Identifying some key characteristics of an integrated approach to teaching modelling
DOI:
https://doi.org/10.48489/quadrante.23595Keywords:
modelling, mathematical knowledge construction, integrated approach, curriculum, teachingAbstract
Proponents of an integrated approach to teaching mathematical modelling recognize that mathematical modelling and applications must be integrated into and contribute to elementary and secondary students’ overall mathematical education. This study seeks to identify the chief characteristics of an integrated approach to teaching modelling by examining a selection of papers published as part of the proceedings of the International Conferences on the Teaching of Mathematical Modelling and Applications beginning in the 1990s, particularly the ICMI 14 study. Two perspectives are identified, each underpinned by a different purpose. One purpose is to solve a real-world problem, where the direction is from the real world to a mathematical world. The second purpose is to deepen students’ understanding of developed representations in a mathematical world, where the direction is from a mathematical world to the real world. This study discusses the rationale for an integrated approach to teaching modelling under four main headings: its significance, key ideas for developing an appropriate primary and junior secondary curriculum, implications for classroom teaching, and the need for further research.
References
Blomhøj, M. (2019). Towards integration of modelling in secondary mathematics teaching. In G. Stillman & J. Brown (Eds.) Lines of inquiry in mathematical modelling research in education (pp. 37-52). Cham: Springer. https://doi.org/10.1007/978-3-030-14931-4_3
Blomhøj, M., & Kjeldsen, T. H. (2013). Students’ mathematical learning in modelling activities. In G. A. Stillman, G. Kaiser, W. Blum, & J. P. Brown (Eds.), Teaching mathematical modelling: Connecting to research and practice (pp. 141-151). Dordrecht: Springer. https://doi.org/10.1007/978-94-007-6540-5_12
Blum, W. (1991). Applications and modelling in mathematics teaching – A review of arguments and instructional aspects. In M. Niss, W. Blum, & I. Huntley (Eds.), Teaching of mathematical modelling and applications (pp. 10-29). Chichester: Ellis Horwood.
Blum, W. (1993). Mathematical modelling in mathematics education and instruction. In T. Breiteig, I. Huntley, & G. Kaiser-Messmer (Eds.), Teaching and learning mathematics in context (pp. 3-14). Chichester: Ellis Horwood.
Blum, W. (1998). On the role of “grundvorstellungen” for reality-related proofs – Examples and reflections. In P. Galbraith, W. Blum, G. Booker, & I. Huntley (Eds.), Mathematical modelling, teaching and assessment in a technology-rich world (pp. 63-74). Chichester: Horwood Publishing.
Blum, W. (2015). Quality teaching of mathematical modelling: What do we know, what can we do? In S. J. Cho (Ed.), The Proceedings of the 12th International Congress on Mathematical Education - Intellectual and Attitudinal Challenges (pp. 73-96). New York: Springer. https://doi.org/10.1007/978-3-319-12688-3_9
Blum, W., & Niss, M. (1989). Mathematical problem solving, modelling, applications, and links to other subjects – State, trends and issues in mathematics instruction. In W. Blum, M. Niss, & I. Huntley (Eds.), Modelling, applications and applied problem solving (pp. 1-21). Chichester: Horwood Publishing.
Borromeo Ferri, R., & Lesh, R. (2013). Should interpretation systems be considered to be models if they only function implicitly? In G. A. Stillman, G. Kaiser, W. Blum, & J. P. Brown (Eds.), Teaching mathematical modelling: Connecting to research and practice (pp. 57-66). Dordrecht: Springer. https://doi.org/10.1007/978-94-007-6540-5
Carreira, S. (2001). The mountain in the utility – On the metaphorical nature of mathematical models. In J. Matos, W. Blum, S. Houston, & S. Carreira (Eds.), Modelling and mathematics education: ICTMA 9: Applications in science and technology (pp. 15-29). Chichester: Horwood Publishing.
Confrey, J. (2007). Epistemology and modelling – Overview. In W. Blum, P. Galbraith, H.-W. Henn, & M. Niss (Eds.), Modelling and applications in mathematics education, The 14th ICMI Study (pp. 125-128). New York: Springer.
