Investigation of Middle School Math Teachers’ Pedagogical Preferences towards Using Models in the Context of Different Fraction Schemes

Year 2025, Volume: 9 Issue: 19, 1 - 22

Yavuz Turan , Hayal Yavuz Mumcu

Abstract

The purpose of this study is to examine middle school mathematics teachers' preferences and performances in using mathematical models in situations involving different fraction schemes and fraction operations. The study, utilizing the case study survey method, involves fifteen mathematics teachers currently working in the Altınordu district of Ordu Province in Turkey. Purposeful sampling methods, including convenience sampling and criterion sampling, were employed in determining the participants of the study. Accordingly, the criteria for selecting teachers in the study were having a minimum of 10 years of professional experience, being stationed in the central district, and volunteering to participate in the study. In this study, the Questionnaire on Model Use Preferences, Open-Ended Questions on Model Use and semi-structured interviews developed by the researchers were used as data collection tools. According to the results of the study, it was observed that the participant teachers generally preferred to use the rectangle model-circle model-number line model and finally the set model when different fraction schemes were considered, and the rectangle model-number line model-circle model and finally the set model when fraction operations were considered. Teachers generally preferred continuous models and did not use discrete models. When the teachers' performances of using models were examined, it was seen that their performance levels were generally adequate except for the cases involving iterative fraction schemes. When the performances for fraction operations were analyzed, it was seen that the teachers generally performed adequately except for multiplication and division operations. In general, teachers used mathematical models not as a tool to support learning, but to complete the tasks assigned to them in the study process. In this context, it can be said that the models used by teachers do not fully include the conceptual meanings and differences related to the current situation in general.

Keywords

Fraction schemes, mathematical models, middle school mathematics teachers, pedagogical preferences

Ethical Statement

İlgili çalışmanın etik kurun belgeleri dosyalar kısmına eklenmiştir.

