The complexity of the epistemological genesis of mathematical proof

Travelling through Tokyo and Singapore, it is a great pleasure to make a stop and meet colleagues and friends, hence one talk and two seminars. First at the Joetsu Seminar of Research on Mathematics Education in Tokyo on September the 13th, then in Singapore for a seminar at the Mathematics and Mathematics Education (MME) laboratory of the National Institute of Education (NIE) on September the 18th.

Abstract
Early learning of mathematics is first rooted in pragmatic evidences or learners’ confidence in the facts and procedures taught. Nonetheless, learners develop a true knowledge which works as a tool in significant problem situations, and which is accessible to falsification and argumentation. As teachers know, they could validate what they claim to be true, but based on means in general not conforming to mathematical standards. Teaching these standards requires an evolution of their understanding of what can count as a proof in the mathematical classroom, as well as an evolution of their mathematical knowing. This claim is discussed from the perspective of modelling the learners ways of knowing (the model cK¢), within the framework of the theory of didactical situations, bridging the semiotic system they use, the type of actions they perform and the controls they implement either to construct or to validate the solutions they propose to a problem.

Although this presentation is self-content, it could be interesting to complement it with the CINVESTAV talk which focused more on the didactical situations of validation [ppt], one of the specific situations of the Theory of didactical situations [ppt]

The complexity of the epistemological and didactical genesis of mathematical proof (2)

Math Ed. Doctoral Colloquium at CINVESTAV ... hereafter an advanced version of the slides in support to my talk (see the post below for a summary). Questions or comments are welcome (in Spanish, French or English), I will consider them for the talk.

Notes from a research journey on learning proof for
the CINVESTAV doctoral colloquium 2015

cKȼ, origine, cadrage théorique, utilisations et questions (la vidéo)

Voir [ici] le résumé de la commande du laboratoire de didactique André Revuz (LDAR) à laquelle répond l'exposé que l'on pourra suivre en visionnant la vidéo ci-dessous, et [là] pour un résumé plus substantiel.

Enregistrement vidéo de l'exposé présenté au séminaire du laboratoire LDAR
Vendredi 10 avril 2015, 14h-17h

Conceptions et situations

La place de la recherche sur les connaissances des élèves n'est pas tout à fait claire en didactique et est parfois contestée. En témoignent les vifs échanges entre psychologues et didacticiens dans les années 80, années fondatrices de la didactique des mathématiques. Pourtant l'étude de ces connaissances pour leur compréhension et leur modélisation est inséparable de celles engagées dans le cadre de la théorie des situations didactiques, c'est dans ces termes que Guy Brousseau l'évoque dans l'article qu'il publie dans le premier numéro de la revue Recherches en Didactique des Mathématiques alors qu'il déplore que les travaux de Diénès ne conduisent pas le didacticien à "questionner les mathématiques pour y chercher, au-delà des structures, les concepts et au-delà des concepts, éventuellement les conceptions qui pourraient se forger chez un sujet dans des situations historiques ou didactiques particulières."
Il poursuit :

"L'analyse de ces conceptions, qu'il faudra que l'élève possède ou évite, est inséparable de celle de la famille des situations spécifiques où elles prennent leur fonction et utilité. Toutes les deux sont inévitables dans toute entreprise qui prétendrait à la fois fournir une théorie dotée de ses méthodes de confrontation (probablement spécifiques aussi) et de techniques didactiques continument contrôlable par les enseignants" (Brousseau 1980 RDM 1.1 p.46)

Dans le même volume (p.80) Régine Douady insiste :

"Le problème didactique est de reconnaitre et décrire, à travers les actions et démarches des enfants placés dans une situation d'apprentissage, les modèles mathématiques qui expliquent, justifient ces actions et démarches."

En d'autres termes, la proposition de Douady est de produire des modèles mathématiques des conceptions dont Brousseau pose qu'elles sont indissociables des situations. Il faut entendre ici situation au sens de ce qui va, dans l'interaction entre l'élève et le milieu, être la source de problèmes mobilisateur des conceptions. Ces conceptions pouvant être, dans une perspective mathématique, erronées ou inadaptées et ce qui fait problème étant finalement largement déterminé par les conceptions initialement disponibles, la production de modèles tels qu'évoqués par Douady est un défi. C'est celui que relève la proposition de modélisation cK¢ notamment en formalisant la dualité entre problèmes et conceptions.

#ocTEL MOOC (week 1 A12) Snapshot on our approach and practice

The second part of the activity of this week focus on our pedagogical approach and our practice. I must say that I have no teaching duty since 1988, when I got a Senior Scientist position at the CNRS. However, I still continued to teach PhD courses and to a certain extent this is not that different from teaching undergraduate. So, let see how I can achieve this A12 task of week 1) From a learner perspective ("My Approach"), we are invited to locate ourself in the following space:

First, I would very much like to balance directivity which would allow me to know where I am going as a learner and whether I am not too far out of the track, and autonomy which would  allow me to experience knowledge and build my own understanding. I imagine that this opinion is very common.

