Skip to main content
Log in

Fostering Transit between Real World and Mathematical World: Some Phases on the Modelling Cycle

  • Published:
International Journal of Science and Mathematics Education Aims and scope Submit manuscript


This study shows how, in the initial training of mathematics teachers, it is possible to promote processes of abstraction and mathematisation through modelling a real situation with the support of auxiliary material to mediate understanding. By adapting elements of the theoretical and methodological framework called Abstraction in Context (AiC), participants’ discussions while building a mathematical model—in a nested epistemic actions—are analysed. Two specific points are discussed in this paper. The first aims to identify how different types of knowledge emerge when an individual is faced with a modelling task. The second is regarding the use of auxiliary material as a means of metaphorising a situation. It was evidenced how the material favours the construction of a mathematical model through the simplification and idealisation that it brings. The meaning constructed for the model is supported in recognising a decreasing behaviour as a part of a whole.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others


  1. Institutional representations are those that are usually accepted and used by the teacher or that appear in text books. A non-institutional representation is one that is related to a personal representation (Hitt & González-Martín, 2015).

  2. The information presented here has been taken from


  • Blomhøj, M., & Jensen, T. H. (2006). What’s all the fuss about competencies? Experiences with using a competence perspective on mathematics education to develop the teaching of mathematical modelling. In W. Blum, P. L. Galbraith, & M. Niss (Eds.), Modelling and applications in mathematics education (pp. 45–56). New York, NY: Springer.

  • Blum, W., & Borromeo, R. (2009). Mathematical modelling: Can it be taught and learnt? Journal of Mathematical Modelling and Application, 1(1), 45–58.

    Google Scholar 

  • Borromeo, F. (2006). Theoretical and empirical differentiations of phases in the modelling process. Zentralblatt für Didaktik der Mathematik, 38(2), 86–95.

    Article  Google Scholar 

  • Busse, A. (2005). Individual ways of dealing with the context of realistic tasks – First steps towards a typology. Zentralblatt für Didaktik der Mathematik, 37(5), 354–360.

    Article  Google Scholar 

  • Carrejo, D., & Marshall, J. (2007). What is mathematical modelling? Exploring prospective teachers’ use of experiments to connect mathematics to the study of motion. Mathematics Education Research Journal, 19(1), 45–76.

    Article  Google Scholar 

  • Doerr, H. M., & Tripp, J. S. (2000). Understanding how students develop mathematical models. Mathematical Thinking and Learning, 1(3), 231–254.

    Article  Google Scholar 

  • Dreyfus, T., Hershkowitz, R., & Schwarz, B. B. (2014). The nested epistemic actions model for abstraction in context: Theory as methodological tool and methodological tool as theory. In A. Bikner-Ahsbahs, C. Knipping, & N. Presmeg (Eds.), Approaches to qualitative research in mathematics education: Examples of methodology and methods (pp. 185–217). Rotterdam, The Netherlands: Springer.

    Google Scholar 

  • English, L. (2009). Promoting interdisciplinarity through mathematical modeling. Mathematics Education, 41, 161–181.

    Google Scholar 

  • Halverscheid, S. (2008). Building a local conceptual framework for epistemic actions in a modelling environment with experiments. Mathematics Education, 40, 225–234.

    Google Scholar 

  • Hernández, R., Fernández, C., & Baptista, P. (2010). Metodología de la investigación [Research methodology]. Ciudad de México: McGraw-Hill.

  • Hitt, F. (2003). Le caractère fonctionnel des représentations [The functional nature of the representations]. Annales de Didactique et des Sciences Cognitives, 8, 255–271.

  • Hitt, F. (2006). Students’ functional representations and conceptions in the construction of mathematical concepts. An example: The concept of limit. Annales de Didactique et des Sciences Cognitives, 11, 253–268.

  • Hitt, F., & González-Martín, A. S. (2015). Covariation between variables in a modelling process: The ACODESA (collaborative learning, scientific debate and self-reflection) method. Educational Studies in Mathematics, 88, 201–219.

    Article  Google Scholar 

  • Justi, R., & Gilbert, J. (2002). Modelling, teachers’ views on the nature of modelling, and implications for the education of modellers. International Journal of Science Education, 24(4), 369–387.

    Article  Google Scholar 

  • Kaiser, G., & Schwarz, B. (2006). Mathematical modelling as bridge between school and university. Zentralblatt für Didaktik der Mathematik, 38(2), 196–208.

