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Trigonometrik Fonksiyonların Toplam ve Fark Formüllerinin Ortaöğretim Düzeyinde Görselleştirilmesi

Year 2010, Volume: 5 Issue: 1, 24 - 37, 01.03.2010

Birol Tekin Alper Cihan Konyalıoğlu

Abstract

İspat matematikçilerce, matematik alanının ve uygulamasının temeli olarak görülür. Birçok öğrenci ise niçin ispat yapmayı öğrenmek zorunda olduklarına anlam verememektedir. Trigonometri ortaöğretim matematik müfredatının önemli bir konusudur. Trigonometriyle çalışırken öğrencilerin çoğu eşitlikleri ezberlemekte ve bunları rutin alıştırmaları çözmek için uygulamaktadırlar. Öğrencilerden trigonometride ispat yapmaları istendiğinde, eşitlikleri taraf tarafa toplamaya dayalı cebirsel dönüşümler yapmaktadırlar. Trigonometri kavramlarını öğrencilerin daha iyi anlamalarını sağlamak için görsel ispatlar yararlı olabilir. Bu çalışmada toplam ve fark formüllerinin görsel şekillerle ispat edilmesi ve bunların diğer araştırmacılara tanıtılması amaçlanmıştır. Bu amaçla görsel ispat ve cebirsel ispat sırasıyla yapılmış ve bunlarla ilgili bazı öneriler verilmiştir.

Keywords

Görselleştirme toplam ve fark formülleri görsel ispat cebirsel ispat

References

  • Alsina, C. & Nelsen, R.B. (2006). Math Made Visual Creating İmages for Understanding Mathematics. Published and distributed by The Mathematical Association of America.
  • Arcavi, A. 2003. The role of visual representations in the learning mathematics. Educational Studies in Mathematics, 5 www.kent.k12.wa.us/KSD/KR/DIGITAL/math2.pdf
  • Bagni, G.T. (1998). Visualization and didactics of mathematics in high school: an experimental reseach. Scientia Paedogogica Experimentalis, 1, 161-180. 2(3), 215-241(27).
  • Borwein, P. & Jörgenson, L. (1997). Visible structures in number theory. Adresi: http://www.cecm.sfu.ca/loki/papers/numbers/node3.html.
  • Brown, J., Stillman, G., Schwarz, B. & Kaiser, G. (2008). The case of mathematical proof in lower secondary school: knowledge and competencies if pre-service teachers, proceedings of the 31st Annual Conference of the Mathematics Education Research Group of
  • Australasia, Hrsg.: M.Goos, R. Brown & K. Makar. (C) Merga Inc., 85-91. Conner, A. (2007). Argumentation in a Geometry Class: Aligned with the Teacher's Conception of Proof,
  • Delice, A. (2004). Trigonometri Sözel Problemlerinde Görselleştirme ve Diyagram Oluşturma, VI. Ulusal Fen Bilimleri ve Matematik Eğitimi Kongresi, Marmara Üni., İstanbul.
  • Delice, A. (2005). Türk ve İngiliz Eğitim sisteminde matematik eğitiminin karşılaştırılması, Milli Eğitim Dergisi, sayı 167. (http://yayim.meb.gov.tr/dergiler/167/index3-delice.htm)
  • Dufour-Janvier, B. (1987). Pedagogical Considerations Concerning the Problem of Representation, in C. Janvier (Ed), Problems of Representation in the Teaching and Learning of Mathematics. Hillsdale, NJ: Lawrence Erlbaum Associates.
  • Duval, R. (2002). Proof understanding in mathematics: what ways for students, proceedings of 2002 International conference on mathematics, http://www.math.ntnu.edu.tw/cyc/_ private/ mathedu/me1/me1_2002_1/duval.doc
  • Gravina, M.A. (2000) The proof in geometry: essays in a dynamical environment. http://www.lettredelapreuve.it/ICME9TG12/ICME9TG12Contributions/GravinaICM E00/GravinaICME00.html
  • Gravina, M.A. (2008). Dynamical visual proof: what does it mean?, ICME 11- TSG 22, http://tsg.icme11.org/document/get/233.
  • Guzman, M., 2002. What is visualization in math? In the teaching and learning of mathematical analysis. http://www.math.uoc.gr/~ictm2/Proceedings/invGuz.pdf.
  • Hanna, G and Villiers, M. (2008). ICMI Study 19: Proof and Proving in mathematics education, ZDM Mathematics Education, 40:329-336.
  • Hanna, G., and Sidoli, N. (2007).Visualization and proof: a brief survey of philosophical perspectives, ZDM Mathematics Education , Heidelberg, Vol. 39, page : 73-78.
  • Karadağ, Z. & McDougall, D. (2009). Visual explorative approaches to learning mathematics, http://www.pmena.org/ 2009/proceedings/workinggroup90649replacement.pdf .
  • Malaty, G. (2009). The Role of Visualization in Mathematics Education: Can visualization Promote the Causal Thinking?
  • MEB, (2006). Orta Öğretim Matematik 10. Sınıf Kitabı, Kelebek Matbaacılık, İstanbul.
  • NCTM (2000). Principles and Standards for School Mathematics, National Council of Teachers of Mathematics Inc.,http://standards.nctm.org/document/chapter7/geom.htm.
  • Nelsen, R.B. (1993). Proof Without Words, Exercises in visual thinking, The Mathematical Association of America Press, Washington, DC.
  • Özer, Ö. ve Arıkan, A. (2002). Lise matematik derslerinde öğrencilerin ispat yapabilme düzeyleri, V. Uluısal Fen ve Matematik Eğitimi Kongresi bildirisi, UFBMEK-5/b_kitabi /PDF/Matematik/Bildiri/t245d.pdf
  • Pettinelli, M. (2008). Visual learning. http://cnx.org/content/m14358/latest/).
  • Presmeg, N. C. (2006). Research on visualization in learning and teaching mathematics in
  • A.Gutierrez, P., Borero (eds.) Handbook of Research on Psychology of Mathematics Education: Past, Present and Future, 205-235. Sense Publishers, Rotterdam /Taipei.
  • Rahim, M.H., & Siddo, R., (2009). The use of visualization in learning and teaching mathematics. In A. Rogerson (Ed.), Proceedings of the 10th International Conference: Models in Developing Mathematics Education. The Mathematics Education into the 21st Century P r o j e c t , D r e s d e n , S a x o n y , G e r m a n y , S e p t 1 1 - 1 7 , 2 0 0 9 . http://math.unipa.it/~grim/21_project/Rahim496-500.pdf
  • Sarı, M. (2008). Undergraduate students' difficulties with mathematical proof and proof teaching, devam eden doktora tezi, Hacettepe Üniversitesi, Fen Bilimleri Enstitüsü. http://yess4.ktu.edu.tr/YermePappers/MeltemSari.pdf
  • Stylianou, D. A. & Silver, E. A. (2004). The role of visual representations in advanced mathematical problem solving: An examination of expert-novice similarities and differences. Journal of Mathematical Thinking and Learning, 6 (4). 353-387.
  • Tall, D., 2004. A versatile theory of visualization and symbolisation in mathematics. Plenary Presentation at the Commission Internationale pour L'Etude et l'Amelioration de l'Ensignement des Mathematiques, Toulouse, France.

