Araştırma Makalesi
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Secondary Mathematics Teacher Candidates’ Geometric Proof Process: A Case Study

Yıl 2022, Cilt: 3 Sayı: 1, 39 - 54, 26.04.2022

Mesut Öztürk Abdullah Kaplan

Öz

This study was carried out in order to examine the geometry proof process of secondary school mathematics teacher candidates. Eight Turkish secondary school mathematics teacher candidates participated in this research, in which the case study model, one of the qualitative research methods, was used. Activities cards and think-aloud protocols were used to collect the data of the study. Two different activity cards were used in the study. The first activity card prepared includes the geometry proposition. Content analysis method was used in the analysis of the data of the study. In the analysis of the data, the data was first analyzed and coded. Then, the acquired skills were categorized according to the common features of the codes. While most of the results reached in the study were supported by the literature, some important and original findings were also reached. One of the most important results reached in this study is that the participants see drawing figures and performing algebraic operations as a necessity while proving geometry.

Anahtar Kelimeler

Activity card Algebraic operation Drawing shape Reading proposition Think aloud protocol

Kaynakça

  • Alcock, L. (2010). Mathematicians’ perspectives on the teaching and learning of proof. In F. Hitt, D. Holton & P. W. Thompson (Eds.), Research in collegiate mathematics education VII (1st ed., pp. 63-91). American Mathematical Society.
  • Alcock, L., & Weber, K. (2005). Proof validation in real analysis: Inferring and checking warrants. Journal of Mathematical Behavior, 24(2), 125–134. https://doi.org/10.1016/j.jmathb. 2005.03.003
  • Alqassab, M., Strijbos, J. W., & Ufer, S. (2018). The impact of peer solution quality on peer-feedback provision on geometry proofs: Evidence from eye-movement analysis. Learning and Instruction, 58, 182-192. https://doi.org/10.1016/j.learninstruc.2018.07.003.
  • Altun, M. (2015). Eğitim fakülteleri ve lise matematik öğretmenleri için liselerde matematik öğretimi (7. baskı). Aktüel Alfa Akademi.
  • Altun, M., & Arslan, Ç. (2006). İlköğretim öğrencilerinin problem çözme stratejilerini öğrenmeleri üzerine bir çalışma. Uludağ Üniversitesi Eğitim Fakültesi Dergisi, 19(1), 1–21.
  • Arslan, S., & Yıldız, C. (2010). 11. Sınıf öğrencilerinin matematiksel düşünmenin aşamalarındaki yaşantılarından yansımalar. Eğitim ve Bilim, 35(156), 17–31.
  • Aydoğdu-İskenderoğlu, T. (2016). Kanıt ve kanıt şemaları. E. Bingölbali, S. Arslan & İ. Ö. Zembat (Eds.) Matematik eğitiminde teoriler (1. baskı, s. 65-84) içinde. Pegem Akademi.
  • Barnard, T., & Tall, D. (1997). Cognitive units, connections and mathematical proof [Conference session]. 21th Conference of the International Group for the Psychology of Mathematics Education, Lahti, Finland.
  • Baykul, Y. (2009). İlköğretimde matematik öğretimi (6-8. sınıflar) (1. baskı). Pegem Akademi.
  • Bloch, E. D. (2011). Proofs and fundamentals: A first course in abstract mathematics (2nd ed.). Springer Science+Business Media.
  • Burton, D. M. (2006). The history of mathematics: An introduction (6nd ed.). The McGraw−Hill Companies.
  • Büyüköztürk, Ş., Kılıç Çakmak, E., Akgün, Ö. E., Karadeniz, Ş. & Demirel, F. (2021). Eğitimde bilimsel araştırma yöntemleri (31. baskı). Pegem Akademi.
