İlköğretim 4-7. Sınıf Öğrencilerinin Denk Kesirlerin Sembolik ve Grafiksel Temsillerini İlişkilendirme Becerilerinin İncelenmesi

Year 2021, Volume: 11 Issue: 3, 613 - 631, 01.12.2021

Levent Ertuna , Zulbiye Toluk Uçar

https://doi.org/10.19126/suje.992377

Cited By: 1

Abstract

Bu çalışmanın amacı, ilköğretim 4-7. sınıf öğrencilerinin denk kesirlerin sembolik ve grafiksel temsillerini ilişkilendirme becerilerinin düzeyini belirlemektir. Araştırmanın örneklemi Sakarya ilinde yer alan ilköğretim okullarında okuyan 4., 5., 6. ve 7. sınıf öğrencileri oluşturmaktadır. Çalışma bu kapsamda yer alan 11 okuldan 1111 öğrenci ile yürütülmüştür. Veri toplama aracı olarak, bu çalışmada, Temsilsel Akıcılık Testi (TAT) kullanılmış olup test, kesirlerin parça-bütün ve ölçme anlamlarını; alan, sayı doğrusu, uzunluk ve küme temsillerini kullanarak ölçmektedir. Normallik varsayımlarının ihlali nedeniyle grupları karşılaştırmak için parametrik olmayan testler kullanılmıştır. Yapılan analizler sonucunda öğrencilerin denk kesirlerin sembolik ve grafiksel temsillerini ilişkilendirme becerilerinin kesirlerin parça-bütün ve ölçme anlamlarında, kesirlerin farklı temsil türlerinde (Alan-Uzunluk, Alan-Sayı D., Küme-Uzunluk, Küme-Sayı D., Uzunluk-Sayı D.) ve kesrin sade ve denk gösterimlerinde farklılaştığı tespit edilmiştir. Bununla birlikte sınıf düzeyi arttıkça genel puanlarda, temsil türlerinde ise alan ve uzunluk temsili hariç diğer temsillerde, kesrin sade ve denk gösterimlerindeki başarının arttığı görülmüştür.

Keywords

Kesirlerin Anlamları, Kesirlerin Denkliği, Çoklu Temsiller, Kesirlerin Temsilleri

An Investigation of Elementary School 4-7th Grade Students' Ability to Link Equivalent Fractions' Symbolic and Graphical Representations

Abstract

The purpose of this study was to determine elementary school 4-7th grade students' ability to link equivalent fractions' symbolic and graphical representations. The design of this research was a survey study. The sample of the study consisted of 4, 5, 6, and 7th-grade elementary school students in the Sakarya province, Turkey. The study was conducted with 1111 students from 11 elementary schools. Representational Fluency Test (RFT) developed by Niemi (1996) was used as a measurement tool. The RFT included items involving regional areas, line segments, and set representations to assess the part-whole meaning and those involving number lines to assess the measure meaning of the rational number. As the normality assumption was violated, non-parametric tests were applied. The results of the analyses showed that the students' performance to link equivalent fractions' symbolic and graphical representations changed significantly with respect to the representation type (region-line segment, region-number line, set-line segment, set-number line, line segment- number line) and with respect to simple and equivalent fractions. Meanwhile, it was seen that as the classroom levels increased, the success rates in overall scores, representation types, except region-line segment, and simple and equivalent fractions increased.

Keywords

Fraction Subconstructs, Equivalence of Fractions, Multiple Representations, Representation of Fractions

