Altıncı Sınıf Öğrencilerinin Matematiksel Modellemede Oluşturdukları Gerçek Yaşam Problem Durumu Modelleri

Çağlar Naci Hıdıroğlu, Yeliz Özkan Hıdıroğlu

Öz


Çalışmanın amacı, altıncı sınıf öğrencilerinin matematiksel modellemede oluşturdukları gerçek yaşam problem durumu modellerinin incelenmesidir. Temel nitel araştırma anlayışıyla yürütülen çalışmanın katılımcıları, Denizli’de ortalama başarıya sahip, daha önce modelleme uygulaması yapılmamış üç ortaokulda öğrenim gören 64 altıncı sınıf öğrencisidir. Nitel veriler dört farklı matematiksel modelleme problemine ilişkin yazılı yanıt kağıtlarından, araştırmacıların çözüm sürecinde aldıkları gözlem notlarından oluşturulmuştur. Nitel veri analizinde içerik analizinden yararlanılmıştır. Gerçek modeller resim, şekil ve tablo/liste türündedir. Modelin oluşturulmasında deneyim, anlama, tahmin/ölçüm ve matematiksel bilgi etkilidir. Gerçek modeller matematiksel çözüm/sonuç ve yorumlama amacıyla kullanılmıştır. Öğrenciler çözümde zihinsel modellerini gerçek yaşam problem durumu modellerine yansıtamamışlar, modellerinde çözüm için gereksiz çizimlere yer vermişler, gerekli stratejik etkenleri belirleyemediklerinden dolayı nitelikli modeller oluşturamamışlardır. Öğrenciler uzunluk, ağırlık gibi kavramları karıştırdıkları ve hatalı tahminlerde bulundukları için gerçekçi bir model elde edememişlerdir. Öğrencilerin modelleri üzerinden yüzeysel işlem/yorumlar yapmaları onların ilerleyen basamaklarda beklenen düzeyde başarılı olmalarını engellemiştir. Atomistik anlayışla, öğrencilerin modellemede gerçek modelleri oluştururlarkenki yeterlikleri geliştirilmelidir.

Tam Metin:

PDF

Referanslar


Abrams, J. P. (2001). Mathematical modeling: Teaching the open-ended application of mathematics. 12 Eylül 2015 tarihinde http://www.meaningfulmath.org/modelingcycle.pdf adresinden erişilmiştir.

Aiken, L. R. (1997). Qustionnaires and inventories: Surveying opinions and assessing personality. New York: John Wiley & Sons.

Ang, K. C. (2006). Mathematical modelling, technology and H3 mathematics. The Mathematics Educator, 9(2), 33-47.

Ang, K. C. (2010). Teaching and learning mathematical modelling with technology. Nanyang Technological University. 20 Mart 2012 tarihinde http://atcm.mathandtech.org/ep2010/invited/3052010_18134.pdf adresinden erişilmiştir.

Ärlebäck, J. B. (2009). On the use of realistic Fermi problems for introducing mathematical modelling in school. The Montana Mathematics Enthusiast, 6(3), 331- 364.

Biccard, P. ve Wessels, D. C. J. (2011). Documenting the development of modelling competencies of grade 7 mathematics students. International Perspectives on the Teaching and Learning of Mathematical Modelling, 1(5), 375-383.

Blomhøj, M. (2008). Different perspectives on mathematical modelling in educational research - Categorising the TSG21 papers. M. Blomhøj ve S. Carreira (Eds), Mathematical applications and modelling in the teaching and learning of mathematics (ss. 1-18) içinde. 8 Ocak 2015 tarihinde http://milne.ruc.dk/imfufatekster/pdf/461.pdf adresinden erişilmiştir.

Blum, W. (2011). Can modelling be taught and learnt? Some answers from empirical research. G. Kaiser, W. Blum, R. Borromeo Ferri ve G. Stillman (Eds), Trends in the teaching and learning of mathematical modelling - Proceedings of ICTMA14 (pp. 15-30) içinde. New York: Springer.

Blum, W. (1985). Anwendungsorientierter mathematikunterricht in der didaktischen diskussion. Mathematische Semesterberichte, 32, 195-232.

Blum, W. (1996). Anwendungsbezüge im mathematikunterricht – Trends und pesrpetiven. Trends und Perspektiven. Schriftenreihe Didaktik Der Mathematik, 23, 15-38.

