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İspat Sürecinden Yansımalar: Öğretmen Adaylarının İleri-Geri Tekniğini Kullanma Durumlarının İncelenmesi

Year 2023, Volume: 40-2 Issue: 2, 227 - 260, 15.10.2023

Neslihan Sönmez Tuğba Öztürk Bülent Güven

https://doi.org/10.52597/buje.1289328

Abstract

Çalışmanın amacı, matematik öğretmen adaylarının ispatlama sürecindeki zihinsel eylemlerini ileri-geri tekniği aracılığıyla resmedebilmektir. Bu amaçla akademik başarı düzeyi iyi, orta ve düşük olarak belirlenen üç matematik öğretmeni adayıyla geometri ve cebir alanından toplam iki soru üzerinden klinik mülakatlar gerçekleştirilmiştir. Öğretmen adaylarının ispat süreci analiz edilerek ileri-geri yöndeki zihinsel eylem haritaları oluşturulmuş ve bu haritaların ispatı tamamlamadaki rolü tartışılmıştır. Araştırma sonucunda öğretmen adaylarının akademik başarı düzeyi ve sorunun alanı fark etmeksizin ispat sürecinde ileri-geri hamlelerini bilinçli olmasa da yoğun bir şekilde gerçekleştirdiği belirlenmiştir. Adayların akademik başarısı arttıkça ispat sürecindeki ileri ve geri hamle sayılarının arttığı belirlenmiştir. Bu bakımdan öğretim üyelerinin ispat yaparken bu teknik aracılığıyla düşüncelerini sesli olarak ifade etmeleri, öğretmen adayları için ispatı yaratıcı bir inşa sürecine dönüştürme imkânı sunabilir.

