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

Tools for Tutoring Theoretical Computer Science Topics

This thesis introduces COMPLEXITY TUTOR, a tutoring system to assist in learning abstract proof-based topics, which has been specifically targeted towards the population of computer science students studying theoretical computer science. Existing literature has shown tremendous educational benefits produced by active learning techniques, student-centered pedagogy, gamification and intelligent tutoring systems. However, previously, there had been almost no research on adapting these ideas to the domain of theoretical computer science. As a population, computer science students receive immediate feedback from compilers and debuggers, but receive no similar level of guidance for theoretical coursework. One hypothesis of this thesis is that immediate feedback while working on theoretical problems would be particularly well-received by students, and this hypothesis has been supported by the feedback of students who used the system. This thesis makes several contributions to the field. It provides assistance for teaching proof construction in theoretical computer science. A second contribution is a framework that can be readily adapted to many other domains with abstract mathematical content. Exercises can be constructed in natural language and instructors with limited programming knowledge can quickly develop new subject material for COMPLEXITY TUTOR. A third contribution is a platform for writing algorithms in Python code that has been integrated into this framework, for constructive proofs in computer science. A fourth contribution is development of an interactive environment that uses a novel graphical puzzle-like platform and gamification ideas to teach proof concepts. The learning curve for students is reduced, in comparison to other systems that use a formal language or complex interface. A multi-semester evaluation of 101 computer science students using COMPLEXITY TUTOR was conducted. An additional 98 students participated in the study as part of control groups. COMPLEXITY TUTOR was used to help students learn the topics of NP-completeness in algorithms classes and prepositional logic proofs in discrete math classes. Since this is the first significant study of using a computerized tutoring system in theoretical computer science, results from the study not only provide evidence to support the suitability of using tutoring systems in theoretical computer science, but also provide insights for future research directions

A Formative Evaluation Research Study to Guide the Design of the Categorization Step Practice Utility (MS-CPU) as an Integral Part of Preparation for the GED Mathematics Test Using the Ms. Stephens Algebra Story Problem-solving Tutor (MSASPT)

abstract: The mathematics test is the most difficult test in the GED (General Education Development) Test battery, largely due to the presence of story problems. Raising performance levels of story problem-solving would have a significant effect on GED Test passage rates. The subject of this formative research study is Ms. Stephens’ Categorization Practice Utility (MS-CPU), an example-tracing intelligent tutoring system that serves as practice for the first step (problem categorization) in a larger comprehensive story problem-solving pedagogy that purports to raise the level of story problem-solving performance. During the analysis phase of this project, knowledge components and particular competencies that enable learning (schema building) were identified. During the development phase, a tutoring system was designed and implemented that algorithmically teaches these competencies to the student with graphical, interactive, and animated utilities. Because the tutoring system provides a much more concrete rather than conceptual, learning environment, it should foster a much greater apprehension of a story problem-solving process. With this experience, the student should begin to recognize the generalizability of concrete operations that accomplish particular story problem-solving goals and begin to build conceptual knowledge and a more conceptual approach to the task. During the formative evaluation phase, qualitative methods were used to identify obstacles in the MS-CPU user interface and disconnections in the pedagogy that impede learning story problem categorization and solution preparation. The study was conducted over two iterations where identification of obstacles and change plans (mitigations) produced a qualitative data table used to modify the first version systems (MS-CPU 1.1). Mitigation corrections produced the second version of the MS-CPU 1.2, and the next iteration of the study was conducted producing a second set of obstacle/mitigation tables. Pre-posttests were conducted in each iteration to provide corroboration for the effectiveness of the mitigations that were performed. The study resulted in the identification of a number of learning obstacles in the first version of the MS-CPU 1.1. Their mitigation produced a second version of the MS-CPU 1.2 whose identified obstacles were much less than the first version. It was determined that an additional iteration is needed before more quantitative research is conducted.Dissertation/ThesisDoctoral Dissertation Educational Technology 201

Mathematics in Undergraduate Study Programs: Challenges for Research and for the Dialogue between Mathematics and Didactics of Mathematics

The topic of undergraduate mathematics is of considerable concern for mathematicians in universities, but also for those teaching mathematics as part of undergraduate studies other than mathematics, for employers seeking to employ a mathematically skilled workforce, and for teacher education. Different countries have made and continue to make massive efforts to improve the quality of mathematics education across all age ranges, with most of the research undertaken particularly at the school level. A growing number of mathematicians and mathematics educators now see the need for undertaking interdisciplinary research and collaborative reflections around issues at the tertiary level. The conference aimed to share research results and experiences as a background to establishing a scientific community of mathematicians and mathematics educators whose concern is the theoretical reflection, the research-based empirical investigation, and the exchange of best-practice examples of mathematics education at the tertiary level. The focus of the conference was mathematics education for mathematics, engineering and economy majors and for future mathematics teachers