Create oral lessons to be passed from student to student
Two years after retiring I returned to my last school to take up the challenge of teaching a blind boy. While I was there I spoke to the mother of one of my former students. She was in charge of the Special Needs Students. I learnt that her son had said to her that I could teach maths to a stone and she told me a story. I’ll tell it to you in a few words.
One day a boy came home from school.
‘What did you do in maths today?’ asked his mother.
‘Mr Grant just told us stories,’ replied her son
‘Tell them to me,’ said his mother, and when he did his mother smiled.
‘Why are you smiling?’ asked the boy.
‘He’s been teaching you maths again,’ was her reply
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Once, I told a year 10 class of boys a story about a young 
cowboy on a cattle drive. There were rustlers, a storm and a cattle stampede. Just before the end of the cattle drive the owner of the cattle, who was called Grandpa, joined the cowboys for a cup of java by their campfire. Now the young cowboy was fascinated by Grandpa’s cattle brand so he asked him about it.
‘Shucks son,’ said Grandpa, ‘My old maths teacher showed me that and I have never forgotten it.You see I had terrible trouble trying to add and subtract fractions and he showed me how to do that by branding them.’
At that point in the story I drew Grandpa’s cattle brand on the board. It consists of two crossed arrows balanced on the edge of a bowl shaped curve taken from a section of a circle. The boys, like the young cowboy, were fascinated by the brand because they strongly identified with him. Although it was still a story, it had seamlessly moved into a mathematical lesson.
‘I’ll show you how it works,’ said Grandpa, and taking a stick he drew a quarter plus two fifths with an equals sign on the ground.
I wrote a quarter plus two fifths on the board with an equal sign, and acted out the part of Grandpa for the rest of the story. I then drew a curve between the 4 and 5 on the bottom lines of the fractions and put a fraction line on the right hand side of the equals sign. ‘Just multiply the bottom numbers together and put the answer on the bottom line of the fraction.’ Then I drew an arrow from the 5 in the bottom line to the one in the top line. ‘Multiply the 5 and the 1 and put the answer on the top of the fractions and the left hand side,’ I said and did it. The next step is to draw an arrow from the 4 on the bottom line to the 2 on the top line. ‘Multiply the 4 and the 2 and put the answer on the top of the fractions and the right hand side,’ I “Sometimes there is magic in words, and stories are full of words. Word – weavers use this magic intuitively to pen powerful stories.“ said and did it. ‘Now put the addition sign between the 5 and the 8. Then just add the numbers in the top line and write down the answer on the next line.
After that the class practised branding fractions so they could add and subtract them.
The lesson worked amazingly well, so well in fact that the lads in this class began retelling the story and teaching other junior students how to use Grandpa’s cattle brand, so that they could add and subtract fractions too. Then the story was retold by them and some teachers were told too.
I tell other mathematical stories as they are an integral part of my teaching. This next story works like a Concrete Concept, linking the development of mathematics to the development of humankind and it will give its readers/listeners an insight into the nature of mathematics. It is a good story to use when you are teaching students about the different number systems. The follow up story on Complex Numbers involves discovery learning and offers students an insight into the creation of new mathematics. It is a Pinnacle Lesson and has a philosophical dimension which is easy for students to understand. It teaches them more than just Complex Numbers. Pinnacle Lessons are multi-dimensional too.
The Number Systems Story
At the beginning of last ice age a tribe lived in caves. The women were food gathers and the men hunted. One day, as the ice advanced closer to theircave and it became colder, the scouts, who had gone out to find the one of the small herds of animals and count them, returned with disastrous news. All the animals were gone and they could give their leader no number, because they didn’t have a number for nothing. They only knew the counting numbers.
Their leader, who was a wise man, just nodded his head. ‘The herds will be moving south, we must follow them,’ he said and they did.
Other animals like wolves also followed the herds to hunt them. When the tribe began to protect the herd from these predators, they began to make the transition to herdsmen. During this time the idea of nothing was assigned a number by the leader of the tribe. That number was zero. When it was added to the Natural or Counting Numbers, the new number system was called the Whole Numbers.
When the tribe became herdsmen, they became more independent and began to trade animals amongst one another. Some herdsmen came to owe animals to others. It was then that the negative numbers came into being and were used for to describe animal debt. These negative numbers were added to the Whole Numbers and the new enlarged number system was called the Integers.
The women were still food gathering while the men hunted and cared for their flocks. At certain times of the year they gathered grain, and enterprising women sowed seed in suitable places in the spring and the herdsmen would return and harvest it in the autumn. They also planted other beneficial plants where they wandered. It was the beginning of farming.
In time, some of the herdsmen settled in fertile areas, grew crops and tended to their animals. They discovered metals, became knowledgeable and began measuring and dividing things. From division came fractions and when the wise men added the fractions to the Integers they called this new numbers system the Rational Numbers. The word ratio means fraction. The wise men just added the letters n, a, and l to the end of the word ratio. At that time the wise men thought they had discovered all the numbers, but they were wrong again.
At this time they were measuring lengths using tape measures and the mathematicians came up with the idea of a number line to help explain the mysterious nature of number to the people so that they could use numbers in their everyday life.
Much later the decimal system was invented and it was much easier for people to use than fractions. Then a mathematician discovered that every fraction has a decimal pattern.
For example:
A half is 0.50000……
One ninth is 0.11111……
It was then that this mathematician realised that there were other numbers like Pi that had no decimal pattern. This was a great revelation. There were numbers between the integers that were not fractions because their decimals had no pattern and the mathematicians decided to call them the Irrational Numbers. The Irrational Numbers were then added to the Rational Numbers and the enlarged number system was called the Real Numbers. Once more the mathematics thought they had discovered all about numbers, but they were wrong again because mathematics is a never ending story.