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Über dieses Buch

This book is a thoroughly revised result, updated to mid-1995, of the NATO Advanced Research Workshop on "Intelligent Learning Environments: the case of geometry", held in Grenoble, France, November 13-16, 1989. The main aim of the workshop was to foster exchanges among researchers who were concerned with the design of intelligent learning environments for geometry. The problem of student modelling was chosen as a central theme of the workshop, insofar as geometry cannot be reduced to procedural knowledge and because the significance of its complexity makes it of interest for intelligent tutoring system (ITS) development. The workshop centred around the following themes: modelling the knowledge domain, modelling student knowledge, design­ ing "didactic interaction", and learner control. This book contains revised versions of the papers presented at the workshop. All of the chapters that follow have been written by participants at the workshop. Each formed the basis for a scheduled presentation and discussion. Many are suggestive of research directions that will be carried out in the future. There are four main issues running through the papers presented in this book: • knowledge about geometry is not knowledge about the real world, and materialization of geometrical objects implies a reification of geometry which is amplified in the case of its implementation in a computer, since objects can be manipulated directly and relations are the results of actions (Laborde, Schumann). This aspect is well exemplified by research projects focusing on the design of geometric microworlds (Guin, Laborde).



A Model of Case-Based Reasoning for Solving Problems of Geometry in a Tutoring System

One of the most important requirements of an intelligent tutoring systems for solving elementary problems in geometry is its ability to guide and follow a human learner. To achieve this task we assume that the teaching aid should operate in a way similar to the learner’s activity. Human learning is based on stepwise cumulative experiments. Therefore, we introduce a model of case-based reasoning — a special analogical reasoning paradigm — that keeps a trace of past experience, so that it can use it for solving new problems analogous to those already memorized. After defining criteria for assessing the analogy of two problems we describe the evolution of Long Term Memory in the tutoring system. This method endows the system with an “apprentice appearance” facilitating its adaptation to human learners.
Eugène Chouraqui, Carlo Inghilterra

Modelling Children’s Informal Arithmetic Strategies

This chapter discusses our approach to modelling, supporting and developing children’s informal arithmetic strategies. The first section of the chapter is an account of a study that was carried out to investigate the development of the arithmetic concepts commutativity and associativity. The implementation of production-rule models of children’s problem-solving strategies is discussed. The last section describes our computer-based package called Shopping on Mars, and reports on some field trials of the package.
Roshni Devi, Tim O’Shea, Sara Hennessy, Ronnie Singer

Cognitive Interpretation of Micro world Operations

Learning environments, or mathematical microworlds, have been claimed to be the prime choice for supporting those learning processes which are aimed at understanding the properties of mathematical objects and the relationships between them which are so important in mathematics. The researcher can investigate such claims by observing the students understanding before and after the learning experience and draw general conclusions from his observations about the overall influence of the microworld. Such investigations may confirm the claim made above, but they will not promote our insight into the mechanisms through which the students acquire their understanding. These mechanisms are linked to cognitive processing on a much more fine-grained level. One way to investigate them, is to establish a more detailed connection between microworld operations and cognitive phenomena.
Tommy Dreyfus, Baruch Schwarz

Calculus Revisited

In the last decade there has been a big debate on the role of computers in mathematics mainly motivated by the technological revolution that allowed to develop numerical experiments and simulations using even very small computers. Infact personal computers have given to all students and teachers great opportunities (reserved before only to experts in computing and computer science) becoming an essential tool for teaching and research. This new and exciting situation has stimulated many experiments in different directions to understand the impact of new technologies in mathematics but has also generated some negative reactions. The supporters of an extensive use of computers in mathematics maintain that by means of computers it is possible to visualize complex concepts and phenomena receiving interesting hints for the theoretical solution of difficult problems, moreover they assert that the application of numerical methods can give “real” solutions instead of pure existence and unicity results and that computers make possible to present in a classroom a variety of examples and situations improving our teaching. This position has been expressed in many articles and reports on experiences which are still going on, more recently the discussion on the increased interaction between computers and mathematics left the circle of experts to involve the whole mathematical community (see, e.g., [13] and the Jon Barwise column in the Notices of the American Mathematical Society)
Maurizio Falcone

Computer Aided Proofs in School Geometry

Proof in geometry problems is an essential feature of secondary school mathematical teaching. This presents difficulties that many may pupils fail to overcome. Experiments with ‘intelligent’ logicials for pupil-aid in geometry, are described here. They are based on research into mathematical teaching and pupil learning patterns, and provide information on pupil spontaneous reasoning applied to geometry.
Régis Gras, Italo Giorgiutti

A Cognitive Analysis of Geometry Proof Focused on Intelligent Tutoring Systems

The elaboration of an Intelligent Tutoring System in geometry proof requires a model of geometry proof problem solving. Actually, the development of these models is still directed by the system’s processes and not by a study of human behaviour. Here we present elements of a cognitive and didactical analysis of proof in geometry that must be taken in account in such a model. This study was carried out with mathematic teachers (GIA: Artificial Intelligence group, IREM Strasbourg).
Dominique Guin

