Article Sidebar

Download
PDF
Statistic
Read Counter : 345 Download : 234

Downloads

Download data is not yet available.

Main Article Content

Abstract

Mathematical arguments are central components of mathematics and play a role in certain types of modelling of potential mathematical giftedness. However, particular characteristics of arguments are interpreted differently in the context of mathematical giftedness. Some models of giftedness see no connection, whereas other models consider the formulation of complete and plausible arguments as a partial aspect of giftedness. Furthermore, longitudinal changes in argumentation characteristics remain open. This leads to the research focus of this article, which is to identify and describe the changes of argumentation products in potentially mathematically gifted children over a longer period. For this purpose, the argumentation products of children from third to sixth grade are collected throughout a longitudinal study and examined with respect to the use of examples and generalizations. The analysis of all products results in six different types of changes in the characteristics of the argumentation products identified over the survey period and case studies are used to illustrate student use of examples and generalizations of these types. This not only reveals the general importance of the use of examples in arguments. For one type, an increase in generalized arguments can be observed over the survey period. The article will conclude with a discussion of the role of argument characteristics in describing potential mathematical giftedness.

Keywords

Examples Longitudinal Study Mathematical Giftedness Mathematical Reasoning Typology

Article Details

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

How to Cite
Jablonski, S., & Ludwig, M. (2022). Examples and generalizations in mathematical reasoning – A study with potentially mathematically gifted children. Journal on Mathematics Education, 13(4), 605–630. https://doi.org/10.22342/jme.v13i4.pp605-630
Download Citation
Endnote/Zotero/Mendeley (RIS)
BibTeX

