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Guy Brousseau Unit

Felix Klein Award 2003


Credits : All modules are presented by Claire Margolinas (Clermont-Auvergne University) and Annie Bessot (grenoble-Alpes University), except Module 0 by Jean-Luc Dorier (University of Geneva). All videos are produced by IPPA (Clermont-Auvergne University).

 

Under construction

Module 0

Career and Background

 

Bibliography

Official citation for the 2003 Felix Klein ICMI Award

A prominent researcher in a central field for mathematics education - A life dedicated to the understanding and improvement of mathematical education and learning by André Rouchier

Gibel, P., Salin, M.H. (réalisateurs). (2016). Entretien avec Guy et Nadine Brousseau [film réalisé pour la Commission Française pour l'Enseignement des Mathématiques]. Interview with Guy and Nadine Brousseau made in the context of the presentation of the French tradition in didactics of mathematics, supervised by Michèle Artigue for ICME-13, in July 2016 in Hamburg (Germany).

Module 1

Theory of didactical situations in mathematics, an epistemological revolution

Claire Margolinas and Annie Bessot, have worked within the theoretical framework of the Theory of didactical situations in mathematics and they have a deep understanding of Brousseau’s works. They have been faced with the very difficult task of selecting some aspects of Brousseau’s work. This selection was necessary because Brousseau’s work is very subtle and has many aspects.
This first module is devoted to the introduction of the principles of Brousseau’s theory: didactics of mathematics as a field of theoretical and experimental scientific research, in relation to engineering as a “phenomenotechnique”.

Guy Brousseau's website: guy-brousseau.com

 

Bibliography

Brousseau G. (2003). Glossary : http://guy-brousseau.com/biographie/glossaires/

Artigue M. (2017). In J. Gascón and P. Nicolás: Can didactics say how to teach? For the Learning of Mathematics, 37, 3.

Bachelard G. (1965). L’activité rationaliste de la physique contemporaine. Paris : PUF.

Bessot A. (2011). L’ingénierie didactique au cœur de la théorie des situations. In C. Margolinas et al. (coordonnés par) En amont et en aval des ingénieries didactiques. Grenoble : La Pensée Sauvage.

Brousseau, G. (1975). Exposé. Colloque « L’analyse de la didactique des mathématiques ». Bordeaux : IREM de Bordeaux.

Brousseau G. (2002). Theory of didactical situations in mathematics. Didactique des mathématiques, 1970 - 1990. Kluwer Academic Publishers : Dordrecht / Boston / London.

Brousseau, G. (2004). Felix Klein Medallist : Research in mathematics education. In M. Niss (Éd.), Proceedings of the Tenth International Congress on Mathematical Education (pp. 244‑254). IMFUFA, Roskilde University.

Brousseau G. (2010). Le cours de Sao Paolo (2009): http://guy-brousseau.com/category/3le-cours-2010/

Brousseau G. (2011). Notes on the observation of classroom practices (V. Warfield, Trad.). http://blog.espe-bretagne.fr/visa/wp-content/uploads/brousseau_2009_3.pdf

International Council for Science (ICSU) (2004) Annual report. https://council.science/publications/annual-report-2004/

Perrin-Glorian, M.-J. (1994). Théorie des situations didactiques : naissance, développements, perspectives. In M. Artigue, R. Gras, C. Laborde, P. Tavignot, (Eds.), Vingt ans de didactique des mathématiques en France (pp. 97-147). Grenoble : La Pensée Sauvage.

Module 2

The dual aspects of knowledge

This second module is devoted to one of the most important concepts of the Theory of mathematical situations, namely the dual aspect of knowledge: situational knowledge (“connaissance” in French) and institutional knowledge (“savoir” in French). This module is the basis for the other modules and can be considered as an introduction.
In this module, we also introduce the situation of “cars and garages”, an example that will be developed in the following modules.

 

Bibliography

Brousseau, G., Brousseau, N., & Warfield, G. (2014). Teaching Fractions through Situations : A Fundamental Experiment. Springer.

Margolinas, C. (2014). Connaissance et savoir. Concepts didactiques et perspectives sociologiques ? Revue Française de Pédagogie, 188, 13-22. https://journals.openedition.org/rfp/4530

Module 3

Mathematical situation

This third module is dedicated to the characterization of mathematical situations. We consider Brousseau’s question: “Why studying situations if what is at stake is to acquire institutional knowledge?”
This module includes the distinction made by Brousseau between ‘problem’ and ‘situation”. The situation of the “race to twenty” is introduced as an example that will be developed in some following modules. We introduce the properties and components of mathematical situations: milieu, stake and action situation as a general model.

 

Bibliography

Brousseau G. (1997). Theory of didactical situations in mathematics. Didactique des mathématiques, 1970 - 1990. Kluwer Academic Publishers : Dordrecht / Boston / London.

Brousseau G. (2010). Le cours de Sao Paolo (2009)  http://guy-brousseau.com/cours-2010-les-situations-mathematiques-proprietes-et-composantes/

http://guy-brousseau.com/587/diaporama-3-ingenierie-des-situations-mathematiques/ 

Module 4

Action situation

This fourth module is dedicated to the concept of action situation, including milieu, stake and situational knowledge studying the “cars and garages” situation introduced in module 2. We develop the first status of knowledge: situational knowledge in action situation.

