Lagrange, J.-B. (2018) Connected working spaces: designing and evaluating modelling-based teaching situations. In E. Bergqvist, M. Österholm, C. Granberg, & L. Sumpter (Eds.). Proceedings of the 42nd Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 291-298). Umeå, Sweden: PME.

This contribution focuses on modelling at upper secondary level. Modelling is considered as a work on various models of a reality, belonging to different scientific fields, with varied mathematizations. The framework of Connected Working Spaces is chosen in order to describe the work on each model, and the connections made along the modelling process. The hypothesis is that these choices allow designing and evaluating situations that help students to understand concepts more comprehensively and to appreciate how scientific fields contribute to understanding the sensible world. This hypothesis is tested by way of an experiment in realistic school settings.

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Lagrange, J.-B. (2016)CONNECTED WORKING SPACES FOR SECONDARY STUDENTS' UNDERSTANDING OF CALCULUS: MODELLING A SUSPENSION BRIDGE THROUGH "JIGSAW" GROUP WORK. Colloque Espaces de Travail Mathématique July 18-22 2016 University of Western Macedonia.

This paper is about the design of classroom situations of modelling real world situations at upper secondary level to confront student to questions and ideas in calculus. At this level, curricula avoid formal approaches of calculus notions, but in current strategies, modelling is used as a motivation for algebraic methods, rather than as a basis for scientific work. The paper proposes the framework of connected working spaces for designing and implementing sustainable situations of modelling helping a wider approach of the curriculum in these classes, and giving sense to the concepts taught by making students understand interactions between these concepts and the interactions between these concepts and other sciences or real world situations. The example of a situation of modelling a suspension bridge is presented on the basis of four working spaces (statics, geometrical, algorithmic, and mathematical functions). The implementation is carried out through "jigsaw group work" that helps students work in these spaces and make connections between them.

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Lagrange, J.-B., Bui, A. K. (2016) Une approche fréquentiste des probabilités et statistiques en première année d'Université au Vietnam dans un cursus non mathématique. Proceedings of INDRUM 2016: First conference of the International Network for Didactic Research in University Mathematics, 31 Mar-2 Apr 2016 Montpellier (France)

Au Vietnam, l'enseignement des probabilités donne une place importante à une approche « classique » basée sur la loi de Laplace, et les statistiques inférentielles sont vues comme une application. De plus, l'enseignement ne tient pas compte de l'utilisation des logiciels, génériques ou consacrés aux statistiques. Ceci entraîne des difficultés ayant une signification particulière pour les étudiants d'autres disciplines que les mathématiques, qui doivent être préparés à la compréhension de phénomènes aléatoires et à une approche statistique dans la vie réelle. Cet article vise à évaluer la possibilité d'introduire des innovations «viables» dans l'enseignement des probabilités et statistiques au Vietnam au niveau universitaire, ainsi que les améliorations que ces innovations apportent à la compréhension des étudiants.

In Vietnam, the teaching of probability favors a "classical" approach based on the law of Laplace, and inferential statistics are seen as an application. Moreover, courses do not take into account the use of software, generic or dedicated to statistics. This causes difficulties especially for not math-major students who have to be prepared for understanding random phenomena and for using statistics in real life. This article aims to evaluate the possibility of introducing 'viable' innovations in the teaching of probability and statistics in Vietnam at university level, as well as the improvements that these innovations contribute to the understanding of students.

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Lagrange, J.-B. (2015) FUNCTIONS IN TECHNOLOGICAL ENVIRONMENTS: FROM MULTI-REPRESENTATIONS TO CONNECTED FUNCTIONAL WORKSPACES in Gómez-Chacón, Mª I., Escribano, J., Kuzniak A., Richard, P. (Eds.). Mathematical Working Space, Proceedings Fourth ETM Symposium. Madrid: Publicaciones del Instituto de Matemática Interdisciplinar, Universidad Complutense de Madrid. ISBN: 978-84-606-9475-5

This paper questions the idea of function on the basis of classroom research at upper secondary level in the Casyopée project (Lagrange, 2010). In this project, students are introduced to functions by working on the same functional dependency in four settings. The underlying theoretical framework is a multi-representation view of functions. In cross-experiments and cross-studies this approach has been confronted with other approaches. This implied to reconsider the idea of function and to examine the potential of the idea of connected functional workspaces for designing and analysing classroom situations around functions especially in technological environments.

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Vandebrouck F., Lagrange J-B., Monaghan J., (2013) Activity theoretical approaches to mathematics classroom practices with the use of technology, Research Forum in  Proceedings of the 37th conference of the International Group for the Psychology of Mathematics Education ", PME 37, Kiel, Germany, July 28‒August 2, 2013. Vol. 1 pp 180-210

This Research Forum seeks to promote further discussion on activity theoretical approaches to mathematics classroom practices with the use of technology. Its origins go back to discussions and collaborations between European mathematics education researchers whose approach to research on the use of technology in mathematics classrooms is ‘informed’ by activity theory (AT). These discussions resulted in a two volume special edition of a journal (see Vandebrouck et al., 2012/13). Our aim at PME is to widen this discussion.

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Lagrange, J.B. (2013). Covariation, Embodied Cognition, Symbolism And Software Design In Teaching/Learning About Functions: The Case Of Casyopée. Proceedings of the Eighth Congress of the European Society for Research in Mathematics Education - CERME 8, Antalya, Turkey, 6 - 10 February, 2013

From the early nineties, most reformed curricula at upper secondary level chose to give functions a major position. The goal of this paper is to introduce key elements of a successful approach to the teaching of functions and to show how functions can influence research work including the design of a software environment and the evaluation of classroom situation. We will discuss how experiencing covariation and using references to body activity are crucial for students’ understanding of functions. An example of a situation based on the use of the Casyopée environment is proposed as an illustration.

