Maths Is Easier When You Understand It.

I sometimes wonder just what we have to do in maths education to demonstrate that things are more easily remembered, used and developed if the learner understands what she/he is learning. This interesting piece of recently published research supports, for at least the millionth time,  that when students learn conceptual knowledge, along with procedural knowledge of maths, it is much, much, much, more effective. The neat thing about this piece of research is that it is for high school maths which has tended to lag behind the research on this topic at the elementary school level.

Probably the best way to distinguish between the two types of knowledge is the idea of pi. If I were to ask you what pi is you would probably say "3.14 but that's all I can remember". If that's all you know about pi then you have a small piece of procedural knowledge that you can plug into equations to find the area or circumference of a circle. Now, imagine that you have some conceptual knowledge to go with this. For example, knowing that pi is a ratio between the diameter and circumference of a circle would be immensely helpful to learning all sorts of more advanced maths. It would also be helpful for doing things in one's daily life. The fact that pi is a ratio means it is just over three times further around a circle than it is across the middle of that circle. If you think of pi as a fraction, 22/7, then the circumference could be 22 inches, or feet, or miles, and the diameter would be 7 inches, feet, or miles.

If I asked you to count by fives you would probably say, "5, 10, 15, 20, 25" etc assuming that I meant you to start at 0. So try counting by fives again starting at 3. The first few will be tough but you'll soon see the repeating pattern of 3s and 8s. Seeing the pattern is a piece of conceptual knowledge because you are applying your conception of counting by 5s and 10s rather than just parroting the numbers.

We have long known that conceptual knowledge of maths is as crucial to learning maths as procedural knowledge. Jo Boaler, the amazing Stanford professor and founder of YouCubed  was one of the first to demonstrate the importance of conceptual knowledge in a wonderful piece of research when she was at Liverpool University in the UK. She discovered that elementary aged students who were taught conceptual knowledge along with procedural knowledge did much better in high school maths that those who were only taught procedural knowledge at the elementary school, One of the interesting findings of her study, if I remember correctly, was that the conceptual/procedural knowledge taught students didn't. do as well on the end of elementary school maths tests as the procedural knowledge only taught students. The reason: the tests only tested the students' procedural knowledge of maths; a problem we still grapple with today.