• Lagrange, J.B. (2021) Les espaces de travail connectés : une perspective nouvelle pour la modélisation dans le secondaire?. Canadian Journal of Science, Mathematics and Technology Education, Springer, 2021,
   
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Our interest is in the development of modeling activities that link knowledge in various mathematical fields and other scientific disciplines. Looking at mathematics education, research we highlight the benefits and limitations of “problem-solving” and of approaches that involve translation between the real world and mathematics. Science education seldom takes mathematics into account. Its contribution is the conception of an “empirical referent”. Using epistemological studies, we highlight examples of modeling situations supported by a plurality of models and an organization of work that allows students to engage with them. To do this we adopt the theoretical framework of connected working spaces. A case study shows how students create linkages between the workspaces and the benefits of the connections on their understanding of concepts at stake.

10.1007/s42330-020-00130-6⟩. ⟨hal-03148894⟩

• Lagrange, J.B., Rogalski, J. (2017) Savoirs, concepts et situations dans les premiers apprentissages en programmation et en algorithmique. ANNALES de DIDACTIQUE et de SCIENCES COGNITIVES
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  • In several countries including France, there is a growing interest for the teaching and learning of algorithmics and programming at school and college level. It is then necessary to question the objectives of this teaching and learning, and to propose controlled implementations. This article, written by a researcher in cognitive ergonomics and a researcher in didactics, aims at assessing some research results in this field, on the basis of research work conducted sporadically since thirty years. It first attempts to show the permanence of questions related to beginners’ conceptual difficulties, and then tackles the issue of learning situations. Then it takes stock of results obtained in psychology of programming, focusing on a conceptual field precisely identified around the concept of computer variable. The conclusion gives evidence of a broad field of research now open.
    The final publication is available at https://mathinfo.unistra.fr/irem/publications/adsc/
• Minh, T. K., Lagrange, J.B. (2016) Connected functional working spaces: a framework for the teaching and learning of functions at upper secondary level. ZDM Mathematics Education on line first.
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  • This paper aims at contributing to remedy the narrow treatment of functions at upper secondary level. Assuming that students make sense of functions by working on functional situations in distinctive settings, we propose to consider functional working spaces inspired by geometrical working spaces. We analyse a classroom situation based on a geometric optimization problem pointing out that no working space has been prepared by the teacher for students’ tasks outside algebra. We specify a dynamic geometry space, a measure space and an algebra space, with artefacts in each space and means for connecting these provided by Casyopée. The question at stake is then the functionality of this framework for implementing and analyzing classroom situations and for analyzing students’ and teachers’ evolution concerning functions, in terms of geneses relative to each space.
    The final publication is available at Springer via http://dx.doi.org//10.1007/s11858-016-0774-z
• Lagrange, J.B. (2014)  Teaching mathematics online: emergent technologies and methodologies. Research in Mathematics Education. 16(2),208-212.
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• Lagrange, J.B. (2014) New representational infrastructures: broadening the focus on functions . Teaching Mathematics and its Applications 33 (3): 179-192 .
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  • For more than 10 years, I had the honour and pleasure to work with Celia Hoyles and Richard Noss. We share a common concern for more learnable mathematics, especially in algebra, and for the need to build new representational infrastructures taking advantage of technology. Beyond this common concern, my choice to work in the French institutional context and paradigms for the Casyopee project led to a different approach. Roughly speaking, in these context and paradigms the stake is to help students access existing representations while Hoyles and Noss privilege building new more learnable representations. The aim of the article is to investigate empirically how, in spite of this gap, Hoyles and Noss’ approach can be an inspiration for enlarging the focus on a given topic, freeing oneself of a too strict dependency to one’s own context.
• Kynigos, C., Lagrange, J.B. (2014) Cross-analysis as a tool to forge connections amongst theoretical frames in using digital technologies in mathematical learning.Educational studies in Mathematics . 85(3), 321-327 (2014).
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  • This is the introductory paper of this special issue which contains a description of and reactions to a sample of cross–case analyses of empirical studies carried out by a group of six European research teams. The cross–case analysis, along with cross-experimentation was a key element of the teams' strategy for integrating theoretical frameworks with a view to also acquiring better focus on context and on the role and uses of the diverse representations afforded by the digital media developed within the project.
• Lagrange, J. B. & Kynigos, C. (2014) Digital technologies to teach and learn mathematics: Context and re- contextualization. Educational studies in Mathematics 85(3), 381-403
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  • The central assumption of this paper is that, especially in the field of digital technologies to teach and learn mathematics, the influence of the context in which research is carried out has not been given enough attention, so that research results are not really useful outside this context. We base our discussion on the work of a group of European teams carrying out research with a special methodology of “cross-studies” and carrying out “cross-analyses” of particular studies. A context for a research study is described as a dynamic construction by researchers, connecting relevant contextual characteristics in the settings (empirical and academic) where research activity takes place and helping to gain insight from the outcomes of the study. Analyzing the design of two “Didactical Digital Artefacts,” and the associated cross- studies involving teams of three countries, we identify more or less conscious influences of characteristics in the researchers' contexts upon research outcomes. Cross-studies and cross-analysis help to go further by making researchers more aware of their context and of its characteristics. It also helps researchers to “re-contextualize,” that is to say to identify new contextual characteristics in the settings they are acting in, to gain insight from research outcomes that emerged in other contexts.
