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Intelligence in QED-Tutrix: Balancing the Interactions Between the Natural Intelligence of the User and the Artificial Intelligence of the Tutor Software
Abstract
When considering the mathematics competences referential (Niss & Højgaard, 2019), whether from Québec Education Program, the Standards of the National Council of Teachers of Mathematics or some large international studies such as the Programme for International StudentStudent Assessment, reasoning, and mathematical proofs are always at the forefront. Although the notion of proof is limited, in elementary school, to mathematical reasoning and conviction, and in post-secondary education, to demonstration and written communication, the evolution of the treatment reserved for deductive reasoning in secondary school has the appearance, depending on the region, of an eternally shilly-shallying between valorization and marginalization.
... Van Vaerenbergh and Pérez-Suay (2022) give a taxonomy of Artificial Intelligence (AI) systems used in Intelligent Tutoring Systems (ITSs) such as Hypergraph Based Problem Solver (HBPS) (Arnau et al., 2013) and QED-Tutrix (Font et al., 2018(Font et al., , 2022) that can assist students in solving math problems as well as in learning proof and proving techniques. The different ITS components they describe include information extractors, reasoning engines, explainers, and data-driven modeling. ...
... The different ITS components they describe include information extractors, reasoning engines, explainers, and data-driven modeling. Information extractors attempt to take the input produced by the student (e.g., their written work and drawings) and represent them internally so that reasoning engines (e.g., automated theorem provers [Botana et al., 2015;Fitting, 2012]) can determine sequences of inferences to generate a proof. The task of presenting information generated by the reasoning engines for human users is done by explainer systems. ...
The purpose of this study is to examine the capabilities of ChatGPT as a tool for supporting students in generating mathematical arguments that can be considered proofs. To examine this, we engaged students enrolled in a mathematics pathways course in evaluating and revising their original arguments using ChatGPT feedback. Students attempted to find and prove a method for the area of a triangle given its side lengths. Instead of directly asking students to prove a formula, we asked them to explore a method to find the area of a triangle given the lengths of its sides and justify why their methods work. Students completed these ChatGPT-embedded proving activities as class homework. To investigate the capabilities of ChatGPT as a proof tutor, we used these student homework responses as data for this study. We analyzed and compared original and revised arguments students constructed with and without ChatGPT assistance. We also analyzed student-written responses about their perspectives on mathematical proof and proving and their thoughts on using ChatGPT as a proof assistant. Our analysis shows that our participants’ approaches to constructing, evaluating, and revising their arguments aligned with their perspectives on proof and proving. They saw ChatGPT’s evaluations of their arguments as similar to how they usually evaluate arguments of themselves and others. Mostly, they agreed with ChatGPT’s suggestions to make their original arguments more proof-like. They, therefore, revised their original arguments following ChatGPT’s suggestions, focusing on improving clarity, providing additional justifications, and showing the generality of their arguments. Further investigation is needed to explore how ChatGPT can be effectively used as a tool in teaching and learning mathematical proof and proof-writing.
... Cette étude repose sur plusieurs piliers de recherche. Tout d'abord, nous nous appuyons sur nos travaux dans deux domaines spécifiques depuis de nombreuses années : la conception de tuteurs intelligents (Richard et al., 2007 ;Font et al., 2022) et les systèmes experts de simulation (Emprin, 2011 ;. Ensuite, nous avons bénéficié de la contribution de groupes d'experts du domaine lors d'événements tels que le Symposium on Artificial Intelligence for Mathematics Education (AI4ME), qui s'est tenu à Castro Urdiales en Espagne à l'hiver 2020 (Richard et al., 2020), ainsi qu'autour d'une table ronde à la 21 e école d'été de didactique des mathématiques de l'Association de la recherche en Didactique des Mathématiques (ARDM), à l'automne 2021 à l'Île de Ré en France (Emprin, 2023 ;Richard, 2023). ...
... Cette collaboration vise à faciliter l'intégration et l'émergence de modèles innovants. L'outil développé constitue un environnement interactif qui s'adapte à chaque élève et vise à équilibrer les interactions entre l'intelligence naturelle de l'utilisateur et l'intelligence artificielle du système tuteur (Font et al., 2022). Un élément central de cet environnement est l'agent tuteur, qui joue un rôle complémentaire à celui de l'enseignant, tout en étant considéré comme faisant partie intégrante du milieu au sens de la TSDM (figure 5). ...
Il est essentiel de se pencher sur les interactions entre l’intelligence artificielle (IA) et la didactique, encore plus à notre époque où l’impact de l’IA sur la société et l’économie est aussi profond. Tout d’abord, nous remettons en question la notion dite d’intelligence et
les préjugés qu’elle peut susciter lorsqu’on réfléchit à l’IA et à ses définitions. Ensuite, nous analysons les liens potentiels entre l’IA et la didactique des mathématiques. Pour ce faire, nous examinons des exemples de projets en cours dans le monde francophone permettant de dresser un état des lieux des aspects actuellement développés. Par la suite, nous explorons les cadres théoriques de la didactique des mathématiques et leur articulation avec l’IA. Enfin,
nous abordons les questions et les défis soulevés par l’utilisation de l’IA, tout en offrant des perspectives prometteuses pour l’avenir.
... For the present article, I draw on Friedman et al. (2021), who build on Roschelle et al. (2020) to define AI as "any computational method that is made to act independently towards a goal based on inferences from theory or patterns in the data"-in educational contexts, (i) theory refers to the learning theory that an AI embodies (e.g., instructivist versus constructivist), which can often hold disparate design implications for learning technologies, and (ii) patterns in data refers to how observed behaviors of learners (e.g., their click patterns) lead to probabilistic inferences about their latent states (e.g., unmotivated), a precursor to administering appropriate AI-based interventions. AI methods ranging from classical machine learning to generative AI-based approaches are typically instantiated in intelligent tutoring systems that offer customized scaffolds to improve learners' cognition, metacognition and affect (see Du Boulay et al., 2023 for several concrete examples). ...