  • Ceylan, T. (2012). Geogebra yazılımı ortamında ilköğretim matematik öğretmen adaylarının geometrik ispat biçimlerinin incelenmesi (Tez No. 302918). [Yüksek lisans tezi, Ankara Üniversitesi-Ankara]. Yükseköğretim Kurulu Başkanlığı Tez Merkezi.
  • Chinnappan, M., Ekanayake, M., & Brown, C. (2012). Knowledge use in the construction of geometry proof by Sri Lankan students. International Journal of Science and Mathematics Education, 10, 865–887. https://doi.org/10.1007/s10763-011-9298-8.
  • Demircioğlu, H., & Polat, K. (2016). Ortaöğretim matematik öğretmeni adaylarının sözsüz ispatlar ile ilgili yaşadıkları zorluklar hakkındaki görüşleri. Uluslararası Türk Eğitim Bilimleri Dergisi, 4(7), 81–89.
  • Doğan, M. (2013). Nokta, doğru, doğru parçası, ışın, düzlem ve uzay kavramları. İ. Ö. Zembat, M. F. Özmantar, E. Bingölbali, H. Şandır & A. Delice (Eds.),Tanımları ve tarihsel gelişimleriyle matematiksel kavramlar (1. baskı, s. 198-221) içinde. Pegem Akademi.
  • Fukawa-Connelly, T. P. (2012). A case study of one instructor’s lecture-based teaching of proof in abstract algebra: Making sense of her pedagogical moves. Educational Studies in Mathematics, 81(3), 325–345. https://doi.org/10.1007/s10649-012-9407-9.
  • Gerstein, L. J. (2012). Introduction to mathematical structures and proofs (2nd ed.). Springer Science+Business Media.
  • Güler, G. (2013). Matematik öğretmeni adaylarının cebir öğrenme alanındaki ispat süreçlerinin incelenmesi (Tez No. 331712) [Doktora tezi, Atatürk Üniversitesi-Erzurum]. Yükseköğretim Kurulu Başkanlığı Tez Merkezi.
  • Güven, B., Çelik, D., & Karataş, İ. (2005). Ortaöğretimdeki çocukların matematiksel ispat yapabilme durumlarının incelenmesi. Çağdaş Eğitim Dergisi, 30, 35–45.
  • Harel, G., & Sowder, L. (1998). Students’ prof schemes: Results from exploratory studies. In J. Kaput, A. H. Schoenfeld & E. Dubinsky (Eds.), Research in collegiate mathematics education III (Cbmsissues in mathematics education) (1st ed., pp. 234-283). American Mathematical Society.
  • Harel, G., & Sowder, L. (2007). Toward comprehensive perspectives on the learning and teaching of proof. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (1st ed., pp. 805-842). NCTM.
  • Hašek, R. (2019). Dynamic geometry software supplementedwith a computer algebra system as a proving tool. Mathematics in Computer Science, 13(13), 95-104. https://doi.org/10.1007/ s11786-018-0369-x
  • Herbst, P. G. (2002). Engaging students in proving: A doublebind on the teacher. Journal for Research in Mathematics Education, 33(3), 176–203. https://doi.org/10.2307/749724.
  • Hızarcı, S., Kaplan, A., İpek, A. S., Işık, C., & Elmas, S. (2009). Düzlem geometri (1. baskı). Palme.
  • Hoffer, A. (1981). Geometry is more than proof. The Mathematics Teacher, 74(1), 11–18. https://doi.org/10.5951/MT.74.1.0011.
  • Kaplan, C. A., & Simon, H. A. (1990). Insearch of insight. Cognitive Psychology, 22(3), 374–419. https://doi.org/10.1016/0010-0285(90)90008-R.
  • Karpuz, Y., & Atasoy, E. (2020). High school mathematics teachers’ content knowledge of the logical structure of proof deriving from figural-concept interaction in geometry. International Journal of Mathematical Education in Science and Technology, 51(4), 585-603. https://doi.org/10.1080/0020739X.2020.1736347
  • Karpuz, Y., Koparan, T., & Güven, B. (2014). Geometride öğrencilerin şekil ve kavram bilgisi kullanımı. Turkish Journal of Computer and Mathematics Education, 5(2), 108–118.
  • Krutetskii V. A. (1976). The psychology of mathematical abilities in schoolchildren (1nd ed.). University of Chicago Press.
