Matematik Öğrenme Güçlüğüne Sahip Öğrencilerin Kesir Kavramına İlişkin Kavrayışlarının İncelenmesi

Yıl 2022, Cilt: 8 Sayı: 3, 333 - 369, 30.11.2022

Cansu Bakırcı Saymaz , Ziya Argün

Öz

Bu çalışmanın amacı, matematik öğrenme güçlüğüne sahip öğrencilerin kesir kavramının oluşumunda yer alan öncül kavramlara ait düşünmelerini, kavrayışlarını ve anlayışlarını derinlemesine incelemektir. Araştırma nitel desene sahip bir durum çalışmasıdır. Çalışmanın katılımcıları, 5., 6. ve 7. sınıflarda öğrenim gören matematik öğrenme güçlüğüne sahip amaçlı örneklem ile seçilmiş yedi öğrencidir. Çalışmanın verileri araştırmacı tarafından hazırlanan kesir kavramının oluşumunda yer alan öncül kavramlar ile ilgili 48 sorunun yer aldığı yarı yapılandırılmış klinik görüşmeler yoluyla toplanmış ve bu veriler betimsel analiz yöntemiyle analiz edilmiştir. Bu analiz sonucu elde edilen bulgulara göre, öğrencilerin, eşit paylaşımın (bir bütün için) bütünün eş parçaya bölünüp parçaların paylaşımı anlamına geldiğinin farkında oldukları, bütünü parçalama ve kontrol için parçayı tekrarlama eylemlerini yapabildikleri, modeller ile temsil eden kesirleri sözel olarak birim kesirle ifade edebildikleri, ekleme yapıldığında basit kesir ile ifade edilen parçalar koleksiyonunu bütün ile karşılaştırabildikleri ve kesirleri sembolik olarak doğru yazdıkları tespit edilmiştir. Bununla birlikte; birim, birimleştirme gibi ölçme kavramlarındaki eksikliklerinin kesrin ölçme anlamının oluşturulması için engel teşkil ettiği, belirli sayıda verilen nesneleri verilen sayıda kişiye/nesneye eş pay düşecek şekilde ayırmada zorluk yaşadıkları, bütüne ait parçalar koleksiyonu verildiğinde eş parça olup olmadıklarına karar verirken parçaların göz önüne alınan niteliğinin miktarlarının eşit olmasından ziyade parçaların fiziki görünüşüne göre karar verdikleri, pay ve paydanın anlamı ile ilgili anlayışlarının sınırlı olduğu, birden büyük sayıları temsil eden kesirleri sözel olarak ifade etmede ve model oluşturmada zorluk yaşadıkları, kesrin sadece parça-bütün anlamına odaklandıkları, kesirlerin denkliği kavramında, kesrin temsil ettiği miktarları karşılaştırmada ve sıralamada zorluk yaşadıkları belirlenmiştir.

Anahtar Kelimeler

Matematik Öğrenme Güçlüğü, Kesirler, Kavrayış, Matematik Eğitimi

Examination of Students with Math Learning Difficulties' Understanding Related to the Concept of Fraction

Öz

This study aims to deeply analyze the thinking, conception, and understanding of the preliminary concepts involved in the formation fraction concept of students with mathematics learning difficulties. The research was designed with a case study, one of the qualitative research methods, and the participants in the study participants consisted of seven students with mathematics learning difficulties. Participants studying in the 5th, 6th, and 7th grades were selected by purposive sampling. The Data was collected through semi-structured clinical interviews with 48 questions about the preliminary concepts involved in the formation fraction concept. Questions were prepared by the researcher and analyzed by the descriptive analysis method. As a result of the analysis, students were found to be aware equal sharing (for a total) means dividing the sum into equal parts and sharing the parts. They can also split a total and repeat the unite part for control, verbally express the fraction represented by the model as a unit fraction, and compare a collection of parts expressed by a simple fraction with the total when adding. Results indicated that participants could compare and correctly write fractions symbolically. In addition, the deficiencies in measurement concepts such as unit and unitization constitute an obstacle to forming the measurement meaning of fractions for them. They also have difficulty in separating a certain number of given objects in a way that equals a given number of people/things, deciding whether the given collection of parts is equal or no, and determining according to the physical appearance of the parts rather than the equivalent amount of the received quantity. Their understanding of the meaning of the numerator and denominator is also limited. Further, they have difficulty verbally expressing the fractions and creating a model of the fractions that represent numbers greater than one, focusing only on the part-sum meaning of the fraction and acquiring the concept of equivalence of fractions in comparing and ordering the quantities represented by the fraction.