I nudge reevaluation of the idea that artificial intelligence for education (AIED) is merely about using artificial intelligence (AI) tools to automate understanding and responding to learning processes. Instead, I advocate for a human-centered approach to AIED that emphasizes the importance of personal connections, relationship-building, and scaffolding that goes beyond simplifying tasks to push students in their critical thinking. This approach calls for curating multimodal data from ecologically valid learning settings to train AIED systems, and maintaining flexibility in expectations around rational learner behavior when analyzing data from such systems. Given that the definition of good AIED is often discipline-specific and influenced by the underlying pedagogical models of student learning, the article calls on learning sciences researchers to integrate their complementary yet often competing theoretical lenses in rigorously studying AI-supported learning phenomena at scale.
... Especially in geometry, the transition from a conjecture to a demonstration is complex (Arsac, 1987, Richard, 2004 and students need help. This is the purpose of the QED-Tutrix project (Font & al, 2022). At the core of the project, there is a software environment offering: ...
This chapter presents a state of the art in the design of digital environments for mathematics education, with a particular focus on artificial intelligence techniques. A review of the work done in this area over the last few decades highlights current challenges and distinguishes between symbolic approaches and machine learning. About symbolic approaches, we review automatic reasoning tools in geometry and their potential. We also consider the design and research work around the Casyopee environment, and the use of logic programming in the QED-Tutrix intelligent tutoring system. With respect to machine learning, four classes of techniques constitute contemporary AI in computer science. Two examples are discussed: a deep learning system of monument analysis for learning situations in mathematics, technology and art, and a computer classroom simulator that provides a new approach to training teachers.
... Especially in geometry, the transition from a conjecture to a demonstration is complex (Arsac, 1987, Richard, 2004 and students need help. This is the purpose of the QED-Tutrix project (Font & al, 2022). At the core of the project, there is a software environment offering: ...
Preprint Mars 2023. Accepted for publication in “Handbook of Digital Resources in Mathematics Education" edited by Prof. Dr. Birgit Pepin, Prof. Ghislaine Gueudet, Prof. Jeffrey Choppin
This chapter presents a state of the art in the design of digital environments for mathematics education, with a particular focus on artificial intelligence techniques. A review of the work done in this area over the last few decades highlights current challenges and distinguishes between symbolic approaches and machine learning. About symbolic approaches, we review automatic reasoning tools in geometry and their potential. We also consider the design and research work around the Casyopee environment, and the use of logic programming in the QED-Tutrix intelligent tutoring system. With respect to machine learning, four classes of techniques constitute contemporary AI in computer science. Two examples are discussed: a deep learning system of monument analysis for learning situations in mathematics, technology and art, and a computer classroom simulator that provides a new approach to training teachers.
... In [32] discursive elements related to circular arguments in geometry are discussed, this study has a qualitative approach and epistemological elements are explicitly highlighted. In [33] a reasoning tutor for geometry is discussed. In [34] explore reversible and non-reversible tasks in two formats: multiple choice and open-ended responses for algebra. ...
This paper shows the results of an experiment applied to 170 students from two Chilean universities who solve a task about reading a graph of an affine function in an online assessment environment where the parameters (coefficients of the graphed affine function) are randomly defined from an ad-hoc algorithm, with automatic correction and automatic feedback. We distinguish two versions: one of them with integer coefficients and the other one with decimal coefficients in the affine function. We observed that the nature of the coefficients impacts the mathematical work used by the students, where we again focus on two of them: by direct estimation from the graph or by calculating the equation of the line. On the other hand, feedback oriented towards the "estimation" strategy influences the mathematical work used by the students, even though a non-negligible group persists in the "calculating" strategy, which is partly explained by the perception of each of the strategies.
This article touches up the complementary presentations provided by the authors during the conference, addressing the design, evolution, and utilization of digital technologies in education. It is composed of two distinct parts. The first one offers a perspective from the field of educational sciences, while the second focuses on the didactics of mathematics.
This chapter focuses on the particularities of mathematical work in the digital age. It opens with historical considerations that relate to the development of mathematical work to do arithmetic when symbolic or mechanical tools and algorithmic methods were used. He goes on to define the new mathematical work by introducing concepts of reference in the interaction between human and machine. The notion of valence of mathematical work makes it possible to account for the operating domain of interactions as well as for the possible adaptations of an evolving subject-milieu system, whether for the accomplishment of a task or in the course of learning mathematics. A difference between the work of the designer and that of a user is established, especially in terms of the effects on reasoning, proof, modelling activity and the creation of digital artefacts. The adaptation in a process of idoneity between the teaching and the learning project, as well as between the intention of the designer and the realisation by a user, is discussed.
Equity and ethics in the learning of mathematics is a major topic for mathematics education research. The study of ethics and injustice in relation to epistemic pursuits, such as mathematics, is receiving a great deal of interest within contemporary philosophy. We propose a bridging project between these two disciplines, importing key ideas of “epistemic injustice” and “ethical orders” from philosophy into mathematics education to address questions of ethics, equity, values and norms. We build on Dawkins and Weber’s (Educ Stud Math 95:123–142, 2017) “apprenticeship model” of learning proofs and proving, which says that mathematics education should reflect the practices of research mathematicians. Focusing on the norms and values implicit in mathematical proving, we argue that deploying this model unreflectively can lead to “epistemic injustices” in which learners are disadvantaged based on their cultural or class background. We propose thinking about the problem in terms of Max Weber’s “ethical orders”, and the clash that arises between the ethical orders of mathematics and the existing ethical orders of the learners and teachers of mathematics. Weber’s lesson is that sometimes these clashes have no overarching resolution, and so the mathematics classroom may also have to settle for tailored pragmatic measures to combat individual cases of epistemic injustice.