Keywords

ispat ileri-geri tekniği öğretmen adayı matematik öğretmen adayı

References

  • Atiyah, M. (2001). Mathematics in the 20th century: Geometry versus algebra. Mathematics Today, 37(2), 46–53.
  • Attouch, H., & Peypouquet, J. (2016). The rate of convergence of nesterov’s accelerated forward-backward method is actually faster than 1/k2. SIAM Journal on Optimization, 26(3), 1824–1834. https://doi.org/10.48550/arXiv.1510.08740
  • Ball, D. L., Hoyles, C., Jahnke, H. N., & Movshovitz-Hadar, N. (2002). The teaching proof. L. I. Tatsien (Haz.), Proceedings of the International Congress of Mathematicians içinde (Cilt. III, s. 907–920). Higher Education Press.
  • Buluş, M., Duru, E., Balkıs, M., & Duru, S. (2011). Öğretmen adaylarında öğrenme stratejilerinin ve bireysel özelliklerin akademik başarıyı yordamadaki rolü. Eğitim ve Bilim, 36(161), 186–197.
  • Chang, S. S., Wen, C. F., Yao, J. C., & Zhang, J. Q. (2017). A generalized forward-backward method for solving split equality quasi inclusion problems in Banach spaces. Journal of Nonlinear Sciences and Application, 10, 4890–4900. http://dx.doi.org/10.22436/jnsa.010.09.29
  • De Guzman, M., Hodgson, B. R., Robert, A., & Villani, V. (1998, August). Difficulties in the passage from secondary to tertiary education. Paper presented at the International Congress of Mathematicians, Berlin.
  • Dreyfus, T. ve Eisenberg, T. (1996). On different facets of mathematical thinking. R. J. Sternberg, & T. Ben-Zeev (Haz.), The nature of mathematical thinking içinde (s. 253-284). Lawrence Erlbaum Associates.
  • Garavalia, L. S. ve Gredler, M. E. (2002). Prior achievement, aptitude, and use of learning strategies as predictors of college student achievement. College Student Journal, 36(4), 616–641.
  • Heinze, A. ve Reiss, K. (2004). The teaching of proof at the lower secondary level – a video study. ZDM Mathematics Education, 36(3), 98–104. https://doi.org/10.1007/BF02652777
  • 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 Mathematics Education, 40, 443–453. https://doi.org/10.1007/s11858-008-0092-1
  • Hine, G., & McNab, N. (2014). Mathematics specialist: Year 11 ATAR course-Units 1 ve 2 (Australian curriculum). Academic Associates.
  • Jones, K. (2000). The student experience of mathematical proof at university level. International Journal of Mathematical Education in Science and Technology, 31(1), 53-60. https://doi.org/10.1080/002073900287381
  • Kankam, K., Pholasa, N., & Cholamjiak, P. (2019). On convergence and complexity of the modified forward-backward method involving new linesearches for convex minimization. Mathematical Methods in Applied Sciences, 42(5), 1352–1362. https://doi.org/10.1002/mma.5420
  • Krutetskii, V. A. (1969). An investigation of mathematical abilities in school children. J. Kilpatrick, ve I. Wirszup (Haz.), Soviet studies in the psychology of learning and teaching mathematics içinde (s. 5–57). University of Chicago Press.
  • Lakatos, I. (1976). Proofs and refutations: The logic of mathematical discovery. Cambridge University Press.
  • Lee, K. (2016). Students’ proof schemes for mathematical proving and disproving of propositions. Journal of Mathematical Behavior, 41, 26–44. https://doi.org/10.1016/j.jmathb.2015.11.005
  • Lin, F. L., & Yang, K. L. (2007). The reading comprehension of geometric proofs: The contribution of knowledge and reasoning. International Journal of Science and Mathematics Education, 5(4), 729–754. https://doi.org/10.1007/s10763-007-9095-6
  • Manin, Y. (1992, August). Contribution in panel discussion on “The theory and practice of proof”. Paper presented at the 7th International Congress on Mathematical Education (ICME-7), Quebec, Canada.
  • Matsuda, N., & VanLehn, K. (2004). GRAMY: A geometry theorem prover capable of construction. Journal of Automated Reasoning, 32, 3–33. https://doi.org/10.1023/B:JARS.0000021960.39761
  • Milli Eğitim Bakanlığı [MEB]. (2018). Ortaöğretim matematik dersi (9, 10, 11 ve 12. sınıflar) öğretim programı. MEB Yayınları.
