Specifying Rewrite Strategies for Interactive Exercises

Abstract

Strategies specify how a wide range of exercises can be solved incrementally, such as bringing a logic proposition to disjunctive normal form, reducing a matrix, or calculating with fractions. In this paper we introduce a language for specifying strategies for solving exercises. This language makes it easier to automatically calculate feedback, for example when a user makes an erroneous step in a calculation. We can automatically generate worked-out examples, track the progress of a student by inspecting submitted intermediate answers, and report back suggestions in case the student deviates from the strategy. Thus it becomes less labor-intensive and less adhoc to specify new exercise domains and exercises within that domain. A strategy describes valid sequences of rewrite rules, which turns tracking intermediate steps into a parsing problem. This is a promising view at interactive exercises because it allows us to take advantage of many years of experience in parsing sentences of context-free languages, and transfer this knowledge and technology to the domain of stepwise solving exercises. In this paper we work out the similarities between parsing and solving exercises incrementally, we discuss generating feedback on strategies, and the implementation of a strategy recognizer.

An Extensible Domain-Specific Language for Describing Problem-Solving Procedures

“…From these traces, alternative next steps can be calculated, and worked-out solutions can be constructed. The technical details can be found elsewhere [10,9].…”

Heuristic Search over Program Transformations

“…Nevertheless, Icarus retains the overall flavour of a production system by following a recognize-act-cycle. A more radical approach to separate logic (domain-specific rules) from control (strategic guidance) is proposed by Heeren et al [5]. They separate: (i) information about the domain (e.g., the subtraction matrix and its place-value system), (ii) rules for manipulating expressions in this domain (e.g., performing a complete borrow-payback operation, or taking the difference in a column), (iii) a strategy for solving the exercise (e.g., performing subtraction from right to the left), and (iv) buggy knowledge for modeling both incorrect expression manipulations and incorrect strategies.…”

Adapting Mathematical Domain Reasoners

“…We have defined such a set of services around rewrite strategies [13,10], which produce step-wise solutions for exercises. With a strategy we can produce worked-out examples (the derivation service), suggest a next step (the allfirsts service), and diagnose a term submitted by a learner (the diagnose service).…”

“…A rewrite strategy [13] guides the process of applying rewrite rules to solve a particular class of problems. Recipes for solving a certain type of problem can be found in textbooks, but they are often not precise enough for the purpose of building a domain reasoner.…”

“…An implicit rule behaves normally, except that it does not show up as a step in a derivation. Implicit rules can be used to perform certain simplification steps automatically, and are comparable to so-called administrative rules [13]. The inverse of hide is the reveal transformation.…”