Anahtar Kelimeler

Mathematics Learning Disability, Fractions, Students' Understanding

Kaynakça

• Barbieri, C. A., Rodrigues, J., Dyson, N., & Jordan, N. C. (2020). Improving fraction understanding in sixth graders with mathematics difficulties: Effects of a number line approach combined with cognitive learning strategies. Journal of Educational Psychology, 112(3), 628–648.
• Barnett-Clarke, C., William, F., Rick, M., Sharon, R., & Rose, M. Z. (2010). Developing essential understanding of rational numbers: Grades 3–5. Reston.
• Bryant, R. A., Mastrodomenico, J., & Felmingham, K. L. (2008). Treatment of acute stress disorder: A randomized controlled trial. Archives of General Psychiatry, 65, 659- 667.
• Bouck, E. C., Satsangi, R., & Park, J. (2018). The concrete–representational–abstract approach for students with learning disabilities: An evidence-based practice synthesis. Remedial and Special Education, 39(4), 211–228.
• Bottge, B. A., Ma, X., Gassaway, L., Toland, M. D., Butler, M., & Cho, S. J. (2014). Effects of blended instructional models on math performance. Exceptional Children, 80, 423-437.
• Büyüköztürk, Ş., Kılıç Çakmak, E., Akgün, Ö.E., Karadeniz, Ş., & Demirel, F. (2009). Bilimsel araştırma yöntemleri. Pegem Yayınları.
• Chmiliar, L. (2010). Multiple-case designs. In A. J. Mills, G. Eurepas & E. Wiebe (Eds.), Encyclopedia of case study research (pp 582-583). Sage.
• Clarke, D. M., Roche, A., & Mitchell, A. (2008). Ten practical tips for making fractions come alive and make sense. Mathematics Teaching in the Middle School, 13(7), 373-380.
• Clement, J. (2000). Analysis of clinical interviews: Foundations and model viability. In R. Lesh & A. E. Kelly (Eds.), Research design in mathematics and science education (pp. 308- 327). Hillsdale.
• Clements, D. H., & Battista, M. T. (2000). Designing effective software. In A. E. Kelly & R. A. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 761–776). Lawrence Erlbaum Associates.
• Clements, D., & Stephan, M. (2004). Measurement in preK-2 mathematics. In D. Clements, J. Sarama, A. M. Di-Biase (Eds.), Engaging young children in mathematics: Standards for early childhood mathematics education (pp. 299- 321). Lawrence Erlbaum.
• Common Core State Standards Initiative (2010). Common core state standards for mathematics. http://www.corestandards.org/assets/CCSSI_Math %20Standards.pdf
• Creswell, J. W. (2007). Qualitative inquiry & research design: Choosing among five approaches (2nd Ed.). Sage.
• Çıkılı, Y., Gönen, A., Aslan Bağcı, Ö., & Kaybar, H. (2020). Özel eğitim alanında görev yapan öğretmenlerin bireyselleştirilmiş eğitim programı (bep) hazırlama konusunda yaşadıkları güçlükler. Uluslararası Toplum Araştırmaları Dergisi, 15, 5123-5148.
• Empson, S. B. (1999). Equal sharing and shared meaning: The development of fraction concepts in a first-grade classroom. Cognition and Instruction, 17(3), 283–342.
• Ennis, R. P., & Losinski, M. (2019). Interventions to improve fraction skills for students with disabilities: A meta-analysis. Exceptional Children, 85, 367-386.
• Fazio, L. K., Kennedy, C. A., & Siegler, R. S. (2016). Improving children's knowledge of fraction magnitudes. Plos One, 11(10), e0165243. https://doi.org/10.1371/journal.pone.0165243
• Fuchs, L. S., Schumacher, R. F., Long, J., Namkung, J., Hamlett, C.L., & Cirino, P.T. (2013). Improving at-risk learners' understanding of fractions. Journal of Educational Psychology, 105(3), 683–700.
• Fuchs, L. S., Schumacher, R. F., Sterba, S. K., Long, J., Namkung, J., & Malone, A. (2014). Does working memory moderate the effects of fraction intervention? An aptitude–treatment interaction. Journal of Educational Psychology, 106, 499–514.
• Fuchs, L. S., Wang, A. Y., & Preacher, K. J. (2020). Addressing challenging mathematics standards with at-risk learners: A randomized controlled trial on the effects of fractions intervention at third Grade. Exceptional Children, 87(2), 163–182.
