How often have you taught a topic, which your students seem to understand, only to be confronted by their inability to apply this learning in a new situation?

In 1976, Richard Skemp published a paper in which he discussed understanding as it applies to mathematics teaching and learning. He identified two types of understanding:

Instrumental Understanding is understanding how to use a rule.

Relational Understanding is knowing why in terms of underlying foundations.

Imagine you are in a new city and you learn from someone else how to get from point A to point B. You might eventually add more points that define locations to which you know how to navigate. However, if you deviate from your known paths, you will be completely lost. You never really develop an overall understanding of what the city looks like, and you might not be aware of other connections between the points that might be more efficient or useful.

This is instrumental understanding.

Imagine instead that you wander the city, sometimes using guides, other times simply exploring on your own. In time you develop on overall picture of the structure of the city. If you were told about a short-cut, you would understand why it worked, and why it was faster or more efficient than the path you discovered. You would not worry about getting lost, because your understanding of the overall structure would guide you to a place you know.

This is relational understanding.

The benefits of relational understanding in mathematics include being able to adapt to new tasks and to reconstruct forgotten rules. And when students internalize the structure of the mathematics, they are able to reason, justify, make sense of math, and become self-reliant, independent mathematical thinkers.

So the next time you plan a lesson, think about the understanding your students will develop from your instruction and consider: what kind of understanding do you want for your students?