of teaching resources. In order to analyze the effects of such opportunities, we focus on two cases: the French Sésamath association, providing OERs at a large scale; and a French mathematics teacher using OERs as a means for accomplishing her teaching. Through the lens of DAD, we investigate the implication of this provision of ‘resources-on-offer’ for teachers’ practices

In this paper we investigate the extent to which the research framework on reasoning developed by Lithner (2008) is adequate for characterizing undergraduate students’ mathematical reasoning. We conducted a small number of individual task-based think-aloud interviews in which students solved integration tasks. Several examples illustrate how we characterized reasoning types by using the framework. However, we found that some reasoning types were not covered by the framework. We propose to extend the framework by introducing a reasoning type that is mathematically founded but not creative, and as a consequence, may be intertwined with imitative reasoning.

This paper reviews the literature on the theme of mathematics teachers’ work and interactions with resources, taking a particular perspective, the so-called ‘collective perspective’ on resources, their use and transformation. The review is presented under three headings: (1) theoretical frameworks commonly used in this area of research; (2) teachers’ interactions with resources in terms of their design and use; and (3) teachers’ interactions with resources in terms of teacher learning and professional development. From the literature, and the collection of papers in this issue, we argue that the collective dimensions play an important role in mathematics teachers’ work with resources and in their professional learning/development. Further empirical investigations are likely to be needed on: how teachers may work in collectives and with resources, and in which ways ‘productive’ collectives may form and work together; which roles particular resources can play in these delicate constellations and how particular resources may support teachers in their work and learning; and which kinds of resources offer opportunities for community building

Working with mathematics curriculum materials to create learning opportunities for pupils, and at the same time to develop mathematical knowledge for teaching is an important part of the work of teaching. In this study I investigate the role of curriculum materials, more precisely a task analysis schedule, as catalyst for teacher learning. I explore the different forms of feedback resulting from developing and working with a "tool" designed to analyse mathematical tasks/curriculum materials for instruction. The results are based on the work with two lower secondary school teachers in England, John and Paul, as they work together, over the period of 9 months, in a group of teachers and with a university academic, on the analysis, development and teaching of mathematics curriculum materials. It is argued that the task analysis "tool" develops, to become a "catalytic tool" to provide feedback for teacher learning. In the process it affords feedback loops and changes its character, from "tool" as artefact to "epistemic object" at the interface between task design and enactment. The results provide deeper insights into the processes of teacher learning with the help of analytic tools and the feedback these may afford.

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In this paper, we will examine the mathematical knowledge that prospective mathematics teachers draw upon when graphing function graphs and curves, with a special focus on the occurrence of asymptotes. Three tasks which involved a graph of a rational and exponential function and a hyperbola as a conic section were designed and administered to students. We performed this study within the framework of Anthropological Theory of the Didactic to examine the relationship of prospective mathematics teachers’ available knowledge with the knowledge to be taught in upper secondary schools and scholarly knowledge relevant for teaching. By studying prospective mathematics teachers’ knowledge, we aim to understand the feasibility of our proposed reference epistemological model for graphing functions and curves in the upper secondary school. Our findings reveal students’ shortcomings with respect to the choice of the appropriate graphing praxeology for given tasks. Students’ graphing strategies relied mostly on plotting points obtained by evaluating a formula, which is a dominant approach in the textbooks we analysed. Plotting points did not lead students to examine asymptotic behaviour, along with the observed monotonicity of a function. Their graphing strategies were found to be predominantly dependent on the particular setting in which the task was presented. Additionally, in our study, the idea of an asymptote as a tangent line at infinity in the geometric setting was questioned.

International audience; After analyzing the relation between mathematics and computer science and the place given to proof, logic and reasoning, we propose and discuss a framework for the study of their interactions based on the ck¢ model. Then, we exemplify this model in the analysis of a problem, making explicita mathematical solution and an algorithmic solution.

This paper reports on a pilot study concerning a first implementation of a collaboration script, aiming at developing students’ argumentative competences in mathematics, as part of an interactive digital storytelling. We discuss the outcomes of the transcripts’ analysis, which seem to show that the collaboration script fosters the introduction of the student to the construction of arguments as cohesive texts, independently on the student’s skill in mathematics, and that the success of the script depends on the learners’ engagement in the story and on the team mood.

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International audience; National French assessment shows difficulties with writing large numbers at the beginning of the 6th grade. But, what do students need to learn and teachers need to teach? What do they actually learn or teach? We investigate these questions at different levels of the didactical transposition: students' knowledge, teaching practices and reference knowledge. We show a lack of mathematical understanding of large numbers and make a proposal for teaching knowledge which provides justifications for the use of large numbers which could foster a 'number sense' understanding of such numbers.

International audience; This article is focused on the teaching and learning of Algorithmics, a discipline at the intersection of Informatics and Mathematics. We focus on the didactic transposition of Algorithmics in secondary school in France and Ukraine. Based on epistemological and didactical frameworks, we identify the general characteristics of the approaches in the two countries, taking into account the organization of the content and the national contexts (in the course of Mathematics in France and in the course of Informatics in Ukraine). Our results give perspectives on understanding the place that Algorithmics can hold in the teaching and learning of Mathematics and Informatics.

