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:

1. One starts with a real world problem which might be clearly formulated or might first need further clarification
2. Then one sets up a real model by identifying the objects that are of interest and their relationships (simplification)
3. This has to be translated to a mathematical model (mathematization)
4. Within the mathematical model a solution of the translated problem must be found
5. This solution is to be interpreted in real world terms
6. The solution and the overall process is validated in the real world (e.g. by comparison with real data obtained by observation or trials).

Having competence to carry out the modelling process can now be split up into having the sub-competencies for each part of the process.

Blomhoj and Jensen (2003) distinguish between a “holistic approach” which claims that all parts of a modelling process should always be present, and an “atomistic approach” which claims that mathematical education should concentrate on the processes of mathematization and analysis of models. They suggest that finding the right balance between these extreme positions is the real challenge.

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:

1. “meta-cognitive modelling competencies”, i.e. being aware of the overall process and the position and meaning of the sub-processes when doing modelling
2. Structuring real world problems and working goal-directed
3. Argumentation and documentation competencies
4. Knowledge of the potential of mathematical modelling for problem solution and positive attitude towards using this potential.

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:

• “degree of coverage”: which sub-competencies are covered?
• “radius of action”: in which contexts and situation can the competency be activated?
•  “technical level”: how advanced are the entities and tools used?

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.

Contributions at SEFI MWG Seminars:

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.

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, accessed 10 March 2010)

Further Literature:

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.

Blomhoj, M., 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 Study, New York: Springer, 45-56.

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 its Applications, London: Kogan Page, 158-174.

Kaiser, G. (2007). Modelling and Modelling Competencies in School. In Haines, Chr., Galbraith, P., Blum, W., Khan, S. (Eds.): Mathematical Modelling, Proc. ICTMA 12, Chichester: Horwood Publishing, 110-119.

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, Athens, Greece: Hellenic Mathematical Society and Cyprus Mathematical Society, 115-124.

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, Pacific Grove: Brooks/Cole-Thomson.

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