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Representations used by mathematics student teachers in mathematical modeling process

Year 2013, Volume: 4 Issue: 2, - , 26.11.2013

Abstract

The purpose of this study is to determine representations used by mathematics student teachers in steps of mathematical modeling process based on their solutions of problems formed in the context of different classification of modeling. The study was conducted with fifteen secondary mathematics student teachers given a Mathematical Modeling course. The participants were separated into five collaboration groups of three students. Data were collected with the detailed written papers given by the groups for the problems and GeoGebra solution files. The groups benefited from verbal, algebraic, figural, tabular and dynamic representations while they were solving the problems. Considering all steps of the process, groups at most used verbal and algebraic representations. While they used only verbal representation in analyzing the problem, they benefited from at most verbal representation and then figural representation in establishing the systematic structure. The most used is algebraic and then verbal representations in the steps of mathematization, meta-mathematization, and mathematical analysis. In the steps of interpretation/evaluation and the model verification, the groups mainly benefited from verbal and then algebraic representations. Further researches towards why representations are preferred in the specific steps of the mathematical modeling process are suggested.


Key Words: Mathematical modeling, modeling problems, mathematics student teachers, representations.

References

  • Cheng, A. K. (2010). Teaching and learning mathematical modelling with technology, Nanyang Technological University. 02012 tarihinde http://atcm.mathandtech.org/ep2010/invited/3052010_18134.pdf adresinden alınmıştır.
  • Arzarello, F., Ferrara, F. & Robutti, O. (2012). Mathematical modelling with technology: The role of dynamic representations. Teaching Mathematics and Its Applications.31. s. 20-30.
  • Berry, J. & Houston, K. (1995).Mathematical modeling. London: Edward Arnold.
  • Blomhİj, M. & Jensen T.H. (2006). What’s all the fuss about competencies? Experiences with using a competence perspective on mathematics education to develop the teaching of mathematical modelling. In W. Blum, P.L. Galbraith and M. Niss: Modelling and Applications in Mathematics Education. New York: Springer, 2(2), 45Blomhİj, M. (2008).Different perspectives on mathematical modelling in educational research-Categorising the TSG21 papers. ICME 11 international Congress on Mathematics Education.1-13.
  • Blum, W. & Leiß, D. (2007). How do students and teachers deal with modelling problems? In C. Haines et al. (Eds), Mathematical Modelling. Education, Engineering and Economics. Chichester: Horwood. 222-231.
  • Borromeo-Ferri, R. (2010). On the influence of mathematical thinking styles on learners’ modeling behaviour. Journal für Mathematik didaktik, 31 (1), 99-118.
  • Borromeo-Ferri, R. B. (2006). Theoretical and empirical differentiations of phases in the modelling process. In Kaiser, G., Sriraman B. ve Blomhoij, M. (Eds.) Zentralblattfür Didaktik der Mathematik.38(2), 86-95.
  • Brenner, M. E., Mayer, R. E., Moseley, B., Brar, T., Duran, R., Reed, B. & Webb, D. (1997). Learning by understanding: The role of multiple representations in learning algebra. American Educational Research Journal, 34(4), 663-689.
  • Corter, J. & Zahrer, D. (2007). Use of external visual representations in probability problem solving. Statistics Education Research Journal, 6 (1), 22-50.
  • DeWindt-King, A.M. ve Goldin, G.A. (2003). Children’s visual imagery: Aspects of cognitive representation in solving problems with fractions. Mediterranean Journal for Research in Mathematics Education. 2(1), 1-42
  • Fox , J. (2006). A justification for mathematical modelling experiences in the preparatory classroom. P. Grootenboer, R., Zevenbergen, ve M. Chinnappan (Eds.) Proceedings 29th annual conference of the Mathematics Education Research Group of Australasia. 1, 21-228.
  • Galbraith, P. & Stillman, G. (2006).A framework for identifying student blockages during transitions in the modelling process. Zentralblattfür Didaktik der MathematikZDM.38(2), 143-162.
  • Galbraith, P., Stillman, G., Brown, J. & Edwards, I. (2007).Facilitating middle secondary modelling competencies. C. Haines, P. Galbraith, W. Blum, S. Khan (Ed.), Mathematical Modelling: ICTMA 12: Education, Engineering an Economics.1301
  • Hıdıroğlu, Ç. N. (2012). Teknoloji destekli ortamda matematiksel modelleme problemlerinin çözüm süreçlerinin analiz edilmesi: Yaklaşım ve düşünme süreçleri üzerine bir açıklama. Yüksek Lisans Tezi. Dokuz Eylül Üniversitesi, İzmir.
  • Kaiser, G. & Sriraman, B. (2006).A global survey of international perspectives on modelling in mathematics education. Zentralblattfür Didaktik der Mathematik. 38(3), 302-310.
  • Kaiser, G., (2005). Introduction to the working group “applications and modelling”. CERME 4 Proceedings, p 1611-1622.
  • King, J. P. (2004). Matematik Sanatı (N. Arık, Çev.). Ankara: Gökçe Ofset. 15. Basım Lesh, R. (1981). Applied mathematical problem solving. Education Studies in Mathematics. 12 235-264.
  • Lesh, R., Landau, M. & Hamilton, E. (1983). Conceptual models in applied mathematical problem solving research. In R. Lesh ve M. Landau (Eds.), Acquisition of mathematics concepts and processes (pp. 263-343). New York: Academic Press. Lesh, R., & Doerr, H. M. (2003). (Eds.). Beyond Constructivism: Models and Modeling Perspectives on Mathematics Problem Solving, Learning and Teaching. Mahwah, NJ: Lawrence Erlbaum.
  • Lowrie, T. (2001).The influence of visual representations on mathematical problem solving and numeracy performance.24th Annual MERGA Conference, Sydney, July 2001(p. 354-361).
  • Maaß, K. (2006). What are modelling competencies? Zentralblattfür Didaktik der Mathematik.38 (2), 113-142.
  • Peter-Koop, A. (2004). Fermi problems in primary mathematics classrooms: pupils’ interactive modelling processes. In I. Putt, R. Farragher, ve M. McLean (Eds.), Mathematics education for the Third Millenium: Towards 2010 (Proceedings of the 27 th Annual Conference of the Mathematics Education Research Group of Australasia, pp. 454-461). Townsville, Queensland: MERGA.
  • Presmeg, N. C. & Nenduradu, R. (2005).An investigation of a preservice teacher’s use of representations in solving algebraic problems involving exponential relationships. In H. Chick ve J. Vincent (Eds.), Proc. 29th Conf. of the Int. Group for the Psychology of Mathematics Education (Vol. 4, pp. 105-112). Melbourne, Australia: PME.
  • Shield, M. & Galbraith, P. (1998).The analysis of student expository writing in mathematics. Educational Studies in Mathematics. 36, 29–52.
  • Siller, H.S. & Greefrath, G. (2010). Mathematical modelling in class regarding to technology. CERME 6 – Proceedings of the sixth Congress of the European Society for Research in Mathematics Education. 108-117.
  • Wu, Z. (2004). The study of middle school teachers’ understanding and use of mathematical representation in relation to teachers’ zone of proximal development in teaching fractions and algebraic functions. Doctoral dissertation. Department of Teaching, Learning and Culture. Texas A&M University, College Station.
  • Thomas, G. B., Weir, M. D., Hass, J. & Giordano, F. R. (2010). Thomas calculus 1 (Baskı, Çeviri: Recep Korkmaz). Beta Basım A.Ş. İstanbul.
  • Tversky, B. (2001). Spatial schemas in depictions. In M. Gattis, Ed. Spatial schemas and abstract thought, pp. 79-111. Cambridge: MIT Press.

