Araştırma Makalesi
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7. Sınıf Öğrencilerinin Kesirlerde Toplama İşlemine Kurdukları Problemlerin Analizi

Yıl 2012, Cilt: 11 Sayı: 4, 1021 - 1035, 26.06.2012

Cemalettin Işık Tuğrul Kar

Öz

Araştırmada, ilköğretim yedinci sınıf öğrencilerinin kesirlerde toplama işlemine yönelik
kurdukları problemlerde karşılaşabilecekleri olası güçlüklerin belirlenmesi amaçlanmıştır. Araştırma,
Erzurum merkezdeki yedi ilköğretim okulunun yedinci sınıflarında öğrenim gören 210 öğrenci ile
yapılmıştır. Kesirlerde toplama işlemine yönelik beş maddeden oluşan Problem Kurma Testi (PKT) veri
toplama aracı olarak kullanılmıştır. Öğrencilerin kurdukları problemlerde; toplanan ikinci kesri bütünün
kalanı üzerinden ifade etme, parça-bütün ilişkisini kuramama, işlem sonucuna doğal sayı anlamı yükleme,
birim kargaşası, toplanan kesir sayılarına doğal sayı anlamı yükleme, işlemi soru köküne yansıtamama ve
tamsayılı kesirlerin tam kısımlarına anlam yükleyememe şeklinde yedi güçlük tespit edilmiştir. En fazla
güçlük sonucun tamsayılı kesir olduğu iki basit kesrin toplamına, en az güçlük ise sonucun basit kesir
olduğu iki basit kesrin toplamına yönelik problem kurmada görülmüştür.

