1,439 research outputs found

    Extracting proofs from documents

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    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..

    Mathematical Simulations in Topology and Their Role in Mathematics Education

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    This thesis presents and discusses several software projects related to the learning of mathematics in general and topological concepts in particular, collecting the results from several publications in this field. It approaches mathematics education by construction of mathematical learning environments, which can be used for the learning of mathematics, as well as by contributing insights gained during the development and use of these learning environments. It should be noted that the presented software environments were not built for the use in schools or other settings, but to provide proofs of concepts and to act as a basis for research into mathematics and its education and communication. The first developed and analyzed environment is Ariadne, a software for the interactive visualization of dots, paths, and homotopies of paths. Ariadne is used as an example of a “mathematical simulation”, capable of supporting argumentation in a way that may be characterized as proving. The software was extended from two to three dimensions, making possible the investigation of two-dimensional manifolds, such as the torus or the sphere, using virtual reality. Another extension, KnotPortal, enables the exploration of three-dimensional manifolds represented as branched covers of knots, after an idea by Bill Thurston to portray these branched covers of knots as knotted portals between worlds. This software was the motivation for and was used in an investigation into embodied mathematics learning, as this virtual reality environment challenges users to determine the structure of the covering by moving their body. Also presented are some unpublished projects that were not completed during the doctorate. This includes work on concept images in topology as well as software for various purposes. One such software was intended for the construction of closed orientable surfaces, while another was focused on the interactive visualization of the uniformization theorem. The thesis concludes with a meta-discussion on the role of design in mathematics education research. While design plays an important role in mathematics education, designing seems to not to be recognized as research in itself, but only as part of theory building or, in most cases, an empirical study. The presented argumentation challenges this view and points out the dangers and obstacles involved

    Foundations of Trusted Autonomy

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    Trusted Autonomy; Automation Technology; Autonomous Systems; Self-Governance; Trusted Autonomous Systems; Design of Algorithms and Methodologie

    Inquiry in University Mathematics Teaching and Learning. The Platinum Project

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    The book presents developmental outcomes from an EU Erasmus+ project involving eight partner universities in seven countries in Europe. Its focus is the development of mathematics teaching and learning at university level to enhance the learning of mathematics by university students. Its theoretical focus is inquiry-based teaching and learning. It bases all activity on a three-layer model of inquiry: (1) Inquiry in mathematics and in the learning of mathematics in lecture, tutorial, seminar or workshop, involving students and teachers; (2) Inquiry in mathematics teaching involving teachers exploring and developing their own practices in teaching mathematics; (3) Inquiry as a research process, analysing data from layers (1) and (2) to advance knowledge inthe field. As required by the Erasmus+ programme, it defines Intellectual Outputs (IOs) that will develop in the project. PLATINUM has six IOs: The Inquiry-based developmental model; Inquiry communities in mathematics learning and teaching; Design of mathematics tasks and teaching units; Inquiry-based professional development activity; Modelling as an inquiry process; Evalutation of inquiry activity with students. The project has developed Inquiry Communities, in each of the partner groups, in which mathematicians and educators work together in supportive collegial ways to promote inquiry processes in mathematics learning and teaching. Through involving students in inquiry activities, PLATINUM aims to encourage students‘ own in-depth engagement with mathematics, so that they develop conceptual understandings which go beyond memorisation and the use of procedures. Indeed the eight partners together have formed an inquiry community, working together to achieve PLATINUM goals within the specific environments of their own institutions and cultures. Together we learn from what we are able to achieve with respect to both common goals and diverse environments, bringing a richness of experience and learning to this important area of education. Inquiry communities enable participants to address the tensions and issues that emerge in developmental processes and to recognise the critical nature of the developmental process. Through engaging in inquiry-based development, partners are enabled and motivated to design activities for their peers, and for newcomers to university teaching of mathematics, to encourage their participation in new forms of teaching, design of teaching, and activities for students. Such professional development design is an important outcome of PLATINUM. One important area of inquiry-based activity is that of „modelling“ in mathematics. Partners have worked together across the project to investigate the nature of modelling activities and their use with students. Overall, the project evaluates its activity in these various parts to gain insights to the sucess of inquiry based teaching, learning and development as well as the issues and tensions that are faced in putting into practice its aims and goals

