## Abstract

Taking its point of departure in the fact that mathematical applications models and models have gained increasing momentum in mathematics education over the last four decades, the present paper attempts to survey and discuss the evolution of research and development of mathematical modelling in mathematics education during those decades, with particular emphasis being placed on last couple of decades.

In a brief historical exposition, the paper describes how, in the early years 1970-1990 the focus was on making a plea for including existing applications and models in mathematics curricula in a number of countries and on presenting cases of experimental teaching and teaching materials, whilst the later decades 1990-2018 have seen a shift in research and development towards focusing on students’ ability to undertake mathematical modelling themselves.

In order to provide a background to more closely considering advances in research and development, the paper offers an account of the basic notions and terms in mathematical modelling, including the modelling cycle and its main constituents: pre-mathematisation, mathematisation, mathematical treatment and problem solving, de-mathematisation, validation of model outcomes and, finally, model evaluation. Furthermore a distinction between descriptive and prescriptive modelling is introduced as is a distinction between mathematical modelling as a goal in itself and as a vehicle for the teaching and learning of mathematics.

Particular attention is being paid to introducing and discussing the notions of modelling competency and modelling (sub-)competencies which have greatly preoccupied the international research community in this domain during the last couple of decades.

During four decades of research and development work on mathematical modelling in mathematics education a number of significant findings have been obtained. Mathematical modelling is difficult and demanding. The ability to undertake it does not automatically follow from being good at pure mathematics. There are severe stumbling blocks along the road, especially connected to the transitions between the stages of the modelling cycle. However, and this is a rather novel finding, also the pre-mathematisation stage, in which an extra-mathematical domain is being prepared for modelling, gives rise to a number of important challenges to students. Another recent finding is that so-called implemented anticipation is a key factor in the success or failure of modelling work. Finally, the good news is that the ability to model can indeed be learnt, but this requires concerted didactical and pedagogical development as well as efforts and investments in human and other resources. But, if you are not willing to pay the expenses entailed by the pursuit of a goal you are not really willing to pursue the goal.

In a brief historical exposition, the paper describes how, in the early years 1970-1990 the focus was on making a plea for including existing applications and models in mathematics curricula in a number of countries and on presenting cases of experimental teaching and teaching materials, whilst the later decades 1990-2018 have seen a shift in research and development towards focusing on students’ ability to undertake mathematical modelling themselves.

In order to provide a background to more closely considering advances in research and development, the paper offers an account of the basic notions and terms in mathematical modelling, including the modelling cycle and its main constituents: pre-mathematisation, mathematisation, mathematical treatment and problem solving, de-mathematisation, validation of model outcomes and, finally, model evaluation. Furthermore a distinction between descriptive and prescriptive modelling is introduced as is a distinction between mathematical modelling as a goal in itself and as a vehicle for the teaching and learning of mathematics.

Particular attention is being paid to introducing and discussing the notions of modelling competency and modelling (sub-)competencies which have greatly preoccupied the international research community in this domain during the last couple of decades.

During four decades of research and development work on mathematical modelling in mathematics education a number of significant findings have been obtained. Mathematical modelling is difficult and demanding. The ability to undertake it does not automatically follow from being good at pure mathematics. There are severe stumbling blocks along the road, especially connected to the transitions between the stages of the modelling cycle. However, and this is a rather novel finding, also the pre-mathematisation stage, in which an extra-mathematical domain is being prepared for modelling, gives rise to a number of important challenges to students. Another recent finding is that so-called implemented anticipation is a key factor in the success or failure of modelling work. Finally, the good news is that the ability to model can indeed be learnt, but this requires concerted didactical and pedagogical development as well as efforts and investments in human and other resources. But, if you are not willing to pay the expenses entailed by the pursuit of a goal you are not really willing to pursue the goal.

Original language | English |
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Title of host publication | Proceedings of the 8th ICMI-East Asia Regional Conference on Mathematics Education - ICMNI-EARCOME8, Tapei, Taiwan, May 7-11, 2018 |

Editors | Feng-Jui Hsieh |

Number of pages | 11 |

Volume | 1 |

Place of Publication | Taipei, Taiwan |

Publisher | EARCOME8 - National University of Taiwan |

Publication date | May 2018 |

Pages | 26-36 |

ISBN (Print) | 978-986-05-5783-1 |

Publication status | Published - May 2018 |

Event | The 8th ICMI-East Asia Regional Conference on Mathematics Education: Flexibility in Mathematics Education - Taipei International Conference Center, Taipei, Taiwan, Province of China Duration: 7 May 2018 → 11 May 2018 Conference number: 8 http://earcome8.math.ntnu.edu.tw/ |

### Conference

Conference | The 8th ICMI-East Asia Regional Conference on Mathematics Education |
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Number | 8 |

Location | Taipei International Conference Center |

Country | Taiwan, Province of China |

City | Taipei |

Period | 07/05/2018 → 11/05/2018 |

Other | “Flexibility in Mathematics Education” has been chosen as the theme of the conference. Flexibility is highly related to creativity, multiplicity, and adaptation. In the current era, rapid changes in economy, environment and society have been facilitated by the rapid development of technology and engineering. Flexibility in mathematical thinking, problem solving, teaching methods, evaluation, teacher education and mathematics education research is a key to empowering learners, teachers, educators and researchers to tackle the complexity and uncertainty, and to giving them the capacity and motive to change in the innovative era. |

Internet address |

## Keywords

- mathematical modelling
- modelling cycle
- modelling competency
- pre-mathematisation
- implemented anticipation