Mathematical applications and modelling in the teaching ...

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Mathematical applications and modelling in the teaching and learning of mathematics

Proceedings from Topic Study Group 21 at the 11th International Congress on Mathematical ducation in Monterrey, Mexico, July 6-13, 2008

Editors: Morten Blomh?j, NSM, Roskilde University, Denmark Susana Carreira, University of Algave, Portugal June 2009

nr. 461 - 2009

Roskilde University, Department of Science, Systems and Models, IMFUFA P.O. Box 260, DK - 4000 Roskilde Tel: 4674 2263 Fax: 4674 3020

Mathematical applications and modelling in the teaching and learning of mathematics

Proceedings from Topic Study Group 21 at the 11th International Congress on Mathematical Education

in Monterrey, Mexico, July 6-13, 2008

IMFUFA tekst nr. 461/ 2009

? 252 pages ?

ISSN: 0106-6242

These proceedings contain the papers reviewed and accepted for Topic Study Group 21 (TSG21) at ICME-11. Prior to acceptance all papers were reviewed by at least two reviewers and revised on the basis of the review reports. Preliminary versions of all the papers were published at the congress web-site before the congress so as to form a common basis for the work of the TSG. The papers presented during the TSG sessions are all marked with a `*' in the table of contents found below. After the congress revised versions of the papers have been submitted for the proceedings.

During the TSG21 sessions the papers were presented and discussed according to a thematic organisation in three themes, namely: Theme 1: Different perspectives on mathematical modelling in educational research; Theme 2: Challenges in international collaboration on the teaching of mathematical modelling, and Theme 3: Didactical reflections on the teaching of mathematical modelling. The intentions of theme 1 were to present and discuss an overall view on different perspectives found in the field of educational research on the teaching and learning of mathematical modelling, and to use this overview to characterise and discuss the research presented in the TSG. The two other themes are more particular foci on the teaching and learning of mathematical modelling dealt with in the papers submitted for TSG21. In these proceedings the papers are presented according to the theme under which they were presented or discussed during the congress.

Morten Blomh?j, July 2009

Roskilde University, Department of Science, Systems and Models, IMFUFA P.O. Box 260, DK - 4000 Roskilde Tel: 4674 2263 Fax: 4674 3020

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Table of contents

Theme 1: Different perspectives on mathematical modelling in educational research

Different perspectives in research on the teaching and learning mathematical modelling

1

? Categorising the TSG21 papers *

Morten Blomh?j

Differential equations as a tool for mathematical modelling in physics and

19

mathematics courses ? A study of high school textbooks and

the modelling processes of senior high students *

Ruth Rodr?guez Gallegos

Mathematical modelling: From Classroom to the real world

35

Denise Helena Lombardo Ferreira & Otavio Roberto Jacobini

On the development of mathematical modelling competencies

47

? The PALMA longitudinal study *

Rudolf vom Hofe, Alexander Jordan, Thomas Hafner, Pascal St?lting,

Werner Blum & Reinhard Pekrun

The teachers' tensions in mathematical modelling practice

61

Andr?ia Maria Pereira de Oliveira & Jonei Cerqueira Barbosa

Teaching to reinforced bonds between modelling and reflecting *

73

Mette Andresen

Applying pastoral metamatism or re-applying grounded mathematics

85

Allan Tarp

A `new' type of diagram to support functional modelling ? PROGRAPH diagrams

101

Hans-Stefan Siller

Mathematical models in the context of sciences *

117

Patrica Camarena Gallardo

Mathematical modelling, the socio-critical perspective and the reflexive discussions *

133

Jonei Cerqueira Barbosa

Mathematical modelling and environmental education

145

Ademir Donizeti Caldeira

Mathematical models in the secondary Chilean education

159

Mar?a D. Aravena & Carlos E. Caama?o

III

Theme 2: Challenges in international collaboration on the teaching of mathematical modelling

Challenges with international collaboration regarding teaching of mathematical modelling * 177 Thomas Lingefj?rd

A comparative study on mathematical modelling competences

197

with German and Chinese students.*

Matthias Ludwig & Binyan Xu

Mathematical modelling in a European context

207

? A European network-project.*

Stefanie Meier

Theme 3: Didactical reflections on the teaching of mathematical modelling

Didactical reflections on the teaching of mathematical modelling

217

? Suggestions from concepts of "time" and "place" *

Toshikazu Ikeda

Formatting real data in mathematical modelling projects *

229

Jussara de Loiola Ara?jo

Simple spreadsheet modelling by first-year business undergraduate students:

241

Difficulties in the transition from real world problem statement to mathematical model *

Djordje Kadijevich

*) Papers presented orally during the TSG21 sessions at ICME-11.

