Options of Discovering and Verifying Mathematical Theorems – Task-design from a Philosophic-logical Point of View

Michael Meyer

doi:10.29333/ejmste/92561

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Michael Meyer ¹ ^*

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¹ University of Cologne, GERMANY^* Corresponding Author

Abstract

Mathematical theorems can be discovered and verified in different ways. The philosophical logic provides a conceptual understanding of these processes and helps to distinguish different options of initiating them. These options are established by analyzing different textbooks and describe the possibilities of the discovery and (empirical) verification of mathematical knowledge.

Keywords

abduction
discoveries
induction
options of task-design
philosophical logic
verifications

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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Article Type: Research Article

EURASIA J Math Sci Tech Ed, Volume 14, Issue 9, September 2018, Article No: em1588

https://doi.org/10.29333/ejmste/92561

Publication date: 25 Jun 2018

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References

Barzel, B., Leuders, T., Prediger, S., & Hußmann, S. (2013). Designing Tasks for Engaging Students in Active Knowledge Organiszation. In C. Margolinas et al. (Ed.), Task Design in Mathematics Education. Proceedings of ICMI Study 22. Oxford.
Bruner, J. S. (1961). The act of discovery. Havard Educational Review, 31, 21-32.
Carrier, M. (2000). Empirische Hypothesenpruefung ohne Felsengrund. In F. Stadler (Ed.), Elemente moderner Wissenschaftstheorie (pp. 43-56). Berlin: Springer.
Eco, U. (1983). Horns, Hooves, Insteps. In U. Eco & T. A. Sebeok (Eds.), The sign of the three. Dupin, Holmes, Peirce (pp. 198-220). Bloomington: Indiana University Press.
Fann, K. T. (1970). Peirce’s theory of abduction. The Hague: Nijhoff. https://doi.org/10.1007/978-94-010-3163-9.
Freudenthal, H. (1991). Revisiting mathematics education. Dodrecht: Kluwer.
Hoffmann, M. (1999). Problems with Peirce’s concept of abduction. Foundations of Science, 4(3), 271-305. https://doi.org/10.1023/A:1009675824079.
Kieran, C., Knott, L., & Yang, Y. (2015). Frameworks and Principles for Task Design. In A. Watson & M. Ohtani, M. (Eds.), Task Design in Mathematics Education. An ICMI Study 22. Cham: Springer. https://doi.org/10.1007/978-3-319-09629-2_2.
Krumsdorf, J. (2015). Beispielgebundenes Beweisen (Dissertation). University of Munster, https://core.ac.uk/download/pdf/56470239.pdf [10.04.2018].
Kunsteller, J. (2018). Developing Rules Due to the Use of Family Resemblances in Classroom Communication. In P. Ernest (Ed.), ICME-13 Monographs. The Philosophy of Mathematics Education Today. Dodrecht: Springer. https://doi.org/10.1007/978-3-319-77760-3_24.
Leung, A., & Bolite-Frant (2015). Theme A: Tools and Representations. In C. Margolinas (Ed.), Task Design in Mathematics Education. Proceedings of ICMI Study 22. 2013. Oxford.
Magnani, L. (2001). Abduction, Reason and Science. New York: Kluwer. https://doi.org/10.1007/978-1-4419-8562-0.
Meyer, M. (2007). Entdecken und Begruenden im Mathematikunterricht. Von der Abduktion zum Argument. Dissertation. Hildesheim: Franzbecker.
Meyer, M. (2010). Abduction – A logical view of processes of discovering and verifying knowledge in mathematics. Educational Studies in Mathematics, 74, 185-205. https://doi.org/10.1007/s10649-010-9233-x.
Meyer, M., & Voigt, J. (2009). Entdecken, Pruefen und Begruenden. Gestaltung von Aufgaben zur Erarbeitung mathematischer Saetze. Mathematica didactica, 32, 31-66.
Meyer, M., & Voigt, J. (2010). Rationale Modellierungsprozesse. In B. Brandt, M. Fetzer & M. Schuette (Eds.), Auf den Spuren Interpretativer Unterrichtsforschung in der Mathematikdidaktik. Munster: Waxmann, 117-148.
Nordsiek, F., & Nordsieck-Schroeer, H. (1969). Aufgabe. In E. Grochla (Ed.), Handwoerterbuch der Organisation. Stuttgart: Poeschel, 191-199.
Oevermann, U. et al. (1987): Structures of meaning and objective Hermeneutics. In: Meha, V. et al. (Eds.), Modern German sociology. (European Perspectives: a Series in Social Thought and Cultural Ctiticism) (pp. 436–447). New York: Columbia University Press.
Olander, H. T., & Robertson, H. C. (1973). The effectiveness of discovery and expository methods in the teaching of fourth-grade mathematics. Journal for Research in Mathematics Education, 4(1), 33-44. https://doi.org/10.2307/749022.
Pedemonte, B. (2007): How can the relationship between argumentation and proof be analysed? Educational Studies in Mathematics, 66, 23-41. https://doi.org/10.1007/s10649-006-9057-x.
Pedemonte, B., & Reid, D. (2011): The role of abduction in proving processes. Educational Studies in Mathematics, 76(3), 281-303. https://doi.org/10.1007/s10649-010-9275-0.
Peirce, Ch. S. CP. (n.d.). Collected Papers of Charles Sanders Peirce (Volumes I-VI, ed. by C. Hartshorne & P. Weiss, 1931-1935; Volumes VII-VIII, ed. by A.W. Burks, 1958 – quotations according to volume and paragraph). Cambridge: Harvard UP.
Polya, G. (1954). Mathematics and Plausible Reasoning. Vol. 1. Princeton: UP.
Reichenbach, H. (1938). Experience and prediction. Chicago: UP.
Schurz, G. (2008). Patterns of abduction. Synthese, 164, 201–234. https://doi.org/10.1007/s11229-007-9223-4.
Söhling, A.-C. (2016). The learning of mathematics via problem solving. In T. Fritzlar (Ed.), Problem solving in mathematics education. Proceedings of the 2015 joint 17th conference of ProMath and the 2nd GDM working group on problem solving (pp. 215-225). Munster: WTM.
Stegmueller, W. (1976). Hauptstroemungen der Gegenwartsphilosophie. Vol. 1. Stuttgart: Koerner.
Voigt, J. (1984). Interaktionsmuster und Routinen im Mathematikunterricht. Theoretische Grundlagen und mikroethnographische Falluntersuchungen. Weihnheim: Beltz.
Watson, A., & Ohtani, M. (2015, Eds.). Task Design in Mathematics Education. An ICMI Study 22. Cham: Springer.
Winch, W. A. (1913). Inductive versus deductive methods of teaching. Baltimore: Warwick.

