All too often I’ll begin tutoring a student and discover that, in the topics they are struggling with, they are lacking a basic understanding of the mathematical principles they are attempting to apply. Take something as simple as multiplication. Many young students learning their multiplication tables don’t know (or remember, if it was mentioned to them) that multiplication is a short-cut for addition. This is an important concept! Not only should it be taught as the student is being introduced to multiplication, but it should be reinforced time and again as it is easy to forget what you’re doing when you’re just rotely reciting and memorizing your times tables.
Teaching and explaining why mathematical definitions, procedures, and algorithms work is just as important as teaching how to do them. It upsets me to hear that students, even when asking directly or expressing to their teachers that they are struggling with a particular topic, aren’t getting the simple explanations they need. I find that when a student is explained and shown why and how a mathematical principle works, it suddenly clicks for them and they no longer find themselves struggling. Practice and repetition are then important afterward, to reinforce the concept and help in the memorization of any steps required. This approach to teaching mathematics is true from the very basics, for example adding and carrying or multiplication, to advanced maths such as calculus and trigonometry.