Blum, W. (2002). ICMI Study 14: Applications and modelling in mathematics education-Discussion document. Educational Studies in Mathematics, 51(1), 149-171.

Blum, W. ve Leiß, D. (2007). How do students and teachers deal with mathematical modelling problems? The example “Sugarloaf”. C. Haines, P. Galbraith, W. Blum, ve Khan, S. (Eds), Mathematical modelling (ICTMA 12): Education, engineering and economics, (ss 222-231) içinde. Chichester: Horwood Publishing.

Borromeo Ferri, R. (2006). Theoretical and empirical differentiations of phases in the modelling process. Zentralblatt für Didaktik der Mathematik, 38(2), 86-95.

Borromeo Ferri, R. (2007). Personal experiences and extra-mathematical knowledge as an influence factor on modelling routes of pupils. D. Pitta-Pantazi ve G. Philippou (Eds), Proceedings of the Fifth Congress of the European Society for Research in Mathematics Education (ss. 2080-2089) içinde. Larnaca: Zypern.

Borromeo Ferri, R. (2010). On the influence of mathematical thinking styles on learners' modeling behaviour. Journal für Mathematikdidaktik, 31(1), 99-118.

Brenner, M. E., Mayer, R. E., Moseley, B., Brar, T., Durán, R., Reed, B. S. ve Webb, D. (1997). Learning by understanding: The role of multiple representations in learning algebra. American Educational Research Journal, 34(4), 663-689.

Cai, J. (2005). US and Chinese teachers’ constructing, knowing and evaluating representations to teach mathematics. Mathematical Thinking and Learning. 7(2), 135-169.

Cai, J. ve Lester, F. K. (2005). Solution representations and pedagogical representations in Chinese and U. S. classrooms. Journal of Mathematical Behavior, 24, 221-237.

Carlson, M., Larsen, S. ve Lesh, R. (2003). Integrating a models and modeling perspective with existing research and practices. R. Lesh ve H. Doerr (Eds), Beyond constructivism: Models and modeling perspectives on mathematics problem solving, learning, and teaching (ss. 465-478) içinde. Hillsdale, NJ: Lawrence Erlbaum Associates.

Charmaz, K. (2006). Constructing grounded theory / A practical guide through qualitative analysis. London: Sage Publications Ltd.,

Cifarelli, V. V. (1998). The development of mental representations as a problem solving activity, Journal of Mathematical Behavior. 17(2), 239-264.

Cox, R. (1999). Representation construction, externalised cognition and individual differences. Learning and Instruction, 9, 343-363.

Crouch, R. ve Haines, C., (2004). Mathematical modeling: Transitions between real world and the mathematical model. Instructional Journal of Mathematical Education in Science and Technology, 35(2), 197-206.

Diezmann, C. M. ve English, L. D. (2001). Promoting the use of diagrams as tools for thinking. A. A. Cuoco ve F. R. Curcio (Eds), The roles of representation in school mathematics: 2001 yearbook (pp. 77–89) içinde. Reston, VA: National Council of Teachers of Mathematics.

Eisner, E. W. (1997). Cognition and representation. Phi Delta Kapan. 78(5), 348-353.

Elia, I, Gagatsis, A. ve Demetriou, A. (2007). The effects of different modes of representation on the solution of one-step additive problems. Learning and Instruction, 17, 658-672.

English, L. D. (2003). Mathematical modelling with young learners. S. J. Lamon, W. A. Parker ve S. K. Houston (Eds), Mathematical modelling: A way of life (ss. 3-18) içinde. Chichester: Horwood Publishing.

Fennell, F. S. ve Rowan T. (2001). Representation: An important process. Teaching and Learning Mathematics, 7(5), 288-292.

Galbraith, P. ve Stillman, G. (2006). A Framework for identifying student blockages during transitions in the modelling process. Zentralblatt für Didaktik der Mathematik-ZDM, 38(2), 143-162.

Galbraith, P., Stillman, G., Brown, J. ve Edwards I. (2007). Facilitating middle secondary modelling competencies. C. Haines, P. Galbraith, W. Blum ve S. Khan (Eds), Mathematical Modelling: ICTMA 12: Education, Engineering an Economics, (ss. 130-140) içinde. Chichester: Horwood Publishing.

