An hommage to Juliana Szendrei

Tomorrow, in Budapest, researchers and teachers from Hungary and abroad will meet to pay hommage to Juliana Szendrei who passed away early January this year. I could not join the conference, so it is with this short post that I will participate and share this moment in memory of Juliana.

Beyond conferences and readings, I came to know Juliana Szendrei from a collaboration within the framework of a Tempus project, thanks to the complicity of Paolo Boero who introduced me. I remember our first meeting in Budapest in the mid-90s. It was in the beginning of the winter, the weather was cold and cloudy, the material conditions a bit limited and the use of the technology somewhat uncertain. But Juliana was there. She was so enthusiastic and eager to facilitate everything that very soon I forgot all these difficulties and enjoyed contributing to her project to enhance teacher training and mathematics learning.

As a research leader in mathematics education, Juliana Szendrei was committed to the international movement to improve the research area and to set firm theoretical foundations, including on a topic which I am specially interested in, the learning of mathematical proof. Actually, she was not only a researcher in mathematics education but also a good mathematician, this shades a very special light on her work. In particular, she was aware of the evolution of her own understanding of what a proof is. She shared this view during one of our working sessions. It has then been published in a book on proof edited by Paolo (see below). She views this evolution as a series of steps, from step 1 to a step 7... at step 1, as a "conformist learner", she saw mathematical proof as a "ceremony" the rules of which she was quite able to follow. The rewards of the teacher led her to step 2: the feeling of being part of a community, something like a community of mathematicians. But, Juliana was concerned by the fact that this could result in a split between this community and the rest of the world. So, she found herself better when teaching probability and coming to the belief that "mathematics is about the theory, not about the real coin". I will not describe here all the steps she told she went through, but notice that her understanding of mathematical proof as a mathematics educator was rooted in this awareness of the role of mathematics as a modelling tool, and the role of proof in making this tool so robust and efficient. This understanding that the meaning of a theory rests in the dialectical relationship between the theory and the concrete world is also a mark of her view on research in mathematics education.

Juliana Szendrei primary objective was the concrete enhancement of mathematics teaching and learning in schools as they were, with the curricula as they were at that time. This pragmatic view of her responsibility as a researcher guided her action. Sure she would smile if I took Tomas Varga words to sum up the lesson I learned from her: teaching and learning problems "cannot be settled without further research and deeper insight into the learning process. But we cannot wait until they are".

Dessin, figure et objet en géométrie

Révisé 08/05/2014

Le problème d'enseignement est connu, probablement aussi ancien que la géométrie elle-même : l'élève raisonne sur ce qu'il voit tracé sur sa feuille comme s'il s'agissait de l'objet géométrique lui-même, aussi lui attribue-t-il souvent des propriétés anecdotiques liées au tracé particulier qu'il a sous les yeux, ou des propriétés de stéréotypes forgés dans les habitudes de représentation silencieusement établies dans la classe. Il faut se méfier de ce que le dessin révèle d'une figure géométrique... aussi, alors que la langue courante considère le plus souvent "dessin" et "figure" comme synonymes pour leur acception scientifique ou technique, les didacticiens proposent-ils de distinguer précisément ces deux termes.

Le CNTRL donne pour "dessin" la définition : "Représentation linéaire de la forme des objets, qui s'exécute à des fins scientifiques, techniques ou industrielles" [*], et pour "figure" la définition : "Ensemble de points, droites, plans, représenté en vraie grandeur ou en perspective, objet d'études mathématiques ou support graphique d'un raisonnement en mathématiques ou dans d'autres sciences" [*]. 

En fait, le mathématicien envisageant une figure géométrique, pense à celle-ci en termes de ses propriétés définitoires et n'évoque le dessin que comme une représentation particulière intéressante pour sa valeur heuristique ou pour exprimer des caractéristiques plus complexes à énoncer en langue naturelle ou symbolique. C'est la raison pour laquelle des chercheurs en didactique ont choisi de forcer la distinction en formulant des définitions différentes, mais reliées, de "dessin" et "figure".

