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Classification Scheme for Modelling Tasks

Klassifikationsschema für Modellierungsaufgaben

Abstract

There are many different types of reality-related and modelling tasks, not to mention the many different classifications for these tasks, so it is difficult to obtain a clear overview of the subject. Therefore, following the idea of design research and based on modelling theory, the aim of this paper is to develop a comprehensive classification scheme that systematizes existing classifications. The scheme is intended to provide an overview of the different features of modelling tasks, thereby offering guidance in the task design and selection processes for specific aims and predefined objectives and target groups. The usefulness of the classification scheme will be shown by means of a concrete example.

Zusammenfassung

Es gibt sehr viele verschiedene Arten von realitätsbezogenen Aufgaben und Modellierungsaufgaben und darüber hinaus auch sehr viele verschiedene Klassifikationen, was zu einem Mangel an Übersicht führt. In diesem Aufsatz soll daher der Idee des Design Research folgend basierend auf der Theorie zum Modellieren ein umfassendes, und bisherige Klassifikationen systematisierende Klassifikationsschema entwickelt werden. Ziel des Schemas soll sein, einen Überblick über die verschiedenen Eigenschaften von Modellierungsaufgaben zu geben und somit den Design- und Auswahlprozess von Modellierungsaufgaben für bestimmte Ziele und Zielgruppen zu steuern. Die Nützlichkeit des Modells wird anschließend exemplarisch an einer Zielgruppe aufgezeigt.

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Fig. 1

Notes

  1. Such definitions may vary according to the modeling cycle chosen.

  2. The concept of metacognition is not defined in a unified way. Definitions range from reflecting on how we think to self-regulation of one’s own work (Schoenfeld 1992, p. 334; Sjuts 2003, p. 271). Within the framework of PISA (Artelt et al., 2001, p. 271) the term “self-regulated thinking” instead of metacognition was used. They define it as the competency of students to set aims, to choose techniques according to content and aim and to correct them if necessary, and to evaluate one’s own proceeding.

  3. In many older classifications not only modeling tasks but all types of reality-related tasks are considered, possibly in order to distinguish between them.

  4. Mental objects are necessary for transitions between reality and mathematics because they mediate between the two worlds. These mental objects are called ‘Grundvorstellungen’ (Kleine et al. 2005). For example, a common ‘Grundvorstellung’ is that a fraction is a part of one whole. A fraction can be also understood as several parts of several wholes.

  5. This does not imply by any means that tasks for assessment do not differ at all from tasks for learning, but nonetheless, there are many tasks which can be used both for assessment and for learning.

  6. Mental objects are necessary for transitions between reality and mathematics because they act as mediators between the two worlds. These mental objects are called ‘Grundvorstellungen’ (Kleine et al. 2005). For example, a common ‘Grundvorstellung’ is that a fraction is a part of one whole. A fraction can also be understood as several parts of several wholes.

  7. Beliefs are composed of a relatively long lasting subjective knowledge of certain objects as well as the attitudes linked to that knowledge. Beliefs can be conscious or unconscious, whereas the latter ones are often distinguished by an affective character (Pehkonen and Törner, 1996, p. 102).

  8. There are also comprehensive schools which combine all three school types, but in Baden-Württemberg there are so few they have little significance here.

  9. 112 Students from different Hauptschulen in Baden-Württemberg were interviewed. See Maaß and Ege (2007) for theoretical background about beliefs and evaluation methods.

  10. This does not mean that we proceed in small steps and work on parts of the modeling process without considering the whole process—like in transmission teaching—but instead we focus on certain steps whilst considering the whole process.

  11. The test items from STRATUM will be presented elsewhere.

  12. All tasks given to the students contained pictures.

  13. Idea: Barbara Schmidt.

  14. Verschaffel et al. (1999, p. 207) provide tasks which ask students to interpret a result differently in three different given situations, thereby enabling students to learn the important lesson that different situations require different interpretations. As the main objective of STRATUM was to prepare for real life, a task with authentic material has been chosen, although only one interpretation is required.

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Maaß, K. Classification Scheme for Modelling Tasks. J Math Didakt 31, 285–311 (2010). https://doi.org/10.1007/s13138-010-0010-2

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Keywords

  • Modelling
  • Modelling tasks
  • Classification of tasks
  • Design of tasks

Mathematics Subject Classification (2000)

  • 97M99
  • 97-02