High School Students’ and Their Teacher’s Experiences with Visual Proofs
Yıl: 2023 Cilt: 25 Sayı: 1 Sayfa Aralığı: 126 - 136 Metin Dili: İngilizce DOI: 10.17556/erziefd.1092716 İndeks Tarihi: 01-06-2023
High School Students’ and Their Teacher’s Experiences with Visual Proofs
Öz: Proof is one of the main components of both mathematics and mathematics education. However, students and teachers at every level have difficulties in proof and proof teaching. Presenting alternative proof methods to students will not only provide a rich learning environment but also allow the students to have different perspectives by providing alternative ways they can apply for constructing a proof. The visual proofs are one of the alternative proof methods. The visual proofs are seen as valuable tools for mathematics education; it is planned to investigate the views of high school students and their teacher about visual proof. Case study method is used in the study. Participants of the study consisted of four high school students who were nineth-graders and their teacher. The teacher said that after the visual proof activities, the students started to wonder where the formulas came from, in other words, they started questioning, their interest levels increased, and the formulas were kept in their minds more. The high school students stated that they found the visual proof activities enjoyable, and they were happy when they understood the visual proofs and unhappy when they could not understand them.
Anahtar Kelime: Lise Öğrencilerinin ve Öğretmenlerinin Görsel İspat ile İlgili Deneyimleri
Öz: İspat, matematiğin ve matematik eğitiminin ana bileşenlerinden biridir. Buna rağmen öğrenciler ve öğretmenler her düzeyde ispat öğreniminde ve öğretiminde güçlük yaşamaktadırlar. Bu bağlamda düşünüldüğünde öğrencilere kullanabilecekleri alternatif ispat yöntemlerinin sunulması ile hem zengin öğrenme ortamları oluşturmuş olacak hem de bir ispatı yapmak için başvurulabilecek alternatif yollar sunulmasıyla öğrenciye farklı bakış açıları kazandırılmış olacaktır. Görsel ispatlar ispat öğretiminde alternatif bir yöntemdir. Bu çalışmada görsel ispat etkinliklerinde yer alan lise öğrencileri ve öğretmenlerinin deneyimlerine dönük görüşleri alınmıştır. Çalışmanın modeli durum çalışmasıdır. Katılımcılar 9. sınıfta öğrenim gören ve uygulama sırasında görsel ispatlar ile deneyim yaşayan dört öğrenci ve onların öğretmenleridir. Öğretmen görsel ispat deneyimi yaşayan öğrencilerin sonraki derslerde formüllerin nereden geldiğini merak ettiklerini başka bir deyişle sorgulamaya başladıklarını, derse ilgilerinin arttığını ve formüllerin akıllarında daha çok kaldığını belirtmiştir. Öğrenciler ise görsel ispat etkinliklerini eğlenceli bulduklarını, ispatı anladıkları zaman mutlu olduklarını buna karşın anlamadıkları zaman mutsuz olduklarını belirtmişlerdir.
Anahtar Kelime: 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.
- 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.
- Bardelle, C. (2010). Visual proofs: An experiment. In V. Durand-Guerrier et al (Eds), Paper presented at the annual meeting of CERME6, Lyon, France. INRP, (pp. 251-260).
- 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
- Borwein, P., & Jörgenson, L. (2002). Visible structures in number theory. The American Mathematical Monthly, 108(5), 897-910. Retrieved from https://www.maa.org/sites/default/files/pdf/upload_library /22/Ford/Borwein897- 910.pdf
- Brown, J. R. (2008). Philosophy of mathematics: A contemporary introduction to the world of proofs and pictures. (2nd ed.). Routledge: Taylor & Francis Group. Retrieved from www.bibotu.com/books
- Creswell, J. W. (2013). Qualitative inquiry & Research design: choosing among five approaches. (Translation editör: M. Bütün & S. B. Demir). Ankara: Siyasal Publishing
- Demircioğlu, 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
- 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)
- Doruk, M., Özdemir, F., & Kaplan, A. (2014). The relationship between prospective mathematics teachers’ conceptions on constructing mathematical proof and their self-efficacy beliefs towards mathematics. Kastamonu Education Journal, 23(2), 861-874.
- Doyle, T., Kutler, L., Miller, R. & Schueller, A. (2014). Visual proofs and beyond. Mathematical Connection of America. https://doi.org/10.4169/convergence20140801
- Flores, A. (2000). Geometric representations in the transition from arithmetic to algebra. In F. Hitt (Ed.), North American chapter of the international group for the psychology of mathematics education. Reprensentation and mathematics visualization (pp. 9-29).