Bernard Parzysz [*] notamment, a proposé de définir "figure" comme étant l'objet géométrique décrit par le texte qui le définit, et "dessin" comme l'une des représentations matérielles possibles de cet objet.  Cette proposition consiste, en fait, à mettre en relation, articulées sur le même objet géométrique invoqué, deux représentations de natures différentes en attribuant à l'une d'entre elles, "le" texte, une fonction définitoire. Pour reprendre les termes de Duval [*], la solution proposée revient en fait à juxtaposer deux représentations sémiotiques, l'une discursive (le texte descriptif) et l'autre non discursive (le dessin matériel). Malheureusement, dans une perspective didactique, le problème reste entier : si les élèves sont confrontés à deux représentations d'un objet géométrique, comment peuvent-ils les situer l'une par rapport à l'autre, et chacune relativement à l'objet géométrique auquel renvoie le problème qui leur est posé. Le texte, comme le dessin, est un signifiant qui dénote un objet géométrique mais ne se confond pas avec lui.

Une solution pour dépasser cette difficulté et rendre compte d'une différence de nature entre "dessin" et "figure", en préservant une relation forte, est proposée par Laborde et Capponi en se plaçant dans le cadre général de la sémiotique saussurienne qui articule référent, signifiant et signifié :

"En tant qu'entité matérielle sur un support, le dessin peut être considéré comme un signifiant d'un référent théorique (objet d'une théorie géométrique comme celle de la géométrie euclidienne, ou de la géométrie projective). La figure géométrique consiste en l'appariement  d'un référent donné à tous ses dessins, elle est alors définie comme l'ensemble des couples formés des deux termes, le premier terme étant le référent, le deuxième étant l'un des dessins qui le représente ; le deuxième terme est pris dans l'univers de tous les dessins possibles du référent. Le terme figure géométrique renvoie dans cette acception à l'établissement d'une relation entre un objet géométrique et ses représentations possibles." (Laborde et Capponi 1994 pp.168-169)

En somme, et c'est là tout l'intérêt de cette idée, la figure est une classe d'équivalence de dessins à laquelle on accèdera par l'un de ses (bons) représentants comme cela se fait classiquement en mathématique. Encore faudra-t-il ne pas confondre la classe et son représentant, ce à quoi on sait bien que nos étudiants sont prompts. Mais, le vrai problème est ailleurs, dans la dernière phrase de la citation précédente et le renvoie, un peu avant, au "référent donné". Quel est ce référent ? Il s'agit, bien sûr, de l'objet géométrique qui a justement bien du mal à s'imposer comme référent parce que, sujet de l'idéalité mathématique, il échappe largement aux tentatives de matérialisation. De plus, invoquer la classe de tous les dessins ne résout pas le problème car cette classe est par nature indéfinie et potentiellement infinie. 

Il faut s'y résoudre, l'objet géométrique échappe à la représentation ou ne s'y soumet que partiellement, en tout cas toujours au risque d'un malentendu. Que faire... une perspective de solution est ouverte par Gilles-Gaston Granger, qui suggère qu'en mathématiques...

"l'objet n'est [...] rien d'autre ni rien de plus que l'invariant, ou le support, d'un système d'opérations. Degré zéro du contenu, cet invariant n'est pas décrit : il n'apparait pour ainsi dire que comme un creux, si l'on tente en vain de le détacher du système opératoire." (Granger 1994 p.41)

