4. Contradictory models

Novices comfortably hold contradictory models. 

Experts do not.


For example:

In a study, children were individually shown the fraction sum $\frac27+\frac17$. Most children evaluated it to $\frac{3}{14}$. The researcher then provided the children with Cuisenaire rods, and they constructed the sum and evaluated it to  $\frac37$. When asked to explain this, some of the children explained, “when you do it in algebra you get  $\frac{3}{14}$ but when you do it with rods, the answer is  $\frac37$”. (The other answers fell into 3 groups: the algebra is wrong; the rods are wrong; and “I don’t know”.)

Our students can comfortably hold contradictions.

This is because students’ models are purely functional – they only need to get to the right answer.

When they meet a new fact, they simply tack it onto what they already know, whether it fits or not.

So they can hold contradictory models and learn which part of the model applies in which situation without ever having to address the contradiction.

If students are not put in situations where they are forced to confront the contradictory models, the contradictions will most likely remain present and unaddressed.