Mathematical Thinking Styles and the features of Modeling Process

Juhaina Awawdeh Shahbari

doi:10.14712/18047106.1579

Vol 11 No 1 (2020): sciED Výzkumné stati

Styly myšlení v matematice a charakteristika procesu modelování

Juhaina Awawdeh Shahbari ¹ http://orcid.org/0000-0002-7361-8069 (neověřeno)

College of Sakhnin for Teacher Education: Sakhnin

DOI: https://doi.org/10.14712/18047106.1579

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Shahbari, J. A. (2020). Styly myšlení v matematice a charakteristika procesu modelování. Scientia in educatione, 11(1), 59-68. https://doi.org/10.14712/18047106.1579

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Abstrakt

Studie se zabývá vztahem mezi stylem myšlení žáků a procesem modelování. Zúčastnilo se jí 35 žáků osmého ročníku. V první fázi žáci řešili úlohy a na základě jejich řešení byli klasifikovány do jednoho ze tří stylů myšlení: vizuálního, analytického a integrativního. V ohnisku našeho zájmu byl vizuální a analytický styl. Vybrali jsme pět žáků z každé skupiny (N = 10), kterým byly zadány tři úlohy vyžadující modelování. Výsledky naznačují rozdíly v modelování mezi oběma skupinami. Hlavní rozdíly se projevily ve zjednodušení, matematizaci a tvorbě matematického modelu. Kromě toho žáci z analytické skupiny přeskočili ve všech úlohách fázi reálného modelu, zatímco žáci z vizuální skupiny reálný model v každé úloze vytvořili.

