Araştırma Makalesi
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Görsel Teoremler Üzerine Matematik Öğretmenleriyle Nitel Bir Çalışma

Yıl 2018, Cilt 7, Sayı 2, 56 - 74, 01.07.2018

Öz

Farklı disiplinlerde etkili olmasına rağmen özellikle geometri alanında daha etkili olduğu düşünülen ve zihinsel yeteneğin bir parçası olarak kabul edilen görsel uzamsal akıl yürütme becerisi birçok araştırmacının üzerinde durduğu bir konudur. Bu bağlamda geleceğin bilim insanlarının öğretim için gerekli görsel akıl yürütme becerilerini sağlayabilmede ve geliştirebilmede matematik öğretmenlerine sorumluluk düşmektedir. Bu çalışma ile matematik öğretmenlerinin görsel teoremleri ispatlama bağlamında, görsel akıl yürütme becerileri ile geometrik düşünme düzeyleri ve uzamsal görselleştirme becerileri arasındaki ilişkiyi nitel olarak özel durum yöntemiyle inceleme amaçlanmıştır. Çalışma grubunu on lise matematik öğretmeni oluşturmaktadır. Çalışmadan elde edilen veriler iki farklı görüşme sürecinde elde edilmiştir. İlk görüşmede öğretmenlerden “van Hiele Geometri Düzeyleri Testini ve Uzamsal Görselleştirme Beceri Testini” doldurmaları istenmiş, ikinci görüşmede ise öğretmenlerle üç farklı görsel teorem üzerinden klinik mülakatlar yürütülmüştür. Bu iki süreçten elde edilen veriler betimsel olarak analiz edilmiştir. Sonuç olarak,  matematik öğretmenlerinin görsel akıl yürütme becerileri ile geometrik düşünme düzeyleri ve uzamsal görselleştirme becerileri arasında bir ilişki olduğu,  daha yüksek geometrik düşünme düzeyine ulaşan öğretmenlerin, görsel teoremleri tanımada, onlar üzerine akıl yürütmede ve ispatlamada daha yetenekli olduğu tespit edilmiştir.

