Profound Understanding of Fundamental Mathematics (PUFM).
This post probably contains the most powerful key to teaching and learning mathematics. So if you are home educating, tutoring or a teacher who has control over what you teach then this post is for you. The key to powerful teaching and learning is PUFM.
PUFM is the thing that separates the people who are good at math from the people who struggle. It separates the kids who never quite 'crack' math, from those who always appear to 'just get it.' It separates the tutors/teachers/parents who are great, from those who are average. It is the "difference that make the difference".
Why? Well, first a little story... Liping Ma is a researcher who wanted to discover why Chinese maths teachers, despite being less qualified and resourced, get far better results than their American counter-parts. That's right, highly qualified American maths teachers (with a significantly larger budgets) get worse results than the Chinese teachers.
Have a think about how and why the Chinese maths teachers get better results?
Perhaps it's culture, family pressure, better behaved children, a natural affinity for mathematics? It isn't. While these account for some small variance none of these explains the quite significant difference.
The answer lies in PUFM. It stands for a 'Profound Understanding of Fundamental Mathematics'. It refers to not only knowing how to 'do' maths, but profoundly understanding the workings of mathematics. Not necessarily understanding higher level mathematics, but how the number system works and how methods work and why.
The Chinese teachers, although less qualified, had a PUFM and they used this to develop a parallel understanding in their students. In one example Liping Ma examined the difference in how the Chinese and Americans taught their students to 'carry' when using an additive algorithm. The American teachers showed the students how it was done using a blackboard and then had their students practice the method by solving additive problems in their workbooks. They did about 30 additive problems every lesson. That's about one every minute and a half.
In contrast the Chinese teachers taught their students why carrying worked. They had the students use equipment to recognise that you could not have more than nine units in a single place, and therefore could carry a ten to the following place. The Chinese students did an average of THREE problems a lesson but explored those three problems in depth. So they spent an average of fifteen minutes per problem.
You see, the Chinese teachers were experts at getting their students to understand why carrying worked. The Americans were experts at 'getting' their student to carry. Understanding fundamental mathematics is necessary to continue to understand mathematics. Unless you understand mathematics you will not be able to use your knowledge to learn more. You may be very good at solving 'canned' problems but you will struggle to learn new maths and apply your skills to unique situations.
So, the trick is to begin to incorporate more of the Chinese philosophy into your current practice. Less of asking students to practice solving similar problems and more digging down into the workings of mathematics. Make the hidden workings of math methods explicit to your learners and you will begin to see drastic changes.
A project for you - if you wish to accept the mission! Why and how does the formula πr2 actually work (that's 'Pi R squared')?
Once you know - then you will begin to have an understanding of its underlying mathematical workings. A little hint is found here.
PUFM is a journey - I began it a while back and have loved every minute of it. The joy of maths is not only found in solving maths problems that no one else can solve - it is found in understanding how simple things actually work.
Long live the joy of maths.