Career and Background

Bibliography

Hodgson, B. R., Kuzniak, A. & Lagrange, J.- B. (Eds.) (2016). The Didactics of Mathematics: Approaches and Issues. A Homage to Michèle Artigue (Epilogue. A didactic adventure, pp. 253-269). New York: Springer.

Karp (Ed.) (2014). Leader in mathematics education: Experience and vision (Chap.1 Interview with Michèle Artigue, pp. 9-28). Rotterdam: Sense Publishers.

Citation by Marie-Jeanne Perrin Glorian for the 2013 Felix Klein ICMI Award

Official citation for the 2013 Felix Klein ICMI Award

The emergence of the Instrumental Approach

In this module, I describe the emergence of the instrumental approach, in the context of the study of the potential of CAS (Computer Algebra System) technology for mathematics learning in France, in the early nineties. I do it answering questions from Jorge Gaona, a Chilean Ph.D student of my research team. I think that knowing this story should help you better understand the ‘raisons d’être’ of this approach. I find also important to tell this story, answering questions of a young researcher who discovered the French didactic culture quite recently.

The key concepts of the Instrumental Approach

These two modules are devoted to the key concepts of the instrumental approach, making clear why and how, respectively cognitive ergonomics and ATD, the anthropological theory of the didactics, have provided the theoretical basis for this approach.

Retrospectively, I see this approach as resulting from a double process of deconstruction and reconstruction: the deconstruction of the dominant discourse regarding the potential and use of digital technologies in mathematics education, and the reconstruction of an alternative discourse thanks to cognitive ergonomics and ATD.

Module 2 is devoted to the ergonomic pillar of the instrumental qpproach, precisely the ergonomic perspective that Pierre Rabardel developed in collaboration with Pierre Vérillon. I introduce this perspective, focusing on the main elements that have been incorporated in the instrumental approach: the distinction between artefacts and instruments, and the concept of instrumental genesis with the dual processes of instrumentalization and instrumentation.

Module 3 is devoted to ATD, the second theoretical pillar of the instrumental approach. I first explain why ATD entered the scene, and then introduce the main elements of this theory incorporated in the instrumental approach: the concept of praxeology and the hierarchy of levels of didactic codetermination. I end this module by showing an interesting phenomenon of hybridization that resulted from this theoretical combination: the distinction between the epistemic and pragmatic values of techniques.

Bibliography

Module 2
Artigue, M. & Robinet, J. (1982). Conceptions du cercle chez des enfants de l’école élémentaire. Recherches en Didactique des Mathématiques, 3.1, 5-64.

Artigue, M. (2002). Learning mathematics in a CAS environment: the genesis of a reflection about instrumentation and the dialectics between technical and conceptual work. The International Journal of Computers for Mathematics Learning, 7 (3), 245-274.

Guin, D. & Trouche, L. (1999). The complex process of converting tools into mathematical instruments : the case of calculators. The International Journal of Computers for Mathematical Learning, 3 (3), 195-227.

Rabardel, P. (1995). Les hommes et les technologies. Paris : Armand Colin.

Rabardel, P. (2002). People and Technologies- a cognitive approach to contemporary instruments. Open access translation by Heidi Wood.

Vérillon, P. & Rabardel, P. (1995). Cognition and artifacts : a contribution to the study of thought in relation to instrumented activity. European Journal of Psychology of Education, 10/1, 77-101.

Module 3
See Yves Chevallard Unit in ICMI AMOR and more especially the modules introducing the concept of praxeology and the hierarchy of levels of didactic codeterminacy.

Artigue, M. (2002). Learning mathematics in a CAS environment: the genesis of a reflection about instrumentation and the dialectics between technical and conceptual work. The International Journal of Computers for Mathematics Learning, 7 (3), 245-274.

Lagrange, J.B. (1999). Complex calculators in the classroom: theoretical and practical reflections on teaching pre-calculus. International Journal of Computers for Mathematical Learning, 4 (1), 51-81.

Lagrange, J.B. (2000). L’intégration d’instruments informatiques dans l’enseignement. Une approche par les techniques. Educational Studies in Mathematics, 43 (1), 1-30.

