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

Polat, Kübra; DEMIRCIOGLU, HANDAN

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

Kübra Polat, (Sivas Cumhuriyet Üniversitesi, Eğitim Fakültesi, Sivas, Türkiye)

Handan Demircioğlu (Sivas Cumhuriyet Üniversitesi, Eğitim Fakültesi, Sivas, Türkiye)

Adıyaman Üniversitesi Eğitim Bilimleri Dergisi

7 2

Yıl: 2021 Cilt: 11 Sayı: 2 Sayfa Aralığı: 93 - 106 Metin Dili: İngilizce İndeks Tarihi: 08-12-2022

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

Öz:

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 preservice 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.

Anahtar Kelime: Proof Proof teaching Proofs without words Visual proofs

Belge Türü: Makale Makale Türü: Araştırma Makalesi Erişim Türü: Erişime Açık

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 processobject 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 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

APA	Polat K, DEMIRCIOGLU H (2021). Contextual Analysis of Proofs Without Words Skills of Pre-service Secondary Mathematics Teachers: Sum of Integers from 1 to n Case. , 93 - 106.
Chicago	Polat Kübra,DEMIRCIOGLU HANDAN Contextual Analysis of Proofs Without Words Skills of Pre-service Secondary Mathematics Teachers: Sum of Integers from 1 to n Case. (2021): 93 - 106.
MLA	Polat Kübra,DEMIRCIOGLU HANDAN Contextual Analysis of Proofs Without Words Skills of Pre-service Secondary Mathematics Teachers: Sum of Integers from 1 to n Case. , 2021, ss.93 - 106.
AMA	Polat K,DEMIRCIOGLU H Contextual Analysis of Proofs Without Words Skills of Pre-service Secondary Mathematics Teachers: Sum of Integers from 1 to n Case. . 2021; 93 - 106.
Vancouver	Polat K,DEMIRCIOGLU H Contextual Analysis of Proofs Without Words Skills of Pre-service Secondary Mathematics Teachers: Sum of Integers from 1 to n Case. . 2021; 93 - 106.
IEEE	Polat K,DEMIRCIOGLU H "Contextual Analysis of Proofs Without Words Skills of Pre-service Secondary Mathematics Teachers: Sum of Integers from 1 to n Case." , ss.93 - 106, 2021.
ISNAD	Polat, Kübra - DEMIRCIOGLU, HANDAN. "Contextual Analysis of Proofs Without Words Skills of Pre-service Secondary Mathematics Teachers: Sum of Integers from 1 to n Case". (2021), 93-106.

APA	Polat K, DEMIRCIOGLU H (2021). Contextual Analysis of Proofs Without Words Skills of Pre-service Secondary Mathematics Teachers: Sum of Integers from 1 to n Case. Adıyaman Üniversitesi Eğitim Bilimleri Dergisi, 11(2), 93 - 106.
Chicago	Polat Kübra,DEMIRCIOGLU HANDAN Contextual Analysis of Proofs Without Words Skills of Pre-service Secondary Mathematics Teachers: Sum of Integers from 1 to n Case. Adıyaman Üniversitesi Eğitim Bilimleri Dergisi 11, no.2 (2021): 93 - 106.
MLA	Polat Kübra,DEMIRCIOGLU HANDAN Contextual Analysis of Proofs Without Words Skills of Pre-service Secondary Mathematics Teachers: Sum of Integers from 1 to n Case. Adıyaman Üniversitesi Eğitim Bilimleri Dergisi, vol.11, no.2, 2021, ss.93 - 106.
AMA	Polat K,DEMIRCIOGLU H Contextual Analysis of Proofs Without Words Skills of Pre-service Secondary Mathematics Teachers: Sum of Integers from 1 to n Case. Adıyaman Üniversitesi Eğitim Bilimleri Dergisi. 2021; 11(2): 93 - 106.
Vancouver	Polat K,DEMIRCIOGLU H Contextual Analysis of Proofs Without Words Skills of Pre-service Secondary Mathematics Teachers: Sum of Integers from 1 to n Case. Adıyaman Üniversitesi Eğitim Bilimleri Dergisi. 2021; 11(2): 93 - 106.
IEEE	Polat K,DEMIRCIOGLU H "Contextual Analysis of Proofs Without Words Skills of Pre-service Secondary Mathematics Teachers: Sum of Integers from 1 to n Case." Adıyaman Üniversitesi Eğitim Bilimleri Dergisi, 11, ss.93 - 106, 2021.
ISNAD	Polat, Kübra - DEMIRCIOGLU, HANDAN. "Contextual Analysis of Proofs Without Words Skills of Pre-service Secondary Mathematics Teachers: Sum of Integers from 1 to n Case". Adıyaman Üniversitesi Eğitim Bilimleri Dergisi 11/2 (2021), 93-106.