Supporting Institution

Ordu Üniversitesi

References

• Acar, N. (2010). Kesir çubuklarının ilköğretim 6. sınıf öğrencilerinin kesirlerde toplama ve çıkarma işlemlerindeki başarılarına etkisi [The effect of fraction rulers on the addition and subtraction of fraction abilities of 6th grade students of elementary school]. Master thesis, Selçuk University, Konya. https://tez.yok.gov.tr/UlusalTezMerkezi/
• Akbaba-Dağ, S. (2014). Mikroöğretim ders imecesi modeli ile sınıf öğretmeni adaylarının kesir öğretim bilgilerinin geliştirilmesine yönelik bir uygulama [A microteaching lesson study practice to improve pre-service teachers' knowledge of teaching fractions]. Master thesis, Dumlupınar University, Kütahya. https://tez.yok.gov.tr/UlusalTezMerkezi/
• Akgün, L., Çiltaş, A., Deniz, D., Çiftçi, Z., & Işık, A. (2013). İlköğretim matematik öğretmenlerinin matematiksel modelleme ile ilgili farkındalıkları [Primary school mathematics teachers’ awareness on mathematical modelling]. Adıyaman University Journal of Social Sciences, 6(12), 1-34. https://doi.org/10.14520/adyusbd.410
• Aktaş, E. (2023). Matematik öğretmenleri ile öğretmen adaylarının kesirlerle bölmeye yönelik öğretimsel açıklamalarının matematiksel modeller bağlamında incelenmesi [Examining in-service and pre-service mathematics teachers' instructional explanations for division by fractions in the context of mathematical models]. Master thesis, Ordu University, Ordu. Retrieved from https://tez.yok.gov.tr/UlusalTezMerkezi/
• Alacacı, C. (2014). Öğrencilerin kesirler konusundaki kavram yanılgıları [Students' misconceptions about fractions]. In E. Bingölbali & M. F. Özmantar (Eds.), İlköğretimde karşılaşılan matematiksel zorluklar ve çözüm önerileri [Mathematical difficulties encountered in primary education and solution suggestions] (pp. 63-94). Pegem Akademi Publishing.
• Arslan-Kılcan, S. (2006). İlköğretim matematik öğretmenlerinin kesirlerle bölmeye ilişkin kavramsal bilgi düzeyleri [The levels of elementary mathematics teachers' conceptual knowledge of the division with fractions]. Master thesis, Abant İzzet Baysal University, Bolu. https://tez.yok.gov.tr/UlusalTezMerkezi/
• Baki, A. (2014). Kuramdan uygulamaya matematik eğitimi (5. Baskı) [Mathematics education from theory to practice (5th edition)]. Harf Yayınları.
• Baştürk, S. (2016). Primary student teachers’ perspectives of the teaching of fractions. Acta Didactica Napocensia, 9(1), 35-44.
• Bayazit, İ., Aksoy, Y., & Kırnap, M. (2011). Öğretmenlerin matematiksel modelleri anlama ve model oluşturma yeterlilikleri [Teachers’ understanding of and proficiency at producing mathematical models]. e-Journal of New World Sciences Academy, 6(4), 2495-2516.
• Baykul, Y. (2009). İlköğretim matematik öğretimi (6-8. sınıflar) [Teaching mathematics in primary school (grades 6-8)]. Pegem Yayıncılık.
• Behr, M. J., Lesh, R., Post, T. R., & Silver, E. A. (1983). Rational-number concepts. In R. Lesh & M. Landau (Eds.), Acquisition of mathematics concepts and processes (pp. 92-126). Academic Pres
• Behr, M.J., Harel, G., Post, T., & Lesh, R. (1993). Rational numbers: Toward a semantic analysis-emphasis on the operator construct. In T.P. Carpenter, E. Fennema, & T.A. Romberg (Eds.), Rational numbers: An integration of research (pp. 13–47). Erlbaum.
• Birgin O., & Gürbüz, R. (2009). İlköğretim II. kademe öğrencilerinin rasyonel sayılar konusundaki işlemsel ve kavramsal bilgi düzeylerinin incelenmesi [Examining the procedural and conceptual knowledge levels of elementary school level II students about rational numbers]. Journal of Uludag University Faculty of Education, 22(2), 529-550.
• Bulgar, S. (2003). Children’s sense-making of division of fractions. The Journal of Mathematical Behavior, 22(3), 319-334. https://doi.org/10.1016/S0732-3123(03)00024-5
• Can, H. N. (2019). Ortaokul matematik öğretmenlerinin kesirlerde işlemler konusu ile ilgili pedagojik alan bilgilerinin öğrenci zorlukları ve kavram yanılgıları bileşeninde incelenmesi [Examination of secondary mathematics teachers' pedagogical content knowledge of fraction operations with regard to students' difficulties and misconceptions]. Master thesis, Marmara University, İstanbul. https://tez.yok.gov.tr/UlusalTezMerkezi/
