Article Sidebar

Download
PDF
Statistic
Read Counter : 744 Download : 301

Downloads

Download data is not yet available.

Main Article Content

Abstract

Failure to deduce false suppositions in proof by contradiction is still considered “more difficult” than proving the conditional to in proof by contraposition. This study aims to identify the types of proof construction failures based on the action steps of proof by contradiction, then offer a framework of construction failure hypothesis specifically used in proof by contradiction. The research data were collected and analyzed from the work of students who have agreed to be research participants, a total of 83 students. The results of the analysis of student work successfully identified four types of failures, namely formulating suppositions, constructing and manipulating suppositions, identifying contradictions, and disproving suppositions. These four types of failures then became the material for the development of the hypothesis framework of a failure to construct proof by contradiction, which consists of 17 hypothesis nodes divided into three main hypotheses, namely: operational (action), affective (emotional), and foundational (logical reasoning). The failure hypothesis framework justifies that the sources of the failure of proof construction in proof by contradiction are understanding of the act of producing a proof by contradiction, emotionality towards the coherence of the construction steps, disproving suppositions, beliefs, use of appropriate definitions-theorems and axioms, and cognitive tension in proof by contradiction; and formal logic of the act of producing a proof by contradiction, as well as differences in the underlying logic with other acts.

Keywords

Causation Failure Proof by Construction Proof Construction The Hypothesis of Contradiction

Article Details

Creative Commons License

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

How to Cite
Hamdani, D., Purwanto, Sukoriyanto, & Anwar, L. (2023). Causes of proof construction failure in proof by contradiction. Journal on Mathematics Education, 14(3), 415–448. https://doi.org/10.22342/jme.v14i3.pp415-448
Download Citation
Endnote/Zotero/Mendeley (RIS)
BibTeX