  • Miyazaki, M., Fujita, T., Jones, K., & Iwanaga, Y. (2017). Designing a web-based learning support system for flow-chart proving in school geometry. Digital Experiences in Mathematics Education, 3, 233–256. https://doi.org/10.1007/s40751-017-0034-z
  • Miyazaki, M., Nagata, J., Chino, K., Fujita, T., Ichikawa, D., Shimizu, S., & Iwanaga, Y. (2016,). Developing a curriculum for explorative proving in lower secondary school geometry. G. Kaiser (Haz.), Proceedings of the 13th International Congress on Mathematical Education içinde (s. 1–4). Springer.
  • Moore, R. C. (1994). Making the transition to formal proof. Educational Studies in Mathematics, 27, 249–266. https://doi.org/10.1007/BF01273731
  • Otani, H. (2019). Comparing structures of statistical hypothesis testing with proof by contradiction: In terms of argument. Hiroshima Journal of Mathematics Education, 2, 1–12.
  • Öztürk, T. (2016). Matematik öğretmeni adaylarının ispatlama becerilerini geliştirmeye yönelik tasarlanan öğrenme ortamının değerlendirilmesi [Yayınlanmamış doktora tezi]. Karadeniz Teknik Üniversitesi.
  • Palatnik, A., & Dreyfus, T. (2019). Students’ reasons for introducing auxiliary lines in proving situations. The Journal of Mathematical Behavior, 55. https://doi.org/10.1016/j.jmathb.2018.10.004
  • Perry, P., Molina, Ó., Camargo, L., & Samper, C. (2011, February). Analyzing the proving activity of a group of three students. Paper presented at Congress of the European Society for Research in Mathematics Education (CERME 7), Poland.
  • Polya, G. (1957). How to solve it (2. baskı). Princeton University Press.
  • Remillard, K. (2014). Identifying discursive entry points in paired-novice discourse as a first step in penetrating the paradox of learning mathematical proof. Journal of Mathematical Behavior, 34, 99–113. https://doi.org/10.1016/j.jmathb.2014.02.002
  • Siegler, R. S., & Wagner Alibali, M. (2005). Children’s thinking (4. baskı). Pearson Prentice Hall.
  • Solow, D. (2014). How to read and do proofs: An introduction to mathematical thought processes (6. baskı). John Wiley ve Sons.
  • Spiro, R. J., & Jehng, J. (1990). Cognitive flexibility and hypertext: Theory and technology for the non-linear and multidimensional traversal of complex subject matter. D. Nix, ve R. Spiro (Haz.), Cognition, education, and multimedia içinde (s. 163–205). Erlbaum.
  • Tall, D. (2002). The psychology of advanced mathematical thinking. D. Tall (Haz.), Advanced mathematical thinking içinde (s. 3–21). Springer.
  • Tall, D. (1995). Cognitive growth in elementary and advanced mathematical thinking. L. Meira, ve D. Carraher (Haz.), Proceedings of the 19th Meeting of the International Group for the Psychology of Mathematics Education içinde (Cilt. I., s. 61–75). Universidade Federal de Pernambuco.
  • Tsujiyama, Y. (2011). On the role of looking back at proving processes in school mathematics: Focusing on argumentation. M. Pytlak, T. Rowland & E. Swoboda (Haz.), Proceedings of the 7th Congress of the European Society for Research in Mathematics Education içinde (s. 161–171). University of Rzeszów.
  • Warner, L. B., Alcock, L. J., Coppolo Jr., J., & Davis, G. E. (2003). How does flexible mathematical thinking contribute to the growth of understanding? N. A. Pateman, B. J. Dougherty, & J. Zillox (Haz.), Proceedings of the Twenty-Seventh Conference of the International Group for the Psychology of Mathematics Education içinde (Cilt. IV., s. 371–378). PME.
  • Yang, K. L., & Lin, F. L. (2012). Effects of reading-oriented tasks on students’ reading comprehension of geometry proof. Mathematics Education Research Journal, 24, 215–238. https://doi.org/10.1007/s13394-012-0039-2
  • Weber, K. (2005). Problem-solving, proving, and learning: The relationship between problem solving processes and learning opportunities in the activity of proof construction. Journal of Mathematical Behavior, 24, 351–360. https://doi.org/10.1016/j.jmathb.2005.09.005

