## Proportional reasoning

Proportional reasoning is a well-defined domain with concrete applications (not too abstract) and therefore seems to lend itself for integration in a Game-Based Learning Environments (GBLE). The domain is also relevant in the Flemish and Dutch prevocational curriculum. Teachers in prevocational education often mention that students have difficulties with proportional reasoning.

Based on the literature we defined three types of proportional reasoning problems.

- In
**missing value****problems**, a missing value in one of two ratios needs to be found. These problems are presented as a/b = ?/d or a/b = c/?. - In
**comparison problems**, students need to determine the relationship between four given values i.c. two ratios. One ratio can be ‘equal to’, ‘less than’ or ‘more than’ the other ratio. **Transformation problems**are problems in which two ratios are given but one (or two) values need to be adapted to create two equivalent ratios. For example, two ratios are given: 3/6 and 4/12. In the second ratio (4/12) 2 needs to be added to the numerator to make this ratio equivalent with the first ratio (3/6 = 6/12).

The latter problem type is the most difficult because the student has to figure out independently how much has to be added and to what amount it has to be added (the numerator or the denominator). This in contrast to the missing value problems where it is clear what number is missing. In addition, the strategies that are used to solve transformation problems require more steps than the strategies used for solving comparison or missing value problems.

For a more extensive description of proportional reasoning see:

##### Vandercruysse, S., ter Vrugte, J., de Jong, T., Wouters, P., van Oostendorp, H., Verschaffel, L., Van Dooren, W., & Elen, J. (2015). “Zeldenrust”: A mathematical game-based learning environment for prevocational students. In J. Torbeyns, E. Lehtinen, & J. Elen (Eds.), *Advances in Game-Based Learning *(pp. 63-81)*.* Berlin: Springer-Verlag.