Matematik Öğretmeni Adaylarının Modelleme Sürecinde Kullandıkları Gösterim Şekilleri

Year 2013, Volume: 4 Issue: 2, - , 26.11.2013

Abstract

Bu çalışmanın amacı, matematik öğretmeni adaylarının farklı modelleme türleri bağlamında oluşturulmuş problemlere ilişkin çözümlerinden yola çıkarak matematiksel modelleme sürecinin basamaklarında kullandıkları gösterim şekillerini belirlemektir. Çalışma, Matematiksel Modelleme dersini alan on beş ortaöğretim matematik öğretmen adayıyla gerçekleştirilmiştir. Katılımcılar kendi istekleri doğrultusunda üçer kişilik beş çalışma grubuna ayrılmışlardır. Veriler, grupların altı matematiksel modelleme problemine ilişkin ayrıntılı çözümlerini içeren yazılı yanıt kağıtları ve GeoGebra çözüm dosyaları yardımıyla toplanmıştır. Grupların modelleme problemlerinin çözümünde sözel, cebirsel, şekilsel, grafiksel, tablo ve dinamiksel gösterim şekillerinden yararlandıkları belirlenmiştir. Sürecin tüm basamaklarına göre gruplar en fazla sözel ve cebirsel gösterimleri kullanmışlardır. Problemin analizi basamağında sadece sözel gösterim kullanılırken, sistematik yapıyı kurma basamağında ise en fazla sözel ardından ise şekilsel gösterimden yararlanılmıştır. Matematikselleştirme, üst matematikselleştirme ve matematiksel analiz basamaklarında en çok kullanılan cebirsel ve ardından sözel gösterimler olmuştur. Yorumlama/değerlendirme ve modelin doğrulanması basamaklarında ise gruplar ağırlıklı olarak sözel ve ardından da cebirsel gösterimlerden yararlanmışlardır. Gösterim şekillerinin matematiksel modelleme sürecinin hangi basamaklarında niçin tercih edildiğine yönelik araştırmaların yapılması önerilmektedir.

Anahtar Kelimeler:   Matematiksel modelleme, modelleme problemleri, matematik öğretmeni adayları, gösterim şekilleri.