References

• Aliustaoğlu, F., Tuna, A., & Biber, A. Ç. (2018). The misconceptions of sixth grade secondary school students on fractions. International Electronic Journal of Elementary Education, 10(5), 591-599.
• Aliustaoğlu, F., Tuna, A., & Biber, A. Ç. (2018). The misconceptions of sixth grade secondary school students on fractions. International Electronic Journal of Elementary Education, 10(5), 591-599.
• Armstrong, B., & Larson, C. (1995). Students' Use of Part-Whole and Direct Comparison Strategies for Comparing Partitioned Rectangles. Journal for Research in Mathematics Education, 26(1), 2-19. doi:10.2307/749225
• Armstrong, B., & Larson, C. (1995). Students' Use of Part-Whole and Direct Comparison Strategies for Comparing Partitioned Rectangles. Journal for Research in Mathematics Education, 26(1), 2-19. doi:10.2307/749225
• Baykul, Y. (2005). İlköğretimde matematik öğretimi (1–5.Sınıflar İçin). Ankara: Pegem Yayıncılık.
• Baykul, Y. (2005). İlköğretimde matematik öğretimi (1–5.Sınıflar İçin). Ankara: Pegem Yayıncılık.
• Behr, M. J., Harel, G., Post, T. R., & Lesh, R. (1992). Rational number, ratio, and proportion. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning: A project of the National Council of Teachers of Mathematics (pp. 296–333). Macmillan Publishing Co, Inc.
• Behr, M. J., Harel, G., Post, T. R., & Lesh, R. (1992). Rational number, ratio, and proportion. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning: A project of the National Council of Teachers of Mathematics (pp. 296–333). Macmillan Publishing Co, Inc.
• Behr, M., Khoury, H., Harel, G., Post, T. & Lesh, R., (1997) Conceptual Units Analysis of Preservice Elementary School Teachers' Strategies on a Rational-Number-as-Operator Task. Journal of Mathematics Education, 28(1), 48-69.
• Behr, M., Khoury, H., Harel, G., Post, T. & Lesh, R., (1997) Conceptual Units Analysis of Preservice Elementary School Teachers' Strategies on a Rational-Number-as-Operator Task. Journal of Mathematics Education, 28(1), 48-69.
• Behr, M., Lesh, R., Post, T., & Silver E. (1983). Rational Number Concepts. In R. Lesh & M. Landau (Eds.), Acquisition of Mathematics Concepts and Processes, (pp. 91-125). New York: Academic Press
• Behr, M., Lesh, R., Post, T., & Silver E. (1983). Rational Number Concepts. In R. Lesh & M. Landau (Eds.), Acquisition of Mathematics Concepts and Processes, (pp. 91-125). New York: Academic Press
• Bright, G., Behr, M., Post, T. & Wachsmuth, I. (1988). Identifying fractions on number lines. Journal for Research in Mathematics Education, 19(3), 215-232.
• Bright, G., Behr, M., Post, T. & Wachsmuth, I. (1988). Identifying fractions on number lines. Journal for Research in Mathematics Education, 19(3), 215-232.
• Castro-Rodríguez, E., Pitta-Pantazi, D., Rico, L., & Gómez, P. (2016). Prospective teachers’ understanding of the multiplicative part-whole relationship of fraction. Educational Studies in Mathematics, 92(1), 129-146.
• Castro-Rodríguez, E., Pitta-Pantazi, D., Rico, L., & Gómez, P. (2016). Prospective teachers’ understanding of the multiplicative part-whole relationship of fraction. Educational Studies in Mathematics, 92(1), 129-146.
• Charalambous, C. Y. & Pitta-Pantazi, D. (2007). Drawing on a theoretical model to study students’ understandings of fractions. Educational Studies in Mathematics, 64(3), 293–316.
• Charalambous, C. Y. & Pitta-Pantazi, D. (2007). Drawing on a theoretical model to study students’ understandings of fractions. Educational Studies in Mathematics, 64(3), 293–316.
• Clarke, D. M., Roche, A., & Mitchell, A. (2008). 10 practical tips for making fractions come alive and make sense. Mathematics Teaching in The Middle School, 13(7), 372-380.
• Clarke, D. M., Roche, A., & Mitchell, A. (2008). 10 practical tips for making fractions come alive and make sense. Mathematics Teaching in The Middle School, 13(7), 372-380.
• Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale, NJ: Lawrence Earlbaum Associates.
• Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale, NJ: Lawrence Earlbaum Associates.
• Contreras, J. N., Martinez-Cruz, A. M. (2000). Prospective elementary teachers' dominant situations and knowledge about representations of rational numbers. Twenty-second Annual Meeting of the International Group for the Psycology of Mathematics Education, (7-10 October), Tucson, Arizona
• Contreras, J. N., Martinez-Cruz, A. M. (2000). Prospective elementary teachers' dominant situations and knowledge about representations of rational numbers. Twenty-second Annual Meeting of the International Group for the Psycology of Mathematics Education, (7-10 October), Tucson, Arizona
• Cramer, K. & Henry, A. (2002). Using manipulative models to build number sense for addition and fractions. In B. Litwiller (Ed.), Making sense of fractions, ratios, and proportions (pp. 41-48). Reston, VA: The National Council of Teachers of Mathematics, Inc.
• Cramer, K. & Henry, A. (2002). Using manipulative models to build number sense for addition and fractions. In B. Litwiller (Ed.), Making sense of fractions, ratios, and proportions (pp. 41-48). Reston, VA: The National Council of Teachers of Mathematics, Inc.
• Cramer, K., Wyberg, T. ve Leavitt, S. (2008). The Role of Representations in Fraction Addition and Subtraction. Mathematics Teaching in the Middle School. 13 (8), 490-496.
• Cramer, K., Wyberg, T. ve Leavitt, S. (2008). The Role of Representations in Fraction Addition and Subtraction. Mathematics Teaching in the Middle School. 13 (8), 490-496.
• Fraenkel, J. R., Wallen, N. E., & Hyun, H. H. (2019). How to design and evaluate research in education (10th ed.). New York: McGraw-Hill.
• Fraenkel, J. R., Wallen, N. E., & Hyun, H. H. (2019). How to design and evaluate research in education (10th ed.). New York: McGraw-Hill.
• Gabriel, F., Coché, F., Szucs, D., Carette, V., Rey, B., & Content, A. (2013). A componential view of children’s difficulties in learning fractions. Frontiers in Psychology, 4, 715.
• Gabriel, F., Coché, F., Szucs, D., Carette, V., Rey, B., & Content, A. (2013). A componential view of children’s difficulties in learning fractions. Frontiers in Psychology, 4, 715.
• Gürbüz, R. & Birgin, O. (2008). Farklı Öğrenim Seviyesindeki Öğrencilerin Rasyonel Sayıların Farklı Gösterim Şekilleriyle İşlem Yapma Performanslarının Karşılaştırılması [The Comparison of Students’ Performance at Different Grades Regarding to Making Operation with Different Types of Representation of the Rational Numbers]. Pamukkale University Journal of Education, 23(1), 85-94
• Gürbüz, R. & Birgin, O. (2008). Farklı Öğrenim Seviyesindeki Öğrencilerin Rasyonel Sayıların Farklı Gösterim Şekilleriyle İşlem Yapma Performanslarının Karşılaştırılması [The Comparison of Students’ Performance at Different Grades Regarding to Making Operation with Different Types of Representation of the Rational Numbers]. Pamukkale University Journal of Education, 23(1), 85-94
• Hart, K.M. (1987). Practical work and formalisation, too great a gap. J.C.Bergeron, N. Herscovics, C. Kieran. Proceedings of the Eleventh International Conference Psychology of Mathematics Education (PME-XI) Vol II, 408-415. Montreal.
• Hart, K.M. (1987). Practical work and formalisation, too great a gap. J.C.Bergeron, N. Herscovics, C. Kieran. Proceedings of the Eleventh International Conference Psychology of Mathematics Education (PME-XI) Vol II, 408-415. Montreal.
• Hull, L. (2005). Fraction Models That Promote Understanding for Elementary Students. [Unpublished master’s thesis]. University of Central Florida.
• Hull, L. (2005). Fraction Models That Promote Understanding for Elementary Students. [Unpublished master’s thesis]. University of Central Florida.
