# Mathematical modelling framework

The IM²C operates on the assumption that mathematics is everywhere in the world around us; the challenge is to identify its presence, access it, and apply it productively. The IM²C exists to help students:

• develop a systematic and successful approach to addressing individual problems located in real-world settings, and
• through this development, enable students cumulatively to become effective solvers of real-world problems.

The desire is to produce students who can not only productively address problems set by others, but become able to identify and address problems themselves.

In order to be useful and applicable in practice (both in the context of the IM²C, and more broadly), the cyclic process of modelling is scaffolded (guided) by a systematic approach to individual problems, consistent with the approach taken by professional modellers when devising solutions to problems in their field.

1. Describe the real-world problem. Identify and understand the practical aspects of the situation.
2. Specify the mathematical problem. Frame the real-world scenario as an appropriate, related mathematical question(s).
3. Formulate the mathematical model. Make simplifying assumptions, choose variables, estimate magnitudes of inputs, justify decisions made.
4. Solve the mathematics.
5. Interpret the solution. Consider mathematical results in terms of their real-world meanings.
6. Evaluate the model. Make a judgment as to the adequacy of the solution to the original question(s). Modify the model as necessary and repeat the cycle until an adequate solution has been found.
7. Report the solution. Communicate clearly and fully your suggestions to address the real-world problem.

The interpretation and evaluation stages indicate the cyclic nature of mathematical modelling.

If the proposed first solution is not an adequate solution to the original question, the problem needs to be readdressed by repeating of earlier stages (stages 3 to 6) in sequence, and this may need to be carried out several times before an adequate solution is found.

Sometimes an extension or refinement of the original problem is suggested by the outcome of a first modelling endeavour. In this instance the question is re-specified, and further cycles of activity are conducted with the new question.

It is also important to note that although the stages are sequential, the cycle is not necessarily smooth, as the constant checking, testing and evaluating contained in each stage means that there is frequent movement within (and between) the stages – potentially making the development of some models a very challenging exercise.

More information is provided in the Guidebook to further explain descriptive and prescriptive modelling.