    Article  Google Scholar 

  • Kelly, C. (2006). Using manipulatives in mathematical problem solving: A performance-bases analysis. The Montana Mathematics Enthusiast, 3(2), 184–193.

    Google Scholar 

  • Lesh, R., & Fennewald, T. (2010). Introduction to part I modeling: What is it? Why do it? In R. Lesh, P. L. Galbraith, C. R. Haines, & A. Hurford (Eds.), Modeling students’ mathematical modeling competencies (pp. 5–10). Dordrecht, The Netherlands: Springer.

    Google Scholar 

  • Lesh, R., Doerr, H., Carmona, G., & Hjalmarson, M. (2003). Beyond constructivism. Mathematical Thinking and Learning, 5(2–3), 211–233.

    Article  Google Scholar 

  • Niss, M., Blum, W., & Galbraith, P. (2007). Introduction. In W. Blum, P. L. Galbraith, & M. Niss (Eds.), Modelling and applications in mathematics education (pp. 3–32). New York, NY: Springer.

  • Poon, C., Yeo, K., & Ahmad, N. (2012). Understand addition through modelling and manipulation of concrete materials. Journal of Education and Practice, 3(8), 55–66.

    Google Scholar 

  • Sandín, M. P. (2003). Investigación cualitativa en educación. Fundamentos y tradiciones [Qualitative research in education. Foundations and traditions]. Madrid, Spain: McGraw-Hill.

  • Sriraman, B., & Kaiser, G. (2006). Theory usage and theoretical trends in Europe: A survey and preliminary analysis of CERME4, research reports. Zentralblatt für Didaktik der Mathematik, 38(1), 22–51.

    Article  Google Scholar 

  • Tabach, M., Hershkowitz, R., Rasmussen, C., & Dreyfus, T. (2014). Knowledge shifts and knowledge agents in the classroom. Journal of Mathematical Behavior, 33, 192–208.

    Article  Google Scholar 

  • Thompson, P. (1994). Concrete materials and teaching for mathematical understanding. Arithmetic Teacher, 41(9), 556–558.

    Google Scholar 

  • Valles, M. (1997). Técnicas cualitativas de investigación social [Qualitative techniques of social research]. Madrid, Spain: Síntesis.

Download references


This work was supported by CONICYT/FONDECYT/POSDOCTORADO/No.3150317.

(Chile), FONDECYT REGULAR, No. 1151093 and PUCV DI 039.330/2016.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Carolina Guerrero-Ortiz.



The movement of marine waters is mainly influenced by oceanic currents2. Oceanic currents arise from differences in temperature and salinity between warmer and colder seas and mainly follow prevailing winds. Water, like the atmosphere, is subject to the Coriolis effect. Winds also have a significant effect on current direction and speed. This circular movement is warm when moving from tropical to polar latitudes, and cold from polar to tropical latitudes. Each branch, warm and cold, is considered a different current and has its own name. Oceanic currents move large amounts of energy from tropical zones to the poles, the movement of currents can be observed in:

The figure shows that currents are like rivers within the sea. The currents generally circulate along the east of the oceans towards lower latitudes. The most significant are Greenland, Labrador, Oyasio, Canaries, Benguela, East Australian, Humboldt and California.

The Humboldt Current runs along the coast of Peru and Chile and is characterised by a system of currents that run almost parallel to the coastline and for their vertical circulation and upwelling due to prevailing trade winds. Events such as El Niño, alter the wind patterns, rainfall and sea currents for months at a time, also affecting sea temperature and oxygen content. In addition to these oceano-climatic oscillations, the ecosystem is affected by decennial and secular variations, as revealed by paleo-climatic studies.

The exceptional productivity of the Humboldt Current, which covers less than 1% of the surface of the world’s oceans and provides more than 10% of fishing catches on the planet, is related to intensive coastal upwelling that bring nutrients to the surface. Cold waters rise along the coasts (around 16° on the surface) favouring the development of animal and plant plankton that provides nutrients to the food chain of many fish species.

Our Problem. We wish to build a laboratory on the Chilean coast within the flow of the Humboldt Current to perform ecological fishing and meteorological studies. It is therefore necessary to have predictive models to determine the degradation of metal materials in the sea.

As mathematicians, assume that you are asked to participate in the planning of experiments to be carried out to gather data and build models to determine the degradation of materials in the sea.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Guerrero-Ortiz, C., Mena-Lorca, J. & Soto, A.M. Fostering Transit between Real World and Mathematical World: Some Phases on the Modelling Cycle. Int J of Sci and Math Educ 16, 1605–1628 (2018).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: