Technology-supported learning of proof in mathematics

Cabri-Euclide, Luengo, 2005

Proof assistants, an automatic theorem proving research track, are reaching a maturity which suggests that is possible to the exploration of their use for the learning of proof in  mathematics first at the level of higher education and at tentatively the upper secondary school. 

It in the context of this pioneer research that is organised the PAT 2023 Thematic School which seeks to offer a broad spectrum of current research in the field of didactic of proof, the impact of the use of proof assistants in education, formalization of mathematics and user interfaces for theorem proving. I will give a lecture which will include  (1) a survey of the evolution of AI research on the learning of proof in mathematics, (2) lessons learned from the past focusing on the relations between reasoning-proving and knowledge representation, and on the problem of feedback, eventually (3) didactic analysis of the teaching of mathematical proof and its implications for the design of learning environments. The introduction will outline the history of the teaching of proof in mathematics, a short epilogue will raise epistemological issues.

Suggested readings:

Arzarello, F. (2007). The proof in the 20th century (From Hilbert to Automatic Theorem Proving Introduction). In P. Boero (Éd.), Theorems in School : From History, Epistemology and Cognition to Classroom Practice (p. 43‑63). BRILL.
Balacheff, N. (2023). Notes for a study of the didactic transposition of mathematical proof. Philosophy of Mathematics Education Journal, 2023 volume
Balacheff, N., & Boy de la Tour, T. (2019). Proof Technology and Learning in Mathematics : Common Issues and Perspectives. In G. Hanna, D. Reid, & M. de Villiers (Éds.), Proof Technology in Mathematics Research and Teaching. Springer.
Czocher, J. A., & Weber, K. (2020). Proof as a Cluster Category. Journal for Research in Mathematics Education, 51(1), 50‑74.
Hanna, G., & Xiaoheng, (Kitty) Yan. (2021). Opening a discussion on teaching proof with automated theorem provers. For the Learning of Mathematics, 41(3), 42‑46.
Luengo, V. (2005). Some didactical and Epistemological Considerations in the Design of Educational Software : The Cabri-euclide Example. International Journal of Computers for Mathematical Learning, 10(1), 1‑29.

Une vie pour l'apprentissage des mathématiques et la pensée informatique

Mathématicien et pionnier des technologies éducatives, Seymour Papert a contribué de façon décisive à l’orientation des recherches sur les environnements informatiques pour l’apprentissage humain (EIAH). Son œuvre est à l’origine d’un courant de recherche international sur l’apprentissage des mathématiques avec lequel la recherche en didactique des mathématiques entretient des relations – que l’on pourrait qualifier de dialectiques – depuis le début des années 80. Victime d’un accident en 2006 à Hanoï, où il était l’invité d’une conférence de la 17° étude ICMI2 sur l'utilisation des technologies numériques dans l'enseignement et l'apprentissage des mathématiques, Seymour Papert a dû brutalement interrompre son activité académique. Il est décédé le 31 juillet 2016.

La revue Recherches en Didactique des mathématiques a souhaité lui rendre hommage, c’est le sens du texte que j'ai proposé pour le numéro 37/2-3 qui vient de paraître. Après avoir évoqué des jalons importants de la vie scientifique de Seymour Papert, ce texte revient sur deux concepts clés, micromonde et constructionisme, qui constituent les piliers fondateurs de son œuvre.

Papert lors de l'ouverture officielle du London Knowledge Lab (début à 14min 15s)

L’héritage de Papert est bien vivant, et l’exploration des voies qu’il a ouvertes est prometteuse. Un Lutin m'a suggéré avec malice de conclure par une citation ; clin d’œil et manière  d’invitation :

"What make our century’s science thinking different from any other century are the ideas associated with computation, computers and information science, and the idea that we should  give children this powerful thing they care about more than anything else, that they ought not to know what goes on inside it - it blows the mind." (Papert 2004)

Design heuristics for authentic simulation-based learning games

Just published in IEEE Transactions on Learning Technologies (TLT) : "Design heuristics for authentic simulation-based learning games", a paper based on the PhD research of Celso Gonçalves supervised by Muriel Ney who created and led the project Laboratorium of epidemiology, to which I was associated.

Here is the abstract :

"Simulation games are games for learning based on a reference in the real world. We propose a model for authenticity in this context as a result of a compromise among learning, playing and realism. In the health game used to apply this model, students interact with characters in the game through phone messages, mail messages, SMS and video. Perceived authenticity is measured after the game in 196 phone interviews
that yield quantitative and qualitative results. We show evidence of relationships between attributes of the game environment and perceived authenticity. This yields a list of parameters that can be adjusted to favour authenticity. We also study three situations of interaction and show when and why they are perceived as authentic, or not. These results lead to recommendations for the design of simulation games that can be perceived as authentic."