Kaynakça

  • Akay, S. (2013). Öğretmen adaylarının geometri düşünme düzeyleri ve beyin baskınlıklarının bazı değişkenler açısından incelenmesi. Yayımlanmamış yüksek lisans tezi, Eskişehir Osmangazi Üniversitesi, Eğitim Bilimleri Enstitüsü, Eskişehir.
  • Altun, M. (2005). İlköğretim ikinci kademede matematik öğretimi. Bursa: Alfa Basım Yayım.
  • Arcavi, A. (1994). Symbol sense: The informal sense-making in formal mathematics. For the Learning of Mathematics, 14(3), 24-35.
  • Aspinwall, L., Shaw, K., & Presmeg, N. (1997). Uncontrollable mental imagery: Graphical connections between a function and its derivative. Educational Studies in Mathematics, 33(3), 301-317.
  • Baki, A. (2008). Kuramdan uygulamaya matematik eğitimi (4. Basım). Ankara: Harf Eğitim Yayıncılığı.
  • Bakker, A., & Hoffmann, M. (2005). Diagrammatic reasoning as the basis for developing concepts: A semiotic analysis of students’ learning about statistical distribution. Educational Studies in Mathematics, 60, 333–358.
  • Ball, D. L. (1990). Prospective elementary and secondary teachers'understanding of division. Journal forResearch in Mathematics Education, 21 (2), 132-144.
  • Ball, D. L., Hill, H.C., & Bass, H. (2005). Knowing mathematics for teaching: Who knows mathematics well enough to teach third grade, and how can we decide? American Educator, 29(1), 14-46.
  • Battista, M. T., & Clements, D. H. (1988). A case for a Logo-based elementary school geometry curriculum. Arithmetic Teacher, 36, 11-17.
  • Baykul, Y. (2002). İlköğretimde matematik öğretimi (6–8. sınıflar). Ankara: Pegem A Yayıncılık.
  • Baykul, Y. (2009). Ortaokulda matematik öğretimi (5-8. sınıflar). Ankara: Pegem Akademi.
  • Carroll, W. M. (1998). Geometric knowledge of middle school students in a reformbased mathematics curriculum. School Science and Mathematics, 98(4), 188-197.
  • Casey, B. J., Jones, R. M., & Hare, T. A. (2008). The adolescent brain. Annals of the New York Academy of Sciences, 1124(1), 111-126.
  • Clements, D. H., & Battista, M. T. (1992). Geometry and spatial reasoning. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning, (pp. 420-464). New York: Macmillan. Cohen, L., Manion, L., & Morrison, K. (2000). Research methods in education (5th ed.). London: Routledge Falmer.
  • Cross, D. I. (2009). Alignment, cohesion, and change: Examining mathematics teachers’ beliefs structures and their influence on instructional practices. Journal Math Teacher Education, 12, 325 – 346.
  • Çepni, S. (2012). Araştırma ve proje çalışmalarına giriş (6. baskı). Trabzon: Celepler Matbaacılık.
  • Duatepe, A. (2000). An investigation of the relationship between van Hiele geometric level of thinking and demographic variable for pre-service elementary school teacher. Yayınlanmamış yüksek lisans tezi, Orta Doğu Teknik Üniversitesi, Ankara.
  • Dursun, Ş. ve Çoban, A. (2006). Geometri dersinin lise programları ve ÖSS soruları açısından değerlendirilmesi. Cumhuriyet Üniversitesi Sosyal Bilimler Dergisi, 30 (2), 213-221.
  • Fischbein, E. (1987). Intuition in science and mathematics. Dodrecht, Holland: Reidel.
  • Fuys, D., Geddes, D., & Tischler, R. (1988). The van Hiele model of thinking in geometry among adolescents. Journal for Research in Mathematics Education, Monograph number 3.
  • Guay, R. B. (1976). Purdue spatial visualization test. West Lafayette, IN: Purdue Research Foundation.
  • Gutierrez, A. (1992). Exploring the links between Van Hiele Levels and 3-dimensional geometry. Structural Topology 18, 31-48.
  • Hızarcı, S. (2004). Sunuş. In S. Hızarcı, A. Kaplan, A. S. İpek ve C. Işık (Eds.). Euclid geometri ve özel öğretimi. Ankara: Öğreti Yayınları.
  • Hill, H., Rowan, B., & Ball, D. L. (2005). Effects of teachers' mathematical knowledge for teaching on student achievement. American Education Research Journal, 42(2), 371-406.
  • Hoffman, D. D. (1998). Visual intelligence: How we create what we see. New York, US: W.W. Norton & Co.
  • Hoffmann, M. H. G. (2007). Cognitive conditions of diagrammatic reasoning. In J. Queiroz & F. Stjernfelt (Eds), Special issue on peircian diagrammatical logic, (pp. 1-28). Georgia Institute of Technology School of Public Policy, Atlanta, USA. [Available online at: http://citeseerx.ist.psu.edu/viewdoc/download?doi= 10.1.1.87.6282&rep=rep1&type=pdf], Retrieved on March 13, 2018.
