The Korea Society Of Educational Studies In Mathematics

Current Issue

Journal of Educational Research in Mathematics - Vol. 30 , No. 2

[ Article ]
Journal of Educational Research in Mathematics - Vol. 30, No. 1, pp.89-110
Abbreviation: JERM
ISSN: 2288-7733 (Print) 2288-8357 (Online)
Print publication date 28 Feb 2020
Received 10 Jan 2020 Revised 10 Feb 2020 Accepted 16 Feb 2020
DOI: https://doi.org/10.29275/jerm.2020.02.30.1.89

Analysis of Abduction in Mathematics Problem Posing and Solving
Lee, Myoung Hwa* ; Kim, Sun Hee**,
*Graduate Student, Kangwon National University, South Korea (hahahoho98@nate.com)
**Professor, Kangwon National University, South Korea (mathsun@kangwon.ac.kr)

Correspondence to : Professor, Kangwon National University, South Korea, mathsun@kangwon.ac.kr


Abstract

The purpose of this study was to analyze the abduction in mathematical problem-posing and problem-solving activities. Abduction is more concerned with creating new things than deduction and inductive reasoning. According to the student's rules, abduction are classified as selective and creative. Selective abduction is again classified into ‘manipulative selective abduction’ and ‘theoretical selective abduction’. Creative abduction is again classified into ‘little creative abduction’ at the level of mathematics in school and ‘big creative abduction’ at the level of academic mathematics. Four middle school sophomore students performed problem-posing activities on four tasks. As for the results of analysis on abduction types by problem-posing stages, all four abduction types were observed. But at the problem-solving stages, manipulative selective abduction and theoretical selective abduction were frequently used, while creative abduction was never used. Thus, for the education of mathematical creativity, deepening and expanding problem-posing is necessary that all the type of abduction has been expressed in the problem-posing activity.