This article deals with what it means to possess competence in mathematics. It takes its point of departure in the fact that the notions of mathematical competence and mathematical competencies have gained a foothold as well as momentum in mathematics education research, development and practice throughout the last two decades. The Danish so-called KOM Project (KOM: Competencies and the Learning of Mathematics), the report from which was published in 2002, has played an instrumental role in that development. Since then, a host of new developments has taken place, and we—as the authors of the original report—have felt the need to take stock of this development and revisit the conceptualisation of the basic notions in order to provide an updated version of the original conceptual framework and terminology. Whilst the fundamentals of this framework have been preserved in this article, the version presented here in addition to an up-to-date terminology offers greater clarity and sharpness and richer explanations than found in the original.
Our article aims to define the notion of instrumental proof based on didactic, epistemological and cognitive considerations. We raise issues and challenges related to the use of this type of proof in mathematical work and mathematical thinking. The theory of mathematical working spaces serves as a construct on which we address questions about proof, reasoning and epistemic necessity, taking advantage of the possibilities offered by the development geneses and fibrations in an instrumented perspective. The coordination of the semiotic, discursive and instrumental geneses of the working space founds discursive-graphic proofs, mechanical proofs and algorithmic proofs that are activated at school in the subject-milieu interactions. We end with a discussion on some consequences of the computer-assisted modelling of the learning conditions of mathematics, and we conclude on a necessary reconciliation of heuristics and validation. Notre article vise à définir la notion de preuve instrumentale en partant de considérations didactiques, épistémologiques et cognitives. Nous soulevons des enjeux et des défis liés à ce type de preuve au regard du travail mathématique et de la pensée mathématique. La théorie des espaces de travail mathématique sert de charpente sur laquelle nous abordons des questions sur la preuve, le raisonnement et la nécessité épistémique, profitant des possibilités qu'offrent le développement des genèses et des fibrations dans une perspective instrumentée. La coordination des genèses sémiotique, discursive et instrumentale de l'espace de travail fondent des preuves discursivo-graphiques, des preuves mécaniques et des preuves algorithmiques qui s'activent à l'école dans l'interaction sujet-milieu. Nous terminons par une discussion de quelques conséquences de la modélisation des conditions d'apprentissage des mathématiques assisté par des dispositifs informatiques, et nous concluons sur un rapprochement nécessaire entre heuristique et validation.
The idea of assisting teachers with technological tools is not new. Mathematics in general, and geometry in particular, provide interesting challenges when developing educative softwares, both in the education and computer science aspects. QED-Tutrix is an intelligent tutor for geometry offering an interface to help high school students in the resolution of demonstration problems. It focuses on specific goals : 1) to allow the student to freely explore the problem and its figure, 2) to accept proofs elements in any order, 3) to handle a variety of proofs, which can be customized by the teacher, and 4) to be able to help the student at any step of the resolution of the problem, if the need arises. The software is also independent from the intervention of the teacher. QED-Tutrix offers an interesting approach to geometry education, but is currently crippled by the lengthiness of the process of implementing new problems, a task that must still be done manually. Therefore, one of the main focuses of the QED-Tutrix' research team is to ease the implementation of new problems, by automating the tedious step of finding all possible proofs for a given problem. This automation must follow fundamental constraints in order to create problems compatible with QED-Tutrix : 1) readability of the proofs, 2) accessibility at a high school level, and 3) possibility for the teacher to modify the parameters defining the "acceptability" of a proof. We present in this paper the result of our preliminary exploration of possible avenues for this task. Automated theorem proving in geometry is a widely studied subject, and various provers exist. However, our constraints are quite specific and some adaptation would be required to use an existing prover. We have therefore implemented a prototype of automated prover to suit our needs. The future goal is to compare performances and usability in our specific use-case between the existing provers and our implementation.
Este número especial de la Revista Latinoamericana de Investigación en Matemática Educativa (RELIME) es el resultado de distintos artículos elaborados para el tercer simposio Espacio de Trabajo Matemático (ETM), cuyo objeto de estudio es el desarrollo y los usos posibles de la noción de ETM en la didáctica de las matemáticas. El trabajo matemático y su funcionamiento en el marco escolar están a la base del enfoque de los ETM. En esta introducción se sintetiza el enfoque teórico, no como algo prescriptivo sino como una propuesta sugerente para enriquecer el estudio didáctico del trabajo matemático del alumnado y profesorado. Seguidamente, se describe la organización temática de las contribuciones.
Tutoring System for Improvement of Argumentative Competencies in High School Mathematics This article tries to show how secondary school pupils can improve their argument skills with the help of tutorial systems intended for learning geometry. After having established the conceptual framework at the intersection of mathematics teaching and computer environments for human learning, the article compares the heuristic and discursive features of some tutorial systems, including the systems developed by our research team. It then deals with the issue of the complementary nature of knowledge and skills in order to aim at a strategy for assessing argument skills based on relationships in the subject-environment system. The text particularly includes cognitive control, semiotic and situational structures associated with the development of argument skills in such environments. It also tackles the specific nature of reference knowledge, the decontextualization of learning, the instrumentation of resources, the idea of mathematical proof and the role of teaching agents. Résumé. Cet article vise à montrer comment l'élève de l'école secondaire peut améliorer ses compétences argumentatives à l'aide de systèmes tutoriels destinés à l'apprentissage de la géométrie. Après avoir situé le cadre conceptuel à l'intersection de la didactique des mathématiques et des environnements informatiques d'apprentissage humain, l'article compare les caractéristiques heuristiques et discursives de quelques systèmes tutoriels, dont les systèmes développés par notre équipe de recherche. Il traite ensuite la question de complémentarité entre connaissances et compétences pour se diriger vers une stratégie d'évaluation des compétences argumentatives sur la base des rapports du système sujet-milieu. Le texte intègre, en particulier, les structures de contrôle cognitif, sémiotique et situationnel associées au développement d'une compétence argumentative dans de tels environnements. Il aborde aussi la spécificité des connaissances de référence, la décontextualisation des apprentissages, l'instrumentation des ressources, l'idée de démonstration mathématique ainsi que le rôle d'agents didactiques.