Ce que nous pourrions reformuler en disant que l'objet géométrique est un référent abstrait (idée, signifié) dont la nature est sans cesse saisie et questionnée par l'ensemble des représentations qui lui sont associées et des actions (système opératoire) mises en œuvres sur ces représentations lors de la résolution de problèmes ou l'accomplissement de tâches l'invoquant. Nous n'avons de l'objet géométrique qu'une conception caractérisée par  la donnée simultanée et reliée des systèmes de représentation, des ensembles d'actions et des problèmes qui l'invoquent. On reconnait là la caractérisation d'un concept de Gérard Vergnaud. Pour ce qui est des solutions proposées initialement par Parzysz, Laborde et Capponi, on peut remarquer que le texte descriptif associé à un objet géométrique est la meilleure caractérisation dont on dispose de la classe des dessins (matériels) qui lui seraient associés. Ces deux caractérisations peuvent donc être rapprochées, ce que nous proposons de faire en les complétant par celles des problèmes dans lesquels les représentations et les actions correspondantes sont opératoires et valides. Le cadre de modélisation cK¢ peut contribuer à mettre en forme cette solution et à la rendre opérationnelle pour fournir des outils pour la conception de situations d'apprentissage en géométrie.

Bridging knowing and proving

The learning of mathematics starts early but remains far from any theoretical considerations: pupils' mathematical knowledge is first rooted in pragmatic evidence or conforms to procedures taught. However, learners develop a knowledge which they can apply in significant problem situations, and which is amenable to falsification and argumentation. They can validate what they claim to be true but using means generally not conforming to mathematical standards. Here, I analyze how this situation underlies the epistemological and didactical complexities of teaching mathematical proof. I show that the evolution of the learners' understanding of what counts as proof in mathematics implies an evolution of their knowing of mathematical concepts. The key didactical point is not to persuade learners to accept a new formalism but to have them understand how mathematical proof and statements are tightly related within a common framework; that is, a mathematical theory. I address this aim by modeling the learners' way of knowing in terms of a dynamic, homeostatic system. I discuss the roles of different semiotic systems, of the types of actions the learners perform and of the controls they implement in constructing or validating knowledge. Particularly with modern technological aids, this model provides a basis designing didactical situations to help learners bridge the gap between pragmatics and theory.

Talk at the conference "Explanation and Proof in Mathematics:
Philosophical and Educational Perspective"
held in Essen in November 2006

Balacheff N. (2010) Bridging knowing and proving in mathematics An essay from a didactical perspective. In Hanna G., Jahnke H. N., Pulte H. (Eds.) Explanation and Proof in Mathematics. pp.115-135. Springer.
Author preprint available from HAL and arXiv.

cK¢ takes up the challenge of modeling learners understanding (a response to Guershon Harel - continued)

Several of the questions Guershon Harel [*] asked after my talk at the PME-NA conference in Chicago concern the scope and objectives of the cK¢ modeling framework. In this post, I take each of these questions and give a short answer leaving for specific posts more elaborated answers when needed. So, here it is:

4. Is the cK¢ a model of learning processes or learning states?

The answer is very simple: cK¢ provides a framework for modeling learning states. Indeed learning processes are of a paramount importance, but they are in my opinion more an object of study for psychology than for mathematics education. Indeed, I don't confuse "learning processes" which are of a mental and intellectual nature, and "problem solving processes" which correspond (to make it simple) to the activity the learner engage when he or she has to solve a problem in a given situation. We need to understand and model these processes, but even if they may inform us about learning processes they are only a dimension of them.

5. Is the cK¢ a model of a learner (period), a model of a learner learning mathematics, or a model of a learner in a mathematics classroom setting?

cK¢ provides a framework to model learners' understanding (learning states, as just said) in mathematics from a situated perspective; situations may be set up within a classroom or in an other context. Actually, the objective is slightly larger, I would claim that cK¢ is a framework to model mathematical understanding taking into account the situational characteristics, not being restricted to learners. A key idea when I started the project was to find a way to model mathematical conceptions with the same tools, be it they conceptions of novices or experts, wrong or correct from whatever knowledgeable perspective.

