Interactive Mathematical Documents on the Web

Cohen, Arjeh M.; Cuypers, Hans; Barreiro, Ernesto Reinaldo; Sterk, H.J.M.

doi:10.1007/978-3-662-05148-1_16

Cited by 13 publications

(18 citation statements)

References 8 publications

(13 reference statements)

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“…Sometimes it is cumbersome to use a domain reasoner to give feedback to students, or there is no domain reasoner available. The DME [19], MathDox [14,15], and Math-Bridge [52] offer the possibility to hardcode custom feedback in an exercise. For a typical multi-step exercise, this sometimes increases the size of the exercise measured in lines of text by a large factor.…”

Section: Design Considerationsmentioning

confidence: 99%

“…The environment provides some reusable components for specifying goals etc., but still requires a substantial amount of programming for developing a domain reasoner. For most domains, developing a domain reasoner is a challenging task and requires learning environments -DME [19] web-based learning environment by the Freudenthal Institute for secondary math education -Math-Bridge [52] e-learning platform for online bridging courses in mathematics, the successor of ActiveMath [38] -MathDox [14,15] software tools for creating interactive mathematical documents by Eindhoven University of Technology prototypes -Logic tool [36,35] bringing propositions into disjunctive normal form, and proving equivalences between logical formulae -Ask-Elle [23] stepwise development of simple functional programs …”

Section: Design Considerationsmentioning

confidence: 99%

“…For example, in a learning environment for mathematics that supports solving an exercise stepwise, an exercise about quadratic equations might be solved as in Figure 1. Stepwise exercises are particularly popular in learning environments for mathematics, such as MathDox [14,15], the Digital Mathematical Environment (DME) of the Freudenthal Institute [19], Math-Bridge [52] (based on ActiveMath [38]), APlusIx [13], the Carnegie Learning Algebra tutor, etc. Environments such as the DME and Math-Bridge offer thousands of stepwise exercises to a student.…”

Section: Introductionmentioning

confidence: 99%

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Feedback services for stepwise exercises

Heeren

Jeuring

2014

Science of Computer Programming

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Advanced learning environments such as intelligent tutoring systems for algebra, logic, programming, physics, etc. let a student practice with stepwise exercises, and support a student solving such exercises by providing feedback. These environments usually provide various types of feedback, for example about the correctness of a step, common errors, hints about how to proceed, or complete worked-out solutions. Calculating feedback is generally delegated to a dedicated expert knowledge module, also known as a domain reasoner. Existing architectural descriptions of learning environments do not precisely specify the interaction between this module and the rest of the learning system. We propose a design based on the stateless client-server architecture that clearly decouples the expert knowledge module from the learning environment. We describe a set of feedback services that support the inner (interactions within an exercise) and outer (over a collection of exercises) loops of a learning system, and that provide meta-information about a class of exercises, such as solving quadratic equations, or performing Gaussian elimination. The feedback services do not depend on a particular domain and are based on the various feedback types described in the literature.The paper analyzes which domain-specific knowledge about an exercise class is needed for implementing the feedback services. Based on this analysis, we developed a framework for implementing domain reasoners that offers generic functionality such as rewriting, simplifying, and comparing terms. We have implemented several domain reasoners in this framework, both for external learning environments and for simple prototypes. The proposed design is evaluated with these implementations, and we reflect on our experience with developing domain reasoners.

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Section: Design Considerationsmentioning

confidence: 99%

Section: Design Considerationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Feedback services for stepwise exercises

Heeren

Jeuring

2014

Science of Computer Programming

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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.…”

Section: Introductionmentioning

confidence: 99%

Adapting Mathematical Domain Reasoners

Heeren¹,

Jeuring²

2010

Lecture Notes in Computer Science

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Abstract. 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 exercisespecific 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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“…A student gets an exercise, for example about solving a system of linear equations, and takes steps towards the solution. Examples of interactive exercise assistants for mathematics are the Digital Mathematics Environment (DWO) of the Freudenthal Institute [5], MathDox [7], Aplusix [6], MathPert [3], WIMS [8], ActiveMath [9], and many more. Here is an example of a series of (correct) steps a student makes when solving a linear equation:…”

Section: Introductionmentioning

confidence: 99%

Canonical Forms in Interactive Exercise Assistants

Heeren

Jeuring

2009

Lecture Notes in Computer Science

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Abstract. Interactive exercise assistants support students in practicing exercises, and acquiring procedural skills. Many mathematical topics can be practiced in such assistants. Ideally, an interactive exercise assistant not only validates final answers, but also comments on intermediate steps submitted by a student, provides hints on how to proceed, and presents worked-out examples. For these purposes, fine control over the symbolic simplification procedures of the underlying mathematical machinery is needed. In this paper, we introduce views for mathematical expressions. A view defines an equivalence relation by choosing a canonical form of mathematical expressions. We use views to track and recognize intermediate answers, to help in presenting expressions to a user, and to control the granularity of the steps in worked-out examples. We develop the concept of a view, discuss the laws it satisfies, and show how views are composed, which means that they can be used for multiple exercise classes.

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Interactive Mathematical Documents on the Web

Cited by 13 publications

References 8 publications

Feedback services for stepwise exercises

Feedback services for stepwise exercises

Adapting Mathematical Domain Reasoners

Canonical Forms in Interactive Exercise Assistants

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