When you are learning maths, there are two types of understanding you can develop: an instrumental understanding and a relational understanding. One is far more beneficial to the learner than the other.
Instrumental understanding: If you have an instrumental understanding of a topic, you have memorised different methods to answer questions. For example, consider the problem of ½ + ¼. Using a method (cross-multiply and add), you can change the fractions to 4/8 + 2/8, then add the numerators which gives 6/8 which is equal to ¾. It might not make sense what is happening, but as long as you can follow the steps you’ll reach an answer.
Relational Understanding: If you have a relational understanding of a topic, you can relate the topic to other areas in mathematics or diagrams. This means your understanding has a context. Reconsider the problem above of ½ + ¼. This time you can think about it in terms of, say, a pizza. Imagine I have a full pizza and cut it into a half and two quarters. I put a half and a quarter on the plate. How much is on the plate? Your answer should be ¾. To reach this answer you have related the question to a diagram of a circle.
So which is better? Education professionals are hesitant when it comes to prescribing ways to teach and learn, but very few would choose an instrumental understanding over a relational one. If you have an instrumental understanding, not only is it difficult to remember the method for the exam, but as soon as the question is slightly changed you need a whole new method to find your answer! This can become long-winded and is the reason a lot of students give up on maths.
Ideally, students will use a method to find the answer to questions and have a relational understanding of the method to keep them in check and give them an idea if something is going wrong.
I hope this article has given some food for thought. If you’d like to talk more about this or about other topics related to maths then I’d be happy to hear from you :)