Verification of Proof Steps for Tutoring Mathematical Proofs

The feedback given by human tutors is strongly based on their evaluation of the correctness of student’s contributions [7]. A typical approach to verification of contributions in ITSs is model tracing, in which contributions are matched against a precomputed solution graph [1,3]. However verifying steps in mathematical theorem proving poses particular problems because there are possibly infinitely many acceptable proofs, steps may be only partially ordered, and the student should be free to build any correct solution. Tracing contributions against static solution graphs is therefore overly restrictive for exercises in typically sized mathematical theories. We propose a flexible domain-independent method of verifying proof steps of varying lengths on-the-fly for a mathematics tutoring system. Our approach can verify steps in unforeseen proofs and incrementally builds a solution state for each possible interpretation of underspecified or ambiguous steps. We utilise an existing mathematical reasoner ΩMEGA [8] and existing mathematical knowledge. Our approach to the verification of proof steps is motivated by our corpus of Wizard-of-Oz tutorial dialogues between students and experienced mathematics teachers [4]. In a proof of the theorem (R∪S)◦T = (R◦T )∪ (S ◦T ), the student’s first proof step “Let (x, y) ∈ (R∪S)◦T ” consists of two definitions: set extensionality and subset. The tutor responds with “Correct!”. This shows the tutor must know the correctness of steps to apply pedagogical strategies and that proof steps can contain applications of many mathematical definitions.