Reference

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Blum, W., & Borromeo Ferri, R. (2009). Mathematical modeling: Can it be taught and learnt? Journal of Mathematical Modeling and Application, 1(1), 45–58. Vyhledej v Google Scholar
Blum, W., & Leiß, D. (2005). “Filling Up” – the problem of independence-preserving teacher interventions in lessons with demanding modeling tasks. In M. Bosch (Ed.), Proceedings of the fourth congress of the European Society for Research in Mathematics Education (CERME 4) (pp. 1623–1633). Sant Feliu de Guíxols, Spain, Fundemi IQS Universitat Ramon Llull. Vyhledej v Google Scholar
Borromeo Ferri, R. (2006). Theoretical and empirical differentiations of phases in the modeling process. ZDM, 38(2), 86–95. https://doi.org/10.1007/bf02655883 Přejít na DOI Vyhledej v Google Scholar
Borromeo Ferri, R. (2007). Modeling problems from a cognitive perspective. In C. P. Haines, P. Galbraith, W. Blum, & S. Khan (Eds.), Mathematical modeling (ICTMA12): Education, engineering and economics (pp. 260–270). Horwood Publishing Limited. Vyhledej v Google Scholar
Borromeo Ferri, R. (2010). On the influence of mathematical thinking styles on learners’ modeling behavior. Journal fur Mathematik-Didaktik, 31(1), 99–118. https://doi.org/10.1007/s13138-010-0009-8 Přejít na DOI Vyhledej v Google Scholar
Borromeo Ferri, R. (2015). Mathematical thinking styles in school and across cultures. In S. Cho (Eds.), Selected regular lectures from the 12th International Congress on Mathematical Education (pp. 153–173). Springer. https://doi.org/10.1007/978-3-319-17187-6 9 Přejít na DOI Vyhledej v Google Scholar
Borromeo Ferri, R., & Kaiser, G. (2003). First results of a study of different mathematical thinking styles of schoolchildren. In L. Burton (Ed.), Which way in social justice in mathematics education? (pp. 209–239). Greenwood. Vyhledej v Google Scholar
Burton, L. (2001). Research mathematicians as learners – and what mathematics education can learn from them. British Educational Research Journal, 27(5), 589–599. https://doi.org/10.1080/01411920120095762 Přejít na DOI Vyhledej v Google Scholar
Campbell, K. J., Collis, K.F., & Watson, J. M. (1995). Visual processing during mathematical problem solving. Educational Studies in Mathematics, 28(2), 177–194. https://doi.org/10.1007/BF01295792 Přejít na DOI Vyhledej v Google Scholar
Cakan, M. (2000). Interaction between cognitive styles and assessment approaches. In LSU Historical dissertations and theses. https://digitalcommons.lsu.edu/gradschool disstheses/7145 Vyhledej v Google Scholar
Dwyer, F. M., & Moore, D. M. (1994). Effect of colour coding and test type (visual/verbal) on students identified as possessing different field dependence levels. British Journal of Educational Technology, 25(3), 217–219. Vyhledej v Google Scholar
English, L.D., & Fox, J. L. (2005). Seventh-graders’ mathematical modeling on completion of a three-year program. In P. Clarkson et al. (Eds.), Building connections: Theory, research and practice (Vol. 1, pp. 321–328). Deakin University Press. Vyhledej v Google Scholar
English, L.D., & Watters, J. J. (2005). Mathematical modeling in the early school years. Mathematics Education Research Journal, 16(3), 58–79. https://doi.org/10.1007/BF03217401 Přejít na DOI Vyhledej v Google Scholar
Huang, C.-H. (2013). Engineering students’ visual thinking of the concept of definite integral. Global Journal of Engineering Education, 15(2), 111–117. Vyhledej v Google Scholar
Glaser, B.G., & Strauss, A.L. (1967). Discovery of grounded theory: Strategies for qualitative research. Aldine Publishing Company. Vyhledej v Google Scholar
Kaiser, G. (2007). Mathematical modeling at schools how to promote modeling competencies. In C.P. Haines, P. Galbraith, W. Blum, & S. Khan (Eds.), Mathematical modeling (ICTMA12): Education, engineering and economics (pp. 110–119). Horwood Publishing Limited. Vyhledej v Google Scholar
Kaiser, G., & Stender, P. (2013). Complex modeling problems in co-operative, self-directed learning environments. In G.A. Stillman, G. Kaiser, W. Blum, & J.P. Brown (Eds.), Teaching mathematical modeling: Connecting to research and practice (pp. 277–294). Springer. https://doi.org/10.1007/978-94-007-6540-5 23 Přejít na DOI Vyhledej v Google Scholar
Kaput, J. J., & Blanton, M. (2001). Algebrafying the elementary mathematics experience: Part 1: Transforming task structures. In H. Chick, K. Stacey, J. Vincent, & J. Vincent (Eds.), Proceedings of the 12th International Commission on Mathematics Instruction Study Conference (pp. 344–351). University of Melbourne, Melbourne, Australia. Vyhledej v Google Scholar
Lean, G., & Clements, M.A. (1981). Spatial ability, visual imagery, and mathematical performance. Educational Studies in Mathematics, 12(3), 267–299. https://doi.org/10.1007/BF00311060 Přejít na DOI Vyhledej v Google Scholar
Lesh, R., & Doerr, H. M. (2003). Foundations of models and modeling perspective on mathematics teaching, learning, and problem solving. In R. Lesh, & H.M. Doerr (Eds.), Beyond constructivism: Models and modeling perspectives on mathematics problem solving, learning, and teaching (pp. 3–33). Erlbaum. Vyhledej v Google Scholar
Lesh, R., Hoover, M., Hole, B., Kelly, A., & Post, T. (2000). Principles for developing thought-revealing activities for students and teachers. In R. Lesh, & A. Kelly (Eds.), Handbook of research design in mathematics and science education (pp. 591–645). Lawrence Erlbaum. Vyhledej v Google Scholar
Lesh, R., & Lehrer, R. (2003). Models and modeling perspectives on the development of students and teachers. Mathematical Thinking and Learning, 5(2–3), 109–129. https://doi.org/10.1080/10986065.2003.9679996 Přejít na DOI Vyhledej v Google Scholar
Lowrie, T., & Clements, M.A. (2001). Visual and nonvisual processes in grade 6 students’ mathematical problem solving. Journal of Research in Childhood Education, 16(1), 77–93. https://doi.org/10.1080/02568540109594976 Přejít na DOI Vyhledej v Google Scholar
Lowrie, T., & Kay, R. (2001). Relationship between visual and nonvisual solution methods and difficulty in elementary mathematics. The Journal of Educational Research, 94(4), 248–255. https://doi.org/10.1080/00220670109598758 Přejít na DOI Vyhledej v Google Scholar
Maaß, K. (2006). What are modeling competencies? ZDM, 38(2), 113–142. https://doi.org/10.1007/BF02655885 Monga, A. B., & John, D.R. (2007). Cultural differences in brand extension evaluation: the influence of analytic versus holistic thinking. Journal of Consumer Research, 33(4), 529–536. https://doi.org/10.1086/510227 Přejít na DOI Vyhledej v Google Scholar
Niss, M., Blum, W., & Galbraith, P. (2007). Introduction. In W. Blum, P. L. Galbraith, H.-W. Henn, & M. Niss (Eds.), Modeling and applications in mathematics education (pp. 3–32). Springer. Vyhledej v Google Scholar
Organization for Economic Cooperation and Development (OECD). (2004). Learning for tomorrow’s world: First results from PISA 2003. OECD. Vyhledej v Google Scholar
Presmeg, N. C. (1986). Visualization in high school mathematics. For the Learning of Mathematics, 6(3), 42–46. Vyhledej v Google Scholar
Shahbari, J.A., & Daher, W. (2016). Mathematical models’ features: Technology and non-technology. European Journal of Science and Mathematics Education, 4(4), 523–533. Vyhledej v Google Scholar
Shahbari, J.A., & Peled, I. (2017). Modeling in primary schools: Constructing a conceptual system and making sense of fractions. International Journal of Science and Mathematics Education, 15(2), 371–391. https://doi.org/10.1007/s10763-015-9702-x Přejít na DOI Vyhledej v Google Scholar
Shahbari, J.A., & Tabach, M. (2016). Different generality levels in the product of a modeling activity. In C. Csikos, A. Rausch, & J. Szitanyi (Eds.), Proceedings of the 40th Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 179–186). PME. Vyhledej v Google Scholar
Shahbari, J. A., & Tabach, M. (2018). Developing prospective mathematics teachers’ knowledge of the modeling approach. Scientia in educatione, 9(2), 46–158. https://doi.org/10.14712/18047106.1183 Přejít na DOI Vyhledej v Google Scholar
Shahbari, J.A., & Tabach, M. (2019). Adopting the modeling cycle for representing prospective and practicing teachers’ interpretations of students’ modeling activities. In G. Stillman, & J. Brown (Eds.), Lines of inquiry in mathematical modeling research in education (pp. 179–196). ICME-13 Monographs, Springer. https://doi.org/10.1007/978-3-030-14931-4 10 Přejít na DOI Vyhledej v Google Scholar
Sternberg, R. J. (1997). Thinking styles. Cambridge University Press. Vyhledej v Google Scholar
Sternberg, R. J., & Zhang, L.–F. (2005). Styles of thinking as a basis of differentiated instruction. Theory into Practice, 44(3), 245–253. https://doi.org/10.1207/s15430421tip4403 9 Přejít na DOI Vyhledej v Google Scholar
Stillman, G., Galbraith, P., Brown, J., & Edwards, I. (2007). A framework for success in implementing mathematical modeling in the secondary school. In J. Watson, & K. Beswick (Eds.), Proceedings of the 30th Annual Conference of the Mathematics Education Research Group of Australasia (MERGA) (Vol. 2, pp. 688–697). MERGA. Vyhledej v Google Scholar

