Proof is epistemologically the fundamental constituent of any mathematical activity. At all levels of schooling and training, it contributes to the process of learning mathematical knowledge. Thanks to the perspective brought by Technology Enhanced Learning (TEL), the international research on technologies in mathematics learning points out the necessity of a multidisciplinary structuring of work to analyze the uses and the benefits of proof assistants (called PA) in the learning of proof. Proof assistants, used by mathematicians, are free open-source software that check a proof. This project aims to fill the gap in research in France on the learning of proof, which remains problematic at the transition from secondary to higher education. It proposes a multidisciplinary approach, combining tools, results and methods from sciences (mathematics and computer science) and from social and human sciences (didactics). The aim is to develop and evaluate a new approach to teaching proof based on the adaptation and the integration of PA at the beginning of university. Three objectives structure this project: to characterize the difficulties of learners and the existing uses of PA in France; to analyze AP in a learning perspective; to experiment and evaluate the benefits of PA at the beginning of university in mathematics learning. The methods used to achieve these objectives mobilize internationally validated epistemological, didactic, and cognitive frameworks on proof and proving, and articulate quantitative and qualitative approaches. The results of this project will inform researchers, teachers, trainers, and educational policy makers about: high school and college students' difficulties with proof, uses and adaptations of PA for educational purposes, evaluation of PA benefits in learning, and didactic recommendations for the teaching and learning of proof in high school and university.

The NéOPRÆVAL project aims to conceive a set of assessment tools in order to help teachers to manage the heterogeneity of learning process in the classroom. This multidisciplinary project addresses both the crucial issue of assessment and the development of various assessment tools. It also addresses the development of assessment software on an online platform that is widely used by teachers, as well as the use of this software in teaching practices. We rely on following research results: - Automatic diagnostic assessment Pepite (Grugeon-Allys, Pilet, Chenevotot & Delozanne 2012; Pilet 2012) in elementary algebra at the end of lower secondary school which is available on the online platform LaboMep; - Teaching practices (Robert & Rogalski 2002; Roditi 2011). The scientific program is organized around three tasks: 1. Gaining expertise in studying the validity of assessment tools and developing assessment tools First, we will analyse various assessment tools (standardized or not, external or internal) from educational, psychological as well as statistical point of view. We will then define the criteria for the validity of such tools, based on the scientific research of the outcomes of numerous external assessments undertaken during the last 30 years, and of the automatic diagnostic assessment Pepite, which was developed for algebraic computation at the end of lower secondary school. We will propose a new methodology to judge the quality of assessment tools and to help to improve them. 2. Using this expertise to expand existing assessments Our goal is to enhance the Pepite assessment. On the one hand, we will expand the automatic diagnosis in algebraic field to different levels of lower secondary school, taking into account the criteria of validity previously defined. On the other hand, we will generalize the design process of assessment tools to the new mathematical field of integer numbers (calculations and numerical problems) at the upper grades of elementary school. It will expand didactical and formal models to develop new prototypes of automatic diagnostic assessment that will enrich the offer on LaboMep platform. These prototypes will be accompanied by pedagogical resources in order to help teachers to regulate their teaching according to the students’ needs which are previously identified. 3. Analysing teachers’ classroom practices: teaching design and learning regulation Beyond analysing practices, this task aims to identify the main elements for organizing practices related to their socio-institutional and personal dimensions and identify changes in use of pedagogical resources dedicated to assessment and regulation. This project will develop: • A new methodology for analysing and improving assessments , • Resources for diagnostic automatic assessment and regulation, implemented on LaboMep platform, • An analysis of assessment practices and formulation of conditions to promote the professional development of teachers in the field of assessment.

Objectives The objective of this proposal is the organization of a multidisciplinary network to finalize a proposal to the H2020 Call Science with and for Society - SWAFS-01-2018-2019. The submitted proposal, called ESMEA, focuses on an embodied teaching method with a “Human Orrery” (a tool that allows learners to enact planetary movements within the Solar System). Through the attractiveness of astronomy for both boys and girls and the innovative potential of enaction, the Human Orrery allows learners to build the knowledge base for SwafS and to enhance their scientific awareness and well-being as citizens. Our current and foreseen network covers different part of Europe, while strong links are expected with third countries, such as Lebanon, Vietnam, Uruguay and USA. This will allow a gender and cross-cultural study of those topics. This network and its diversity of expertise also makes the dissemination of pedagogical sequences using the Human Orrery a Measurable and Achievable goal through national network of schools and science centres and links with stakeholders. Attention will be given to set Realistic and Time-Bound objectives. Constructing and using a Human Orrery An Orrery is a mechanical device illustrating the circular orbits of the planets. On a human Orrery, the orbits of planets and comets are drawn at a human scale allowing movements in the Solar System to be enacted by the learners. The implementation of Human Orrery, as learning experience, involves topics from mathematics and science and illustrates an example of a STEM approach of learning about concepts perceived as abstracts by students and teachers. Astronomy provides a highly motivating context for learners to develop observational skills, discover methods of scientific inquiry, and explore some of the fundamental laws of physics and concepts of mathematics in both an attractive and meaningful way. In the course of the project, a selection of schools (or science centres) in each of the involved countries are going to build their own Human Orrery. Those will then be used as starting hubs to disseminate this new teaching-learning method through local training and demonstration inside and outside schools. In the meantime, we shall inquire about the creation of a “teaching kit”. Theoretical background and methods Our ambition is to serve the co-emergence of a coherent theoretical approach of enaction in mathematics and science education, overcoming the risks of the actual theoretical fragmentation between disciplines, which undermines the strengths and scope of their results. Indeed, the use of a Human Orrery in education is based on the assumption that bodily perceptions help the learning of abstract concepts. Cognitive science theory of enaction as well as science and mathematics education research provide theoretical foundations and empirical results to this claim, by showing the role of gestures, signs, as well as that of artefacts, in learning processes, leading to consider “multimodal learning”. At the heart of research is the question of how learners’ conceptions are transformed to increasingly approximate conceptions shared by scientists. Networking expertise in science and mathematics education will be used to design learning sequences that account for known conceptual obstacles to this transformation. We will refer to the Activity Theory as developed currently in mathematics education to study those sequences. In a broader perspective, our approach intends to make the learning process more engaged and meaningful and to promote an equity among scientific, literary and artistic disciplines. Those topics are related to the notions of gender on one side and of awareness and motivation on the other side. We will examine the relationship between learning environments and students’ creative performances, as well as its correlations with motivation