International Electronic Journal of Mathematics Education

How do the Undergraduate Students Use an Example in Mathematical Proof Construction: A Study based on Argumentation and Proving Activity
AMA 10th edition
In-text citation: (1), (2), (3), etc.
Reference: Laamena CM, Nusantara T, Irawan EB, Muksar M. How do the Undergraduate Students Use an Example in Mathematical Proof Construction: A Study based on Argumentation and Proving Activity. Int Elect J Math Ed. 2018;13(3), 185-198. https://doi.org/10.12973/iejme/3836
APA 6th edition
In-text citation: (Laamena et al., 2018)
Reference: Laamena, C. M., Nusantara, T., Irawan, E. B., & Muksar, M. (2018). How do the Undergraduate Students Use an Example in Mathematical Proof Construction: A Study based on Argumentation and Proving Activity. International Electronic Journal of Mathematics Education, 13(3), 185-198. https://doi.org/10.12973/iejme/3836
Chicago
In-text citation: (Laamena et al., 2018)
Reference: Laamena, Christina M., Toto Nusantara, Edy Bambang Irawan, and Makbul Muksar. "How do the Undergraduate Students Use an Example in Mathematical Proof Construction: A Study based on Argumentation and Proving Activity". International Electronic Journal of Mathematics Education 2018 13 no. 3 (2018): 185-198. https://doi.org/10.12973/iejme/3836
Harvard
In-text citation: (Laamena et al., 2018)
Reference: Laamena, C. M., Nusantara, T., Irawan, E. B., and Muksar, M. (2018). How do the Undergraduate Students Use an Example in Mathematical Proof Construction: A Study based on Argumentation and Proving Activity. International Electronic Journal of Mathematics Education, 13(3), pp. 185-198. https://doi.org/10.12973/iejme/3836
MLA
In-text citation: (Laamena et al., 2018)
Reference: Laamena, Christina M. et al. "How do the Undergraduate Students Use an Example in Mathematical Proof Construction: A Study based on Argumentation and Proving Activity". International Electronic Journal of Mathematics Education, vol. 13, no. 3, 2018, pp. 185-198. https://doi.org/10.12973/iejme/3836
Vancouver
In-text citation: (1), (2), (3), etc.
Reference: Laamena CM, Nusantara T, Irawan EB, Muksar M. How do the Undergraduate Students Use an Example in Mathematical Proof Construction: A Study based on Argumentation and Proving Activity. Int Elect J Math Ed. 2018;13(3):185-98. https://doi.org/10.12973/iejme/3836

Abstract

Various studies have been conducted regarding the use of examples in a mathematical proof. This study aims to describe how the students use the example in the proof analyzed by argumentation and proving activity. Qualitative methods are used to explain the phenomena that arise in the use of examples on mathematical proof. The data collected is the result of student work, think aloud, field notes and interview results. The results show that the example is used as an exploratory tool, an example as an investigative tool for justification and an example as a conviction tool. In Toulmin’s view, examples as explorers serve as data, examples as an instrument of investigation for justification serve as warrant and backing while the example as a conviction tool serves as a qualifier to convince them of the resulting claim. An example as an investigative tool for justification produces two types of argumentation structures: argument structures consisting of one cycle and two cycles. An example of an exploratory tool serves as a data form of simple argumentation. Examples play an important role in mathematical proof although the example is not deductive proof. Examples can be used with various functions depending on the student’s needs.