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DIFFERENT PERSPECTIVES IN RESEARCH ON THE TEACHING AND LEARNING MATHEMATICAL MODELLING

- CATEGORISING THE TSG21 PAPERS Morten Blomh?j

NSM, Roskilde University, Denmark

Introduction

We have a large and growing collection of didactical research on mathematical modelling. Moreover this research even seems to have had a serious impact on the practices of mathematics teaching at least on curricula level. During the last couple of decades the introduction of mathematical modelling and applications is probably - together with the introduction of information technology - the most prominent common features in mathematics curricula reforms around the world (Kaiser, Blomh?j and Sriraman, 2006, p. 82). Curricula reforms in many western countries, especially at secondary level have emphasised mathematical modelling as an important element in an up-to date mathematics secondary curricula preparing generally for further education. Didactical research has undoubtedly played an important role in this development. The fundamental goals in the teaching of mathematical modelling and the reasons for pursuing these goals developed and analysed in research can be pinpointed in the guidelines for mathematics teaching in many countries. Also, the general understanding of the model concept and of a modelling process expressed in many mathematics curricula is clearly influenced by didactical research. (Blum et al, 2007) and (Haines et al, 2006).

However, the way mathematical modelling and applications is organised in curricula and, especially, how these parts of the curricula are assessed reveal only a very limited influence from research. And when it comes to the level of teaching practice in the classroom it is still a pending question to which degree the many developmental modelling projects carried out and analysed in research have actually influenced the practices of teaching mathematical modelling.

Influencing practices of mathematics teaching are not the only criteria for progress in the didactical research on mathematical modelling. It is also relevant to try to evaluate the coherency of the theories developed. In the editorial Towards a didactical theory for mathematical modelling of ZDM (2, vol. 38), we argued that at a general level it is possible to identify in the field of research

... a global theory for teaching and learning mathematical modelling, in the sense of a system of connected viewpoints covering all didactical levels: learning goals, fundamental reasons for pursuing these goals at different levels of the educational systems, tested ideas about how to support teachers in implementing learning goals and recognised didactical challenges and dilemmas related to different ways of organising

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Morten Blomh?j

the teaching, theoretically and empirically based analyses of learning difficulties connected to modelling, and ideas about different ways of assessing students' learning in modelling activities and related pitfalls. (Kaiser, Blomh?j and Sriraman, 2006, p. 82)

However, this "global theory" is not based on a single strong research paradigm. On the contrary, in fact, it is possible to identify a number of different approaches and perspectives in mathematics education research on the teaching and learning of modelling. This is, precisely, the reason for choosing Conceptualizations of mathematical modelling in different theoretical frameworks and for different purposes as one of the themes for the Topic Study Group on Mathematical applications and modelling in the teaching and learning of mathematics at ICME-11 (TSG21). We intended to provide a background for in-depth discussions of the theoretical basis of the different approaches within the field.

Kaiser & Sriraman (2006) report about the historical development of different research perspectives and identify seven main perspectives describing the current trends in the research field.

These perspectives may have overlaps and also they do not necessarily cover the entire research area. Nevertheless, they all represent distinctive perspectives of research on the teaching and learning of mathematical modelling, and they have been developed in particular research milieus over a long period of time and all of them have produced a considerable number of research publications. The main rationale for developing a categorisation of research perspectives is of course to deepen our mutual understanding of the individual perspective and to recognise similarities and differences amongst these. The idea is not to try to judge about their relevance or their relative importance.

