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10th January 2003 at 00:00
Children learn much faster if they understand the thinking behind mathematical concepts, says Colin Hannaford

What is the magic formula that will guarantee any pupil a happier, healthier, wealthier future? Teach them to understand their maths book. There. Did the earth move? The heavens explode? No? Then let's start again.

In The TES three years ago, I reported on an important discovery by Professor Eva V s rhelyi of the Eotvos Lor nd University of Budapest. I was especially excited, because I had recently made the same discovery myself. Talking to students about mathematics will lead to greater insights and a real understanding of the subject. Since I started using the method in the classroom, the number of students at my school taking the stronger maths option in the baccalaureate has increased significantly.

Children who fail in maths are commonly believed to lack natural aptitude. To make an omelette, a child must follow instructions such as "butter the pan" or "beat the eggs". But what if they never understand why they need to butter the pan, or beat the eggs? What if they can never make anything except an omelette? Isn't this what you might also call "lacking natural aptitude"?

The methods I use to break down barriers to understanding don't require computers or any other expensive equipment; they simply require time, patience, and textbooks.

It's the beginning of term. Thirty solemn faces wait for their first maths lesson in big school. If they are statistically normal, a quarter failed their literacy target, and in two out of five homes no parent has read a single book. These are the children who are disadvantaged as soon as they walk into school. It is your job to prove to them that their intelligence is not correlated with their parents' social status.

The following exchange is typical of my classroom at the start of each autumn term.

"Open your book at the first page." I request. "Will someone please read what it says at the top of that page?"

A clear voice begins: "The addition and subtraction of whole numbers."

"Stop!" I insist to their surprise. "I suppose you all did a lot of adding and subtracting last year." All nod.

"So, will someone tell me just what adding whole numbers actually means?

"It means putting numbers together."

"To do what?"

"Well, to add them."

"Which means?"

"You get a bigger number."

"You mean actually a BIGGER number."

"No, no; it should be the same size, but, well, just larger."

"But now just one number."


"So you start with two numbers, and a sign, the plus sign, and even another sign, the equals, and you end up with just one number, not bigger, only - did you say larger?"

"No, I mean it isn't larger. It means something larger!"

"But what is it that is larger?"


Nearly every year I have a conversation like this. These children are doing their best to explain what they have been taught to do and yet they do not know the necessary words.

They cannot say that a number is the simplest name of a group and that addition is giving a group of groups its new simplest name.

I do not make fun of these children but unless we teach them to talk about maths with understanding, they will never be able to think with understanding. To teach what the words in a maths textbook mean is immensely interesting and rewarding. It is also tiring.

But the effect is miraculous. Pupils who have the least experience of reading at home and the least experience of logical argument benefit the most. It also benefits teachers with less experience and advantages are soon apparent for experienced teachers. They soon find pupils display increasing confidence, self-reliance, enthusiasm and trust for each other. And children who are trained like this, to be self-reliant and self-directed, will continue to learn far more independently for the rest of their lives.

Shortly after I began to teach like this, a boy asked one day if he could tell me a secret. Waiting until all his classmates had gone, he whispered:

"You know this method you are teaching us" I nodded. He leant even closer:

"I have found out: it works in other subjects too!"

Colin Hannaford is head of mathematics for junior secondary classes in the English section at the European School, Culham, near Oxford. His workbook, Socrates' Method for 9 to 19 Year Olds, Learning Mathematics in 10 Minutes a Day, pound;5 inc pamp;p, is available from the Institute for Democracy in Mathematics, 10 Marlborough Court, Oxford OX2 0QTTel: 01865 793752 Email: Colindemocracy

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