• Lagrange, J.B. (2014). A Functional Perspective on the Teaching of Algebra: Current Challenges and the Contribution of Technology. International Journal for Technology in Mathematics Education 21(1), 3-10.
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  • From the early nineties, most reformed curricula at upper secondary level choose to give functions a major position and a priority over rational expressions and equations of traditional algebra. The goal of this paper is to introduce key challenges resulting from this choice and to discuss the contribution that software environments associating dynamic geometry and algebra can bring to the teaching learning of functions. Two examples of situations based on the use of the Casyopée environment are proposed. They illustrate how educational design can handle key questions: experiencing co- variation and using references to bodily activity is crucial for students' understanding of functions, making sense of the independent variable is a major difficulty that needs to be addressed by special situations, and understanding the structure of the algebraic formula in a function is critical.
• Lagrange, J. B. & Psycharis, G. (2013) Investigating the Potential of Computer Environments for the Teaching and Learning of Functions: A Double Analysis from Two Research Traditions. Technology, Knowledge and Learning. 19(3), 255-286.
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  • The general goal of this paper is to explore the potential of computer environments for the teaching and learning of functions. To address this, different theoretical frameworks and corresponding research traditions are available. In this study, we aim to network different frameworks by following a ‘double analysis’ method to analyse two empirical studies based on the use of computational environments offering integrated geometrical and algebraic representations. The studies took place in different national and didactic contexts and constitute cases of Constructionism and Theory of Didactical Situations. The analysis indicates that ‘double analysis’ resulted in a deepened and more balanced understanding about knowledge emerging from empirical studies as regards the nature of learning situations for functions with computers and the process of conceptualisation of functions by students. Main issues around the potential of computer environments for the teaching and learning of functions concern the use of integrated representations of functions linking geometry and algebra, the need to address epistemological and cognitive aspects of the constructed knowledge and the critical role of teachers in the design and evolution of students’ activity. We also reflect on how the networking of theories influences theoretical advancement and the followed research approaches
• Lagrange, J.B. (2013). Anthropological Approach and Activity Theory: Culture, Communities and Institutions. International Journal for Technology in Mathematics Education, 20(1), 33-38.
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  • The goal of this paper is to evaluate the contribution of the anthropological approach (AA) concurrently to Activity Theory (AT) in view of overarching questions about classroom use of technology for teaching and learning mathematics. I will do it first from a philosophical point of view, presenting the main notions of AA that have been used to address these questions, and then consider the conceptual roots and development of AA in comparison with those of AT. Then I will consider a particular research study for which a specific AT framework has been used, together with the AA notion of instrumented technique.
• Lagrange, J.B., (2010). Teaching and learning about functions at upper secondary level: designing and experimenting the software environment Casyopée.International Journal of Mathematical Education in Science and Technology, 41(2),243-255.
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  • Casyopée is an evolving project focusing on the development of both software and classroom situations to teach algebra and analysis at upper secondary level. In this article, we sketch the rationales for the Casyopée project in relationship with the focus on functions in upper secondary curricula. To evaluate Casyopée's contribution, we present and analyse an experimental teaching unit carried out in the ReMath European project focusing on the approach to functions via multiple representations for the 11th grade.
• Lagrange, J.B. (2010). Innovations technologiques dans l’enseignement des mathématiques : paradigmes et changement de la professionnalité de l’enseignant. Quadrante, Revista Teórica e de Investigação em Educação Matemática. XVIII, 29-51.
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  • Taking Mathematics as an exemplar, this paper addresses the processes of dissemination, regression and resurgence in the development of technological innovations for teaching/learning. Starting from the notion of technological revolution, the concepts of paradigm, instrumented action schemes and professionality are used to analyse these processes, and particularly the position of a key actor: the teacher. The example of the blackboard helps to specify how the paper looks to evolutions resulting from the use of artefacts for teaching/learning. It shows how the above concepts are relevant to analyse a century long process.Then the paper considers technological revolutions that proposed their paradigms in the last thirty years. These new paradigms demand deep changes in teachers’ professionality as well as instrumental genesis particular to the associated tools. These changes and new genesis can only develop on the long run.
    Mathematics teaching/learning was particularly affected by two overlapping revolutions: programming and symbolic computation on one side, and visualisation and interactivity on the other side. The underlying paradigms did not really permeate teaching practices. Didactical research worked to reconstruct these paradigms especially by taking the teacher into account. But, meanwhile, some of them lost their topicality. The Internet revolution deeply impinges on the teachers’ professionality. The paper offers two potential models of evolution and points out entries for research studies.
• Lagrange, J.B., Erdogan, E. (2009). Teacher’s emergent goals in spreadsheet based lessons: Analysing the complexity of technology integration. Educational Studies in Mathematics, 71(1), 65- 84.