• Hunting, R. P. (1984). Understanding equivalent fractions. Journal of Science and Mathematics Education in Southeast Asia, 7, 26-33
• Hunting, R. P. (1984). Understanding equivalent fractions. Journal of Science and Mathematics Education in Southeast Asia, 7, 26-33
• Kamii, C., & Clark, F. B. (1995). Equivalent fractions: Their difficulty and educational implications. Journal of Mathematical Behavior, 14, 365-378.
• Kamii, C., & Clark, F. B. (1995). Equivalent fractions: Their difficulty and educational implications. Journal of Mathematical Behavior, 14, 365-378.
• Kılıç, Ç. (2009). İlköğretim beşinci sınıf öğrencilerinin matematiksel problemlerin çözümlerinde kullandıkları temsiller [Representations the Fifth Grade Primary School Students Used in The Process of Solving Mathematical Problems]. (Doctoral Dissertation). Retrieved from Turkish Council of Higher Education Thesis Database (Thesis Number: 235250).
• Kılıç, Ç. (2009). İlköğretim beşinci sınıf öğrencilerinin matematiksel problemlerin çözümlerinde kullandıkları temsiller [Representations the Fifth Grade Primary School Students Used in The Process of Solving Mathematical Problems]. (Doctoral Dissertation). Retrieved from Turkish Council of Higher Education Thesis Database (Thesis Number: 235250).
• Kieren, T. E. (1988). Personal knowledge of rational numbers: Its intuitive and formal development. In M. J. Behr & J. Hiebert (Eds), Number Concepts and Operations in the Middle Grades (pp. 162–181). Reston, VA: National Council of Teachers of Mathematics
• Kieren, T. E. (1988). Personal knowledge of rational numbers: Its intuitive and formal development. In M. J. Behr & J. Hiebert (Eds), Number Concepts and Operations in the Middle Grades (pp. 162–181). Reston, VA: National Council of Teachers of Mathematics
• Larson, N. C. (2000). The Development of The Part-Whole And Measure Subconstructs Of Four Sixth-Graders During An Instructional Unit. Twenty-second Annual Meeting of the International Group for the Psycology of Mathematics Education, (7-10 October), Tucson, Arizona.
• Larson, N. C. (2000). The Development of The Part-Whole And Measure Subconstructs Of Four Sixth-Graders During An Instructional Unit. Twenty-second Annual Meeting of the International Group for the Psycology of Mathematics Education, (7-10 October), Tucson, Arizona.
• Lesh, R. (1979). Mathematical learning disabilities: Considerations for identification, diagnosis, and remediation. In R. Lesh, D. Mierkiewicz, & M. G. Kantowski (Eds.), Applied mathematical problem solving (pp. 111-180). Columbus, OH: ERIC/SM.
• Lesh, R. (1979). Mathematical learning disabilities: Considerations for identification, diagnosis, and remediation. In R. Lesh, D. Mierkiewicz, & M. G. Kantowski (Eds.), Applied mathematical problem solving (pp. 111-180). Columbus, OH: ERIC/SM.
• Mısral, M. (2009). Kesrin farklı anlamlarına göre yapılan öğretimin İlköğretim 6. Sınıf öğrencilerinin kesirlerde toplama çıkarma ve çarpma işlemlerinde kavramsal ve işlemsel bilgi düzeylerine etkisi [The Effect of The Education Which Is Done by The Different Subconstructs of Fractions on The Conceptual and Operational Knowledge Levels of Primary School 6th Grade Students About Adding Subtraction and Multiplication in Fraction]. (Masters’ Thesis). Retrieved from Turkish Council of Higher Education Thesis Database (Thesis Number: 237470).
• Mısral, M. (2009). Kesrin farklı anlamlarına göre yapılan öğretimin İlköğretim 6. Sınıf öğrencilerinin kesirlerde toplama çıkarma ve çarpma işlemlerinde kavramsal ve işlemsel bilgi düzeylerine etkisi [The Effect of The Education Which Is Done by The Different Subconstructs of Fractions on The Conceptual and Operational Knowledge Levels of Primary School 6th Grade Students About Adding Subtraction and Multiplication in Fraction]. (Masters’ Thesis). Retrieved from Turkish Council of Higher Education Thesis Database (Thesis Number: 237470).