  • Ko, Y. Y., & Rose, M. K. (2021). Are self-constructed and student-generated arguments acceptable proofs? Pre-service secondary mathematics teachers’ evaluations. Journal of Mathematical Behavior, 64, 1-15.. https://doi.org/10.1016/j.jmathb.2021.100912
  • Komatsu, K., & Jones, K. (2021). Generating mathematical knowledge in the classroom through proof, refutation, and abductive reasoning. Educational Studies in Mathematics, 109(3), 1-25. https://doi.org/10.1007/s10649-021-10086-5
  • Llinares, S., & Clemente, F. (2019) Characteristics of the shifts from configural reasoning to deductive reasoning in geometry. Mathematics Education Research Journal, 31(31), 259-277. https://doi.org/10.1007/s13394-018-0253-7
  • Merriam, S. B. (2013). Nitel araştırma: Desen ve uygulama için bir rehber (S. Turan, Çev.; 3. baskı). Nobel. (Orijinal çalışmanın basımı 2009)
  • Nool, N. R. (2012). Exploring the metacognitive processes of prospective mathematics teachers during problem solving. International Proceedings of Economics Development and Research, 30, 302–306.
  • Öztürk, M. (2021). Cognitive and metacognitive skills performed by math teachers in the proving process of number theory. Athens Journal of Education, 8(1), 53–71.
  • Öztürk, M., Akkan, Y., & Kaplan, A. (2019). Sınıf öğretmenliği öğrencilerinin temel matematik ispatlarını yapma sürecindeki bilişsel yapılar ve argümanları. Cumhuriyet Uluslararası Eğitim Dergisi, 8(2), 429–452. http://dx.doi.org/10.30703/cije.490887.
  • Öztürk, M., & Kaplan, A. (2019). Cebirsel ispat yapma sürecinin bilişsel açıdan incelenmesi: Bir karma yöntem araştırması. Eğitim ve Bilim, 44(197), 25–64. https://doi.org/ 10.15390/EB.2018.7504
  • Regier, P., & Savic, M. (2020). How teaching to foster mathematical creativity may impact student self-efficacy for proving. The Journal of Mathematical Behavior, 57, 1-18. https://doi.org/10.1016/j.jmathb.2019.100720.
  • Schraw, G., & Dennison, R. S. (1994). Assessing metacognitive awareness. Contemporary Educational Psychology, 19(4), 460–475. https://doi.org/10.1006/ceps.1994.1033.
  • Senk, S. L. (1985). How well do students write geometry proofs? The Mathematics Teacher, 78(6), 448–456. https://doi.org/10.5951/MT.78.6.0448.
  • Shigematsu, K., & Sowder, L. (1994). Drawings for story problems: Practices in Japan and the United States. Arithmetic Teachers, 41(9), 544–547. https://doi.org/10.5951/AT.41.9.0544.
  • Shongwe, B., & Mudaly, V. (2021). Introducing a measure of perceived self-efficacy for proof (PSEP): Evidence of validity. Journal of Research and Advances in Mathematics Education, 6(3), 260-276. https://doi.org/10.23917/jramathedu.v6i3.14138.
  • Smith, E. E., & Kosslyn, S. M. (2014). Bilişsel psikoloji: Zihin ve beyin (M. Şahin, Çev.; 1. baskı). Nobel Akademik. (Orijinal çalışmanın basımı 2007)
  • Stylianides, A. J., & Stylianides, G. J. (2009). Proof constructions and evaluations. Educational Studies in Mathematics, 72(2), 237–253. https://doi.org/10.1007/s10649-009-9191-3
  • The National Council of Teachers of Mathematics [NCTM]. (2000). Principles and standards for school mathematics. NCTM.
  • Uğurel, I., Moralı, S., Koyunkaya, M. Y., & Karahan, Ö. (2016). Pre-service secondary mathematics teachers’ behaviors in the proving process. Eurasia Journal of Mathematics, Science & Technology Education, 12(2), 203–231. https://doi.org/10.12973/eurasia.2016.1523a.
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Ortaöğretim Matematik Öğretmeni Adaylarının Geometrik İspat Yapma Süreci: Bir Durum Çalışması