  • Karrass, M. (2012). Diagrammatic reasoning skills of pre-service mathematics teachers. The Graduate School of Arts and Sciences, Columbia University, (Order No. 3493651). [Available online at: http://search. proquest.com/docview/919522981?accountid=15333], Retrieved on March 02, 2018.
  • Kösa, T. (2011). Ortaöğretim öğrencilerinin uzamsal yeteneklerinin incelenmesi. Yayınlanmamış doktora tezi, Karadeniz Teknik Üniversitesi Eğitim Bilimleri Enstitüsü, Trabzon.
  • Kümbetoğlu, B. (2005). Sosyoloji ve antropolojide niteliksel yöntem ve araştırma. İstanbul: Bağlam Yayıncılık.
  • Lakoff, G., & Núñez, R. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. New York: Basic Books.
  • Lawton, C. A. (2010). Gender, spatial abilities, and way finding. In J. C. Chrisler & D. R. Mc Creary (Eds.), Handbook of gender research in psychology, (pp.317-341). Springer New York.
  • Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States. Mahwah, NJ: Lawrence Erlbaum Associates.
  • Meng, C. C., Lian, L. H., & Idris, N. (2009). Pre service secondary mathemetics teachers’ geometric thinking and course grade. [Available online at: http://ftp.recsam.edu.my/cosmed/cosmed09/AbstractsFullPapers 2009/Abstract/Mathematics%20Parallel%20PDF/Full%20Paper/M07.pdf], Retrieved on March 07, 2018.
  • Monk, D. H. (1994). Subject area preparation of secondary mathematics and science teachers and student achievement. Economics of Education Review, 13(2), 125–145.
  • National Council of Teachers of Mathematics [NCTM]. (2000). Principles and standarts for school mathematics. Reston, VA: National Council of Teachers of Mathematics.
  • Nelsen, R. B. (1993). Proofs without words: Exercises in visual thinking. The Mathematical Association of America.
  • Pinto, M., & Tall, D. (2002). Building formal mathematics on visual imagery: a case study and a theory. For the Learning of Mathematics, 22(1), 2–10.
  • Presmeg, N.C. (1992). Prototypes, metaphors, metonymies and imaginative rationality in high school mathematics. Educational Studies in Mathematics, 23, 595-610.
  • Senk, S. (1989). Van Hiele levels and achievement in writing geometry proofs. Journal for Research in Mathematics Education. 20(3), 309-321.
  • Sevimli, E., Yıldız, Ç. ve Delice, A. (2008). Geometri sorularında görselleme sürecine bir bakış: Nereden çizeyim? 8. Ulusal Fen Bilimleri ve Matematik Eğitimi Kongresinde sunulan bildiri, Abant İzzet Baysal Üniversitesi, Bolu.
  • Terao, A., Koedinger K., Sohn, M-H., Anderson, J. R., & Carter, C. S. (2004). An fMRI study of the interplay of visual-spatial systems in mathematical reasoning. In Proceedings of the 26th Annual Conference of the Cognitive Science Society (pp. 1327-1332). August 4-7, Chicago, USA.
  • Turgut, M. (2007). İlköğretim II. kademede öğrencilerin uzamsal yeteneklerinin incelenmesi. Yayınlanmamış yüksek lisans tezi, Dokuz Eylül Üniversitesi, İzmir, Türkiye.
  • Türnüklü, E., Gündoğdu Alaylı, F., Simge Ergin, A. ve Baştürk Şahin, B.N. (2016). İlköğretim matematik öğretmen adaylarının şekil oluşturma düzeylerinin bazı değişkenlerle ilişkisi. Necatibey Eğitim Fakültesi Elektronik Fen ve Matematik Eğitimi Dergisi, 10 (1), 281-312.
  • Usiskin, Z. (1982). Van Hiele levels and achievement in secondary school geometry. Chicago: University of Chicago, Department of Education, (ERIC Document Reproduction Service No. ED 220 288).
  • Van De Walle, J. A. (2001). Elementary and middle school mathematics: Teaching developmentally. Boston: Allyn and Bacon.
  • Van Hiele, P. M. (1986). Structure and insight. New York: Academic Press.
  • Whiteley, W. (2004). Visualization in mathematics: Claims and questions towards a research program. Paper presented at the 10 International Congress on Mathematics Education, Copenhagen, Denmark, Cambridge, England: Cambridge University Press.
  • Yıldırım, A. ve Şimşek, H. (2013). Sosyal bilimlerde nitel araştırma yöntemleri. (9. Baskı). Ankara: Seçkin Yayıncılık.
  • Zaskis, R., & Hazzan, O. (1999). Interviewing in mathematics education research: Choosing the questions. Journal of Mathematical Behaviour, 17(4), 429-439.
  • Zhang, D., Ding, Y., Stegall, J., & Mo, L. (2012). The effect of visual‐chunking‐representation accommodation on geometry testing for students with math disabilities. Learning Disabilities Research & Practice, 27(4), 167-177.
  • Zimmerman, W., & Cunningham, S. (1991). Editor’s introduction: What is mathematical visualization? In W. Zimmerman and S. Cunningham (Eds.), Visualization in teaching and learning mathematics, (pp.1–8). Mathematical Association of America.

