Proof step analysis for proof tutoringa learning approach to granularity (original) (raw)
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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.
Towards computer-assisted proof tutoring
2007
Abstract We present a recent application area of the proof assistant Ωmega, the teaching of mathematical proofs within an environment for tutorial dialog. We discuss the design of our dialog system prototype for proof tutoring in the light of the requirements imposed by its potential users. Empirical studies investigating those requirements guide the development of the system.
Granularity judgments in proof tutoring
2006
The SFB 378 project Dialog[2] investigates natural tutorial dialog between a student and an assistance system for mathematics (ASM). The aim is to mechanize the tutoring of mathematical proofs. At the present time, the (simplified) approach in the Dialog system is: (1) The student inputs a proof via the user interface using a mixture of natural language and formulas.
Towards a Principled Approach to Tutoring Mathematical Proofs
Proceedings of the …, 2003
Studies comparing human and computer-based tutoring have identified natural language communication about the tutorial goal as a major source for the increased learning success in human tutoring. However, due to the inherent difficulty of natural language processing in non-tightly restricted tasks, which tutoring is even in simple domains, only few state-of-the-art systems use natural-language style interaction. Addressing tutoring in a principled way, we are developing a system that teaches proving skills in mathematics, by incrementally enhancing a testable Wizard-of Oz environment with tutoring components, existing domain reasoning systems and natural language processing components. The first component integrated is an elaborate hinting algorithm. Experiments carried out recently provided valuable insights in relevant phenomena and demands for tutorial and task-specific enhancements of available mathematical reasoning and natural language processing components. 1 The DIALOG project is part of the Collaborative Research Center on Resource-Adaptive Cognitive Processes (SFB 378) at University of the Saarland [17].
An Interactive Proof Development Environment+ Anticipation= A Mathematical Assistant?
International Journal of Computing Anticipatory Systems (CASYS), 1999
Current semi-automated theorem provers are often advertised as “mathematical assistant systems”. However, these tools behave too passively and in a stereotypic way to meet this ambitious goal because they lack the capability to adequately take into account requirements on proof search control and user demands for their own actions. Motivated by this deficit, we have incorporated several facilities into the ªMEGA proof development system that anticipate a number of divergent factors, based on mathematical knowledge, proof ...
HAL (Le Centre pour la Communication Scientifique Directe), 2022
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An Interactive Driver for Goal-directed Proof Strategies
Electronic Notes in Theoretical Computer Science, 2009
Interactive Theorem Provers (ITPs) are tools meant to assist the user during the formal development of mathematics. Automatic proof searching procedures are a desirable aid, and most ITPs supply the user with an extensive set of facilities to improve automation. However, the black-box nature of most automatic procedure conflicts with the interactive nature of these tools: a newcomer running an automatic procedure learns nothing by its execution (especially in case of failure), and a trained user has no opportunities to interactively guide the procedure towards the solution, e.g. pruning wrong or not promising branches of the search tree. In this paper we discuss the implementation of the resolution based automatic procedure of the Matita ITP, explicitly conceived to be interactively driven by the user through a suitable, simple graphical interface.
Tutorial dialogs on mathematical proofs
IJCAI Workshop on Knowledge Representation and Automated Reasoning for E-Learning Systems, 2003
The representation of knowledge for a mathematical proof assistant is generally used exclusively for the purpose of proving theorems. Aiming at a broader scope, we examine the use of mathematical knowledge in a mathematical tutoring system with flexible natural language dialog. Based on an analysis of a corpus of dialogs we collected with a simulated tutoring system for teaching proofs in naive set theory, we identify several interesting problems which lead to requirements for mathematical knowledge representation. This includes resolving reference between natural language expressions and mathematical formulas, determining the semantic role of mathematical formulas in context, and determining the contribution of inference steps specified by the user.
Learning from experts to aid the automation of proof search
2009
Most formal methods give rise to proof obligations which are putative lemmas that need proof. Discharging these POs can become a bottleneck in the use of formal methods in practical applications. Some techniques for reducing this bottleneck are known—it is our aim to increase the repertoire of techniques by tackling learning from proof attempts. Even after obvious fixed heuristics are used, there remains the problem of what to do with POs that are not discharged automatically.