• Muzheve, M. T. (2014). Pre-service teachers’ conceptions of representations of equivalent fractions and of fraction units. In Matney, G. T. and Che, S. M. (Eds.) (pp. 57-63). Proceedings of the 41st Annual Meeting of the Research Council on Mathematics Learning. San Antonio, TX.
• Muzheve, M. T. (2014). Pre-service teachers’ conceptions of representations of equivalent fractions and of fraction units. In Matney, G. T. and Che, S. M. (Eds.) (pp. 57-63). Proceedings of the 41st Annual Meeting of the Research Council on Mathematics Learning. San Antonio, TX.
• Ni, Y. (2001). Semantic domains of rational numbers and the acquisition of fraction equivalence. Contemporary Educational Psychology, 26, 400–417.
• Ni, Y. (2001). Semantic domains of rational numbers and the acquisition of fraction equivalence. Contemporary Educational Psychology, 26, 400–417.
• Niemi, D. (1996). Instructional influence on content area explanations and represenational knowledge: Evidence for the construct validity of measures of principled understanding. CSE Technical Report 403. National Center for Research on Evaluation, Standards, and Student Testing, Los Angeles, CA: University of California.
• Niemi, D. (1996). Instructional influence on content area explanations and represenational knowledge: Evidence for the construct validity of measures of principled understanding. CSE Technical Report 403. National Center for Research on Evaluation, Standards, and Student Testing, Los Angeles, CA: University of California.
• Olkun S. ve Toluk Uçar Z. (2004). İlköğretimde Etkinlik Temelli Matematik Öğretimi. Ankara: Anı Yayıncılık. Park, J., Güçler, B. ve McCrory, R. (2013). Teaching Prospective Teachers about Fractions: Historical and Pedagogical Perspectives, Educational Studies in Mathematics, 82(3), 455-479
• Olkun S. ve Toluk Uçar Z. (2004). İlköğretimde Etkinlik Temelli Matematik Öğretimi. Ankara: Anı Yayıncılık. Park, J., Güçler, B. ve McCrory, R. (2013). Teaching Prospective Teachers about Fractions: Historical and Pedagogical Perspectives, Educational Studies in Mathematics, 82(3), 455-479
• Pesen, C. (2003). Eğitim Fakülteleri ve Sınıf Öğretmenleri için Matematik Öğretimi. Ankara: Nobel Yayın Dağıtım
• Pesen, C. (2003). Eğitim Fakülteleri ve Sınıf Öğretmenleri için Matematik Öğretimi. Ankara: Nobel Yayın Dağıtım
• Pesen, C. (2008). Kesirlerin Sayı Doğrusu Üzerindeki Gösteriminde Öğrencilerin Öğrenme Güçlükleri ve Kavram Yanılgıları [Students’ Learning Difficulties and Misconceptions in Pointing the Fractions on The Number Line]. Inonu University Journal of the Faculty of Education, 9(15), 157–168.
• Pesen, C. (2008). Kesirlerin Sayı Doğrusu Üzerindeki Gösteriminde Öğrencilerin Öğrenme Güçlükleri ve Kavram Yanılgıları [Students’ Learning Difficulties and Misconceptions in Pointing the Fractions on The Number Line]. Inonu University Journal of the Faculty of Education, 9(15), 157–168.
• Piaget, J. (1983). Piaget’s Theory. In P. Mussen (Ed.) Handbook of child psychology. Wiley.
• Piaget, J. (1983). Piaget’s Theory. In P. Mussen (Ed.) Handbook of child psychology. Wiley.
• Piaget, J. (1987). Possibility and necessity. Vol. II: The role of necessity in cognitive development. Minneapolis: University of Minnesota Press.
• Piaget, J. (1987). Possibility and necessity. Vol. II: The role of necessity in cognitive development. Minneapolis: University of Minnesota Press.
• Piaget, J., Inhelder, B. ve Szeminska, A. (1960). The child’s conception of geometry. (E.A. Lunzer, Trans.) New York: Basic Books. (Original work published 1948)
• Piaget, J., Inhelder, B. ve Szeminska, A. (1960). The child’s conception of geometry. (E.A. Lunzer, Trans.) New York: Basic Books. (Original work published 1948)
• Putra, Z. H. (2018). A praxeological analysis of pre-service elementary teachers’ knowledge of rational numbers. Recherches En Didactique Des Mathematiques, 38(3), 315-364.