• Ginsburg, H. P. (1997). Entering the child's mind: The clinical interview in psychological research and practice. Cambridge University Press.
• Güven Akdeniz, D., & Argün, Z. (2019). İlköğretim 5. sınıf öğrencilerinin uzunluk kavrayışlarına dair bir durum çalışması. Necatibey Eğitim Fakültesi Elektronik Fen ve Matematik Eğitimi Dergisi, 13(2), 807-836.
• Hacker, D., Kiuhara, S. A., & Levin, J. R. (2019). A metacognitive intervention for teaching fractions to students with or at-risk for learning disabilities in mathematics. ZDM– The International Journal on Mathematics Education, 51, 601–612.
• Hancock, R.D., & Algozzine, B. (2006). Doing case study research. Teachers College Press.
• Haser, Ç., & Ubuz, B. (2002). Kesirlerde kavramsal ve işlemsel performans. Eğitim ve Bilim, 27, 53-61.
• Hughes, E. (2019). Point of view video modeling to teach simplifying fractions to middle school students with mathematical learning disabilities. Learning Disabilities: A Contemporary Journal, 17, 41–57.
• Hunt, J. H., Silva, J., & Lambert, R. (2019). Empowering students with specific learning disabilities: Jim's concept of unit fraction. Journal of Mathematical Behaviour, 56(2), 100738.
• Hwang, J. (2016). An examination of error patterns associated with middleschool students' solution pathways when solving fraction computationinvolving addition. Unpublished Doctoral Dissertation, The Pennsylvania State University, Pennsylvania, PA.
• Hwang, J., Riccomini, P. J., Hwang, S. Y., & Morano, S. (2019). A systematic analysis of experimental studies targeting fractions for studentswith mathematics difficulties. Learning Disabilities Research & Practice, 34(1), 47–61.
• Ikhwanudin, T., & Suryadi, D. (2018). How students with mathematics learning disabilities understands fraction: A case from the indonesian inclusive school. International Journal of Instruction, 11(3), 309-326.
• Ikhwanudin, T., & Wahyudin, S. P. (2019). The error pattern of students with mathematics learning disabilities in the inclusive school on fractions learning. International Journal of Learning, Teaching and Educational Research, 18(3), 75-95.
• Jordan, N. C., Hansen, N., Fuchs, L. S., Siegler, R. S., Gersten, R., & Micklos, D. (2013). Developmental predictors of fraction concepts and procedures. Journal of Experimental Child Psychology, 116(1), 45-58.
• Lamon, S. (1999). Teaching fractions and ratios for understanding. Essential content knowledge and instructional strategies for teachers. Erlbaum.
• Lamon, S. (2007). Rational numbers and proportional reasoning: Towards a theoretical framework for research. In F. Lester (Ed.). Second handbook of research on mathematics teaching and learning (pp. 629-667). Information Age Publishing.
• Learning Disabilities Association (2005). Types of learning disabilities. https://ldaamerica.org/disabilities/dyscalculia/?audience=Parents
• Lehrer, R. (2003). Developing understanding of measurement. In J. Kilpatrick, W.G. Martin, & D. Schifer (Eds.), A research companion to principles and standards for school mathematics (pp. 179-192). Reston: National Council of Teachers of Mathematics.
• Lewis, K. E. (2014). Difference not deficit: Reconceptualizing mathematical learning disabilities. Journal for Research in Mathematics Education, 45(3), 351-396.
• Lewis, K. E. (2016). Understanding mathematical learning disabilities as developmental difference: A fine-grained analysis of one student's partitioning strategies for fractions. Infancia y Aprendizaje, 39(4), 812-857.
• Lewis, K. E., Thompson, G. M., & Tov, S. A. (2021). Screening for characteristics of dyscalculia: Identifying unconventional fraction understandings. International Electronic Journal of Elementary Education, 14(3), 243-267.