• Castro, B. (2008). Cognitive models: the missing link to learning fraction multiplication and division. Asia Pacific Education Review, 9(2), 101-112. https://doi.org/10.1007/BF03026491
• Çelik, B., & Çiltaş, A. (2015). Beşinci sınıf kesirler konusunun öğretim sürecinin matematiksel modeller açısından incelenmesi [Investigation of the teaching process of 5th grade-fractions subject in terms of mathematical models]. Journal of Bayburt Education Faculty, 10(1), 180-204.
• Çiltaş, A., & Işık, A. (2012). Matematiksel modelleme yönteminin akademik başarıya etkisi [The effect of mathematical modeling method on academic achievement], Journal of Contemporary Education Academic, 1(2), 57-67.
• Çiltaş, A., & Yılmaz, K. (2013). İlköğretim matematik öğretmeni adaylarının teoremlerin ifadeleri için kurmuş oldukları matematiksel modeller [Mathematical models formed by prospective elementary mathematics teachers for the expressions of theorems]. Journal of Research in Education and Teaching, 2(2), 107-115.
• Cluff, J. J. (2005). Fraction multiplication and division image change in pre-service elementary teachers. Unpublished doctoral dissertation, Brigham Young University, USA. https://www.proquest.com/docview/2452091802?pq
• Cramer, K., & Henry, A. (2002). Using manipulative models to build number sense for addition of fractions. In Making sense of fractions, ratios, and proportions (pp. 41-48). National Council of Teachers of Mathematics.
• Cramer, K., Wyberg, T., & Leavitt, S. (2008). The role of representations in fraction addition and subtraction. Mathematics Teaching in the Middle School, 13(8), 490-496. https://doi.org/10.5951/MTMS.13.8.0490
• Deniz, D., & Akgün, L. (2017). Ortaöğretim matematik öğretmenlerinin matematiksel modelleme yöntemi ve uygulamalarına yönelik görüşleri [High school mathematic teachers’ views about mathematical modelling method and applications]. Journal of Social Sciences of Muş Alparslan University, 5(1), 95-117. https://doi.org/10.18506/anemon.272677
• Doğan-Temur, Ö. (2011). Dördüncü ve beşinci sınıf öğretmenlerinin kesir öğretimine ilişkin görüşleri: Fenomenografik araştırma [Opinions of teachers of fourth and fifth grade about teaching fractions: A phenomenograhic research]. Dumlupınar University Journal of Social Sciences, 29, 203-212.
• Duran, N. B. (2017). Ortaokul matematik öğretmen adaylarının alan ve pedagojik alan bilgileri çerçevesinde kesirlerle çarpma ve bölme işlemlerinin öğretimine ilişkin kullandıkları modeler [Models used by preservice middle school mathematics teachers for teaching multiplication and division of fractions within the scope of content knowledge and pedagogical content knowledge]. Master thesis, Pamukkale University, Denizli. https://tez.yok.gov.tr/UlusalTezMerkezi/
• Durmuş, S. (2005). İlköğretim öğretmen adaylarının rasyonel sayıları anlama düzeylerinin belirlenmesi [Determination of primary teacher candidates' level of understanding of rational numbers]. Educational Sciences in Theory and Practice, 5(2), 639-666.
• Duzenlı-Gokalp, N., & Sharma, M. D. (2010). A study on addition and subtraction of fractions: The use of Pirie and Kieren model and hands- on activities. Procedia Social and Behavioral Sciences 2(2), 5168- 5171. https://doi.org/10.1016/j.sbspro.2010.03.840
• Ertem-Akbaş, E. (2019). Eğitim bilişim ağı (EBA) destekli matematik öğretiminin 5. sınıf kesir konusunda öğrenci başarılarına etkisi [The impact of eba (educational informatics network) assisted mathematics teaching in 5th grade fractions on students’ achievements]. Journal of Computer and Education Research, 7 (13), 120-145. https://doi.org/10.18009/jcer.531953
• Gökkurt, A. G. B., Şahin, A. G. Ö., & Soylu, Y. (2012). Matematik öğretmenlerinin matematiksel alan bilgileri ile pedagojik alan bilgileri arasındaki ilişkinin incelenmesi [An analysis on the relationship between the pedagogical and mathematical content knowledge of mathematics teachers]. The Journal of Academic Social Science Studies, 5(8), 997-1012.
• Gökkurt, B., Soylu, Y., & Demir, Ö. (2015). Ortaokul matematik öğretmenlerinin kesirlerin öğretimine yönelik görüşlerinin incelenmesi [Examining the opinions of secondary mathematics teachers on teaching fractions]. Necatibey Faculty of Education Electronic Journal of Science and Mathematics Education, 9 (2), 230-251. https://doi.org/10.17522/nefefmed.23191