References

  • Aberdein, A. (2012). The parallel structure of mathematical reasoning. In A. Aberdein & J. Dove (Eds.), The argument of mathematics (pp. 351–370).
  • Alcock, L., & Inglis, M. (2008). Doctoral students’ use of examples in evaluating and proving conjectures. Educational Studies in Mathematics, 69(2), 111–129. https://doi.org/10.1007/s10649-008-9149-x
  • Balacheff, N. (1988). Aspects of proof in pupils’ practice of school mathematics. In Mathematics, Teachers and Children (pp. 216–235). Hoodes & Stoughton: Londers.
  • Boero, P., Garuti, R., & Mariotti M, A. (1996). Some dynamic mental processes underlying producing and proving conjectures. In O. Figueras, J. L. Cortina, S. Alatorre, T. Rojano, & A. Sepúlveda (Eds.), Proceedings of the 20th Conference of the International Group for the Psychology of Mathematics Education PME-XX, vol. 2 (pp. 121 – 128). Valencia.
  • Bromley, D. B. (1986). The case-study method in psychology and related disciplines. New Jersey, USA: John Wiley & Sons.
  • Buchbinder, B., & Pedemonte O. (2011). Examining the role of examples in proving processes through a cognitive lens : the case of triangular numbers, 257–267. https://doi.org/10.1007/s11858-011-0311-z
  • Cañadas, M. C., Castro, E., & Castro, E. (2009). Using a model to describe students’ inductive reasoning in problem solving.
  • Conner, A., Singletary, L. M., Smith, R. C., Wagner, P. A., & Francisco, R. T. (2014). Identifying Kinds of Reasoning in Collective Argumentation. Mathematical Thinking and Learning, 16(3), 181–200. https://doi.org/10.1080/10986065.2014.921131
  • De Villiers, M. D. (1990). The role and function of proof in mathematics. Pythagoras, 24, 17–24.
  • Douek, N. (1999). Some Remarks About Argumentation And Mathematical Proof And Their Educational Implications. In I. Schwank (Ed.), European Research in Mathematics Education I (pp. 125–139). Forschungsinstitut fur mathematikdidaktik, Osnabruck.
  • Ellis, A. E., Lockwood, E., Williams, C. C. W., Dogan, M. F., & Knuth, E. (2012). Middle school students’ example use in conjecture exploration and justification. In Proceedings of the 34th Annual Meeting of the North American Chapter of the Psychology of Mathematics Education.
  • Fraenkel, J. R., & Wallen, N. E. (2003). How to design and evaluate research in education. New York, USA: McGraw-Hill Higher Education.
  • Harel, G., & Sowder, L. (1998). Students’ proof schemes: Results from exploratory studies. Research in Collegiate Mathematics Education III, 234–283. https://doi.org/10.1090/cbmath/007/07
  • Healy, L., & Hoyles, C. (2000). A study of proof conceptions in algebra. Journal for Research in Mathematics Education, 396–428. https://doi.org/10.2307/749651
  • Iannone, P., Inglis, M., Mejia-Ramos, J. P., Simpson, A., & Weber, K. (2012). Does Generating Examples Aid Proof Production? Educational Studies in Mathematics, 77(1), 1-14. https://doi.org/10.1007/s10649-011-9299-0
  • Inglis, M., & Mejia-Ramos, J. P. (2010). The Effect of Authority on the Persuasiveness of Mathematical Arguments. Cognition and Instruction, 27(1), 25-50. https://doi.org/10.1080/07370000802584513
  • Inglis, M., Mejia-Ramos, J. P., & Simpson, A. (2007). Modelling Mathematical Argumentation : The Importance. Educational Studies in Mathematics, 66(1), 3–21. https://doi.org/10.1007/s10649-006-9059-8
  • 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
  • Knipping, C. (2003). Argumentation structures in classroom proving situations. In M. Hanna & G. Villiers (Eds.), Proceedings of the third congress of the european society for research in mathematics education. Bellaria, Italy, ERME.
  • Knipping, C. (2004). Argumentation structures in classroom proving situations’. In M. Mariotti (Ed.), Proceedings of CERME (Vol. 3).