- 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
- Hanna, G. (1989). More than formal proof. For the Learning of Mathematics, 9(1), 20-23.
- Hanna, G. (2000). Proof, explanation and exploration: an overview. Educational Studies in Mathematics, 44, 5-23. https://doi.org/10.1023/A:1012737223465
- Hanna, G., & Sidoli, N. (2007). Visualisation and proof: A brief survey of philosophical perspectives. ZDM Mathematics Education, 39(1–2), 73–78. https://doi.org/10.1007/s11858-006-0005-0
- Harel, G., & Sowder, L. (1998). Students’ proof schemes: Results from exploratory studies. CBMS Issues in Mathematics Education, 7, 234-282. https://doi.org/10.1090/cbmath/007/07
- Hawro, J. (2007). University students’ difficulties with formal proving and attempts to overcome them. CERME-5, 2290- 2299. http://www.erme.tu- dortmund.de/~erme/CERME5b/WG14.pdf#page=72 Accessed 15 February 2017.
- 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
- Herbert, S., Colleen, V., Bragg, L.A., & Widjaja, W. (2015). A framework for primary teacher’s perceptions of mathematical reasoning. International Journal of Educational Research, 74, 26-37. https://doi.org/10.1016/j.ijer.2015.09.005
- İpek, S. (2010). The investigation of preservice elementary mathematics teachers? geometric and algebraic proof processes by using dynamic geometry software [Master thesis]. YÖK. (Thesis Number. 265219)
- Jamnik, M., Bundy, A., & Green, I. (1997). Automation of diagrammatic reasoning. In: 15th International Joint Conference on Artificial Intelligence, San Mateo, CA 1, 528–533. https://www.cl.cam.ac.uk/~mj201/publications/pub873.dr ii-ijcai1997.pdf Accessed 10 February 2017
- 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. https://doi.org/10.1088/1742- 6596/1097/1/012144
- Knuth, E. (2002). Secondary school mathematics teachers' conceptions of proof. Journal for Research in Mathematics Education 33(5), 379-405. https://doi.org/10.2307/4149959
- Lockhart, P. (2009). A Mathematician’s Lament. https://www.maa.org/external_archive/devlin/LockhartsL ament.pdf Accessed 13 February 2017.
- Marco, N., Palatnik, A., & Schwarz, B. B. (2022). When is less more? Investigating gap-filling in proofs without words activities. Educational Studies in Mathematics, 111(2), 271-297.
- Merriam, S. B. (2013). Qualitative research A guide to design and implementation (S. Turan Translation editor.). Ankara: Nobel Publishing.
- Miller R. L. (2012). On proofs without words. http://www.whitman.edu/mathematics/SeniorProjectArchi ve/2012/Miller.pdf Accessed 10 February 2015
- Ministry of National Education (2013). Curriculum of Secondary School Mathematics (9,10,11 and 12.) MNE Ankara.
- Ministry of National Education (2018). Curriculum of Secondary School Mathematics (9,10,11 and 12.) MNE Ankara.
- Morash, R. P. (1987). Bridge to abstract mathematics: Mathematical proof and structures, New York: Random House.
- Mudaly, V. (2013). Is proving a visual act? Mevlana International Journal of Education. Special Issue: Dynamic and Interactive Mathematics Learning Environments, 3(3), 36-44. https://doi.org/10.13054/mije.si.2013.04
- National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.
- Nelsen, R. (1993). Proofs without words: Exercises in visual thinking. Washington: Mathematical Connection of America.
- Nelsen, R (2000). Proofs without words II: More exercises in visual thinking. Washington: Mathematical Connection of America.
- Nirode, W. (2017). Proofs without words in geometry. The Mathematics Teacher, 110(8), 580-586.
- Polat, K., & Demircioğlu, H. (2016). Proofs without words in Mathematics Education: A Theoretical Study. Journal of Ziya Gökalp Faculty of Education, 28, 129-140. https://doi.org/10.14582/DUZGEF.686
- Polat, K. & Demircioğlu, H. (2021). Contextual analysis of proofs without words skills of pre-service secondary mathematics teachers: sum of integers from 1 to n case. Adiyaman Univesity Journal of Educational Sciences, 11(2), 93-106.
- Reid, D. & Vallejo Vargas (2016). When Is a Generic Argument a Proof? 13th International Congress on Mathematical Education Hamburg, 24-31 July.
- Rinvold R. A., & Lorange A. (2013). Multimodal proof in arithmetic. Proceedings of CERME 8, Antalya, 16-225. www.mathematik.unidortmund.de/~erme/doc/CERME8/ CERME8_2013_Proceedings.pdf Accessed 10 September 2017.