The first outcomes of research based on the Instrumental Approach

This fourth module is devoted to the first research works engaging the instrumental approach: the research carried out in the frame of a national project mentioned in Module 1, the doctoral theses by Badr Defouad in Paris and Luc Trouche in Montpellier. These made clear the complexity of instrumental geneses of CAS technology and the necessity of their institutional management.  I synthesize the main empirical and theoretical outcomes of these research works, using selected examples. I end the module by evoking the rapid international dissemination of this approach, which induced new and fruitful theoretical combinations.

Bibliography

Artigue, M. (2002). Learning mathematics in a CAS environment: the genesis of a reflection about instrumentation and the dialectics between technical and conceptual work. The International Journal of Computers for Mathematics Learning, 7 (3), 245-274.

Artigue, M. (2005). The integration of symbolic calculators into secondary education: some lessons from didactic engineering. In D. Guin, K. Ruthven & L. Trouche (Eds.), The didactic challenge of symbolic calculators. Turning a computational device into a mathematical instrument (pp. 231-294). Dordrecht : Kluwer Academic Publishers.

Artigue, M. (2014). Perspectives on Design Research : The Case of Didactical Engineering. In A. Bikner-Ahsbahs, C. Knipping & N. Presmeg (Eds.), Approaches to Qualitative Research in Mathematics Education (pp. 467-496). New York : Springer.

Defouad, B. (2000). Étude de genèses instrumentales liées à l’utilisation d’une calculatrice symbolique en classe de première S. Thèse de doctorat. Université Paris 7.

Drijvers, P. (2003). Learning algebra in a computer algebra environment. Design research on the understanding of the concept of parameter (Doctoral dissertation). Utrecht: CD-β press.

Drijvers, P. (2004). Learning Algebra in a Computer Algebra Environment. International Journal for Technology in Mathematics Education, 11 (3), 77-90.

Guin, D., Ruthven, K., & Trouche, L. (Eds.) (2005). The didactic challenge of symbolic calculators. Turning a computational device into a mathematical instrument. Dordrecht : Kluwer Academic Publishers.

Guin, D. & Trouche, L. (1999). The complex process of converting tools into mathematical instruments : the case of calculators. The International Journal of Computers for Mathematical Learning, 3 (3), 195-227.

Maschietto, M. (2002). L’enseignement de l’analyse au lycée. Les débuts du jeyu local-global dans l’environnement de calculatrices. Thèse de doctorat. Université Paris 7 & Università degli Studi Torino.

Maschietto, M. (2008). Graphic Calculators and Micro-Straightness: Analysis of a Didactic Engineering. The International Journal of Computers for Mathematics Learning, 13 (3), 207-230.

Trouche, L. (1997). À propos de l’apprentissage de fonctions dans un environnement de calculatrices, étude des rapports entre processus de conceptualisation et processus d’instrumentation (Thèse de doctorat). Université de Montpellier.

Trouche, L. (2003). From Artifact to Instrument: Mathematics Teaching Mediated by Symbolic Calculators. In P. Rabardel, & Y. Waern (Eds.), Special issue of Interacting with Computers, 15(6), 783-800.

Kieran, C., & Drijvers, P. (2006). The  co-emergence of machine techniques, paper-and-pencil techniques, and theoretical reflection: A study of CAS use in secondary school algebra. The International Journal of Computers for Mathematics Learning, 11 (2), 205-263.

Testing the instrumental approach with new artefacts: the case of spreadsheet

In this fifth module, I present and discuss the first test of the pertinence of the instrumental approach outside the CAS context where it emerged. This occurred with the doctoral thesis of a student of mine, Mariam Haspekian, with whom I have prepared this module. The thesis, defended in 2005, used the instrumental approach to study the integration of spreadsheet technology in middle school mathematics, with a particular focus on the use of spreadsheet for the introduction of algebra. I describe how this test showed the pertinence of the instrumental approach in this new context, led to question the educational discourse regarding spreadsheet and algebra, and also produced theoretical outcomes with the notion of instrumental distance.