Confrey, J., & Maloney, A. (2007). A theory of mathematical modelling in technological settings. In W. Blum, P. Galbraith, H.-W. Henn, & M. Niss (Eds.), Modelling and applications in mathematics education, The 14th ICMI Study (pp. 57-68). New York: Springer.
English, L. (2003). Mathematical modelling with young learners. In S. Lamon, W. Parker, & K. Houston (Eds.), Mathematical modelling: A way of life, ICTMA 11 (pp. 3-17). Chichester: Horwood Publishing.
Freudenthal, H. (1973). Mathematics as an educational task. Dordrecht: Reidel. https://doi.org/10.1007/978-94-010-2903-2
Freudenthal, H. (1983). Didactical phenomenology of mathematics structures. Dordrecht: Reidel. https://doi.org/10.1007/0-306-47235-X
Freudenthal, H. (1991). Revisiting mathematics education. Dordrecht: Kluwer Academic Publishers. https://doi.org/10.1007/0-306-47202-3
Garfunkel, S. (1993). Future of mathematical modelling in the classroom. In T. Breiteig, I. Huntley, & G. Kaiser-Messmer (Eds.), Teaching and learning mathematics in context (pp. 241-249). Chichester: Ellis Horwood.
Gravemeijer, K. (1993). Modelling two-digit addition and subtraction with an empty line. In T. Breiteig, I. Huntley, & G. Kaiser-Messmer (Eds.), Teaching and learning mathematics in context (pp. 51-61). Chichester: Ellis Horwood.
Gravemeijer, K. (2007). Emergent modelling as a precursor to mathematical modelling. In W. Blum, P. Galbraith, H.-W. Henn, & M. Niss (Eds.), Modelling and applications in mathematics education, The 14th ICMI Study (pp. 137-144). New York: Springer.
Hanna, G. (2003). Using ideas from physics in teaching mathematical proof. In Q.-X. Ye, W. Blum, K. Houston, & Q.-Y. Jiang (Eds.), Mathematical modelling in education and culture: ICTMA 10 (pp. 31-40). Chichester: Horwood Publishing.
Hanna, G., & Janke, H. N. (2007). Proving and modelling. In W. Blum, P. Galbraith, H.-W. Henn, & M. Niss (Eds.), Modelling and applications in mathematics education, The 14th ICMI Study (pp. 145-152). New York: Springer.
Hesse, M. B. (1966). Models and analogies in science. Notre Dame, IN: University of Notre Dame Press.
Ikeda, T., & Stephens, M. (2011). Making connections between modelling and constructing mathe-matical knowledge: An historical perspective. In G. Kaiser, W. Blum, R. B. Ferri, & G. Stillman (Eds.), Trends in teaching and learning of mathematical modelling, ICTMA14 (pp. 669-678). New York: Springer. https://doi.org/10.1007/978-94-007-0910-2_64
Ikeda, T., & Stephens, M. (2015). Reconsidering the roles and characteristics of models in mathematics education. In G. A. Stillman, W. Blum, & M. S. Biembengut (Eds.), Mathematical modelling in education research and practice - Cultural, social and cognitive influences (pp. 351-361). Switzerland: Springer. https://doi.org/10.1007/978-3-319-18272-8_29
Ikeda, T., & Stephens, M. (2017). Modelling as interactive translations among plural worlds – Experimental teaching using the night-time problem. In G. A. Stillman, W. Blum, & G. Kaiser (Eds.), Mathematical modelling and applications – Crossing and researching boundaries in mathematics education (pp. 349-358). Switzerland: Springer.
Ikeda, T., & Stephens, M. (2020). Using a mathematical modelling activity to assist students to make sense of a limit theorem in trigonometry. In G. A. Stillman, G. Kaiser, & C. E. Lampen (Eds.), Mathematical Modelling Education and sense-making (pp. 287-298). Switzerland: Springer. https://doi.org/10.1007/978-3-030-37673-4_25
Julie, C., & Mudaly, V. (2007). Mathematical modelling of social issues in school mathematics in South Africa. In W. Blum, P. Galbraith, H.-W. Henn, & M. Niss (Eds.), Modelling and applications in mathematics education, The 14th ICMI Study (pp. 503-510). New York: Springer.