Anahtar Kelimeler

Problem kurma kesirler kesirlerde toplama işlemi

Kaynakça

  • Abu-Elwan, R. (2002). Effectiveness of problem posing strategies on prospective mathematics teachers‟ problem solving performance. Journal of Science and Mathematics Education in S.E. Asia, 25(1), 56-69.
  • Ahmad, W. F. B. W. & Latih, N. H. B. A. (2010). Development of a mathematics courseware: Fractions. Paper presented at the Asian Technology Conference in Mathematics, Malaysia.
  • Akay, H. (2006). Problem kurma yaklaşımı ile yapılan matematik öğretiminin öğrencilerin akademik başarısı, problem çözme becerisi ve yaratıcılığı üzerindeki etkisinin incelenmesi (Yayınlanmamış Doktora Tezi). Gazi Üniversitesi, Ankara.
  • Alacacı, C. (2009). Öğrencilerin kesirler konusundaki kavram yanılgıları. Bingölbali, E. & Özmantar, M. F. (Eds.), İlköğretimde karşılaşılan matematiksel zorluklar ve çözüm önerileri. Pegem Akademi Yayınları, Ankara.
  • Amato, A. S. (2005). Developıng students‟ understandıng of the concept of fractıons as numbers. Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education, Australia.
  • Birgin, O. ve Gürbüz, R. (2009). İlköğretim II. Kademe Öğrencilerinin Rasyonel Sayılar Konusundaki İşlemsel Ve Kavramsal Bilgi Düzeylerinin İncelenmesi. Uludağ Üniversitesi Eğitim Fakültesi Dergisi, XXII(2), 529-550.
  • Brueckner, L. J. (1928). Analysis of errors in fractıon. The Elementary School Journal, 28(10), 760- 770.
  • Bulgar, S. (2003). Children's sense-making of division of fractions. The Journal of Mathematical Behavior, 20, 319-334.
  • Cai, J. (2003). Singaporean students‟ mathematical thinking in problem solving and problem posing: an exploratory study. International Journal of Mathematical Education in Science and Technology, 34(5), 719-737.
  • Charalambous, C. Y., Delaney, S., Hsu, H. Y. & Mesa, V. (2010). A comparative analysis of the addition and subtraction of fractions in textbooks from three countries. Mathematical Thinking and Learning, 12, 117–151.
  • Charalambous C. Y. & Pantazi, D. P. (2005). Revisiting a theoretical Model on Fractions: Implications for teaching and Research. In Chick, H.L. & Vincent, J. L. (Eds.), Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education, (Vol 2, pp. 233 – 240).
  • Crespo, S. & Sinclair, N. (2008). What makes a problem mathematically interesting? Inviting prospective teachers to pose better problems. Journal Mathematics Teacher Education, 11, 395–415.
  • Darley, J. W. (2005). Ninth grader‟s ınterpretations and use of contextualized models of fractions and algebraic properties: A classroom-based approach (Unpublished doctoral dissertation). University of South Carolina.
  • Dickerson, V. M. (1999). The impact of problem-posing instruction on the mathematical problem- solving achievement of seventh graders (Unpublished doctoral dissertation). University of Emory, Atlanta.
  • Doğan, M. ve Yeniterzi, B. (2011). İlköğretim 7. Sınıf Öğrencilerinin Rasyonel Sayılar Konusundaki Hazır Bulunuşlukları. Selçuk Üniversitesi Ahmet Keleşoğlu Eğitim Fakültesi Dergisi, 31,217-237.
  • English, L. D. (1997). The development of fifth-grade children‟s problem-posing abilities. Educational Studies in Mathematics, 34, 183-217.
  • English, L. D. (1998). Children‟s problem posing within formal and informal contexts. Journal for Research in Mathematics Education, 29(1), 83-106.
  • English, L. D. & Halford, G. S. (1995). Mathematics education: Models and processes. New Jersey: Lawrence Erlbaum Associates.
  • Frankenstein, M. (1998). Reading the world with maths: Goals for a critical mathematical literacy curriculum. In P. Gates (Ed.), Proceedings of the First International Mathematics Education and Society Conference (pp. 180-189). Nottingham: Centre for the Study of Mathematics Education, Nottingham University.
  • Gallistel, C. R. & Gelman, R . (1992). Preverbal and verbal counting and computation. Cognition, 44(1–2), 43–74.
  • Hasemann, K. (1981). On difficulties with fractions. Educational Studies in Mathematics 12, 71-87.
  • Herman, J., Ilucova, L., Kremsova, V., Pribyl, J., Ruppeldtova, J., Simpson, A., Stehlikova, N., Sulista, M. & Ulrychova, M. (2004). “Images of fractions as process and images of fractions in processes.” Proceedings of the 28th Conference of the International Group of the Psychology of Mathematics Education, Bergen.
  • Işık, C. (2011). İlköğretim Matematik Öğretmeni Adaylarının Kesirlerde Çarpma ve Bölmeye Yönelik Kurdukları Problemlerin Kavramsal Analizi. Hacettepe Üniversitesi Eğitim Fakültesi Dergisi, 41, 231-243.
  • Işık, C., Işık, A. ve Kar, T. (2011). Öğretmen Adaylarının Sözel ve Görsel Temsillere Yönelik Kurdukları Problemlerin Analizi. Pamukkale Üniversitesi Eğitim Fakültesi Dergisi, 30, 39- 49.
  • Işık, C. ve Kar, T. (2012). Matematik Dersinde Problem Kurmaya Yönelik Öğretmen Görüşleri Üzerine Nitel Bir Çalışma. Milli Eğitim Dergisi, 194, 199-215.
  • Işıksal, M. (2006). A study on pre-service elementary mathematics teachers‟ subject matter knowledge and pedagogical content knowledge regarding the multiplicatıon and division of fractions (unpublished doctoral dissertation). Middle East Technical University, Turkey.
