Mathematical learning environments help students in mastering mathematical knowledge. Mature environments typically offer thousands of interactive exercises. Providing feedback to students solving interactive exercises requires domain reasoners for doing the exercise-specific calculations. Since a domain reasoner has to solve an exercise in the same way a student should solve it, the structure of domain reasoners should follow the layered structure of the mathematical domains. Furthermore, learners, teachers, and environment builders have different requirements for adapting domain reasoners, such as providing more details, disallowing or enforcing certain solutions, and combining multiple mathematical domains in a new domain. In previous work we have shown how domain reasoners for solving interactive exercises can be expressed in terms of rewrite strategies, rewrite rules, and views. This paper shows how users can adapt and configure such domain reasoners to their own needs. This is achieved by enabling users to explicitly communicate the components that are used for solving an exercise.
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Deep Dive into Adapting Mathematical Domain Reasoners.
Mathematical learning environments help students in mastering mathematical knowledge. Mature environments typically offer thousands of interactive exercises. Providing feedback to students solving interactive exercises requires domain reasoners for doing the exercise-specific calculations. Since a domain reasoner has to solve an exercise in the same way a student should solve it, the structure of domain reasoners should follow the layered structure of the mathematical domains. Furthermore, learners, teachers, and environment builders have different requirements for adapting domain reasoners, such as providing more details, disallowing or enforcing certain solutions, and combining multiple mathematical domains in a new domain. In previous work we have shown how domain reasoners for solving interactive exercises can be expressed in terms of rewrite strategies, rewrite rules, and views. This paper shows how users can adapt and configure such domain reasoners to their own needs. This is ac
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Mathematical learning environments and intelligent tutoring systems such as MathDox [8], the Digital Mathematics Environment (DWO) of the Freudenthal Institute [9], and the ActiveMath system [14], help students in mastering mathematical knowledge. All these systems manage a collection of learning objects, and offer a wide variety of interactive exercises, together with a graphical user interface to enter and display mathematical formulas. Sophisticated systems also have components for exercise generation, for maintaining a student model, for varying the tutorial strategy, and so on. Mathematical learning environments often delegate dealing with exercise-specific problems, such as diagnosing intermediate answers entered by a student and providing feedback, to external components. These components can be computer algebra systems (CAS) or specialized domain reasoners.
The wide range of exercise types in a mathematical learning environment is challenging for systems that have to construct a diagnosis from an intermediate
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student answer to an exercise. In general, CAS will have no problem calculating an answer to a mathematics question posed at primary school, high school, or undergraduate university level. However, CAS are not designed to give detailed diagnoses or suggestions to intermediate answers. As a result, giving feedback using CAS is difficult. Domain reasoners, on the other hand, are designed specifically to give good feedback.
Developing, offering, and maintaining a collection of domain reasoners for a mathematical learning environment is more than just a software engineering problem applied to domain reasoners. Mathematical learning environments usually offer topics incrementally, building upon prior knowledge. For example, solving linear equations is treated before and used in solving quadratic equations. Following Beeson’s principles [4] of cognitive fidelity (the software solves the problem as a student does) and glassbox computation (you can see how the software solves the problem), domain reasoners should be organized with the same incremental and layered organization. Structuring domain reasoners should therefore follow the organization of mathematical knowledge. Domain reasoners are used by learners, teachers, and developers of mathematical environments. Users should be able to customize a domain reasoner [16]. The different groups of users have various requirements with respect to customization. For example, a learner might want to see more detail at a particular point in an exercise, a teacher might want to enforce that an exercise is solved using a specific approach, and a developer of a mathematical environment might want to compose a new kind of exercise from existing parts. Meeting these requirements is challenging in the development of domain reasoners. It is our experience that users request many customizations, and it is highly unlikely that a static collection of domain reasoners offering exercises at a particular level will be sufficient to satisfy everyone. Instead, we propose a dynamic approach that enables the groups of users to customize the domain reasoners to their needs.
In this paper we investigate how we can offer users the possibility to adapt and configure domain reasoners. In the first part of the paper we identify the problems associated with managing a wide range of domain reasoners for mathematics, and we argue why allowing configuration and adaptation of the concepts describing domain reasoners is desirable. This is the paper’s first contribution. Section 2 further motivates our research question. We then give a number of case studies in Section 3 that illustrate the need for adaptation and configuration. Most of these case studies are taken from our work on developing domain reasoners for about 150 applets from the DWO of the Freudenthal Institute.
The second part starts with an overview of the fundamental concepts by means of which we describe mathematical knowledge for solving exercises in domain reasoners. We show how these concepts interoperate, and how they are combined (Section 4). Next, we present a solution for adapting and configuring domain reasoners in Section 5, which is our second contribution. In particular, we show how our solution helps in solving the case studies. The techniques that are proposed in this paper have been implemented in our framework for developing domain reasoners 1 , and we are currently changing the existing domain reasoners accordingly. We evaluate the advantages and disadvantages of our approach, and draw conclusions in the final section.
Computer algebra systems (CAS) are designed specifically for solving complex mathematical tasks, and performing symbolic computations. CAS are often used in intelligent tutoring systems as a back-end for assessing the correctness of an answer. In general, they are suitable for such a task, although different normal f
