Advertisement

Can Modelling Be Taught and Learnt? Some Answers from Empirical Research

  • Werner BlumEmail author
Conference paper
Part of the International Perspectives on the Teaching and Learning of Mathematical Modelling book series (IPTL, volume 1)

Abstract

This chapter deals with empirical findings on the teaching and learning of mathematical modelling, with a focus on grades 8–10, that is, 14–16-year-old students. The emphasis lies on the actual behaviour of students and teachers in learning environments with modelling tasks. Most examples in this chapter are taken from our own empirical investigations in the context of the project DISUM. In the first section, the terms used in this chapter are recollected from a cognitive point of view by means of examples, and reasons are summarised why modelling is an important and also demanding activity for students and teachers. In the second section, examples are given of students’ difficulties when solving modelling tasks, and some important findings concerning students dealing with modelling tasks are presented. The third section concentrates on teachers; examples of successful interventions are given, as well as some findings concerning teachers treating modelling examples in the classroom. In the fourth section, some implications for teaching modelling are summarised, and some encouraging (though not yet fully satisfying) results on the advancement of modelling competency are presented.

Keywords

Word Problem Modelling Task Solution Plan Modelling Competency Mathematical Competency 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. Abrantes, P. (1993). Project work in school mathematics. In J. De Lange et al. (Eds.), Innovation in maths education by modelling and applications (pp. 355–364). Chichester: Horwood.Google Scholar
  2. Aebli, H. (1985). Zwölf Grundformen des Lehrens. Stuttgart: Klett-Cotta.Google Scholar
  3. Alsina, C. (2007). Less chalk, less words, less symbols … More objects, more context, more actions. In W. Blum et al. (Eds.), Modelling and applications in mathematics education (pp. 35–44). New York: Springer.CrossRefGoogle Scholar
  4. Antonius, S., et al. (2007). Classroom activities and the teacher. In W. Blum et al. (Eds.), Modelling and applications in mathematics education (pp. 295–308). New York: Springer.CrossRefGoogle Scholar
  5. Baruk, S. (1985). L‘age du capitaine. De l‘erreur en mathematiques. Paris: Seuil.Google Scholar
  6. Blomhøj, M., & Jensen, T. H. (2007). What’s all the fuss about competencies? In W. Blum et al. (Eds.), Modelling and applications in mathematics education (pp. 45–56). New York: Springer.CrossRefGoogle Scholar
  7. Blum, W. (1998). On the role of “Grundvorstellungen” for reality-related proofs – Examples and reflections. In P. Galbraith et al. (Eds.), Mathematical modelling – Teaching and assessment in a technology-rich world (pp. 63–74). Chichester: Horwood.Google Scholar
  8. Blum, W., & Leiß, D. (2008). Investigating quality mathematics teaching – The DISUM project. In C. Bergsten et al. (Eds), Proceedings of MADIF-5, Malmö.Google Scholar
  9. Blum, W., & Leiß, D. (2006). Filling up – In the problem of independence-preserving teacher interventions in lessons with demanding modelling tasks. M. Bosch (Ed.), CERME-4–Proceedings of the Fourth Conference of the European Society for Research in Mathematics Education. Guixol.Google Scholar
  10. 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 economic (pp. 222–231). Chichester: Horwood.CrossRefGoogle Scholar
  11. Blum, W., & Niss, M. (1991). Applied mathematical problem solving, modelling, applications, and links to other subjects – State, trends and issues in mathematics instruction. Educational Studies in Mathematics, 22(1), 37–68.CrossRefGoogle Scholar
  12. Blum, W., et al. (2002). ICMI Study 14: applications and modelling in mathematics education – Discussion document. Educational Studies in Mathematics, 51(1/2), 149–171.CrossRefGoogle Scholar
