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Communication: The Essential Difference Between Mathematical Modeling and Problem Solving

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Modeling Students' Mathematical Modeling Competencies

Abstract

In this chapter, I discuss the formulation of tasks used as a communicative tool for developing someone’s mathematical modeling competency and mathematical problem solving competency. These two competencies are characterized and their different crux is highlighted. This is exemplified by the formulation of different kind of tasks, and two hypotheses are offered for further debate and investigation concerning the kind of tasks that dominate in mathematics education and why.

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References

  • Blomh0j, M., and Jensen, T. H. (2003). Developing mathematical modeling competence: Conceptual clarification and educational planning. Teaching Mathematics and Its Applications, 22, 123-139.

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  • Blomh0j, M., and 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 modeling. In W. Blum, P. Galbraith, H. Henn, and M. Niss (Eds.), Applications and Modeling in Mathematics Education: The 14th IMCI Study (pp. 45-56). New York: Springer.

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  • Haines, C., Galbraith, P., Blum, W., and Khan, S. (Eds.) (2007). Mathematical Modeling (ICTMA 12): Education, Engineering and Economics. Chichester, UK: Horwood.

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  • Jensen, T. H., Larsen, L. H., Pedersen, B. B., and Sonne, H. (2002). Matematrix 9. Copenhagen, Denmark: Alinea. Mathematics textbook for grade 9.

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  • Jensen, T. H. (2007a). Udvikling af matematisk modelleringskompetence som matematikundervisningens omdrejningspunkt – hvorfor ikke? (Developing mathematical modeling competency as the hub of mathematics education – why not?). Textsfrom IMFUFA 458. Roskilde University, Denmark: IMFUFA. Ph.d. Dissertation (in Danish with English summary). To be ordered from imfufa@ruc.dk.

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  • Jensen, T. H. (2007b). Assessing mathematical modeling competency. In C. Haines, P. Galbraith, W. Blum, and S. Khan (Eds.) (2007). Mathematical Modeling (ICTMA 12): Education, Engineering and Economics (pp. 141-148). Chichester, UK: Horwood.

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  • Niss, M., and Jensen, T. H. (Eds.) (2002). Kompetencer og matematiktering – Ideer og inspiration til udvikling af matematikundervisning i Danmark (‘the KOM report’). Uddannelsesstyrelsens temahtzfteserie 18. Copenhagen, Denmark: The Ministry of Education.

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  • Niss, M., and Jensen, T. H. (Eds.) (to appear). Competencies and mathematical learning – Ideas and inspiration for the development of mathematics teaching and learning in Denmark. English translation of part I-VI of Niss and Jensen (2002). Under preparation for publication in the series Texts from IMFUFA. Roskilde University, Denmark: IMFUFA. To be ordered from imfufa@ruc.dk.

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Author information

Authors and Affiliations

  1. The Danish School of Education, Århus University, Århus, Denmark

    Tomas Højgaard

Authors
  1. Tomas Højgaard
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Correspondence to Tomas Højgaard .

Editor information

Editors and Affiliations

  1. School of Education, Indiana University, N. Rose Ave. 201, Bloomington, 47405-1006, Indiana, USA

    Richard Lesh

  2. Graduate School of Education, University of Queensland, Brisbane, 4074, Queensland, Australia

    Peter L. Galbraith

  3. Dept. Continuing Education, City University, Northampton Square, London, EC1V 0HB, United Kingdom

    Christopher R. Haines

  4. School of Education, University of Utah, Campus Center Drive 1705, Salt Lake City, 84112, Utah, USA

    Andrew Hurford

Appendix: Examples of Tasks Developed for and Used in Grades 9-12 with the Specific Aim of Promoting the Development of Mathematical Modeling Competency (Jensen, 2002, 2007a)

Appendix: Examples of Tasks Developed for and Used in Grades 9-12 with the Specific Aim of Promoting the Development of Mathematical Modeling Competency (Jensen, 2002, 2007a)

Invitations to develop mathematical modeling competency – long duration (2-4 weeks):

  1. 1.

    What is the relation between one’s income and the tax paid?

  2. 2.

    What is the cost of me?

  3. 3.

    Which means of transportation is the best?

  4. 4.

    How can one navigate?

  5. 5.

    Can one become slim by exercising?

  6. 6.

    How many windmills should Denmark have?

  7. 7.

    What is the best shape of a tin can?

Invitations to develop mathematical modeling competency – short duration (within a lesson):

  1. 8.

    How much fabric does one need to make a cloth for the dinner table?

  2. 9.

    How many times can one brush one’s teeth with a tube of toothpaste?

  3. 10.

    Draw a sketch of a 135 m2 house.

  4. 11.

    How far away is the horizon?

  5. 12.

    How far ahead must the road be clear for you to make a safe overtaking?

  6. 13.

    At what angle of incline does a tower topple?

  7. 14.

    What are the maximum sizes of a board if one is to turn a corner?

Invitations to develop mathematization competency – short duration (within a lesson):

  1. 15.

    How does the tax one pays depend on the income tax and the VAT?

  2. 16.

    When you buy something, is it better to get a percentage of the price in discount before or after the VAT has been added?

  3. 17.

    Which savings account do you prefer: The one that pays 8 % in annual interest or the one that pays DKK 110 in annual interest?

  4. 18.

    A theater increases the ticket price by 30 %, which causes the income from the sale of tickets to go up by 17 %. By how many percentages has the size of the audience changed?

  5. 19.

    Between three cities of the same size, where should the only high school in the area be?

  6. 20.

    A liqueur glass is cone-shaped. What height of the liqueur served in the glass makes it halfway full?

  7. 21.

    An enclosure must have the shape of a rectangle with a semicircle at one end. How much land can you enclose with a given length of fence?

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Højgaard, T. (2013). Communication: The Essential Difference Between Mathematical Modeling and Problem Solving. In: Lesh, R., Galbraith, P., Haines, C., Hurford, A. (eds) Modeling Students' Mathematical Modeling Competencies. International Perspectives on the Teaching and Learning of Mathematical Modelling. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-6271-8_22

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