    Inquiry in University Mathematics Teaching and Learning

    Get PDF
    The book presents developmental outcomes from an EU Erasmus+ project involving eight partner universities in seven countries in Europe. Its focus is the development of mathematics teaching and learning at university level to enhance the learning of mathematics by university students. Its theoretical focus is inquiry-based teaching and learning. It bases all activity on a three-layer model of inquiry: (1) Inquiry in mathematics and in the learning of mathematics in lecture, tutorial, seminar or workshop, involving students and teachers; (2) Inquiry in mathematics teaching involving teachers exploring and developing their own practices in teaching mathematics; (3) Inquiry as a research process, analysing data from layers (1) and (2) to advance knowledge inthe field. As required by the Erasmus+ programme, it defines Intellectual Outputs (IOs) that will develop in the project. PLATINUM has six IOs: The Inquiry-based developmental model; Inquiry communities in mathematics learning and teaching; Design of mathematics tasks and teaching units; Inquiry-based professional development activity; Modelling as an inquiry process; Evalutation of inquiry activity with students. The project has developed Inquiry Communities, in each of the partner groups, in which mathematicians and educators work together in supportive collegial ways to promote inquiry processes in mathematics learning and teaching. Through involving students in inquiry activities, PLATINUM aims to encourage students` own in-depth engagement with mathematics, so that they develop conceptual understandings which go beyond memorisation and the use of procedures. Indeed the eight partners together have formed an inquiry community, working together to achieve PLATINUM goals within the specific environments of their own institutions and cultures. Together we learn from what we are able to achieve with respect to both common goals and diverse environments, bringing a richness of experience and learning to this important area of education. Inquiry communities enable participants to address the tensions and issues that emerge in developmental processes and to recognise the critical nature of the developmental process. Through engaging in inquiry-based development, partners are enabled and motivated to design activities for their peers, and for newcomers to university teaching of mathematics, to encourage their participation in new forms of teaching, design of teaching, and activities for students. Such professional development design is an important outcome of PLATINUM. One important area of inquiry-based activity is that of “modelling” in mathematics. Partners have worked together across the project to investigate the nature of modelling activities and their use with students. Overall, the project evaluates its activity in these various parts to gain insights to the sucess of inquiry based teaching, learning and development as well as the issues and tensions that are faced in putting into practice its aims and goals

    Re-examining the impact of analogies on ideation search patterns: Lessons from an in vivo study in engineering design

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    Decades of research on the cognitive science of innovation have consistently implicated the importance of analogy during creative ideation. While the association of analogies with innovative design concepts is clear, more work is needed to understand the specific mechanisms by which analogy might help designers generate such concepts. The present work employed detailed analysis of the temporal interplay between analogy use and ideation in the naturalistic brainstorming conversations of a real-world professional design team to test between competing hypotheses in the literature: (1) analogy supports innovation primarily via large steps in design spaces during concept generation (jumps), and (2) analogy supports innovation primarily via small steps (incremental search). In Study 1, self-generated analogies (including distant ones) were not systematically associated with jumps; on the contrary, concepts tended to be more similar to their precedents after analogy use in comparison to baseline situations (i.e., without analogy use). Study 2 found that the rate of concept generation was greater when associated with analogy in comparison to baseline conditions, suggesting that the effects observed in Study 1 were not due to an overall fixating effect of analogies. Overall, these results challenge the view that analogies help designers generate innovative concepts mainly via jumps in design spaces, and instead suggests that analogies primarily support incremental search. Theoretical implications and future directions for the cognitive science of analogy and innovation are discussed

    User Interaction in Deductive Interactive Program Verification

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