Year 2010, Volume: 5 Issue: 1, 24 - 37, 01.03.2010

Birol Tekin Alper Cihan Konyalıoğlu

Abstract

Proof is viewed by mathematicians as central to the discipline and practice of mathematics. Most students are unable to understand why they must learn how to write a proof. Trigonometry is an important subject in the secondary school mathematics curriculum. When studying trigonometry, most students just have to memorize a set of identities and to apply them to solve routine exercises. When students are asked to do proofs in trigonometry, they do some proofs usually consist of algebraic transformations linking a side of an identity to the other side. To promote students' understanding of trigonometric concepts, visual proofs may be beneficial. In this study, it is aimed to prove the sum and difference formulas in trigonometry by visual figures and to introduce them to other researchers. For this goal, visual proof and algebraic proof processes were made respectively and some suggestions were given about them

Keywords

Visualization sum and difference formulas visual proof algebraic proof

References

  • Alsina, C. & Nelsen, R.B. (2006). Math Made Visual Creating İmages for Understanding Mathematics. Published and distributed by The Mathematical Association of America.
  • Arcavi, A. 2003. The role of visual representations in the learning mathematics. Educational Studies in Mathematics, 5 www.kent.k12.wa.us/KSD/KR/DIGITAL/math2.pdf
  • Bagni, G.T. (1998). Visualization and didactics of mathematics in high school: an experimental reseach. Scientia Paedogogica Experimentalis, 1, 161-180. 2(3), 215-241(27).
  • Borwein, P. & Jörgenson, L. (1997). Visible structures in number theory. Adresi: http://www.cecm.sfu.ca/loki/papers/numbers/node3.html.
  • Brown, J., Stillman, G., Schwarz, B. & Kaiser, G. (2008). The case of mathematical proof in lower secondary school: knowledge and competencies if pre-service teachers, proceedings of the 31st Annual Conference of the Mathematics Education Research Group of
  • Australasia, Hrsg.: M.Goos, R. Brown & K. Makar. (C) Merga Inc., 85-91. Conner, A. (2007). Argumentation in a Geometry Class: Aligned with the Teacher's Conception of Proof,
  • Delice, A. (2004). Trigonometri Sözel Problemlerinde Görselleştirme ve Diyagram Oluşturma, VI. Ulusal Fen Bilimleri ve Matematik Eğitimi Kongresi, Marmara Üni., İstanbul.
  • Delice, A. (2005). Türk ve İngiliz Eğitim sisteminde matematik eğitiminin karşılaştırılması, Milli Eğitim Dergisi, sayı 167. (http://yayim.meb.gov.tr/dergiler/167/index3-delice.htm)
  • Dufour-Janvier, B. (1987). Pedagogical Considerations Concerning the Problem of Representation, in C. Janvier (Ed), Problems of Representation in the Teaching and Learning of Mathematics. Hillsdale, NJ: Lawrence Erlbaum Associates.
  • Duval, R. (2002). Proof understanding in mathematics: what ways for students, proceedings of 2002 International conference on mathematics, http://www.math.ntnu.edu.tw/cyc/_ private/ mathedu/me1/me1_2002_1/duval.doc
  • Gravina, M.A. (2000) The proof in geometry: essays in a dynamical environment. http://www.lettredelapreuve.it/ICME9TG12/ICME9TG12Contributions/GravinaICM E00/GravinaICME00.html
  • Gravina, M.A. (2008). Dynamical visual proof: what does it mean?, ICME 11- TSG 22, http://tsg.icme11.org/document/get/233.
