Browse > Article
http://dx.doi.org/10.7468/mathedu.2020.59.1.81

Analysis of abduction and thinking strategies by type of mathematical problem posing  

Lee, Myoung Hwa (Graduate School of Kangwon National University)
Kim, Sun Hee (Kangwon National University)
Publication Information
The Mathematical Education / v.59, no.1, 2020 , pp. 81-99 More about this Journal
Abstract
This study examined the types of abduction and the thinking strategies by the mathematics problems posed by students. Four students who were 2nd graders in middle school participated in problem posing on four tasks that were given, and the problems that they posed were classified into equivalence problem, isomorphic problem, and similar problem. The type of abduction appeared were different depending on the type of problems that students posed. In case of equivalence problem, the given condition of the problems was recognized as object for posing problems and it was the manipulative abduction. In isomorphic problem and similar problem, manipulative abduction, theoretical abduction, and creative abduction were all manifested, and creative abduction was manifested more in similar problem than in isomorphic problem. Thinking strategies employed at abduction were examined in order to find out what rules were presumed by students across problem posing activity. Seven types of thinking strategies were identified as having been used on rule inference by manipulative selective abduction. Three types of knowledge were used on rule inference by theoretical selective abduction. Three types of thinking strategies were used on rule inference by creative abduction.
Keywords
problem posing; abduction type; thinking strategy;
Citations & Related Records
연도 인용수 순위
  • Reference
1 Conner, A., Singletary, L. M., Smith, R. C., Wagner, P. A., & Francisco, R. T. (2014). Identifying kinds of reasoning in collective argumentation. Routledge, Mathematical Thinking and Learning, 16(3), 181-200.   DOI
2 Creswell, J. W. (2009). Research design: Qualitative, quantitative, and mixed methods approaches (3rd ed.). Thousand Oaks, CA: Sage Publications.
3 Eco, U. (1983). Horns, hooves, insteps: Some hypotheses on three types of abduction. In U. Eco & T. Sebeol(eds.), The sign of three : Dupin, Holmes and Peirce (pp. 198-220). Bloomington, IN : Indiana University Press.
4 Inglis, M., Mejia-Ramos, J. P., & Simpson, A. (2007). Modelling mathematical argumentation: the importance of qualification. Educational Studies in Mathematics, 66(1), 3-21.   DOI
5 Jung, S. M. (1993). Three concepts of discovery. The Korean Society for Cognitive Science, 4(1), 25-49.
6 Kim, P. S. (2005). Analysis of thinking process and steps in problem posing of the mathematically gifted children. The Journal of Elementary Education, 18(2), 303-334.
7 Kim, S. H. & Lee, C. H. (2002). Abduction as a mathematical resoning. The Journal of Educational Research in Mathematics, 12(2), 275-290.
8 Kim, S. H. (2004). Semiotic consideration on the appropriation of the mathematical knowledge. Doctoral dissertation. Ehwa Womans University, Korea.
9 Kim, W. K., Jo, M. S., Bang, G. S., Bae, S. K., Ji, E. J., Im, S. H., Kim, D. H., Kang, S. J., & Kim, Y. H. (2015). Mathematic 1, Seoul: visang.
10 Brown, S. I., & Walter, M. I. (2005). The art of problem posing(3rd. Ed. e-book). Lawrence Erlbaum Associates Publisher.
11 Krulik, S. & Rudnick, J. A. (1992). Reasoning and problem solving: A handbook for elementary school teachers. the United States: NCTM.
12 Lee, Y. H. & Kahng, M. J. (2013). An analysis of problems of mathematics textbooks in regards of the types of abductions to be used to solve, The Journal of Educational Research in Mathematics, 23(3), 335-351.
13 Kwon, Y. J., Jeong, J. S., Kang, M. J. & Kim, Y. S. (2003). Research article : A grounded theory on the process of generating hypothesis-knowledge about scientific episodes. Journal of the Korean Association for Research in Science Education, 23(5), 458-469.
