Mathematical Thinking Styles and the features of Modeling Process

Juhaina Awawdeh Shahbari

doi:10.14712/18047106.1579

Vol. 11 No. 1 (2020): sciED Research Reports

Mathematical Thinking Styles and the features of Modeling Process

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

College of Sakhnin for Teacher Education: Sakhnin

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

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Shahbari, J. A. (2020). Mathematical Thinking Styles and the features of Modeling Process. Scientia in Educatione, 11(1), 59-68. https://doi.org/10.14712/18047106.1579

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Abstract

The current study investigated the relationship between students’ mathematical thinking style and their modeling processes and routes. Thirty-five eighth-grade students were examined. In the first stage, the students solved questions, and according to their solutions, they were assigned to one of two thinking style groups: visual and analytic. The two groups engaged in three modeling activities. Findings indicated differences in the groups’ modeling processes in performing the three activities. The primary differences in the modeling processes were manifested in simplifying, mathematizing, and eliciting a mathematical model. In addition, the analytic thinking group skipped the real-model phase in the three activities, while the visual group built a real model for each activity.

References

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. Search in Google Scholar
Blum, W., & Borromeo Ferri, R. (2009). Mathematical modeling: Can it be taught and learnt? Journal of Mathematical Modeling and Application, 1(1), 45–58. Search in 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. Search in 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 Open DOI Search in 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. Search in 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 Open DOI Search in 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 Open DOI Search in 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. Search in 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 Open DOI Search in 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 Open DOI Search in 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 Search in 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. Search in 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. Search in 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 Open DOI Search in 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. Search in Google Scholar
Glaser, B.G., & Strauss, A.L. (1967). Discovery of grounded theory: Strategies for qualitative research. Aldine Publishing Company. Search in 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. Search in 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 Open DOI Search in 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. Search in 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 Open DOI Search in 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. Search in 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. Search in 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 Open DOI Search in 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 Open DOI Search in 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 Open DOI Search in 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 Open DOI Search in 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. Search in Google Scholar
Organization for Economic Cooperation and Development (OECD). (2004). Learning for tomorrow’s world: First results from PISA 2003. OECD. Search in Google Scholar
Presmeg, N. C. (1986). Visualization in high school mathematics. For the Learning of Mathematics, 6(3), 42–46. Search in 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. Search in 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 Open DOI Search in 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. Search in 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 Open DOI Search in 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 Open DOI Search in Google Scholar
Sternberg, R. J. (1997). Thinking styles. Cambridge University Press. Search in 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 Open DOI Search in 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. Search in 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. Search in 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. Search in 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. Search in 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 Open DOI Search in 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. Search in 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 Open DOI Search in 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 Open DOI Search in 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. Search in 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 Open DOI Search in 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 Open DOI Search in 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 Search in 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. Search in 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. Search in 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 Open DOI Search in 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. Search in Google Scholar

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

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

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

[35] Sternberg, R. J. (1997). Thinking styles. Cambridge University Press. Search in 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 Open DOI Search in 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. Search in Google Scholar

Scientia in educatione

Mathematical Thinking Styles and the features of Modeling Process

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