Abstract
The present study is about high school graduates’ basis for argumentation in elementary arithmetic. Besides knowing the elements of the basis for argumentation, the question arises in how far individual understandings of these components differ. We conducted task-based interviews focussing on learners’ usage and meaning of statements in terms of their embeddedness in a local organisation, the epistemic values assigned to them, and respective effects on the conclusion’s modal qualifier. We want to highlight the following results: While all graduates accept definitions and rules for term manipulation, there is no consensus concerning the statements involved. Furthermore, the individuals’ epistemic values concerning the statements involved affect their usage in a chain of arguments and the individuals’ evaluation of the conclusion. Although the assessments of a local organisation of mathematical content differ, the epistemic values seem to be decisive for the individual evaluation of the conclusion. Thus, we extend the existing theory by investigating the meaning of epistemic value in the context of the basis for argumentation and its effects on the individual’s proof constructions. For practice-oriented research, we contribute to the ongoing discussion about the learning of proof in school mathematics by investigating the basis for argumentation of high school graduates in arithmetic.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
We use the term “elementary arithmetic” to summarize the mathematical content covering properties of the natural numbers and divisibility issues.
- 2.
Marks are scaled from 1 to 4, 1 being the best mark.
- 3.
Marks in school subjects are scaled from 0 to 15, 15 being the best mark.
References
Baumert, J., Bos, W., & Watermann, R. (1998). TIMSS/III. Schülerleistungen in Mathematik und den Naturwissenschaften am Ende der Sekundarstufe 2 im internationalen Vergleich. Max-Planck-Institut für Bildungsforschung.
Biehler, R., & Kempen, L. (2013). Students` use of variables and examples in their transition from generic proof to formal proof. In B. Ubuz, C. Haser, & M. A. Mariotti (Eds.), Proceedings of the Eighth Congress of the European Society for Research in Mathematics Education (pp. 86–95). Middle East Technical University.
Brunner, E. (2014). Mathematisches Argumentieren, Begründen und Beweisen. Grundlagen, Befunde und Konzepte. Springer Spektrum.
Bürger, H. (1979). Beweisen im Mathematikunterricht – Möglichkeiten der Gestaltung in der Sekundarstufe I und II. [Proving in schoolmathematics – possibilites for sedoncdary school.] In W. Dörfler & R. Fischer (Eds.), Beweisen im Mathematikunterricht. Vorträge des 2. internationalen Symposiums für “Didaktik der Mathematik” in Klagenfurt (pp. 103–134). Hölder-Pichler-Tempsky.
Coe, R., & Ruthven, K. (1994). Proof practices and constructs of advanced mathematics students. British Educational Research Journal, 20(1), 41–53. https://doi.org/10.1080/0141192940200105
Duval, R. (1991). Structure du raisonnement deductif et apprentissage de la demonstration. Educational Studies in Mathematics, 22(3), 233–261. https://doi.org/10.2307/3482298
Duval, R. (2007). Cognitive functioning and the understanding of mathematical processes of proof. In P. Boero (Ed.), Theorems in school: From history, epistemology and cognition to classroom practice (pp. 137–161). Sense Publishers.
Edwards, L. D. (1998). Odds and evens: Mathematical reasoning and informal proof among high school students. The Journal of Mathematical Behavior, 17(4), 489–504. https://doi.org/10.1016/S0732-3123(99)00002-4
Freudenthal, H. (1973). Mathematics as an educational task. D. Reidel Publishing Company.
Goldin, G. (2000). A scientific perspective on structured, task-based interviews in mathematics education research. In A. E. Kelly & R. A. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 517–546). Erlbaum.
Inglis, M., Mejia-Ramos, J. P., & Simpson, A. (2007). Modelling mathematical argumentation: The importance of qualification. Educational Studies in Mathematics, 66(1), 3–21. https://doi.org/10.1007/s10649-006-9059-8
Kempen, L., & Biehler, R. (2019a). Fostering first-year pre-service teachers’ proof competencies. ZDM, 51(5), 731–746. https://doi.org/10.1007/s11858-019-01035-x
Kempen, L., & Biehler, R. (2019b). Pre-service teachers’ benefits from an inquiry-based transition-to-proof course with a focus on generic proofs. International Journal of Research in Undergraduate Mathematics Education, 5(1), 27–55. https://doi.org/10.1007/s40753-018-0082-9
Kempen, L., Krämer, S., & Biehler, R. (2020). Investigating high school graduates’ personal meaning of the notion of “mathematical proof”. In T. Hausberger, M. Bosch, & F. Chellougui (Eds.), Proceedings of the third conference of the International Network for Didactic Research in University Mathematics (pp. 358–367). University of Carthage and INDRUM.
Kitcher, P. (1984). The nature of mathematical knowledge. Oxford University Press.
Knipping, C. (2003). Beweisprozesse in der Unterrichtspraxis. Vergleichende Analysen von Mathematikunterricht in Deutschland und Frankreich. Franzbecker Verlag.
Knipping, C. (2008). A method for revealing structures of argumentations in classroom proving processes. ZDM, 40(3), 427. https://doi.org/10.1007/s11858-008-0095-y
Knuth, E., Choppin, J., Slaughter, M., & Sutherland, J. (2002). Mapping the conceptual terrain of middle school students’ competencies in justifying and proving. In Proceedings of the 24th annual meeting for the psychology of mathematics education – North America, v.4 (pp. 1693–1700). GA.
Krämer, S. (2019). Beweisvorstellungen von Abiturientinnen und Abiturienten – Eine qualitative Interviewstudie auf der Grundlage individueller Beweiskonstruktionen. [high school graduates’ personal meaning of the notion of “mathematical proof” – a qualitative interview study based on individual proof constructions.] [master thesis]. Paderborn University.
Mayring, P. (2014). Qualitative content analysis. Theoretical foundation, basic procedures and software solution. Beltz.
Reiss, K., & Heinze, A. (2000). Begründen und Beweisen im Verständnis von Abiturienten. In Beiträge zum Mathematikunterricht 2000 (pp. 520–523). Franzbecker.
Stylianides, A. J. (2007). Proof and proving in school mathematics. Journal for Research in Mathematics Education, 38(3), 289–321. https://doi.org/10.2307/30034869
Tietze, U.-P., Klika, M., & Wolpers, H. (1997). Mathematikunterricht in der Sekundarstufe II. [mathematics teaching in upper secondary school.] Vieweg + Teubner Verlag.
Toulmin, S. (1958). The uses of argument. Cambridge University Press.
Weber, K., Lew, K., & Mejía-Ramos, J. P. (2020). Using expectancy value theory to account for individuals’ mathematical justifications. Cognition and Instruction, 38(1), 27–56. https://doi.org/10.1080/07370008.2019.1636796
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Kempen, L. (2022). Investigating High School Graduates’ Basis for Argumentation: Considering Local Organisation, Epistemic Value, and Modal Qualifier When Analysing Proof Constructions. In: Biehler, R., Liebendörfer, M., Gueudet, G., Rasmussen, C., Winsløw, C. (eds) Practice-Oriented Research in Tertiary Mathematics Education. Advances in Mathematics Education. Springer, Cham. https://doi.org/10.1007/978-3-031-14175-1_10
Download citation
DOI: https://doi.org/10.1007/978-3-031-14175-1_10
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-14174-4
Online ISBN: 978-3-031-14175-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)