• Losinski, M., Ennis, R. B., Shaw, A., & Gage, N. A. (2021). Supporting students within an MTSS framework using SRSD fractions: Results of a regression discontinuity design. Learning Disabilities Research & Practice, 36(3), 213–223.
• Mazzocco, M. M., Myers, G. F., Lewis, K. E., Hanich, L. B., & Murphy, M. M. (2013). Limited knowledge of fraction representations differentiates middle school students with mathematics learning disability (dyscalculia) versus low mathematics achievement. Journal of Experimental Child Psychology, 115(2), 371–387.
• Milli Eğitim Bakanlığı (1973, Haziran). Milli Eğitim Temel Kanunu. https://www.mevzuat.gov.tr/MevzuatMetin/1.5.1739.pdf
• Milli Eğitim Bakanlığı (2021a). Öğrenme güçlüğü olan bireyler için öğretim programı. https://orgm.meb.gov.tr/meb_iys_dosyalar/2021_05/21130110_Ogrenme_Guclugu.pdf
• Milli Eğitim Bakanlığı (2021b). Özel öğrenme güçlüğü olan bireyler: Aileler için rehber kitapçık. https://orgm.meb.gov.tr/meb_iys_dosyalar/2021_02/04102620_OYRENME_GUCLUYU_OLAN_BYREYLER__TR.pdf
• Mills, J. (2016). Developing conceptual understanding of fractions with year five and six students. Mathematics Education Research Group of Australasia, Paper presented at the Annual Meeting of the Mathematics Education Research Group of Australasia (MERGA) (39th, Adelaide, South Australia, 2016).
• Mumcu, H. Y. (2018). Matematiksel ilişkilendirme becerisinin kuramsal boyutta incelenmesi: Türev kavramı örneği. Turkish Journal of Computer and Mathematics Education, 9(2), 211-248.
• Namkung, J., & Fuchs, L. (2019). Remediating difficulty with fractions for students with mathematics learning difficulties. Learning Disabilities: A Multidisciplinary Journal, 24(2), 36-48.
• Namkung, J. M., Fuchs, L. S., & Koziol, N. (2018). Does initial learning about the meaning of fractions challenging for students with adequate whole-number skill? Learning and Individual Differences, 61, 151–157.
• National Center for Education Statistics (2017). The condition of education 2017. https://nces.ed.gov/pubsearch/pubsinfo.asp?pubid=2017144.
• Nelson, G., Crawford, A., Hunt, J., Park, S., Leckie, E., Duarte, A., Brafford, T., Ramos-Duke, M. & Zarate, K. (2022). A systematic review of research syntheses on students with mathematics learning disabilities and difficulties. Learning Disabilities Research & Practice, 37(1), 18–36.
• Newman, R. M. (1998). Gifted & math learning disabled: The dyscalculia syndrome. https://www.dyscalculia.org/dyscalculia/dyscalculia-syndrome
• Newton, K. J., Jansen, A., & Puleo, P. (2022). Elements of instruction that motivate students with learning disabilities to learn fractions. Mathematical Thinking and Learning, 24(4), 1-20.
• Özan, S., & Sarıca, A. D. (2021). Bireyselleştirilmiş eğitim programı: Sınıf ve rehberlik öğretmenlerinin deneyimleri. Ankara Üniversitesi Eğitim Bilimleri Fakültesi Özel Eğitim Dergisi, 22(1), 147-174.
• Patton, M. Q. (2005). Qualitative research. Wiley Online Library.
• Peltier, C., Sinclair, T. E., Pulos, J. M., & Suk, A. (2020). Effects of schema-based instruction on immediate, generalized, and combined structured word problems. The Journal of Special Education, 54(2), 101–112.
• Post, T. R., Wachsmuth, I., Lesh, R., & Behr, M. J. (1985). Order and equivalence of rational number: A cognitive analysis. Journal for Research in Mathematics Education, 16(1), 18-36.
• Proffessional Development Service for Teachers (2014). Fractions's teachers handbook. https://pdst.ie/sites/default/files/Fractions%20Teacher%20Handbook%20FINAL.pdf
• Satsangi, R. & Raines, A. R. (2022). Examining virtual manipulatives for teaching computations with fractions to children with mathematics difficulty. Journal of Learning Disabilities, Advance online publication. https://doi.org/10.1177/00222194221097710