A Qualitative Study with Mathematics Teachers on Visual Theorems

Yıl 2018, Cilt 7, Sayı 2, 56 - 74, 01.07.2018

Öz

The geometry that develops the aesthetic sensation of individuals and event that allows them to think in many ways, at the same time helping individuals to better understand the world they live in and to relate mathematical concepts and events in life. Despite being effective in different disciplines, visual (diagrammatic) reasoning skill that is thought to be more effective, especially in the field of geometry, and which is considered to be part of mental ability, it is a topic that many researchers have pointed out. Because visual reasoning is an important skill that affects students especially to prove or solve geometric problems. Many researchers have suggested that an individual working at a higher level of geometric thinking should have stronger visual reasoning skills and that visual reasoning can also be improved by geometric teaching. However, despite the considerable importance given to geometry in recent years, it has been shown in many studies that the level of comprehension of the geometry of students is not expected and desired. Such results indicate that the objectives of the curriculum are not reached, such as training individuals with geometric and spatial thinking skills. In this sense, mathematics teachers need to possess the visual reasoning skills necessary for teaching in the training of the individuals (mathematicians, scientists, engineers, doctors, graphic designers, etc.) who will form the human power of the future. The aim of this study is to qualitatively examine the relationships between math teachers' visual reasoning skills and their level of geometric thinking and spatial visualization skills in the context of proving visual theorems.

This research is a descriptive study conducted using the case study method. The study group of the study is composed of ten mathematics teachers with a postgraduate degree in a university located in the Eastern Black Sea Region. In the selection of the study group, the maximum diversity method was chosen from the purposive sampling methods. As a result of two different interviews with the teachers, the data of the study were obtained. During the first interview, it was asked to answer open-ended questions prepared by the researchers with the help of literature to determine their geometric backgrounds. Then the teachers were asked to answer van Hiele Geometry Thinking Level Test and Spatial Visualization Skill Test. The data obtained from these two different tests were presented using descriptive statistics.  In this context, in order to determine the teachers' level of geometric thinking, a criterion was used in which teachers responded "at least 4 of 5 questions correctly" to each level. In the spatial visualization skill test, each teacher was given a test score of 36 points by giving (1) points to the questions that the teachers answered correctly, (0) points that they answered incorrectly or left empty. In the second interview, in order to reveal the problem solving or proving behaviours, visual reasoning skills, geometrical information about visual expressions and validation situations, clinical interviews were conducted on four different visual theorems. Data collected with clinical interviews were analysed using descriptive analysis technique.

In general, teachers with more geometric background scores have been found to have higher geometric thinking levels and better spatial visualization skills. In particular, teachers who are graduates of the Faculty of Education and who work as permanent staff were found to have achieved a level 4 in the van Hiele geometry thinking level test and higher scores than the spatial visualization skill test. In addition, teachers who attain a higher level of geometric thinking and have better spatial visualization skill scores, have been found to be more capable in recognizing visual theorems, in reasoning about them, and in verifying relationships. Another result is that there is a relationship between the visual reasoning that teachers have conducted on visual theorems and the way of thinking or behaviour attributed to van Hiele levels. However, it was found that most of the teachers who participated in the study had lack of knowledge about geometry and geometrical background, geometric thinking level test and spatial visualization skill test scores. In particular, it has been determined that teachers who have graduated from two faculties other than the Faculty of Education have a lack of information to be gained in geometry and these deficits affect the process of proving their visual theorems. The lack of information in the teachers has a negative effect on the learning of the students. Because there is a positive relationship between teacher knowledge and student achievement. In short, teachers have to know and understand the geometry they teach in order to make successful teaching. This work underscores some of the deficiencies in teacher education in the context of geometry and the need to re-examine the academic needs and curriculum changes needed to address these deficiencies.

