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.

💡 Research Summary

This paper presents the design, implementation, and evaluation of an intelligent tutoring system (ITS) specifically aimed at teaching textbook‑style mathematical proofs. The authors begin by highlighting the unique challenges of proof instruction: learners must not only apply definitions and theorems correctly but also construct logical arguments and articulate them in natural language. Traditional ITS models—model‑tracing, constraint‑based, and cognitive‑diagnostic—are examined and found insufficient for the non‑linear, abstract nature of proof tasks. Consequently, the authors introduce a “assertion‑level” abstraction that treats each logical statement as a unit of reasoning, enabling the system to reason about partial proofs and to extend them toward complete solutions.

To ground their design in real tutoring practice, the researchers conducted a Wizard‑of‑Oz study with 20 university students, collecting a corpus of 1,200 dialogue turns. Analysis of this corpus revealed frequent learner requests for “why” and “how” explanations, the need for adaptive hint timing, and the importance of error‑specific feedback. These insights guided the construction of a hybrid ITS architecture that combines strengths from existing models while addressing proof‑specific requirements.

The core of the system is an adapted assertion‑level proof assistant originally developed for automated and interactive theorem provers (e.g., Coq, Isabelle). The proof assistant stores mathematical definitions and theorems as uniquely identified logical formulas enriched with metadata. When a learner submits a partial proof, the system’s proof‑search engine decomposes the goal into assertion‑level subgoals and applies a library of strategy templates—derived from automated theorem proving heuristics—to explore possible completions. Simultaneously, an error‑diagnosis module compares the learner’s steps with the logical structure of the goal, classifying mistakes into categories such as misuse of a definition, omitted premises, or logical discontinuities. For each category, the system generates a concise, natural‑language explanation and a context‑appropriate hint.

A dialogue management component, trained on the Wizard‑of‑Oz corpus, handles intent recognition, response selection, and state transitions. It ensures that the tutor’s interventions are timely and pedagogically sound, mirroring the behavior of a human tutor who can ask probing questions, provide scaffolding, or let the learner explore independently.

The prototype was evaluated with 30 students across five proof problems drawn from set theory, real analysis, and elementary algebra. Participants were split between the new system (ITMP) and a conventional online proof tutor lacking assertion‑level reasoning. Results showed a statistically significant improvement in success rate (78 % vs. 60 %), reduced average solution time (4.3 min vs. 5.7 min), and higher satisfaction scores (85 % positive responses regarding clarity and usefulness of feedback). System logs indicated that the error‑diagnosis module correctly identified the type of mistake in 92 % of cases, while the dialogue manager accurately inferred learner intent 88 % of the time. Moreover, the proof‑search component responded within an average of 1.2 seconds, supporting real‑time interaction.

The authors argue that reusing representations and search strategies from theorem proving not only accelerates development but also yields a more robust reasoning engine than ad‑hoc rule‑based tutors. They acknowledge limitations: the current proof library focuses on introductory textbook material, and the learner model does not yet capture long‑term knowledge growth or meta‑cognitive strategies. Future work will expand the strategy repository to handle higher‑level mathematics (e.g., topology, abstract algebra), integrate more sophisticated learner modeling for personalized feedback, and explore multimodal interfaces (speech, diagrammatic input) to enrich the tutoring experience.

In summary, the paper demonstrates that an ITS built on an assertion‑level proof assistant can effectively tutor students in constructing mathematical proofs, offering precise error diagnostics, adaptive hints, and natural‑language explanations. Empirical evaluation confirms both learning gains and user satisfaction, positioning this approach as a promising direction for computer‑supported proof education.