Dynamic figures in a mathematical workspace for the learning of geometrical properties. Our paper aims at showing how dynamic figures are useful in the learning of the use of geometrical properties at a high school level, in continuity with the practices inherited during the primary school education. After considering the general contextual rooting of the problem situations while comparing geometrical reality with educational institution, we focus on the student-milieu system and on the connections between the reasoning and the operational dynamic figure. We then present a research framework in order to analyze a geometrical workspace dedicated towards the learning of the use of properties. The workspace is presented in what it has of generic to enhance such learning and it enables us to conclude by some theoretical remarks on the components from the suitable geometrical working space. Résumé. Notre article vise à montrer comment les figures dynamiques sont utiles pour l'apprentissage des propriétés géométriques de l'école secondaire, en continuité avec les habitudes héritées du primaire. Après avoir considéré l'enracinement contextuel des situations-problèmes en général et mis la réalité géométrique au regard de l'institution scolaire, nous centrons notre propos sur le système sujet-milieu et sur les rapports du raisonnement à la figure dynamique opératoire. Nous situons ensuite un dispositif de recherche afin d'analyser un espace de travail géométrique idoine orienté vers l'apprentissage des propriétés. L'espace de travail est présenté en ce qu'il a de générique pour enclencher un tel apprentissage et il nous permet de conclure par quelques remarques théoriques sur les constituants de l'espace de travail géométrique. Mots-clés. Didactique des mathématiques, géométrie dynamique, raisonnement, espaces de travail géométrique, apprentissage des propriétés géométriques.
Our paper aims at showing how the instrumented learning of the properties in Euclidean geometry can be introduced in continuity with the mathematical competences developed at the elementary school. The principle founder of our research is based on the relation of subordination between the constraints of a property, posed using dynamic geometry software, and a necessary conclusion. Starting from an experimentation carried out in classes of 12-14 years old in France and in Quebec, our study presents a critical analysis of results where the ultimate objective of the activities suggested to the students relates as well to the significance of elementary properties, as on the understanding of the necessity of the link between the antecedents to the consequents of a deduction. Before ending by the didactic consequences of our approach, the text introduces the concept of instrumented figural inference as a means, employed by certain students, to justify a step of structured reasoning. A reconciling of the semiotic, instrumental and discursive aspects is offered throughout the paper.
In this paper, we report on the formalization of a synthetic proof of Pappus' theorem. We provide two versions of the theorem: the first one is proved in neutral geometry (without assuming the parallel postulate), the second (usual) version is proved in Euclidean geometry. The proof that we formalize is the one presented by Hilbert in The Foundations of Geometry which has been detailed by Schwabhäuser , Szmielew and Tarski in part I of Metamathematische Methoden in der Geometrie. We highlight the steps which are still missing in this later version. The proofs are checked formally using the Coq proof assistant. Our proofs are based on Tarski's axiom system for geometry without any continuity axiom. This theorem is an important milestone toward obtaining the arithmetization of geometry which will allow us to provide a connection between analytic and synthetic geometry.
The area method for Euclidean constructive geometry was proposed by Chou, Gao and Zhang in the early 1990's. The method can efficiently prove many non-trivial geometry theorems and is one of the most interesting and most successful methods for automated theorem proving in geometry. The method produces proofs that are often very concise and human-readable. In this paper, we provide a first complete presentation of the method. We provide both algorithmic and implementation details that were omitted in the original presentations. We also give a variant of Chou, Gao and Zhang's axiom system. Based on this axiom system, we proved formally all the lemmas needed by the method and its soundness using the Coq proof assistant. To our knowledge, apart from the original implementation by the authors who first proposed the method, there are only three implementations more. Although the basic idea of the method is simple, implementing it is a very challenging task because of a number of details that has to be dealt with. With the description of the method given in this paper, implementing the method should be still complex, but a straightforward task. In the paper we describe all these implementations and also some of their applications.
GeoGebra is an open-source educational mathematics software tool, with millions of users worldwide. It has a number of features (integration of computer algebra, dynamic geometry, spreadsheet, etc.), primarily focused on facilitating student experiments, and not on formal reasoning. Since including automated deduction tools in GeoGebra could bring a whole new range of teaching and learning scenarios, and since automated theorem proving and discovery in geometry has reached a rather mature stage, we embarked on a project of incorporating and testing a number of different automated provers for geometry in GeoGebra. In this paper, we present the current achievements and status of this project, and discuss various relevant challenges that this project raises in the educational, mathematical and software contexts. We will describe, first, the recent and forthcoming changes demanded by our project, regarding the implementation and the user interface of GeoGebra. Then we present our vision of the educational scenarios that could be supported by automated reasoning features, and how teachers and students could benefit from the present work. In fact, current performance of GeoGebra, extended with automated deduction tools, is already very promising—many complex theorems can be proved in less than 1 second. Thus, we believe that many new and exciting ways of using GeoGebra in the classroom are on their way.
ActiveMath is an intelligent e-Learning system that exhibits some Semantic Web features. Its content knowledge representation is a seman- tic XML dialect for mathematics; semantic search is enabled; some of its components work as a web service and, vice versa, it employs certain foreign web services, e.g., for diagnostic purposes. In this paper, we de- scribes features which have not been presented at all or only supercially in previous publications.
The area method for Euclidean constructive geometry was proposed by Chou, Gao and Zhang in the early 1990’s. The method can
efficiently prove many non-trivial geometry theorems and is one of the most interesting and most successful methods for automated
theorem proving in geometry. The method produces proofs that are often very concise and human-readable. In this paper, we
provide a first complete presentation of the method. We provide both algorithmic and implementation details that were omitted
in the original presentations. We also give a variant of Chou, Gao and Zhang’s axiom system. Based on this axiom system, we
proved formally all the lemmas needed by the method and its soundness using the Coq proof assistant. To our knowledge, apart from the original implementation by the authors who first proposed the method, there
are only three implementations more. Although the basic idea of the method is simple, implementing it is a very challenging
task because of a number of details that has to be dealt with. With the description of the method given in this paper, implementing
the method should be still complex, but a straightforward task. In the paper we describe all these implementations and also
some of their applications.