  13. Borromeo Ferri, R. (2004). Mathematische Denkstile. Ergebnisse einer empirischen Studie. Hildesheim: Franzbecker.Google Scholar
  14. Borromeo Ferri, R. (2006). Theoretical and empirical differentiations of phases in the modelling process. Zentralblatt für Didaktik der Mathematik, 38(2), 86–95.CrossRefGoogle Scholar
  15. Borromeo Ferri, R. (2007). Modelling problems from a cognitive perspective. In C. Haines et al. (Eds.), Mathematical modelling: education, engineering and economics (pp. 260–270). Chichester: Horwood.Google Scholar
  16. Borromeo Ferri, R., & Blum, W. (2010). Insights into teachers’ unconscious behaviour in modeling contexts. In R. Lesh et al. (Eds.), Modeling students’ mathematical modeling competencies (pp. 423–432). New York: Springer.Google Scholar
  17. Brown, J. S., Collins, A., & Duguid, P. (1989). Situated cognition and the culture of learning. Educational Researcher, 18, 32–42.Google Scholar
  18. Burghes, D. (1986). Mathematical modelling – Are we heading in the right direction? In J. Berry et al. (Eds.), Mathematical modelling methodology, models and micros (pp. 11–23). Chichester: Horwood.Google Scholar
  19. Burkhardt, H. (2004). Establishing modelling in the curriculum: Barriers and levers. In H. W. Henn & W. Blum (Eds.), ICMI Study 14: Applications and modelling in mathematics education pre-conference volume (pp. 53–58). Dortmund: University of Dortmund.Google Scholar
  20. Burkhardt, H. (2006). Functional mathematics and teaching modelling. In C. Haines et al. (Eds.), Mathematical modelling: Education, engineering and economics (pp. 177–186). Chichester: Horwood.Google Scholar
  21. Burkhardt, H., & Pollak, H. O. (2006). Modelling in mathematics classrooms: Reflections on past developments and the future. Zentralblatt für Didaktik der Mathematik, 38(2), 178–195.CrossRefGoogle Scholar
  22. DaPonte, J. P. (1993). Necessary research in mathematical modelling and applications. In T. Breiteig et al. (Eds.), Teaching and learning mathematics in context (pp. 219–227). Chichester: Horwoood.Google Scholar
  23. De Corte, E., Greer, B., & Verschaffel, L. (1996). Mathematics teaching and learning. In D. C. Berliner & R. C. Calfee (Eds.), Handbook of educational psychology (pp. 491–549). New York: Macmillan.Google Scholar
  24. DeLange, J. (1987). Mathematics, insight and meaning. Utrecht: CD-Press.Google Scholar
  25. Doerr, H. (2007). What knowledge do teachers need for teaching mathematics through applications and modelling? In W. Blum et al. (Eds.), Modelling and applications in mathematics education (pp. 69–78). New York: Springer.CrossRefGoogle Scholar
  26. Freudenthal, H. (1973). Mathematics as an educational task. Dordrecht: Reidel.Google Scholar
  27. Galbraith, P., & Clathworthy, N. (1990). Beyond standard models – Meeting the challenge of modelling. Educational Studies in Mathematics, 21(2), 137–163.CrossRefGoogle Scholar
  28. Galbraith, P., & Stillman, G. (2006). A framework for identifying student blockages during transitions in the modelling process. Zentralblatt für Didaktik der Mathematik, 38(2), 143–162.CrossRefGoogle Scholar
  29. Haines, C., & Crouch, R. (2001). Recognizing constructs within mathematical modelling. Teaching Mathematics and Its Applications, 20(3), 129–138.CrossRefGoogle Scholar
  30. Henn, H.-W. (2007). Modelling pedagogy – Overview. In W. Blum et al. (Eds.), Modelling and applications in mathematics education (pp. 321–324). New York: Springer.CrossRefGoogle Scholar
  31. Hiebert, J., & Carpenter, T. P. (1992). Learning and teaching with understanding. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 65–97). New York: Macmillan.Google Scholar
  32. Hofe, R. V. (1998). On the generation of basic ideas and individual images: Normative, descriptive and constructive aspects. In J. Kilpatrick & A. Sierpinska (Eds.), Mathematics education as a research domain: A search for identity (pp. 317–331). Dordrecht: Kluwer.Google Scholar
  33. Houston, K. (2007). Assessing the “phases” of mathematical modelling. In W. Blum et al. (Eds.), Modelling and applications in mathematics education (pp. 249–256). New York: Springer.CrossRefGoogle Scholar