Keywords: teaching of functions; design of a software environment; understanding of functions.

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Lagrange, J. B. & Psycharis, G. (2013) Exploring The Potential Of Computer Environments For The Teaching And Learning of Functions: A Double Analysis From Two Traditions Of Research  Proceedings of the Eighth Congress of the European Society for Research in Mathematics Education - CERME 8, Antalya, Turkey, 6 - 10 February, 2013

In this paper we aim to address the potential of computational environments offering integrated geometrical and algebraic representations for the teaching and learning of functions. We follow a ‘double analysis’ method to analyse learning situations of an experiment that took place in the French context through the lens of the original research tradition (Theory of Didactical Situations) and an ‘alien’ one (Constructionism). The analysis indicates that this method enhances our efficiency to capture aspects of research traditions which influence knowledge concerning the nature of learning situations for functions with computers.

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Lagrange, J.B. (2012). Opportunities for Learning with Digital Technologies: A Question of Recontextualisation. In Proceedings of the 36th Conference of the International Group for the Psychology of Mathematics Education. Taipei, Taiwan: Tai-Yih Tso, 2012. v. 1. p. 113-120.

The panel members were invited to structure their contributions around three questions:
• Research has shown that many of the conditions that characterize the context in which learning occurs contribute to mediating the opportunities that different groups of learners have to engage with mathematics. How we look at these conditions deeply influences the ways we think of learning opportunities. Which of these conditions do you privilege in your work?
• What tools (theoretical frameworks and research methods) do you adopt to explore the role of these conditions in mediating access to mathematical knowledge and to focus on the questions of empowerment or disempowerment?
• Can widening our views on mediation help to identify previously hidden opportunities for learning?

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Lagrange, J.B. (2011). Working with teachers: Innovative software at the boundary between research and classroom. In B. Ubuz,  ed., Proceedings of 35th Conference of the International Group for the Psychology of Mathematics Education (Vol. 3,). Ankara, Turkey: PME, 113-120.

This paper is based on an experiment of a small scale community around Casyopée, an innovative software for the learning of functions. A first assumption was that dissemination of research outcomes is possible through the development of communities involving researchers and teachers. Another assumption was that all teachers are not to be considered at the same level. “First-adopters” are teachers that chose to be associated with the project development from the beginning. “Mid-adopters” are teachers that can be interested by using innovative software in the classroom, but will choose to do it only when it brings a real added value. All other teachers potentially users of Casyopée make a third layer. This paper is a first step in the study of communication and collaboration in a “multi-layer” community.

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Lagrange, J.-B., Artigue, M., Healy, L., Kynigos, K., Morgan, C., & Sacristan, A.I. (2010). Research Forum: The Conceptualisation and Role of Context In Research With Digital Technologies. In M.M. F. Pinto, & T. F. Kawasaki (Eds.), Proceedings of the 34th Conference of the International Group for the Psychology of Mathematics Education. (pp. 283- 312) vol. 1. Belo Horizonte Brazil: PME.

Over the last two decades, substantial international research into the use of digital technologies in the teaching and learning of mathematics has yielded powerful results. At the same time educators and software developers in many countries across the world have attempted to develop and introduce technological artefacts into classrooms with the aim of improving mathematics education. In spite of the massive investment of resources, both financial and human, in pursuit of this aim, impact on the reality of school practices has been limited. How are we to understand, explain and attempt to overcome this disjunction between effort and impact?

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Lagrange, J.B., & Artigue, M. (2009). Students’ activities about functions at upper secondary level: a grid for designing a digital environment and analysing uses. In  M.  Tzekaki,  M.  Kaldrimidou,  & H. Sakonidis (Eds.) Proceedings of the 33rd Conference of the International Group for the Psychology of Mathematics Education (pp. 465-472) vol. 3. Thessalonique: Aristotle University of Thessaloniki & University of Macedonia.

We present a grid that organises and connects various students’ activities about functions at upper secondary level. We built this grid to overcome what we saw as a fragmentation of research, and to work on the design and experimentation of one of the digital environments of the ReMath European project, Casyopée. Analysing classroom use of the environment, we show how the grid helps to make sense of its potentialities, especially as a tool for functional modeling. Finally, we situate this outcome inside the theoretical ReMath work that aims to progress in connecting and integrating theoretical frames in technology enhanced mathematics learning.

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Lagrange, J.B., Artigue, M., Laborde, C., Trouche, L. (2001) A meta study on IC Technologies in Education. Towards a multidimensional framework to tackle their integration into the teaching of mathematics. In M. van den Heuvel-Panhuizen (ed.) Proceedings of PME 25, University of Utrecht, July 2001 Vol 1, 111-122.

A survey of literature about educational uses of IC Technologies in mathematics education was done by a team issuing from five French laboratories working in various fields. A quantitative analysis of a corpus of 662 papers and two qualitative analyses of a sub-corpus helped to specify “dimensions” for the analysis of this very varied mass of innovation and research. Then, cluster analysis performed on each of the dimensions led to informative partitions. The method of analysis and the picture resulting from the partitions are offered as means to tackle the complex integration of IC Technologies into teaching and learning.

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