Examining the Processes of High School Students to do Proof without Words

Abstract

There are opinions about approaching the proof in mathematics education need to change as providing students to understand the mathematical proof rather than developing formal mathematical proof skills. In this context, proofs without words described as informal proofs can be used in proof teaching. The purpose of this study is to investigate the processes of high school students to do proofs without words. This study was designed as the case study. This study was conducted in a high school in Turkey. Consequently, activities of proof without words that allow the student to take an active role in the proof process can be presented as an alternative method in proof teaching and the proof without words can be used for teaching students the stages of proof process. It is necessary for teachers to be aware of the proof process stages and guide the students.

Keywords

• Proof without words,proof process

References

1. Alibert, D., & Thomas, M. (1991). Research on mathematical proof. In D. Tall (Ed.), Advanced mathematical thinking (pp. 215-230). Dordrecht, The Netherlands: Kluwer.
2. Alsina, C., & Nelsen R. (2010). An invitation to proofs without words. European Journal of Pure and Applied Mathematics, 3(1), 118-127. Retrieved from http://www.labjor.unicamp.br/comciencia/files/matematica/ar_roger/ar_roger.pdf
3. Ball, D. L., Hoyles, C., Jahnke, H. N., & Movshovitz-Hadar, N. (2002). The teaching of proof. In L. I. Tatsien (Ed.), Proceedings of the International Congress of Mathematicians, Beijing, 3, 907–920. Retrieved from https://arxiv.org/pdf/math/0305021.pdf
4. Bell, C. J. (2011). Proof without words: A visual application of reasoning. Mathematics Teachers, 104(9), 690–695. Retrieved from http://is234mathforum.webs.com
5. Boero, P. (1999). Argumentation and mathematical proof: a complex, productive, unavoidable relationship in mathematics and mathematics education. International Newsletter on The Teaching and Learning of Mathematical Proof, 7,8. Retrieved from http://www.lettredelapreuve.org/OldPreuve/Newsletter/990708Theme/990708ThemeUK.html
6. Coe, R., & Ruthven, K. (1994). Proof practices and constructs of advanced mathematics students. British Educational Research Journal, 20(1), 41-53. doi:10.1080/0141192940200105
7. Çalışkan, Ç. (2012). The interrelations between 8th grade class students' mathematics success and proving levels. (Master’s thesis). Available from Council of Higher Education Thesis Center. (UMI No. 328655)
8. Davis, P.J. ( 1993). Visual theorems. Educational Studies in Mathematics, 24(4), 333–344. doi:10.1007/BF01273369