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[1] Ben-Chaim, D., Keret, Y., & Ilany, S.-B. (2012). Ratio and proportion: Research and teaching in mathematics teachers’ education (Pre- and in-service mathematics teachers of elementary and middle school classes). Sense Publisher. Vyhledej v Google Scholar

[2] Blum, W., & Borromeo Ferri, R. (2009). Mathematical modeling: Can it be taught and learnt? Journal of Mathematical Modeling and Application, 1(1), 45–58. Vyhledej v Google Scholar

[3] Blum, W., & Leiß, D. (2005). “Filling Up” – the problem of independence-preserving teacher interventions in lessons with demanding modeling tasks. In M. Bosch (Ed.), Proceedings of the fourth congress of the European Society for Research in Mathematics Education (CERME 4) (pp. 1623–1633). Sant Feliu de Guíxols, Spain, Fundemi IQS Universitat Ramon Llull. Vyhledej v Google Scholar

[4] Borromeo Ferri, R. (2006). Theoretical and empirical differentiations of phases in the modeling process. ZDM, 38(2), 86–95. https://doi.org/10.1007/bf02655883 Přejít na DOI Vyhledej v Google Scholar

[5] Borromeo Ferri, R. (2007). Modeling problems from a cognitive perspective. In C. P. Haines, P. Galbraith, W. Blum, & S. Khan (Eds.), Mathematical modeling (ICTMA12): Education, engineering and economics (pp. 260–270). Horwood Publishing Limited. Vyhledej v Google Scholar

[6] Borromeo Ferri, R. (2010). On the influence of mathematical thinking styles on learners’ modeling behavior. Journal fur Mathematik-Didaktik, 31(1), 99–118. https://doi.org/10.1007/s13138-010-0009-8 Přejít na DOI Vyhledej v Google Scholar

[7] Borromeo Ferri, R. (2015). Mathematical thinking styles in school and across cultures. In S. Cho (Eds.), Selected regular lectures from the 12th International Congress on Mathematical Education (pp. 153–173). Springer. https://doi.org/10.1007/978-3-319-17187-6 9 Přejít na DOI Vyhledej v Google Scholar

[8] Borromeo Ferri, R., & Kaiser, G. (2003). First results of a study of different mathematical thinking styles of schoolchildren. In L. Burton (Ed.), Which way in social justice in mathematics education? (pp. 209–239). Greenwood. Vyhledej v Google Scholar

[9] Burton, L. (2001). Research mathematicians as learners – and what mathematics education can learn from them. British Educational Research Journal, 27(5), 589–599. https://doi.org/10.1080/01411920120095762 Přejít na DOI Vyhledej v Google Scholar

[10] Campbell, K. J., Collis, K.F., & Watson, J. M. (1995). Visual processing during mathematical problem solving. Educational Studies in Mathematics, 28(2), 177–194. https://doi.org/10.1007/BF01295792 Přejít na DOI Vyhledej v Google Scholar

[11] Cakan, M. (2000). Interaction between cognitive styles and assessment approaches. In LSU Historical dissertations and theses. https://digitalcommons.lsu.edu/gradschool disstheses/7145 Vyhledej v Google Scholar

[12] Dwyer, F. M., & Moore, D. M. (1994). Effect of colour coding and test type (visual/verbal) on students identified as possessing different field dependence levels. British Journal of Educational Technology, 25(3), 217–219. Vyhledej v Google Scholar

[13] English, L.D., & Fox, J. L. (2005). Seventh-graders’ mathematical modeling on completion of a three-year program. In P. Clarkson et al. (Eds.), Building connections: Theory, research and practice (Vol. 1, pp. 321–328). Deakin University Press. Vyhledej v Google Scholar

[14] English, L.D., & Watters, J. J. (2005). Mathematical modeling in the early school years. Mathematics Education Research Journal, 16(3), 58–79. https://doi.org/10.1007/BF03217401 Přejít na DOI Vyhledej v Google Scholar

[15] Huang, C.-H. (2013). Engineering students’ visual thinking of the concept of definite integral. Global Journal of Engineering Education, 15(2), 111–117. Vyhledej v Google Scholar

[16] Glaser, B.G., & Strauss, A.L. (1967). Discovery of grounded theory: Strategies for qualitative research. Aldine Publishing Company. Vyhledej v Google Scholar

[17] Kaiser, G. (2007). Mathematical modeling at schools how to promote modeling competencies. In C.P. Haines, P. Galbraith, W. Blum, & S. Khan (Eds.), Mathematical modeling (ICTMA12): Education, engineering and economics (pp. 110–119). Horwood Publishing Limited. Vyhledej v Google Scholar

[18] Kaiser, G., & Stender, P. (2013). Complex modeling problems in co-operative, self-directed learning environments. In G.A. Stillman, G. Kaiser, W. Blum, & J.P. Brown (Eds.), Teaching mathematical modeling: Connecting to research and practice (pp. 277–294). Springer. https://doi.org/10.1007/978-94-007-6540-5 23 Přejít na DOI Vyhledej v Google Scholar

[19] Kaput, J. J., & Blanton, M. (2001). Algebrafying the elementary mathematics experience: Part 1: Transforming task structures. In H. Chick, K. Stacey, J. Vincent, & J. Vincent (Eds.), Proceedings of the 12th International Commission on Mathematics Instruction Study Conference (pp. 344–351). University of Melbourne, Melbourne, Australia. Vyhledej v Google Scholar

[20] Lean, G., & Clements, M.A. (1981). Spatial ability, visual imagery, and mathematical performance. Educational Studies in Mathematics, 12(3), 267–299. https://doi.org/10.1007/BF00311060 Přejít na DOI Vyhledej v Google Scholar

[21] Lesh, R., & Doerr, H. M. (2003). Foundations of models and modeling perspective on mathematics teaching, learning, and problem solving. In R. Lesh, & H.M. Doerr (Eds.), Beyond constructivism: Models and modeling perspectives on mathematics problem solving, learning, and teaching (pp. 3–33). Erlbaum. Vyhledej v Google Scholar

