1.4 The consequences

We should not underestimate the significance of the development of computers for mathematical modeling. Their capacity to solve mathematical problems has already changed the way in which we deal with and teach applied mathematics. The relative importance of skills for arithmetic and symbolic manipulation will further decrease. It will be more important to understand the concept of a derivative than to calculate it. And we will need more persons qualified to translate real-life problems into formal language, and what activity is more rewarding and intellectually more challenging in applied mathematics than just that – namely modeling? While relatively fewer mathematicians are needed to solve a system of differential equations or to manipulate certain mathematical objects – these are tasks better done by computers – more and more persons are needed who are skilled and expert in modeling – in capturing the essence, the pattern and the structure of a problem and to formulate it in a precise way.

This development is by no means confined to science. Since World War II, a growing interest has been shown in formulating mathematical models representing physical processes. These kinds of models are also beginning to pervade our industrial and economical processes. In several key industries, such as chip production or flight traffic, optimizing software is an integral part of their daily activities. In many other industrial sectors, companies are beginning to formalize their operations: Planning the workforce, sport scheduling, production planning, route planning for transportation, just to mention a few. Mathematical models are beginning to be an essential part of our highly developed society.

An important condition for the widespread use of mathematical modeling tools is a change in the mathematical curriculum in school: A greater part of the manipulation of mathematical structures should be left to the machine, but more has to be learnt about how to recognize a mathematical structure when analyzing a particular problem. It should be an important goal in applied mathematics to foster creative attitudes towards solving problem and to encourage the students’ acquisition and understanding of mathematical concepts rather than drumming purely mechanical calculation into their heads. Only in this way can the student be prepared for practical applications and modeling. Science is modeling! The ability to solve problems is modeling!

But how can modeling be learnt? Problems, in practice, do not come neatly packaged and expressed in mathematical notation; they turn up in messy, confused ways, often expressed, if at all, in somebody else’s terminology. Therefore, a modeler needs to learn a number of skills. She must have a good grasp of the system or the situation which she is trying to model; she has to choose the appropriate mathematical methods and tools to represent the problem formally; she must use software tools to formulate and solve the models; and finally, she should be able to communicate the solutions and the results to an audience, who is not necessarily skilled in mathematics.

It is often said that modeling skills can only be acquired in a process of learning-by-doing; like learning to ride a bike can only be achieved by getting on the saddle. It is true that the case study approach is most helpful, and many university courses in (mathematical) modeling use this approach. But it is also true – once some basic skills have been acquired – that theoretical knowledge about the mechanics of bicycles can deepen our understanding and enlarge our faculty to ride it. This is even more important in modeling industrial processes. It is not enough to exercise these skills, one should also acquire methodologies and the theoretical background to modeling. In applied mathematics, more time than is currently spent, should be given to the study of discovery, expression and formulation of the problem, initially in non-mathematical terms.

So, the novice needs first to be an observer and then, very quickly, a do-er. Modeling is not learnt only by watching others build models, but also by being actively and personally involved in the modeling process.