Exploring the Potential Role of Reversible Reasoning: Cognitive Research on Inverse Function Problems in Mathematics

Abstract

Researchers have argued that reversible reasoning is involved in all topics in mathematics. The study employed an qualitative research approach, consisted of three sessions (pre-assessment, thinking-aloud, and interview), and involved eight participants enrolled in Algebra class. The aim was to explore the potential role of reversible reasoning on students’ inverse functions. The result of study indicated that there three categories of reversible reasoning that refer to the consistency of students in completing inverse function tasks, which are relational-harmonic, relational-visual, and relational-identity. Mental activities performed by the students in constructing and reasoning inverse functions were also explained. In addition, potential aspects of the students’ reversible reasoning created during the process of constructing meaning were highlighted. These findings provide perspectives on reversible reasoning, students’ understanding of inverse functions, and areas of future research.

Keywords

• reversibility,reversible reasoning,inverse function,problem-solving

References

1. Anderson Norton, & Jesse L. M. Wilkins. (2012). The Splitting Group. Journal for Research in Mathematics Education, 43(5), 557. https://doi.org/10.5951/jresematheduc.43.5.0557
2. As’ari, A. R., Kurniati, D., Abdullah, A. H., Muksar, M., & Sudirman, S. (2019). Impact of infusing truth-seeking and open-minded behaviors on mathematical problem-solving. Journal for the Education of Gifted Young Scientists, 7(4), 1019–1036. https://doi.org/10.17478/jegys.606031
3. Bagley, S., Rasmussen, C., & Zandieh, M. (2015). Inverse, composition, and identity: The case of function and linear transformation. Journal of Mathematical Behavior, 37, 36–47. https://doi.org/10.1016/j.jmathb.2014.11.003
4. Byerley, C., & Thompson, P. W. (2017). Secondary mathematics teachers’ meanings for measure, slope, and rate of change. Journal of Mathematical Behavior, 48(February), 168–193. https://doi.org/10.1016/j.jmathb.2017.09.003
5. Carlson, M. P., Madison, B., & West, R. D. (2015). A Study of Students ’ Readiness to Learn Calculus. International Journal of Research in Undergraduate Mathematics, 1(2), 209–233. https://doi.org/10.1007/s40753-015-0013-y
6. Carlson, M. P., & Thompson, P. W. (2017). Variation , covariation , and functions : Foundational ways of thinking mathematically. In Compendium for research in mathematics education (pp. 421–456).
7. Chun, J. (2017). Construction of the Sum of Two Covarying Oriented Quantities. Potential Analysis. University of Georgia. https://doi.org/10.1007/s11118-013-9365-6
8. Creswell, J. W., & Guetterman, T. C. (2018). Educational Research: planning, conducting, and evaluating quantitative and qualitative research, 6th Edition. Boston, United States of America: Pearson Education.