Kaynakça

  • Akay, S. (2013). Öğretmen adaylarının geometri düşünme düzeyleri ve beyin baskınlıklarının bazı değişkenler açısından incelenmesi. Yayımlanmamış yüksek lisans tezi, Eskişehir Osmangazi Üniversitesi, Eğitim Bilimleri Enstitüsü, Eskişehir.
  • Altun, M. (2005). İlköğretim ikinci kademede matematik öğretimi. Bursa: Alfa Basım Yayım.
  • Arcavi, A. (1994). Symbol sense: The informal sense-making in formal mathematics. For the Learning of Mathematics, 14(3), 24-35.
  • Aspinwall, L., Shaw, K., & Presmeg, N. (1997). Uncontrollable mental imagery: Graphical connections between a function and its derivative. Educational Studies in Mathematics, 33(3), 301-317.
  • Baki, A. (2008). Kuramdan uygulamaya matematik eğitimi (4. Basım). Ankara: Harf Eğitim Yayıncılığı.
  • Bakker, A., & Hoffmann, M. (2005). Diagrammatic reasoning as the basis for developing concepts: A semiotic analysis of students’ learning about statistical distribution. Educational Studies in Mathematics, 60, 333–358.
  • Ball, D. L. (1990). Prospective elementary and secondary teachers'understanding of division. Journal forResearch in Mathematics Education, 21 (2), 132-144.
  • Ball, D. L., Hill, H.C., & Bass, H. (2005). Knowing mathematics for teaching: Who knows mathematics well enough to teach third grade, and how can we decide? American Educator, 29(1), 14-46.
  • Battista, M. T., & Clements, D. H. (1988). A case for a Logo-based elementary school geometry curriculum. Arithmetic Teacher, 36, 11-17.
  • Baykul, Y. (2002). İlköğretimde matematik öğretimi (6–8. sınıflar). Ankara: Pegem A Yayıncılık.
  • Baykul, Y. (2009). Ortaokulda matematik öğretimi (5-8. sınıflar). Ankara: Pegem Akademi.
  • Carroll, W. M. (1998). Geometric knowledge of middle school students in a reformbased mathematics curriculum. School Science and Mathematics, 98(4), 188-197.
  • Casey, B. J., Jones, R. M., & Hare, T. A. (2008). The adolescent brain. Annals of the New York Academy of Sciences, 1124(1), 111-126.
  • Clements, D. H., & Battista, M. T. (1992). Geometry and spatial reasoning. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning, (pp. 420-464). New York: Macmillan. Cohen, L., Manion, L., & Morrison, K. (2000). Research methods in education (5th ed.). London: Routledge Falmer.
  • Cross, D. I. (2009). Alignment, cohesion, and change: Examining mathematics teachers’ beliefs structures and their influence on instructional practices. Journal Math Teacher Education, 12, 325 – 346.
  • Çepni, S. (2012). Araştırma ve proje çalışmalarına giriş (6. baskı). Trabzon: Celepler Matbaacılık.
  • Duatepe, A. (2000). An investigation of the relationship between van Hiele geometric level of thinking and demographic variable for pre-service elementary school teacher. Yayınlanmamış yüksek lisans tezi, Orta Doğu Teknik Üniversitesi, Ankara.
  • Dursun, Ş. ve Çoban, A. (2006). Geometri dersinin lise programları ve ÖSS soruları açısından değerlendirilmesi. Cumhuriyet Üniversitesi Sosyal Bilimler Dergisi, 30 (2), 213-221.
  • Fischbein, E. (1987). Intuition in science and mathematics. Dodrecht, Holland: Reidel.
  • Fuys, D., Geddes, D., & Tischler, R. (1988). The van Hiele model of thinking in geometry among adolescents. Journal for Research in Mathematics Education, Monograph number 3.
  • Guay, R. B. (1976). Purdue spatial visualization test. West Lafayette, IN: Purdue Research Foundation.
  • Gutierrez, A. (1992). Exploring the links between Van Hiele Levels and 3-dimensional geometry. Structural Topology 18, 31-48.
  • Hızarcı, S. (2004). Sunuş. In S. Hızarcı, A. Kaplan, A. S. İpek ve C. Işık (Eds.). Euclid geometri ve özel öğretimi. Ankara: Öğreti Yayınları.
  • Hill, H., Rowan, B., & Ball, D. L. (2005). Effects of teachers' mathematical knowledge for teaching on student achievement. American Education Research Journal, 42(2), 371-406.
  • Hoffman, D. D. (1998). Visual intelligence: How we create what we see. New York, US: W.W. Norton & Co.
  • Hoffmann, M. H. G. (2007). Cognitive conditions of diagrammatic reasoning. In J. Queiroz & F. Stjernfelt (Eds), Special issue on peircian diagrammatical logic, (pp. 1-28). Georgia Institute of Technology School of Public Policy, Atlanta, USA. [Available online at: http://citeseerx.ist.psu.edu/viewdoc/download?doi= 10.1.1.87.6282&rep=rep1&type=pdf], Retrieved on March 13, 2018.