How to cite this article

APA

Meyer, M. (2018). Options of Discovering and Verifying Mathematical Theorems – Task-design from a Philosophic-logical Point of View. Eurasia Journal of Mathematics, Science and Technology Education, 14(9), em1588. https://doi.org/10.29333/ejmste/92561

Vancouver

Meyer M. Options of Discovering and Verifying Mathematical Theorems – Task-design from a Philosophic-logical Point of View. EURASIA J Math Sci Tech Ed. 2018;14(9):em1588. https://doi.org/10.29333/ejmste/92561

AMA

Meyer M. Options of Discovering and Verifying Mathematical Theorems – Task-design from a Philosophic-logical Point of View. EURASIA J Math Sci Tech Ed. 2018;14(9), em1588. https://doi.org/10.29333/ejmste/92561

Chicago

Meyer, Michael. "Options of Discovering and Verifying Mathematical Theorems – Task-design from a Philosophic-logical Point of View". Eurasia Journal of Mathematics, Science and Technology Education 2018 14 no. 9 (2018): em1588. https://doi.org/10.29333/ejmste/92561

Harvard

Meyer, M. (2018). Options of Discovering and Verifying Mathematical Theorems – Task-design from a Philosophic-logical Point of View. Eurasia Journal of Mathematics, Science and Technology Education, 14(9), em1588. https://doi.org/10.29333/ejmste/92561

MLA

Meyer, Michael "Options of Discovering and Verifying Mathematical Theorems – Task-design from a Philosophic-logical Point of View". Eurasia Journal of Mathematics, Science and Technology Education, vol. 14, no. 9, 2018, em1588. https://doi.org/10.29333/ejmste/92561