## Adapting Mathematical Domain (2010)

### BibTeX

@MISC{Heeren10adaptingmathematical,

author = {Bastiaan Heeren and Johan Jeuring and Bastiaan Heeren and Johan Jeuring},

title = {Adapting Mathematical Domain},

year = {2010}

}

### OpenURL

### Abstract

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. 1

### Citations

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Term Rewriting and All That
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(Show Context)
Citation Context ...to specify rules with a left-hand side and a right-hand side, and to rely on unification and substitution of terms to do the transformation [15]. This is common practice in term rewrite systems (TRS) =-=[3]-=-. We allow rewrite rules to yield multiple results. 4.2 Rewrite strategies Simple problems can be solved by applying a set of rules exhaustively (for instance, when the set of rules is confluent), but... |

265 | The use of explicit plans to guide inductive proofs
- Bundy
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(Show Context)
Citation Context ...we have designed our own XML language for specifying rewrite strategies, and transformations on these strategies. Our strategy language is very similar to the tactic languages used in theorem proving =-=[6, 2]-=-, and has the same expressive power. Several authors discuss adaptation of various aspects in learning environments [16, 19], but we are not aware of previous work on configuring and adapting domain r... |

28 | PDS — A Three-Dimensional Data Structure for Proof Plans
- Cheikhrouhou, Sorge
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(Show Context)
Citation Context ...power. Several authors discuss adaptation of various aspects in learning environments [16, 19], but we are not aware of previous work on configuring and adapting domain reasoners. Hierarchical proofs =-=[7, 2]-=-, which represent proofs at different levels of abstraction, are related to turning a strategy into a rule and vice versa. As far as we found, hierarchical proofs are not used to recognize proving ste... |

23 | A lightweight approach to datatype-generic rewriting
- Noort, Yakushev, et al.
- 2010
(Show Context)
Citation Context ...r of the underlying programming language. An alternative is to specify rules with a left-hand side and a right-hand side, and to rely on unification and substitution of terms to do the transformation =-=[15]-=-. This is common practice in term rewrite systems (TRS) [3]. We allow rewrite rules to yield multiple results. 4.2 Rewrite strategies Simple problems can be solved by applying a set of rules exhaustiv... |

21 |
Design principles of Mathpert: software to support education in algebra and calculus
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(Show Context)
Citation Context ...ments 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 sam... |

19 | Managing evolution and change in web-based teaching and learning environments
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(Show Context)
Citation Context ...re 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... |

12 | Arrows and computation
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(Show Context)
Citation Context ... be idempotent. A view is assumed to preserve a term’s semantics. Primitive views can be composed into compound views, in two different ways. Firstly, views are closely related to the arrow interface =-=[17]-=-, and its bidirectional variant. The combinators of this interface can be used for combining views, such as using views in succession. Secondly, views can be parameterized with another view. Consider ... |

10 | MathDox : mathematical documents on the web
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(Show Context)
Citation Context ...ed by enabling users to explicitly communicate the components that are used for solving an exercise. 1 Introduction 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 o... |

10 | Specifying rewrite strategies for interactive exercises
- Heeren, Jeuring, et al.
(Show Context)
Citation Context ...s of exercises, it is important that the reasoners share a set of feedback services, and that these services are exercise independent. 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 submitt... |

9 | ACTIVEMATH: An Intelligent Tutoring System for Mathematics
- Melis, Siekmann
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(Show Context)
Citation Context ...oduction 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 interf... |

8 | Feedback services for exercise assistants
- Gerdes, Heeren, et al.
- 2008
(Show Context)
Citation Context ...s of exercises, it is important that the reasoners share a set of feedback services, and that these services are exercise independent. 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 submitt... |

6 |
Representation for interactive exercises
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- 2009
(Show Context)
Citation Context ... (the findbuggyrules service). Other services we offer are variations of the ones listed above. All services calculate feedback automatically from a strategy specification and rewrite rules. Goguadze =-=[11]-=- describes a set of feedback services used by the ActiveMath learning environment to serve as an interface for calling external domain reasoners. His services are similar to ours, and also assume the ... |

4 |
Canonical forms in interactive exercise assistants
- Heeren, Jeuring
- 2009
(Show Context)
Citation Context ...ression 1x 2 + (−3)x is rather atypical. These implicit assumptions make that standard rewriting techniques do not apply directly. Canonical forms and notational conventions can be captured in a view =-=[12]-=-, which consists of a partial function for matching, and a (complete) functionfor building. Matching may result in a value of a different type, such as the pair (−3, 5) for the expression −(3 − 5). I... |

2 | A comparison of equality in computer algebra and correctness in mathematical pedagogy
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- 2009
(Show Context)
Citation Context ...gent 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 forms can have subtle effects on an assessment =-=[5]-=-. CAS are less suitable for supporting more advanced tutoring functionality, such as suggesting a meaningful next step, showing a worked-out example, or discovering a common misconception: they have n... |

2 |
Design and implementation of a computer supported learning environment for mathematics
- Doorman, Drijvers, et al.
- 2009
(Show Context)
Citation Context ...for solving an exercise. 1 Introduction 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, toget... |

1 |
et al. Getal & Ruimte
- Admiraal
- 2009
(Show Context)
Citation Context ...revisit the cases in Section 5.6. 3.1 Case study: controlling the solutions for an exercise A quadratic equation can be solved in many ways. For example, the Dutch mathematics textbook Getal & Ruimte =-=[1]-=-, used in more than half of the high schools in the Netherlands, gives many techniques to solve an equation of the form ax 2 + bx + c = 0. It considers the case of a binomial (b = 0 or c = 0) and the ... |

1 | Tactics for hierarchical proof
- Aspinall, Denney, et al.
(Show Context)
Citation Context ...we have designed our own XML language for specifying rewrite strategies, and transformations on these strategies. Our strategy language is very similar to the tactic languages used in theorem proving =-=[6, 2]-=-, and has the same expressive power. Several authors discuss adaptation of various aspects in learning environments [16, 19], but we are not aware of previous work on configuring and adapting domain r... |

1 | A new framework for dynamic adaptations and actions
- Ullrich, Lu, et al.
- 2009
(Show Context)
Citation Context ...different classes of exercises. Advanced environments include tools for authoring exercises (for teachers), they maintain a model of a learner, and can have a component for adaptive course generation =-=[19]-=-. All these aspects are related to domain reasoners, and the facilities they offer for customization. Environments are the primary clients of a domain reasoner. – Domain reasoners. From within a domai... |