• Sharp, E., & Shih Dennis, M. (2017). Model drawing strategy for fraction word problem solving of fourth-grade students with learning disabilities. Remedial and Special Education, 38(3), 181-192.
• Shin, M., & Bryant, D. P. (2017). Fraction interventions for students struggling to learn mathematics: A research synthesis. Remedial and Special Education, 1-14.
• Siebert, D., & Gaskin, N. (2006). Creating, naming and justifying fractions. Teaching Children Mathematics, 12, 394-400.
• Simon, M. A., Placa, N., Kara, M., & Avitzur, A. (2018). Empirically-based hypothetical learning trajectories for fraction concepts: Products of the Learning Through Activity research program. Journal of Mathematical Behavior, 52, 188-200.
• Söğüt, D. A., & Deniz, S. (2018). Sınıf öğretmenlerinin bireyselleştirilmiş eğitim programı (bep) hazırlamada karşılaştıkları güçlükler ve kaynaştırma uygulamalarına ilişkin görüşlerinin değerlendirilmesi. Erzincan Üniversitesi Eğitim Fakültesi Dergisi, 20(2), 423-443.
• Stavy, R., & Tirosh, D. (2000). How students (mis)understand science and mathematics: intuitive rules. Teachers College Press.
• Steffe, L. P., & Thompson, P. W. (2000). Teaching experiment methodology: Underlying principles and essential elements. In A. E. Kelly & R. A. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 267- 306). Lawrence Erlbaum.
• Şahin, A., & Gürler, B. (2018). Destek eğitim odasında ve kaynaştırma ortamlarında çalışan öğretmenlerin bireyselleştirilmiş eğitim programı hazırlama sürecinde yaşadıkları güçlüklerin belirlenmesi. Adıyaman Üniversitesi Sosyal Bilimler Enstitüsü Dergisi, 29, 594-625.
• Thompson, P. W., & Saldanha, L. A. (2003). Fractions and multiplicative reasoning. J. Kilpatrick, G. Martin, & D. Schifter (Eds.), Research companion to the Principles and Standards for School Mathematics (pp. 95-114). Reston.
• Toprak, Ö. F. (2018). Bireyselleştirilmiş eğitim programı hazırlama sürecine ilişkin ekip üyelerinin deneyimleri: bir ortaokul örneği. Yayınlanmamış yüksek lisans tezi. Anadolu Üniversitesi, Eskişehir.
• Torbeyns, J., Schneider, M., Xin, Z., & Siegler, R. S. (2015). Bridging the gap: Fraction understanding is central to mathematics achievement in students from three different continents. Learning and Instruction, 37, 5-13.
• Tzur, R. (1999). An integrated study of children's construction of improper fractions and the teacher's role in promoting that learning. Journal for Research in Mathematics Education, 30(4), 390–416.
• Van De Walle, J. A., Karp-Karen, S., & Bay-Williams, J.M. (2009). Elementary an middle school mathematics: teaching developmentally (7th Ed). Pearson Education.
• Wang, A. Y., Fuchs, L. S., Fuchs, D., Gilbert, J. K., Krowka, S., & Abramson, R. (2019). Embedding self-regulation instruction within fractions intervention for third graders with mathematics difficulties. Journal of Learning Disabilities, 52(4), 337-348.
• Widodo, S., & Ikhwanudin, T. (2019). Students with mathematics learning disabilities and their ways of thinking in fraction learning. In Misciagna S. (Eds.), Learning Disabilities - Neurological Bases, Clinical Features and Strategies of Intervention, 1-13.
• Wijaya, A. (2017). The relationships between Indonesian fourth graders' difficulties in fractions and the opportunity to learn fractions: A snapshot of TIMSS results. International Journal of Instruction, 10(4), 221-236.
• Wilkins, J. L. M., & Norton, A. (2018). Learning progression toward a measurement concept of fractions. International Journal of STEM Education, 5(27), 1-11.
• Yazıcıoğlu, T. (2019). Rehberlik öğretmenlerinin bireyselleştirilmiş eğitim programı (bep) biriminin işleyişine ilişkin görüşleri. Anemon Muş Alparslan Üniversitesi Sosyal Bilimler Dergisi, 7(5) 225–234.
• Yıldırım, A., & Şimşek, H. (2016). Sosyal bilimlerde nitel araştırma yöntemleri (8. baskı). Seçkin Yayınevi.
• Yin, R. (1984). Case study research: Design and methods (3rd Ed.). Sage.