Development Cognitive Mathematical Modeling Competencies of a Fourth-Grade Elementary School Student: Elaborated Feedback

Year 2024, , 29 - 71, 23.09.2024

H. Beyza Canbazoğlu Albayrak , Esra Bukova Güzel

https://doi.org/10.9779/pauefd.1290603

Abstract

This study aimed to examine the effect of the elaborated feedback on developing the mathematical modeling competencies of a fourth-grade student. In the research, holistic single case design was employed and task-based interview method was used as a data collection tool. The rubric for the Assessment of Modeling Skills (RAMS) developed by Tekin Dede and Bukova Güzel (2018) was used to detect the level of the fourth-grade elementary school student both at the beginning and at end of the research and to explain the development of the student’s modeling competencies throughout the research process. When the student’s cognitive mathematical modeling competencies were evaluated at the beginning of the research, it was found that the student had difficulties in understanding the problem, simplifying, interpreting and validating competencies. When the mathematical modeling competencies of the student were evaluated at the end of the process, it was found that there were a number of improvements in the competencies of understanding the problem, simplifying the problem, mathematizing, working mathematically, interpreting and validating. In other words, thanks to the elaborated feedback provided to the fourth-grade elementary school student regarding each mathematical modeling competence, it was observed that the student’s modeling competencies displayed progress.

Keywords

Cognitive mathematical modeling, Elementary school, Modeling competence development, Mathematical modeling

Bir İlkokul Dördüncü Sınıf Öğrencisinin Bilişsel Matematiksel Modelleme Yeterliklerinin Geliştirilmesi: Ayrıntılı Geri Bildirim

Abstract

Bu çalışmada, bir ilkokul dördüncü sınıf öğrencisinin matematiksel modelleme yeterliklerinin değerlendirilmesine yönelik sağlanan ayrıntılı geri bildirimlerin, öğrencinin modelleme yeterliklerinin gelişimine etkisini incelemek amaçlanmıştır. Araştırmada, bütüncül tek durum deseni kullanılmıştır. Araştırmada veri toplama aracı olarak göreve dayalı mülakat (task-based interview) yöntemi kullanılmıştır. İlkokul dördüncü sınıf öğrencisinin hem araştırmanın başlangıcındaki ve sonundaki düzeyini belirlemede hem araştırma sürecince öğrencinin modelleme yeterliklerinin gelişimini açıklamada Tekin Dede ve Bukova Güzel (2018) tarafından oluşturulan Modelleme Yeterlikleri Değerlendirme Rubriği (MYDR) kullanılmıştır. Araştırmanın başlangıcında öğrencinin bilişsel matematiksel modelleme yeterlikleri değerlendirildiğinde, problemi anlama, sadeleştirme, yorumlama ve doğrulama yeterliklerini sergilemede güçlükler yaşadığı belirlenmiştir. Süreç sonunda öğrencinin matematiksel modelleme yeterlikleri değerlendirildiğinde problemi anlama, sadeleştirme, matematikselleştirme, matematiksel olarak çalışma, yorumlama ve doğrulama yeterliklerinde gelişimler olduğu belirlenmiştir. Bir başka deyişle ilkokul dördüncü sınıf öğrencisine her bir matematiksel modelleme yeterliklerine yönelik sağlanan ayrıntılı geri bildirimler sayesinde öğrencinin modelleme yeterliklerinde gelişim olduğu görülmüştür.

Keywords

bilişsel matematiksel modelleme, ilkokul, modelleme yeterliği geliştirme, matematiksel modelleme