Five of the research perspectives pinpointed by Kaiser & Sriraman (2006) are ? according to my analysis ? represented among the sixteen papers accepted for TSG21. As an introduction to our work in Topic Study Group it is therefore relevant to characterise and briefly discuss these five research perspectives. For each perspective I give a short presentation of the TSG21 papers that I found can be said to representing the perspective. The aim is to provide a background for discussing the many interesting papers of TSG21 in relation to their research perspective and theoretical foundation. Hopefully, the categorisation can also facilitate discussions of similarities and differences among the perspectives. It goes without saying that the categorisation in itself should made object for discussion and debated. At the end of the paper, I summarise, in the form of a template, the descriptions of the perspectives in few words together with a list of the TSG-papers, which I have categorised under the individual perspectives. In the following I refer to the TSG21 papers included in the proceedings by the authors' names and (TSG21).

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Different perspectives in research on the teaching and learning mathematical modelling

The realistic perspective

The realistic perspective on the teaching and learning of mathematical modelling takes its point of departure in the fact that mathematical models are being extensively used in very many different scientific and technological disciplines and in many societal contexts. In this perspective, mathematical modelling is viewed as applied problem solving and a strong emphasis is put on the real life situation to be modelled and on the interdisciplinary approaches.

According to this perspective, in order to really support the students' development of a mathematical modelling competence that is relevant for their further education and for their subsequent professions, it is essential that the students work with realistic and authentic real life modelling. The students' modelling work should be supported by the use of relevant technology, such as for example advanced computer programmes for setting up and analysing mathematical models. The modelling process and the model results should be assessed through validation against real or realistic data. Therefore, in this perspective it is important to study in depth mathematical modelling processes in different professions and areas of societal applications of mathematical models. Such studies should inform the design of modelling courses in schools in order for the teaching in modelling to as realistic as possible.

The main criterion for progress in the students' learning is the students' success with solving real life problems by the means of mathematical modelling. Pollak (1969) can be regarded as a prototype of the realistic perspective.

Often physicists and sometimes also researchers from other natural sciences argue that what we in mathematics education calls mathematical modelling in their subject area should be thought of as physics modelling (or just physics ? because modelling is what physicists do all the time ? they say) or biological modelling. Nevertheless as mathematics educators we focus on the general elements in the teaching and learning and not on the differences of modelling in different areas. Should we need to defend ourselves, we could argue that so far not much educational attention or research has been directed towards (mathematical) modelling outside mathematics education research. However, the realistic perspective is really taking the subject area of the application of mathematics very seriously, and actually in this perspective is seen as an interdisciplinary problem solving activity in which, of course, mathematics is playing a very important role.

The paper by Rodr?guez (TSG21) is an example of how the conceptualisation of a mathematical modelling process may be influenced by the subject area in which the modelling takes place. The paper reports from a developmental project carried out at a French University context where mathematical modelling was used as a didactical means for supporting the students' learning of mathematics and physics in a calculus and a physics course, respectively. A

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Morten Blomh?j

physics domain was "inserted" in a six phased model of the modelling process in order to distinguish the physical elements in the systematisation and mathematisation processes and in the interpretation of the model results. In the physics course the modelling process was conceptualised as illustrated in figure 1. However, since the learning goal in this project was to support the students' learning of mathematics and physics by means of mathematical modelling and not to model real life situations, in my opinion, this paper does not belong to the realistic perspective, but rather to the educational perspective discussed below.

Figure 1: The modelling process in a physics course (Rodr?guez, TSG21).

The paper by Kadijevich (TSG21) is properly the closest we get to a paper within the realistic perspective among the TSG21 papers. The paper reports and analyses the experiences from a developmental project for undergraduate business students in Serbia. The students were to build and analyse a total financial balance model in the form of a spreadsheet for a business activity of their own choice. The use of technology in the form of a spreadsheet is an important and integrated element in this approach. The success criterion for the students' modelling work was to apply the model for deciding whether or not their business activity was a profitable one and to make suggestions on how to make more profit. This type of pragmatic criteria for solving authentic real life problems or realistic problems by means of mathematical modelling is, I think, characteristics for the realistic perspective. I consider it also as a characteristic element in this approach that Kadijevich in his design builds on the heuristics for technology-supported modelling of real life situations developed by Galbraith & Stillman (2006).

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