  • Kerslake, D. (1986). Fractions: Children‟s strategies and errors. A report of the strategies and errors in secondary mathematics project. Windsor, England: NFER-Nelson.
  • Kocaoğlu, T. ve Yenilmez, K. (2010). Beşinci Sınıf Öğrencilerinin Kesir Problemlerinde Yaptıkları Hatalar ve Kavram Yanılgıları. Dicle Üniversitesi Ziya Gökalp Eğitim Fakültesi Dergisi, 14, 71-85.
  • Küçük, A. ve Demir, B. (2009). İlköğretim 6–8. Sınıflarda Matematik Öğretiminde Karşılaşılan Bazı Kavram Yanılgıları Üzerine Bir Çalışma. Dicle Üniversitesi Ziya Gökalp Eğitim Fakültesi Dergisi, 13, 97-112.
  • Lamon, S. (1999). Teaching fractions and ratios for understanding. Essential content knowledge and instructional strategies for teachers. London: Erlbaum.
  • Lowrie, T. (2002). Young children posing problems: the influence of teacher intervention on the type of problems children pose. Mathematics Education Research Journal,14(2), 87-98.
  • Mack, N. K. (1990). Learning fractions with understanding: Building on informal knowledge. Journal for Research in Mathematics Education, 21, 16-32.
  • Mack, N. K. (1995). Confounding whole-number and fraction concepts when building on informal knowledge. Journal for Research in Mathematics Education, 26(5), 422-441.
  • Mcmillan, H. J. & Schumacher, S. (2010). Research in education. Boston, USA: Pearson Education.
  • MEB (2009). İlköğretim Matematik Dersi 1-5. Sınıflar Öğretim Programı. Ankara: MEB Basımevi.
  • MEB (2009). İlköğretim Matematik Dersi 6-8. Sınıflar Öğretim Programı. Ankara: MEB Basımevi.
  • Misquitta, R. (2011). A review of the literature: Fraction instruction for struggling learners in mathematics. Learning Disabilities Research & Practice, 26(2), 109–119.
  • Mok, I., Cai, J. & Fong-Fung, A. (2008). Missing learning opportu-nities in classroom instruction: evidence from an analysis of a well-structured lesson on comparing fractions. The Mathematics Educator, 11(1-2), 111-126.
  • Ni, Y. & Zhou, Y. D. (2005). Teaching and learning fraction and rational numbers: The origins and implications of whole number bias. Educational Psychologist , 40(1), 27–52.
  • Olkun, S. ve Toluk, Z. (2003). İlköğretimde etkinlik temelli matematik öğretimi. Ankara, Anı yayıncılık.
  • Pantziara, M. & Philippou, G. (2011). Levels of students‟ “conception” of fractions. Educational Studies in Mathematics, 79(1), 61-83.
  • Pitkethly, A. & Hunting, R. (1996). A review of recent research in the area of initial fraction concepts. Educational Studies in Mathematics, 30, 5-38.
  • Redmond, A. (2009). Prospective elementary teachers‟ division of fractions understanding: A mixed methods study (Unpublished doctoral thesis). Oklahoma State University.
  • Redmond, A. & Utley, J. (2007). “Prospective elementary teachers understanding of and attitudes towards the division of fractions”. Paper presented at the Research Council on Mathematics Learning Annual Convention, Oklahoma City.
  • Rule, A. C. & Hallagan, J. E. (2006). “Preservice elementary teachers use drawings and make sets of materials to explain multiplication and division by fractions.” Paper Presented at the 2nd Annual Preparing Mathematicians to Educate Teachers (PMET) Conference at Oswego, New York.
  • Sharp, J. & Adams, B. (2002). Children's constructions of knowledge for fraction division after solving realistic problems. The Journal of Educational Research, 95(6), 333-347.
  • Siegler, R. S. (2003). Implications of cognitive science research for mathematics education. In Kilpatrick, J., Martin, W. B., & Schifter, D. E. (Eds.), A research companion to principles and standards for school mathematics (pp. 219-233). Reston, VA: National Council of Teachers of Mathmatics.
  • Soylu, Y. ve Soylu, C. (2005). İlköğretim Beşinci Sınıf Öğrencilerinin Kesirler Konusundaki Öğrenme Güçlükleri: Kesirlerde Sıralama, Toplama, Çıkarma, Çarpma Ve Kesirlerle İlgili Problemler. Erzincan Eğitim Fakültesi Dergisi. 7(2), 101-117.
  • Ticha, M. & Hošpesová, A. (2009). Problem posing and development of pedagogical content knowledge in pre-service teacher training. Paper presented at the meeting of CERME 6, Lyon.
  • Tirosh, D. (2000). Enhancing prospective teachers‟ knowledge of children‟s conception: The case of division of fraction. Journal for Research in Mathematics Education, 31(1), 5-25.
  • Toluk-Uçar, Z. (2009). Developing pre-service teachers understanding of fractions through problem posing. Teaching and Teacher Education, 25, 166–175.
  • Van De Walle, J. A. (2004). Elementary and middle school mathematics: teaching developmentally (5th ed.). Boston, MA: Pearson.
  • Yanik, H. B., Helding, B. & Flores, A. (2008). Teaching the Concept of Unit in Measurement Interpretation of Rational Numbers. Elementary Education Online, 7(3), 693-705, [Online]: http://ilkogretim-online.org.tr adresinden 10 Aralık 2011 tarihinde indirilmiştir.
  • Zembat, İ.Ö. (2007). Sorun Aynı – Kavramlar; Kitle Aynı - Öğretmen Adayları. Elementer Education Online, 6(2), 305-312, [Online]: http://ilkogretim-online.org.tr adresinden 12 Aralık 2011 tarihinde indirilmiştir.