References

  1. Alacaci, C., & Pasztor, A. (2005). On People’s Incorrect Either-Or Patterns in Negating Quantified Statements: A Study. Proceedings of the Annual Meeting of the Cognitive Science Society, 27(27), 1714–1719.
  2. Alcock, L., & Weber, K. (2010). Undergraduates’ Example Use in Proof Construction: Purposes and Effectiveness. Investigations in Mathematics Learning. https://doi.org/10.1080/24727466.2010.11790298
  3. Andrew, L. (2007). Reasons Why Students Have Difficulties with Mathematical Induction. 1–21. https://files.eric.ed.gov/fulltext/ED495959.pdf
  4. Antonini, S. (2019). Intuitive acceptance of proof by contradiction. ZDM - Mathematics Education, 51(5), 793–806. https://doi.org/10.1007/s11858-019-01066-4
  5. Antonini, S., & Mariotti, M. A. (2006). Reasoning in an Absurd World: Difficulties With Proof By Contradiction. Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education, 2(1945), 65–72.
  6. Antonini, S., & Mariotti, M. A. (2008). Indirect proof: What is specific to this way of proving? ZDM - International Journal on Mathematics Education, 40(3), 401–412. https://doi.org/10.1007/s11858-008-0091-2
  7. Antonini, S., & Mariotti, M. A. (2009). Abduction and the explanation of anomalies: The case of proof by contradiction. In V. Durand-Guerrier, S. Soury-Lavergne, & F. Arzarello (Eds.), Proceedings of the Sixth Congress of the European Society for Research in Mathematics Education (pp. 322–331). INSTITUT NATIONAL DE RECHERCHE PÉDAGOGIQUE.
  8. Baker, J. D. (1996). Students’ Difficulties with Proof by Mathematical Induction. Annual Meeting of the American Educational Research Association, 21.
  9. Balacheff, N. (1991). Treatment of Refutations: Aspects of the complexity of a constructivist approach to mathematics learning. In E. von Glasersfeld (Ed.), Radical Constructivism in Mathematics Education (pp. 89–110). Kluwer Academic.
  10. Barnard, T., & Tall, D. (1997). Cognitive Units, Connections, and Mathematical Proof. Proceedings of the 21st Conference of the International Group for the Psychology of Mathematics Education, 2, 41–28.
  11. Bleiler, S. K., Thompson, D. R., & Krajčevski, M. (2014). Providing written feedback on students’ mathematical arguments: proof validations of prospective secondary mathematics teachers. Journal of Mathematics Teacher Education, 17(2), 105–127. https://doi.org/10.1007/s10857-013-9248-1
  12. Brown, S. (2013). Partial Unpacking and Indirect Proofs: a Study of Students’ Productive Use of the Symbolic Proof Scheme. https://www.researchgate.net/publication/301674910
  13. Brown, S. (2016). When nothing leads to everything: Novices and experts working at the level of a logical theory. In D. Stalvey, Harrison E; Vidakovic (Ed.), Proceedings of the 19th Annual Conference on Research in Undergraduate Mathematics Education (pp. 579–587).
  14. Brown, S. A. (2018). Are indirect proofs less convincing? A study of students’ comparative assessments. Journal of Mathematical Behavior, 49, 1–23. https://doi.org/10.1016/j.jmathb.2016.12.010
  15. Chamberlain, D., & Vidakovic, D. (2016). Developing Student Understanding: The Case of Proof by Contradiction. 2003, 1–15.
  16. Chamberlain, D., & Vidakovic, D. (2021). Cognitive trajectory of proof by contradiction for transition-to-proof students. Journal of Mathematical Behavior, 62(February), 100849. https://doi.org/10.1016/j.jmathb.2021.100849
  17. Dawkins, P. C. (2017). On the Importance of Set-Based Meanings for Categories and Connectives in Mathematical Logic. International Journal of Research in Undergraduate Mathematics Education, 3(3), 496–522. https://doi.org/10.1007/s40753-017-0055-4
  18. Dawson, J. W. (2006). Why do mathematicians re-prove theorems? Philosophia Mathematica, 14(3), 269–286. https://doi.org/10.1093/philmat/nkl009
  19. Doruk, M. (2019). Preservice Mathematics Teachers’ Determination Skills of Proof Techniques: The Case of Integers. International Journal of Education in Mathematics Science and Technology, 7(4), 335–348. https://www.ijemst.net/index.php/ijemst/article/view/729
  20. Doruk, M., & Kaplan, A. (2018). Prospective Mathematics Teachers’ Strategies for Evaluating the Accuracy of Proofs in the Field of Analysis. Çukurova Üniversitesi Eğitim Fakültesi Dergisi, 47(2), 623–666. https://doi.org/10.14812/cuefd.358017
  21. Dubinsky, E. (1986). Teaching Mathematical Induction I. In Journal of Mathematical Behavior (Vol. 5, pp. 305–317).
  22. Dubinsky, E., Elterman, F., & Gong, C. (1988). The Student’s Construction of Quantification. For the Learning of Mathematics, 8(2), 44–51.
  23. Dubinsky, E., & Yiparaki, O. (2000). On student understanding of AE and EA quantification. Research in Collegiate Mathematics Education, IV, Providence: American Mathematical Society., 239–289. https://doi.org/10.1090/cbmath/008/11