  • Laamena, C. M. (2017). Karakteristik Warrant Dalam Menemukan Counter Example. In Prosiding Seminar Nasional Hasil Penelitian dan Pengabdian kepada Masyarakat. Senasif 2017 (pp. 212-222).
  • Laamena, C. M., Nusantara, T., Irawan, E. B., & Muksar, M. (2018). Analysis of the Students’ Argumentation based on the level of Ability: Study on the Process of Mathematical Proof Analysis of the Students’ Argumentation based on the level of Ability: Study on the Process of Mathematical Proof. Journal of Physics: Conference Series, 1028(1), 0–7.
  • Lockwood, E., Ellis, A., Knuth, E., Dogan, M. F., & Williams, C. (2013). Strategically Chosen Examples Leading to Proof Insight: A Case Study of a Mathematician’s Proving Process. In Proceedings of the 35th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 245–252).
  • Mason, J., Burton, L., & Stacey, K. (2010). Thinking mathematically (2nd ed). Harlow: Pearson.
  • Mason, J., Watson, A., Mason, J., & Watson, A. (1984). Getting Students to Create Boundary Examples. Retrieved from http://mrbartonmaths.com/resourcesnew/8.%20Research/Inquries/Getting%20
  • students%20to%20create%20Boundary%20Examples%20-%20Watson%20and%20Mason.pdf
  • Mejía-Ramosa, J. P., & Inglish, M. (2008). What are the argumentative activities associated with proof? In M. Joubert (Ed.), Proceedings of the British Society for Research into Learning Mathematics 28(2) June 2008 (pp. 67–72).
  • Musser, G. L., Burger, W. F., & Peterson, B. E. (2011). Mathematics for Elementary Teachers a Contemporary Approach (NINTH EDIT). United States of America: John Wiley & Sons, Inc.
  • Nardi, E., Biza, I., & Watson, S. (2014). What makes a claim an acceptable mathematical argument in the secondary classroom? A preliminary analysis of teachers’ warrants in the context of an Algebra Task. In S. Pope (Ed.), British Congress of Mathematics Education (pp. 247–254).
  • Nardi, E., Biza, I., & Zachariades, T. (2012). “Warrant” revisited: Integrating mathematics teachers’ pedagogical and epistemological considerations into Toulmin’s model for argumentation. Educational Studies in Mathematics, 79(2), 157–173. https://doi.org/10.1007/s10649-011-9345-y
  • Pedemonte, B. (2003). What kind of proof can be constructed following an abductive argumentation. In Proceedings of the Third Conference on European Research in Mathematics Education.
  • Pedemonte, B. (2007). How can the relationship between argumentation and proof be analysed? Educational Studies in Mathematics, 66(1), 23–41. https://doi.org/10.1007/s10649-006-9057-x
  • Reid, D. A. (2002). Conjectures and refutations in grade 5 mathematics. Journal for Research in Mathematics Education, 5–29. https://doi.org/10.2307/749867
  • Toulmin, S. E. (1958). The uses of argument. Cambridge: Cambridge University Press.
  • Toulmin, S. E. (2003). The Uses of Argument (Second Ed.). Cambridge, UK: Cambridge University Press. https://doi.org/10.1017/CBO9780511840005
  • Ubuz, B., Dincer, S., & Bulbul, A. (2012). Argumentation in Undergraduate Math Courses: A Study on Proof Generation. Proceedings of the 36th Conference of the International Group for the Psychology of Mathematics Education, 4, 163–170.
  • Ubuz, B., Dincer, S., & Bülbül, A. (2013). Argumentation in Undergraduate Math Courses: A Study on Definition Construction. In A. M. Lindmeier & A. Heinze (Eds.), Proceedings of the 37th Conference of the International Group for the Psychology of Mathematics Education (pp. 313–320). Kiel, Germany: PME.
  • Viholainen, A. (2011). The view of mathematics and argumentation. In E. S. Marta Pytlak, T. Rowland, E. Swoboda Pytlak (Eds.), European Society for Research in Mathematics 7. Rzeszów, Poland.
  • Watson, A., & Mason, J. (2005). Mathematics as a constructive activity: Learners generating examples. Mahwah, NJ: Erlbaum.

License

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.