Arithmetic problems at school: when there is an apparent contradiction between the situation model and the problem...

Understanding and solving problems involves different levels of representation. On the one hand, there are logico-mathematical representations, or problem models (PMs), which contain information such as 'the size of the flock changed from 31 sheep to 42' while, on the other hand, there are more qualitative representations, or qualitative situation models (QSMs), comprising events expressed in everyday terms, such as 'a lot of lambs had been born'.

We set out to show that an apparent contradiction between a PM and a QSM can result in poorer performances, especially when the activation of formal representations is not fully automatic.

We tested 44 third years (mean age 8;6) and 46 fourth years (mean age 9; 8) from two primary schools in France.

Change problems and compare problems were drafted in two versions: one where the QSM and PM were consistent and one where they appeared to be contradictory but were not (referred to hereafter as the inconsistent version). For example, if the numerical data proved that the size of a flock of sheep had increased (PM), the consistent version would state that a lot of lambs had been born (QSM), whereas the inconsistent version would state that the wolf had devoured some of the sheep (QSM). Each pupil was given 7 consistent problems and 7 'inconsistent' ones to solve.

For the change problems, errors were more frequent on the inconsistent versions, especially among the younger subjects. For the compare problems, there were more errors on the inconsistent versions regardless of the subjects' age.

It is in situations where the problem schema is not automatically activated (younger subjects or compare problems) that the QSM plays its role of intermediate representation and causes performances to deteriorate if it does not correspond to the PM.