Bibliography

Arzarello, F., Bazzini, L. & Chiappini, G. (2001). A model for analysing algebraic processes of thinking. In R. Sutherland, T. Assude, A. Bell and R. Lins (Eds.), Perspectives on school algebra, vol. 22, (pp. 61-81). Dordrecht: Kluwer Academic Publishers.

Balacheff, N. (1994). Didactique et intelligence artificielle. Recherches en Didactique des Mathématiques, 14 (1-2), 9-42.

Capponi, B. (1990). Calcul algébrique et programmation dans un tableur : le cas de Multiplan. Doctoral thesis. Université Joseph Fourier – Grenoble 1.

Capponi, B. (1999). Le tableur pour le collège, un outil pour l’enseignement des mathématiques. Petit x, 52, 5-42.

Haspekian, M. (2005a). Intégration d’outils informatiques dans l’enseignement des mathématiques, étude du cas des tableurs. Doctoral thesis. Université Paris 7.

Haspekian, M. (2005b). An “Instrumental Approach” to Study the Integration of a Computer Tool Into Mathematics Teaching: the Case of Spreadsheets. The International Journal of Computers for Mathematics Learning, 10 (2), 109-141.

Sutherland, R., & Rojano, T. (1993). A spreadsheet approach to solving algebra problems. Journal of Mathematical Behaviour, 12, 353-383.

Haspekian M., & Artigue M. (2007) L'intégration de technologies professionnelles à l'enseignement dans une perspective instrumentale : le cas des tableurs. In G. Baron, D. Guin et L. Trouche (Eds), Environnements informatisés et ressources numériques pour l’apprentissage (pp. 37-63). Paris : Hermès. English version in preparation (will be available here soon)

The extension of the instrumental approach to dynamic geometry

This sixth module is devoted to the extension of the Instrumental Approach to dynamic geometry. I prepared it with Colette Laborde, a well-known researcher in the field of dynamic geometry who has been directly involved in this extension, through her investment in the MAGI project, “Learning better geometry with computers”. Relying on the research carried out in this project and the associated doctoral thesis by Angela Restrepo focusing on the development of dragging schemes by 6 graders, I show how this research made clear the complexity of the instrumental genesis of dragging, and proved the pertinence of the instrumental approach also for technologies designed with educational purposes.

Bibliography

Arzarello, F., Olivero, F., Paola, D., & Robutti, O. (2002). A cognitive analysis of dragging practises in Cabri environments. Zentralblatt fur Didaktik der Mathematik, 34 (3), 66-72.

Duval, R. (2020). Registers of semiotic representation. In S. Lerman (Ed.), Encyclopedia of Mathematics Education (Second Edition). New York: Springer.

Healy, L. (2000) Identifying and explaining geometrical relationship: Interactions with robust and soft Cabri constructions. In Proceedings of the 24th Conference of the International Group for the Psychology of Mathematics Education (Vol. 1 , pp. 103-117). Hiroshima: Tadao Nakahara, Masataka Koyama.

Laborde, C., Assude, T., Grugeon, B., & Soury-Lavergne, S. (2006).  Study of a teacher professional problem: how to take into account the instrumental dimension when using Cabri-geometry? In C. Hoyles, J.-B. Lagrange, L.-H. Son, & N. Sinclair (Eds.), Proceedings of  the Seventeenth ICMI Study Conference “Technology Revisited” (pp. 317-325). Hanoi Institute of Technology.

Laborde, C., & Laborde, J.-M. (2014). Dynamic and Tangible Representations in Mathematics Education. In S. Rezat, M. Hattermann, & A. Peter-Koop (Eds.), Transformation – A Fundamental Idea of Mathematics Education (pp.187-202). New-York: Springer.

Restrepo, A. (2008). Genèse instrumentale du déplacement en géométrie dynamique chez des élèves de 6ème. Doctoral thesis. Université Joseph Fourier Grenoble 1.