Kaiser, G. (2015). Modelling competencies: Past development and further perspectives. In G. A. Stillman, W. Blum, & M. S. Biembengut (Eds.), Mathematical modelling in education research and practice – Cultural, social and cognitive influences (pp. 351-361). Switzerland: Springer.
Lakoff, G., & Johnson, M. (1980). Metaphors we live by. London: The University of Chicago Press.
Lamon, S. J. (1998). Algebra, modelling, and achievement. In P. Galbraith, W. Blum, G. Booker, & I. Huntley (Eds.), Mathematical modelling, teaching and assessment in a technology-rich world (pp. 307-315). Chichester: Horwood Publishing.
Lehrer, R., & Schauble, L. (2007). A developmental approach for supporting the epistemology of modelling. In W. Blum, P. Galbraith, H.-W. Henn, & M. Niss (Eds.), Modelling and applications in mathematics education, The 14th ICMI Study (pp. 153-160). New York: Springer.
Lesh, R. A. (2003). How mathematizing reality is different from realizing mathematics. In S. Lamon, W. Parker, & K. Houston (Eds.), Mathematical modelling: A way of life, ICTMA 11 (pp. 37-52). Chichester: Horwood Publishing.
Lesh, R., & Yoon, C. (2007). What is distinctive in (our views about) model and modelling perspectives on mathematics problem solving, learning, and teaching? In W. Blum, P. Galbraith, H.-W. Henn, & M. Niss (Eds.), Modelling and applications in mathematics education, The 14th ICMI Study (pp. 161-170). New York, NY: Springer.
Matos, J. F. (1998). Mathematics learning and modelling: Theory and practice. In P. Galbraith, W. Blum, G. Booker, & I. Huntley (Eds.), Mathematical modelling, teaching and assessment in a technology-rich world (pp. 20-27). Chichester: Horwood Publishing.
Matos, J. F., & Carreira, S. (1997). The Quest for meaning in students’ mathematical modelling activity. In S. Houston, W. Blum, I. Huntley, & N. Neill (Eds.), Teaching and learning mathematical modelling (pp. 63-75). Chichester: Albion Publishing.
Niss, M. (2013). Modelling a crucial aspect of students’ mathematical modelling. In R. Lesh, P. L. Galbraith, C. R. Haines, & A. Hurford (Eds.), Modelling students’ mathematical modelling compe-tencies: ICTMA13 (pp. 43-59). Dordrecht: Springer.
Niss, M., Blum, W., & Galbraith, P. (2007). Introduction. In W. Blum, P. Galbraith, H.-W. Henn, & M. Niss (Eds.). Modelling and applications in mathematics education, The 14th ICMI Study (pp. 161-170). New York: Springer.
Ponte, J. P. (1993). Necessary research in mathematical modelling and applications. In T. Breiteig, I. Huntley, & G. Kaiser-Messmer (Eds.), Teaching and learning mathematics in context (pp. 219-227). Chichester: Ellis Horwood.
Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and ob¬jects as different sides of the same coin. Educational Studies in Mathematics, 22, 1-36. https://doi.org/10.1007/BF00302715
Swan, M. (1991). Mathematical modelling for all abilities. In M. Niss, W. Blum, & I. Huntley (Eds.), Teaching of mathematical modelling and applications (pp. 137-146). Chichester: Ellis Horwood.
Usiskin, Z. (1989). The sequencing of applications and modelling in the University of Chicago School Mathematics Project 7-12 curriculum. In W. Blum, J. S. Berry, & R. Biehler (Eds.), Applications and modelling in learning and teaching mathematics (pp. 176-181). Chichester: Horwood.
Usiskin, Z. (1991). Building mathematics curricula with applications and modelling. In M. Niss, W. Blum, & I. Huntley (Eds.), Teaching of mathematical modelling and applications (pp. 30-45). Chichester: Ellis Horwood.
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