3.5 Modelling skills

Purposeful modelling requires students to have an adequate level of mathematical skills and abstraction level.² Points of attention for modelling instruction are:

• Variables
  When students must define a variable by means of a formula, it often turns out that they do not know what the concept of variable means and what a formula actually is. This concerns not only students in lower secondary education, but also those in upper secondary education. The reason for this may be that mathematics and physics textbooks usually pay little attention to an explanation of these essential concepts.³
• Formulas
  Many formulas that appear in syllabi and textbooks are given there in the simplest form. To be able to use these formulas in a model, an adjustment often has to be made.
  • An example is the formula for the spring force of a vibrating mass-spring system: $F_{\text{v}}=-C\cdot u$ , where $u=0$ is the equilibrium position. In the case of a mass hanging on a spring, however, the stretching of the spring in this equilibrium position is not zero, but is determined by the mass and the spring constant. It becomes even more complicated if u is also used to denote the position of the mass. Students must learn to deal with this and will sometimes have to adapt formulas or even have create new ones.
• Difference equations
  The core of dynamic models is formed by differential or difference equations.⁴ This has various consequences:
  • For example, it is wise in a modelling learning path to use from the beginning the formal $\Delta$-notation for 'change of'. Students find a change in notation from $s=v\cdot t$ into $\Delta x=v\cdot \Delta t$ unnecessarily confusing.
  • A second aspect is the shape of the formulas. Dynamic modelling concerns the integration of difference equations, not differentiation. This means that, for example, the equation for displacement, velocity and duration in a model is used in the form $\Delta x=v\cdot \Delta t$, but not in the form $v=\Delta x/\Delta t$. For students this is not obvious; after all, they normally use a formula in any form within ordinary tasks. This can cause problems when students must construct or expand a model themselves.