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Measuring in dynamic geometry environments as a tool for conjecturing and proving

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Abstract

This paper sits within the research on the affordances of new technologies in the mathematics classroom and focuses on a specific feature that is available in dynamic geometry environments, i.e. measuring tools, within the context of conjecturing and proving in open geometry problems. We develop a classification of different modalities of measuring, based on our previous work on dragging. The modalities are illustrated through the analysis of 15–16 year-old students’ proving processes, which focuses on how these modalities relate to the moves between the spatio-graphical field and the theoretical field and may either support or hinder the proving process. The classification of the modalities of measuring enables researchers to access students’ cognitive processes and teachers to be aware of the different possible uses and interpretation of measuring, giving them tools to support students when difficulties arise.

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Notes

  1. This discussion is referred to Cabri because it is the software we used in our research but similar issues could apply to other dynamic geometry environments.

  2. Measurements are updated in real time as the figure is dragged.

  3. Boieri (1995) refers to Cabri I but the method of computation of measurements performed by the subsequent versions of Cabri is still the same.

  4. It can be verified (Boieri 1995) in practice that the relative uncertainty for measurements in Cabri is less than 2%, and it can be reduced according to the choice of the measurement unity.

  5. The same thing may happen with other geometric properties, as for example angles in circles, Pythagoras theorem, etc.

  6. It is interesting to notice that this problem sits within the bigger issue of the trust that users (students in particular) normally have in what is calculated/produced/shown by the computer.

  7. Every theorem is a statement B for which there is another statement A such that B is a logical consequence of A.

  8. We use the word ‘field’, rather than ‘level’ as originally used by (Laborde 1998), in order to stress the fact that spatio-graphical and theoretical field may coexist.

  9. For more information on the methodology of the project see (Olivero 2002).

  10. Here and elsewhere we refer to the description of the different dragging modalities given in (Arzarello et al. 1998a).

  11. The students of a Liceo Psico-Pedagogico attend 4 mathematics classes per week. This type of school does not have a strong mathematic-scientific orientation.

  12. The same sort of conflict could be originated with younger students by using other measuring tools, as for example a ruler.

  13. In this case, not only measurements but also the use of constructions to check a property.

  14. Similarly to the previous example, also in this case the figure on the screen is not a ‘constructed’ square.

  15. The students of a Liceo Scientifico attend 5 mathematics classes per week. This type of school has a strong mathematic-scientific orientation.

  16. We analysed this activity from a cognitive point of view in other works (see for example Arzarello 2000; Arzarello et al. 1998a,b).

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Acknowledgments

Research program supported by MIUR, Università di Torino and Università di Modena e Reggio Emilia (PRIN Contract n. 2005019721), and by an ESRC studentship at the University of Bristol.

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Authors and Affiliations

  1. Graduate School of Education, University of Bristol, 35 Berkeley Square, Bristol, BS8 1JA, UK

    Federica Olivero

  2. Dipartimento di Matematica, Università di Torino, Via Carlo Alberto, 10, 10123, Turin, Italy

    Ornella Robutti

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  1. Federica Olivero
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  2. Ornella Robutti
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Correspondence to Federica Olivero.

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Olivero, F., Robutti, O. Measuring in dynamic geometry environments as a tool for conjecturing and proving. Int J Comput Math Learning 12, 135–156 (2007). https://doi.org/10.1007/s10758-007-9115-1

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