6. What exactly are the challenging aspects of modeling learning relative to modeling content and pedagogy? 

Anyone will expect the content to be in some sense "correct" and explicit enough to be defined precisely as a content to be taught. Still, there are challenging aspects related to its nature; for example, to model algebra or geometry from an epistemic and teaching perspective is not of the same level of difficulty.
Concerning pedagogy, which is in the first place the product of a practice, the related knowledge is largely implicit. Practitioners can share and discuss their expertise within their community or with teacher students in an apprenticeship approach, but the communication register is largely based on a pragmatic co-construction of meaning referring to a shared practice; as a result it is difficult to model. The challenge is to make explicit what is largely knowledge in action. However, there are progresses as witnessed by research on tutoring and adaptive learning systems.
In line with what I wrote before, I will not consider learning but "the best conditions for learning" (best or optimal). So, I would consider the question: what are the challenging aspects of modeling (determining) the best conditions for learning compared to those of modeling content and pedagogy? I would say that these challenges are very close the one to the other. First, part of the challenge comes from the nature of the content at stake, in particular the role of representations and the complexity of validating the related mathematical statements. Arithmetic, algebra, geometry, calculus, probability, discrete mathematics raise their own specific learning challenges within mathematics. Then, determining the best conditions for learning requires knowing some critical things about the initial state of knowledge of the learners (as a matter of fact, we can only teach people who know); in other words it depends on our knowledge of the conceptions which have to evolve or to be rejected. Eventually, one can say that the challenge is of knowing the knowledge at stake from a learning perspective, understanding learners initial understanding, and being able to design situations likely to stimulate, support and validate the construction of new knowledge. Eventually, modeling pedagogy consists in stating principles for designing situations which implements the "best conditions for learning" under multiple constraints: curricula, institutional standard, cultural and economical environment, time and all material means to make the class working properly.

7. What are the interdependent relationships among these three models? 

Indeed, as the above answers suggests it, the three models are tightly related. In particular, from an educational perspective modeling knowledge is under learning constraints because what we need is not a "knowledge model" for itself but a model of the intended learning outcomes. This knowledge (intended learning outcomes) must be learnable (accessible to learners) and teachable (manageable by teachers); actually, the objective of the didactical transposition is exactly to produce this knowledge which is always at a distance from a knowledge of reference which to some extend justify it. 

8. What is the efficacy of such models if they are constructed independently from each other? In particular, can models of content and pedagogy be viable without the presence of a learning model?

Be it explicitly the case or not, any pedagogical model includes a learning model; I mean a model of the (claimed) best conditions for learning. 

Then, a more philosophically oriented, yet critical, question is

9. Are cognitive models of thinking possible?

Once we have agreed on what means "cognitive", "model" and "thinking" my answer would be: yes... but a discussion of this answer may go far beyond my field of expertise and beyond the scope of this blog as well.

A decade after, what is left from Kaleidoscope?

Ten years ago, on March 2004 the 9th, we held the kick-off meeting of Kaleidoscope, a FP6 network of excellence, in the Castle of Sassenage, near Grenoble. A great day for a great ambition. The network initially gathered 76 research teams in Technology Enhanced Learning (TEL), what meant about 850 researchers and PhD students ; by the end of the EC contract we were about an hundred research teams associated in some way, and more than a thousands researchers and PdD students.

The aim of Kaleidoscope was to foster integration of different research disciplines relevant to TEL, bridging educational, cognitive and social sciences, and emerging technologies. To bring this ambition to reality, in a very fragmented European TEL research area, we chosen to involve a large number of contributors of which only a small number were already collaborating, and a large range of different research themes. Hence a very high level challenge. A set of instruments (focussed joint projects, virtual doctoral school, common platform, etc.) was planned to support the integration process at both the content and the infrastructure level (cf. the technical annex of the project [here], and the slides of the general presentation at the kick-off meeting [there]).

In my opinion, situated at equal distance from success and failure, Kaleidoscope was both a human and a scientific venture. Writing a report on the lessons learned with Sten Ludvigsen, scientific director of the network during the last period of the contract, we noted that "the history of these four years is that of the construction of the network in interaction with a process for understanding what to be a Network of Excellence means, and what integration means in the TEL research area. It is also the history of the interactions between the consortium and the reviewers team and the project officers."

