Abstract
This chapter focuses on four topics: how exploration, especially hands-on exploration with concrete materials, is integrated into the teaching of proof in Taiwan; different ways of interpreting exploration; exploration’s intrinsic features and value in the context of proving; and a proposed conceptual model, the ‘EP-spectrum’, for the relationships amongst exploration, problem-solving, proving and proof in the classroom. Furthermore, the chapter raises some ideas requiring future research, such as the primary factors contributing to the progression from exploration to proof production and the integration of exploration into the context of proving.
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Notes
- 1.
This spectrum was initially developed by the working group WG5. It was then revised by the authors.
- 2.
The term additive model is borrowed from colour theory. In contrast to a subtractive model, which employs an assumption of a minimum whole requirement to be successful in the process of proving, there is no requirement in an additive model for a fixed minimum whole. We have chosen the additive model since its combinations – produced by the intertwining of varying capacities in each of the three primary factors – have potentially unlimited results, many of which work well in the process of proving. Conversely, according to the subtractive model, lacking a certain capacity in a primary factor will inhibit an individual from being able to successfully complete the proving process, due to the minimum whole requirement.
- 3.
The students’ academic competence was measured by their performance in the simulated national senior high school entrance examinations for ninth-graders. Often, the results of these examinations can accurately predict students’ performance in actual national entrance examinations.
- 4.
Although the affective domain is not our focus, it is relevant to the effectiveness of this stage in the EP-spectrum.
- 5.
Some literature or studies do not distinguish argumentation from proof.
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Acknowledgement
We are grateful to Gila Hanna, John Holt, Roza Leikin, Elena Nardi, and Sarah-Jane Patterson for reviewing and editing our paper. We wish to thank Man Keung Siu, Chia-Jui Hsieh, Ting-Ying Wang, Shia-Jei Tang and Guoheng Chao for providing ideas on the EP-spectrum.
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NB: References marked with * are in F. L. Lin, F. J. Hsieh, G. Hanna, & M. de Villiers (Eds.) (2009). ICMI Study 19: Proof and proving in mathematics education. Taipei, Taiwan: The Department of Mathematics, National Taiwan Normal University.
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Hsieh, FJ., Horng, WS., Shy, HY. (2012). From Exploration to Proof Production. In: Hanna, G., de Villiers, M. (eds) Proof and Proving in Mathematics Education. New ICMI Study Series, vol 15. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-2129-6_12
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