Modelling competencies and their teaching
Blomhoj and Jensen (2007, p.47) define a mathematical competency as “someone’s insightful readiness to act in response to a certain kind of mathematical challenge of a given situation” (see also Niss 2003). Correspondingly, they define mathematical modelling competency as “someone’s insightful readiness to carry through all parts of a mathematical modelling process”.
In order to elaborate the term further by identifying sub-competencies, one must therefore look at the parts of a modelling process. It seems to be agreed in the community of researchers on teaching mathematical modelling (cf. Maaß 2006, Kaiser 2007) that the modelling process can be split up into the following steps:
Having competence to carry out the modelling process can now be split up into having the sub-competencies for each part of the process.
Maaß (2006, p.139) provides an elaboration of the sub-competencies listed above according to Blum and Kaiser and she adds further competencies that are important for carrying out the overall process:
Niss (2003) (see also Blomhoj and Jensen 2007) identified three dimensions of the competence space where students can make progress in their overall modelling competency. The dimensions can also guide the teaching process when they serve to specify in more detail which kind of progress is intended in a certain learning scenario. The dimensions are:
Blomhoj and Jensen (2007) emphasize that there is no dichotomy between competence and incompetence but rather a continuum and the dimensions of Niss reflect this.
Engineers often work within already existing models (cf. Bissell and Dillon, 2000, for control engineers). They then need in particular the sub-competencies 4 and 5 stated above. This is also reflected in Niss (2003) who distinguishes between “active modelling” and dealing with existing models. But Gainsburg (2006) also observed the other phases of the modelling process when investigating the work of structural engineers. So, it is an interesting question for a certain branch of engineering or for certain job profiles within a branch what the balance between active modelling and work within existing models looks like. Workplace studies are required for this (cf. the respective item in the list of important topics).
Engineering education is full of mathematical modelling from the beginning. For example, engineering mechanics introduces the central modelling concepts like force and momentum (torque) and in statics, the sum of all forces and torques must equal 0 which leads to a system of equations. So, modelling concepts and standard models are already provided in the teaching of application subjects. It remains open, whether mathematics education should also engage in teaching mathematical modelling. And what is the difference between teaching mathematical modelling within an application subject and within mathematics, if any? This also needs further elaboration.
If mathematical modelling is to be integrated into the mathematical education of engineers, then this could be done within the regular lectures (e.g. using mini-projects) or within separate modelling courses (cf. Blomhoj and Jensen 2003). For both, the literature on modelling courses given below could be used.
Literature and Links
Kleiza, V., Purvinis, O. (2004). Teaching of Mathematics and Mathematical Modelling using Computer applications, Demlova, M., Lawson, D. (Eds.) Proc. 12th SEFI Maths Working Group Seminar in Vienna, Prague: Vydavatelstvi CVUT, 77-85.
(available as download:here, accessed 10 March 2010)
Klymchuk, S. et al. (2008). Increasing Engineering
Students’ Awareness to Environmental Issues through Innovative Teaching of
Mathematical Modelling, Proc. of the 14th SEFI MWG seminar joint
with IMA (eds. B. Alpers et al.), Loughborough 2008. (available as download: here,
here,accessed 10 March 2010)
Bissell, C., Dillon, C. (2000): Telling tales: Models, stories, and meanings. For the Learning of Mathematics 20, 3-11.
Blomhoj, M., Jensen, T.H. (2003). Developing mathematical modelling competence: Conceptual clarification and educational planning. Teaching Mathematics and its Applications 22(3), 123-139.
Jensen, T.H. (2007). What’s all the fuss about competencies? Experiences
with using a competence perspective on mathematics education to develop the
teaching of mathematical modelling. In Blum, W., Galbraith, P.L., Henn, H.-W.,
Niss, M. (Eds.) Modelling and Applications in Mathematics Education. 14th ICMI
Gainsburg, J. (2006). The mathematical modeling of structural engineers, Mathematical Thinking and Learning 8: 3-36.
Hibberd, S.: Mathematical Modelling Skills. In Kahn,
P., Kyle, J. (Eds.) Effective learning & teaching in Mathematics and
Kaiser, G. (2007). Modelling and Modelling Competencies
in School. In Haines, Chr., Galbraith, P., Blum, W., Khan, S. (Eds.): Mathematical
Modelling, Proc. ICTMA 12,
Maaß, K. (2006). What are modelling competencies? Zentralblatt für Didaktik der Mathematik (ZDM) 38(2), 113-142.
Niss, M. (2003). Mathematical competencies and the learning
of mathematics: The Danish KOM project. In Gagatsis, A., Papastavridis, S.
(Eds.), 3rd Mediterranean Conference on Mathematical Education,
General books on mathematical modelling and modelling courses:
Edwards, D., Hamson, M. (2001). Guide to Mathematical Modelling, 2nd Edition, Houndmills/Basingstoke: Palgrave.
Giordano, F.R., Weir, M.D., Fox, W.P.(2003). A First
Course in Mathematical Modeling, 3rd Edition,
An abundance of references and links can be found on the web site of ICTMA, the International Community of Teachers of Mathematical Modelling and Applications: http://www.ictma.net