This contribution, examining the increasing presence at school of East-Asian students (Chinese in particular) and the related didactical complexity rising from the contemporary presence of Western and non Western cultural heritages, discusses some historical, epistemological and linguistic aspects of Chinese culture and the related Shuxue (Mathematics) education. These were presented to Italian teachers and educators during a training course organized by the authors with the aim of answering teachers’ needs (K-12) for training about the didactical problematic of teaching in multicultural milieu (in particular with Chinese students). The laboratory activities designed and implemented in an ad hoc way by the authors, gave trainers the opportunity to construct a general epistemological framework about Chinese culture and to “compare” it with their own cultural reference. This chance offered them some key lens useful to critically reread and better understand the different cognitive styles of their Chinese and Italian students in mathematics. Thanks to the training course they learned how to refer to cultural diversity as a resource for their teaching activities and not as an obstacle.

In this paper we take a critical look at the state-of-the-art in the research domain of early mathematical development and education. We start with a brief review of the influential and successful (neuro)cognitive research in this domain - which is heavily focused on the development and teaching of children’s (non-symbolic and symbolic) magnitude representation and strongly dominated by the theory of an approximate number system (ANS). We confront and complement this (neuro)cognitive approach with various other lines of research that may help to provide a more comprehensive picture of the development and stimulation of children’s early mathematical competence and how it relates to their later mathematical proficiency at school. Plenary lecture ispartof: pages:31-52 ispartof: Proceedings of the Tenth Congress of the European Society for Research in Mathematics Education (CERME10) pages:31-52 ispartof: Tenth congress of the European Society for Research in Mathematics Education location:Dublin, Ireland date:1 Feb - 5 Feb 2017 status: Published online

The problem of taking a data set and separating it into subgroups, where the members of each subgroup are more similar to each other than they are to members outside the subgroup, has been extensively studied in science and mathematics education research. Student responses to written questions and multiple-choice tests have been characterised and studied using several qualitative and/or quantitative analysis methods. However, there are inherent difficulties in the categorisation of student responses in the case of open-ended questionnaires. Very often, researcher bias means that the categories picked out tend to find the groups of students that the researcher is seeking out. In our contribution, we discuss an example of application of hierarchical and non-hierarchical analysis method, to interpret the answers given by 118 Tenth Grade students in Palermo (Italy), to six open-ended questions about algebraic thinking. We show that the parallel use of the two quantitative analyses allows us to interpret in deep way the reasoning of students solving different mathematical problems using Algebra. These clustering methods also allow us to highlight different students groups, that can be recognized and characterized by common traits in their answers, without any prior knowledge on the part of the researcher.

We present a study about the development of interpretative skills in prospective teachers. In particular we discuss a kind of tasks designed by us for teacher education, containing the request of interpreting students’ answers. The task in this study was built on an item concerning the sum of powers of 10 and was proposed to a group of prospective secondary teachers who were attending a Math Education course. The task was first faced individually by them and then discussed in group. We present the interpretations proposed by two prospective teachers before and after the collective discussion, in order to reflect on the differences in terms both of mathematical knowledge put in play and of attitudes exhibited.

10th Congress of the European-Society-for-Research-in-Mathematics-Education (CERME) -- FEB 01-05, 2017 -- Dublin, IRELAND, WOS: 000467053304043, The purpose of this study was to investigate middle school mathematics teachers' knowledge for teaching algebra. The participants of the study were 48 mathematics teachers from various middle schools. A questionnaire was conducted in order to collect data about the teachers' knowledge related to teaching of algebra. The results showed that the participating teachers were competent in making transitions among different algebraic representations. However, they had difficulties in explaining the conceptual bases of some of the algebraic concepts and procedures. In addition, results indicated that some of the teachers had difficulties and misconceptions similar to those of students as depicted in the scenarios provided in the questionnaire., European Soc Res Math Educ

In this paper we discuss a teacher education task centred on the request to interpret the reasoning of students addressing a ‘throw of the dice’ task. The task for teachers was designed after an analysis of class excerpts, carefully selected by our research group because of the interesting educational reflections they gave rise. The goal of the task is to support prospective teachers to develop interpretive ability in order to use errors as a didactic resource and to help students build their mathematical knowledge starting from their reasoning.

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International audience; Cette contribution se propose de présenter une analyse didactique de séances d’enseignement mises en œuvre par des professeurs stagiaires en sciences économiques et sociales. L’analyse mettra notamment en évidence les points forts, en termes d’efficacité de l’apprentissage des élèves, de dispositifs visant à initier les élèves à la démarche scientifique, mais aussi les difficultés et contraintes rencontrées tant par les élèves que par les jeunes professeurs. Il s’agit ici de mettre en évidence le fait que la démarche consistant à prendre au sérieux les savoirs savants, à confronter les élèves à des énigmes et à favoriser les interactions verbales pour progresser vers la résolution de l’énigme est extrêmement féconde.La communication s’appuie sur des retranscriptions de séances vidéos dans les classes de jeunes professeurs stagiaires, séances qui font l’objet d’une analyse didactique menée d’un triple point de vue : par le stagiaire filmé (qui avait pris connaissance du montage réalisé avant la séance) par les autres professeurs stagiaires et par le chercheur animant la séance de « didactique et analyse des pratiques ».La communication permettra de mettre en lumière, à partir de plusieurs exemples précis, comment des séquences pédagogiques de type « débat scientifique dans la classe » sont de nature à favoriser l’apprentissage des élèves, mais aussi combien sont grandes les difficultés auxquelles peuvent se heurter les professeurs stagiaires désireux de mettre en œuvre de telles stratégies didactiques : les exemples étudiés montrent en effet que la gestion de ce type de séance est particulièrement contraignante et que certaines « maladresses » conduisent à des difficultés très claires d’apprentissage de la part des élèves. Pour autant, l’exemple d’une séance « réussie » montre à quel point les élèves peuvent apprendre efficacement en se posant des problèmes dont ils cherchent eux-mêmes la solution.

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