  • Guzman, M., 2002. What is visualization in math? In the teaching and learning of mathematical analysis. http://www.math.uoc.gr/~ictm2/Proceedings/invGuz.pdf.
  • Hanna, G and Villiers, M. (2008). ICMI Study 19: Proof and Proving in mathematics education, ZDM Mathematics Education, 40:329-336.
  • Hanna, G., and Sidoli, N. (2007).Visualization and proof: a brief survey of philosophical perspectives, ZDM Mathematics Education , Heidelberg, Vol. 39, page : 73-78.
  • Karadağ, Z. & McDougall, D. (2009). Visual explorative approaches to learning mathematics, http://www.pmena.org/ 2009/proceedings/workinggroup90649replacement.pdf .
  • Malaty, G. (2009). The Role of Visualization in Mathematics Education: Can visualization Promote the Causal Thinking?
  • MEB, (2006). Orta Öğretim Matematik 10. Sınıf Kitabı, Kelebek Matbaacılık, İstanbul.
  • NCTM (2000). Principles and Standards for School Mathematics, National Council of Teachers of Mathematics Inc.,http://standards.nctm.org/document/chapter7/geom.htm.
  • Nelsen, R.B. (1993). Proof Without Words, Exercises in visual thinking, The Mathematical Association of America Press, Washington, DC.
  • Özer, Ö. ve Arıkan, A. (2002). Lise matematik derslerinde öğrencilerin ispat yapabilme düzeyleri, V. Uluısal Fen ve Matematik Eğitimi Kongresi bildirisi, UFBMEK-5/b_kitabi /PDF/Matematik/Bildiri/t245d.pdf
  • Pettinelli, M. (2008). Visual learning. http://cnx.org/content/m14358/latest/).
  • Presmeg, N. C. (2006). Research on visualization in learning and teaching mathematics in
  • A.Gutierrez, P., Borero (eds.) Handbook of Research on Psychology of Mathematics Education: Past, Present and Future, 205-235. Sense Publishers, Rotterdam /Taipei.
  • Rahim, M.H., & Siddo, R., (2009). The use of visualization in learning and teaching mathematics. In A. Rogerson (Ed.), Proceedings of the 10th International Conference: Models in Developing Mathematics Education. The Mathematics Education into the 21st Century P r o j e c t , D r e s d e n , S a x o n y , G e r m a n y , S e p t 1 1 - 1 7 , 2 0 0 9 . http://math.unipa.it/~grim/21_project/Rahim496-500.pdf
  • Sarı, M. (2008). Undergraduate students' difficulties with mathematical proof and proof teaching, devam eden doktora tezi, Hacettepe Üniversitesi, Fen Bilimleri Enstitüsü. http://yess4.ktu.edu.tr/YermePappers/MeltemSari.pdf
  • Stylianou, D. A. & Silver, E. A. (2004). The role of visual representations in advanced mathematical problem solving: An examination of expert-novice similarities and differences. Journal of Mathematical Thinking and Learning, 6 (4). 353-387.
  • Tall, D., 2004. A versatile theory of visualization and symbolisation in mathematics. Plenary Presentation at the Commission Internationale pour L'Etude et l'Amelioration de l'Ensignement des Mathematiques, Toulouse, France.
There are 28 citations in total.

Details

Other ID JA55CT76VT
Journal Section Research Article
Authors

Birol Tekin This is me

Alper Cihan Konyalıoğlu This is me

Publication Date March 1, 2010
Submission Date March 1, 2010
Published in Issue Year 2010 Volume: 5 Issue: 1

Cite

APA Tekin, B., & Konyalıoğlu, A. C. (2010). Trigonometrik Fonksiyonların Toplam ve Fark Formüllerinin Ortaöğretim Düzeyinde Görselleştirilmesi. Bayburt Eğitim Fakültesi Dergisi, 5(1), 24-37.