References

  • Cheng, A. K. (2010). Teaching and learning mathematical modelling with technology, Nanyang Technological University. 02012 tarihinde http://atcm.mathandtech.org/ep2010/invited/3052010_18134.pdf adresinden alınmıştır.
  • Arzarello, F., Ferrara, F. & Robutti, O. (2012). Mathematical modelling with technology: The role of dynamic representations. Teaching Mathematics and Its Applications.31. s. 20-30.
  • Berry, J. & Houston, K. (1995).Mathematical modeling. London: Edward Arnold.
  • Blomhİj, M. & Jensen T.H. (2006). What’s all the fuss about competencies? Experiences with using a competence perspective on mathematics education to develop the teaching of mathematical modelling. In W. Blum, P.L. Galbraith and M. Niss: Modelling and Applications in Mathematics Education. New York: Springer, 2(2), 45Blomhİj, M. (2008).Different perspectives on mathematical modelling in educational research-Categorising the TSG21 papers. ICME 11 international Congress on Mathematics Education.1-13.
  • Blum, W. & Leiß, D. (2007). How do students and teachers deal with modelling problems? In C. Haines et al. (Eds), Mathematical Modelling. Education, Engineering and Economics. Chichester: Horwood. 222-231.
  • Borromeo-Ferri, R. (2010). On the influence of mathematical thinking styles on learners’ modeling behaviour. Journal für Mathematik didaktik, 31 (1), 99-118.
  • Borromeo-Ferri, R. B. (2006). Theoretical and empirical differentiations of phases in the modelling process. In Kaiser, G., Sriraman B. ve Blomhoij, M. (Eds.) Zentralblattfür Didaktik der Mathematik.38(2), 86-95.
  • Brenner, M. E., Mayer, R. E., Moseley, B., Brar, T., Duran, R., Reed, B. & Webb, D. (1997). Learning by understanding: The role of multiple representations in learning algebra. American Educational Research Journal, 34(4), 663-689.
  • Corter, J. & Zahrer, D. (2007). Use of external visual representations in probability problem solving. Statistics Education Research Journal, 6 (1), 22-50.
  • DeWindt-King, A.M. ve Goldin, G.A. (2003). Children’s visual imagery: Aspects of cognitive representation in solving problems with fractions. Mediterranean Journal for Research in Mathematics Education. 2(1), 1-42
  • Fox , J. (2006). A justification for mathematical modelling experiences in the preparatory classroom. P. Grootenboer, R., Zevenbergen, ve M. Chinnappan (Eds.) Proceedings 29th annual conference of the Mathematics Education Research Group of Australasia. 1, 21-228.
  • Galbraith, P. & Stillman, G. (2006).A framework for identifying student blockages during transitions in the modelling process. Zentralblattfür Didaktik der MathematikZDM.38(2), 143-162.
  • Galbraith, P., Stillman, G., Brown, J. & Edwards, I. (2007).Facilitating middle secondary modelling competencies. C. Haines, P. Galbraith, W. Blum, S. Khan (Ed.), Mathematical Modelling: ICTMA 12: Education, Engineering an Economics.1301
  • Hıdıroğlu, Ç. N. (2012). Teknoloji destekli ortamda matematiksel modelleme problemlerinin çözüm süreçlerinin analiz edilmesi: Yaklaşım ve düşünme süreçleri üzerine bir açıklama. Yüksek Lisans Tezi. Dokuz Eylül Üniversitesi, İzmir.
  • Kaiser, G. & Sriraman, B. (2006).A global survey of international perspectives on modelling in mathematics education. Zentralblattfür Didaktik der Mathematik. 38(3), 302-310.
  • Kaiser, G., (2005). Introduction to the working group “applications and modelling”. CERME 4 Proceedings, p 1611-1622.
  • King, J. P. (2004). Matematik Sanatı (N. Arık, Çev.). Ankara: Gökçe Ofset. 15. Basım Lesh, R. (1981). Applied mathematical problem solving. Education Studies in Mathematics. 12 235-264.
  • Lesh, R., Landau, M. & Hamilton, E. (1983). Conceptual models in applied mathematical problem solving research. In R. Lesh ve M. Landau (Eds.), Acquisition of mathematics concepts and processes (pp. 263-343). New York: Academic Press. Lesh, R., & Doerr, H. M. (2003). (Eds.). Beyond Constructivism: Models and Modeling Perspectives on Mathematics Problem Solving, Learning and Teaching. Mahwah, NJ: Lawrence Erlbaum.
  • Lowrie, T. (2001).The influence of visual representations on mathematical problem solving and numeracy performance.24th Annual MERGA Conference, Sydney, July 2001(p. 354-361).
  • Maaß, K. (2006). What are modelling competencies? Zentralblattfür Didaktik der Mathematik.38 (2), 113-142.
  • Peter-Koop, A. (2004). Fermi problems in primary mathematics classrooms: pupils’ interactive modelling processes. In I. Putt, R. Farragher, ve M. McLean (Eds.), Mathematics education for the Third Millenium: Towards 2010 (Proceedings of the 27 th Annual Conference of the Mathematics Education Research Group of Australasia, pp. 454-461). Townsville, Queensland: MERGA.
  • Presmeg, N. C. & Nenduradu, R. (2005).An investigation of a preservice teacher’s use of representations in solving algebraic problems involving exponential relationships. In H. Chick ve J. Vincent (Eds.), Proc. 29th Conf. of the Int. Group for the Psychology of Mathematics Education (Vol. 4, pp. 105-112). Melbourne, Australia: PME.
  • Shield, M. & Galbraith, P. (1998).The analysis of student expository writing in mathematics. Educational Studies in Mathematics. 36, 29–52.
  • Siller, H.S. & Greefrath, G. (2010). Mathematical modelling in class regarding to technology. CERME 6 – Proceedings of the sixth Congress of the European Society for Research in Mathematics Education. 108-117.
  • Wu, Z. (2004). The study of middle school teachers’ understanding and use of mathematical representation in relation to teachers’ zone of proximal development in teaching fractions and algebraic functions. Doctoral dissertation. Department of Teaching, Learning and Culture. Texas A&M University, College Station.
  • Thomas, G. B., Weir, M. D., Hass, J. & Giordano, F. R. (2010). Thomas calculus 1 (Baskı, Çeviri: Recep Korkmaz). Beta Basım A.Ş. İstanbul.
  • Tversky, B. (2001). Spatial schemas in depictions. In M. Gattis, Ed. Spatial schemas and abstract thought, pp. 79-111. Cambridge: MIT Press.
There are 27 citations in total.

