Towards an Intelligent Tutor for Mathematical Proofs

Computer-supported learning is an increasingly important form of study since it allows for independent learning and individualized instruction. In this paper, we discuss a novel approach to developing an intelligent tutoring system for teaching textbook-style mathematical proofs. We characterize the particularities of the domain and discuss common ITS design models. Our approach is motivated by phenomena found in a corpus of tutorial dialogs that were collected in a Wizard-of-Oz experiment. We show how an intelligent tutor for textbook-style mathematical proofs can be built on top of an adapted assertion-level proof assistant by reusing representations and proof search strategies originally developed for automated and interactive theorem proving. The resulting prototype was successfully evaluated on a corpus of tutorial dialogs and yields good results.Comment: In Proceedings THedu'11, arXiv:1202.453

MENON : automating a Socratic teaching model for mathematical proofs

This thesis presents an approach to adaptive pedagogical feedback for arbitrary domains as an alternative to resource-intensive pre-compiled feedback, which represents the state-of-the-art in intelligent tutoring systems today. A consequence of automatic adaptive feedback is that the number of tasks with pedagogical feedback that can be offered to the student increases, and with it the opportunity for practice. We focus on automating different aspects of teaching that together are primarily responsible for learning and can be integrated in a unified natural-language output. The automatic production and natural-language generation of feedback enables its personalisation both at the pedagogical and the natural-language dialogue level. We propose a method for automating the production of domain-independent adaptive feedback. The proof- of-concept implementation of the tutorial manager Menon is carried out for the domain of set-theory proofs. More specifically, we define a pedagogical model that abides by schema and cognitive load theory, and by the synergistic approach to learning. We implement this model in a Socratic teaching strategy whose basic units of feedback are dialogue moves. We use empirical data from two domains to derive a taxonomy of tutorial-dialogue moves, and define the most central and sophisticated move hint. The formalisation of the cognitive content of hints is inspired by schema theory and is facilitated by a domain ontology.Die vorliegende Arbeit präsentiert eine Annäherung an adaptives pädagogisches Feedback für beliebige Domäne. Diese Herangehensweise bietet eine Alternative zu ressource-intensivem, vorübersetztem Feedback, dass das heutige "state-of-the-art'; in intelligenten tutoriellen Systemen ist. Als Folge können zahlreiche Aufgaben mit pädagogischem Feedback für die Praxis angeboten werden. Der Schwerpunkt der Arbeit liegt auf der Automatisierung verschiedener Aspekte des Lehrprozesses, die in ihrer Gesamtheit wesentlich den Lernprozess beeinflussen, und in einer einheitlichen Systemausgabe Natürlicher Sprache integriert werden können. Die automatische Produktion und die Systemgenerierung von Feedback in Natürlicher Sprache ermöglichen eine Individualisierung des Feedback auf zwei Ebenen: einer pädagogischen und einer dialogischen Ebene. Dazu schlagen wir eine Methode vor, durch die adaptives Feedback automatisiert werden kann, und implementieren den tutoriellen Manager Menon als "proof-of-concept'; beispielhaft für die Domäne von Beweisen in der Mengentheorie. Konkret definieren wir ein pädagogisches Modell, das sich auf Schema- und Kognitionstheorie sowie auf die synergetische Herangehensweise an Lernen stützt. Dieses Modell wird in einer Sokratischen Lehrmethode implementiert, deren basale Feedback-Elemente aus Dialogakten bestehen. Zur Bestimmung einer Taxonomie Tutorielle-Dialogakte sowie des zentralen und komplexen Dialogakts hint (Hinweis) wenden wir empirische Daten aus zwei Domänen an. Die Formalisierung des kognitiven Inhaltes von Hinweisen folgt der Schematheorie und basiert auf einer Domänenontologie

Students´ language in computer-assisted tutoring of mathematical proofs

Truth and proof are central to mathematics. Proving (or disproving) seemingly simple statements often turns out to be one of the hardest mathematical tasks. Yet, doing proofs is rarely taught in the classroom. Studies on cognitive difficulties in learning to do proofs have shown that pupils and students not only often do not understand or cannot apply basic formal reasoning techniques and do not know how to use formal mathematical language, but, at a far more fundamental level, they also do not understand what it means to prove a statement or even do not see the purpose of proof at all. Since insight into the importance of proof and doing proofs as such cannot be learnt other than by practice, learning support through individualised tutoring is in demand. This volume presents a part of an interdisciplinary project, set at the intersection of pedagogical science, artificial intelligence, and (computational) linguistics, which investigated issues involved in provisioning computer-based tutoring of mathematical proofs through dialogue in natural language. The ultimate goal in this context, addressing the above-mentioned need for learning support, is to build intelligent automated tutoring systems for mathematical proofs. The research presented here has been focused on the language that students use while interacting with such a system: its linguistic propeties and computational modelling. Contribution is made at three levels: first, an analysis of language phenomena found in students´ input to a (simulated) proof tutoring system is conducted and the variety of students´ verbalisations is quantitatively assessed, second, a general computational processing strategy for informal mathematical language and methods of modelling prominent language phenomena are proposed, and third, the prospects for natural language as an input modality for proof tutoring systems is evaluated based on collected corpora

