According to Skemp (1989) there are two kinds of learning in mathematics; Instrumental or relational understanding. This is my understanding of the two kinds.
Instrumental understanding – having a mathematical rule and being able to apply and manipulate it.
Relational understanding – having a mathematical rule, knowing how to use it AND knowing why it works.
From this, I can see that relational understanding is a deeper, more complex understanding of instrumental understanding. While instrumental understanding is knowing and applying the rule, relational understanding is the same but also knowing why it works and how it connects to other rules. Using both of these understanding you will arrive at the same correct answer but relational understanding is way more extensive. Of course each has its own strengths for example, Instrumental understanding is often much easier to comprehend; some concepts in maths are difficult to follow but can be grasped quicker through the use of rules rather than knowing the ‘ins and outs’ of why it works. Results are instant; once you have learned the rule, it can be applied to many mathematical concepts in the same format to achieve correct answers. Relational understanding is pretty much the opposite of this as it is much more difficult to understand and it is so much more time consuming that just applying a rule. However, because relational understanding is already present, the learner can take what they have learned and easily adjust it to a new mathematical concept when a new task is introduced. This approach can also be its own goal since having the understanding of complex mathematical concepts can be rewarding in the first instance.