Conceptual models

A conceptual model is a visual representation that facilitates comprehension. In maths, these are typically diagrams.

We routinely use such diagrams to illustrate concepts, and to succinctly synthesize a set of information in a way that shows the connections between the components.

Many teachers expect that their use of instructive conceptual models will allow students to learn and develop an understanding of the concepts under instruction.

“When we teach, it is common to assume that students have acquired or constructed mental models that are copies of the conceptual models [diagrams] that have been presented to them.”

Greca, I. & Moreira, M. (2000). Mental models, conceptual models, and modelling. International Journal of Science Education, 22(1), 1–11.

However,

“Regardless of the effort that teachers invest on their task …  students limit themselves to learning by heart long lists of formulae and definitions, which they do not understand, because the phenomena described there are not being interpreted according to the mental models the students should be constructing. “

Greca, I. & Moreira, M. (1997). The kinds of mental representations‐models, propositions and images‐used by college physics students regarding the concept of field. International Journal of Science Education, 19(6), 711–724.

Teaching with conceptual models is excellent practice, but is not sufficient for the development of good mental models. If it was, then all of our students would naturally develop good mental models, which they unfortunately don’t.

“It seems reasonable that there should be a simple and direct relation between a conceptual model and a mental model, but this does not seem to be the case.”

Norman, D. 1983. “Some observations on mental models”. In Mental models, Edited by: Centnerend, D. and Stevens, A. 6–14. N.J.: Lawrence Erlbeum Associetes (p12).

This is a somewhat unexpected result.

Moreover, it is not only students whose mental models don’t match up with conceptual models.

The mental models of active physicists have been found not to match the conceptual  (textbook) models either.

But, when these same scientists present their findings, they present them via conceptual diagrams and equations (just like in a textbook) but they don’t disclose the process that they used to get to these conceptions.

Typically, students never see the internal mental models of the scientists, (or mathematicians in our case) and they never see how to construct good mental models (from conceptual models, or otherwise). Students are meant to acquire this skill implicitly. A minority of students somehow manage to achieve this - these are the students who become capable mathematics students.