The extension of the Instrumental Approach to the teacher

This seventh module is devoted to the extension of the Instrumental Approach to the teacher. It has been prepared jointly with Maha Abboud, Mariam Haspekian and Luc Trouche, whose research has substantially contributed to this extension. I begin by the first step towards this extension: the introduction of the idea of instrumental orchestration by Luc Trouche in 2003. I move then to a complementary dimension of this extension with the idea of double instrumental genesis I introduced jointly with Mariam Haspekian, to account for the fact that, for a teacher, two distinct but intertwined instrumental geneses are at stake: a personal and a professional genesis, transforming the artefact respectively into a mathematical and a didactic instrument. I describe then how the combination with the Double Approach of teaching practices in the GUPTEN project led to a more global vision in terms of geneses of use, especially developed by Maha Abboud, Jean-Baptiste Lagrange and Fabrice Vzandebrouck.

Bibliography

Abboud-Blanchard, M. (2013). Technology in mathematics education. Study of teachers’ practices and teacher education. Syntheses and new perspectives.  Synthesis for HDR, University of Paris Diderot.

Abboud-Blanchard, M. (2014). Teachers and technologies: shared constraints, common responses. In A. Clark-Wilson, O. Robutti & N. Sinclair (Eds.), The Mathematics Teacher in the Digital Era: An International Perspective on Technology Focused Professional Development (pp. 297-318). London: Springer.

Abboud-Blanchard, M. (2016). The teacher perspective in mathematics education research: a long and slow journey still unfinished. In B.R. Hodgson, A. Kuzniak & J.-B. Lagrange (Eds), The didactics of mathematics: Approaches and issues. A homage to Michèle Artigue (pp.143-153). New York: Springer.

Abboud-Blanchard, M., & Lagrange, J.-B. (2006). Uses of ICT by pre-service teachers: Towards a professional instrumentation? International Journal for Technology in Mathematics Education, 13 (4), 183-191.

Abboud, M., & Rogalski, J. (2017). Des outils conceptuels pour analyser l'activité de l'enseignant "ordinaire" utilisant les technologies. Recherches en Didactique des Mathématiques, 37 (2-3), 161-216.

Abboud, M., & Vandebrouck,  F. (2013). Geneses of technology  uses: A theoretical model to study  the development of  teachers’ practices in technology environments.  In B. Ubuz, C.  Haser, & M.A. Mariotti (Eds.), Proceedings of CERME 8 (pp 2504-2513).

Artigue M., & Groupe TICE IREM Paris 7 (2006). L’utilisation de ressources en ligne pour l’enseignement des mathématiques au lycée : du suivi d’une expérimentation régionale à un objet de recherche. In N. Bednarz & C. Mary (Eds.), Actes du Colloque EMF 2006 (pp. 1-11). Sherbrooke : Université de Sherbrooke.

Drijvers, P. (2012). Teachers transforming resources into orchestrations. In G. Gueudet, B. Pepin, & L. Trouche (Eds.), From texts to ‘lived’ resources: Mathematics curriculum materials and teacher development (pp. 265-281). New York: Springer.

Gueudet, G., & Trouche, L., (2009). Towards new documentation systems for mathematics teachers. Educational Studies in Mathematics, 71, 199-218.

Haspekian, M. (2011). The co-construction of a mathematical and a didactical instrument. In M. Pytak, E. Swoboda, & T. Rowland (Eds.), Proceedings of CERME 7 (pp. 2298-2307).

Haspekian, M. (2014). Teachers' instrumental geneses when integrating spreadsheet software. In A. Clark-Wilson, O. Robutti, & N. Sinclair (Eds.), The Mathematics Teacher in the Digital Era (pp.241-275). New-York: Springer.

Robert, A., & Rogalski, J. (2002). Le système complexe et cohérent des pratiques des enseignants de mathématiques : une double approche. Revue Canadienne de l’Enseignement des Sciences, des Mathématiques et des Technologies, 2 (4), 505-528.

Robert, A., & Rogalski, J. (2005). A cross-analysis of the mathematics teacher’s activity. An example in a French 10th grade. Educational Studies in Mathematics, 59, 269-298.

Thomas, M.O.J. , & Palmer, J. M. (2014). Teaching with digital technologies: obstacles and opportunities. In A. Clark-Wilson, O. Robutti, & N. Sinclair (Eds.), The mathematics teacher in the digital era (pp. 71-89). New-York: Springer.

