Modelling in Mathematics Classrooms: reflections on past developments and the future

This paper describes the development of mathematical modelling as an element in school mathematics curricula and assessments. After an account of what has been achieved over the last forty years, illustrated by the experiences of two mathematician-modellers who were involved, I discuss the implications for the future – for what remains to be done to enable modelling to make its essential contribution to the "functional mathematics", the mathematical literacy, of future citizens and professionals. What changes in curriculum are likely to be needed? What do we know about achieving these changes, and what more do we need to know? What resources will be needed? How far have they already been developed? How can mathematics teachers be enabled to handle this challenge which, scandalously, is new to most of them? These are the overall questions addressed. The lessons from past experience on the challenges of large-scale of implementation of profound changes, such as teaching modelling in school mathematics, are discussed. Though there are major obstacles still to overcome, the situation is encouraging. ZDM-classification: D30, M10 The last 40 years have seen the explicit teaching of modelling with mathematics move forward, from small-scale explorations, through developments in typical classrooms, to established courses, albeit in a small minority of mathematics classrooms worldwide. Two kinds of student learning activity are essential for mathematics to be functional in everyday life and work: learning illustrative applications and standard models; active modelling by students, using mathematics to tackle problems that are new to them. There are now some examples of well-engineered materials to support both activities. However, there are important gaps. We need to know more about how best to enable all teachers of mathematics to acquire the extra mathematical and pedagogical skills needed to teach modelling. Further, while policy makers regularly stress the importance of students "being able to use their mathematics", their actual policy decisions rarely support effective ways to bring this about. This paper will review the current situation and outline a programme for further progress. 1. Modelling with mathematics For this issue of ZDM, I do not need to describe the processes of modelling in detail; it is common ground. However, discussion of the nature of modelling and its role, both in mathematics and in mathematics education, is often oversimplified. For mathematics education this arises partly because the dominant intellectual influences on school mathematics have come from pure mathematicians. Understandably, they regard it as 'their subject'; yet their attitudes to mathematics differ fundamentally from the far greater number of those who use mathematics in life and work, as citizens or as professional users of mathematics in engineering, science, economics and other fields where mathematics is a key language. These all see mathematics as primarily a powerful toolkit to help them understand and solve problems from the real world. That will be the perspective of this paper. While the 'pure' and 'applied' viewpoints have many things in common, including delight in the elegance of mathematics, they differ in others that are important for the design of school curricula – notably the central importance of teaching modelling. To bring out the aims of modelling in mathematics education, I will begin with personal views on modelling as an activity from two mathematician-modellers who work in mathematics education, Henry Pollak and myself. These comments reflect our different experience of 'doing mathematics', but illustrate the attitudes 1 I am grateful to Henry, not only for his substantial contributions to this paper through 'private communications' but for illuminating discussions on the subject over many years. ZDM 2006 Vol. 38 (2) Analyses 179 to mathematics we seek to develop in students, and the adults they become. HOP: My problem has always been that I like too many different areas of mathematics. As an undergraduate at Yale, I liked everything, but particularly analysis and topology of the oldfashioned point-set kind, and any combinatorics and number theory I saw, even though we had almost no courses in those kinds of areas. My enjoyment for topology came from Ed Begle. Of course, that isn't the way you get a doctorate – there you have to find a hole and dig and dig until it's yours and yours alone. I went to graduate school at Harvard, where I thought that I could do more topology with Hassler Whitney – but he soon left and I got into Complex Variables, and did a thesis in Geometric Function Theory with Lars Ahlfors. I also worked on a Navy Contract under Stefan Bergman in the Engineering School during most of my time there. In my last year of graduate work, I taught. When it came time to find a job, it was clear to me that, even though everybody said that you were going to be paid to do good teaching, don't let anybody kid you – you were really going to be paid for your research. I thought that was kind of dishonest. A recruiter from Bell Laboratories came by and I went to see him. He made it clear that if I were going to be hired, it would be to do research, and that sounded good. I was probably the first mathematician hired there in a long time who was trained as a pure mathematician. But it worked out OK. Alongside helping with various people's math problems, mostly in classical analysis, I started working in the area of balancing and optimizing a defensive missile system. I made a simplified mathematical model of it, which in retrospect I know now was a dynamic programming model. I didn't know about Richard Bellman's work (I don't even know if that came earlier); I therefore didn't know that nobody ever solved dynamic programming problems analytically – and so I did just that. Bell Labs appreciated it. I also learnt that I wasn't supposed to write papers as tersely as possible, in math journal style; I was writing for people who could use the understanding and techniques I might have developed, and how I got there. So I learnt to take the time to document it in such a way that they could understand all of that – another link to teaching. Bell Labs in its heyday was full of interesting people, with exciting things going on. Everybody was encouraged to talk to everybody else. The most interesting things I did might come from anywhere – from a specific practical question, or from some idea in the atmosphere that seemed not to be fully understood. Broad interests and good communication were at the heart of it all. HB: My perspective on modelling comes both from my experience in using mathematics in tackling diverse problems from everyday life and from my university experience as a theoretical physicist – we are all modellers, particularly in forefront fields of physics like my own, elementary particles, where there is no established theory. Even where there is an underlying theory that is true, the complexity of a system may make it insufficient. Turbulent flow of liquids and gases is a classical example where you cannot calculate directly the consequences of Newton's Laws; you have to build models. Quantum mechanics provides others. It works well for atoms, where there is a heavy central nucleus with a large dominant electric charge around which the light electrons move, essentially independently. Neither of these conditions applies inside nuclei; though quantum mechanics applies there too, there is no dominant centre and the particles interact strongly with each other. This situation makes the calculations too complex for current methods. Nuclear theorists developed a set of models, each of which accounts for important features of the data; however, their assumptions seem contradictory. The 'Shell Model' assumes that the particles move independently around a common force centre, as in an atom. How can this be so? Yet it explains the quantum states 2 A theory is a model that has been shown to work well across a well-defined field of phenomena: Newton's Laws of Motion, or Gravitation, or Maxwell's Electromagnetic Theory, Darwin's Theory of Evolution, and Quantum Mechanics are giant examples; there are many more limited theories – of gas pressure, circulation of the blood, or elasticity, for example – effective within smaller well-defined domains of phenomena. Analyses ZDM 2006 Vol. 38 (2) 180 found in light nuclei. In contrast, the 'Liquid Drop Model' treats the nucleus as a liquid with every particle interacting strongly with its neighbours alone; this works well for some properties of heavy nuclei. The reconciliation of these and other apparently incompatible models is an interesting story – but it is not for here. The point from our perspective is that limited precision and inconsistency are characteristic of modelling in all fields, profound or familiar, abstract or concrete. Economics is equally full of different perspectives that fit some features of the system but not others; the inconsistencies stimulate deeper analysis. People seek models that seem promising to explain features of the data, but they are often incomplete. When Watson and Crick were trying to work out the structure of DNA, Franklin's X-ray pictures suggested to them a helical structure. They then experimented with various wire models of the molecular structure (literal modelling, indeed) until they found one that would reproduce that X-ray data and fit the laws of chemical bonding. Their creation-discovery of the double helix model has been a beacon of scientific research, and medical advance, ever since but, though the essence of the structure was correct, there were complexities in the way it works that stimulated decades of further modelling by molecular biologists. How does all this relate to everyday life problem solving of ordinary people – to the student wondering whether she should go to university or straight into employmen