9. Delice, A., & Kertil, M. (2013). Service Mathematics Teachers in a Modelling. International Journal of Science and Mathematics Education, 2013(February 2012), 631–657.
10. Dougherty, B., Bryant, D. P., Bryant, B. R., & Shin, M. (2017). Helping Students With Mathematics Difficulties Understand Ratios and Proportions Ratios and Proportions. TEACHING Exceptional Children, 49(2), 96–105. https://doi.org/10.1177/0040059916674897
11. Dougherty, B. J., Bryant, D. P., Bryant, B. R., Darrough, R. L., & Pfannenstiel, K. H. (2015). Developing Concepts and Generalizations to Build Algebraic Thinking: The Reversibility, Flexibility, and Generalization Approach. Intervention in School and Clinic, 50(5), 273–281. https://doi.org/10.1177/1053451214560892
12. Dubinsky, E. (2002). Reflective Abstraction in Advanced Mathematical Thinking. Advanced Mathematical Thinking, 95–126. https://doi.org/10.1007/0-306-47203-1_7
13. Ellis, A. B., Ozgur, Z., Kulow, T., Dogan, M. F., & Amidon, J. (2016). An Exponential Growth Learning Trajectory: Students’ Emerging Understanding of Exponential Growth Through Covariation. Mathematical Thinking and Learning, 18(3), 151–181. https://doi.org/10.1080/10986065.2016.1183090
14. García-García, J., & Dolores-Flores, C. (2018). Intra-mathematical connections made by high school students in performing Calculus tasks. International Journal of Mathematical Education in Science and Technology, 49(2), 227–252. https://doi.org/10.1080/0020739X.2017.1355994
15. González-Calero, J. A., Arnau, D., & Laserna-Belenguer, B. (2015). Influence of additive and multiplicative structure and direction of comparison on the reversal error. Educational Studies in Mathematics, 89(1), 133–147. https://doi.org/10.1007/s10649-015-9596-0
16. Gray, E. M., & Tall, D. O. (1994). Duality, Ambiguity, and Flexibility: A “Proceptual” View of Simple Arithmetic. Journal for Research in Mathematics Education, 25(2), 116. https://doi.org/10.2307/749505
17. Haciomeroglu, E. S., Aspinwall, L., & Presmeg, N. (2009). The role of reversibility in the learning of the calculus derivative and antiderivative graphs. Proceedings of the 31st Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Vol. 5, 5, 81–88.
18. Haciomeroglu, E. S., Aspinwall, L., & Presmeg, N. C. (2010). Contrasting cases of calculus students’ understanding of derivative graphs. Mathematical Thinking and Learning, 12(2), 152–176. https://doi.org/10.1080/10986060903480300
19. Hackenberg, A. J. (2010). Students’ reasoning with reversible multiplicative relationships. Cognition and Instruction, 28(4), 383–432. https://doi.org/10.1080/07370008.2010.511565
20. Hackenberg, A. J., & Lee, M. Y. (2015). Relationships between students’ fractional knowledge and equation writing. Journal for Research in Mathematics Education, 46(2), 196–243. https://doi.org/10.5951/jresematheduc.46.2.0196
21. Hackenberg, A. J., & Lee, M. Y. (2016). Students’ distributive reasoning with fractions and unknowns. Educational Studies in Mathematics, 93(2), 245–263. https://doi.org/10.1007/s10649-016-9704-9
22. Hoffkamp, A. (2011). The use of interactive visualizations to foster the understanding of concepts of calculus: Design principles and empirical results. ZDM - International Journal on Mathematics Education, 43(3), 359–372. https://doi.org/10.1007/s11858-011-0322-9
23. Ikram, M., Purwanto, Parta, I. N., & Susanto, H. (2018). Students ’ Reversible Reasoning on Function Composition Problem : Reversible on Function and Subtitution. International Journal of Insights for Mathematics Teaching, 01(1), 9–24.
24. Inhelder, & Piaget. (1958). The Growth of Logical Thinking From Child to Adolecence. New York: Basic Books, Inc.
25. Kontorovich, I. (2017). Students’ confusions with reciprocal and inverse functions. International Journal of Mathematical Education in Science and Technology, 48(2), 278–284. https://doi.org/10.1080/0020739X.2016.1223361
26. Marmur, O., & Zazkis, R. (2018). Space of fuzziness : avoidance of deterministic decisions in the case of the inverse function. In M. P. Carlson & C. Rasmussen (Eds.). In Making the connection: Research and teaching in undergraduate mathematics education (pp. 27–42). Washington D.C: MAA.
27. Martínez-planell, R., & Delgado, A. C. (2016). The unit circle approach to the construction of the sine and cosine functions and their inverses : An application of APOS theory. Journal of Mathematical Behavior, 43, 111–133. https://doi.org/10.1016/j.jmathb.2016.06.002
28. Moore, K. C. (2014). Quantitative Reasoning and the Sine Function: The Case of Zac. Journal for Research in Mathematics Education, 45(1), 102–138. https://doi.org/10.5951/jresematheduc.45.1.0102
29. Natsheh, I., & Karsenty, R. (2014). Exploring the potential role of visual reasoning tasks among inexperienced solvers. ZDM - International Journal on Mathematics Education, 46(1), 109–122. https://doi.org/10.1007/s11858-013-0551-1
30. Oehrtman, M., Carlson, M., & Thompson, P. W. (2008). Foundational reasoning abilities that promote coherence in students’ understandings of function. Research and Teaching in Undergraduate Mathematics Education, 27(4).
31. Paoletti, T. (2015). Pre-Service Teachers’ Development of Bidirectional Reasoning. Dissertation. University of Georgia.
32. Paoletti, T., Stevens, I. E., Hobson, N. L. F., Moore, K. C., & LaForest, K. R. (2018). Inverse function: Pre-service teachers’ techniques and meanings. Educational Studies in Mathematics, 97(1), 93–109. https://doi.org/10.1007/s10649-017-9787-y