References

• Asempapa, R. S., & Foley, G. D. (2018). Classroom assessment of mathematical modeling tasks. Education Research Highlights in Mathematics, Science and Technology, 6, 1-20.
• Besser, M., Blum, W., & Klimczak, M. (2013). Formative assessment in every-day teaching of mathematical modelling: implementation of written and oral feedback to competency-oriented tasks. G. Stillman, W. Blum, J. Brown ve G. Kaiser (Eds.), Teaching mathematical modelling: Connecting to research and practice (pp. 469-478) içinde. Springer. https://doi.org/10.1007/978-94-007-6540-5_40
• Besser, M., Blum, W., & Leiss, D. (2015). How to support teachers to give feedback to modelling tasks effectively? Results from a teacher-training-study in the Co2CA project. In G. Stillman, W. Blum ve M. S. Biembengut (Eds.), Mathematical modelling in education research and practice: Cultural, social and cognitive influences (pp. 151-160). Springer. https://doi.org/10.1007/978-3-319-18272-8
• Black, P., & Wiliam, D. (1998). Inside the black box: Raising standards through classroom assessment. Phi Delta Kappan 80(2), 139-148.
• Black, P., Harrison, C., Lee, C., Marshall, B., & Wiliam, D. (2004). Working inside the black box: Assessment for learning in the classroom. Phi Delta Kappan, 86(1), 9-21.
• Bliss, K., & Libertini, J. (2016). What is mathematical modeling? InS. Garfunkel & M. Montgomery (Eds.), GAIMME: Guidelines for assessment & instruction in mathematical modeling education (pp. 7-21). Society for Industrial and Applied Mathematics SIAM.
• Blomhøj, M. (2011). Modelling competency: Teaching, learning and assessing competencies-Overview. In Kaiser, G., Blum, W., Borromeo Ferri, R., & Stillman, G. (Eds.), Trends in Teaching and Learning of Mathematical Modelling (pp. 343-347). Springer. https://doi.org/10.1007/978-94-007-0910-2_34
• Blomhøj, M., & Jensen, T. H. (2003). Developing mathematical modelling competence: Conceptual clarification and educational planning. Teaching Mathematics and its Applications, 22(3), 123-139. https://doi.org/10.1093/teamat/22.3.123
• Blum, W. (2015). Quality teaching of mathematical modelling: What do we know, what can we do? In S. J. Cho (Ed.), Proceedings of the 12th international congress on mathematical education: Intellectual and attitudinal challenges (pp. 73-96). Springer. https://doi.org/10.1007/978-3-319-12688-3
• Borromeo Ferri, R. (2006). Theoretical and empirical differentiations of phases in the modelling process. ZDM, 38(2), 86-95. https://doi.org/10.1007/BF02655883
• Blum, W., & Borromeo Ferri, R. (2009). Mathematical modelling: Can it be taught and learnt?. Journal of Mathematical Modelling and Application, 1(1), 45-58.
• Canbazoğlu, H. B., & Tarım, K. (2021). A teaching process for mathematical modeling in primary school. The Journal of Buca Faculty of Education, (51), 210-225. https://doi.org/10.53444/deubefd.825361
• Canbazoğlu Albayrak, H. B., & Tarım, K. (2023). Cognitive mathematical modelling competencies of elementary school students. SDU International Journal of Educational Studies, 10(1), 1-21. https://doi.org/10.33710/sduijes.1191490
• Carlson, M. A., Wickstrom, M. H., Burroughs, E. A., & Fulton, E. W. (2016). A case for mathematical modeling in the elementary school classroom. In C. R. Hirsch & A. R. McDuffie (Eds.), Mathematical modeling and modeling mathematics (pp. 121-129). National Council of Teachers of Mathematics.
• Carlson, M., Larsen, S., & Lesh, R. (2003). Integrating a models and modeling perspective with existing research and practices. In R. Lesh & H. Doerr (Eds), Beyond constructivism: Models and modeling perspectives on mathematics problem solving, learning, and teaching (pp. 465-478). Lawrence Erlbaum Associates.
• Chan, C. M. E., Ng, K. E. D., Widjaja, W., & Seto, C. (2012). Assessment of primary 5 students' mathematical modelling competencies. Journal of Science and Mathematics Education in Southeast Asia, 35(2), 146-178.
• Corbett, A. T., & Anderson, J. R. (2001). Locus of feedback control in computer-based tutoring: Impact on learning rate, achievement and attitudes. In Jacko, J., Sears, A., Beaudouin-Lafon, M. & Jacob, R. (Eds.), Proceedings of ACM CHI’2001 Conference on Human Factors in Computing Systems (pp. 245-252). ACM Press. https://doi.org/10.1145/365024.365111
• Diefes‐Dux, H. A., Zawojewski, J. S., Hjalmarson, M. A., & Cardella, M. E. (2012). A framework for analyzing feedback in a formative assessment system for mathematical modeling problems. Journal of Engineering Education, 101(2), 375-406. https://doi.org/10.1002/j.2168-9830.2012.tb00054.x