14 Lee, H. (1989). Syllogism and dialectic. Kyungnam Journal of Philosophy, 5, 3-30.
15 Lee, M. H. (2020). Analysis of Abduction Types and Thinking Strategies on Mathematics Problem Posing. Doctoral dissertation, Kangwon National University.
16 Na, G. S. (2017). Examining the problem making by mathematically gifted students. School Mathematics, 19(1), 77-93.
17 Magnani, L. (2001). Abduction, reason, and science: Process of discovery and explanation. New York: Kluwer Academic/Plenum Publishers.
18 Ministry of Education (2015). Mathematical curriculum. Seoul: Author.
19 Merriam, S. B. (1988). Qualitative research and case study applications in education. San Francisco: Jossy-Bass.
20 National Council of Teachers of Mathematics (NCTM, 1989). Curriculum and evaluation standards for school mathematics. Reston, Va.
21 Oh, P. S. & Kim, C. J. (2005). A theoretical study on abduction as an inquiry method in earth science. Journal of the Korean Association for Science Education, 25(5), 610-623.
22 Oh, P. S. (2006). Rule-inferring strategies for abductive reasoning in the process of solving an earth-environmental problem. Journal of the Korean Association for Science Education, 26(4), 546-558.
23 Paik, S. Y. (2016). Teaching & learning of mathematical problem-solving. Seoul: Kyungmoon.
24 Polya, G. (1957), How to solve it(2nd ed.). NY.: Doubleday & Company, Inc.
25 Pease, A., & Aberdein, A. (2011). Five theories of reasoning: Interconnections and applications to mathematics. Logic and Logical Philosophy, 20(1-2), 7-57.
26 Pedemonte, B. (2007). How can the relationship between argumentation and proof be analysed? Educational Studies in Mathematics, 66(1), 23-41.   DOI
27 Pedemonte, B., & Reid, D. (2011). The role of abduction in proving processes. Educational Studies in Mathematics, 76(3), 281-303.   DOI
28 Peirce, C. S. (1958). Collected papers of Charles Sanders Peirce Vols I-VI. C. Hartshorne & P. Weiss (Eds.). Cambridge, MA: Harvard University Press.
29 Peirce, C. S. (1980). Collect papers of Charles Sanders Peirce Vols VII-VIII. In B. Arthur (Ed.). Cambridge, MA: Harvard University Press.
30 Psillos, S. (2000). Abduction: Between conceptual richness and computational complexity. Retrieved Jan. 20, 2018, from http://users.uoa.gr/-psillos/PapersI/85-Abduction%20(Kakas%20&%20Flach)%20chapter.pdf
31 Reed, S. K. (1989). Constraints on the abstraction of solutions. Journal of Educational Psychology, 81, 532-540.   DOI
32 Reys, R. E., Lindquist, M. M., Lambdin, D. V., Smith, N. L., Rogers, A., Falle, J., Frid, S. & Bennett, S.(2012). Helping children learn mathematics. (1st Australian Ed.). Australia. Milton, Qld: John Wiley & Sons.
33 Rivera, F. D., & Rossi Becker, J. (2007). Abduction in pattern generalization. In Woo, J. H., Lew, H. C., Park, K. S. & Seo, D. Y. (Eds.). Proceedings of the 31st Conference of the international group for the Psychology of Mathematics Education(vol.4, pp. 97-104), Seoul, Korea.
34 Silver, E. A., Mamona-Dows, J., Leung, S. S., & Kenney, P. A. (1996). Posing mathematical problems. Journal for Research in Mathematical Education, 27(3), 293-309.   DOI
35 Weber, K., & Alcock, L. (2005). Proof validation in real analysis: Inferring and checking warrants. The Journal of Mathematical Behavior, 24(2), 125-134.   DOI
36 Stickles, P. R. (2006). An analysis of secondary and middle school teacher's mathematical problem posing. Doctoral dissertation, Indiana University. the United States.
37 Toulmin, S. E. (1958). The uses of argument. Cambridge: Cambridge University Press.