Keywords: problem posing, problem solving, abduction, Toulmin model

Acknowledgments

이 논문은 이명화의 박사학위논문의 내용 일부를 요약, 정리한 것임


References
1. Brown, S. I., & Walter, M. I. (2005). The art of problem posing(3rd. Ed. e-book). Lawrence Erlbaum Associates Publisher.
2. Cho, J. Y. & Paik, S. Y. (2009). The effects of mathematical problem-posing activities on elementary students’ problem-solving understanding and mathematics learning attitudes. The Journal of Korea Elementary Education, 19(2), 71-88.
3. Choi, S. K. & Mok, Y. H. (2011). Subject, structure, discourse, and the learning of mathematics. Journal of the Korean school mathematics society, 14(2), 163-178.
4. 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.
5. Creswell, J. W. (2009). Research design: Qualitative, quantitative, and mixed methods approaches (3rd ed.). Thousand Oaks, CA: Sage Publications.
6. Eco, U. (1983). Horns, hooves, insteps : Some hypotheses on three types of abduction. In U. Eco & T. Sebeok(Eds.), The sign of three : Dupin, Holmes and Peirce(pp.198-220). Bloomington, IN : Indiana University Press.
7. Eco, U., & Sebeok, T. A. (1983). The sign of three: Dupin, Holmes, Peirce. United States of America: indiana university. 김주환, 한은경 역(2015). 셜록 홈스, 기호학자를 만나다. 경기: 이마.
8. Eilbert, K. & Lafronza, V. (2005). Working together for community health- a model and case studies. Evaluation and Program Planning, 28, 185-199.
9. Furtak, E. M., Hardy, I., Beinbrech, C., Shavelson, R. J. & Shemwell, J. T. (2010). A framework for analyzing evidence-based reasoning in science classroom discourse, Educational Assessment, 15(3), 175-196.
10. Goldberg, T., Cai, W., Peppa, M., Dardaine, V., Baliga, B.S., Uribarri, J., Vlassara, H., (2004). Advanced glycoxidation end products in commonly consumed foods. Journal of the American Dietetic Association. 104, 1287-1291.
11. Haig, B. D. (2005). Exploratory factor analysis, theory generation, and scientific method. Multivariate Behavioral Research, 40, 303-329.
12. Hanna, R. (2005). Kant and nonconceptual content. European Journal of Philosophy, 13(2), 247-290.
13. Huh, N. & Shin, H. C. (2013). Analysis on sentence error types of mathematical problem posing of pre-service elementary teachers. Journal of the Korean School Mathematics Society, 16(4), 797-820.
14. Inglis, M., Mejía-Ramos, J. P., & Simpson, A. (2007). Model ling mathematical argumentation: The importance of qualification. Educational Studies in Mathematics, 66, 3-21.
15. Jimenez-Aleixandre, M., Rodriguez, A. B. & Duschlet, R. A. (2000). "Doing the lesson" or "doing science": Argument in high school genetics. Science Education, 84(6), 757-792.
16. Jung, S. G. & Park, M. G. (2010). The effects of the mathematical problem generating program on problem solving ability and learning attitude. Journal of Elementary Mathematics Education in Korea, 14(2), 315-335.
17. Kang, M. J. & Lee, S. J. (2014). Abduction and design thinking -Theory and practice of creative ideation-. Korean Association for Semiotic Studies, 38, 7-35.
18. Kim, M. Y. (2013). Developing teaching materials for elementary school students to create a matchstick puzzle. Proceedings of the 2013 Joint Conference on Mathematics Education, 2013(2), 281-285.
19. 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.
20. Kim, S. H. & Lee, C. H. (2002). Abduction as a mathematical reasoning. The Journal of Educational Research in Mathematics, 12(2), 275-290.
21. Kim, Y. M. & Seo, H. A. & Park, J. S. (2012). Problem finding and abduction in science and science abduction. Korean Research Memory (National Research Foundation of Korea (NRF)).
22. Ko, S. S. & Jeon, S. H. (2009). A case study on students’ problem solving in process of problem posing for equation at the middle school level. The Korea Society of Mathematical Education, Communications of mathematical Education, 23(1), 109-128.
23. Ko, J. H. (2014). The analysis of problem posing activities and students’ performance in the 4-1 textbook and workbook. Journal of the Korean School Mathematics Society, 18(1), 103-122.
24. Lee, D. H. & Song, S. H. (2013). The case analysis of Rummikub game redeveloped by gifted class using What-If-Not strategy. Journal of Elementary Mathematics Education in Korea, 17(2), 285-299.
25. Lee, D. H. (2017). The analysis of problem posing cases of pre-service primary teachers. School Mathematics, 19(1), 1-18.
26. Lee, G. M. & Bang, S. J. (2015). Method of problem posing to increase problem solving power. The Korean Society of Mathematical Education, 31(4), 158, 19.
27. Lee. D. H. (2012). A study on the factors of mathematical creativity and teaching and learning models to enhance mathematical creativity. Journal of Elementary Mathematics Education in Korea, 16(1), 39-61.
28. Lee, J. H. & Kim, M. K. (2016). The development and application of posing open-Ended problems program with Renzulli’s enrichment triad model for mathematics-gifted elementary students. The Mathematical Education, 55(2), 209-232.