Yıl 2022, Cilt: 3 Sayı: 1, 39 - 54, 26.04.2022

Mesut Öztürk Abdullah Kaplan

Öz

Bu çalışma ortaöğretim matematik öğretmeni adaylarının geometrik ispat yapma sürecini incelemek amacıyla yürütülmüştür. Nitel araştırma yöntemlerinden durum çalışması modelinin kullanıldığı bu araştırmaya sekiz ortaöğretim matematik öğretmeni adayı katılmıştır. Çalışmanın verilerinin toplanmasında etkinlik kartları ve sesli düşünme protokolleri kullanılmıştır. Çalışmada iki farklı etkinlik kartı kullanılmıştır. Hazırlanan ilk etkinlik kartı geometri önermesini içermektedir. Çalışmanın verilerinin analizinde içerik analizi yöntemi kullanılmıştır. Verilerin analizinde ilk olarak veriler çözümlenerek kodlanmıştır. Ardından kodların ortak özelliklerine göre elde edilen beceriler kategorilendirilmiştir. Çalışmada ulaşılan sonuçların birçoğu alan yazınla desteklenirken, bazı önemli ve özgün bulgulara da ulaşılmıştır. Bu çalışmada ulaşılan en önemli sonuçlardan biri katılımcıların geometrik ispat yaparken ispat yapmada şekil çizmeyi ve cebirsel işlem yapmayı bir zorunluluk olarak görmeleridir.

