Mental Models and External Representations

Imagine holding a ball in your outstretched hand, palm facing up, and then opening your hand. What would happen? Presumably the ball would stay in your hand. What if your palm was facing down when you opened it? You're probably imagining that the ball would fall. But, what if the ball was very light, like a ping-pong ball, and you're standing over an air vent? It might fall, at a slower rate, or it might be suspended by the flowing air, jiggling around. When you imagine these events and make predictions of what the ball would do, you are constructing and "running" a mental model of a situation.

We use mental models all the time to predict and to explain what is happening. In some sense, we only have mental models. We don't have direct access to the external world, exactly. Our senses recieve data from the external world, but then our brain constructs a model from that data. The model might be mistaken. Optical illusions are one way of showing that our mental models do not always correspond to reality. (One of my favorite model that our brain uses is that of chronostasis, leading to the stopped-clock illusion.)

So, our mental models, like all models, are only useful approximations. Of course, most of the time our mental models of everyday physical reality are quite adequate for getting around and building civilization.

Thus far when discussing scientific models we have emphasized the external representations: diagrams, equations, computer code, etc. But, as we mentioned earlier, every scientific model has a mental model component as well. The external representation has to be interpretted as corresponding to something in the world in order to be a model of that thing and this is done with the help of mental models. So, every scientific model includes both a mental model and an external model or representation. A mental model by itself cannot really be scientific. To be scientific, a model needs to be shareable and that requires external representations.

External representations are useful for more than just communication. They can help us to reason and change how we think. Powerful external representations literally make us smarter. For example, doing long division and multiplication with Roman numerals is so difficult (see cartoon below) that there were professionals to do these types of calculations for people who needed it, like merchants (the professionals probably used abacuses). Now that we have Hindu-Arabic numerals, elementary school students routinely learn how to do long division and multiplication.

Powerful representations are not always intuitive. Mathematical representations, for example, are usually not immediately intuitive. It is not at all obvious to a novice that $x(t)=a\cos{(\sqrt{\frac{k}{m}}t-\phi)}$ is a model of the position of an oscillating mass attached to a spring. To be useful as modeling tools, we need to build up intuitions about the external representations we use and link the representations to our intuitive mental models about the phenomenon being modeled.

Both of these steps can be challenging. We don't naturally have mental models of many phenomena of scientific interest because we do not encounter them in our regular lives. For example, we spend a lot of time building up structural and mathematical models of the atom because the quantum mechanical nature of the object is so foreign to us. Indeed, most physicists declare that at some point the weirdness of quantum mechanics requires them to give up on structural models of atoms and proceed only with mathematical ones.