[22] Lesh, R., Hoover, M., Hole, B., Kelly, A., & Post, T. (2000). Principles for developing thought-revealing activities for students and teachers. In R. Lesh, & A. Kelly (Eds.), Handbook of research design in mathematics and science education (pp. 591–645). Lawrence Erlbaum. Vyhledej v Google Scholar

[23] Lesh, R., & Lehrer, R. (2003). Models and modeling perspectives on the development of students and teachers. Mathematical Thinking and Learning, 5(2–3), 109–129. https://doi.org/10.1080/10986065.2003.9679996 Přejít na DOI Vyhledej v Google Scholar

[24] Lowrie, T., & Clements, M.A. (2001). Visual and nonvisual processes in grade 6 students’ mathematical problem solving. Journal of Research in Childhood Education, 16(1), 77–93. https://doi.org/10.1080/02568540109594976 Přejít na DOI Vyhledej v Google Scholar

[25] Lowrie, T., & Kay, R. (2001). Relationship between visual and nonvisual solution methods and difficulty in elementary mathematics. The Journal of Educational Research, 94(4), 248–255. https://doi.org/10.1080/00220670109598758 Přejít na DOI Vyhledej v Google Scholar

[26] Maaß, K. (2006). What are modeling competencies? ZDM, 38(2), 113–142. https://doi.org/10.1007/BF02655885 Monga, A. B., & John, D.R. (2007). Cultural differences in brand extension evaluation: the influence of analytic versus holistic thinking. Journal of Consumer Research, 33(4), 529–536. https://doi.org/10.1086/510227 Přejít na DOI Vyhledej v Google Scholar

[27] Niss, M., Blum, W., & Galbraith, P. (2007). Introduction. In W. Blum, P. L. Galbraith, H.-W. Henn, & M. Niss (Eds.), Modeling and applications in mathematics education (pp. 3–32). Springer. Vyhledej v Google Scholar

[28] Organization for Economic Cooperation and Development (OECD). (2004). Learning for tomorrow’s world: First results from PISA 2003. OECD. Vyhledej v Google Scholar

[29] Presmeg, N. C. (1986). Visualization in high school mathematics. For the Learning of Mathematics, 6(3), 42–46. Vyhledej v Google Scholar

[30] Shahbari, J.A., & Daher, W. (2016). Mathematical models’ features: Technology and non-technology. European Journal of Science and Mathematics Education, 4(4), 523–533. Vyhledej v Google Scholar

[31] Shahbari, J.A., & Peled, I. (2017). Modeling in primary schools: Constructing a conceptual system and making sense of fractions. International Journal of Science and Mathematics Education, 15(2), 371–391. https://doi.org/10.1007/s10763-015-9702-x Přejít na DOI Vyhledej v Google Scholar

[32] Shahbari, J.A., & Tabach, M. (2016). Different generality levels in the product of a modeling activity. In C. Csikos, A. Rausch, & J. Szitanyi (Eds.), Proceedings of the 40th Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 179–186). PME. Vyhledej v Google Scholar

[33] Shahbari, J. A., & Tabach, M. (2018). Developing prospective mathematics teachers’ knowledge of the modeling approach. Scientia in educatione, 9(2), 46–158. https://doi.org/10.14712/18047106.1183 Přejít na DOI Vyhledej v Google Scholar

[34] Shahbari, J.A., & Tabach, M. (2019). Adopting the modeling cycle for representing prospective and practicing teachers’ interpretations of students’ modeling activities. In G. Stillman, & J. Brown (Eds.), Lines of inquiry in mathematical modeling research in education (pp. 179–196). ICME-13 Monographs, Springer. https://doi.org/10.1007/978-3-030-14931-4 10 Přejít na DOI Vyhledej v Google Scholar

[35] Sternberg, R. J. (1997). Thinking styles. Cambridge University Press. Vyhledej v Google Scholar

[36] Sternberg, R. J., & Zhang, L.–F. (2005). Styles of thinking as a basis of differentiated instruction. Theory into Practice, 44(3), 245–253. https://doi.org/10.1207/s15430421tip4403 9 Přejít na DOI Vyhledej v Google Scholar

[37] Stillman, G., Galbraith, P., Brown, J., & Edwards, I. (2007). A framework for success in implementing mathematical modeling in the secondary school. In J. Watson, & K. Beswick (Eds.), Proceedings of the 30th Annual Conference of the Mathematics Education Research Group of Australasia (MERGA) (Vol. 2, pp. 688–697). MERGA. Vyhledej v Google Scholar

Scientia in educatione

Styly myšlení v matematice a charakteristika procesu modelování

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