No abstract

No abstract

This article describes an experiment with LogEx, an e‐learning environment that supports students in learning how to prove the equivalence between two logical formulae, using standard equivalences such as DeMorgan. In the experiment, we compare two groups of students. The first group uses the complete learning environment, including hints, next steps, worked solutions, and informative timely feedback. The second group uses a version of the environment without hints or next steps, but with worked solutions, and delayed flag feedback. We use pretest and posttest to measure the performance of both groups with respect to error rate and completion of the exercises. We analyse the loggings of the student activities in the learning environment to compare its use by the different groups. Both groups score significantly better on the posttest than on the pretest. We did not find significant differences between the groups in the posttest, although the group using the full learning environment performed slightly better than the other group. In the examination, which took place 5 weeks after the experiment, the group of students who used the complete learning environment scored significantly better than a control group of students who did not participate in the experiment.

This paper describes an algorithm to generate Hilbert-style axiomatic proofs. Based on this algorithm we develop logax, a new interactive tutoring tool that provides hints and feedback to a student who stepwise constructs an axiomatic proof. We compare the generated proofs with expert and student solutions, and conclude that the quality of the generated proofs is comparable to that of expert proofs. logax recognizes most steps that students take when constructing a proof. If a student diverges from the generated solution, logax can still provide hints and feedback.

Structural induction is a proof technique that is widely used to prove statements about discrete structures. Students find it hard to construct inductive proofs, and when learning to construct such proofs, receiving feedback is important. In this paper we discuss the design of a tutoring system, LogInd, that helps students with constructing stepwise inductive proofs by providing hints, next steps and feedback. As far as we know, this is the first tutoring system for structural induction with this functionality. We explain how we use a strategy to construct proofs for a restricted class of problems. This strategy can also be used to complete partial student solutions, and hence to provide hints or next steps. We use constraints to provide feedback. A pilot evaluation with a small group of students shows that LogInd indeed can give hints and next steps in almost all cases.