Goldin, G.ve Shteingold, N. (2001). Systems of representations and the development of mathematical concepts. A. Cuoco ve F. Curcio (Eds), The roles of representation in school mathematics (ss. 1-23) içinde. Va: Reston Virginia, NCTM Yearbook.

Greeno, G. ve Hall, R. P. (1997). Practicing Representation: Learning with and about Representational Forms. Phi Delta Kappan, 78(5) , 361-367.

Hıdıroğlu, Ç. N. (2012). Teknoloji destekli ortamda matematiksel modelleme problemlerinin çözüm süreçlerinin analiz edilmesi: Yaklaşım ve düşünme süreçleri üzerine bir açıklama. Yayımlanmamış yüksek lisans tezi, Dokuz Eylül Üniversitesi, İzmir.

Hıdıroğlu, Ç. N. (2015). Teknoloji destekli ortamda matematiksel modelleme problemlerinin çözüm süreçlerinin analizi: Bilişsel ve üstbilişsel yapılar üzerine bir açıklama. Yayımlanmamış doktora tezi, Dokuz Eylül Üniversitesi, İzmir.

Hıdıroğlu, Ç. N. ve Bukova Güzel, E. (2013). Teknoloji destekli ortamda matematiksel modellemede modelin doğrulanmasındaki yaklaşımların ve düşünme süreçlerinin kavramsallaştırılması. Kuram ve Uygulamada Eğitim Bilimleri Dergisi (KUYEB), 13(4), 2487-2508.

Hıdıroğlu, Ç. N. ve Bukova Güzel, E. (2014). Matematiksel modellemede GeoGebra kullanımı: Boy-Ayak uzunluğu problemi. Pamukkale Üniversitesi Eğitim Fakültesi Dergisi, 36(2), 29-44.

Hıdıroğlu, Ç. N. ve Bukova Güzel, E. (2015). Teknoloji destekli ortamda matematiksel modellemede ortaya çıkan üst bilişsel yapılar. Turkish Journal of Computer and Mathematics Education, 6(2), 179-208.

Hıdıroğlu, Ç. N. ve Bukova Güzel, E. (2016). Transitions between cognitive and metacognitive activities in mathematical modelling process within a technology enhanced environment. Necatibey Faculty of Education Electronic Journal of Science and Mathematics Education, 10(1), 313-350.

Hıdıroğlu, Ç. N. and Bukova Güzel, E. (2017-Basımda). The Conceptualization of the Mathematical Modelling Process in Technology-Aided Environment. The International Journal for Technology in Mathematics Education, 24(1), ?-?.

Ikeda, T. ve Stephens M. (2001). The Effects of students’ discussion in mathematics modelling. J.P. Matos, W. Blum, K. Houston ve S.P. Carriera (Eds), Modelling and mathematics education: ICTMA 9: Applications in science and techonology (ss. 381-390) içinde. Chichester: Horwood Publishing.

Janvier, B. D. ve Bednarz, N. (1987). Pedagogical considerations concerning the problem of representation. C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics (ss. 67-72) içinde. New Jersey: Lawrence Erlbaum Associates.

Ji, X. (2012). A quasi-experimental study of high school students’ mathematics modelling competence. Paper presented at 12th International Congress on Mathematical Education Program, COEX, Seoul, Korea. 27 Haziran 2015 tarihinde http://www.icme12.org/ adresinden erişilmiştir.

Kaiser Meßmer, G. (1986). Anwendungen im mathematikunterricht. Bad Salzdetfurth: Franzbecker.

Kaiser, G. (2005). Introduction to the working group “applications and modelling”. M. Bosch (Ed.), Proceedings of the Fourth Congress of the European Society for Research in Mathematics Education (ss. 1613-1622) içinde. Universitat Ramon Llull: FUNDEMI IQS.

Kaiser, G. (2007). Modelling and modelling competencies in school. C. Haines, P. Galbraith, W. Blum, ve S. Khan (Eds), Mathematical Modelling: ICTMA 12: Education, Engineering an Economics, (ss. 110-119) içinde. Chichester: Horwood Publishing.

Kaiser, G. ve Sriraman, B. (2006). A global survey of ınternational perspectives on modelling in mathematics education. Zentralblatt für Didaktik der Mathematik, 38(3), 302-310.

Lamberts, K. (2005). Mathematical modelling of cognition. K, Lamberts ve R. L. Goldstone (Eds), Handbook of cognition (ss. 407-421) içinde. London: Sage.