33. Pino-Fan, L. R., Font, V., Gordillo, W., Larios, V., & Breda, A. (2017). Analysis of the Meanings of the Antiderivative Used by Students of the First Engineering Courses. International Journal of Science and Mathematics Education, 1–23. https://doi.org/10.1007/s10763-017-9826-2
34. Ramful, A. (2009). Reversible Reasoning In Multiplicative Situations: Conceptual Analysis, Affordances And Constraints. Dissertation. University of Georgia.
35. Ramful, A. (2014). Reversible reasoning in fractional situations: Theorems-in-action and constraints. Journal of Mathematical Behavior, 33, 119–130. https://doi.org/10.1016/j.jmathb.2013.11.002
36. Ramful, A. (2015). Reversible reasoning and the working backwards problem solving strategy. Australian Mathematics Teacher, 71(4), 28–32.
37. Ramful, A., & Olive, J. (2008). Reversibility of thought: An instance in multiplicative tasks. Journal of Mathematical Behavior, 27(2), 138–151. https://doi.org/10.1016/j.jmathb.2008.07.005
38. Ron Tzur. (2011). Can Dual Processing Theories of Thinking Inform Conceptual Learning in Mathematics? The Mathematics Enthusiast, 8(3), 597–636.
39. Sangwin, C. J., & Jones, I. (2017). Asymmetry in student achievement on multiple-choice and constructed-response items in reversible mathematics processes. Educational Studies in Mathematics, 94(2), 205–222. https://doi.org/10.1007/s10649-016-9725-4
40. Simon, M. A., Kara, M., Placa, N., & Sandir, H. (2016). Categorizing and promoting reversibility of mathematical concepts. Educational Studies in Mathematics, 93(2), 137–153. https://doi.org/10.1007/s10649-016-9697-4
41. Simon, M. A., Placa, N., & Avitzur, A. (2016). Participatory and anticipatory stages of mathematical concept learning: Further empirical and theoretical development. Journal for Research in Mathematics Education, 47(1), 63–93. https://doi.org/10.5951/jresematheduc.47.1.0063
42. Soneira, C., González-Calero, J. A., & Arnau, D. (2018). An assessment of the sources of the reversal error through classic and new variables. Educational Studies in Mathematics, 99(1), 43–56. https://doi.org/10.1007/s10649-018-9828-1
43. Steffe, L., & Olive, J. (2010). Children´s Knowledge, Fractional. London: Springer Science & Business Media.
44. Sun, R. (2006). Cognition and Multi-Agent Interactions From Cognitive Modeling to Social Simulation. Communicating and Collaborating with Robotic Agents. Cambridge University Press.
45. Thahir, A., Komarudin, Hasanah, U. N., & Rahmahwaty. (2019). MURDER learning models and self efficacy: Impact on mathematical reflective thinking ability. Journal for the Education of Gifted Young Scientists, 7(4), 1120–1133. https://doi.org/10.17478/jegys.594709
46. Törner, G., Potari, D., & Zachariades, T. (2014). Calculus in European classrooms: curriculum and teaching in different educational and cultural contexts. ZDM - International Journal on Mathematics Education, 46(4), 549–560. https://doi.org/10.1007/s11858-014-0612-0
47. Tunç-Pekkan, Z. (2015). An analysis of elementary school children’s fractional knowledge depicted with circle, rectangle, and number line representations. Educational Studies in Mathematics, 89(3), 419–441. https://doi.org/10.1007/s10649-015-9606-2
48. Tzur, R. (2007). Fine grain assessment of students’ mathematical understanding: Participatory and anticipatory stagesin learning a new mathematical conception. Educational Studies in Mathematics, 66(3), 273–291. https://doi.org/10.1007/s10649-007-9082-4
49. Vilkomir, T., & O’Donoghue, J. (2009). Using components of mathematical ability for initial development and identification of mathematically promising students. International Journal of Mathematical Education in Science and Technology, 40(2), 183–199. https://doi.org/10.1080/00207390802276200
50. Von Glasersfeld, E. (1995). Radical Constructivism: A Way of Knowing and Learning. Studies in Mathematics Education Series. https://doi.org/10.4324/9780203454220
51. Wasserman, N. H. (2017). Making Sense of Abstract Algebra: Exploring Secondary Teachers’ Understandings of Inverse Functions in Relation to Its Group Structure. Mathematical Thinking and Learning, 19(3), 181–201. https://doi.org/10.1080/10986065.2017.1328635
52. Weber, E., & Thompson, P. W. (2014). Students’ images of two-variable functions and their graphs. Educational Studies in Mathematics, 87(1), 67–85. https://doi.org/10.1007/s10649-014-9548-0
53. Wong, B. (1977). The Relationship between Piaget’s Concept of Reversibility and Arithmetic Performance among Second Grades. The Annual Meetiny of the American Educational Research Association. Retrieved from https://files.eric.ed.gov/fulltext/ED136962.pdf
54. Yasin, M., Jauhariyah, D., Madiyo, M., Rahmawati, R., Farid, F., Irwandani, I., & Mardana, F. F. (2019). The guided inquiry to improve students mathematical critical thinking skills using student’s worksheet. Journal for the Education of Gifted Young Scientists, 7(4), 1345–1360. https://doi.org/10.17478/jegys.598422
55. Yin, R. K. (2014). Case study research: Design and methods (5th ed.). SAGE publication.
56. Zazkis, R., & Kontorovich, I. (2016). A curious case of superscript ( − 1 ): Prospective secondary mathematics teachers explain. Journal of Mathematical Behavior, 43, 98–110. https://doi.org/10.1016/j.jmathb.2016.07.001
57. Zazkis, R., & Zazkis, D. (2011). The significance of mathematical knowledge in teaching elementary methods courses : perspectives of mathematics teacher educators. Educational Studies in Mathematics, 76(3), 247–263. https://doi.org/10.1007/s10649-010-9268-z