Relational Understanding and Instrumental Understanding

A major question in mathematics education today is what qualifies truly understanding the material. There are generally two different approaches to understanding: instrumental understanding and relational understanding. Instrumental understanding is having a mathematical rule and being able to use and manipulate it. Relational understanding is having a mathematical rule, knowing how to use it, and knowing why it works.

From the definitions given, relational understanding includes instrumental understanding and more. Instrumental is simply knowing and applying the rule, while relational is knowing and applying the rule while also being able to know why a rule works and connect one rule with another. Both types of understanding give the correct answers, but relational is much more extensive.

Although relational understanding is often thought of to be a better alternative to instrumental understanding, there are advantages and disadvantages of both. Often the advantages of one type of understanding are the disadvantages of the other. Instrumental has three main advantages. The first is that it is easier to understand, often to a great extent. Some topics are difficult to grasp and can much sooner be learned through just using rules and set computations than through knowing why something works the way it does. The second is that the positive results are instantaneous. Once the rule or algorithm is learned, the student can use it to do many problems in that format and get the correct answers. The third advantage of instrumental learning goes along with the second in that the correct answers can be obtained very quickly and consistently. Instead of relational learning, where the thought process is longer in trying to understand the problem, in instrumental learning, once the student is able to follow a rule or algorithm, they can do problems that apply to that rule rapidly and always get the right answers. All three of these are disadvantages of relational understanding. Having to think about why something works the way it does or how it works is much more difficult than using rules. It is also more time consuming and involves a lot more thought. This applies to both teaching relationally and actually doing the problems. For each problem, the student is relying on his or her understanding of the idea to solve the problem, which may be correct or incorrect. When the students simply learn and use rules that are always there for them (instrumental), they will always get the right answer. There are four general advantages to relational understanding that are disadvantages to instrumental understanding. The first is that relational is easily adjusted when a new task is introduced. The students can take what they have already learned and apply and adjust it to the new idea because they understand why it works. This does not work with instrumental because the students can only do problems that fit within the rule they learned. Although a rule may be very similar to one they already learned, they will not discover it on their own because they don't know why the first rule works. The second advantage to relational is that students can more easily remember what they've been taught. If they know general ideas about why computations work the way they do, they can connect them with other ideas and therefore remember them more easily. On the other hand, in instrumental understanding, a student has to remember many separate rules that seem unconnected from each other. The third is that relational learning turns into its own goal. External benefits are not needed as much as they are with instrumental because just through fully understanding the idea the student is rewarded. The last advantage is relational knowledge naturally grows. Once a student understands relationally what has been taught, he or she will want to expand that knowledge and look for new concepts to apply it to. This does not happen in instrumental where the student learns a new rule and is content until given the next rule because he or she does not know how to make the connections to expand his or her knowledge.