Contextual Analysis of Proofs Without Words Skills of Pre-service Secondary Mathematics Teachers: Sum of Integers from 1 to n Case

Year 2021, Volume: 11 Issue: 2, 93 - 106, 30.12.2021

Kübra Polat , Handan Demircioğlu

https://doi.org/10.17984/adyuebd.589464

Cited By: 1

Abstract

Recently, studies have been carried out on alternative proof methods due to the change in the perspective of teaching proof and the difficulties of learners in proof. In this context, proof without words, which are presented as an alternative to proof teaching, defined by diagrams or visual representations and require the student to explain how proof is, are discussed in this study. The aim of this study is to examine pre-service mathematics teachers' explanations of proof without words about the sum of consecutive numbers from 1 to n. The data were collected by the proof of the sum of consecutive integers. 27 pre-service teachers from a university in the Middle Anatolia region participated in this study, which was conducted using a basic qualitative research design. At the end of the study, it was seen that most of the pre-service teachers were unable to explain the proof without words of the sum of integers from 1 to n. One of the reason for this may be related to the spatial thinking skills of pre-service teachers. However, there are pre-service teachers who can interpret the visual given in the proof correctly, use the necessary mathematical knowledge, but cannot generalize using the given visual. The reasons why the pre-service teachers could not express the general situation are considered as the lack of algebraic thinking.

Keywords

Proof, Proof teaching, Proof without words, Visual proof

Ortaöğretim Matematik Öğretmen Adaylarının Sözsüz İspat Becerilerinin Bağlamsal İncelenmesi: 1’den n’ye Kadar Olan Tamsayıların Toplamı Durumu

Abstract

İspata yönelik algının son dönemlerde değişmesi ve öğrenenlerin ispatta zorlanmalarından dolayı alternatif ispat yolları ile ilgili araştırmalar yapılmaktadır. Bu bağlamda, ispat öğretimine alternatif bir yol olarak sunulan, ispatın doğruluğunu göstermekle beraber öğrencinin ispatı açıklamasını gerektiren diyagram veya resimler olarak tanımlanan sözsüz ispatlardan biri bu çalışmada ele alınmıştır. Çalışmanın amacı öğretmen adaylarının 1'den n'ye kadar olan ardışık tamsayıların toplamının sözsüz ispatına yönelik açıklamalarını incelemektir. Veriler, ardışık tamsayılar toplamının sözsüz ispatı ile toplanmıştır. Temel nitel araştırma deseninde gerçekleştirilen bu araştırmaya İç Anadolu Bölgesi'ndeki bir üniversiteden 27 öğretmen adayı katılmıştır. Araştırmanın sonucunda matematik öğretmen adaylarının çoğunun, ardışık tamsayıların toplamı ile ilgili verilen sözsüz ispatı açıklayamadıkları görülmüştür. Bu durumun nedenlerinden birisi öğretmen adaylarının uzamsal düşünme yetenekleridir. Bununla birlikte öğretmen adaylarından sözsüz ispattaki görseli doğru biçimde yorumlayabilen, gerekli olan matematiksel bilgiyi kullanabilen ancak verilen görseli genelleyemeyenler de bulunmaktadır. Öğretmen adaylarının genel durumu ifade edememelerinin nedenleri cebirsel düşünme eksikliği olarak değerlendirilebilir.