The Analysis of the Employment of the Forward-Backward Technique by Mathematics Pre-service Teachers

Year 2023, Volume: 40-2 Issue: 2, 227 - 260, 15.10.2023

Neslihan Sönmez Tuğba Öztürk Bülent Güven

https://doi.org/10.52597/buje.1289328

Abstract

The study aims to depict the mental actions of mathematics pre-service teachers during the process of proof, with the employment of the forward-backward method. Clinical interviews were conducted with three pre-service mathematics teachers, whose academic achievements were good, moderate and low, based on one geometry and one algebra questions. Mental maps for the forward-backward technique were developed based on the analysis of the proof processes adopted by each pre-service teacher, and the role of these maps in the achievement of the proof was discussed. The results demonstrated that three pre-service teachers intensively employed the forward-backward technique in proof, albeit not always consciously, regardless of their academic achievement levels and the field of the question. It was determined that the number of forward-backward steps increased with the increase in academic achievements of the pre-service teachers. Since the technique allows the faculty members to express their ideas verbally, this could provide the opportunity to turn proof into a creative construction process for the pre-service teachers.

Keywords

proof forward-backward technique pre-service mathematics teacher

References

  • Atiyah, M. (2001). Mathematics in the 20th century: Geometry versus algebra. Mathematics Today, 37(2), 46–53.
  • Attouch, H., & Peypouquet, J. (2016). The rate of convergence of nesterov’s accelerated forward-backward method is actually faster than 1/k2. SIAM Journal on Optimization, 26(3), 1824–1834. https://doi.org/10.48550/arXiv.1510.08740
  • Ball, D. L., Hoyles, C., Jahnke, H. N., & Movshovitz-Hadar, N. (2002). The teaching proof. L. I. Tatsien (Haz.), Proceedings of the International Congress of Mathematicians içinde (Cilt. III, s. 907–920). Higher Education Press.
  • Buluş, M., Duru, E., Balkıs, M., & Duru, S. (2011). Öğretmen adaylarında öğrenme stratejilerinin ve bireysel özelliklerin akademik başarıyı yordamadaki rolü. Eğitim ve Bilim, 36(161), 186–197.
  • Chang, S. S., Wen, C. F., Yao, J. C., & Zhang, J. Q. (2017). A generalized forward-backward method for solving split equality quasi inclusion problems in Banach spaces. Journal of Nonlinear Sciences and Application, 10, 4890–4900. http://dx.doi.org/10.22436/jnsa.010.09.29
  • De Guzman, M., Hodgson, B. R., Robert, A., & Villani, V. (1998, August). Difficulties in the passage from secondary to tertiary education. Paper presented at the International Congress of Mathematicians, Berlin.
  • Dreyfus, T. ve Eisenberg, T. (1996). On different facets of mathematical thinking. R. J. Sternberg, & T. Ben-Zeev (Haz.), The nature of mathematical thinking içinde (s. 253-284). Lawrence Erlbaum Associates.
  • Garavalia, L. S. ve Gredler, M. E. (2002). Prior achievement, aptitude, and use of learning strategies as predictors of college student achievement. College Student Journal, 36(4), 616–641.
  • Heinze, A. ve Reiss, K. (2004). The teaching of proof at the lower secondary level – a video study. ZDM Mathematics Education, 36(3), 98–104. https://doi.org/10.1007/BF02652777
  • 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 Mathematics Education, 40, 443–453. https://doi.org/10.1007/s11858-008-0092-1
  • Hine, G., & McNab, N. (2014). Mathematics specialist: Year 11 ATAR course-Units 1 ve 2 (Australian curriculum). Academic Associates.
  • Jones, K. (2000). The student experience of mathematical proof at university level. International Journal of Mathematical Education in Science and Technology, 31(1), 53-60. https://doi.org/10.1080/002073900287381
  • Kankam, K., Pholasa, N., & Cholamjiak, P. (2019). On convergence and complexity of the modified forward-backward method involving new linesearches for convex minimization. Mathematical Methods in Applied Sciences, 42(5), 1352–1362. https://doi.org/10.1002/mma.5420
  • Krutetskii, V. A. (1969). An investigation of mathematical abilities in school children. J. Kilpatrick, ve I. Wirszup (Haz.), Soviet studies in the psychology of learning and teaching mathematics içinde (s. 5–57). University of Chicago Press.
  • Lakatos, I. (1976). Proofs and refutations: The logic of mathematical discovery. Cambridge University Press.
  • Lee, K. (2016). Students’ proof schemes for mathematical proving and disproving of propositions. Journal of Mathematical Behavior, 41, 26–44. https://doi.org/10.1016/j.jmathb.2015.11.005
  • Lin, F. L., & Yang, K. L. (2007). The reading comprehension of geometric proofs: The contribution of knowledge and reasoning. International Journal of Science and Mathematics Education, 5(4), 729–754. https://doi.org/10.1007/s10763-007-9095-6
  • Manin, Y. (1992, August). Contribution in panel discussion on “The theory and practice of proof”. Paper presented at the 7th International Congress on Mathematical Education (ICME-7), Quebec, Canada.
