
In recent years, our computerbased ability to connect and manipulate representations of knowledge in mathematics has become ever more powerful and flexible. The reification of mathematical knowledge in computerbased learning environments, and accompanying enrichment of mathematical experience due to progress in interface design and knowledge representation, widen and deepen access to experiential learning. Some of these environments even involve tools or models that adapt to the learner or guide their learning. Such developments raise questions regarding the use of these tools in the classroom: How can teachers and others assess and make sense of what students learn? How can they manage computerintensive classroom situations? How is mathematical knowledge transformed when instantiated in such computational environments? How can teachers bridge between low technology and high technology approaches to teaching and learning? Dealing with such questions is essential to the productive use of technology by teachers and designers of instruction.It was the intention in this Topic Group to examine these questions from the research point of view as well as from practice. Since the design and implementation of computerbased interactive learning environments results from the collaboration of at least two communities, computer scientists and mathematics educators, TG 19 addressed each of them regarding the difficulties and success in the past and the problems to be investigated by research and development in the future.
The Topic Group programme was developped around keynote addresses, short communications, and a panel session, focussing on the state of the art relative to the guiding questions. A special evening session provided the opportunity to demonstrate some of the software underlying the presentation.
An electronic poster session had been organised with the help of the MathForum on a site which is still accessible as of this writing /mathed/seville/.
The outcome of the activity of the Topic Group can be summarized along the three following issues: CBILEs and the Construction of Meaning, Designing CBILEs, Effective Use of CBILEs.
CBILEs and the construction of meaning
Because mathematics learning is at the core of our motivation to explore the potential of CBILEs, we addressed first the construction of meaning. Richard Noss explored two theoretical problems, both of which contain important psychological and epistemological kernels. The first problem is that it has long been underscored that mathematical knowledgelike any otheris intimately bound into its setting. The second problem concerns how knowledge is built into mathematical CBILEsand, critically, how it can be dug out.Research tells us how particular systems mediate mathematical expression are highly specific. For example, Baruch Schwartz showed, in the case of Algebra, how collaborative construction of meaning could result in the development of the concept of operator (with the OPERA system, which implements a conceptual model), and the concept of algebraic formula (with the spreadsheetbased formal model). Along the same lines, John Olive noted, in the context of the project "Tools for an Interactive Mathematical Activity" (TIMA), that children were able to construct complex operations involving multiplication and division of fractions in TIMA microworlds, but their intuitions differed from the symbolic operations they had learned through their classroom experiences.
This evidence supports the question addressed by Richard Noss: How, exactly, can we systematically specify the relationship between knowledge placed within a system by a designer, and knowledge constructed by a learner as she or he interacts with it?
Designing CBILEs
The collaboration of computer scientists, researchers in mathematics education and teachers is highly needed for the design and the development of CBILEs. Felisa Verdejo, a computerscientist from Madrid, gave an idea of the richness of the tools offered by AI. But, despite the great efforts to bridge computerscienceespecially Artificial Intelligenceand education, the relevance of the tool involved in the development of CBILEs dedicated to mathematics learning is an open question. As JeanFrancois Nicaud emphasized, there is a necessity to build a theoretical framework in order to support the design of CBILEs, starting from elements taken from psychology, didactics, computer science and artificial intelligence.The main challenge of CBILEs design is to offer students a space in which they can explore freely a virtual world designed to support the construction of some mathematical knowledge. This idea has its early roots in the Logo project and has developed since with a stronger focus on the specific needs of mathematics. A panel on "microworlds and interactive learning" discussed this issue. John Olive discussed the issue with reference to the fractions microworlds TIMA. Nick Jackiw and JeanMarie Laborde illustrated how geometry may be the domain in which the most impressive progress has been made, with the development of the concept of Dynamic Geometry best examplified by Cabrigˇom¸tre and The Geometer's Sketchpad. The key factor is direct exploration, where the standard paperpencil environment is replaced with a much more powerfull medium in which geometrical drawings can be seamlessly reshaped via mousedragging. This new generation of microworlds offers a novel approach to developing geometric reasoning and analysis skills.