• Gümüş, İ., Demir, Y., Koçak, E., Kaya, Y., & Kırıcı, M. (2008). Modelle öğretimin öğrenci başarısına etkisi [The effects of model-teaching on students’ success]. Erzincan University Journal of Education Faculty, 10(1), 65-90.
• Gürbüz, R., & Birgin, O. (2008). Farklı öğrenim seviyesindeki öğrencilerin rasyonel sayıların farklı gösterim şekilleriyle işlem yapma becerilerinin 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(23), 85-94.
• Işık, C. (2011). İlköğretim matematik öğretmeni adaylarının kesirlerde çarpma ve bölmeye yönelik kur-dukları problemlerin kavramsal analizi [Conceptual analysis of multiplication and division problems in fractions posed by preservice elementary mathematics teachers]. Hacettepe University Journal of Education, 41, 231-243.
• Jung, H., Stehr, E. M., & He, J. (2019). Mathematical modeling opportunities reported by secondary mathematics preservice teachers and instructors. School Science and Mathematics, 119(6), 353-365. https://doi.org/10.1111/ssm.12359
• Kieren, T. E. (1976). On the mathematical, cognitive and instructional foundations of rational numbers. In R. A. Lesh (Ed.), Number and measurement: Papers from a research workshop (pp. 101–144). Georgia Center for the Study of Learning and Teaching Mathematics.
• Kutluca, T., & Kaya, D. (2023). Mathematical modelling: A retrospective overview. Journal of Computer and Education Research, 11 (21), 240-274. https://doi.org/10.18009/jcer.1242785
• Lesh, R., & Doerr, H. M. (2000). Symbolizing, communicating and mathematizing: Key components of models and modeling. In P. Cobb, E. Yackel & K. McClain (Eds.), Symbolizing and communicating in mathematics classrooms: Perspectives on discourse, tools and instructional design. Lawrence Erlbaum.
• Lo, J. J., & Luo, F. (2012). Prospective elementary teachers's knowledge of fraction division. Journal of Mathematics Teacher Education, 15, 481-500. https://doi.org/10.1007/s10857-012-9221-4
• Mills, A. J., Durepos, G., & Wiebe, E. (2009). Case study surveys. In Sage encyclopedia of case study research (pp. 124-126). Sage.
• Ministry of National Education [MoNE] (2018). Matematik dersi öğretim programları-İlkokul ve ortaokul 1., 2., 3., 4.,5.,6., 7. ve 8. sınıflar [Mathematics curricula-primary and secondary school 1st, 2nd, 3rd, 4th, 5th, 6th, 7th and 8th grades)]. MEB Yayınları.
• National Council of Teachers of Mathematics [NCTM] (2000). Principles and standards for school mathematics. National Council of Teachers of Mathematics.
• Nemirovsky, R. (1994). On ways of symbolizing: The case of Laura and velocity sign. The Journal of Mathematical Behavior 13, 389–422. https://doi.org/10.1016/0732-3123(94)90002-7
• Newstead, K., & Murray, H. (1998). Young students’ constructions of fractions. In A. Olivier & K. Newstead (Eds.), Proceedings of the 22nd Conference of the International Group for the Psychology of Mathematics Education, 3, 295-303. Stellenbosch, South Africa.
• Niss, M. (1987). Applications and modelling in the mathematics curriculum—state and trends. International Journal of Mathematical Education in Science and Technology, 18(4), 487-505.
• Norton, A. H., & McCloskey, A. (2008). Teaching experiments and professional development. Journal of Mathematics Teacher Education, 11, 285-305. https://doi.org/10.1007/s10857-008-9076-x
• Norton, A., & Wilkins, J. L. M. (2009). A quantitative analysis of children’s splitting operations and fraction schemes. Journal of Mathematical Behavior, 28, 150–161.
• Olive, J., & Steffe, L. P. (2002). The construction of an iterative fractional scheme: The case of Joe. The Journal of Mathematical Behavior, 20(4), 413-437. https://doi.org/10.1016/S0732-3123(02)00086-X
• Parmar, R. (2003). Understanding the concept of “division”: Assessment considerations. Exceptionality, 11(3), 177-189. https://doi.org/10.1207/S15327035EX1103_05
• Patton, M. Q. (1987). How to use qualitative methods in evaluation. Sage.
• Piaget, J. (1964). Part 1: Cognitive development in children: Development and learning. Journal of Research in Science Teaching, 2, 176–186.
• Şahin, E. (2019). Ortaokul öğrencilerinin kesirler konusunda temsiller arası geçişleri [Transition between the represantions of middle school students in term of fractions]. Master thesis, Zonguldak. https://tez.yok.gov.tr/UlusalTezMerkezi/