Although important, the social dimension was not the main thing, apart from the joy of collectively arguing. Actually it depends on the content at stake. In mathematics and natural sciences learning collaboratively is quite productive thanks to the fact that the disciplines clearly gives the rules to solve conflict. In literature and several other topics, this is more difficult and the benefit of social interaction is less clear; indeed it brings the context to shape arguments and learn how to manage contradictions. It is a case where "reflective communication with the instructor" is really beneficial.

Hence, I would not fill one graph, but one for each discipline.

From a teacher perspective ("My course"), my first concern once I know what I want to teach is to find a way to pass to students the understanding that there is somewhere a problem and that the knowledge I claim to bring to them is the optimal one (possibly the not the only one) to solve this problem. For this, I start by a situation which allow students to express views, opinion, conceptions about a situation which later on will appear to be problematic in the sense I need in order to teach. If this is successful, for example (A11) having shaped a variety of evidence based opinions on behaviourism, I would stimulate the formulation of the problem(s) which will be the best to justify the knowledge I target, for example (A11) the problem of nature of the meaning built at an outcome come of the learning situation and the problem of its assessment. We understand that these situations blend individual, social and with-the-teacher situations.

Actually, this view is substantiated by the Theory of Didactical Situation, which provides the tools to assess continuously the relations between the activity, the situation and knowledge (to be learned, as it were).

A didactical view on authenticity

Retrieved from the TEL opinion blog, August the 27th, 2008

The search for authenticity of learning situations is a concern for most designers of TEL environments. Most of them realise soon that this is a desperate project since any environment is a representation of some kind of a reference, often called "reality", which keeps staying at a distance. To be as close as possible to reality does not mean much, unless we can qualify or quantify the closeness. Indeed, this is a challenge and we are not be well equipped today to take it up. A solution might be to find a theoretical framework within which we can formulate the problem, and then search for a solution within this framework. This first step will put limits on this solution, but it will make it much more tangible and so accessible to further progress. Currently we too much lack definitions and references to ensure that we can seriously discuss the issue. But, let's try something...

 First, we may agree that a learning environment becomes such if it is embedded in a situation which can contextualise the learner activity and hopefully stimulate, support and validate his or her successful learning. Be they formal or informal, these situations have an objective which can be made explicit in learning terms at least from the point of view of their designers; the fact that this objective is explicit for learners is another story. Following Brousseau(*), let's call "didactical" these situations. Didactical situations can be distinguished from other situations by their explicit intention to "teach". And here is the problem! As soon as the learner identifies this intention and bases on it his or her activity, it is very likely that the learning outcome will not have the expected "authenticity". All the search for authenticity of learning situations (and learning environments) is dedicated to the overcoming of this difficulty.

Second, let's consider the limit case of a didactical situation which didactical intention is completely transparent. If learning occurs in such a situation, we could ascertain that its outcome has the expected "authenticity": it does not owe the didactical intention (in other words the reasons for the activity of the learners are in the knowledge at stake not due to any guessing of the teacher or trainer expectations) These are "adidactical" situations, let's quote Brousseau:

"[The student must know that] this knowledge is entirely justified by the internal logic of the situation and that she can construct it without appealing to didactical reasoning. Not only can she do it, but she must do it because she will have truly acquired this knowledge only when she is able to put it to use by herself in situtations which she will come across outside any teaching context and in the absence of any intentional direction. Such situation is called an adidactical situation. Each item of knowledge can be characterized by a (or some) adidactical situation(s) which preserve(s) meaning; we shall call this a fundamental situation." (Brousseau p.30)

These three concepts: didactical situation, adidactical situation and fundamental situation will allow us to locate our problem of authenticity, and to formulate it.

So, designing an authentic learning situation depends on our capacity to characterize the related fundamental situation in relation to the piece of knowledge which learning is at stake. The problem is then not the closeness to reality, but the fact that the situation has the epistemic properties specific to this piece of knowledge. The specification of a so-called authentic environment requires first the expression of these epistemic properties and of the way they can be "translated" in the tangible world. However, once we have such a situation, there may be still a long way to designing an adidactical situation likely to be made available to learners. The design of this adidactical situation and the related environment is the challenge of designers of authentic TEL environments. After that, there is still one issue for the teacher: bring to life this adidactical situation in the classroom without damaging its "authenticity", what is the didactical challenge!

A quick example to conclude this post: the concept of "angle" in mathematics finds its full meaning when linear measurement is not possible or too "expensive" (for example when sailing on the Atlantic). Let say that the fundamental situation for "angle" is the problem of locating a point in the macro-space. We can realize that the classroom can hardly host a macro-space, we have then to find a situation which has the characteristics of the macrospace (linear measurement being impossible or too "expensive") and can be implemented in a classroom. If the problem were presented in the frame of a piece of paper (micro-space), the situation may appear quite artificial in the students eyes (a ruler is enough), then the corresponding didactical situation would be delicate to negotiate and in the end fragile. Technology can offer a solution, opening the window of the computer on the macro-space...

Brousseau G. (1997) Theory of Didactical Situations in Mathematics . Springer (Kluwer Acad. Pub.)