• Putra, Z. H. (2018). A praxeological analysis of pre-service elementary teachers’ knowledge of rational numbers. Recherches En Didactique Des Mathematiques, 38(3), 315-364.
• Putra, Z. H. (2019). Praxeological change and the density of rational numbers: The case of pre-service teachers in Denmark and Indonesia. Eurasia Journal of Mathematics, Science and Technology Education, 15(5), em1711.
• Putra, Z. H. (2019). Praxeological change and the density of rational numbers: The case of pre-service teachers in Denmark and Indonesia. Eurasia Journal of Mathematics, Science and Technology Education, 15(5), em1711.
• Sidney, P. G., Thompson, C. A., & Rivera, F. D. (2019). Number lines, but not area models, support children’s accuracy and conceptual models of fraction division. Contemporary Educational Psychology, 58, 288-298.
• Sidney, P. G., Thompson, C. A., & Rivera, F. D. (2019). Number lines, but not area models, support children’s accuracy and conceptual models of fraction division. Contemporary Educational Psychology, 58, 288-298.
• Siebert, D. & Gaskin, N. (2006). Creating, naming, and justifying fractions. Teaching Children Mathematics, 12(8), 394-400.
• Siebert, D. & Gaskin, N. (2006). Creating, naming, and justifying fractions. Teaching Children Mathematics, 12(8), 394-400.
• Streefland, L. (1991). Fractions in Realistic Mathematics Education: A Paradigm of Developmental Research, The Netherlands, Kluwer Academic Publishers.
• Streefland, L. (1991). Fractions in Realistic Mathematics Education: A Paradigm of Developmental Research, The Netherlands, Kluwer Academic Publishers.
• Toluk, Z. (2002). İlkokul Öğrencilerinin Bölme İşlemi ve Rasyonel Sayıları İlişkilendirme Süreçleri. Boğaziçi Üniversitesi Eğitim Dergisi, 19(2), 81-101.
• Toluk, Z. (2002). İlkokul Öğrencilerinin Bölme İşlemi ve Rasyonel Sayıları İlişkilendirme Süreçleri. Boğaziçi Üniversitesi Eğitim Dergisi, 19(2), 81-101.
• Toluk-Uçar, Z. (2009). Developing pre-service teachers understanding of fractions through problem posing. Teaching and Teacher Education, 25(1), 166-175.
• Toluk-Uçar, Z. (2009). Developing pre-service teachers understanding of fractions through problem posing. Teaching and Teacher Education, 25(1), 166-175.
• Wong, M., Evans, D. & Anderson, J. (2006). Developing a Diagnostic Assessment Instrument for Identifying Students’ Understanding of Fraction Equivalence. The University of Sydney. ACSPRI Conference.( December) Sydney, Australia.
• Wong, M., Evans, D. & Anderson, J. (2006). Developing a Diagnostic Assessment Instrument for Identifying Students’ Understanding of Fraction Equivalence. The University of Sydney. ACSPRI Conference.( December) Sydney, Australia.
• Wong, M. & Evans, D. (2011). Assessing Students’ Understanding of Fraction Equivalence. In J. Way & J. Bobis (Eds.), Fractions: Teaching for Understanding (pp. 81-90). Adelaide: The Australian Association of Mathematics Teachers Inc.
• Wong, M. & Evans, D. (2011). Assessing Students’ Understanding of Fraction Equivalence. In J. Way & J. Bobis (Eds.), Fractions: Teaching for Understanding (pp. 81-90). Adelaide: The Australian Association of Mathematics Teachers Inc.
• Yazgan, Y. (2007). 10-11 yaş Grubundaki Öğrencilerin Kesirleri Kavramaları üzerine Deneysel Bir Çalışma [An Experimental Study on Fraction Understanding of Children at The Age Of 10 and 11]. (Doctoral Dissertation). Retrieved from Turkish Council of Higher Education Thesis Database (Thesis Number: 220989).
• Yazgan, Y. (2007). 10-11 yaş Grubundaki Öğrencilerin Kesirleri Kavramaları üzerine Deneysel Bir Çalışma [An Experimental Study on Fraction Understanding of Children at The Age Of 10 and 11]. (Doctoral Dissertation). Retrieved from Turkish Council of Higher Education Thesis Database (Thesis Number: 220989).