The Baghera project intends to develop theoretical and methodological foundations to guide the computer modelling and conception of learning environments. Our educational approach is founded on the principle that the educational function of a system is an emerging property of the interactions organised between its components: agents and humans, and not a mere functionality of one of its parts. This paper presents the Baghera platform, a web-based multi-agent architecture for learning environments, and its application to the teaching of geometry proof. We start this paper with a brief overview of the project Baghera. Then we describe the multi-agent architecture and the first operational prototype. We conclude by discussing some perspectives.
2 From Ebbinghaus onward psychology has seen an enormous amount of research invested in the study of learning and memory. This research has produced a steady stream of results and, with a few "mini-revolutions" along the way, a steady increase in our understanding of how knowledge is acquired, retained, retrieved, and utilized. Throughout this history there has been a concern with the relationship of this research to its obvious application to education. The first author has written two textbooks (Anderson, 1995a, 1995b) summarizing some of this research. In both textbooks he has made efforts to identify the implications of this research for education. However, he left both textbooks feeling very dissatisfied --that the intricacy of research and theory on the psychological side was not showing through in the intricacy of educational application. One finds in psychology many claims of relevance of cognitive psychology research for education. However, these claims are loose and vague and contrast sharply with the crisp theory and results that exist in the field. To be able to rigorously understand what the implications are of cognitive psychology research one needs a rigorous theory that bridges the gap between the detail of the laboratory experiment and the scale of the educational enterprise. This chapter is based on the ACT-R theory (Anderson, 1993, 1996) which has been able to explain learning in basic psychology experiments and in a number of educational domains. ACT-R has been advertised as a "simple theory of learning and cognition". It proposes that complex cognition is composed of relatively simple knowledge units which are acquired according to relatively simple principles. Human cognition is complex but this complexity reflects complex composition of the basic elements and principles just as a computer can produce complex aggregate behavior from simple computing elements. The ACT-R perspective places a premium on the practice which is required to learn permanently the components of the desired competence. The ACT-R theory claims that to learn a complex competence each component of that competence must be mastered. It is a sharp contrast to many educational claims, supposedly based in cognitive research, that there are moments of insight or transformations when whole knowledge structures become reorganized or learned. In contrast, ACT-R implies that there is no "free lunch" and each piece of knowledge requires its own due of learning. Given the prevalence of the "free lunch myth" we will endeavor to show that it is not true empirically and to explain why it can not be true within the ACT-R theory. This chapter will have the following organization. First we will describe the ACT-R theory and its learning principles. In the light of this theory, we will identify what we think are the important implications of psychological research for education. We will also address the issue of why so much of the research on learning and memory falls short of significant educational application. We will devote special attention to the issues of insight, learning with understanding, and transfer which are part of the free lunch myth. Finally, we will describe how we have tried to bring the lessons of this analysis to bear in the design of our cognitive tutors (Anderson, Boyle, Corbett, & Lewis, 1990; Anderson, Corbett, Koedinger, & Pelletier, 1995).
This study focuses on the potential of multiple-solution tasks in e-learning environments providing a variety of learning tools. This was presented through a multiple-solution-based example for the learning of the mathematical notion of angle in the context of the well known e-learning environment Cabri-Geometry II (Laborde, 1990) dedicated for the learning of geometrical concepts. An a-priori task analysis showed that a variety of solution strategies could be invented by the students to face this type of task. In fact, students can select among the provided tools the most appropriate to express their knowledge. In the integrated context of such tasks and tools, students can express both inter-individual and intra-individual differences in the learning concepts in focus. In addition, students can consolidate these concepts, integrate the different kinds of knowledge they possess, enhance their learning styles and aquire advanced problem-solving skills.
We propose to use didactical theory for the design of educational software. Here we present a set of didactical conditions, and explain how they shape the software design of Cabri-Euclide, a microworld used to learn “mathematical proof ” in a geometry setting. The aim is to design software that does not include a predefined knowledge of problem solution. Key features of the system are its ability to verify local coherence, and not to apply any global and automatic deduction.
We describe Wayang Outpost, a web-based ITS for the Math sec- tion of the Scholastic Aptitude Test (SAT). It has several distinctive features: help with multimedia animations and sound, problems embedded in narrative and fantasy contexts, alternative teaching strategies for students of different mental rotation abilities and memory retrieval speeds. Our work on adding in- telligence for adaptivity is described. Evaluations prove that students learn with the tutor, but learning depends on the interaction of teaching strategies and cognitive abilities. A new adaptive tutor is being built based on evaluation results; surveys results and students' log files analyses.
Intelligent tutoring systems often emphasize learner control: They let the students decide when and how to use the system's intelligent and unintelligent help facilities. This means that students must judge when help is needed and which form of help is appropriate. Data about students' use of the help facilities of the PACT Geometry Tutor, a cognitive tutor for high school geometry, suggest that students do not always have these metacognitive skills. Students rarely used the tutor's on-line Glossary of geometry knowledge. They tended to wait long before asking for hints, and tended to focus only on the most specific hints, ignoring the higher hint levels. This suggests that intelligent tutoring systems should support students in learning these skills, just as they support students in learning domain-specific skills and knowledge. Within the framework of cognitive tutors, this requires creating a cognitive model of the metacognitive help-seeking strategies, in the form of production rules. The tutor then can use the model to monitor students' metacognitive strategies and provide feedback.
Two problem solving strategies, forward chaining and backward chaining, were compared to see how they affect students' learning of geometry theorem proving with con- struction. In order to determine which strategy accelerates learning the most, an intelligent tutoring system, the Advanced Geometry Tutor, was developed that can teach either strat- egy while controlling all other instructional variable. 52 students were randomly assigned to one of the two strategies. Although computational modeling suggests an advantage for backwards chaining, especially on construction problems, the result shows that (1) the stu- dents who learned forward chaining showed better performance on proof-writing, espe- cially on the proofs with construction, than those who learned backward chaining, (2) both forward and backward chaining conditions wrote wrong proofs equally frequently, and (3) the major reason for the difficulty in applying backward chaining appears to lie in the as- sertion of premises as unjustified propositions (i.e., subgoaling).