Learning and Education in and through Modelling and Applications [LEMA], (2009). Booklet: Reality-based tasks for school. 08 Şubat 2017 tarihinde www.primas-project.eu/servlet/supportBinaryFiles?referenceId=3&supportId=1245 adresinden erişilmiştir.

Lesh, R. ve Caylor, B. (2007). Introduction to special issue: Modeling as application versus modeling as a way to create mathematics. International Journal of Computers for Mathematical Learning, 12(3), 173-194.

Lesh, R. ve Doerr, H. M. (2003). Foundations of a models and modeling perspective on 3 mathematics teaching, learning, and problem solving. R. Lesh ve H. M. Doerr (Eds), Beyond constructivism: Models and modeling perspectives on mathematics problem solving, learning and teaching (ss. 3-34) içinde. Mahwah, NJ: Lawrence Erlbaum.

Lesh, R., Post T. ve Behr, M. (1987). Representation and Translations among Representations in Mathematics Learning and Problem Solving. C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics (ss. 33-40). New Jersey: Lawrence Erlbaum Associates.

Lingefjärd, T. (2012). Learning mathematics through mathematical modelling. Journal of Mathematical Modelling and Application, 1(5), 41-49.

Maaß, K. (2006) What are modelling competencies?. Zentralblatt für Didaktik der Mathematik, 38(2),113-142.

Magiera, M. T. ve Zawojewski, J. (2011). Characterizations of social-based and self-based contexts associated with students’ awareness, evaluation, and regulation of their thinking during small-group mathematical modeling. Journal for Research in Mathematics Education, 42(5), 486-520.

Maki, D. ve Thompson, M. (2010). The mathematical modeling cycle. 25 Temmuz 2016 tarihinde http://www.indiana.edu/~hmathmod/modelmodel.html adresinden erişilmiştir.

Merriam, S. B. (2013). Nitel araştırma desen ve uygulama için bir rehber (S. Turan, Çev. Ed.). Ankara: Nobel Yayıncılık.

Miles, H. B. ve Huberman, A.M. (1994). Qualitative data analysis (2. Baskı). Thousand Oaks, CA: Sage.

Milli Eğitim Bakanlığı [MEB], (2013). Ortaöğretim Matematik Dersi (9-12. Sınıflar) Öğretim Programı. Ankara: MEB Basımevi.

Montague, M. (2008). Math problem solving for middle school students with disabilities. 8 Mayıs 2016 tarihinde http://165.139.150.129/intervention/Math%20Problem%20Solving%20for%20Middle%20School%20Students%20with%20Disabilities.pdf adresinden erişilmiştir.

Mousoulides, N., Christou, C. ve Sriraman, B. (2006). From problem solving to modelling- A meta analysis. 26 Kasım 2016 tarihinde https://pdfs.semanticscholar.org/5ec1/10b0416eb038850354228eefb358fa715c85.pdf adresinden erişilmiştir.

National Council of Teachers of Mathematics [NCTM]. (2000). Principles and standards for school mathematics. 21 Eylül 2016 tarihinde http://www.nctm.org/Standards-and-Positions/Principles-and-Standards/ adresinden erişilmiştir.

Neria, D. ve Amit, M. (2004). Students preference of non-algebraic representations in mathematical communication. Proceedings of the 28th Conference of the International Group for the Pyschology of Mathematics Education, 3, 409-416. 21 Ekim 2016 tarihinde https://www.emis.de/proceedings/PME28/RR/RR222_Neria.pdf adresinden erişilmiştir.

Niss, M. (2001). Issues and problems of research on the teaching and learning of applications and Modelling. J. F. Matos, W. Blum, S. K. Houston ve S. P. Carrera (Eds), Modelling and mathematics education (pp. 73-88) içinde. Chichester. Horwoord Publishing.

Organisation for Economic Co-operation and Development [OECD] (2003). Education at a glance: OECD indicators 2003. 11 Ekim 2016 tarihinde http://www.oecd.org/site/worldforum/33703760.pdf adresinden erişilmiştir.

Organisation for Economic Co-operation and Development [OECD] (2016). Education at a glance 2016: OECD indicators. 11 Ekim 2016 tarihinde http://download.ei-ie.org/Docs/WebDepot/EaG2016_EN.pdf adresinden erişilmiştir.

