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

MENON : automating a Socratic teaching model for mathematical proofs

This thesis presents an approach to adaptive pedagogical feedback for arbitrary domains as an alternative to resource-intensive pre-compiled feedback, which represents the state-of-the-art in intelligent tutoring systems today. A consequence of automatic adaptive feedback is that the number of tasks with pedagogical feedback that can be offered to the student increases, and with it the opportunity for practice. We focus on automating different aspects of teaching that together are primarily responsible for learning and can be integrated in a unified natural-language output. The automatic production and natural-language generation of feedback enables its personalisation both at the pedagogical and the natural-language dialogue level. We propose a method for automating the production of domain-independent adaptive feedback. The proof- of-concept implementation of the tutorial manager Menon is carried out for the domain of set-theory proofs. More specifically, we define a pedagogical model that abides by schema and cognitive load theory, and by the synergistic approach to learning. We implement this model in a Socratic teaching strategy whose basic units of feedback are dialogue moves. We use empirical data from two domains to derive a taxonomy of tutorial-dialogue moves, and define the most central and sophisticated move hint. The formalisation of the cognitive content of hints is inspired by schema theory and is facilitated by a domain ontology.Die vorliegende Arbeit präsentiert eine Annäherung an adaptives pädagogisches Feedback für beliebige Domäne. Diese Herangehensweise bietet eine Alternative zu ressource-intensivem, vorübersetztem Feedback, dass das heutige "state-of-the-art'; in intelligenten tutoriellen Systemen ist. Als Folge können zahlreiche Aufgaben mit pädagogischem Feedback für die Praxis angeboten werden. Der Schwerpunkt der Arbeit liegt auf der Automatisierung verschiedener Aspekte des Lehrprozesses, die in ihrer Gesamtheit wesentlich den Lernprozess beeinflussen, und in einer einheitlichen Systemausgabe Natürlicher Sprache integriert werden können. Die automatische Produktion und die Systemgenerierung von Feedback in Natürlicher Sprache ermöglichen eine Individualisierung des Feedback auf zwei Ebenen: einer pädagogischen und einer dialogischen Ebene. Dazu schlagen wir eine Methode vor, durch die adaptives Feedback automatisiert werden kann, und implementieren den tutoriellen Manager Menon als "proof-of-concept'; beispielhaft für die Domäne von Beweisen in der Mengentheorie. Konkret definieren wir ein pädagogisches Modell, das sich auf Schema- und Kognitionstheorie sowie auf die synergetische Herangehensweise an Lernen stützt. Dieses Modell wird in einer Sokratischen Lehrmethode implementiert, deren basale Feedback-Elemente aus Dialogakten bestehen. Zur Bestimmung einer Taxonomie Tutorielle-Dialogakte sowie des zentralen und komplexen Dialogakts hint (Hinweis) wenden wir empirische Daten aus zwei Domänen an. Die Formalisierung des kognitiven Inhaltes von Hinweisen folgt der Schematheorie und basiert auf einer Domänenontologie

Advances in Intelligent Tutoring Systems: Problem-solving Modes and Model of Hints

The paper focuses on the issues of providing an adaptive support for learners in intelligent tutoring systems when learners solve practical problems. The results of the analysis of support policies of learners in the existing intelligent tutoring systems are given and the revealed problems are emphasized. The concept and the architectural parts of an intelligent tutoring system are defined. The approach which provides greater adaptive abilities of systems of such kind offering two modes of problem-solving and using a two-layer model of hints is described. It is being implemented in the intelligent tutoring system for the Minimax algorithm at present. In accordance with the proposed approach the learner solves problems in the mode which is the most appropriate for him/her and receives the most suitable hint

SUBTLE CUES AND HIDDEN ASSUMPTIONS: AN ACTION RESEARCH STUDY OF TEACHER QUESTIONING PATTERNS IN 7TH AND 8TH GRADE MATHEMATICS CLASSROOMS

This action research project explores the link between a teacher's questioning patterns and the modes of thinking, analyzing, evaluating and communicating that are developed in his 7th and 8th grade math students. The highly qualitative analysis focuses on three videotaped lessons from his 7th and 8th grade classrooms, and evaluates the lessons according to four categories or "lenses": cognitive demand, task completion, self-efficacy, and metacognitive activity. It then seeks to identify and codify the predominant questioning pattern used in each lesson, and connect this pattern to the levels of success exhibited in each of the four categories. Four principal patterns are observed and discussed in the lessons: Unilateral Inquiry Response Evaluation, Multilateral Inquiry Response Evaluation, Inquiry Response Collection, and Inquiry Response Revoicing Controversy. The fourth pattern is proposed as a tool for managing classroom discourse that involves a variety of (sometimes competing) student opinions

Observational Equivalence and Full Abstraction in the Symmetric Interaction Combinators

The symmetric interaction combinators are an equally expressive variant of Lafont's interaction combinators. They are a graph-rewriting model of deterministic computation. We define two notions of observational equivalence for them, analogous to normal form and head normal form equivalence in the lambda-calculus. Then, we prove a full abstraction result for each of the two equivalences. This is obtained by interpreting nets as certain subsets of the Cantor space, called edifices, which play the same role as Boehm trees in the theory of the lambda-calculus