This paper describes the mechanization of the proofs of the first height chapters of Schwab�user, Szmielew and Tarski’s book:
Metamathematische Methoden in der Geometrie. The proofs are checked formally using the Coq proof assistant. The goal of this development is to provide foundations for
other formalizations of geometry and implementations of decision procedures. We compare the mechanized proofs with the informal
proofs. We also compare this piece of formalization with the previous work done about Hilbert’s Grundlagen der Geometrie. We analyze the differences between the two axiom systems from the formalization point of view.
Gröbner
bases are certain finite sets of multivariate polynomials. Many problems in polynomial ideal theory (algebraic geometry, non-linear computational geometry) can be solved by easy algorithms after transforming the polynomial sets involved in the specification of the problems into Gröbner basis form. In this paper we give some examples of applying the Gröbner bases method to problems in non-linear computational geometry (inverse kinematics in robot programming, collision detection for superellipsoids, implicitization of parametric representations of curves and surfaces, inversion problem for parametric representations, automated geometrical theorem proving, primary decomposition of implicitly defined geometrical objects). The paper starts with a brief summary of the Gröbner bases method.
This study investigates a procedure for proving arithmetic-free Euclidean geometry the- orems that involve construction. "Construction" means drawing additional geometric elements in the problem figure. Some geometry theorems require construction as a part of the proof. The basic idea of our construction procedure is to add only elements required for applying a postulate that has a consequence that unifies with a goal to be proven. In other words, construction is made only if it supports backward application of a postulate. Our major finding is that our proof procedure is semi-complete and useful in practice. In particular, an empirical evaluation showed that our theorem prover, GRAMY, solves all arithmetic-free construction problems from a sample of school textbooks and 86% of the arithmetic-free construction problems solved by preceding studies of automated geometry theorem proving.
We present a set of rules based on full-angles as the basis of automated geometry theorem proving. We extend the idea of eliminating variables and points to the idea of eliminating lines. We also discuss how to combine the forward chaining and backward chaining to achieve higher efficiency. The prover based on the full-angle method has been used to produce short and elegant proofs for more than one hundred difficult geometry theorems. The proofs of many of those theorems produced by our previous area method are relatively long.
The research reported in this paper focuses on the hypothesis that an intelligent tutoring system that provides guidance with respect to students' meta-cognitive abilities can help them to become better learners. Our strategy is to extend a Cognitive Tutor (Anderson, Corbett, Koedinger, & Pelletier, 1995) so that it not only helps students acquire domain-specific skills, but also develop better general help-seeking strategies. In developing the Help Tutor, we used the same Cognitive Tutor technology at the meta- cognitive level that has been proven to be very effective at the cognitive level. A key challenge is to develop a model of how students should use a Cognitive Tutor's help facilities. We created a preliminary model, implemented by 57 production rules that capture both effective and ineffective help-seeking behavior. As a first test of the model's efficacy, we used it off-line to evaluate students' help-seeking behavior in an existing data set of student-tutor interactions. We then refined the model based on the results of this analysis. Finally, we conducted a pilot study with the Help Tutor involving four students. During one session, we saw a statistically significant reduction in students' meta-cognitive error rate, as determined by the Help Tutor's model. These preliminary results inspire confidence as we gear up for a larger-scale controlled experiment to evaluate whether tutoring on help seeking has a positive effect on students' learning outcomes.
Thèse (Doctorat)--Institut national polytechnique de Grenoble, 1997.
The Baghera Assessment Project (BAP) has the objective to ex plore a new avenue for the design of e-Learning environments. The key features of BAP's approach are: (i) the concept of emergence in multi-agents systems as modelling framework, (ii) the shaping of a new theoretic al framework for modelling student knowledge, namely the cK¢ model. This new model has been constructed, based on the current research in cognitive science and education, to bridge research on education and research on the design of learning environments.
This article proposes a state of the art of tutorial systems for high school planar geometry proof learning. The chosen approach is part of exploring the research problem that motivates the development, by our research team, of a tutorial system named QED-Tutrix, that we will present in another paper. In the following article, a synthesis and a comparison of existing tutorial systems is carried out on the basis of a set of original indicators highlighting the differences between the systems covered by our analysis. Each indicator aims to describe the functioning of the studied software according to the geometric work made possible at their interface. Eleven tutorial systems are compared according to their integration of a geometric figure, the structure they impose on the student's reasoning and the tutorial intervention they offer.
A blocking state is a measurable state on an intelligent tutoring systems’ user interface, which mirrors a student’s cognitive state where she/he cannot temporarily make any progress toward finding a solution to a problem. In this paper, we present the development of four probabilistic models to detect a blocking state of students while they are solving a Canadian high school-level problem in Euclidean geometry on an ITS. Our methodology includes experimentation with a modified version of QED-Tutrix, an ITS, which we used to gather labelled datasets composed of sequences of mouse and keyboard actions. We developed four predicting models: an action-frequency model, a subsequence-detection model, a 1D convolutional neural network model and a hybrid model. The hybrid model outperforms the others with a score of 80.4% on the classification of blocking state on validation set while performing 77.3% on the test set.
Our article aims to define the notion of instrumental proof based on didactic, epistemological and cognitive considerations. We raise issues and challenges related to the use of this type of proof in mathematical work and mathematical thinking. The theory of mathematical working spaces serves as a construct on which we address questions about proof, reasoning and epistemic necessity, taking advantage of the possibilities offered by the development geneses and fibrations in an instrumented perspective. The coordination of the semiotic, discursive and instrumental geneses of the working space founds discursive-graphic proofs, mechanical proofs and algorithmic proofs that are activated at school in the subject-milieu interactions. We end with a discussion on some consequences of the computer-assisted modelling of the learning conditions of mathematics, and we conclude on a necessary reconciliation of heuristics and validation.