• Doerr, H. M., & English, L. D. (2003). A modeling perspective on students’ mathematical reasoning about data. Journal for Research in Mathematics Education, 34(2), 110-136. https://doi.org/10.2307/30034902
• English, L. (2012). Data modelling with first-grade students. Educational Studies in Mathematics, 81(1), 15-30. https://doi.org/10.1007/s10649-011-9377-3
• English, L. D. (2006). Introducing young children to complex systems through modeling. In M. Chinnappan, P. Grootenboer & R. Zevenbergen (Eds.), Proceedings of the 29th Annual Conference of the Mathematics Education Research Group of Australasia (pp. 195-202). MERGA Inc.
• English, L. D. (2007). Interdisciplinary modelling in the primary mathematics curriculum. In J. Watson & K. Beswick (Eds.), Mathematics: Essential Research, Essential Practice (pp. 275-284). MERGA Inc.
• English, L. D., & Watters, J. J. (2004). Mathematical modelling with young children. International Group for the Psychology of Mathematics Education, 2, 335-342.
• English, L. D., & Watters, J. J. (2005). Mathematical modeling in third-grade classrooms. Mathematics Education Research Journal, 16, 59-80. https://doi.org/10.1007/BF03217401
• Goldin, G. (2000). A scientific perspective on structures, task-based interviews in mathematics education research. In R. Lesh & A. E. Kelly (Eds.), Research design in mathematics and science education (pp. 547-590). Lawrence Erlbaum Associates, Inc., Publishers.
• Greefrath, G., & Vorhölter, K. (2016). Teaching and learning mathematical modelling: Approaches and developments from German speaking countries. Springer Nature. https://doi.org/10.1007/978-3-319-45004-9
• Haas, B., Kreis, Y., & Lavicza, Z. (2020). Connecting the real world to mathematical models in elementary schools in Luxemburg. Proceedings of the British Society for Research into Learning Mathematics, 40(2), 1-6.
• Haines C., & Crouch R. (2007). Mathematical modelling and applications: Ability and competence frameworks. In W. Blum, P. L. Galbraith, H. W. Henn, & M. Niss (Eds.), Modelling and applications in mathematics education (pp. 417-424). Springer. https://doi.org/10.1007/978-0-387-29822-1_46
• Hattie, J., & Timperley, H. (2007). The power of feedback. Review of Educational Research, 77(1), 81-112. https://doi.org/10.3102/003465430298487
• Kaiser, G., & Sriraman, B. (2006). A global survey of international perspectives on modelling in mathematics education. ZDM, 38(3), 302-310. https://doi.org/10.1007/BF02652813
• Kaiser, G., Schwarz, B., & Tiedemann, S. (2010). Future teachers’ professional knowledge on modeling. In R. Lesh, P. L. Galbraith, C. R. Haines & A. Hurford (Eds.), Modeling students’ mathematical modeling competencies (pp. 433-444). Springer. https://doi.org/10.1007/978-1-4419-0561-1_37
• Kulhavy, R. W., & Stock, W. (1989). Feedback in written instruction: The place of response certitude. Educational Psychology Review, 1(4), 279–308. https://doi.org/10.1007/BF01320096
• Lesh, R. A., & Doerr, H. M. (2003). Foundations of a models and modeling perspective on mathematics teaching and learning. In R. A. Lesh & H. Doerr (Eds.), Beyond constructivism: Models and modeling perspectives on mathematics problem solving, learning, and teaching (pp. 3-34). Lawrence Erlbaum Associates Publishers.
• Levy, R., Zbiek, R. M., Galluzzo, B., & Long, M. (2016). Mathematical modeling in the early and middle grades: Prekindergarten through grade 8. In S. Garfunkel & M. Montgomery (Eds.), GAIMME: Guidelines for assessment & instruction in mathematical modeling education (pp. 23-43). Society for Industrial and Applied Mathematics SIAM.
• Maaß, K. (2006). What are modelling competencies? ZDM, 38(2), 113-142. https://doi.org/10.1007/BF02655885 Maaß, K., & Mischo, C. (2011). Implementing modelling into day-to-day teaching practice-The project STRATUM and its framework. Journal Für Mathematik-Didaktik, 1(32), 103-131. https://doi.org/10.1007/s13138-010-0015-x
• Mason, B. J., & Bruning, R. (2001). Providing feedback in computer-based instruction: What the research tells us. Center for Instructional Innovation, University of Nebraska-Lincoln.
• Miles, M. B., & Huberman, A. M. (2016). Qualitative data analysis: An expanded sourcebook. Sage Publications. Ministry of National Education [MNE]. (2018). Elementary school mathematics (grades 1-4) curriculum. Ankara: Talim Terbiye Başkanlığı Yayınları.