  34. Ikeda, T. (2007). Possibilities for, and obstacles to teaching applications and modelling in the lower secondary levels. In W. Blum et al. (Eds.), Modelling and applications in mathematics education (pp. 457–462). New York: Springer.CrossRefGoogle Scholar
  35. Jensen, T. H. (2007). Assessing mathematical modelling competencies. In C. Haines et al. (Eds.), Mathematical modelling: Education, engineering and economics (pp. 141–148). Chichester: Horwood.Google Scholar
  36. Kaiser, G. (2007). Modelling and modelling competencies in school. In C. Haines et al. (Eds.), Mathematical modelling: Education, engineering and economics (pp. 110–119). Chichester: Horwood.Google Scholar
  37. Kaiser, G., & Maaß, K. (2007). Modelling in lower secondary mathematics classroom – Problems and opportunities. In W. Blum et al. (Eds.), Modelling and applications in mathematics education (pp. 99–108). New York: Springer.CrossRefGoogle Scholar
  38. Kaiser, G., & Schwarz, B. (2006). Mathematical modelling as bridge between school and university. Zentralblatt für Didaktik der Mathematik, 38(2), 196–208.CrossRefGoogle Scholar
  39. Kaiser, G., & Schwarz, B. (2010). Authentic modelling problems in mathematics education – Examples and experiences. Journal für Mathematik-Didaktik, 31, 51–76.CrossRefGoogle Scholar
  40. Kaiser, G., Blomhøj, M., & Sriraman, B. (2006). Mathematical modelling and applications: Empirical and theoretical perspectives. Zentralblatt für Didaktik der Mathematik, 38(2), 178–195.CrossRefGoogle Scholar
  41. Kaiser-Messmer, G. (1987). Application-oriented mathematics teaching. W. Blum et al. (Eds.), Applications and modelling in learning and teaching mathematics (pp. 66–72). Chichester: Horwood.Google Scholar
  42. Kintsch, W., & Greeno, J. (1985). Understanding word arithmetic problems. Psychological Review, 92(1), 109–129.CrossRefGoogle Scholar
  43. Krainer, K. (1993). Powerful tasks: A contribution to a high level of acting and reflecting in mathematics instruction. Educational Studies in Mathematics, 24, 65–93.CrossRefGoogle Scholar
  44. Kramarski, B., Mevarech, Z. R., & Arami, V. (2002). The effects of metacognitive instruction on solving mathematical authentic tasks. Educational Studies in Mathematics, 49(2), 225–250.CrossRefGoogle Scholar
  45. Krauss, S., Baumert, J., & Blum, W. (2008). Secondary mathematics teachers’ pedagogical content knowledge and content knowledge: Validation of the COACTIV constructs. Zentralblatt für Didaktik der Mathematik, 40(5), S 873–892.CrossRefGoogle Scholar
  46. Leikin, R., & Levav-Waynberg, A. (2007). Exploring mathematics teacher knowledge to explain the gap between theory-based recommendations and school practice in the use of connecting tasks. Educational Studies in Mathematics, 66, 349–371.CrossRefGoogle Scholar
  47. Leiß, D. (2007). Lehrerinterventionen im selbständigkeitsorientierten Prozess der Lösung einer mathematischen Modellierungsaufgabe. Hildesheim: Franzbecker.Google Scholar
  48. Lesh, R. A., & Doerr, H. M. (2003). Beyond constructivism: A models and modelling perspective on teaching, learning, and problem solving in mathematics education. Mahwah: Lawrence Erlbaum.Google Scholar
  49. Lingefjaerd, T. (2007). Modelling in teacher education. In W. Blum et al. (Eds.), Modelling and applications in mathematics education (pp. 475–482). New York: Springer.CrossRefGoogle Scholar
  50. Lipowsky, F. (2006). Auf den Lehrer kommt es an. Zeitschrift für Pädagogik, 51. Beiheft. Weinheim: Beltz, 47–70.Google Scholar
  51. Maaß, K. (2006). What are modelling competencies? Zentralblatt für Didaktik der Mathematik, 38(2), 113–142.CrossRefGoogle Scholar
  52. Maaß, K. (2007). Modelling in class: What do we want the students to learn? In C. Haines et al. (Eds.), Mathematical modelling: Education, engineering and economics (pp. 63–78). Chichester: Horwood.Google Scholar