Anahtar Kelimeler

Cebirsel işlem Etkinlik kartı Önermeyi okuma Sesli düşünme protokolü Şekil çizme

Kaynakça

  • Alcock, L. (2010). Mathematicians’ perspectives on the teaching and learning of proof. In F. Hitt, D. Holton & P. W. Thompson (Eds.), Research in collegiate mathematics education VII (1st ed., pp. 63-91). American Mathematical Society.
  • Alcock, L., & Weber, K. (2005). Proof validation in real analysis: Inferring and checking warrants. Journal of Mathematical Behavior, 24(2), 125–134. https://doi.org/10.1016/j.jmathb. 2005.03.003
  • Alqassab, M., Strijbos, J. W., & Ufer, S. (2018). The impact of peer solution quality on peer-feedback provision on geometry proofs: Evidence from eye-movement analysis. Learning and Instruction, 58, 182-192. https://doi.org/10.1016/j.learninstruc.2018.07.003.
  • Altun, M. (2015). Eğitim fakülteleri ve lise matematik öğretmenleri için liselerde matematik öğretimi (7. baskı). Aktüel Alfa Akademi.
  • Altun, M., & Arslan, Ç. (2006). İlköğretim öğrencilerinin problem çözme stratejilerini öğrenmeleri üzerine bir çalışma. Uludağ Üniversitesi Eğitim Fakültesi Dergisi, 19(1), 1–21.
  • Arslan, S., & Yıldız, C. (2010). 11. Sınıf öğrencilerinin matematiksel düşünmenin aşamalarındaki yaşantılarından yansımalar. Eğitim ve Bilim, 35(156), 17–31.
  • Aydoğdu-İskenderoğlu, T. (2016). Kanıt ve kanıt şemaları. E. Bingölbali, S. Arslan & İ. Ö. Zembat (Eds.) Matematik eğitiminde teoriler (1. baskı, s. 65-84) içinde. Pegem Akademi.
  • Barnard, T., & Tall, D. (1997). Cognitive units, connections and mathematical proof [Conference session]. 21th Conference of the International Group for the Psychology of Mathematics Education, Lahti, Finland.
  • Baykul, Y. (2009). İlköğretimde matematik öğretimi (6-8. sınıflar) (1. baskı). Pegem Akademi.
  • Bloch, E. D. (2011). Proofs and fundamentals: A first course in abstract mathematics (2nd ed.). Springer Science+Business Media.
  • Burton, D. M. (2006). The history of mathematics: An introduction (6nd ed.). The McGraw−Hill Companies.
  • Büyüköztürk, Ş., Kılıç Çakmak, E., Akgün, Ö. E., Karadeniz, Ş. & Demirel, F. (2021). Eğitimde bilimsel araştırma yöntemleri (31. baskı). Pegem Akademi.
  • Ceylan, T. (2012). Geogebra yazılımı ortamında ilköğretim matematik öğretmen adaylarının geometrik ispat biçimlerinin incelenmesi (Tez No. 302918). [Yüksek lisans tezi, Ankara Üniversitesi-Ankara]. Yükseköğretim Kurulu Başkanlığı Tez Merkezi.
  • Chinnappan, M., Ekanayake, M., & Brown, C. (2012). Knowledge use in the construction of geometry proof by Sri Lankan students. International Journal of Science and Mathematics Education, 10, 865–887. https://doi.org/10.1007/s10763-011-9298-8.
  • Demircioğlu, H., & Polat, K. (2016). Ortaöğretim matematik öğretmeni adaylarının sözsüz ispatlar ile ilgili yaşadıkları zorluklar hakkındaki görüşleri. Uluslararası Türk Eğitim Bilimleri Dergisi, 4(7), 81–89.
  • Doğan, M. (2013). Nokta, doğru, doğru parçası, ışın, düzlem ve uzay kavramları. İ. Ö. Zembat, M. F. Özmantar, E. Bingölbali, H. Şandır & A. Delice (Eds.),Tanımları ve tarihsel gelişimleriyle matematiksel kavramlar (1. baskı, s. 198-221) içinde. Pegem Akademi.
  • Fukawa-Connelly, T. P. (2012). A case study of one instructor’s lecture-based teaching of proof in abstract algebra: Making sense of her pedagogical moves. Educational Studies in Mathematics, 81(3), 325–345. https://doi.org/10.1007/s10649-012-9407-9.
  • Gerstein, L. J. (2012). Introduction to mathematical structures and proofs (2nd ed.). Springer Science+Business Media.
  • Güler, G. (2013). Matematik öğretmeni adaylarının cebir öğrenme alanındaki ispat süreçlerinin incelenmesi (Tez No. 331712) [Doktora tezi, Atatürk Üniversitesi-Erzurum]. Yükseköğretim Kurulu Başkanlığı Tez Merkezi.
  • Güven, B., Çelik, D., & Karataş, İ. (2005). Ortaöğretimdeki çocukların matematiksel ispat yapabilme durumlarının incelenmesi. Çağdaş Eğitim Dergisi, 30, 35–45.
  • Harel, G., & Sowder, L. (1998). Students’ prof schemes: Results from exploratory studies. In J. Kaput, A. H. Schoenfeld & E. Dubinsky (Eds.), Research in collegiate mathematics education III (Cbmsissues in mathematics education) (1st ed., pp. 234-283). American Mathematical Society.
  • Harel, G., & Sowder, L. (2007). Toward comprehensive perspectives on the learning and teaching of proof. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (1st ed., pp. 805-842). NCTM.
  • Hašek, R. (2019). Dynamic geometry software supplementedwith a computer algebra system as a proving tool. Mathematics in Computer Science, 13(13), 95-104. https://doi.org/10.1007/ s11786-018-0369-x