References

  1. Amielia, S. D., Suciati, S., & Maridi, M. (2018). Enhancing students' argumentation skill using an argument driven inquiry-based module. Journal of Education and Learning (EduLearn), 12(3), 464–471. https://doi.org/10.11591/edulearn.v12i3.8042
  2. Assmus, D., & Fritzlar, T [Torsten] (2022). Mathematical creativity and mathematical giftedness in the primary school age range: an interview study on creating figural patterns. ZDM – Mathematics Education, 54(1), 113–131. https://doi.org/10.1007/s11858-022-01328-8
  3. Baker, M. (2003). Computer-Mediated Argumentative Interactions for the Co-Elaboration of Scientific Notions. In J. Andriessen, M. Baker, & D. Suthers (Eds.), Arguing to Learn: Confronting Cognitions in Computer-Supported Collaborative Learning environments (pp. 47–78). Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0781-7_3
  4. Bezold, A. (2009). Förderung von Argumentationskompetenzen durch selbstdifferenzierende Lernangebote [Promotion of argumentation skills through self-differentiating learning opportunities]. Kovac.
  5. Brunner, E. (2019). Förderung mathematischen Argumentierens im Kindergarten: Erste Erkenntnisse aus einer Pilotstudie [Promoting Mathematical Reasoning in Kindergarten: Initial Findings from a Pilot Study]. Journal für Mathematik-Didaktik, 40(2), 323–356. https://doi.org/10.1007/s13138-019-00146-y
  6. Durak, T., & Tutak, F. A. (2019). Comparison of Gifted and Mainstream 9th Grade Students’ Statistical Reasoning Types. In M. Nolte (Ed.), Including the Highly Gifted and Creative Students – Current Ideas and Future Directions (pp. 136–143). WTM. https://doi.org/10.37626/GA9783959871327.0
  7. Fritzlar, T [T.], & Nolte, M. (2019). Research in mathematical giftedness in Germany-Looking back and ahead. In M. Nolte (Ed.), Including the Highly Gifted and Creative Students – Current Ideas and Future Directions (pp. 8–20). WTM. https://doi.org/10.37626/GA9783959871327.0
  8. Fuchs, M. (2006). Procedures of mathematically potentially talented third and fourth classes in problem-free empirical studies on typing specific problem-handling styles. Journal für Mathematik-Didaktik, 27(2), 165–166. https://doi.org/10.1007/BF03339035
  9. Fuchs, M., & Käpnick, F. (2009). Mathe für kleine Asse. Klasse 3/4. Band 2 [Math for little aces. Grade 3/4. volume 2]. Cornelsen.
  10. Gutierrez, A., Benedicto, C., Jaime, A., & Arbona, E. (2018). The Cognitive Demand of a Gifted Student’s Answers to Geometric Pattern Problems. In Mathematical Creativity and Mathematical Giftedness (pp. 169–198). Springer, Cham. https://doi.org/10.1007/978-3-319-73156-8_7
  11. Habermas, J. (1984-87). The theory of communicative action: Reason and the rationalization of society. Polity; Beacon Press.
  12. Hanna, G. (2000). Proof, explanation and exploration: An overview. Educational Studies in Mathematics, 44(1/2), 5–23. https://doi.org/10.1023/A:1012737223465
  13. Harel, G., & Sowder, L. (1998). Students‘ proof schemes: Results from exploratory studies. In E. Dubinsky, A. Schoenfeld, & J. Kaput (Eds.), Research in Collegiate Mathematics Education (pp. 234–283). American Mathematical Society.
  14. Heinze, A. (2005). Lösungsverhalten mathematisch begabter Grundschulkinder - aufgezeigt an ausgewählten Problemstellungen [Solving behavior of mathematically gifted elementary school children - demonstrated by selected problems]. LIT.
  15. Heinze, A. (2006). Lösungsverhalten mathematisch begabter Grundschulkinder — aufgezeigt an ausgewählten Problemstellungen [Solving behavior of mathematically gifted elementary school children - demonstrated by selected problems]. Journal Für Mathematik-Didaktik, 27(1), 79–80. https://doi.org/10.1007/BF03340105
  16. Jablonski, S., & Ludwig, M. (2021). Changes in mathematical reasoning - A longitudinal study with mathematically gifted children. In M. Inparsitha, N. Changsri, & N. Boonsena (Eds.), Proceedings of the 44th Conference of the International Group for Psychology of Mathematics Education (Vol. 3, pp. 91–100). PME.
  17. Joklitschke, J., Rott, B., & Schindler, M. (2022). Notions of creativity in mathematics education research: A systematic literature review. International Journal of Science and Mathematics Education, 20(6), 1161–1181. https://doi.org/10.1007/s10763-021-10192-z
  18. Käpnick, F. (1998). Mathematisch begabte Kinder. Modelle, empirische Studien und Förderungsprojekte für das Grundschulalter [Mathematically gifted children. Models, empirical studies and support projects for the primary school age]. Peter Lang GmbH.
  19. Kieran, C., Pang, J., Schifter, D., & Ng, S. F. (2016). Early Algebra. Springer International Publishing. https://doi.org/10.1007/978-3-319-32258-2
  20. Kluge, S. (2000). Empirically grounded construction of types and typologies in qualitative social research. Forum Qualitative Sozialforschung / Forum: Qualitative Social Research, 1(1). https://doi.org/10.17169/fqs-1.1.1124
  21. Koleza, E., Metaxas, N., & Poli, K. (2017). Primary and secondary students’ argumentation competence: a case study. In T. Dooley & G. Gueudet (Eds.), Proceedings of the Tenth Congress of the European Society for Research in Mathematics Education (pp. 179–186). DCU Institute of Education and ERME.
  22. Komatsu, K. (2017). Fostering empirical examination after proof construction in secondary school geometry. Educational Studies in Mathematics, 96(2), 129–144. https://doi.org/10.1007/s10649-016-9731-6
  23. Maddocks, D. L. S. (2018). The identification of students who are gifted and have a learning disability: A comparison of different diagnostic criteria. Gifted Child Quarterly, 62(2), 175–192. https://doi.org/10.1177/0016986217752096
  24. Mayring, P. (2014). Qualitative content analysis: Theoretical foundation, basic procedures and software solution. SSOAR. https://nbn-resolving.org/urn:nbn:de:0168-ssoar-395173
  25. Moor, E. de. (1980). Wiskobas bulletin. Leerplanpublikatie 11. IOWO.
  26. Nussbaum, E. M. (2011). Argumentation, dialogue theory, and probability modeling: Alternative frameworks for argumentation research in education. Educational Psychologist, 46(2), 84–106. https://doi.org/10.1080/00461520.2011.558816
  27. Pedemonte, B. (2002). Relation between argumentation and proof in mathematics: cognitive unity or break? In J. Novotná (Ed.), Proceedings of the 2nd Conference of the European Society for Research in Mathematics Education (Vol. 2, pp. 70–80). Charles University, Faculty of Education.
  28. Piaget, J. (1928). Judgment and reasoning in the child. Routledge. https://doi.org/10.4324/9780203207260
  29. Reid, D., & Knipping, C. (2010). Proof in mathematics education. Research, learning and teaching. Sense Publisher.
  30. Sari, P., & Ng, S. F. (2022). Exploring quantitative relationship through area conservation activity. Journal on Mathematics Education, 13(1), 31–50. https://doi.org/10.22342/jme.v13i1.pp31-50
  31. Schneider, W., & Bullock, M. (Eds.). (2009). Human development from early childhood to early adulthood: Findings from a 20 year longitudinal study. Psychology Press.
  32. Sjuts, B. (2017). Mathematisch begabte Fünft- und Sechstklässler [Mathematically gifted fifth and sixth graders]. WTM.
  33. Sowell, E. J., Zeigler, A. J., Bergwall, L., & Cartwright, R. M. (1990). Identification and description of mathematically gifted students: A review of empirical research. Gifted Child Quarterly, 34(4), 147–154. https://doi.org/10.1177/001698629003400404
  34. Sriraman, B. (2004). Gifted ninth graders' notions of proof: Investigating parallels in approaches of mathematically gifted students and professional mathematicians. Journal for the Education of the Gifted, 27(4), 267–292. https://doi.org/10.4219/jeg-2004-317
  35. Toulmin, S. E. (2003). The uses of argument. Cambridge University Press. https://doi.org/10.1017/CBO9780511840005
  36. Ufer, S., Heinze, A., & Reiss, K. (2008). Individual predictors of geometrical proof competence. In O. Figueras, J. L. Cortina, S. Alatorre, T. Rojano, & A. Sepúlveda (Eds.), Proceedings of the Joint Meeting of the 32nd Conference of the International Group for the Psychology of Mathematics Education and the XX North American Chapter (pp. 361–368). PME.
  37. van Eemeren, F. H., Grootendorst, R., Snoeck Henkemans, A. F., & Blair, J. A. (1996). Fundamentals of argumentation theory: A handbook of historical backgrounds and contemporary developments. L. Erlbaum. https://doi.org/10.4324/9780203811306
  38. Wittmann, E., & Müller, G. (1990). Handbuch produktiver Rechenübungen. Vom Einspluseins zum Einmaleins [Handbook of productive arithmetic exercises. From the multiplication table to the multiplication table]. Klett.