Analyzing Problems Posed by 7th Grade Students for Addition Operation with Fractions

Yıl 2012, Cilt: 11 Sayı: 4, 1021 - 1035, 26.06.2012

Cemalettin Işık Tuğrul Kar

Öz

The aim of the present study was to determine the potential difficulties which might be
experienced by elementary school 7th grade students related to the problems posed by them on addition
operation with fractions. The study was conducted with 210 students studying at seven elementary
schools located in Erzurum city center. The Problem Posing Test consisted of five items related to
addition operation with fractions was used as the data collection tool. Seven difficulties were determined
in the problems posed by the students. The most significant difficulty was observed in posing problems
requiring addition of two simple fractions to produce a fraction with integer, while the least significant
one was experienced in posing problems requiring addition of two simple fractions to produce a simple
fraction.

Anahtar Kelimeler

Problem posing fractions addition operation with fractions.

Kaynakça

  • Abu-Elwan, R. (2002). Effectiveness of problem posing strategies on prospective mathematics teachers‟ problem solving performance. Journal of Science and Mathematics Education in S.E. Asia, 25(1), 56-69.
  • Ahmad, W. F. B. W. & Latih, N. H. B. A. (2010). Development of a mathematics courseware: Fractions. Paper presented at the Asian Technology Conference in Mathematics, Malaysia.
  • Akay, H. (2006). Problem kurma yaklaşımı ile yapılan matematik öğretiminin öğrencilerin akademik başarısı, problem çözme becerisi ve yaratıcılığı üzerindeki etkisinin incelenmesi (Yayınlanmamış Doktora Tezi). Gazi Üniversitesi, Ankara.
  • Alacacı, C. (2009). Öğrencilerin kesirler konusundaki kavram yanılgıları. Bingölbali, E. & Özmantar, M. F. (Eds.), İlköğretimde karşılaşılan matematiksel zorluklar ve çözüm önerileri. Pegem Akademi Yayınları, Ankara.
  • Amato, A. S. (2005). Developıng students‟ understandıng of the concept of fractıons as numbers. Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education, Australia.
  • Birgin, O. ve Gürbüz, R. (2009). İlköğretim II. Kademe Öğrencilerinin Rasyonel Sayılar Konusundaki İşlemsel Ve Kavramsal Bilgi Düzeylerinin İncelenmesi. Uludağ Üniversitesi Eğitim Fakültesi Dergisi, XXII(2), 529-550.
  • Brueckner, L. J. (1928). Analysis of errors in fractıon. The Elementary School Journal, 28(10), 760- 770.
  • Bulgar, S. (2003). Children's sense-making of division of fractions. The Journal of Mathematical Behavior, 20, 319-334.
  • Cai, J. (2003). Singaporean students‟ mathematical thinking in problem solving and problem posing: an exploratory study. International Journal of Mathematical Education in Science and Technology, 34(5), 719-737.
  • Charalambous, C. Y., Delaney, S., Hsu, H. Y. & Mesa, V. (2010). A comparative analysis of the addition and subtraction of fractions in textbooks from three countries. Mathematical Thinking and Learning, 12, 117–151.
  • Charalambous C. Y. & Pantazi, D. P. (2005). Revisiting a theoretical Model on Fractions: Implications for teaching and Research. In Chick, H.L. & Vincent, J. L. (Eds.), Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education, (Vol 2, pp. 233 – 240).
  • Crespo, S. & Sinclair, N. (2008). What makes a problem mathematically interesting? Inviting prospective teachers to pose better problems. Journal Mathematics Teacher Education, 11, 395–415.
  • Darley, J. W. (2005). Ninth grader‟s ınterpretations and use of contextualized models of fractions and algebraic properties: A classroom-based approach (Unpublished doctoral dissertation). University of South Carolina.