Keywords

İspat, İspat öğretimi, Sözsüz ispatlar, Görsel ispatlar

References

• Almeida, D. (1996). Variation in proof standards: Implication for mathematics education. International Journal of Mathematical Education in Science and Technology, 27, 659–665. https://doi.org/10.1080/0020739960270504
• Alsina, C., & Nelsen R. (2010). An invitation to proofs without words. European Journal of Pure and Applied Mathematics, 3(1), 118-127. http://www.labjor.unicamp.br/comciencia/files/matematica/ar_roger/ar_roger.pdf
• Arslan, C. (2007). The development of elementary school students on their reasoning and proof ideas. (Publication No. 210145) [Doctoral dissertation, Uludag University]. Available from Council of Higher Education Thesis Center.
• Altun, M. (2014). Liselerde matematik öğretimi. (5. ed). Aktüel Yayıncılık.
• Balacheff, N. (1988). Aspects of proof in pupils' practice of school mathematics (D. En Pimm, Ed.). London: Hodder & Stoughton.
• 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, (pp. 907–920). Beijing. https://arxiv.org/pdf/math/0305021.pdf
• Bardelle, C. (2010). Visual proofs: An experiment. In V. Durand-Guerrier et al (Eds.), CERME6, (251-260). Lyon. France. http://ife.ens-lyon.fr/publications/edition-electronique/cerme6/wg2-08- bardelle.pdf
• Bell, C. J. (2011). Proof without words: A visual application of reasoning. Mathematics Teachers, 104(9), 690–695. https://doi.org/10.5951/MT.104.9.0690 Birinci, K. S. (2010). Investigation of mathematics teacher candidates’ proof processes in terms of process-object relationship (Publication No.279855) [Master’s thesis, Marmara University]. Available from Council of Higher Education Thesis Center.
• Britz, T., Mammoliti, A. & Sørensen, H. K. (2014). Proof by picture: A selection of nice picture proofs. Parabola, 50(3), 1-8.
• Brown, J. R. (1997). Proofs and pictures. British Journal for the Philosophy of Science, 48, 161-180. https://faculty.unlv.edu/jwood/unlv/Articles/Brown.pdf
• Cain, A.J. (2019). Visual thinking and simplicity of proof. Philosophical Transactions A. 377, 1-13. https://doi.org/10.1098/rsta.2018.0032
• Davis, P.J. (1993). Visual theorems. Educational Studies in Mathematics, 24(4), 333–344. https://doi.org/10.1007/BF01273369
• Dove, I. (2002). Can pictures prove? Logique & Analyze, 45(179), 309-340.
• Demircioglu, H., & Polat, K. (2015). Secondary mathematics pre-service teachers’ opinions about proof without words. The Journal of Academic Social Science Studies, 41, 233-254. https://doi.org/10.9761/JASSS3171
• Demircioglu, 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.
• Dogan, M. F. (2019). The nature of middle school ın-service teachers’ engagements in proving-related activities. Cukurova University Faculty of Education Journal, 48(1), 100-130. https://doi.org/10.14812/cufej.442893
• Dogan, M.F. (2020). Pre-service teachers’ criteria for evaluating mathematical arguments that include generic examples. International Journal of Contemporary Educational Research, 7(1), 267-279. https://doi.org/10.33200/ijcer.721136
• Dogan, M. F., & Williams-Pierce, C. (2021). The role of generic examples in teachers’ proving activities. Educational Studies in Mathematics,106, 133–150. https://doi.org/10.1007/s10649-020-10002-3
• Doyle, T., Kutler, L., Miller, R., Schueller, A. (2014). Proof without words and beyond. Mathematical Association of America. https://doi:10.4169/convergence20140801
• Duval R. (1999). Representation, vision and visualization: Cognitive functions in mathematical thinking. Basic issues for learning. In F. Hitt & M. Santos (Eds.). Proceedings of the twenty-first annual meeting of the north American chapter of the international group for the psychology of mathematics education, 1, 3-26. https://files.eric.ed.gov/fulltext/ED466379.pdf
• Flores, A. (2000). Geometric representations in the transition from arithmetic to algebra. In F. Hitt (Ed.). Representation and mathematics Visualization (pp. 9-29).
• Giaquinto, M. (2007). Visual thinking in mathematics: An epistemological study. Oxford, UK: Oxford University Press. https://doi.org/10.1093/acprof:oso/9780199285945.001.0001
• Gierdien, F. (2007). From “Proofs without words” to “Proofs that explain” in secondary mathematics. Pythagoras, 65, 53 – 62. https://doi.org/10.4102/pythagoras.v0i65.92
• Güler, G., & Ekmekci, S. (2016). Examination of the proof evaluation skills of the prospective mathematics teachers: The example of sum of successive odd numbers. Journal of Bayburt Education Faculty, 11(1), 60-81.
• Hanna, G. (1990). Some pedagogical aspects of proof. Interchange, 21(1), 6-13.
• Hanna, G. (2000). Proof, Explanation and Exploration: An Overview. Educational Studies in Mathematics, 44, 5-23. https://doi.org/10.1023/A:1012737223465
• 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. https://doi.org/10.1007/BF02652777
• Jamnik, M., Bundy, A., & Green, I. (1997). Automation of diagrammatic reasoning. 15th international joint conference on artificial intelligence, 1, 528–533. San Mateo, CA. https://www.cl.cam.ac.uk/~mj201/publications/pub873.drii-ijcai1997.pdf
• 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. https://doi.org/10.1088/1742-6596/1097/1/012144
• Kulpa, Z. (2009). Main problems with diagrammatic reasoning. Part 1: The generalization problem. Foundations of Science, 14(1–2), 75–96. https://doi.org/10.1007/s10699-008-9148-5
• Lam, T. T. (2007) Contextual approach in teaching mathematics: An example using the sum of series of positive integers, International Journal of Mathematical Education in Science and Technology, 38(2), 273-282, https://doi.org/10.1080/00207390600913376
• Larson, L.C. (1985). A discrete look at 1+2+ ...+n. The College Mathematics Journal, 16, 369-382. https://doi.org/10.2307/2686996
• 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. https://doi.org/10.1023/A:1012785106627
• Mason, J. (1996). Expressing generality and roots of algebra. In N. Bernarz, C. Kieran, & L. Lee (Eds.), Approaches to algebra: Perspectives for research and teaching (pp. 65-86). Dordrecht: Kluwer Academic Publishers. https://doi.org/10.1007/978-94-009-1732-3_5
• Merriam, S. B. (2013). Nitel araştırma desen ve uygulama için bir rehber (Çev. Ed. Selahattin Turan). Nobel Yayıncılık.
• Miller, R. L. (2012). On proofs without words. http://www.whitman.edu/mathematics/SeniorProjectArchive/2012/Miller.pdf
• Nelsen. R. (1993). Proofs without words: Exercises in visual thinking. Mathematical Association of America.
• Nelsen. R (2000). Proofs without words II: More exercises in visual thinking. Mathematical Association of America.
• Ugurel, I., Morali, 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. https://doi.org/10.18404/ijemst.61686
• Polat, K. (2018). Proofs without words as an alternative proof method: Investigating of high school students’ proof skills. (Publication No.503666) [Doctoral dissertation, Ataturk University]. Available from Council of Higher Education Thesis Center.
• Polat, K. & Akgün, L. (2020). Examining the processes of high school students to do proof without words. Education Reform Journal, 5(1), 8-26. https://doi.org/10.22596/erj2020.05.01.8.26
• Presmeg, N., C. (1986). Visualization in high school mathematics. For the Learning of Mathematics, 6(3), 42-46.
• Rinvold, R. A., & Lorange, A. (2011). Multimodal derivation and proof in algebra. In M. Pytlak, T. Rowland & E. Swoboda (Eds.). Proceedings of CERME 7, 233-242, Poland. http://cerme8.metu.edu.tr/wgpapers/WG1/WG1_Rinvold.pdf
• 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. https://doi.org 10.1207/S15327833MTL0203_4
• Thornton, S. (2001). A picture is worth a thousand words. http://math.unipa.it/grim/AThornton251.PDF