 

Bibliography

Brousseau, G. (1972). Processus de mathématisation. In La mathématique à l’Ecole Elémentaire (p. 428-442). APMEP. http://guy-brousseau.com/wp-content/uploads/2010/09/Processus_de_mathematisationVO.pdf

Briand, J., Loubet, M., & Salin, M.-H. (2004). Apprentissages mathématiques en maternelle. Hatier.

Margolinas, C., & Wozniak, F. (2012). Le nombre à l’école maternelle. Une approche didactique. De Boeck.

Module 5

Formulation situation

This fifth module is dedicated to the concept of formulation situation.
Implicit situational knowledge has been recognized as useful for action. In order to produce and use a formulation, one has to encounter a new situation during which formulation is necessary. The status of the situational knowledge changes when some aspects of knowledge are formulated, as we will explain it in this module.

 

Bibliography

Brousseau, G. (1997). Theory of didactical situations in mathematics. Didactique des mathématiques, 1970 - 1990. Kluwer Academic Publishers : Dordrecht / Boston / London.

Briand, J., Loubet, M. & Salin, M.-H. (2004). Apprentissages mathématiques en maternelle. Paris : Hatier.

Module 6

Validation situation

This sixth module is dedicated to the concept of validation situation.
Implicit situational knowledge and explicitly formulated situational knowledge has been encountered in the previous situations. However, in mathematics, validation is an essential dimension: formulations have to become statements and then conjectures, which then have to resist to contradiction, in order to be validated. We thus explore the conditions and constraints of the validation situations.

 

Bibliography

Brousseau, G. (1997). Theory of didactical situations in mathematics. Didactique des mathématiques, 1970 - 1990. Kluwer Academic Publishers : Dordrecht / Boston / London.

Margolinas, C. (1993). De l’importance du vrai et du faux dans la classe de mathématiques. Grenoble : La pensée sauvage.

Margolinas, C. (2009). La importancia de lo Verdadero y de lo Falso en la clase de matemàtematicas (J. E. Fiallo Leal, Éd.). Ediciones Universidad Industrial de Santander.

Module 7

Fundamental situation. An epistemological point of view

In this module and the following three, we present a very important concept of Theory of Mathematical Situations, i.e. the notion of fundamental situation. This concept is characteristic of the epistemological point of view. We first synthesize the previous modules in order to introduce the concept of fundamental situation. Looking for a fundamental situation mainly means seeking how controlling, from an epistemological point of view, that there is a necessity to consider some meanings of an institutional knowledge as different.

Brousseau G. (2010) De l’art d’enseigner à l’étude des situations .  Le cours 2010. http://guy-brousseau.com/wp-content/uploads/2020/02/de-l-art-d-enseigner-%C3%A0-l-etude-des-situationsGB33.ppt
Powerpoint presenattion of the teh race to 20 http://guy-brousseau.com/587/diaporama-3-lingenierie-des-situations-mathematiques/
Brousseau G. (2005) Situations fondamentales et processus génétiques de la statistique. In Mercier A. et Margolinas C. (coordonné par) Balises en didactique des Mathématiques, cours de la XIIe école d’été de DDM. La Pensée Sauvage : Grenoble. 165-194
AMOR Yves Chevallard unit. Module 2 Didactic transposition: a tool for analysis.

Module 8

In search of a fundamental situation concerning cardinal numbers

We chose to start with natural numbers for a number of reasons: consistency with axiomatics within mathematics, convergence with anthropology... human history and psychogenetic development... Hence our first question is to find what could be a fundamental situation for natural numbers.
We are looking for a situation that can generate at least the two basic meanings of natural number, i.e. cardinal and ordinal. The question thus becomes two folded. We will only examine the first here: is there a fundamental situation for the "cardinal" meaning of natural numbers?

Briand, J., Loubet, M., & Salin, M.-H. (2004). Apprentissages mathématiques en maternelle. Paris: Hatier.
Margolinas, C., & Wozniak, F. (2012). Le nombre à l’école maternelle. Une approche didactique. De Boeck.

Module 9

In search of a fundamental situation concerning ordinal numbers

Here, we examine the second question: Is there a fundamental situation for the "ordinal" meaning of natural numbers?

Margolinas, C., & Wozniak, F. (2014). Early construction of number as position with young children : A teaching experiment. ZDM - The International Journal of Mathematics Education, 46(1), 29‑44. https://doi.org/10.1007/s11858-013-0554-y

Module 10

In search of a fundamental situation concerning rational numbers

We continue our search for a fundamental situation for numbers by briefly mentioning the didactic engineering emblematic of Guy and Nadine Brousseau's work on rational numbers and decimals.