A recent trend of research is to link powerful tools such as theorem provers, to microworlds in order to support students exploration of mathematical properties, testing of conjectures, and searching for counterexamples. Tomas Recio presented the use of the computer algebra software CoCoA to support the exploration of elementary Euclidean geometry theorems, suggesting that this program could be thought as the core of a future intelligent, interactive, learning environment linked to a sketch tool such as CabriGˇom¸tre. Philippe Bernat illustrated this trend in development of CBILEs, which consist of augmenting a microworld with "reasoning tools", with the project CHYPRE which aims to give freedom to explore a problem in any way and to test any plan of problemsolving.
The trend in design is clearly to develop environments specific to mathematics and provide means for students to express their ideas about objects and relations, and possibly their reasoning as well. Some participants expressed their worry that all these developments may be technology pushed, whereas the Panel argued on the contrary that they are user&mathematics driven. Mathematics is at the core of modern CBILEs, but the complexity of their contribution to learning is questionable to teachers considering their everyday practice. This issue has also been addressed in a pragmatic way.
The effective use of CBILEs
There are currently no theoretically based tools to address the question of CBILEs use in mathematics classrooms. Actionresearch is the main answer to current needs, and in some projects it has been systematized so as to provide practical means which can be easily disseminated for preservice or inservice purposes.Richard Allen insisted that the principal vehicle through which teachers could reconstruct their pedagogies is the writing and use of teaching scenarios that integrate CBILEs into their teaching These scenarios create inquirybased, exploratoryoriented teaching situations with a major emphasis on bringing to the forefront mathematical connections. He placed special emphasis on creating laboratorylike teaching environments where both CBILEs and manipulatives are used together to connect and reinforce different ways of teaching old and new curriculum topics.
A similar strategy was presented by Doug McDougall who said that teachers' accounts would be useful to other teachers who may find themselves attempting to integrate computers into their classrooms  they would be helped to understand the supports that other teachers have needed in similar situations.
Teachers' professional knowledge related to the use of education technologies, and especially CBILEs, was aknowleged by several participants as being a key issue for further research and development, especially in relation to preservice teacher training.
We are already in the future
The Topic Group exhibited the great progress in the field as well as the rise of complexity, especially in the construction of meaning and teachers' practice. But this is just the indication that we are at the gate of the future. As Jim Kaput, drawing our attention to larger trends, argued in the closing session, we are yet in the early days and have little firm knowledge to guide us, especially since the technology changes the subject matter and all aspects of education so deeply. In this context he also noted the trend of several centuries of needing to teach much more mathematics to many more people, a trend that is accelerating as more mathematics lives in the computational medium, e.g., dynamical systems. He then identified several educational trends: increasing integration of technology in the larger educational enterprise, systematic support for teachers (not only learners), increasing representational pluralism and realism, deeper epistemological penetration by the technology, and integration between levels of technology such as calculators and computers. Finally, he predicted a continuing transition from Doing (old) Things Better to Doing Better Things. Let us take this last sentence as a challenge for teachers and researchers for the coming decade.Further readings:
N. Balacheff, J. Kaput (1997) "ComputerBased Learning Environments in Mathematics", in : A. Bishop et al. (eds.) International Handbook in Mathematics Education, (pp.469501), Dordrecht: Kluwer Academic Publisher.Keitel C., Ruthven K. (eds.) Learning from computers, mathematics education and technology. Berlin: Springer Verlag.
Laborde J.M. (ed.) (1996) Intelligent Learning Environments, the case of geometry. Berlin: Springer Verlag.
Noss R., Hoyles C. (1997) Windows on mathematical meaning. Learning culture and computers. Dordrecht: Kluwer Academic Publishers.
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