Trouche, L. (2004). Managing complexity of human/machine interactions in computerized learning environments. International Journal of Computers for Mathematics Learning, 9 (3), 281-307.

Trouche, L. (2005a). Construction et conduite des instruments dans les apprentissages mathématiques : nécessité des orchestrations. Recherches en Didactique des Mathématiques, 25 (1), 91-138.

Trouche, L, & Drijvers, P. (2010). Handled technology : Flashback into the future. ZDM. The International Journal on Mathematics Education, 42 (7), 667-681.

Reflecting on the whole story (1)

In the previous modules, I have introduced the instrumental approach, explaining its “raison d’être”, tracing its progressive development since the mid-nineties, and showing some of its main outcomes. In the eighth and ninth modules, I reflect on the whole story. Since this approach has benefited from the contributions of many researchers from different countries and cultures, with different theoretical backgrounds and research interests, I find interesting to rely for this reflection on the concept of research praxeology I introduced jointly with Marianna Bosch and Josep Gascón in 2011. Doing so, I connect this reflection with a more recent dimension of my research activity, today known as the networking of theories. In this module, thus, I present the concept of research praxeology, as I envisage it now, using examples from previous modules to help make sense of this construction.

Bibliography

Yves Chevallard’s Unit

Artigue, M. (Ed.) (2009). Connecting approaches to technology enhanced learning in mathematics: The TELMA experience. International Journal of Computers for Mathematics Learning, 14 (3).

Artigue, M. & Bosch, M. (2014). Reflection on Networking through the praxeological lens. In A. Bikner-Ahsbahs & S. Prediger (Eds.), Networking of Theories as a Research Practice in Mathematics Education (pp. 249-266). New York : Springer.

Artigue, M., Bosch, M., & Gascón, J. (2011). Research praxeologies and networking theories. In M. Pytlak, T. Rowlad, E. Swoboda (Eds), Proceedings of the Seventh Congress of the European Society for Research in Mathematics Education (pp. 2381-2390). University of Rzeszów, Poland.

Defouad B. (2000). Étude de genèses instrumentales liées à l’utilisation d’une calculatrice symbolique en classe de première S. Doctoral thesis. Université Paris 7.

Haspekian, M. (2005a). Intégration d’outils informatiques dans l’enseignement des mathématiques, étude du cas des tableurs. Doctoral thesis. Université Paris 7.

Kynigos, C., & Lagrange, J.-B. (Eds.) (2014). The ReMath enterprise. Educational Studies in Mathematics, 85 (3).

Trouche, L. (1997). À propos de l’apprentissage de fonctions dans un environnement de calculatrices, étude des rapports entre processus de conceptualisation et processus d’instrumentation. Thèse de doctorat. Université de Montpellier.

Reflecting on the whole story (2)

In this ninth module I continue the reflection undertaken in Module 8, drawing some lessons from the whole story. I do this by reviewing the different components of research praxeologies relying on the instrumental approach: the questions addressed, the techniques used to answer them, the theoretical discourse. This reflection makes clear substantial and convergent outcomes in terms of the understanding of instrumental geneses processes and their complexity; the production of many engineering designs; and also theoretical outcomes. There is no doubt that the praxeological dynamics of the instrumental approach has benefitted from the diversity of theoretical combinations it has generated. However, theoretical combinations and diversity are also source of questions. For instance, I show how the combination of cognitive ergonomics and ATD has been the source of vivid discussions regarding the relationships between schemes and techniques. And, concluding this series of modules, I insist on the fact that what has been shown is the collective work of a community, the empirical and theoretical advances that this community has made possible.

Bibliography

Artigue M. (2007). Digital technologies: a window on theoretical issues in mathematics education. In, D. Pitta-Oantazi & G. Philippou (Eds.), Proceedings of CERME 5 (pp. 68-82). Cyprus University Editions.

