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

Instructional strategies in explicating the discovery function of proof for lower secondary school students

In this paper, we report on the analysis of teaching episodes selected from our pedagogical and cognitive research on geometry teaching that illustrate how carefully-chosen instructional strategies can guide Grade 8 students to see and appreciate the discovery function of proof in geometr

LAI 531 Curriculum Prospectus: Fundamentals of Logic, Reasoning, and Argumentation for Public Engagement with Science

LAI 531 Curriculum Prospectus: Fundamentals of Logic, Reasoning, and Argumentation for Public Engagement with Scienc

Designing Normative Theories for Ethical and Legal Reasoning: LogiKEy Framework, Methodology, and Tool Support

A framework and methodology---termed LogiKEy---for the design and engineering of ethical reasoners, normative theories and deontic logics is presented. The overall motivation is the development of suitable means for the control and governance of intelligent autonomous systems. LogiKEy's unifying formal framework is based on semantical embeddings of deontic logics, logic combinations and ethico-legal domain theories in expressive classic higher-order logic (HOL). This meta-logical approach enables the provision of powerful tool support in LogiKEy: off-the-shelf theorem provers and model finders for HOL are assisting the LogiKEy designer of ethical intelligent agents to flexibly experiment with underlying logics and their combinations, with ethico-legal domain theories, and with concrete examples---all at the same time. Continuous improvements of these off-the-shelf provers, without further ado, leverage the reasoning performance in LogiKEy. Case studies, in which the LogiKEy framework and methodology has been applied and tested, give evidence that HOL's undecidability often does not hinder efficient experimentation.Comment: 50 pages; 10 figure

Studying disciplinary corpora to teach the craft of discussion

Producing publishable quality research articles is a difficult task for novice scholarly writers. Particularly challenging is writing the Discussion/Conclusion section, which requires taking evaluative and interpretive stances on obtained results and substantiating claims regarding the worth of the scholarly contribution of the article to scientific knowledge. Conforming to the expectations of the target disciplinary community adds another dimension to the challenge. Corpus-based genre analysis can foster postgraduate writing instruction by providing insightful descriptions of rhetorical patterns and variation in disciplinary discourse. This paper introduces a pedagogically-oriented cross-disciplinary model of moves and steps devised through top-down corpus analysis. The model was applied to pedagogical materials and tasks designed to enhance genre and corpus-based teaching of Discussion/ Conclusions with an explicit focus on rhetorical conventions

Attaining Scientific Literacy through a Rhetoric of Science Composition Pedagogy

For the past sixty years, acquiring scientific literacy has proven to be a daunting task in education, especially for undergraduate nonscience majors. Although some education scholars have recognized the importance and use of arguments to teach science, these pedagogical practices often are aimed at primary school children rather than college students or involve reporting science experiments rather than actually studying or constructing arguments about issues related to science.However, in this dissertation, I contend that educators often neglect a more available tool that not only examines arguments concerning scientific issues but also demonstrates the very heart of scientific literacy: critical thinking. In this project, I argue that a rhetoric of science course that teaches undergraduate nonscience majors to assess and engage the rhetorical components of scientific arguments provides a more pedagogically sound means of helping these students attain scientific literacy than the course designs in popular Introduction to Biology textbooks and their related course syllabi. In order to support this claim, I define and focus on the relationship of five main terms throughout this project as they pertain to the teaching of scientific literacy: science, scientific literacy, critical thinking, argumentation, and rhetoric. Science is a rhetorical practice through argumentation, and as I explain, argumentation fulfills the process of critical thinking, utilizing the analysis, evaluation, and creation of arguments. Applying an Aristotelian understanding of the term, rhetoric involves both the employment and discernment of the means of persuasion. From these terms, I establish that the attainment of scientific literacy is a rhetorical endeavor that necessitates the use of rhetoric of science, which is the analysis, evaluation, and creation of scientific arguments through composition.Examining three major Introduction to Biology textbooks and nine related course syllabi, I demonstrate that these courses generally emphasize learning facts rather than how to think critically about issues pertaining to science, and any writing required in these courses do not include composing arguments concerning issues related to science. In contrast, I demonstrate how a rhetoric of science that uses rhetorical criticism can be used to help nonscience majors attain scientific literacy as I define the term

Notes for a study of the didactic transposition of mathematical proof

It is nowadays common to consider that proof must be part of the learning of mathematics from Kindergarten to University1. As it is easy to observe, looking back to the history of mathematical curricula, this has not always been the case either because following an old pedagogical tradition of rote learning proof was reduced to the formalism of a text and deprived from its meaning or, despiteits acknowledged presence anywhere in mathematics, proof did not get the status of something to learn for what it is. On the long way from its absence as such in the past to its contemporary presence as a content to be taught at all grades, proof has had to go through a process of didactical transposition to satisfy a number of different constraints either of an epistemic, didactical, logical ormathematical nature. I will follow a chronological order to outline the main features of this process with the objective to better understand the didactical problem that our current research is facing.Comment: ISSN 1465-2978 (online). Philosophy of Mathematics Education Journal, In pres

Trialing project-based learning in a new EAP ESP course: A collaborative reflective practice of three college English teachers

Currently in many Chinese universities, the traditional College English course is facing the risk of being ‘marginalized’, replaced or even removed, and many hours previously allocated to the course are now being taken by EAP or ESP. At X University in northern China, a curriculum reform as such is taking place, as a result of which a new course has been created called ‘xue ke’ English. Despite the fact that ‘xue ke’ means subject literally, the course designer has made it clear that subject content is not the target, nor is the course the same as EAP or ESP. This curriculum initiative, while possibly having been justified with a rationale of some kind (e.g. to meet with changing social and/or academic needs of students and/or institutions), this is posing a great challenge for, as well as considerable pressure on, a number of College English teachers who have taught this single course for almost their entire teaching career. In such a context, three teachers formed a peer support group in Semester One this year, to work collaboratively co-tackling the challenge, and they chose Project-Based Learning (PBL) for the new course. This presentation will report on the implementation of this project, including the overall designing, operational procedure, and the teachers’ reflections. Based on discussion, pre-agreement was reached on the purpose and manner of collaboration as offering peer support for more effective teaching and learning and fulfilling and pleasant professional development. A WeChat group was set up as the chief platform for messaging, idea-sharing, and resource-exchanging. Physical meetings were supplementary, with sound agenda but flexible time, and venues. Mosoteach cloud class (lan mo yun ban ke) was established as a tool for virtual learning, employed both in and after class. Discussions were held at the beginning of the semester which determined only brief outlines for PBL implementation and allowed space for everyone to autonomously explore in their own way. Constant further discussions followed, which generated a great deal of opportunities for peer learning and lesson plan modifications. A reflective journal, in a greater or lesser detailed manner, was also kept by each teacher to record the journey of the collaboration. At the end of the semester, it was commonly recognized that, although challenges existed, the collaboration was overall a success and they were all willing to continue with it and endeavor to refine it to be a more professional and productive approach