A generic architecture for interactive intelligent tutoring systems

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University, 07/06/2001.This research is focused on developing a generic intelligent architecture for an interactive tutoring system. A review of the literature in the areas of instructional theories, cognitive and social views of learning, intelligent tutoring systems development methodologies, and knowledge representation methods was conducted. As a result, a generic ITS development architecture (GeNisa) has been proposed, which combines the features of knowledge base systems (KBS) with object-oriented methodology. The GeNisa architecture consists of the following components: a tutorial events communication module, which encapsulates the interactive processes and other independent computations between different components; a software design toolkit; and an autonomous knowledge acquisition from a probabilistic knowledge base. A graphical application development environment includes tools to support application development, and learning environments and which use a case scenario as a basis for instruction. The generic architecture is designed to support client-side execution in a Web browser environment, and further testing will show that it can disseminate applications over the World Wide Web. Such an architecture can be adapted to different teaching styles and domains, and reusing instructional materials automatically can reduce the effort of the courseware developer (hence cost and time) in authoring new materials. GeNisa was implemented using Java scripts, and subsequently evaluated at various commercial and academic organisations. Parameters chosen for the evaluation include quality of courseware, relevancy of case scenarios, portability to other platforms, ease of use, content, user-friendliness, screen display, clarity, topic interest, and overall satisfaction with GeNisa. In general, the evaluation focused on the novel characteristics and performances of the GeNisa architecture in comparison with other ITS and the results obtained are discussed and analysed. On the basis of the experience gained during the literature research and GeNisa development and evaluation. a generic methodology for ITS development is proposed as well as the requirements for the further development of ITS tools. Finally, conclusions are drawn and areas for further research are identified

Investigation of methods for user adapted visualisation of information in a hypermedia generation system

A literature review of user interaction to support creative processes is given. A design for an authoring system for semi-automatically generated hypermedia presentations is developed. The system designed is called SampLe (a Semi-Automatic Multimedia Presentation generation Environment

A metacognitive feedback scaffolding system for pedagogical apprenticeship

This thesis addresses the issue of how to help staff in Universities learn to give feedback with the main focus on helping teaching assistants (TAs) learn to give feedback while marking programming assignments. The result is an innovative approach which has been implemented in a novel computer support system called McFeSPA. The design of McFeSPA is based on an extensive review of the research literature on feedback. McFeSPA has been developed based on relevant work in educational psychology and Artificial Intelligence in EDucation (AIED) e.g. scaffolding the learner, ideas about andragogy, feedback patterns, research into the nature and quality of feedback and cognitive apprenticeship. McFeSPA draws on work on feedback patterns that have been proposed within the Pedagogical Patterns Project (PPP) to provide guidance on structuring the feedback report given to the student by the TA. The design also draws on the notion of andragogy to support the TA. McFeSPA is the first Intelligent Tutoring System (ITS) that supports adults learning to help students by giving quality feedback. The approach taken is more than a synthesis of these key ideas: the scaffolding framework has been implemented both for the domain of programming and the feedback domain itself; the programming domain has been structured for training TAs to give better feedback and as a framework for the analysis of students’ performance. The construction of feedback was validated by a small group of TAs. The TAs employed McFeSPA in a realistic situation that was supported by McFeSPA which uses scaffolding to support the TA and then fade. The approach to helping TAs become better feedback givers, which is instantiated in McFeSPA, has been validated through an experimental study with a small group of TAs using a triangulation approach. We found that our participants learned differently by using McFeSPA. The evaluation indicates that 1) providing content scaffolding (i.e. detailed feedback about the content using contingent hints) in McFeSPA can help almost all TAs increase their knowledge/understanding of the issues of learning to give feedback; 2) providing metacognitive scaffolding (i.e. each level of detailed feedback in contingent hint, this can also be general pop-up messages in using the system apart from feedback that encourage the participants to give good feedback) in McFeSPA helped all TAs reflect on/rethink their skills in giving feedback; and 3) when the TAs obtained knowledge about giving quality feedback, providing adaptable fading of TAs using McFeSPA allowed the TAs to learn alone without any support

Extracting proofs from documents

Often, theorem checkers like PVS are used to check an existing proof, which is part of some document. Since there is a large difference between the notations used in the documents and the notations used in the theorem checkers, it is usually a laborious task to convert an existing proof into a format which can be checked by a machine. In the system that we propose, the author is assisted in the process of converting an existing proof into the PVS language and having it checked by PVS. 1 Introduction The now-classic ALGOL 60 report [5] recognized three different levels of language: a reference language, a publication language and several hardware representations, whereby the publication language was intended to admit variations on the reference language and was to be used for stating and communicating processes. The importance of publication language ---often referred to nowadays as "pseudo-code"--- is difficult to exaggerate since a publication language is the most effective way..