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  • We examine teachers’ classroom activities with the spreadsheet, focusing especially on episodes marked by improvisation and uncertainty. The framework is based on Saxe’s cultural approach to cognitive development. The study considers two teachers, one positively disposed towards classroom use of technology, and the other not, both of them experienced and in a context in which spreadsheet use was compulsory: a new curriculum in France for upper secondary non- scientific classes. The paper presents and contrasts the two teachers in view of Saxe’s parameters, and analyzes their activity in two similar lessons. Goals emerging in these lessons show how teachers deal with instrumented techniques and the milieu under the influence of cultural representations. The conclusion examines the contribution that the approach and the findings can bring to understanding technology integration in other contexts, especially teacher education..
• Lagrange, J.B. & Caliskan, N. (2009). Usages de la technologie dans des conditions ordinaires : le cas de la géométrie dynamique au collège. Recherches en Didactique des Mathématiques, 29(2), 189–226.
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  • This article studies the actual uses of digital technology by teachers under “ordinary” conditions. These uses do not reveal a high degree of integration but rather the expectations that teachers have regarding technology. Our hypothesis is that discrepancies exist, first, between those expectations and the potentialities that a didactical analysis assigns to technology and, second, between those expectations and didactical phenomena in actual classrooms. We study the case of dynamic geometry in the middle grades in France. We observe successive gaps in the discourses of didactical research, official curricular directives, and textbooks. In the textbooks’ propositions, we distinguish two types of uses, and we report on observations of two teachers, each practicing one type. Interpreting those observations by means of Ruthven and Hennessy’s model, we show how teachers’ expectations work in the classroom. Our study helps distinguish two worlds : the “world of expectations” and the “world of potentialities”. It also contributes to new directions for the professional development of teachers in technology use.
• Lagrange J.-B., Grugeon B.(2003)Vers une prise en compte de la complexité de l’usage des TIC dans l’enseignement. Revue française de pédagogie 143, 101-111.
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  • Selon de nombreux travaux d'innovation et de recherche les TICE sont susceptibles d'apporter une contribution intéressante aux apprentissages scolaires. Les institutions scolaires mènent des politiques volontaristes en faveur de leur utilisation. Pourtant, les utilisations réellement constatées dans les classes demeurent souvent limitées.
    Pour comprendre les raisons de ce décalage, nous avons considéré un corpus international quasi exhaustif dans une discipline­ ­– les mathématiques – sur une durée de cinq ans et nous lui avons appliqué des traitements qualitatifs et quantitatifs dans l'esprit d'une "méta-étude". Ces traitements ont fait apparaître des facteurs relatifs à l’utilisation des TICE très inégalement pris en compte dans les publications et une lente évolution vers une reconnaissance de la complexité de l'enseignement et de l'apprentissage avec les TICE. Nous avons organisé ces facteurs en "dimensions d'analyse" qui devraient aider les innovateurs et chercheurs dans cette évolution en leur donnant les moyens d'appréhender cette complexité.
• Lagrange, J.B. (2000). L'Integration d'Instruments Informatiques dans l'Enseignement: Une Approche par les Techniques (The Integration of technological Instruments in teaching/learning: An Approach from the techniques). Educational Studies in Mathematics, 43(1), 1-30.
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  • The use of graphical and symbolic facilities in the teaching and learning of algebra and calculus will soon be a reality. Authors who write about the introduction of these instruments often claim that new technology is able to redress the imbalance between skill-dominated conceptions of school mathematics in favour of understanding. More recently some have stressed that 'experimental mathematics' traditionally the reserve of mathematical research may be incorporated into the teaching and learning of mathematics. This paper looks into these two ideas and shows that they conceal an essential dimension: techniques play an important role in mathematical activity, intermediate between tasks and theories. This paper draws on research studies on the introduction of symbolic systems on computers and calculators and considers 'new' techniques that accompany new technological instruments, their role in conceptualising and their links with 'usual' paper/pencil techniques, as a key to analyse the role of technology in education. This view implies non obvious tasks for the teacher in the introduction of technology: the design of praxeologies adapted to new instrumental settings and everyday action on students' techniques.
• Lagrange, J.B. (1999) Complex calculators in the classroom : theoretical and practical reflections on teaching precalculus. International Journal of Computers for Mathematical Learning 1999 4(1) 51-81.
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  • University and older school students following scientific courses now use complex calculators with graphical, numerical and symbolic capabilities. In this context, the design of lessons for 11th grade pre-calculus students was a stimulating challenge. In the design of lessons, emphasising the role of mediation of calculators and the development of schemes of use in an 'instrumental genesis' was productive. Techniques, often discarded in teaching with technology, were viewed as a means to connect task to theories. Teaching techniques of use of a complex calculator in relation with 'traditional' techniques was considered to help students to develop instrumental and paper/pencil schemes, rich in mathematical meanings and to give sense to symbolic calculations as well as graphical and numerical approaches. The paper looks at tasks and techniques to help students to develop an appropriate instrumental genesis for algebra and functions, and to prepare for calculus. It then focuses on the potential of the calculator for connecting enactive representations and theoretical calculus. Finally, it looks at strategies to help students to experiment with symbolic concepts in calculus.