  • Dickerson, V. M. (1999). The impact of problem-posing instruction on the mathematical problem- solving achievement of seventh graders (Unpublished doctoral dissertation). University of Emory, Atlanta.
  • Doğan, M. ve Yeniterzi, B. (2011). İlköğretim 7. Sınıf Öğrencilerinin Rasyonel Sayılar Konusundaki Hazır Bulunuşlukları. Selçuk Üniversitesi Ahmet Keleşoğlu Eğitim Fakültesi Dergisi, 31,217-237.
  • English, L. D. (1997). The development of fifth-grade children‟s problem-posing abilities. Educational Studies in Mathematics, 34, 183-217.
  • English, L. D. (1998). Children‟s problem posing within formal and informal contexts. Journal for Research in Mathematics Education, 29(1), 83-106.
  • English, L. D. & Halford, G. S. (1995). Mathematics education: Models and processes. New Jersey: Lawrence Erlbaum Associates.
  • Frankenstein, M. (1998). Reading the world with maths: Goals for a critical mathematical literacy curriculum. In P. Gates (Ed.), Proceedings of the First International Mathematics Education and Society Conference (pp. 180-189). Nottingham: Centre for the Study of Mathematics Education, Nottingham University.
  • Gallistel, C. R. & Gelman, R . (1992). Preverbal and verbal counting and computation. Cognition, 44(1–2), 43–74.
  • Hasemann, K. (1981). On difficulties with fractions. Educational Studies in Mathematics 12, 71-87.
  • Herman, J., Ilucova, L., Kremsova, V., Pribyl, J., Ruppeldtova, J., Simpson, A., Stehlikova, N., Sulista, M. & Ulrychova, M. (2004). “Images of fractions as process and images of fractions in processes.” Proceedings of the 28th Conference of the International Group of the Psychology of Mathematics Education, Bergen.
  • Işık, C. (2011). İlköğretim Matematik Öğretmeni Adaylarının Kesirlerde Çarpma ve Bölmeye Yönelik Kurdukları Problemlerin Kavramsal Analizi. Hacettepe Üniversitesi Eğitim Fakültesi Dergisi, 41, 231-243.
  • Işık, C., Işık, A. ve Kar, T. (2011). Öğretmen Adaylarının Sözel ve Görsel Temsillere Yönelik Kurdukları Problemlerin Analizi. Pamukkale Üniversitesi Eğitim Fakültesi Dergisi, 30, 39- 49.
  • Işık, C. ve Kar, T. (2012). Matematik Dersinde Problem Kurmaya Yönelik Öğretmen Görüşleri Üzerine Nitel Bir Çalışma. Milli Eğitim Dergisi, 194, 199-215.
  • Işıksal, M. (2006). A study on pre-service elementary mathematics teachers‟ subject matter knowledge and pedagogical content knowledge regarding the multiplicatıon and division of fractions (unpublished doctoral dissertation). Middle East Technical University, Turkey.
  • Kerslake, D. (1986). Fractions: Children‟s strategies and errors. A report of the strategies and errors in secondary mathematics project. Windsor, England: NFER-Nelson.
  • Kocaoğlu, T. ve Yenilmez, K. (2010). Beşinci Sınıf Öğrencilerinin Kesir Problemlerinde Yaptıkları Hatalar ve Kavram Yanılgıları. Dicle Üniversitesi Ziya Gökalp Eğitim Fakültesi Dergisi, 14, 71-85.
  • Küçük, A. ve Demir, B. (2009). İlköğretim 6–8. Sınıflarda Matematik Öğretiminde Karşılaşılan Bazı Kavram Yanılgıları Üzerine Bir Çalışma. Dicle Üniversitesi Ziya Gökalp Eğitim Fakültesi Dergisi, 13, 97-112.
  • Lamon, S. (1999). Teaching fractions and ratios for understanding. Essential content knowledge and instructional strategies for teachers. London: Erlbaum.
  • Lowrie, T. (2002). Young children posing problems: the influence of teacher intervention on the type of problems children pose. Mathematics Education Research Journal,14(2), 87-98.
  • Mack, N. K. (1990). Learning fractions with understanding: Building on informal knowledge. Journal for Research in Mathematics Education, 21, 16-32.
  • Mack, N. K. (1995). Confounding whole-number and fraction concepts when building on informal knowledge. Journal for Research in Mathematics Education, 26(5), 422-441.
  • Mcmillan, H. J. & Schumacher, S. (2010). Research in education. Boston, USA: Pearson Education.