- 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
- Sigler, A., Segal R, & Stupel M. (2016). The standard proof, the elegant proof, and the proofs without words of tasks in geometry, and their dynamic investigation. International Journal of Mathematics Education Scientific Technology, 47(8), 1226–1243. https://doi.org/10.1080/0020739X.2016.1164347
- Strausova, I. & Hasek, R. (2012). “Dynamic visual proofs” using DGS. The Electronic Journal of Mathematics and Technology, 7(2), 130-143.
- Tekin, B. & Konyalıoğlu, A. C. (2010). Visualization of sum and difference formulas of trigonometric functions at secondary level. Bayburt University Journal of Education Faculty, 5(1-2), 24-37.
- Uğurel, I., & Moralı, S. (2010). A close view on the discussion in relation to an activity about proving a theorem in a high school mathematics lesson via students’ discourse. Journal of Buca Education Faculty, 28, 135-154.
- 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. https://doi.org/10.18404/ijemst.61686
- 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.
- Weber, K. (2001). Student difficulty in constructing proof: The need for strategic knowledge. Educational Studies in Mathematics, 48(1), 101–119. https://doi.org/10.1023/A:1015535614355
| APA | Polat K, Akgun L (2023). High School Students’ and Their Teacher’s Experiences with Visual Proofs. , 126 - 136. 10.17556/erziefd.1092716 |
| Chicago | Polat Kübra,Akgun Levent High School Students’ and Their Teacher’s Experiences with Visual Proofs. (2023): 126 - 136. 10.17556/erziefd.1092716 |
| MLA | Polat Kübra,Akgun Levent High School Students’ and Their Teacher’s Experiences with Visual Proofs. , 2023, ss.126 - 136. 10.17556/erziefd.1092716 |
| AMA | Polat K,Akgun L High School Students’ and Their Teacher’s Experiences with Visual Proofs. . 2023; 126 - 136. 10.17556/erziefd.1092716 |
| Vancouver | Polat K,Akgun L High School Students’ and Their Teacher’s Experiences with Visual Proofs. . 2023; 126 - 136. 10.17556/erziefd.1092716 |
| IEEE | Polat K,Akgun L "High School Students’ and Their Teacher’s Experiences with Visual Proofs." , ss.126 - 136, 2023. 10.17556/erziefd.1092716 |
| ISNAD | Polat, Kübra - Akgun, Levent. "High School Students’ and Their Teacher’s Experiences with Visual Proofs". (2023), 126-136. https://doi.org/10.17556/erziefd.1092716 |
| APA | Polat K, Akgun L (2023). High School Students’ and Their Teacher’s Experiences with Visual Proofs. Erzincan Üniversitesi Eğitim Fakültesi Dergisi, 25(1), 126 - 136. 10.17556/erziefd.1092716 |
| Chicago | Polat Kübra,Akgun Levent High School Students’ and Their Teacher’s Experiences with Visual Proofs. Erzincan Üniversitesi Eğitim Fakültesi Dergisi 25, no.1 (2023): 126 - 136. 10.17556/erziefd.1092716 |
| MLA | Polat Kübra,Akgun Levent High School Students’ and Their Teacher’s Experiences with Visual Proofs. Erzincan Üniversitesi Eğitim Fakültesi Dergisi, vol.25, no.1, 2023, ss.126 - 136. 10.17556/erziefd.1092716 |
| AMA | Polat K,Akgun L High School Students’ and Their Teacher’s Experiences with Visual Proofs. Erzincan Üniversitesi Eğitim Fakültesi Dergisi. 2023; 25(1): 126 - 136. 10.17556/erziefd.1092716 |
| Vancouver | Polat K,Akgun L High School Students’ and Their Teacher’s Experiences with Visual Proofs. Erzincan Üniversitesi Eğitim Fakültesi Dergisi. 2023; 25(1): 126 - 136. 10.17556/erziefd.1092716 |
| IEEE | Polat K,Akgun L "High School Students’ and Their Teacher’s Experiences with Visual Proofs." Erzincan Üniversitesi Eğitim Fakültesi Dergisi, 25, ss.126 - 136, 2023. 10.17556/erziefd.1092716 |
| ISNAD | Polat, Kübra - Akgun, Levent. "High School Students’ and Their Teacher’s Experiences with Visual Proofs". Erzincan Üniversitesi Eğitim Fakültesi Dergisi 25/1 (2023), 126-136. https://doi.org/10.17556/erziefd.1092716 |