Artigue, M., & Mariotti, M.A. (2014). Networking theoretical frames: the ReMath enterprise. Educational Studies in Mathematics, 85 (3), 329-356. https://hal.archives-ouvertes.fr/hal-02368186

Caliskan-Dedeoglu, N. (2006). Usages de la géométrie dynamique par des enseignants de collège. Des potentialités à la mise en oeuvre : quelles motivations, quelles pratiques ? Doctoral thesis. Université Paris Diderot-Paris 7. https://tel.archives-ouvertes.fr/tel-00152076/document

Chaachoua H., Bessot A., Romo A., Castela  C. (2019). Developments and functionalities in the praxeological model. In M. Bosch, Y. Chevallard, F. Javier Garcia & J. Monaghan (Eds.), Working with the anthropological theory of the didactic in mathematics education:  A comprehensive casebook (pp.41-60). Routledge.

Croset, M.-C. (2009). Modélisation des connaissances des élèves au sein d'un logiciel d'algèbre. Études des erreurs stables inter-élèves et intra-élève en termes de praxis-en-acte. Doctoral thesis. Université Joseph Fourier- Grenoble 1.

Croset, M.-C., & Chaachoua, H. (2016). Une réponse à la prise en charge de l’apprenant dans la TAD : la praxéologie personnelle. Recherches en Didactique des Mathématiques, 36 (2), 161-196.

Defouad B. (2000). Étude de genèses instrumentales liées à l’utilisation d’une calculatrice symbolique en classe de première S. Thèse de doctorat. Université Paris 7.

Drijvers, P., & Gravemeijer, K. P. E. (2004). Computer algebra as an instrument: Examples of algebraic schemes. In D. Guin, K. Ruthven, & L. Trouche (Eds.), The didactical challenge of symbolic calculators: turning a computational device into a mathematical instrument (pp. 163-196). Dordrecht: Kluwer Academic Publishers.

Haspekian, M. (2005a). Intégration d’outils informatiques dans l’enseignement des mathématiques, étude du cas des tableurs. Doctoral thesis. Université Paris 7. (especially Chapter 2)

Hoyles, C., Kent, P., & Noss, R. (2004). On the integration of digital technologies into mathematics classrooms. International Journal of Computers for Mathematics Learning, 9 (3), 309-326.

Lagrange, J.-B. (2005). Using symbolic calculators to study mathematics. The case of tasks and techniques. In D. Guin, K. Ruthven & L. Trouche (Eds.), The didactic challenge of symbolic calculators. Turning a computational device into a mathematical instrument (pp. 113-135). Dordrecht: Kluwer Academic Publishers.

Monaghan, J. (2004). Teachers’ activities in technology-based mathematics lessons. International Journal of Computers for Mathematics Learning, 9 (3), 327-357.

Monaghan, J. (2007). Computer algebra, instrumentation and the anthropological approach. The International Journal for Technology in Mathematics Education, 14 (2), pp. 63-71.

Ozdemir Erdogan, E. (2006). Pratiques d'enseignants de mathématiques en environnement technologique : l'intégration du tableur dans l'enseignement des suites en première littéraire. Doctoral thesis. Université Paris Diderot - Paris 7. https://tel.archives-ouvertes.fr/tel-00419671/document

Thomas, M.O.J. , & Palmer, J. M. (2014). Teaching with digital technologies: obstacles and opportunities. In A. Clark-Wilson, O. Robutti, & N. Sinclair (Eds.), The mathematics teacher in the digital era (pp. 71-89). New-York: Springer.

Trouche, L. (1997). À propos de l’apprentissage de fonctions dans un environnement de calculatrices, étude des rapports entre processus de conceptualisation et processus d’instrumentation. Thèse de doctorat. Université de Montpellier.

Trouche, L., & Drijvers, P. (2014). Webbing and orchestration. Two interrelated views on digital tools in mathematics education, Teaching Mathematics and Its Applications. International Journal of the Institute of Mathematics and its Applications, 33 (3), 193-209.

Trouche, L., Gitirana, V., Miyakawa, T., Pepin, B., & Wang, C. (2019). Studying mathematics teachers interactions with curriculum materials through different lenses: towards a deeper understanding of the processes at stake. International Journal of Educational Research, 93, 53-67.    https://doi.org/10.1016/j.ijer.2018.09.002

Trouche, L., Gueudet, G., & Pepin, B. (2019). The resource approach to mathematics education. New York : Springer.