Interestingly, this difference in the views about Kaleidoscope may be illustrated with a certain sense of humour by this picture. Above the head of Jens Christensen, our founding project officer, the portrait of Gaspar Baron de Sassenage, above myself the image of a character taking off supported by angels in a blue sky... Ten years after the character has landed. He is back with ideas still ambitious but probably better shaped by experience and a certain sense of pragmatism which he learned in particular in an other TEL network of excellence from the FP7, STELLAR. Some outcomes of this joint academic venture are still there, as the TeLearn Open archive, the TEL dictionary, and the largely disseminated book synthesizing the Kaleidoscope scientific legacy. TELEARC, the association which has taken the challenge of keeping alive and building on Kaleidoscope legacy has organised a new Alpine Rendez-vous conference in collaboration with STELLAR, and may organize an other one. But all this does not really account for what the Kaleidoscope network has changed in the TEL research area, to understand this change the best data we could have is that from your own view and experience, hence the question:

As a participant in the Kaleidoscope network of excellence, either contractor or associated, what in your opinion can be considered as a legacy? What is left or what you miss when looking back to what we did?

You can respond by leaving a commentary on this post. If there are enough comments, I will make a synthesis of your views and publish it on this blog (let's say in a month or two) and possibly find a way to share it with the project officers and the reviewers who have looked after us during these years.

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.

cK¢ is not a cognitive model (a response to Guershon Harel)

The first question Guershon Harel [*] asked about cK¢ is

3. To what extent is the cK¢ a cognitive model?

Actually, this question comes after a more general one: (1) "What is a cognitive model and what are its purposes?" and a more direct one (2) "Is the cK¢ a cognitive model?
My response is very simple and direct:

cK¢ does not propose a framework to construct cognitive models. It does not pretend to model an "approximation to processes of humans’ mental activities" and do not ambition to be "capable of explaining mental processes or interactions among them", eventually it does not aim at answering a specific question such as "how do we learn to categorize perceptual objects?"

Yet, cK¢ has a very strong relation to the learner by being focused on his or her interaction with a learning environment (more precisely the "milieu"). Indeed, cK¢ could contribute to a cognitive approach, but it is not its objective in the first place.

Based on the evidence we can get from the learner's activity, the objective is to characterize it in terms intelligible from a mathematical perspective and which can serve as inputs to take teaching decisions. Two types of evidence are easy to get: representations manipulated by the learner and operators he or she uses in order to achieve a task or to solve a problem. Actually, these operators are not always explicit but it is not impossible to have an interpretation of the learner's behaviours which makes sense from a mathematical perspective (this corresponds to the Vergnaud coup de force when he coined the concept of "theorem in action"). It is then reasonable to claim that we have a picture of the learner understanding when these representations and operators are stable within a problem-space. This has to be completed by a description of the means the learner uses to take a decision about the validity of his or her activity and the related outcomes. It is the idea of the control structure. Once we have a characterization along these four dimensions, we can conjecture a mathematical meaning, but this does not tell what are the related mental activities or cognitive structures as psychology or neuroscience would understand them. It is very likely that different learning theories would shade different lights on these characterizations. However, my claim is that such characterizations are sufficient to assess the so-called mathematical understanding, and to take teaching decisions or  to design learning environments.

For the rest, cK¢ shares many of the scientific characteristics of "cognitive models": it is based on "rigorous methods", it is "capable of generating testable predictions" and of generating descriptions in "formal, mathematical or computer, languages". It does not describe processes but  nothing prevents it a priori to contribute to such descriptions, this is something to explore.

Eventually, it is important to realize that cK¢ does not ambition to construct models to respond to the question "How does a child transition from additive reasoning to multiplicative reasoning?" but to the question "What are the optimal conditions to initiate and support the child transition from additive reasoning to multiplicative reasoning?"