• Moreno, R. (2004). Decreasing cognitive load for novice students: Effects of explanatory versus corrective feedback in discovery-based multimedia. Instructional Science, 32, 99-113. https://doi.org/10.1023/B:TRUC.0000021811.66966.1d
• Mory, E. H. (2004). Feedback research review. In D. Jonassen (Ed.), Handbook of research on educational communications and technology (pp. 745-783). Lawrence Erlbaum.
• Narciss, S., & Huth, K. (2004). How to design informative tutoring feedback for multimedia learning. H. M. Niegemann, D. Leutner & R. Brunken (Eds.), Instructional design for multimedia learning (pp. 181-195). Waxmann.
• National Research Council [NRC] (2001). Adding it Up: Helping Children Learn Mathematics. Washington, DC: National Academy Press.
• Organisation for Economic Co-operation and Development [OECD]. (2023a). PISA 2022 assessment and analytical framework. Paris: OECD Publishing.
• Organisation for Economic Co-operation and Development [OECD]. (2023b). PISA 2022 results (Volume I): The state of learning and equity in education. Paris: OECD Publishing.
• Paas, F., Renkl, A., & Sweller, J. (2003). Cognitive load theory and instructional design: Recent developments. Educational Psychologist, 38, 1-4. https://doi.org/10.1207/S15326985EP3801_1
• Shute, V. J. (2008). Focus on formative feedback. Review of Educational Research, 78(1), 153-189. https://doi.org/10.3102/0034654307313795
• Shute, V. J., Hansen, E. G., & Almond, R. G. (2007). An assessment for learning system called ACED: Designing for learning effectiveness and accessibility. ETS Research Report Series, 2007(2), i-45. https://doi.org/10.1002/j.2333-8504.2007.tb02068.x
• Steen, L. A., Turner, R., & Burkhardt, H. (2007). Developing mathematical literacy. Modelling and applications in mathematics education: The 14th ICMI study (pp. 285-294). Springer US.
• Suh, J., Matson, K., Seshaiyer, P., Jamieson, S., & Tate, H. (2021). Mathematical modeling as a catalyst for equitable mathematics ınstruction: Preparing teachers and young learners with 21st century skills. Mathematics, 9(2), 162. https://doi.org/10.3390/math9020162
• Şahin, N. (2019). Determining and evaluating of primary 4th-grade school students‘ cognitive modelling competencies. [Doctoral dissertation, Ondokuz Mayıs University]. Ulusal Tez Merkezi.
• Şahin, N., & Eraslan, A. (2016). Modeling processes of primary school students: The crime problem. Education and Science, 41(183), 47-67. http://dx.doi.org/10.15390/EB.2016.6011
• Şahin, N., & Eraslan, A. (2017). Fourth-grade primary school students’ thought processes and challenges encountered during the butter beans problem. Educational Sciences: Theory & Practice, 17(1), 105-127. https://doi.org/10.12738/estp.2017.1.0038
• Tekin Dede, A. (2015). Developing students' modeling competencies in mathematics lessons: An action research study. [Doctoral dissertation, Dokuz Eylül University]. Ulusal Tez Merkezi.
• Tekin-Dede, A., & Bukova-Güzel, E. (2018). A rubric development study for the assessment of modeling skills. The Mathematics Educator, 27(2), 33-72.
• Tran, D., & Dougherty, B. J. (2014). Authenticity of mathematical modeling. The Mathematics Teacher, 107(9), 672-678. https://doi.org/10.5951/mathteacher.107.9.0672
• Ulu, M. (2017). Examining the mathematical modeling processes of primary school 4th-grade students: Shopping problem. Universal Journal of Educational Research, 5(4), 561-580. https://doi.org/10.13189/ujer.2017.050406
• Verschaffel, L., De Corte, E., & Vierstraete, H. (1999). Upper elementary school pupils’ difficulties in modeling and solving nonstandard additive word problems involving ordinal numbers. Journal for Research in Mathematics Education, 30(3), 265-285. https://doi.org/10.2307/749836
• Wake, G. (2010). Modelling and formative assessment pedagogies mediating change in actions of teachers and learners in mathematics classrooms. In V. Durand-Guerrier, S. Soury-Lavergne & F. Arzarello (Eds.), Proceedings of CERME 6 (pp. 2086-2095). Institut Français de Éducation.
• Watters, J. J., English, L. D., & Mahoney, S. (2004). Mathematical modeling in the elementary school. American Educational Research Association Annual Meeting (pp. 1-12). San Diego. Yamane, T. (2009). Temel örnekleme yöntemleri. Literatür Yayıncılık.
• Yıldırım, A., & Şimşek, H. (2016). Qualitative research methods in social sciences. (10. Baskı). Seçkin Yayıncılık.
• Yin, R. K. (2017). Applications of case study research. Nobel Akademik Yayıncılık.