  • Herbst, P. G. (2002). Engaging students in proving: A doublebind on the teacher. Journal for Research in Mathematics Education, 33(3), 176–203. https://doi.org/10.2307/749724.
  • Hızarcı, S., Kaplan, A., İpek, A. S., Işık, C., & Elmas, S. (2009). Düzlem geometri (1. baskı). Palme.
  • Hoffer, A. (1981). Geometry is more than proof. The Mathematics Teacher, 74(1), 11–18. https://doi.org/10.5951/MT.74.1.0011.
  • Kaplan, C. A., & Simon, H. A. (1990). Insearch of insight. Cognitive Psychology, 22(3), 374–419. https://doi.org/10.1016/0010-0285(90)90008-R.
  • Karpuz, Y., & Atasoy, E. (2020). High school mathematics teachers’ content knowledge of the logical structure of proof deriving from figural-concept interaction in geometry. International Journal of Mathematical Education in Science and Technology, 51(4), 585-603. https://doi.org/10.1080/0020739X.2020.1736347
  • Karpuz, Y., Koparan, T., & Güven, B. (2014). Geometride öğrencilerin şekil ve kavram bilgisi kullanımı. Turkish Journal of Computer and Mathematics Education, 5(2), 108–118.
  • Krutetskii V. A. (1976). The psychology of mathematical abilities in schoolchildren (1nd ed.). University of Chicago Press.
  • Ko, Y. Y., & Rose, M. K. (2021). Are self-constructed and student-generated arguments acceptable proofs? Pre-service secondary mathematics teachers’ evaluations. Journal of Mathematical Behavior, 64, 1-15.. https://doi.org/10.1016/j.jmathb.2021.100912
  • Komatsu, K., & Jones, K. (2021). Generating mathematical knowledge in the classroom through proof, refutation, and abductive reasoning. Educational Studies in Mathematics, 109(3), 1-25. https://doi.org/10.1007/s10649-021-10086-5
  • Llinares, S., & Clemente, F. (2019) Characteristics of the shifts from configural reasoning to deductive reasoning in geometry. Mathematics Education Research Journal, 31(31), 259-277. https://doi.org/10.1007/s13394-018-0253-7
  • Merriam, S. B. (2013). Nitel araştırma: Desen ve uygulama için bir rehber (S. Turan, Çev.; 3. baskı). Nobel. (Orijinal çalışmanın basımı 2009)
  • Nool, N. R. (2012). Exploring the metacognitive processes of prospective mathematics teachers during problem solving. International Proceedings of Economics Development and Research, 30, 302–306.
  • Öztürk, M. (2021). Cognitive and metacognitive skills performed by math teachers in the proving process of number theory. Athens Journal of Education, 8(1), 53–71.
  • Öztürk, M., Akkan, Y., & Kaplan, A. (2019). Sınıf öğretmenliği öğrencilerinin temel matematik ispatlarını yapma sürecindeki bilişsel yapılar ve argümanları. Cumhuriyet Uluslararası Eğitim Dergisi, 8(2), 429–452. http://dx.doi.org/10.30703/cije.490887.
  • Öztürk, M., & Kaplan, A. (2019). Cebirsel ispat yapma sürecinin bilişsel açıdan incelenmesi: Bir karma yöntem araştırması. Eğitim ve Bilim, 44(197), 25–64. https://doi.org/ 10.15390/EB.2018.7504
  • Regier, P., & Savic, M. (2020). How teaching to foster mathematical creativity may impact student self-efficacy for proving. The Journal of Mathematical Behavior, 57, 1-18. https://doi.org/10.1016/j.jmathb.2019.100720.
  • Schraw, G., & Dennison, R. S. (1994). Assessing metacognitive awareness. Contemporary Educational Psychology, 19(4), 460–475. https://doi.org/10.1006/ceps.1994.1033.
  • Senk, S. L. (1985). How well do students write geometry proofs? The Mathematics Teacher, 78(6), 448–456. https://doi.org/10.5951/MT.78.6.0448.
  • Shigematsu, K., & Sowder, L. (1994). Drawings for story problems: Practices in Japan and the United States. Arithmetic Teachers, 41(9), 544–547. https://doi.org/10.5951/AT.41.9.0544.
  • Shongwe, B., & Mudaly, V. (2021). Introducing a measure of perceived self-efficacy for proof (PSEP): Evidence of validity. Journal of Research and Advances in Mathematics Education, 6(3), 260-276. https://doi.org/10.23917/jramathedu.v6i3.14138.
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  • Zaimoğlu, Ş. (2012). 8. sınıf öğrencilerinin geometrik ispat süreci ve eğilimleri (Tez No. 307591) [Yüksek lisans tezi, Kastamonu Üniversitesi-Kastamonu]. Yükseköğretim Kurulu Başkanlığı Tez Merkezi.
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Toplam 59 adet kaynakça vardır.

Ayrıntılar

Birincil Dil Türkçe
Konular Alan Eğitimleri
Bölüm Araştırma Makaleleri
Yazarlar

Mesut Öztürk 0000-0002-2163-3769

Abdullah Kaplan 0000-0001-6743-6368

Yayımlanma Tarihi 26 Nisan 2022
Kabul Tarihi 7 Şubat 2022
Yayımlandığı Sayı Yıl 2022 Cilt: 3 Sayı: 1

Kaynak Göster

APA Öztürk, M., & Kaplan, A. (2022). Ortaöğretim Matematik Öğretmeni Adaylarının Geometrik İspat Yapma Süreci: Bir Durum Çalışması. Eurasian Journal of Teacher Education, 3(1), 39-54.
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