  24. Dumas, B. A., & McCarthy, J. E. (2015). Transition to Higher Mathematics: Structure and Proof (Second Edition). In Creative Commons Attribu- tion, NonCommercial License. https://doi.org/10.7936/K7Z899HJ
  25. Epp, S. S. (2003). The Role of Logic in Teaching Proof. The American Mathematical Monthly, 110(10), 886–899. https://doi.org/10.1080/00029890.2003.11920029
  26. Epp, S. S. (2020). A Unified Framework for Proof and Disproof. The Mathematics Teacher, 91(8), 708–740. https://doi.org/10.5951/mt.91.8.0708
  27. Hanna, G. (2000). Proof, explanation and exploration: An overview. Educational Studies in Mathematics, 44, 5–23. https://doi.org/10.1023/A:1012737223465
  28. Hanna, G., & de Villiers, M. (2021). Aspects of Proof in Mathematics Education. In G. Hanna & M. de Villiers (Eds.), Proof and Proving in Mathematics Education: The 19th ICMI Study (Vol. 15, pp. 1–12). International Commission on Mathematical Instruction. https://doi.org/10.1007/978-94-007-2129-6_16
  29. Harel, G. (2001). The Development of Mathematical Induction as a Proof Scheme : A Model for DNR-Based Instruction. Learning and Teaching Number Theory, Journal of Mathematical Behavior, 185–212.
  30. Harel, G., & Sowder, L. (1998). Students’ proof schemes: Results from exploratory studies. CBMS Issues in Mathematics Education_American Mathematical Society, 7, 234–283. https://doi.org/10.1090/cbmath/007/07
  31. Harel, G, & Sowder, L. (2007). Toward comprehensive perspectives on the learning and teaching of proof. Second Handbook of Research on Mathematics Teaching and Learning. https://doi.org/10.4324/9780203882009
  32. Hine, G. (2019). Proof by contradiction: Teaching and learning considerations in the secondary mathematics classroom. Australian Mathematics Education Journal, 1(3), 24–28. https://researchonline.nd.edu.au/edu_article/227
  33. Hoyles, C. (1997). The Curricular Shaping of Students’ Approaches to Proof. For the Learning of Mathematics, 17(1), 7–16.
  34. Inglis, M., & Simpson, A. (2008). Conditional inference and advanced mathematical study. Educational Studies in Mathematics, 67(3), 187–204. https://doi.org/10.1007/s10649-007-9098-9
  35. Jourdan, N., & Yevdokimov, O. (2016). On the analysis of indirect proofs: Contradiction and contraposition. Australian Senior Mathematics Journal, 30(1), 55–64.
  36. Knipping, C. (2008). A method for revealing structures of argumentations in classroom proving processes. ZDM - International Journal on Mathematics Education, 40(3), 427–441. https://doi.org/10.1007/s11858-008-0095-y
  37. Ko, Y. Y. (2010). Mathematics Teachers’ Conceptions of Proof: Implications for Educational Research. International Journal of Science and Mathematics Education, 8(6), 1109–1129. https://doi.org/10.1007/s10763-010-9235-2
  38. Koichu, B. (2012). Enhancing an intellectual need for defining and proving: A case of impossible objects. For the Learning of Mathematics, 32(1), 2–7.
  39. Lee, H. (2019). Notes on Proof by Contrapositive and Proof by Contradiction. 1–2.
  40. Leron, U. (1985). A Direct Approach to Indirect Proofs. Educational Studies in Mathematics, 16(3), 321–325.
  41. Lew, K., & Zazkis, D. (2019). Undergraduate mathematics students’ at-home exploration of a prove-or-disprove task. Journal of Mathematical Behavior, 54(March 2017), 0–1. https://doi.org/10.1016/j.jmathb.2018.09.003
  42. Lin, F.-L., Lee, Y.-S., & Yu, J.-Y. W. (2003). Students’ Understanding of Proof By Contradiction. In & J. Z. Pateman, B. J. Dougherty (Ed.), Proceedings of the 2003 Joint Meeting of PME and PMENA (Issue August 2015, pp. 443–450).
  43. Lin, F., Lee, Y., & Yu, J. W. (1998). Students ’ Understanding of Proof By Contradiction. January, 443–450.
  44. Mejía-Ramos, J. P., Weber, K., & Fuller, E. (2015). Factors Influencing Students’ Propensity for Semantic and Syntactic Reasoning in Proof Writing: a Case Study. International Journal of Research in Undergraduate Mathematics Education, 1(2), 187–208. https://doi.org/10.1007/s40753-015-0014-x
  45. Moore, R. C. (1994). Making the Transition to Formal Proof. Educational Studies in Mathematics, 27(3), 249–266.
  46. Netti, S., & Herawati, S. (2019). Characteristics of Undergraduate Students’ Mathematical Proof Construction on Proving Limit Theorem. KnE Social Sciences, 3(15), 153–163. https://doi.org/10.18502/kss.v3i15.4362
  47. Otani, H. (2019). Comparing Structures of Statistical Hypothesis Testing With Proof By Contradiction : in Terms of Argument. 1–12.
  48. Ozgur, Z., Ellis, A. B., Vinsonhaler, R., Dogan, M. F., & Knuth, E. (2019). From examples to proof: Purposes, strategies, and affordances of example use. Journal of Mathematical Behavior, 53, 284–303. https://doi.org/10.1016/j.jmathb.2017.03.004