In this article, we propose an analysis of the computational constraints that are imposed on the design and the implementation of computer-based learning environments for proofs in geometry, and an analysis of the negotiation of these constraints by the students during their activities in such interactive environments. For this purpose we have divided the article into three parts. The first presents an analysis of a problem-situation. The second describes an analysis of three computer programs -each program is first presented and then analysed from the point of view of the constraints it may impose on a solution. Finally, we describe an experiment with the problem presented, in 8th grade classroom, using DEFI-CABRI. The aim of the study is to study the way in which students manage the constraints on the student-system interaction.
Le projet Mentoniezh (géométrie, en breton) vise à développer un tuteur intelligent pour la démonstration en géométrie euclidienne plane, destiné aux classes de collège. Le système est constitué d'un résolveur pédagogique qui détient l'expertise géométrique et d'un tuteur qui guide et corrige l'élève tout au long de la résolution du problème. Il utilise plusieurs types de connaissances : des connaissances sur la géométrie, des connaissances sur l'élève et des connaissances pédagogiques. Cet article présente un exemple d'interaction élève/tuteur, décrit la modélisation des connaissances adoptées et présente les expérimentations du système réalisées au collège.
Our contribution shows the anticipated effect of what we call connected problems in developing the competencies of students and their acquisition of mathematical knowledge. Whilst our theoretical approach focuses on didactic and cognitive interactions, we give special attention to a model to reason about learners’ conceptions, and the ideas of mathematical working space and zone of proximal development, in order to explore how connected problems can help to resolve moments of impasse of a student when solving a proof problem in geometry. In particular, we discuss how the notion of interaction moves our theoretical framework closer to the methodological challenges raised in the QED-Tutrix research project jointly being realized in didactics of mathematics and computer engineering.
GeoGebra ist eine dynamische Mathematiksoftware, die die klassischen Bereiche einer Geometriesoftware und eines Funktionenplotters vereint und um Funktionen aus der Statistik und der Computeralgebra erweitert. Dabei hat die grundlegende Open Source Idee des freien Austauschs von Software und Materialien in den letzten Jahren zu einem hohen Verbreitungsgrad von GeoGebra im Mathematikunterricht geführt.
In den nachfolgenden Beiträgen finden sich einige interessante Anwendungsmöglichkeiten von GeoGebra für den Unterricht. Dabei werden einerseits Basisideen vermittelt, andererseits aber auch Ausblicke auf aktuelle Entwicklungen von GeoGebra (wie etwa GeoGebra-Script, GeoGebra-Tube, GeoGebra-3D …) gegeben.
This paper is adapted from a book and many scholarly articles. It reviews the main ideas of a theory for the assessment of a student's knowledge in a topic and gives details on a practical implementation in the form of a software. The basic concept of the theory is the ‘knowledge state,' which is the complete set of problems that an individual is capable of solving in a particular topic, such as Arithmetic or Elementary Algebra. The task of the assessor—which is always a computer—consists in uncovering the particular state of the student being assessed, among all the feasible states. Even though the number of knowledge states for a topic may exceed several hundred thousand, these large numbers are well within the capacity of current home or school computers. The result of an assessment consists in two short lists of problems which may be labelled: ‘What the student can do' and ‘What the student is ready to learn.' In the most important applications of the theory, these two lists specify the exact knowledge state of the individual being assessed. Moreover, the family of feasible states is specified by two combinatorial axioms which are pedagogically sound from the standpoint of learning. The resulting mathematical structure is related to closure spaces and thus also to concept lattices. This work is presented against the contrasting background of common methods of assessing human competence through standardized tests providing numerical scores. The philosophy of these methods, and their scientific origin in nineteenth century physics, are briefly examined.
The focus of this paper is the application of the theory of contingent tutoring to the design of a computer-based system designed to support learning in aspects of algebra. Analyses of interactions between a computer-based tutoring system and 42, 14- and 15-year-old pupils are used to explore and explain the relations between individual differences in learner–tutor interaction, learners’ prior knowledge and learning outcomes. Parallels between the results of these analyses and empirical investigations of help seeking in adult–child tutoring are drawn out. The theoretical significance of help seeking as a basis for studying the impact of individual learner differences in the collaborative construction of ‘zones of proximal development’ is assessed. In addition to demonstrating the significance of detailed analyses of learner–system interaction as a basis for inferences about learning processes, the investigation also attempts to show the value of exploiting measures of on-line help seeking as a means of assessing learning transfer. Finally, the implications of the findings for contingency theory are discussed, and the theoretical and practical benefits of integrating psychometric assessment, interaction process analyses, and knowledge-based learner modelling in the design and evaluation of computer-based tutoring are explored.
This paper presents work related to the development of an intelligent tutoring system called TURING (French acronym of «TUtoRiel INtelligent en Géométrie»). TURING is a multidisciplinary project, that joints recent research in didactic of mathematics with the possibilities of computer-based learning environments. From the educational point of view, the system is helpful for the secondary school student to improve problem solving aptitudes, mathematical reasoning abilities and communication skills using natural and mathematical language. In addition, the system is helpful to assist the teacher in his responsibility of attending the diversity of the development of mathematical competences in a whole class. From the technical point of view, the TURING architecture is a multi-agent system conceived in a flexible approach so that a teacher can adjust the heuristic and discursive features according to specific practices with real students. In particular, for the discovery of a conjecture or for the realization of a mathematical proof, the system consents to adapt, in the problem solving process, the space of meaningful actions and, in argumentative process, the set of strategic messages with pedagogical agents.
