Learning to Prove Theorems via Interacting with Proof Assistants

Humans prove theorems by relying on substantial high-level reasoning and problem-specific insights. Proof assistants offer a formalism that resembles human mathematical reasoning, representing theorems in higher-order logic and proofs as high-level tactics. However, human experts have to construct proofs manually by entering tactics into the proof assistant. In this paper, we study the problem of using machine learning to automate the interaction with proof assistants. We construct CoqGym, a large-scale dataset and learning environment containing 71K human-written proofs from 123 projects developed with the Coq proof assistant. We develop ASTactic, a deep learning-based model that generates tactics as programs in the form of abstract syntax trees (ASTs). Experiments show that ASTactic trained on CoqGym can generate effective tactics and can be used to prove new theorems not previously provable by automated methods. Code is available at https://github.com/princeton-vl/CoqGym.Comment: Accepted to ICML 201

Exploring grade 11 learners’ use of the geogebra programme when learning euclidean geometry.

Masters Degree. University of Kwa-Zulu Natal, Durban.The GeoGebra programme is a free computer application programme that provides an algebra view, Geometry view, spreadsheet view and an input bar. This study explored how the GeoGebra programme contributed to learners’ learning and understanding of Euclidean Geometry. The research focused on participants’ experiences as they used the GeoGebra programme to support their understanding of Euclidean Geometry. It highlighted learners’ perspectives on the role of the GeoGebra programme in supporting an exploration of Euclidean Geometry in particular and mathematical ideas in general. The focus of the study was to explore the way in which the GeoGebra programme is used, as a learning tool and mediating artefact in the learning of Euclidean Geometry in Grade 11 Mathematics. This study also aimed to explore learners’ experiences and perceptions when the GeoGebra programme is used to support the learning of Grade 11 Euclidean Geometry. The main research questions that guided this study focused on how learners used the GeoGebra programme Euclidean Geometry to support their understanding and why the GeoGebra programme is used in the way that it is when learning Grade 11 Euclidean Geometry. The study is rooted within a Constructivist view of learning and mediated learning and the approach used is a case study. The research was carried out in a public school that involved 16 learners. Data was generated by using tasks, lesson observations and interviews. Based on a qualitative analysis of the data generated, the findings indicate that the introduction of the GeoGebra programme did have an influence on the learning practice in three dimensions, namely: (1) the GeoGebra programme provided a medium for visualisation that linked the development of mathematical ideas and concepts through computer-based learning, (2) the GeoGebra programme created an independent constructive learning environment and (3) the utilisation of the GeoGebra programme as a learning tool enhanced learners’ conceptual understanding of Euclidean Geometry understanding

The Effects of the Geometer\u27s Sketchpad Software on Achievement of Geometric Knowledge of High School Geometry Students

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Effects of Dynamic Geometry Software on Secondary Students’ Understanding of Geometry Concepts

The purpose of this study was to determine the effects of dynamic geometry software (DGS) on secondary students’ understanding of geometry concepts. This paper examines how dynamic geometry software (DGS) is used in the secondary mathematics classroom and its effects on student understanding and achievement through a review of the literature. The use of technology in mathematics education is well documented and its effects are noted within this paper. The focus of the literature review in Chapter 2 includes: 1) how the computer technology is used in the classroom; 2) the use of DGS to promote student understanding of inductive and deductive reasoning; 3) creation of proof and the discovery of geometric theorems; and 4). the role of the software on student performance. In Chapter 3 I conclude by reviewing the research findings and giving recommendations

Learning-Assisted Automated Reasoning with Flyspeck

The considerable mathematical knowledge encoded by the Flyspeck project is combined with external automated theorem provers (ATPs) and machine-learning premise selection methods trained on the proofs, producing an AI system capable of answering a wide range of mathematical queries automatically. The performance of this architecture is evaluated in a bootstrapping scenario emulating the development of Flyspeck from axioms to the last theorem, each time using only the previous theorems and proofs. It is shown that 39% of the 14185 theorems could be proved in a push-button mode (without any high-level advice and user interaction) in 30 seconds of real time on a fourteen-CPU workstation. The necessary work involves: (i) an implementation of sound translations of the HOL Light logic to ATP formalisms: untyped first-order, polymorphic typed first-order, and typed higher-order, (ii) export of the dependency information from HOL Light and ATP proofs for the machine learners, and (iii) choice of suitable representations and methods for learning from previous proofs, and their integration as advisors with HOL Light. This work is described and discussed here, and an initial analysis of the body of proofs that were found fully automatically is provided

The technological mediation of mathematics and its learning

This paper examines the extent to which mathematical knowledge, and its related pedagogy, is inextricably linked to the tools – physical, virtual, cultural – in which it is expressed. Our goal is to focus on a few exemplars of computational tools, and to describe with some illustrative examples, how mathematical meanings are shaped by their use. We begin with an appraisal of the role of digital technologies, and our rationale for focusing on them. We present four categories of digital tool-use that distinguish their differing potential to shape mathematical cognition. The four categories are: i. dynamic and graphical tools, ii. tools that outsource processing power, iii. new representational infrastructures, and iv. the implications of highbandwidth connectivity on the nature of mathematics activity. In conclusion, we draw out the implications of this analysis for mathematical epistemology and the mathematical meanings students develop. We also underline the central importance of design, both of the tools themselves and the activities in which they are embedded

Dagstuhl News January - December 2005

"Dagstuhl News" is a publication edited especially for the members of the Foundation "Informatikzentrum Schloss Dagstuhl" to thank them for their support. The News give a summary of the scientific work being done in Dagstuhl. Each Dagstuhl Seminar is presented by a small abstract describing the contents and scientific highlights of the seminar as well as the perspectives or challenges of the research topic

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

Proof and Proving in Mathematics Education

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