  49. Piatek-Jimenez, K. (2010). Students’ interpretations of mathematical statements involving quantification. Mathematics Education Research Journal, 22(3), 41–56. https://doi.org/10.1007/BF03219777
  50. Quarfoot, D., & Rabin, J. M. (2021). A Hypothesis Framework for Students’ Difficulties with Proof By Contradiction. In International Journal of Research in Undergraduate Mathematics Education. International Journal of Research in Undergraduate Mathematics Education. https://doi.org/10.1007/s40753-021-00150-z
  51. Rabin, J. M., & Quarfoot, D. (2021). Sources of Students’ Difficulties with Proof By Contradiction. International Journal of Research in Undergraduate Mathematics Education, July. https://doi.org/10.1007/s40753-021-00152-x
  52. Rav, Y. (1999). Why Do We Prove Theorems? Philosophia Mathematica, 7, 5–41. https://doi.org/10.1093/philmat/7.1.5
  53. Reid, D., & Dobbin, J. (1998). Why is proof by contradiction difficult? Proceedings of the 22nd Conference of the International Group for the Psychology of Mathematics Education, 4(May), 41–48.
  54. Reid, D., Knipping, C., & Crosby, M. (2008). Refutations and the logic of practice. PME 32 and PME-NA XXX, 1–9. https://doi.org/10.30827/pna.v6i1.6148
  55. Relaford-Doyle, J. (2020). Characterizing students ’ conceptual difficulties with mathematical induction using visual proofs. International Journal of Research in Undergraduate Mathematics Education, 1–20.
  56. Relaford-Doyle, J., & Núñez, R. (2021). Characterizing students’ conceptual difficulties with mathematical induction using visual proofs. International Journal of Research in Undergraduate Mathematics Education, 7(1), 1–20. https://doi.org/10.1007/s40753-020-00119-4
  57. Schüler-Meyer, A. (2022). How transition students relearn school mathematics to construct multiply quantified statements. Educational Studies in Mathematics, 110(2), 291–311. https://doi.org/10.1007/s10649-021-10127-z
  58. Selden, A., & Selden, J. (2008). Overcoming students’ difficulties in learning to understand and construct proofs. In Making the Connection: Research and Teaching in Undergraduate Mathematics Education. https://doi.org/10.5948/UPO9780883859759.009
  59. Selden, J., & Selden, A. (2009). Understanding the proof construction process. Proceedings of the ICMI Study 19 Conference: Proof and Proving in Mathematics Education, 2(November), 196–201.
  60. Sellers, M. E. (2018). When “Negation” Impedes Argumentation : The Case of Dawn. 21st Annual Conference on Research in Undergraduate Mathematics Education, September, 242–256.
  61. Sellers, M. E., Roh, K. H., & Parr, E. D. (2021). Student Quantifications as Meanings for Quantified Variables in Complex Mathematical Statements. Journal of Mathematical Behavior, 61(100802), 1–16. https://doi.org/10.1016/j.jmathb.2020.100802
  62. Shipman, B. A. (2016). Subtleties of hidden quantifiers in implication. Teaching Mathematics and Its Applications, 35(1), 41–49. https://doi.org/10.1093/teamat/hrv007
  63. Stavrou, S. G. (2014). Common Errors and Misconceptions in Mathematical Proving by Education Undergraduates. IUMPST: The Journal, 1(March), 1–8.
  64. Stylianides, A. J., Stylianides, G. J., & Philippou, G. N. (2004). Undergraduate students’ understanding of the contraposition equivalence rule in symbolic and verbal contexts. Educational Studies in Mathematics, 55(1–3), 133–162. https://doi.org/10.1023/B:EDUC.0000017671.47700.0b
  65. Tall, D. (1980). Cognitive aspects of proof, with special reference to the irrationality of √2. In In H. Athen & K. Heinz (Eds.) (Ed.), Proceedings of the Third International Conference for the Psychology of Mathmatics Education (Vol. 176, pp. 170–176).
  66. Tall, D., & Mejia-Ramos, J. P. (2010). The Long-Term Cognitive Development of Reasoning and Proof: Explanation and Proof Mathematics. In G. Hanna, H. N. Jahnke, & H. Pulte (Eds.), Explanation and Proof in Mathematics: Philosophical and Educational Perspectives. https://doi.org/10.1007/978-1-4419-0576-5_8
  67. Tall, D., Yevdokimov, O., Koichu, B., Whiteley, W., Kondratieva, M., & Cheng, Y. H. (2012). Cognitive Development of Proof. In New ICMI Study Series 15 (Vol. 15). https://doi.org/10.1007/978-94-007-2129-6_2
  68. Thompson, D. R. (1966). Learning and Teaching Indirect Proof. The Mathematics Teacher, 89(6), 474–482.
  69. Weber, K. (2001). Student difficulty in constructing proofs: The need for strategic knowledge. Educational Studies in Mathematics, 48(1), 101–119. https://doi.org/10.1023/A:1015535614355
  70. Weber, K. (2004). A Framework for Describing the Processes That Undergraduate use to Construct Proofs. Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education, 4, 425–432.
  71. Weber, K. (2008). How mathematicians determine if an argument is a valid proof. Journal for Research in Mathematics Education, 39(4), 431–459.