  • Karrass, M. (2012). Diagrammatic reasoning skills of pre-service mathematics teachers. The Graduate School of Arts and Sciences, Columbia University, (Order No. 3493651). [Available online at: http://search. proquest.com/docview/919522981?accountid=15333], Retrieved on March 02, 2018.
  • Kösa, T. (2011). Ortaöğretim öğrencilerinin uzamsal yeteneklerinin incelenmesi. Yayınlanmamış doktora tezi, Karadeniz Teknik Üniversitesi Eğitim Bilimleri Enstitüsü, Trabzon.
  • Kümbetoğlu, B. (2005). Sosyoloji ve antropolojide niteliksel yöntem ve araştırma. İstanbul: Bağlam Yayıncılık.
  • Lakoff, G., & Núñez, R. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. New York: Basic Books.
  • Lawton, C. A. (2010). Gender, spatial abilities, and way finding. In J. C. Chrisler & D. R. Mc Creary (Eds.), Handbook of gender research in psychology, (pp.317-341). Springer New York.
  • Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States. Mahwah, NJ: Lawrence Erlbaum Associates.
  • Meng, C. C., Lian, L. H., & Idris, N. (2009). Pre service secondary mathemetics teachers’ geometric thinking and course grade. [Available online at: http://ftp.recsam.edu.my/cosmed/cosmed09/AbstractsFullPapers 2009/Abstract/Mathematics%20Parallel%20PDF/Full%20Paper/M07.pdf], Retrieved on March 07, 2018.
  • Monk, D. H. (1994). Subject area preparation of secondary mathematics and science teachers and student achievement. Economics of Education Review, 13(2), 125–145.
  • National Council of Teachers of Mathematics [NCTM]. (2000). Principles and standarts for school mathematics. Reston, VA: National Council of Teachers of Mathematics.
  • Nelsen, R. B. (1993). Proofs without words: Exercises in visual thinking. The Mathematical Association of America.
  • Pinto, M., & Tall, D. (2002). Building formal mathematics on visual imagery: a case study and a theory. For the Learning of Mathematics, 22(1), 2–10.
  • Presmeg, N.C. (1992). Prototypes, metaphors, metonymies and imaginative rationality in high school mathematics. Educational Studies in Mathematics, 23, 595-610.
  • Senk, S. (1989). Van Hiele levels and achievement in writing geometry proofs. Journal for Research in Mathematics Education. 20(3), 309-321.
  • Sevimli, E., Yıldız, Ç. ve Delice, A. (2008). Geometri sorularında görselleme sürecine bir bakış: Nereden çizeyim? 8. Ulusal Fen Bilimleri ve Matematik Eğitimi Kongresinde sunulan bildiri, Abant İzzet Baysal Üniversitesi, Bolu.
  • Terao, A., Koedinger K., Sohn, M-H., Anderson, J. R., & Carter, C. S. (2004). An fMRI study of the interplay of visual-spatial systems in mathematical reasoning. In Proceedings of the 26th Annual Conference of the Cognitive Science Society (pp. 1327-1332). August 4-7, Chicago, USA.
  • Turgut, M. (2007). İlköğretim II. kademede öğrencilerin uzamsal yeteneklerinin incelenmesi. Yayınlanmamış yüksek lisans tezi, Dokuz Eylül Üniversitesi, İzmir, Türkiye.
  • Türnüklü, E., Gündoğdu Alaylı, F., Simge Ergin, A. ve Baştürk Şahin, B.N. (2016). İlköğretim matematik öğretmen adaylarının şekil oluşturma düzeylerinin bazı değişkenlerle ilişkisi. Necatibey Eğitim Fakültesi Elektronik Fen ve Matematik Eğitimi Dergisi, 10 (1), 281-312.
  • Usiskin, Z. (1982). Van Hiele levels and achievement in secondary school geometry. Chicago: University of Chicago, Department of Education, (ERIC Document Reproduction Service No. ED 220 288).
  • Van De Walle, J. A. (2001). Elementary and middle school mathematics: Teaching developmentally. Boston: Allyn and Bacon.
  • Van Hiele, P. M. (1986). Structure and insight. New York: Academic Press.
  • Whiteley, W. (2004). Visualization in mathematics: Claims and questions towards a research program. Paper presented at the 10 International Congress on Mathematics Education, Copenhagen, Denmark, Cambridge, England: Cambridge University Press.
  • Yıldırım, A. ve Şimşek, H. (2013). Sosyal bilimlerde nitel araştırma yöntemleri. (9. Baskı). Ankara: Seçkin Yayıncılık.
  • Zaskis, R., & Hazzan, O. (1999). Interviewing in mathematics education research: Choosing the questions. Journal of Mathematical Behaviour, 17(4), 429-439.
  • Zhang, D., Ding, Y., Stegall, J., & Mo, L. (2012). The effect of visual‐chunking‐representation accommodation on geometry testing for students with math disabilities. Learning Disabilities Research & Practice, 27(4), 167-177.
  • Zimmerman, W., & Cunningham, S. (1991). Editor’s introduction: What is mathematical visualization? In W. Zimmerman and S. Cunningham (Eds.), Visualization in teaching and learning mathematics, (pp.1–8). Mathematical Association of America.

