Nicolas Balacheff

Computer-Based Learning Environments in Mathematics

This chapter attempts to set a perspective on where interactive technologies have taken us and where they seem to be headed. After briefly reviewing their impact in different mathematical domains, including arithmetic, algebra, geometry, statistics, and calculus, we examine what we believe to be the sources of technology’s power, which we feel is primarily epistemological. While technology’s impact on daily practice has yet to match expectations from two or three decades ago, its epistemological impact is deeper than expected. This impact is based in a reification of mathematical objects and relations that students can use to act more directly on these objects and relations than ever before. This new mathematical realism, when coupled with the fact that the computer becomes a new partner in the didactical contract, forces us to extend the didactical transposition of mathematics to a computational transposition. This new realism also drives ever deeper changes in the curriculum, and it challenges widely held assumptions about what mathematics is learnable by which students, and when they may learn it. We also examine the limits of Artificial Intelligence and microworlds and how these may be changing. We close by considering the newer possibilities offered by the Internet and its dramatic impact on connections among learners, teachers, and the immense resources that are becoming available to both. Our conclusion is that we are very early in the technological transformation and that we desperately need research in all aspects of teaching and learning with technology.

Didactical Complexity of Computational Environments for the Learning of Mathematics

This revised version was published online in September 2005 with corrections to the Cover Date.

Artificial Intelligence and Real Teaching

Yesterday’s classroom could be roughly viewed as the economic answer to the 19th century society’s need for more educated people, for both industrial purposes and democratic development. The gathering in the same place of a teacher as representative of the knowing society and students as the ones who must be educated in order to become full members of this society, was probably the best answer at the time, given the technological means for communication available.

Bridging Knowing and Proving in Mathematics: A Didactical Perspective

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.

Advanced Educational Technology: Knowledge Revisited

AET R&D cannot avoid the question of the nature of knowledge which is at the core of both learning and teaching or training. The way this problem can be handled for the purpose of design and implementation of systems supporting human learning, the question of knowledge representations for the purpose of computational models as well as the question of the place of knowledge in person/machine interactions suggest that knowledge should be revisited in the light of the AET research programme. In this chapter I consider this question from the point of view of computational modeling and situated AET.

Construction et analyse d’une situation didactique Le cas de «la somme des angles d’un triangle»

Using the case of the sum of the angles of a triangle, we analyse the constraints which apply to the specification and application of a situation in which students, who have not yet studied the concept of mathematical proof, have to produce a conjecture and consider the problem of proving it. The general framework for the study is the theory of didactical situations (BROUSSEAU 1986), which we have shown to be pertinent to research on the teaching of mathematical proof at these level of schooling. With respect to the relationships between proof, refutation and knowledge we have relied to a large extent on the thesis of LAKATOS (1976).

Quelle est l’influence du contexte sur les raisonnements d’étudiants sur la mesure en physique ?

Through the use of written work produced in three distinct situations, this study looks at the steps students go through during measuring exercises and the reasoning behind their decisions. Methods used in previous studies concerning how students reason during such exercises were consulted and applied to the 3 following situations used in this study. A questionnaire linked to the carrying out of an experiment in a tutorial, The carrying out of an experiment in laboratory work, An experiment during laboratory work which was graded. Three types of student strategies were discerned. Point reasoning: in the students eyes there is a true value which can be measured after only one reading, Mixed reasoning: the student seeks to take a reading which is closest to a reference value. Global reasoning: the student understands the reading based on a value and an uncertainty factor linked to the value. What comes to the fore is the link between situation and reasoning and their contingence on context. Furthermore, the reasoning adapted by each student in the laboratory in situations 2 and 3 appear related whilst those applied in situations 1 and 2 appear to be mutually exclusive.

The Grand Challenge of Diversity—A Comment from a Researcher Perspective

This new series of Grand Challenge Problems (GCPs) addresses cognitive as well as emotional issues, individual as well as societal issues, epistemic as well as ethical issues. This diversity is associated with a movement tending to blow up the borders between the cognitive and the social, the formal and the informal, the school and the workplace. But the newest and most remarkable fact is that this set of research propositions is at the same time on the edge of the disappearance of institutions as the unique site for teaching and learning, with an increasing emphasis on self‐regulated learning and learning communities, and of the rise of technologies empowering institutions and teachers with more efficient tools and means to support, drive, and assess learning.

Grand Challenge Problem 10: TELEARC an Agile and Productive Networking of TEL Small and Medium Sized Research Labs

Small and medium sized research labs (SMLs) are dominating European TEL research. This is justified by the great numbers of countries and regions in Europe needing to develop a research and innovation competence to facilitate the diverse educational systems contextualized in various institutional settings across Europe. However, to strengthen the various research practices and to develop a common scientific language on TEL research the Grand Challenge Problem is to establish a vivid network and a community of practice among the research labs. TELEARC (Technology Enhanced Learning European Advanced Research Consortium) has been established to realize such a network. The chapter presents the framework of TELEARC.

For a dictionary of research on Technology Enhanced Learning

Questo articolo presenta le principali linee di azione di un meta-progetto iniziato nell’ambito della Rete di Eccellenza STELLAR con l’obiettivo di creare e dare struttura a un Thesaurus e a un Dizionario dei termini e delle espressioni in uso nell’ambito della ricerca sul Technology Enhanced Learning (TEL). Questo meta-progetto intende fornire gli strumenti per superare le difficolta dovute alla ricchezza e alla rapida evoluzione di quest’area di ricerca multidisciplinare, allo scopo di facilitare il coinvolgimento dei ricercatori giovani e di quelli che si accostano per la prima volta a questo settore, nonche di tutti coloro che a vario titolo si interessano al TEL. Il meta-progetto in questione ha anche l’ambizione di facilitare la comunicazione tra individui appartenenti a diverse culture scientifiche e tra le lingue nazionali. Dopo una presentazione delle motivazioni, vengono discussi gli attuali sviluppi del lavoro e presentati i criteri seguiti per le principali scelte effettuate in merito ai metodi e alle tecnologie utilizzate.