Details

Primary Language Turkish
Journal Section Research Articles
Authors

Aytuğ Özaltun

Çağlar Hıdıroğlu

Semiha Kula

Esra Bukova Güzel

Publication Date November 26, 2013
Published in Issue Year 2013 Volume: 4 Issue: 2

Cite

APA Özaltun, A., Hıdıroğlu, Ç., Kula, S., Bukova Güzel, E. (2013). Matematik Öğretmeni Adaylarının Modelleme Sürecinde Kullandıkları Gösterim Şekilleri. Turkish Journal of Computer and Mathematics Education (TURCOMAT), 4(2). https://doi.org/10.16949/turcomat.65070
AMA Özaltun A, Hıdıroğlu Ç, Kula S, Bukova Güzel E. Matematik Öğretmeni Adaylarının Modelleme Sürecinde Kullandıkları Gösterim Şekilleri. Turkish Journal of Computer and Mathematics Education (TURCOMAT). November 2013;4(2). doi:10.16949/turcomat.65070
Chicago Özaltun, Aytuğ, Çağlar Hıdıroğlu, Semiha Kula, and Esra Bukova Güzel. “Matematik Öğretmeni Adaylarının Modelleme Sürecinde Kullandıkları Gösterim Şekilleri”. Turkish Journal of Computer and Mathematics Education (TURCOMAT) 4, no. 2 (November 2013). https://doi.org/10.16949/turcomat.65070.
EndNote Özaltun A, Hıdıroğlu Ç, Kula S, Bukova Güzel E (November 1, 2013) Matematik Öğretmeni Adaylarının Modelleme Sürecinde Kullandıkları Gösterim Şekilleri. Turkish Journal of Computer and Mathematics Education (TURCOMAT) 4 2
IEEE A. Özaltun, Ç. Hıdıroğlu, S. Kula, and E. Bukova Güzel, “Matematik Öğretmeni Adaylarının Modelleme Sürecinde Kullandıkları Gösterim Şekilleri”, Turkish Journal of Computer and Mathematics Education (TURCOMAT), vol. 4, no. 2, 2013, doi: 10.16949/turcomat.65070.
ISNAD Özaltun, Aytuğ et al. “Matematik Öğretmeni Adaylarının Modelleme Sürecinde Kullandıkları Gösterim Şekilleri”. Turkish Journal of Computer and Mathematics Education (TURCOMAT) 4/2 (November 2013). https://doi.org/10.16949/turcomat.65070.
JAMA Özaltun A, Hıdıroğlu Ç, Kula S, Bukova Güzel E. Matematik Öğretmeni Adaylarının Modelleme Sürecinde Kullandıkları Gösterim Şekilleri. Turkish Journal of Computer and Mathematics Education (TURCOMAT). 2013;4. doi:10.16949/turcomat.65070.
MLA Özaltun, Aytuğ et al. “Matematik Öğretmeni Adaylarının Modelleme Sürecinde Kullandıkları Gösterim Şekilleri”. Turkish Journal of Computer and Mathematics Education (TURCOMAT), vol. 4, no. 2, 2013, doi:10.16949/turcomat.65070.
Vancouver Özaltun A, Hıdıroğlu Ç, Kula S, Bukova Güzel E. Matematik Öğretmeni Adaylarının Modelleme Sürecinde Kullandıkları Gösterim Şekilleri. Turkish Journal of Computer and Mathematics Education (TURCOMAT). 2013;4(2).