  • MEB (2009). İlköğretim Matematik Dersi 1-5. Sınıflar Öğretim Programı. Ankara: MEB Basımevi.
  • MEB (2009). İlköğretim Matematik Dersi 6-8. Sınıflar Öğretim Programı. Ankara: MEB Basımevi.
  • Misquitta, R. (2011). A review of the literature: Fraction instruction for struggling learners in mathematics. Learning Disabilities Research & Practice, 26(2), 109–119.
  • Mok, I., Cai, J. & Fong-Fung, A. (2008). Missing learning opportu-nities in classroom instruction: evidence from an analysis of a well-structured lesson on comparing fractions. The Mathematics Educator, 11(1-2), 111-126.
  • Ni, Y. & Zhou, Y. D. (2005). Teaching and learning fraction and rational numbers: The origins and implications of whole number bias. Educational Psychologist , 40(1), 27–52.
  • Olkun, S. ve Toluk, Z. (2003). İlköğretimde etkinlik temelli matematik öğretimi. Ankara, Anı yayıncılık.
  • Pantziara, M. & Philippou, G. (2011). Levels of students‟ “conception” of fractions. Educational Studies in Mathematics, 79(1), 61-83.
  • Pitkethly, A. & Hunting, R. (1996). A review of recent research in the area of initial fraction concepts. Educational Studies in Mathematics, 30, 5-38.
  • Redmond, A. (2009). Prospective elementary teachers‟ division of fractions understanding: A mixed methods study (Unpublished doctoral thesis). Oklahoma State University.
  • Redmond, A. & Utley, J. (2007). “Prospective elementary teachers understanding of and attitudes towards the division of fractions”. Paper presented at the Research Council on Mathematics Learning Annual Convention, Oklahoma City.
  • Rule, A. C. & Hallagan, J. E. (2006). “Preservice elementary teachers use drawings and make sets of materials to explain multiplication and division by fractions.” Paper Presented at the 2nd Annual Preparing Mathematicians to Educate Teachers (PMET) Conference at Oswego, New York.
  • Sharp, J. & Adams, B. (2002). Children's constructions of knowledge for fraction division after solving realistic problems. The Journal of Educational Research, 95(6), 333-347.
  • Siegler, R. S. (2003). Implications of cognitive science research for mathematics education. In Kilpatrick, J., Martin, W. B., & Schifter, D. E. (Eds.), A research companion to principles and standards for school mathematics (pp. 219-233). Reston, VA: National Council of Teachers of Mathmatics.
  • Soylu, Y. ve Soylu, C. (2005). İlköğretim Beşinci Sınıf Öğrencilerinin Kesirler Konusundaki Öğrenme Güçlükleri: Kesirlerde Sıralama, Toplama, Çıkarma, Çarpma Ve Kesirlerle İlgili Problemler. Erzincan Eğitim Fakültesi Dergisi. 7(2), 101-117.
  • Ticha, M. & Hošpesová, A. (2009). Problem posing and development of pedagogical content knowledge in pre-service teacher training. Paper presented at the meeting of CERME 6, Lyon.
  • Tirosh, D. (2000). Enhancing prospective teachers‟ knowledge of children‟s conception: The case of division of fraction. Journal for Research in Mathematics Education, 31(1), 5-25.
  • Toluk-Uçar, Z. (2009). Developing pre-service teachers understanding of fractions through problem posing. Teaching and Teacher Education, 25, 166–175.
  • Van De Walle, J. A. (2004). Elementary and middle school mathematics: teaching developmentally (5th ed.). Boston, MA: Pearson.
  • Yanik, H. B., Helding, B. & Flores, A. (2008). Teaching the Concept of Unit in Measurement Interpretation of Rational Numbers. Elementary Education Online, 7(3), 693-705, [Online]: http://ilkogretim-online.org.tr adresinden 10 Aralık 2011 tarihinde indirilmiştir.
  • Zembat, İ.Ö. (2007). Sorun Aynı – Kavramlar; Kitle Aynı - Öğretmen Adayları. Elementer Education Online, 6(2), 305-312, [Online]: http://ilkogretim-online.org.tr adresinden 12 Aralık 2011 tarihinde indirilmiştir.
Toplam 54 adet kaynakça vardır.