Patton, M. Q. (1987). How to use qualitative methods in evaluation. Newbury Park, CA: Sage.

Patton, M. Q. (2002). Qualitative research & evaluation methods (Third edition). Thousand Oaks, CA: Sage.

Peter Koop, A. (2004). Fermi problems in primary mathematics classrooms: Pupils’ interactive modelling processes. I. Putt, R. Farragher ve M. McLean (Eds), Mathematics education for the third millenium: Towards 2010, Proceedings of the 27th Annual Conference of the Mathematics Education Research Group of Australasia, (pp. 454-461) içinde. Townsville, Queensland: MERGA.

Pólya, G. (1945). How to solve it: A new aspect of mathematical method. Princeton, USA, Princeton University Press.

Preston, R. ve Garner, A. S. (2003). Representation as a vehicle for solving and communication. Mathematics Teaching in The Middle School, 9(1), 38-44.

Punch, K. F. (2013). Sosyal araştırmalara giriş nicel ve nitel yaklaşımlar (2. Baskı). (D. Bayrak, H.B. Arslan ve Z. Akyüz, Çev.). Siyasal Kitabevi, Ankara.

Saeki, A. ve Matsuzaki, A. (2011). Dual modelling cycle framework for responding to the diversities of modellers. G. A. Stillmann, G. Kaiser ve W. Blum (Eds), Proceedings of ICTMA15, Teaching mathematical modelling: Connecting to research and practice (pp. 89–99). Dordrecht, NLD: Springer.

Saeki, A. ve Matsuzaki, A. (2013). Dual modelling cycle framework for responding to the diversities of modellers. G. Stillman, G. Kaiser, W. Blum, ve J. Brown (Eds), Teaching Mathematical Modelling: Connecting to Research and Practice, (pp. 89-99) içinde. New York, USA: Springer.

Schoenfeld, A. H. (1985). Mathematical problem solving. Orlando: Academic Press.

Schwarz, B., Wissmach, B. ve Kaiser, G. (2008). ‘Last curves not quite correct’: Diagnostic compentences of future teachers with regard to modeling and graphical representations. ZDM Mathematics Education, 40(5), 777-790.

Stark, R. M. ve Nichols, R. L. (2005). Mathematical foundations for design: Civil engineering systems. Dover Publications.

Strauss, A. ve Corbin, J. (1990). Basics of qualitative research (1. Baskı). Thousand Oaks, CA: Sage.

Şen Zeytun, A. (2013). An investigation of prospective teachers’ mathematical modeling processes and their views about factors affecting these processes. Unpublished doctoral dissertation, Middle East Technical University, Ankara, Turkey.

Tatsis, K. (2010). Assessing in-service teachers’ modeling activities: Issues of content and complexity. The 17th annual conference of ALM adults learning mathematics – A research forum. Oslo, 28 June 2010.

Treilibs, V., Burkhardt, H. ve Low, B. (1980). Formulation processes in mathematical modelling. Nottingham, England: Shell Centre for Mathematical Education.Ang, K. C. (2001). Teaching mathematical modelling in Singapore schools. The Mathematics Educator, 6(1), 63-75.

Turner, R. (2007). Modelling and applications in PISA. W. Blum, P. Galbraith, H.-W. Henn ve M. Niss (Eds), Modelling and Applications in Mathematics Education, The 14. ICMI Study (ss. 433-440) içinde. New York: Springer.

Verschaffel, L., De Corte, E. ve Borghart, I. (1997). Pre-service teachers’ conceptions and beliefs about the role of real-world knowledge in mathematical modeling of school word problems. Learning and Instruction, 7(4), 339-359.

Zawojewski, J. S., ve Lesh, R. (2003). A models and modelling perspective on problem solving. R. A. Lesh, ve H. Doerr (Eds.), Beyond constructivism: Models and modeling perspectives on mathematics problem solving, learning, and teaching (ss. 317-336) içinde. Mahwah, NJ: Lawrence Erlbaum.


Refback'ler

  • Şu halde refbacks yoktur.




Telif Hakkı (c) 2017 İlköğretim Online

Creative Commons License
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

 

 

 

Creative Commons Lisansı

İlköğretim Online Dergisi Creative Commons Alıntı-Gayriticari 4.0 Uluslararası Lisansı ile lisanslanmıştır.

ISSN: 1305-3515