Teaching Numbers(i)

Early numbers

Teachers tend to rush early number work in order that their pupils can add up as soon as possible.
This is often a false achievement that will make future progress much harder.

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Children should be able to:

A brief history of number
Early man used the fingers for counting. This was the origin of the decimal system with ten used as the base for counting.

A tally stick was developed by cutting notches, or scoring marks, onto a piece of wood.
The English word score is still used to mean twenty (i.e. 10 fingers and 10 toes), and it also relates to the number of points, runs, goals scored in a game (i.e. the score is 3-1).

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The ancient Babylonians used six and sixty for their number base.
There are still traces of this system when we use 60 minutes and 60 seconds in the measurement of time, and in 360 degrees in a circle (6 segments of 60).

Some cultures used a base 12 system, and we have been left with 12 hours in a day and 12 months in a year, and there used to be 12 pence in a British shilling.
The English word dozen shows the importance of this number.

Writing on papyrus, bark and clay tablets still kept the marks from the tally stick and even today the Chinese and Roman scripts show 1, 2 and 3 by strokes like I, II and III.
The introduction of separate symbols for individual numbers was a big advance so that IIIIIII can be expressed either by a word, seven or a symbol 7.

Egyptian hieroglyphics, or pictorial symbols, evolved over 5000 years ago, and they represented the numbers from 1 to 9 with straight strokes, the number 10 was a reversed U representing a hobble for restraining the leg of a cow, and 100 was represented by a curled measuring rope.

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The Greeks used alphabetical symbols for their numerals. Their first two numbers were called, alpha and beta, and the word alphabet was derived by joining these two words.
Roman numerals are still used on clock faces and for page numbering, and the system was based on seven letters: I, V, X, L, C, D and M.

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The now internationally common Arabic numerals were adopted by most European countries only in about 1500. Surprisingly, they are not Arabic in origin, but Hindu, as they evolved in India some time between the 2nd and 6th century AD, and are named Arabic as Arabic writers and travellers spread the knowledge of them across Europe.

A calculating device that has served for many centuries as a halfway house between human fingers and the modern computer is the abacus. Its name derives from Greek abax, a term used for a board covered in sand on which calculations could be traced, and itself borrowed from the Hebrew word, abhaq, meaning dust.
The frame and beads, which make up the abacus, can often be seen in Chinese shops, and it is an ideal instrument for showing the place value system to children. A simple abacus can be made on the ground outside the classroom, using stones instead of beads. In fact, the Latin word, calculi means a small stone and the word calculate derives from using stones for calculations long ago.

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Pre-Number Activities
It is vital that teachers see pre-number work as an essential stage in the development of numeracy. It should not be rushed.

Group Work
Children should work in small groups. Groups are essential for good, effective mathematical teaching and learning. It should be stressed that children are at different levels of mathematical ability and readiness, and some sort of grouping should take place; not for all the time, but for much of the time during mathematics lessons.

Number Facts
Some memory work is necessary in order to gain numeracy, but the activities to help children remember the number facts must be interesting, motivating and effective. Children start to think about numbers when they handle and sort concrete objects.
This sorting will help children to:

They can make these comparisons without actually counting during activities like sorting, ordering and matching.

Question: Why is sorting an important pre-number activity?

Teachers will observe how their own classes improve when sorting activities are encouraged.
Children can sort such material as leaves, seeds, flowers, stones, sticks, nails, string, wool, geometrical shapes, bottle tops, etc. The sorting can be performed on the floor, on desks or table tops, or on the ground outside the classroom.

There are various ways to organise the sorting activities in your classroom:

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Ordering - Vocabulary:
Ordering - putting in order.
Seriate - putting in series.

Make about six collections of material that can be put in some order, i.e. stones, twigs, cartons, strips of paper, string, bottles, books, leaves, etc.
Give out a collection to each group of children. Encourage them to put the collections in order.

It is important that children recognise similarities and differences between various objects and pictures.
These are skills which are essential for the recognition of numbers and number patterns and the symbols used for representing them.

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One-to-One Correspondence
Children use one-to-correspondence naturally. Select two groups of children containing the same number in each group. Let the two groups join into pairs, and because each pupil has a partner we can say that there is an equal number in each group, without counting. Each pupil in the group has a corresponding child in the other group.

Do the same activity with two unequal groups so that one or two pupils do not have a corresponding partner.
What can be said about the groups now?
Which is the larger group?

Before children can count they use this method of one-to-one correspondence to compare groups. Do the same exercise as above, but use objects instead.
Find different ways for linking the objects in one group with the objects in the other group (i.e. chalk lines, paper strips, string, etc.)

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It is important that children compare various groups of materials before they can actually use numbers to count them.

Conservation of Number
Young children do not conserve (keep or store) numbers in their minds. Children find conservation of number difficult. Do not expect them to do complicated numerical exercises before their conservation skills are mature enough.

One can test conservation skills in children by putting out two groups of five objects arranged like this:

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Before children can conserve numbers they think that there are more objects in the spaced-out group of objects on the left than in the group where the objects are closer together on the right.

Children see what their eyes tell them:

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Give children many activities related to one-to-one correspondence and this will lead them to be able to conserve numbers.
Without these activities children will find early number work very difficult.

Number Recognition
Teachers and parents are often impressed with a child who can count from an early age. This reciting of the counting words from 1 to 10 has little to do with real counting, but the recognition of five objects, and naming that group five, is an important step in early number work. Children must know the real values of 1, 2, 3, etc., before they start more formal number work.
Common numbers are used to let children appreciate the numbers around them like '2 eyes, 2 feet, 1 nose, 4 chair legs', etc.

Mathematical games

Playing mathematical games is important at all levels of mathematical development. Games should not just be given to children after they have finished their work or to make them keep quiet.
Most teachers realise that games are a vital part of mathematical teaching and learning, which help them to gain mathematical concepts and skills, and challenges them to practise their number facts with greater speed and accuracy.

Games enrich the teaching of mathematics by stimulating interest and adding enjoyment to the learning process, as well as giving the important repetition of number bonds which children need for their foundation work.

Mathematical games are important because they:

Here are a few suggestions for introducing games to children:

Some teachers may fear that number games are not on the Syllabus so they should not be taught. Here are a few extracts from a typical Syllabus which would be relevant for introducing number games...

The child should be able to:

Games also promote the use of language between pupils while engaged in purposeful and enjoyable activities. Mathematical development is equally rooted in language development, and the ability to talk about mathematics is vital to being able to read and write about it.

Children need constant classroom experiences of talking about mathematics. The best way of organising classroom talking is to allow children to work and play in groups.
Give these groups simple problems, games and challenges.
You will find that pupils will gain confidence as they are allowed to talk about their mathematics.


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