Marvin R. G. Schiller

DiaWOz-II: a tool for wizard-of-Oz experiments in mathematics

We present DiaWOz-II, a configurable software environment for Wizard-of-Oz studies in mathematics and engineering. Its interface is based on a structural wysiwyg editor which allows the input of complex mathematical formulae. This allows the collection of dialog corpora consisting of natural language interleaved with non-trivial mathematical expressions, which is not offered by other Wizard-of-Oz tools in the field. We illustrate the application of DiaWOz-II in an empirical study on tutorial dialogs about mathematical proofs, summarize our experience with DiaWOz-II and briefly present some preliminary observations on the collected dialogs.

Proof Step Analysis for Proof Tutoring -- A Learning Approach to Granularity

We present a proof step diagnosis module based on the mathematical assistant system Ωmega. The task of this module is to evaluate proof steps as typically uttered by students in tutoring sessions on mathematical proofs. In particular, we categorise the step size of proof steps performed by the student, in order to recognise if they are appropriate with respect to the student model. We propose an approach which builds on reconstructions of the proof in question via automated proof search using a cognitively motivated proof calculus. Our approach employs learning techniques and incorporates a student model, and our diagnosis module can be adjusted to different domains and users. We present a first evaluation based on empirical data.

Deep Inference for Automated Proof Tutoring

i¾?MEGA [7], a mathematical assistant environment comprising an interactive proof assistant, a proof planner, a structured knowledge base, a graphical user interface, access to external reasoners, etc., is being developed since the early 90s at Saarland University. Similar to HOL4, Isabelle/HOL, Coq, or Mizar, the overall goal of the project is to develop a system platform for formal methods (not only) in mathematics and computer science. In i¾?MEGA, user and system interact in order to produce verifiable and trusted proofs. By continously improving (not only) automation and interaction support in the system we want to ease the usually very tedious formalization and proving task for the user.

A comparison between cognitive and AI models of blackjack strategy learning

Cognitive models of blackjack playing are presented and investigated. Blackjack playing is considered a useful test case for theories on human learning. Curiously, despite the existence of a relatively simple, well-known and optimal strategy for blackjack, empirical studies have found that casino players play quite differently from that strategy. The computational models presented here attempt to explain this result by modelling blackjack playing using the cognitive architecture CHREST. Two approaches to modeling are investigated and compared; (i) the combination of classical and operant conditioning, as studied in psychology, and (ii) SARSA, as studied in AI.

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.

Resource-Bounded Modelling and Analysis of Human-Level Interactive Proofs

Mathematics is the lingua franca of modern science, not least because of its conciseness and abstractive power. The ability to prove mathematical theorems is a key prerequisite in many fields of modern science, and the training of how to do proofs therefore plays a major part in the education of students in these subjects. Computer-supported learning is an increasingly important form of study since it allows for independent learning and individualised instruction.

Presenting proofs with adapted granularity

When mathematicians present proofs they usually adapt their explanations to their didactic goals and to the (assumed) knowledge of their addressees. Modern automated theorem provers, in contrast, present proofs usually at a fixed level of detail (also called granularity). Often these presentations are neither intended nor suitable for human use. A challenge therefore is to develop user- and goal-adaptive proof presentation techniques that obey common mathematical practice. We present a flexible and adaptive approach to proof presentation based on classification. Expert knowledge for the classification task can be hand-authored or extracted from annotated proof examples via machine learning techniques. The obtained models are employed for the automated generation of further proofs at an adapted level of granularity.

Proof Granularity as an Empirical Problem

Even in introductory textbooks on mathematical proof, intermediate proof steps are generally skipped when this seems appropriate. This gives rise to different granularities of proofs, depending on the intended audience and the context in which the proof is presented. We have developed a mechanism to classify whether proof steps of different sizes are appropriate in a tutoring context. The necessary knowledge is learnt from expert tutors via standard machine learning techniques from annotated examples. We discuss the ongoing evaluation of our approach via empirical studies.

Cognitive Models of Gambling and Problem Gambling

Current research paints the picture of problem gambling as a multifaceted phenomenon, for which there is not one single explanation. A wealth of factors are implied in the development and maintenance of problem gambling, including biological mechanisms of rewardprocessing (e.g. Linnet et al., 2010a), cognitive processes of attention (e.g. Brevers et al., 2011), implicit memory (e.g. McCusker & Gettings, 1997), decision-making (e.g. Brevers et al., 2013) and beliefs (e.g. Myrseth et al., 2010), mechanisms underlying mood regulation (Brown et al., 2004) and coping styles (e.g. Gupta et al., 2004). Individual factors are thought to interact with the gambling environment and the larger social, professional and familial environment, adding to the complexity. Integrated models of problem gambling, such as the pathways model of Blaszczynski and Nower (2002), attempt to (re-)establish a holistic view in a research field that resorts to increasingly specific and intricate research designs. The underlying mechanisms and their interactions, however, are still not well understood (Gobet & Schiller, 2011).