Modelling Geometrical Knowledge: The Case of the Student

Geometry provides a domain in which to study and operationalise deductive methods and at the same time a means by which space can be explored inductively. These opportunities arise from two characteristics of geometry, namely its logical structure and its potential for modelling the real world. The tension inherent in endeavouring to preserve a balance between these twin features is evident in the debates over many decades about the place of geometry in the school curriculum. A report on the teaching of geometry in schools in the UK in the 1960s suggested that “neither the subject matter to which attention is invited nor the operation to which the name of proof is given should retain a uniform character throughout the school age” (Mathematical Association 1963, p. 7).
Celia Hoyles

Intelligent Microworlds and Learning Environments

Recent developments in computerised learning environments have given the concept of microworld a privileged status. In this paper we look at the conception of learning environments based on microworlds which manage knowledge explicitly.
Jean-Marie Laborde

A Constructivist Model for Redesigning AI Tutors in Mathematics

This chapter suggests a new approach to designing AI tutors that grew primarily out of results from a study that is reported in greater detail in a paper entitled The development of tutoring capabilities in middle school mathematics teachers (Lesh & Kelly, in press). That study focused on a ten-week project in which real human tutors provided the intelligence behind a computer-based tutoring system. Our goal was to investigate the understandings, assumptions, and procedures that were really used in tutoring by teachers who were at various levels of expertise — and to observe the evolution of these tutoring abilities over time, as tutoring effectiveness gradually improved.
Richard Lesh, Anthony E. Kelly

The Influence of Interactive Tools in Geometry Learning

An essential procedural aim in secondary education is the learning to solve problems. Problem solving its to be seen as the most complex form of learning, it can include concept learning and rule learning. We hope and believe that the abilities and skills developed through problem solving in secondary schools can be transfered to extra school activities. In geometry teaching we can differentiate between the following typical kinds or ideal types of problems: construction problems, calculation problems, theorem finding problems, proving problems...
Heinz Schumann

Socratic Tutoring with Software: An Example and a Prospectus

This paper describes a procedure for algorithmically generating a connected semantic graph that captures the structure of any algebraic modeling problem as it is being solved. The generated graph then allows a microcomputer-based tutor to guide a student in traversing any logically permissible path through the space of semantic relations of the problem situation. The generalizability of this approach to ICAI in geometry and other mathematical domains is discussed.
Judah L. Schwartz

Students’ Constructions and Proofs in a Computer Environment — Problems and Potentials of a Modelling Experience

Starting from the complementarity of empirical concepts (related to the physical, material world) and theoretical (sometimes logical) concepts pertinent to geometry, the paper reports on an experience with the software tool CABRI-géomètre. The software-representation of the use of empirical and/or theoretical concepts will be described, analysed and confronted to the students’ handling of the tool in a construction task. Consequences for the modelling of constructions and proofs as part of the student knowledge in geometry focus on the computer-evaluation of empirical approaches and the complementarity of the visual representation on the screen (biased to empirical concepts) and the internal representation of a solution (biased to theoretical concepts).
Rudolf Strässer

Some Hyperbolic Geometry with CABRI-Géomètre

We consider the teaching of Euclidean and non-Euclidean geometries to students enrolled in a university program in Mathematics Education. We point out several problems encountered by students during this teaching which provide information about student’s knowledge in geometry. We then propose a partial solution to these problems by using the CABRI-Géomètre software to illustrate several properties of hyperbolic geometry. Finally, we discuss the interest of this type hyperbolic geometry software and we propose desirable features for such a software.
Marie-France Thibault, Robert La Barre

Micro-Robots as a Source of Motivation for Geometry

We describe some robots based micro-worlds and we show how they can be useful to present some geometrical problems. The basic ideas about modelling, the need for formal systems can be examinated here. The different types of models: ‘models by physicists’ based upon laws built from observation and measures or ‘models by mathematicians’ based upon geometrical descriptions needing the choice of representations can be presented. The roles of a model to represent a real world are underlined and this can lead to the necessity of proving properties in the geometrical descriptions of a given robot. The work done shows that we can find tools to motivate children for geometry, to exhibit the need for formal descriptions to be able to predict phenomena, events,... The need for proofs in geometry can be shown in very motivating environments based upon trucks, cranes, arms,... We know that the perception of the needed for a proof is crucial for beginners. The micro-worlds we have designed appear to us as a required pre-culture useful to reach better basic attitudes in geometry. In a second part, we describe the work we have done to design a shell to write intelligent tutoring systems (ITS).
Martial Vivet

Complex Factors of Generalization Within a Computerized Microworld: The Case of Geometry

The rationale for teaching geometry as part of the standard high school curriculum is twofold: 1) to teach students about the measurement, properties, and relationships of points, lines, angles, surfaces, and solids; and 2) to teach students deductive reasoning by exposing them to classical Euclidian geometry, the archetype deductive system. Most geometry courses come up short on both counts.
Michal Yerushalmy


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