Ayrıntılar

Birincil Dil Türkçe
Bölüm Makaleler
Yazarlar

Cemalettin Işık

Tuğrul Kar Bu kişi benim

Yayımlanma Tarihi 26 Haziran 2012
Yayımlandığı Sayı Yıl 2012 Cilt: 11 Sayı: 4

Kaynak Göster

APA Işık, C., & Kar, T. (2012). 7. Sınıf Öğrencilerinin Kesirlerde Toplama İşlemine Kurdukları Problemlerin Analizi. İlköğretim Online, 11(4), 1021-1035.
AMA Işık C, Kar T. 7. Sınıf Öğrencilerinin Kesirlerde Toplama İşlemine Kurdukları Problemlerin Analizi. İOO. Aralık 2012;11(4):1021-1035.
Chicago Işık, Cemalettin, ve Tuğrul Kar. “7. Sınıf Öğrencilerinin Kesirlerde Toplama İşlemine Kurdukları Problemlerin Analizi”. İlköğretim Online 11, sy. 4 (Aralık 2012): 1021-35.
EndNote Işık C, Kar T (01 Aralık 2012) 7. Sınıf Öğrencilerinin Kesirlerde Toplama İşlemine Kurdukları Problemlerin Analizi. İlköğretim Online 11 4 1021–1035.
IEEE C. Işık ve T. Kar, “7. Sınıf Öğrencilerinin Kesirlerde Toplama İşlemine Kurdukları Problemlerin Analizi”, İOO, c. 11, sy. 4, ss. 1021–1035, 2012.
ISNAD Işık, Cemalettin - Kar, Tuğrul. “7. Sınıf Öğrencilerinin Kesirlerde Toplama İşlemine Kurdukları Problemlerin Analizi”. İlköğretim Online 11/4 (Aralık 2012), 1021-1035.
JAMA Işık C, Kar T. 7. Sınıf Öğrencilerinin Kesirlerde Toplama İşlemine Kurdukları Problemlerin Analizi. İOO. 2012;11:1021–1035.
MLA Işık, Cemalettin ve Tuğrul Kar. “7. Sınıf Öğrencilerinin Kesirlerde Toplama İşlemine Kurdukları Problemlerin Analizi”. İlköğretim Online, c. 11, sy. 4, 2012, ss. 1021-35.
Vancouver Işık C, Kar T. 7. Sınıf Öğrencilerinin Kesirlerde Toplama İşlemine Kurdukları Problemlerin Analizi. İOO. 2012;11(4):1021-35.