# What is a mental model?

When you see or hear a word, for example, “dog”, your mind springs to action and provides you with a whole lot of associations.

You may see an image of a dog.

You may hear the dog barking or panting.

You may smell the dog.

You may see a mental movie of the dog doing something: maybe chasing a ball or playing.

You may recall the feel of the dog’s fur.

The word “dog” may arouse emotional associations – good ones if you have a pet dog that you love; or bad ones if you don’t like dogs.

A whole lot of other, more abstract associations will also spring to mind: mammal, four-legged, loyal companion, and whatever else you associate with dogs.

This complete set of ideas and associations form your mental model for “dog”. The exact nature of your mental model is personalised to you and is dependent on many things, including your experiences, they way your brain interprets sensory inputs, and your education.

#### Mental models in mathematics

We also form mental models when we are learning mathematics.

As mathematicians and maths educators, our mathematical mental models are not dissimilar to our models for things like “dog”: they are accurate, detailed, highly interconnected, and useful.

For example:

The word “integers” may conjure up thoughts of:

• …, -3, -2, -1, 0, 1, 2, 3, …;
• one-to-one correspondences;
• the number line;
• induction;
• $\aleph_0$;
• $\mathbb{Z}$; subsets of $\mathbb{R}$;
• rings; and so on.

When we learn maths, we, consciously or unconsciously, form some sort of a mental model that lets us synthesize and understand all the information – all the words, pictures, formulae, relationships, and so on – in a way that’s consistent and makes sense. As our mathematical understanding develops, our models move from concrete to abstract and from simple to sophisticated.

Consequently, we have a good understanding of maths, and enjoy doing maths.

Not so for most of our students.