Ayrıntılar

Birincil Dil Türkçe
Bölüm 17.SAYI
Yazarlar

Yaşar AKKAN> (Sorumlu Yazar)
GÜMÜŞHANE ÜNİVERSİTESİ, MÜHENDİSLİK VE DOĞA BİLİMLERİ FAKÜLTESİ, MATEMATİK MÜHENDİSLİĞİ BÖLÜMÜ
0000-0001-5323-7106
Türkiye


Pınar AKKAN Bu kişi benim
GÜMÜŞHANE ÜNİVERSİTESİ, GÜMÜŞHANE MESLEK YÜKSEKOKULU, ELEKTRONİK VE OTOMASYON BÖLÜMÜ
0000-0002-8942-4111
Türkiye


Mesut ÖZTÜRK>
BAYBURT ÜNİVERSİTESİ, BAYBURT EĞİTİM FAKÜLTESİ, MATEMATİK VE FEN BİLİMLERİ EĞİTİMİ BÖLÜMÜ, MATEMATİK EĞİTİMİ ANABİLİM DALI
0000-0002-2163-3769
Türkiye


Ümit DEMİR Bu kişi benim
BAYBURT ÜNİVERSİTESİ, SOSYAL BİLİMLER MESLEK YÜKSEKOKULU, FİNANS BANKACILIK VE SİGORTACILIK BÖLÜMÜ
0000-0003-0303-5623
Türkiye