A domain reasoner for geometry exercises

With advances in ICT around the world, digital tutors are an increasingly attractive option to provide education to a large audience inexpensively. Important components of a digital tutor include the exercises or a method to generate exercises, an algorithm for finding and verifying solutions to exercises and a method of providing hints to a student while the student is working on an exercise. The domain reasoner models all paths from the proposition(s) of an exercise to the solution(s). This enables the digital tutor to provide hints from any situation the student might encounter. This thesis contributes a method to generate a domain reasoner from an exercise solution in the domain of high school level geometry exercises. The program representing a solution to an exercise is first represented as a directed acyclic graph of which the edges represent steps in the solution. Each edge uses a formal rule to execute a step and human-readable hints are attached to these rules. Since an algorithm for generating solutions from the formal description of an exercises currently exists, this domain reasoner enables the generation of hints from just that formal description of an exercise. The exercise-specific domain reasoner enables feedforward, feedback and worked-out examples as well as a degree of adaptivity to a student's knowledge by providing hints for steps composed of smaller steps

The Rationality of College Mathematics Instructors: The Choice to Use Inquiry-Oriented Instruction

This study of inquiry-oriented instruction (IOI) explores what inquiry-oriented practices are used by college mathematics instructors, and what relationships there are between their use of those practices, their beliefs about students’ mathematics learning, and their recognition of professional obligations. I offer a conceptualization of inquiry-oriented instruction in which IOI practices documented in the literature are organized by the theory of the instructional triangle (Cohen, Raudenbush, & Ball, 2003), which pays particular attention to instruction as transactions of content between teacher and students. The INQUiry-Oriented Instructor REview (INQUIRE) instrument was developed on this conceptualization and used to gather data on the frequency that instructors report using inquiry-oriented practices. Professional obligations of mathematics teaching include the responsibilities that instructors have towards various stakeholders, including the institution, the individual student, mathematics as a discipline, and society (Herbst & Chazan, 2012) and instructors’ recognition of these obligations was hypothesized as playing a role in explaining the use of IOI practices. A modified version of the PRofessional Obligations Scenario Evaluation (PROSE) instrument, a scenario-based assessment, was created for this study to be used with college mathematics instructors. In addition to developing the INQUIRE and PROSE instruments, this study incorporated an existing beliefs instrument (Clark et al., 2014) to measure instructors’ beliefs on students’ mathematics learning. I used factor analyses to confirm the hypothesized inquiry-oriented practices in the instructional triangle framework and hierarchical cluster modeling to reveal patterns of inquiry-oriented practices among instructors. I found that instructors reported using seven distinct sets of practices, and instructors grouped into four different clusters based on their pattern of use of these practices – revealing different characterizations of IOI. This finding has implications for future research of IOI, showing that characterizing IOI as a singular pedagogy is problematic; rather, there are different types of IOI that are grounded in content-specific interactions. The first cluster includes participants that report the highest use of teacher-student and student-student interactions, but not the highest use of the student-content practices of giving students opportunities to construct and critique claims or write proofs. The second cluster includes participants that report the highest use of the aforementioned student-content practices, and second highest use of all five other inquiry-oriented practices. The third cluster included participants that reported low use of all inquiry-oriented practices except the teacher-content ones of interactive lecture and hinting without telling, which they use at levels comparable to other clusters. These three clusters or characterizations of IOI are all juxtaposed against the fourth cluster, which had the lowest reported use of all seven practices. I used structural equation modeling to explore the hypothesized relationships. Past studies have reported inconsistencies between beliefs and practice; instructors’ degree of recognition of the professional obligations helped explain why instructors may not always actualize their beliefs in the classroom. I found that learner-focused beliefs often predict the use of inquiry-oriented practices, but recognition of the disciplinary and interpersonal obligations can work in direct opposition of those beliefs – helping to explain why instructors sometimes do not instruct with IOI even if they believe it would be beneficial. These findings have practical implications for those wishing to shift trends in college mathematics instruction. Future work could use the INQUIRE instrument to link inquiry-oriented practices to student experiences.PHDEducational StudiesUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/155118/1/mollee_1.pd

A study of the classification of low-dimensional data with supervised manifold learning

Supervised manifold learning methods learn data representations by preserving the geometric structure of data while enhancing the separation between data samples from different classes. In this work, we propose a theoretical study of supervised manifold learning for classification. We consider nonlinear dimensionality reduction algorithms that yield linearly separable embeddings of training data and present generalization bounds for this type of algorithms. A necessary condition for satisfactory generalization performance is that the embedding allow the construction of a sufficiently regular interpolation function in relation with the separation margin of the embedding. We show that for supervised embeddings satisfying this condition, the classification error decays at an exponential rate with the number of training samples. Finally, we examine the separability of supervised nonlinear embeddings that aim to preserve the low-dimensional geometric structure of data based on graph representations. The proposed analysis is supported by experiments on several real data sets