Skemp suggests that there are 2 types of understanding: instrumental and relational.

Instrumental understanding – Not quite about playing the piano, but I think the analogy applies. Instrumental understanding is like knowing the notes to play, but not knowing the tune – in maths terms, it is about knowing the rules, formulas and processes to get an answer, but not knowing the underlying concepts. This method is easier to understand and children get to see the results of their learning quicker, giving them a sense of success that children, like all of us, are excited by. These reasons are why it is understandable that teachers use instrumental understanding in the classroom. If there are upcoming tests or exams, it is quicker and easier to teach this way if it is a subject they simply need to know. A teacher may also feel that the children have not developed skills that are needed to understand relational thinking, the other type of understanding described by Skemp.

Relational understanding is a more complex affair, however, the long term effects are substantially worthwhile. It is about knowing why you are using a certain rule and the concepts beneath the strategy, i.e. why two negatives make a positive. This approach is more adaptable to new tasks and is easier to remember in the long term. Relational understanding is knowing all of the connections across mathematical topics.

Skemp uses the analogy of a town to explain the difference between these two approaches. You can walk through town knowing your route from A to B and a few other routes nearby to get to the essential places you need (instrumental) or you can create a “cognitive map” of the town in your head, knowing all the routes and which is best for your journey (relational). If you can master the second approach, then you will never be lost.

References

Skemp, R. (1976) Relational Understanding and Instrumental Understanding. Mathematics Teaching. Available at https://alearningplace.com.au/wp-content/uploads/2016/01/Skemp-paper1.pdf (Accessed 23/09/16)