  • Matsuda, N., & VanLehn, K. (2004). GRAMY: A geometry theorem prover capable of construction. Journal of Automated Reasoning, 32, 3–33. https://doi.org/10.1023/B:JARS.0000021960.39761
  • Milli Eğitim Bakanlığı [MEB]. (2018). Ortaöğretim matematik dersi (9, 10, 11 ve 12. sınıflar) öğretim programı. MEB Yayınları.
  • Miyazaki, M., Fujita, T., Jones, K., & Iwanaga, Y. (2017). Designing a web-based learning support system for flow-chart proving in school geometry. Digital Experiences in Mathematics Education, 3, 233–256. https://doi.org/10.1007/s40751-017-0034-z
  • Miyazaki, M., Nagata, J., Chino, K., Fujita, T., Ichikawa, D., Shimizu, S., & Iwanaga, Y. (2016,). Developing a curriculum for explorative proving in lower secondary school geometry. G. Kaiser (Haz.), Proceedings of the 13th International Congress on Mathematical Education içinde (s. 1–4). Springer.
  • Moore, R. C. (1994). Making the transition to formal proof. Educational Studies in Mathematics, 27, 249–266. https://doi.org/10.1007/BF01273731
  • Otani, H. (2019). Comparing structures of statistical hypothesis testing with proof by contradiction: In terms of argument. Hiroshima Journal of Mathematics Education, 2, 1–12.
  • Öztürk, T. (2016). Matematik öğretmeni adaylarının ispatlama becerilerini geliştirmeye yönelik tasarlanan öğrenme ortamının değerlendirilmesi [Yayınlanmamış doktora tezi]. Karadeniz Teknik Üniversitesi.
  • Palatnik, A., & Dreyfus, T. (2019). Students’ reasons for introducing auxiliary lines in proving situations. The Journal of Mathematical Behavior, 55. https://doi.org/10.1016/j.jmathb.2018.10.004
  • Perry, P., Molina, Ó., Camargo, L., & Samper, C. (2011, February). Analyzing the proving activity of a group of three students. Paper presented at Congress of the European Society for Research in Mathematics Education (CERME 7), Poland.
  • Polya, G. (1957). How to solve it (2. baskı). Princeton University Press.
  • Remillard, K. (2014). Identifying discursive entry points in paired-novice discourse as a first step in penetrating the paradox of learning mathematical proof. Journal of Mathematical Behavior, 34, 99–113. https://doi.org/10.1016/j.jmathb.2014.02.002
  • Siegler, R. S., & Wagner Alibali, M. (2005). Children’s thinking (4. baskı). Pearson Prentice Hall.
  • Solow, D. (2014). How to read and do proofs: An introduction to mathematical thought processes (6. baskı). John Wiley ve Sons.
  • Spiro, R. J., & Jehng, J. (1990). Cognitive flexibility and hypertext: Theory and technology for the non-linear and multidimensional traversal of complex subject matter. D. Nix, ve R. Spiro (Haz.), Cognition, education, and multimedia içinde (s. 163–205). Erlbaum.
  • Tall, D. (2002). The psychology of advanced mathematical thinking. D. Tall (Haz.), Advanced mathematical thinking içinde (s. 3–21). Springer.
  • Tall, D. (1995). Cognitive growth in elementary and advanced mathematical thinking. L. Meira, ve D. Carraher (Haz.), Proceedings of the 19th Meeting of the International Group for the Psychology of Mathematics Education içinde (Cilt. I., s. 61–75). Universidade Federal de Pernambuco.
  • Tsujiyama, Y. (2011). On the role of looking back at proving processes in school mathematics: Focusing on argumentation. M. Pytlak, T. Rowland & E. Swoboda (Haz.), Proceedings of the 7th Congress of the European Society for Research in Mathematics Education içinde (s. 161–171). University of Rzeszów.
  • Warner, L. B., Alcock, L. J., Coppolo Jr., J., & Davis, G. E. (2003). How does flexible mathematical thinking contribute to the growth of understanding? N. A. Pateman, B. J. Dougherty, & J. Zillox (Haz.), Proceedings of the Twenty-Seventh Conference of the International Group for the Psychology of Mathematics Education içinde (Cilt. IV., s. 371–378). PME.
  • Yang, K. L., & Lin, F. L. (2012). Effects of reading-oriented tasks on students’ reading comprehension of geometry proof. Mathematics Education Research Journal, 24, 215–238. https://doi.org/10.1007/s13394-012-0039-2
  • Weber, K. (2005). Problem-solving, proving, and learning: The relationship between problem solving processes and learning opportunities in the activity of proof construction. Journal of Mathematical Behavior, 24, 351–360. https://doi.org/10.1016/j.jmathb.2005.09.005
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Details

Primary Language Turkish
Subjects Mathematics Education
Journal Section Original Articles
Authors

Neslihan Sönmez 0000-0003-1631-9510

Tuğba Öztürk 0000-0003-1599-8574

Bülent Güven 0000-0001-8767-6051

Publication Date October 15, 2023
Published in Issue Year 2023 Volume: 40-2 Issue: 2

Cite

APA Sönmez, N., Öztürk, T., & Güven, B. (2023). İspat Sürecinden Yansımalar: Öğretmen Adaylarının İleri-Geri Tekniğini Kullanma Durumlarının İncelenmesi. Bogazici University Journal of Education, 40-2(2), 227-260. https://doi.org/10.52597/buje.1289328
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