## 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 πr

^{2}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.

I've got this book... I think I'll have to read it now.

ReplyDeleteHmmm, Its good in a researchy way. Great overall info, but its not going to get the adrenaline pumping.

ReplyDelete