9. Demircioğlu, H., & Polat, K. (2016). Secondary mathematics pre-service teachers’ opinions about the difficulties with “proof without words”. International Journal of Turkish Education Sciences, 4(7), 82-99.Doruk, M. (2016). Investigation of preservice elementary mathematics teachers' argumentation and proof processes in domain of analysis (Doctoral dissertation). Available from Council of Higher Education Thesis Center. (UMI No. 433823)
10. Doyle, T., Kutler, L., Miller, R. & Schueller, A. (2014). Proof without words and beyond. Mathematical Association of America. doi:10.4169/convergence20140801Gierdien, F. (2007). From “Proofs without words” to “Proofs that explain” in secondary mathematics. Pythagoras, 65, 53 – 62. doi:10.4102/pythagoras.v0i65.92 Hanna, G. (2000). Proof, Explanation and Exploration: An Overview. Educational Studies in Mathematics, 44, 5-23. doi:10.1023/A:1012737223465
11. Hanna, G., & de Villiers, M. (Eds.). (2012). Proof and proving in mathematics education: The 19th ICMI study. New York: Springer.
12. Harel, G. & Sowder, L. (2007). Toward comprehensive perspectives on the learning and teaching of proof [DX Reader version]. Retrieved from http://www.math.ucsd.edu/~harel/publications/Downloadable/TowardComprehensivePerspective.pdf
13. Hauben, M. (2018). A visual aid for teaching the Mann-Whitney U formula. Teaching Statistics, 4(2), 60-63. doi.org/10.1111/test.12155
14. Heinze, A., Cheng, Y. H., Ufer, S., Lin, F.L., & Reiss, K. (2008). Strategies to foster students’ competencies in constructing multi-steps geometric proofs: Teaching experiments in Taiwan and Germany. ZDM International Journal on Mathematics Education, 40, 443-453. doi:10.1007/s11858-008-0092-1
15. Heinze, A., & Reiss, K. (2004). The teaching of proof at lower secondary level—a video study. ZDM International Journal on Mathematics Education, 36(3), 98–104. doi: 10.1007/BF02652777
16. Inam, B., Uğurel, I., & Boz Yaman, B. (2018). High school students’ performances on proof comprehension tests. International Journal of Assessment Tools in Education, 5(2), 339-369. doi: 10.21449/ijate.416261
17. Karras, M. (2012). Diagrammatic Reasoning Skills of Pre-Service Mathematics Teachers. (Doctoral dissertation). Retrieved from ProQuest LLC.Kristiyajati, A. & Wijaya, A. (2018). Teachers' perception on the use of "Proof without Words (PWWs)" visualization of arithmetic sequences. Journal of Physics: Conference Series, 1097, 1-7.
18. Marrades R., & Gutierrez A. (2000). Proofs produced by secondary school students learning geometry in a dynamic computer environment. Educational Studies in Mathematics 44, 87–125. doi:10.1023/A:1012785106627
19. Miller, D., Infante, N. & Weber, K. (2018). How mathematicians assign points to students proofs. The Journal of Mathematical Behavior, 49, 24-34. https://doi.org/10.1016/j.jmathb.2017.03.002
20. Moore, R. C. (1994). Making the transition to formal proof. Educational Studies in Mathematics, 27,249-266. doi:10.1007/BF01273731
21. Nelsen. R. (1993). Proofs without words: Exercises in visual thinking. Washington: Mathematical Association of America.
22. Öztürk, T. (2016). The evaluation of the learning environment designed for improving pre-service mathematics teachers' proving skills (Doctoral dissertation). Available from Council of Higher Education Thesis Center. (UMI No. 448319)
23. Reiss, K. M., Heinze, A. Renkl, A., & Gross, C. (2008). Reasoning and Proof in Geometry: Effects Of A Learning Environment Based On Heuristic Worked-Out Examples, ZDM Mathematics Education, 40, 455-467. doi:10.1007/s11858-008-0105-0
24. Reiss, K., Hellmich, F., & Reiss, M. (2002). Reasoning and proof in geometry: Prerequisites of knowledge acquisition in secondary school students. In A. D. Cockburn & E. Nardi (Eds.), Proceedings of The 26th Conference of The International Group For The Psychology of Mathematics Education, Norwich: University of East Anglia, 4, 113–120. doi: 10.1007/s11858-008-0105-0
25. Rodd, M. M. (2000). On mathematical warrants: Proof does not always warrant, and a warrant may be other than a proof. Mathematical Thinking and Learning, 2(3), 221–244. doi:10.1207/S15327833MTL0203_4
26. Sigler A, Segal R, & Stupel M. (2016).The standard proof, the elegant proof, and the proof without words of tasks in geometry, and their dynamic investigation. International Journal of Mathematics Education Scientific Technology, 47(8), 1226–1243. doi:10.1080/0020739X.2016.1164347
27. Strausova, I. & Hasek, R. (2012). “Dynamic visual proofs” using DGS. The Electronic Journal of Mathematics and Technology, 7(2), 130-143. Retrieved from http://eds.b.ebscohost.com
28. Tall, D., Yevdokimov, O. Koichu, B., Whiteley, W., Kondratieva, M., & Cheng, Y.H. (2011). Cognitive development of proof. Proof and Proving in Mathematics Education, 13-49. doi:10.1007/978-94-007-2129-6_2
29. Uğurel, I., & Moralı , S. (2010). A close view on the dissussion in relation to a activity about proving a theorem in a high school mathematics lesson via students’ discourse. Buca Eğitim Fakültesi Dergisi, 28, 135-154. Retrieved from http://acikerisim.deu.edu.tr/xmlui/bitstream/handle/12345/117/pdf_91.pdf?sequence=1&isAllowed=y
30. Uğurel, I., Moralı, H. S., Karahan, Ö., & Boz, B. (2016). Mathematically gifted high school students’ approaches to developing visual proofs (vp) and preliminary ideas about VP. International Journal of Education in Mathematics, Science and Technology, 4(3), 174-197. doi:10.18404/ijemst.61686
31. Urhan, S., & Bülbül, A. (2016). The Relationship is between argumentation and mathematical proof processes. Necatibey Faculty of Education Electronic Journal of Science and Mathematics Education, 10,(1), 351-373. Retrieved fromhttp://dergipark.ulakbim.gov.tr/balikesirnef/article/view/5000194112
32. Yin, R. K. (2003). Case study research design and methods (3rd edit.). London: Sage Publication.