[Get the paper]

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.

Why do learners (not) blog?

Retrieved from Nicolas Balacheff  (2010) comments on papers available on the SOA scientific portal

We have heard here and there claims and expectations about the so-called Learning 2.0 revolution. The rational is that learners will be able to share, collaborate, exchange in a more open and dynamic way, blowing the barriers that formal education and training may have raised on the way towards knowledge. Among learning 2.0 tools, there are blogs. All of us, I mean the blogers, know that bloging is not such an easy thing and having the tools is not enough. So a paper like the one of Behringer and colleagues is especially interesting in that it explores in a pragmatic and rigorous way the students motivation or lack of motivation to use blogs; among the latter there is a preference for direct communication and fear of a loss of privacy. May be not a surprise… a question one may have is that of knowing how far this is intrinsic or witnesses the weight of a culture and a lack of experience of these tools.

A note after the reading of: Andergassen, M, Behringer, R, Finlay, J, Gorra, A, and Moore, D. “Weblogs in Higher Education – why do Students (not) Blog?” Electronic Journal of e-Learning Volume 7 Issue 3 2009, (pp203 -215)

Could designed-based research become the TEL research standard?

Retrieved from Nicolas Balacheff  (2010) comments on papers available on the SOA scientific portal

Design-based research is a rather interesting framework for TEL research project,which although not new (the seminal papers go back to the early 90s) does not seem to have deserved all the attention it should in the European research area. Taking as an indicator the references of the EduTech wiki from TECFA (*), one may conjecture that it reached us in the early 2000, but since then I have not the feeling that it has spread very much within our research community. I don't see clear reasons for that. In my opinion we must spend some efforts, especially in a network like Stellar to consider this approach. From the presentation of Wang and Hannafin (ETR&D 53/4 2005), design-based research seems to be especially adapted to multidisciplinary research as well as to research which must be carried out in close connection to the field it explores. It may be the case that some researchers think that they are working in this paradigm while actually they do not, since there are at least two occasions of misunderstanding. The first source of misunderstanding is the emphasis of design-based research on iteration, an emphasis which is reminiscent of the life cycle of technology design. But here iteration is not only aiming at the improvement of the design, but also at critically revisiting theories to develop or refine them. What is valued is the practical use of theories (p.13), and the fact that theoretical and practical issues are tightly related. The second source of misunderstanding is that because of its close relationship to the field, design-based research may be confounded with action-research. This is missing the priority of design-based research, while acknowledging its situatedness, to transcend the particularities of the context in which the experiment is been carried out. This difficulty is very well identified in this paper, and addresses directly the main concern of our field which is of understanding what results we produce which could be of a general value beyond the specific project in which it has been obtained: "Generalizability […] must be verified according to the theory goals of the design and discipline requirements of the research. Researchers need to optimize a local design without decreasing its generalizability […]" (p.19). So design-based research is not a sophisticated conceptualisation of the life cycle of a technology, it is of a different nature and objective than action-research. To some extend it can be seen as a proposal for a new discipline with original problems of methods and rigour.

There is one point on which this paper passes a bit too fast. It is the meaning of "real", what could count as "real-word context". Any experiment or observation carried out in a classroom changes what it is as opposed to its normal functioning. This is well-known but  as a common sense fact and scientifically not enough documented. The authors acknowledge the complexity of reality, but do not catch the need to model it in an explicit way (with all the constraints of something which is out of reach of an exhaustive description, indeed). Any experimental science is faced to this need including natural science. The place where an experiment is carried out is an experimental apparatus, it has to be described as such even if it is embedded in a so called real context. (it means a context which is largely out of control but if which many features have no impact). This is a condition to be later on able to discriminate between generalizable results and "idiosyncratic" (p.19) adaptations or observations, and to sort out which of the data are relevant for further analysis. This dimension of design-research may well be the missing element to succeed in becoming the scientific standard it is ambitions.

Blog post based on the reading of: Wang F., Hannafin M.J. (2005) Design-based research and technology-enhanced learning environments.  Educational Technology Research and Development 53 (4) 5-23, DOI: 10.1007/BF02504682

Teachers adopt technology, learners should

As Rosamund Sutherland notices, educational technology has entered the classroom, but in many cases in support of the teacher activity (e.g. the interactive white board used as an enhanced black board) and not so much in support of learners activity. But let's hear her view:
But may be this is the result of the incapacity yet for technology to respond to the proper ecology of the classroom which is complex either from a spatial, social or intellectual perspective. The classroom as a technology enhanced learning space, still a challenge.