Brousseau G. (1981). Problèmes de didactique des décimaux : deuxième partie. Recherches en Didactique des Mathématiques, 2(1),. La Pensée Sauvage. 37–127. https://revue-rdm.com/1981/problemes-de-didactique-des/
Brousseau G., Brousseau N. (1987). Rationnels et décimaux dans la scolarité obligatoire. IREM de Bordeaux. https://hal.science/hal-00610769
Brousseau, G., Brousseau, N., & Warfield, V. (2004). Rationals and decimals as required in the school curriculum. Part 1 : Rationals as measurement. The Journal of Mathematical Behavior, 23(1), 1-20. https://www.sciencedirect.com/science/article/abs/pii/S0732312303000683?via%3Dihub
Brousseau, G., Brousseau, N., & Warfield, V. (2007). Rationals and decimals as required in the school curriculum : Part 2 : From rationals to decimals. The Journal of Mathematical Behavior, 26(4), 281‑300. http://www.sciencedirect.com/science/article/B6W5B-4R5G38N-1/2/89f8fa55b19b428152a164aa16a029bb
Brousseau, G., Brousseau, N., & Warfield, V. (2008). Rationals and decimals as required in the school curriculum : Part 3. Rationals and decimals as linear functions. The Journal of Mathematical Behavior, 27(3), 153‑176.
http://www.sciencedirect.com/science/article/B6W5B-4TN5MH1-1/2/8b7c9f4fd5b0182ea2ce54a8657b515b
Brousseau, G., Brousseau, N., & Warfield, V. (2009). Rationals and decimals as required in the school curriculum : Part 4 : Problem solving, composed mappings and division. The Journal of Mathematical Behavior, 28(2‑3), 79‑118.
 http://www.sciencedirect.com/science/article/B6W5B-4XBWW66-1/2/e8ed54e6e9f5ab39f8c336c70805a027
Brousseau G., Brousseau N., Warfield V. (2014). Teaching Fractions through Situations: A Fundamental Experiment. Springer.
Ratsimba-Rajohn, H. (1982). Eléments d’étude de deux méthodes de mesures rationnelles. Recherches en Didactique des Mathématiques, 3(1), 65‑113.

Module 11

 

Theory of didactical situations in mathematics

To conclude this unit, in this module we establish a link between the theory of mathematical situations and the theory of didactic situations in mathematics within the theory of situations.

Bessot A. (2024). Introduction to the theory of situations: Fundamental Concepts of the Didactics of Mathematics. https://hal.science/hal-04500947
https://hal.science/hal-04473846 for the original in French
https://hal.science/hal-04500955 for the German translation
https://hal.science/hal-04523563 for the Italian translation
Guy Brousseau's website:    guy-brousseau.com
Brousseau G. (1978). Monographie d’un enfant en difficulté, l’enfant Gaël.
https://guy-brousseau.com/2432/la-monographie-d%e2%80%99un-enfant-en-difficulte-l%e2%80%99enfant-gael-1978/#more-2432 
Brousseau, G. (1980). L’échec et le contrat. La politique de l’ignorance. Mathématiques enseignement et société, Recherches, 41,177‑182
Brousseau, G., & Warfield, V. M. (1999). The Case of Gaël. The Journal of Mathematical Behavior, 18(1), 7‑52. http://www.sciencedirect.com/science/article/B6W5B-3Y2NBSY-K/2/aebd19d0e81ab68e152cdf430309082b
(Brousseau (Mexico) 2000, p. 24)
Brousseau G. (1999) Education et didactique des mathématiques. Educacion y didactica de las matematicas, Mexique. https://hal.science/hal-00466260
Brousseau, G. (2000) Education et Didactique des mathématiques. Version française de l’auteur. Educaciòn matemàtica, v. 12, n. 1. 5–39.
Brousseau G. (2002). Theory of didactical situations in mathematics. Didactique des mathématiques, 1970 - 1990. Kluwer Academic Publishers : Dordrecht / Boston / London.
Brousseau G. (2010) Les situations mathématiques : propriétés et composantes. Diaporama: cours de Sao Paolo. http://guy-brousseau.com/cours-2010-les-situations-mathematiques-proprietes-et-composantes  
Margolinas C. (2005). Essai de généalogie en didactique des mathématiques. Schweizerische Zeitschrift für Bildungswissenschaften (Revue Suisse des Sciences de l'Education). 27(3). 343-360.
Margolinas C. (2014). Construire des points de vue d’élèves : des défis théoriques et méthodologiques pour la recherche en didactique des mathématiques. Chaachoua, H., Bessot, A. et al. (dir.) (2021). Nouvelles perspectives en didactique : le point de vue de l’élève, questions curriculaires, grandeur et mesure. La pensée sauvage : Grenoble. 19-47.
Margolinas C. (2015). Exercices, problèmes, situations et tâches comme lieux de Rencontre. Revue des HEP et institutions assimilées de Suisse romande et du Tessin, 19. 31-49. https://hal.archives-ouvertes.fr/hal-01165559
Margolinas C. (2021). Construire des points de vue d’élèves : des défis théoriques et méthodologiques pour la recherche en didactique des mathématiques. Chaachoua, Bessot & al. (éds). Nouvelles perspectives en didactique :Le point de vue de l’élève, questions curriculaires, grandeur et mesure. La pensée sauvage sauvage : Grenoble. 19-48. https://hal.science/hal-03824848