  53. Matos, J. F., & Carreira, S. (1997). The quest for meaning in students’ mathematical modelling activity. In S. K. Houston et al. (Eds.), Teaching & leaning mathematical modelling (pp. 63–75). Chichester: Horwood.Google Scholar
  54. Niss, M. (Ed.). (1993). Investigations into assessment in mathematics education. Dordrecht: Kluwer.Google Scholar
  55. Niss, M. (1996). Goals of mathematics teaching. In A. Bishop et al. (Eds.), International handbook of mathematical education (pp. 11–47). Dordrecht: Kluwer.Google Scholar
  56. Niss, M. (1999). Aspects of the nature and state of research in mathematics education. Educational Studies in Mathematics, 40, 1–24.CrossRefGoogle Scholar
  57. Niss, M. (2001). Issues and problems of research on the teaching and learning of applications and modelling. In J. F. Matos et al. (Eds.), Modelling and mathematics education: ICTMA-9 (pp. 72–88). Chichester: Ellis Horwood.Google Scholar
  58. Niss, M. (2003). Mathematical competencies and the learning of mathematics: the Danish KOM project. In A. Gagatsis & S. Papastavridis (Eds.), 3rd Mediterranean conference on mathematical education (pp. 115–124). Athens: The Hellenic Mathematical Society.Google Scholar
  59. Niss, M., Blum, W., & Galbraith, P. (2007). Introduction. In W. Blum et al. (Eds.), Modelling and applications in mathematics education (pp. 3–32). New York: Springer.CrossRefGoogle Scholar
  60. OECD (2005). PISA 2003 Technical Report. Paris: OECD.Google Scholar
  61. OECD. (2007). PISA 2006 – Science competencies for tomorrow’s world (Vol. 1&2). Paris: OECD.Google Scholar
  62. Palm, T. (2007). Features and impact of the authenticity of applied mathematical school tasks. In W. Blum et al. (Eds.), Modelling and applications in mathematics education (pp. 201–208). New York: Springer.CrossRefGoogle Scholar
  63. Pauli, C., & Reusser, K. (2000). Zur Rolle der Lehrperson beim kooperativen Lernen. Schweizerische Zeitschrift für Bildungswissenschaften, 3, 421–441.Google Scholar
  64. Pollak, H. O. (1979). The interaction between mathematics and other school subjects. In UNESCO (Ed.), New Trends in Mathematics Teaching IV (pp. 232–248). UNESCO: Paris.Google Scholar
  65. Polya, G. (1957). How to solve it. Princeton: Princeton University Press.Google Scholar
  66. Schoenfeld, A. H. (1988). When good teaching leads to bad results: The disasters of “well-taught” mathematics courses. Educational Psychologist, 23, 145–166.CrossRefGoogle Scholar
  67. Schoenfeld, A. H. (1994). Mathematical thinking and problem solving. Hillsdale: Erlbaum.Google Scholar
  68. Staub, F. C., & Reusser, K. (1995). The role of presentational structures in understanding and solving mathematical word problems. In C. A. Weaver, S. Mannes, & C. R. Fletcher (Eds.), Discourse comprehension. Essays in honor of Walter Kintsch (pp. 285–305). Hillsdale: Lawrence Erlbaum.Google Scholar
  69. Stillman, G., & Galbraith, P. (1998). Applying mathematics with real world connections: Metacognitive characteristic of secondary students. Educational Studies in Mathematics, 36(2), 157–195.CrossRefGoogle Scholar
  70. Tanner, H., & Jones, S. (1993). Developing metacognition through peer and self assessment. In T. Breiteig et al. (Eds.), Teaching and learning mathematics in context (pp. 228–240). Chichester: Horwoood.Google Scholar
  71. Turner, R. et al. (in press). Using mathematical competencies to predict item difficulty in PISA: A MEG study 2003–2009. To appear in: Proceedings of the PISA Research Conference, Kiel, 2009.Google Scholar
  72. Verschaffel, L., Greer, B., & DeCorte, E. (2000). Making sense of word problems. Lisse: Swets&Zeitlinger.Google Scholar
  73. Vos, P. (2007). Assessment of applied mathematics and modelling: Using a laboratory-like environment. In W. Blum et al. (Eds.), Modelling and applications in mathematics education (pp. 441–448). New York: Springer.CrossRefGoogle Scholar
  74. Zöttl, L., Ufer, S., & Reiss, K. (this volume). Assessing modelling competencies using a multidimensional IRT approach.Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of KasselKasselGermany

Personalised recommendations