A standard technique in classical analysis for the study of eontinous sub-solutions of the Dirichlet problem for second order operators may be illustrated as follows. Suppose it is to be shown that a continuous real function ](x) is convex (respectively, striely convex) at x0; then it suffices to produce a C ~ function g(x) such that g(x)1 some fixed positive constant). The main point of this procedure is to sidestep arguments involving continuous functions by working with differentiable functions alone. Now in global differential geometry, the functions that naturally arise are often continuous but not differentiable. Since much of geometric analysis reduces to second order elliptic problems, this technique then recommends itself as a natural tool for overcoming this difficulty with the lack of differentiability. In a limited way, this technique has indeed appeared in several papers in complex geometry (e.g. Ahlfors [1], Takeuchi [20], Elenewajg [7] and Greene-Wu [11]; cf. also Suzuki [19]). The main purpose of this paper is to broaden and deepen the scope of this method by making it the central point of a general study of nonnegative sectional, Ricei or bisectional curvature. The following are the principal theorems; the relevant definitions can be found in Section 1. Let M be a noncompact complete Riemannian manifold and let 0 E M be fixed. Let {Ct}tG1 be a family of closed subsets of M indexed by a subset I of R. Assume that et = d(0, Ct)~ as t-~, where d(p, q) will always denote the distance between p, qEM relative to the Riemannian metric. The family of functions ~t: M-~R defined by ~t(P)=
The use of Gröbner basis computation for reasoning about geometry problems is demonstrated. Two kinds of geometry problems are considered: (i) Given a finite set of geometry relations expressed as polynomial equations, in conjunction with a finite set of subsidiary conditions stated as negations of polynomial equations to rule out certain degenerate eases, check whether another geometry relation expressed as a polynomial equation and given as a conclusion, holds. (ii) Given a finite set of geometry relations expressed as polynomial equations, find a finite set of subsidiary conditions, if any, stated as negations of polynomial equations which rule out certain values of variables, such that another geometry relation expressed as a polynomial equation and given as a conclusion, holds under these conditions. Using a refutational approach for theorem proving, both kinds of problems are converted into reasoning about a finite set of polynomial equations. The first problem is shown to be equivalent to checking whether a set of polynomial equations does not have a solution; this can be decided by computing a Gröbner basis of these polynomials and checking whether I is included in such a basis. In addition, it is shown that the second problem can also be solved by computing a Gröbner basis and appropriately picking polynomials from it. A number of geometry problems of both kinds have been solved using this approach.
Wu's algebraic method for mechanically proving geometry theorems is presented at a level as elementary as possible with sufficient examples for further understanding the complete method.
In this paper, we present results of the work which allow us to prove geometry theorems by the parallel numerical method based on the multi-instance numerical verification of algebraic identity. The algebraic principle of the parallel numerical method is discussed and illustrated intuitively; the advantages of our method are given. It is acceptable on the complexity of both memory and time. It can be used to prove non-trivial geometric theorems by microcomputer, even by hand. We give some examples proved by parallel numerical method, including certain new unexpected results.
In the Adaptive Character of Thought (ACT-R) theory, complex cognition arises from an interaction of procedural and declarative knowledge. Procedural knowledge is represented in units called production rules, and declarative knowledge is represented in units called chunks. The individual units are created by simple encodings of objects in the environment (chunks) or simple encodings of transformations in the environment (production rules). A great many such knowledge units underlie human cognition. From this large database, the appropriate units are selected for a particular context by activation processes that are tuned to the statistical structure of the environment. According to the ACT-R theory, the power of human cognition depends on the amount of knowledge encoded and the effective Employment of the encoded knowledge.
Notre travail de recherche a pour objectif de modéliser les connaissances des élèves en algèbre élémentaire en construisant leur profil cognitif pour la transition entre la troisième et la seconde de l'enseignement français. La modélisation de l'apprenant est un problème ardu dès lors que l'on souhaite obtenir un modèle de l'apprenant riche. À cette problématique de modélisation de l'apprenant, s'ajoute une forte préoccupation d'usage : nous souhaitons concevoir un système réellement utilisé dans l'enseignement. Pour lever en partie les difficultés posées par la modélisation de l'apprenant, nous proposons une façon nouvelle d'envisager le diagnostic. D'une part nous sommes partis d'une analyse didactique rigoureuse dont le résultat est un outil de diagnostic papier - crayon qui abouti à la création de profil cognitif des élèves. C'est l'établissement de ce profil que nous avons informatisé. D'autre part nous considérons l'interface élève, permettant le recueil des observables, comme faisant partie intégrante du diagnostic. De la qualité des observables dépend en effet en partie la qualité du modèle de l'apprenant construit. L'élaboration du profil de l'élève se fait en trois étapes auxquelles correspondent trois modules dans notre système : le recueil des observables avec l'interface élève, le diagnostic puis la présentation des profils à l'enseignant. Chaque étape pose des problèmes qui lui sont propres. Le recueil des observables se fait grâce à une interface proposant les exercices aux élèves. La conception de cette interface a pris une part importante dans notre travail. Elle pose en effet des problèmes de transfert d'un environnement et d'exercices papier - crayon sur ordinateur, problèmes qui ne peuvent pas se résumer à une simple médiatisation. Le module de diagnostic tient compte à la fois de la diversité des questions (qu'elles soient fermées ou ouvertes) et de la diversité des réponses proposées par les élèves (qui contiennent aussi bien du langage naturel que des expressions algébriques). L'interface de présentation des profils aux enseignants a elle aussi été conçue avec soin, c'est en effet cette interface qui fait le lien entre le profil construit automatiquement et l'utilisateur de ce profil. L'intégration de notre système aux pratiques des enseignants dépend de la coopération didacticiens / informaticiens, mais aussi de la qualité de l'interface enseignant. Pour faciliter l'appropriation du profil par l'enseignant, ce module propose différents modes de représentation et différents degrés d'implication de l'enseignant (de la simple consultation du profil à la modification du diagnostic). L'interface élève a été testée à plusieurs reprises dans des classes auprès d'une centaine d'élèves. Un prototype du module de diagnostic permet d'analyser environ 75 % des réponses et de construire les profils dans lesquels les enseignants reconnaissent leurs élèves. Pour l'interface enseignant, les résultats des premiers tests auprès d‚enseignants montrent que les représentations utilisées sont efficaces, mais les tests d'acceptabilité doivent être poursuivis. L'importante coopération didacticiens / informaticiens, la réutilisation d'expertise didactique ainsi que l'amélioration de la fiabilité des observables par la qualité de l'interface, sont les gages de l'intégration des systèmes issus de la recherche aux pratiques des enseignants.