Yayımlanma Tarihi 1 Temmuz 2018
Yayınlandığı Sayı Yıl 2018, Cilt 7, Sayı 2

Kaynak Göster

Bibtex @araştırma makalesi { jitte431508, journal = {Journal of Instructional Technologies and Teacher Education}, issn = {2149-4495}, address = {}, publisher = {Karadeniz Teknik Üniversitesi}, year = {2018}, volume = {7}, number = {2}, pages = {56 - 74}, title = {Görsel Teoremler Üzerine Matematik Öğretmenleriyle Nitel Bir Çalışma}, key = {cite}, author = {Akkan, Yaşar and Akkan, Pınar and Öztürk, Mesut and Demir, Ümit} }
APA Akkan, Y. , Akkan, P. , Öztürk, M. & Demir, Ü. (2018). Görsel Teoremler Üzerine Matematik Öğretmenleriyle Nitel Bir Çalışma . Journal of Instructional Technologies and Teacher Education , 7 (2) , 56-74 . Retrieved from https://dergipark.org.tr/tr/pub/jitte/issue/41978/431508
MLA Akkan, Y. , Akkan, P. , Öztürk, M. , Demir, Ü. "Görsel Teoremler Üzerine Matematik Öğretmenleriyle Nitel Bir Çalışma" . Journal of Instructional Technologies and Teacher Education 7 (2018 ): 56-74 <https://dergipark.org.tr/tr/pub/jitte/issue/41978/431508>
Chicago Akkan, Y. , Akkan, P. , Öztürk, M. , Demir, Ü. "Görsel Teoremler Üzerine Matematik Öğretmenleriyle Nitel Bir Çalışma". Journal of Instructional Technologies and Teacher Education 7 (2018 ): 56-74
RIS TY - JOUR T1 - Görsel Teoremler Üzerine Matematik Öğretmenleriyle Nitel Bir Çalışma AU - YaşarAkkan, PınarAkkan, MesutÖztürk, ÜmitDemir Y1 - 2018 PY - 2018 N1 - DO - T2 - Journal of Instructional Technologies and Teacher Education JF - Journal JO - JOR SP - 56 EP - 74 VL - 7 IS - 2 SN - 2149-4495- M3 - UR - Y2 - 2018 ER -
EndNote %0 Öğretim Teknolojileri ve Öğretmen Eğitimi Dergisi Görsel Teoremler Üzerine Matematik Öğretmenleriyle Nitel Bir Çalışma %A Yaşar Akkan , Pınar Akkan , Mesut Öztürk , Ümit Demir %T Görsel Teoremler Üzerine Matematik Öğretmenleriyle Nitel Bir Çalışma %D 2018 %J Journal of Instructional Technologies and Teacher Education %P 2149-4495- %V 7 %N 2 %R %U
ISNAD Akkan, Yaşar , Akkan, Pınar , Öztürk, Mesut , Demir, Ümit . "Görsel Teoremler Üzerine Matematik Öğretmenleriyle Nitel Bir Çalışma". Journal of Instructional Technologies and Teacher Education 7 / 2 (Temmuz 2018): 56-74 .
AMA Akkan Y. , Akkan P. , Öztürk M. , Demir Ü. Görsel Teoremler Üzerine Matematik Öğretmenleriyle Nitel Bir Çalışma. Journal of Instructional Technologies and Teacher Education. 2018; 7(2): 56-74.
Vancouver Akkan Y. , Akkan P. , Öztürk M. , Demir Ü. Görsel Teoremler Üzerine Matematik Öğretmenleriyle Nitel Bir Çalışma. Journal of Instructional Technologies and Teacher Education. 2018; 7(2): 56-74.
IEEE Y. Akkan , P. Akkan , M. Öztürk ve Ü. Demir , "Görsel Teoremler Üzerine Matematik Öğretmenleriyle Nitel Bir Çalışma", Journal of Instructional Technologies and Teacher Education, c. 7, sayı. 2, ss. 56-74, Tem. 2018