• Şen, C. (2021). Assessment of a middle-school mathematics teacher’s knowledge for teaching the 5th-grade subject of fractions. Turkish Journal of Computer and Mathematics Education, 12(1), 96-138. https://doi.org/10.16949/turkbilmat.742136
• Siebert, D., & Gaskin, N. (2006). Creating, naming, and justifying fractions. Teaching Children Mathematics, 12(8), 394-400.
• Soylu, Y., & Soylu, C. (2005). Learning difficulties of 5fh class in primary education at fraction: Ordering, adding, subtraction, multiplication in fraction and problems related to fraction. Erzincan University Journal of Education Faculty, 7(2), 101–118.
• Steffe, L. P., & Olive, J. (Eds.). (2010). Children’s fractional knowledge. Springer.
• Tekin-Dede, A., & Bukova-Güzel, E. (2013). Mathematics teachers’ views concerning model eliciting activities, developmental process and the activities themselves. Bartın University Journal of Faculty of Education, 2(1), 300-322.
• Thompson, P. W., & Saldanha, L. A. (2003). Fractions and multiplicative reasoning. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.), A research companion to principles and standards for school mathematics (pp. 95-113). National Council of Teachers of Mathematics.
• Topçu, M., & Gürefe, N. (2020). 7. sınıf öğrencilerinin kesir şemalarının belirlenmesi [Determination of fraction schemas of 7th grade students]. The Journal of International Education Science, 22 (7), 97-118.
• Topcu, Y. (2019). Ortaokul öğrencilerinin kesir şemalarının incelenmesi [Investigation of fraction schemes of middle school students]. Master thesis, Anadolu University, Eskişehir. Retrieved from https://tez.yok.gov.tr/UlusalTezMerkezi/
• Toptaş, V., Han, B., & Akın, Y. (2017). Sınıf öğretmenlerinin kesirlerin farklı anlam ve modelleri konusunda görüşlerinin incelenmesi [Primary school teachers’ opinions about different meanings of fractions and models of fractions]. Sakarya University Journal of Education Faculty, 33, 49-67.
• Van de Walle, J. A., Karp, K. S., & Bay-Williams, J. M. (Eds.). (2014). Elementary and middle school mathematics: Teaching developmentally. Pearson Education.
• von Glasersfeld, E. (1995). A constructivist approach to teaching. In L. P. Steffe & J. Gale, Constructivism in education (pp. 3-16). Erlbaum.
• Wilkie, K. J., & Roche, A. (2023). Primary teachers’ preferred fraction models and manipulatives for solving fraction tasks and for teaching. Journal of Mathematics Teacher Education, 26(6), 703-733. https://doi.org/10.1007/s10857-022-09542-7
• Yanık, H. B. (2016). Kavramsal ve işlemsel anlama [Conceptual and procedural understanding]. In E. Bingölbali, S. Arslan, & İ. Ö. Zembat (Eds.), Matematik eğitiminde teoriler [Theories in mathematics education] (pp. 101-116). Pegem Akademi.
• Yavuz-Mumcu, H. (2018). Kesir işlemlerinde model kullanma: Bir durum çalışması [Using mathematical models in fraction operations: A case study]. Necatibey Faculty of Education Electronic Journal of Science and Mathematics Education, 12(1), 122-151.
• Yavuz-Mumcu, H. (2023). Farklı temsiller arası ilişkilendirme [Making connections between different representations]. In H. Yavuz Mumcu, A. Osmanoğlu, & H. Korkmaz (Eds.), Matematik eğitiminde ilişkilendirme [Connections in mathematics education] (pp. 72-119). Pegem A.
• 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, Uludağ University, Bursa. Retrieved from https://tez.yok.gov.tr/UlusalTezMerkezi/
• Yılmaz, G. (2016). Ortaokul matematik öğretmenlerinin çoklu temsilleri kullanarak kesirlerle toplama ve çıkarma işlemlerini öğretme yaklaşımlarının incelenmesi [Investigating middle scholl tearchers' use of multiple represantations regarding addition and substraction of fracti̇ons]. Master thesis, Dokuz Eylül University, İzmir. https://tez.yok.gov.tr/UlusalTezMerkezi/
• Yılmaz, Z., & Yenilmez, K. (2008). İlköğretim 7. ve 8. sınıf öğrencilerinin ondalık sayılar konusundaki kavram yanılgıları (Uşak İli Örneği) [7th and 8th grades students’ misconceptions about decimal numbers (The case of Uşak)]. Afyon Kocatepe University Journal of Science and Engineering, 8(1), 291-312.
• Zembat, İ. Ö. (2007). Working on the same problem–concepts; with the usual subjects–prospective elementary teachers. Elementary Education Online, 6(2), 305-312.