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

Bringing Nordic mathematics education into the future : Preceedings of Norma 20 : The ninth Nordic conference on mathematics education Oslo, 2021

publishedVersio

Bringing Nordic mathematics education into the future : Preceedings of Norma 20 : The ninth Nordic conference on mathematics education Oslo, 2021

This volume presents Nordic mathematics education research, which will be presented at the Ninth Nordic Conference on Mathematics Education, NORMA 20, in Oslo, Norway, in June 2021. The theme of NORMA 20 regards what it takes or means to bring Nordic mathematics education into the future, highlighting that mathematics education is continuous and represents stability just as much as change.publishedVersio

Assertion level proof planning with compiled strategies

This book presents new techniques that allow the automatic verification and generation of abstract human-style proofs. The core of this approach builds an efficient calculus that works directly by applying definitions, theorems, and axioms, which reduces the size of the underlying proof object by a factor of ten. The calculus is extended by the deep inference paradigm which allows the application of inference rules at arbitrary depth inside logical expressions and provides new proofs that are exponentially shorter and not available in the sequent calculus without cut. In addition, a strategy language for abstract underspecified declarative proof patterns is developed. Together, the complementary methods provide a framework to automate declarative proofs. The benefits of the techniques are illustrated by practical applications.Die vorliegende Arbeit beschäftigt sich damit, das Formalisieren von Beweisen zu vereinfachen, indem Methoden entwickelt werden, um informale Beweise formal zu verifizieren und erzeugen zu können. Dazu wird ein abstrakter Kalkül entwickelt, der direkt auf der Faktenebene arbeitet, welche von Menschen geführten Beweisen relativ nahe kommt. Anhand einer Fallstudie wird gezeigt, dass die abstrakte Beweisführung auf der Fakteneben vorteilhaft für automatische Suchverfahren ist. Zusätzlich wird eine Strategiesprache entwickelt, die es erlaubt, unterspezifizierte Beweismuster innerhalb des Beweisdokumentes zu spezifizieren und Beweisskizzen automatisch zu verfeinern. Fallstudien zeigen, dass komplexe Beweismuster kompakt in der entwickelten Strategiesprache spezifiziert werden können. Zusammen bilden die einander ergänzenden Methoden den Rahmen zur Automatisierung von deklarativen Beweisen auf der Faktenebene, die bisher überwiegend manuell entwickelt werden mussten

Assertion level proof planning with compiled strategies

This book presents new techniques that allow the automatic verification and generation of abstract human-style proofs. The core of this approach builds an efficient calculus that works directly by applying definitions, theorems, and axioms, which reduces the size of the underlying proof object by a factor of ten. The calculus is extended by the deep inference paradigm which allows the application of inference rules at arbitrary depth inside logical expressions and provides new proofs that are exponentially shorter and not available in the sequent calculus without cut. In addition, a strategy language for abstract underspecified declarative proof patterns is developed. Together, the complementary methods provide a framework to automate declarative proofs. The benefits of the techniques are illustrated by practical applications.Die vorliegende Arbeit beschäftigt sich damit, das Formalisieren von Beweisen zu vereinfachen, indem Methoden entwickelt werden, um informale Beweise formal zu verifizieren und erzeugen zu können. Dazu wird ein abstrakter Kalkül entwickelt, der direkt auf der Faktenebene arbeitet, welche von Menschen geführten Beweisen relativ nahe kommt. Anhand einer Fallstudie wird gezeigt, dass die abstrakte Beweisführung auf der Fakteneben vorteilhaft für automatische Suchverfahren ist. Zusätzlich wird eine Strategiesprache entwickelt, die es erlaubt, unterspezifizierte Beweismuster innerhalb des Beweisdokumentes zu spezifizieren und Beweisskizzen automatisch zu verfeinern. Fallstudien zeigen, dass komplexe Beweismuster kompakt in der entwickelten Strategiesprache spezifiziert werden können. Zusammen bilden die einander ergänzenden Methoden den Rahmen zur Automatisierung von deklarativen Beweisen auf der Faktenebene, die bisher überwiegend manuell entwickelt werden mussten

Proceedings of the Seventh Congress of the European Society for Research in Mathematics Education

International audienceThis volume contains the Proceedings of the Seventh Congress of the European Society for Research in Mathematics Education (ERME), which took place 9-13 February 2011, at Rzeszñw in Poland

Structures, commitments and games in strategic conversations

Les effets d'une action linguistique dépendent du contexte. Cela pose plusieurs questions, auxquelles il est essentiel qu'un modèle linguistique réponde: comment représenter le contexte conversationnel, la façon dont celui-ci influe sur le "sens" de chaque contribution et sur le choix rationnel fait par un agent de ce qu'il va dire ensuite? Il existe plusieurs théories de la structure du discours, mais celles-ci ne s'accordent pas sur un ensemble précis de contraintes structurelles régissant le contexte conversationnel. Nous proposons un formalisme unifié pour traduire et comparer entre théories et représentations distinctes et nous étudions les fondations axiomatiques d'une mesure de déviation 'sémantique' entre deux contextes conversationnels, et de l'impact de l'entrée de nouveaux éléments dans le contexte sur cette deviation. Un second travail porte sur l'interaction entre forme logique et rationnalité dans les conversations, plus spécifiquement, lorsque les intérêts des participants divergent. Nous proposons un modèle en théorie des jeux, dans lequel une conversation est une séquence infinie de coups linguistiques. Dans ce cadre nous formalisons certaines contraintes linguistiques génériques comme des conditions nécessaires au succès d'un agent (rester consistant, cohérent, crédible). Les préférences des agents sont décrites par des contenus auxquels ceux-ci souhaitent, ou ne souhaitent pas s'engager. Crucialement, on peut justifier et expliquer via des considérations sémantiques le choix des objectifs conversationnels des agents, et montrer quand et comment certaines inferences (implicatures) survivent ou disparaissent. Cela nécessite une sémantique adéquate, pour l'obtenir nous définissons une logique modale dynamique des engagements publics. Celle-ci permet de représenter les déclaration des participants vis-à-vis de leur propre engagements et de ceux de leurs interlocuteurs. Cela permet enfin un modèle de "grounding" a granularité plus fine que les approches existantes, qui demeurent cependant axiomatisable comme des cas particulier.The effects of a linguistic action depend on its context of use. This raises a certain number of issues for a model of language use: how to represent the conversational context, its relation to the meanings that agents convey, and how to model agents' rational choice of the next thing to say, in context? There is, between existing theories of discourse structure, no general agreement on a precise set of structural constraints governing the conversational context. To remedy this, we propose a unified framewok to translate and compare between distinct theories and representations. We then lay the axiomatic foundations of metrics measuring the semantic deviation between two conversational contexts, and the changes brought into such a deviation, as new moves enter the context. A second body of work focuses on the modeling of conversational meaning and its interaction with that of rationnality in conversations, more specifically strategic dialogs, where the interest of the participants diverge. We propose a game theoretic account of such conversations, as infinite sequences of linguistic moves. We formalize linguistic constraints that are generic necessary conditions on successful plays (staying coherent, consistent, credible), and describe agents' preferences in terms of the contents that agents commit to. Crucially, we can describe a player's objective and explain why it is adopted on semantic grounds. We show on this basis how and when inferences to non-litteral meaning survives or are cancelled. As this requires a semantics expressive enough, we define a dynamic logic of public commitments to represent participants' commitments about the content of theirs, or their opponent's moves, and keep those representations subject to a sound notion of logical consequence (and hence, of consistency). This yields an account of acknowledgment and grounding more formal and fine-grained than traditional approaches, recoverable as particular cases