29. Lee, Y. Ha. & 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.
30. Lee, M. H. & Kim, S. H. (2018). The problem posing types and the recognition of teaching and leaning of pre-service mathematics teachers. School Mathematics, 20(3), 395-408.
31. Lee, M. H. (2020). Analysis of abduction types and thinking strategies in mathematics problem posing. Unpublished doctoral dissertation, Kangwon National University.
32. Magnani, L. (2001). Abduction, reason, and science: Process of discovery and explanation. New York: Kluwer Academic/Plenum Publishers.
33. Magnani, L. (2004). Model-based and manipulative abduction in science. Foundation of Science, 9, 219-247.
34. Magnani, L. (2006). Multimodal abduction: External semiotic anchors and hybrid representations. Logic Journal of the IGPL, 14(2), 107-136.
35. Mann, E. (2006). The essence of mathematics. Journal for the Education of the Gifted, 30, 236-260.
36. Merrifield, P. R, Guilford, J. P., Christensen, P. R., & Frick, J. W. (1962). The role of intellectual factors in problem solving. Psychological Monographs: General and Applied, 76(10), 1-21.
37. Mumford , M. D., Mobley, M. I., Reiter-Palmon, R., Uhlman, C. E., & Doares, L. M. (1991). Process analytic models of creative capacities. Creativity Research Journal, 4(2), 91-122.
38. Na, G. S. (2017). Examining the problem making by mathematically gifted students. School Mathematics, 19(1), 77-93.
39. Noh, J. H. & Ko, H. K. & Huh, N. (2012). An analysis of pre-service teachers’ pedagogical content knowledge about story problem for division of fractions. Education of Primary School Mathematics, 19(1), 19-30.
40. Noh, J. H. (2017). Teachers’ decision and enactment of their content knowledge assessed through problem posing - A U.S. case. Journal of Mathematics Education in Korea, 31(2), 153-166.
41. 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.
42. Oh, P. S. (2016). Roles of models in abductive reasoning: A schematization through theoretical and empirical studies. Journal of the Korean Association for Science Education, 36(4), 551-561.
43. Osborne, J. F., Erduran, S., & Simon, S. (2004). Ideas, evidence and argument in science. In-service Training Pack, Resource Pack and Video. London: Nuffield Foundation.
44. Pease, A., & Aberdein, A. (2011). Five theories of reasoning: Interconnections and applications to mathematics. Logic and Logical Philosophy, 20(1-2), 7-57.
45. Pedemonte, B., & Reid, D. (2011). The role of abduction in proving processes. Educational Studies in Mathematics, 76(3), 281-303.
46. Pedemonte, B. (2007). Structural relationships between argumentation and proof in solving open problems in algebra. In Proceedings of the V Congress of the European Society for Research in Mathematics Education CERME 5, (pp. 643-652). Larnaca, Cyprus.
47. Peirce, C. S. (1958). Collected papers of Charles Sanders Peirce Vols Ⅰ-Ⅵ. In C. Hartshorne & P. Weiss (Eds.). Cambridge, MA: Harvard University Press. [Reference to Peirce’s papers will be designated CP.]
48. Pólya, G. (1957), How to Solve it(2nd ed.). NY: Doubleday & Company, Inc.
49. Psillos, S. (2000). Abduction: Between conceptual richness and computational complexity. In P. A. Flach, & A. C. Kakas (Eds.), Abduction and Induction, 59-74.
50. Rubinshtein, S. L. (1989). The principle of creative self-activity (philosophical foundations of modern pedagogy). Journal Soviet Psychology, 27(2), 6-21.
51. Schoenfeld, A. H. (1980). Heuristics in the classroom. in NCTM 1980 yearbook. Reston, Va.: NCTM.
52. Silver, E. A. (1994). On mathematical problem posing. For the Learning of Mathematics, 14(1), 19-28.
53. Silver, E. A., & Cai, J. (1996). An analysis of arithmetic problem posing by middle school students. Journal for Research in Mathematics Education, 27.
54. Stephan, M., & Rasmussen, C. (2002). Classroom mathematical practices in differential equations. Journal of Mathematical Behavior, 21, 459-490.
55. Stickles, P. R. (2006). An analysis of secondary and middle school teacher’s mathematical problem posing. Unpublished Doctoral Dissertation, Indiana University.
56. Strauss, A., & Corbin, J. (1998). Basics of qualitative research: Techniques and procedures for developing grounded theory(2nd ed.). Sage Publication, California.
57. Thagard, P. (2010). How brains make mebtal models. In L. Magnani, W. Carnielli, & C. Pizzi (Eds.), Model-based reasoning in science and technology: Abduction, logic, and computational discovery (pp. 447-461). Berlin: Springer.
58. Weber, K., & Alcock, L. (2005). Using warranted implications to understand and validate proofs. For the Learning of Mathematics, 25(1), 34-38.
59. Wolcott, H. F. (1994). Transforming qualitative data: Description, analysis, and interpretation. Thousand Oaks, Calif. : Sage Publications.
60. Yackel, E. (2001). Explanation, justification and argumentation in mathematics classrooms. Proceedings of the Conference of the International Group for the Psychology of Mathematics Education, 1-4, 065-634.