Abstract. The feedback given by e-learning tools that support incrementally solving problems in mathematics, logic, physics, etc. is limited, or laborious to specify. In this paper we introduce a language for specifying strategies for solving exercises. This language makes it easier to automatically calculate feedback when users make erroneous steps in a calculation. Although we need the power of a full programming language to specify strategies, we carefully distinguish between context-free and non-contextfree sublanguages of our strategy language. This separation is the key to automatically calculating all kinds of desirable feedback. 1

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Immediate feedback has a positive effect on the performance of a student practising a procedural skill in exercises. Giving feedback to a number of students is labour-intensive for a teacher. To alleviate this, many electronic exercise assistants have been developed. However, many of the exercise assistants have some limitations in the feedback they offer. We have a feedback engine that gives semantically rich feedback for several domains (like logic, linear algebra, arithmetic), and that can be relatively easy extended with new domains. Our feedback engine needs to have knowledge about the domain, how to reason with that knowledge (i.e. a set of rules), and a specified strategy. We offer the following types of feedback: correct/incorrect statements, distance to the solution, rulebased feedback, buggy rules, and strategy feedback. We offer the feedback functionality in the form of light-weight web services. These services are offered using different protocols, for example

We address the general problem of determining the validity of boolean combinations of equalities and inequalities between real-valued expressions. In particular, we consider methods of establishing such assertions using only restricted forms of distributivity. At the same time, we explore ways in which “local ” decision or heuristic procedures for fragments of the theory of the reals can be amalgamated into global ones. Let Tadd[Q] be the first-order theory of the real numbers in the language with symbols 0, 1, +, −, <,..., fa,... where for each a ∈ Q, fa denotes the function fa(x) = ax. Let Tmult[Q] be the analogous theory for the language with symbols 0, 1, ×, ÷, <,..., fa,.... We show that although T [Q] = Tadd[Q]∪Tmult[Q] is undecidable, the universal fragment of T [Q] is decidable. We also show that terms of T [Q] can fruitfully be put in a normal form. We prove analogous results for theories in which Q is replaced, more generally, by suitable subfields F of the reals. Finally, we consider practical methods of establishing quantifier-free validities that approximate our (impractical) decidability results. 1

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 ad-hoc 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.

We use lters of open sets to provide a semantics justifying the use of in nity in informal limit calculations in calculus, and in the same kind of calculations in computer algebra. We compare the behavior of these lters to the way Mathematica behaves when calculating with in nity.

Abstract Exercises in mathematics are often solved using a standard procedure, such as for example solving a system of linear equations by subtracting equations from top to bottom, and then substituting variables from bottom to top. Students have to practice such procedural skills: they have to learn how to apply a particular strategy to an exercise. E-learning systems offer excellent possibilities for practicing procedural skills. The first explanations and motivation for a procedure that solves a particular kind of problems are probably best taught in a class room, or studied in a book, but the subsequent practice can often be done behind a computer. There exist many e-learning systems or intelligent tutoring systems that support practicing procedural skills. The tools vary widely in breadth, depth, user-interface, etc, but, unfortunately, almost all of them lack sophisticated techniques for providing immediate feedback. If feedback mechanisms are present, they are hard coded in the tools, often even with the exercises. This situation hampers the usage of e-learning systems for practicing mathematical skills. This paper introduces a formalism for specifying strategies for solving exercises. It shows how a strategy can be viewed as a language in which sentences consist of transformation steps. Furthermore, it discusses how we can use advanced techniques from computer science, such as term rewriting, strategies, error-correcting parsers, and parser combinators to provide feedback at each intermediate step from the start towards the solution of an exercise. Our goal is to obtain e-learning systems that give immediate and useful feedback. 1

Abstract. Rewrite strategies are used to specify how mathematical exercises are solved in interactive learning environments, and to provide feedback to students solving such exercises. We have developed a generic strategy language with which we can specify rewrite strategies in many (mathematical) domains. Although our strategy language is quite powerful, it lacks an essential component for specifying strategies, namely the interleaving of two strategies. Often students have to perform multiple subtasks, but the order in which these tasks are performed is irrelevant, and steps of solutions may be interleaved. We show the need for combinators that support interleaving by means of several examples. We extend our strategy language with different combinators for interleaving, define the semantics of the extension, and show how the interleaving combinators are implemented in the parsing framework we use for recognizing student behavior and providing hints.

interfaces. We will specify and implement abstract interfaces for the student model, the dialogue history and the problem state. 4 Diagnosis and Therapy. We will view the diagnosis task as a plan recognition problem. We will explore the possibilities (i) of using proof plans and Oyster/Clam's proof planning facility to support the diagnosis and therapy task; (ii) of adapting a probabilistic plan recognition approach using Bayes's belief networks. This is the approach taken in Andes, a physics tutoring system [GCV98] with similar domain properties. Since specifying and implementing the diagnosis and therapy module of Intuition will be a significant project in itself, we will recruit a PhD student who will concentrate his research efforts solely on these components: 4.1 Knowledge acquisition (6pm), literature survey (3pm), and summarisation of intermediate results (1pm). 4.2 Diagnosis module (9pm), and summarisation of intermediate results (1pm). 4.3 Therapy module (9pm), and summar...

Math-Bridge is a European project which aims to provide facilities for bridging the mathematics gap between schools and higher education in Europe. The Open Universiteit Nederland is responsible for the interactive exercise player for Math-Bridge. This paper discusses the various forms interactions can take when solving mathematical exercises, the kind of feedback an exercise player should give according to teachers and developers of learning environments, and how strategies can be used to automatically calculate many of these kinds of feedback. Furthermore, it discusses some of the peculiarities of mathematical exercises that challenge our strategy framework. 1