It can be shocking to discover just how little some children understand about how numbers work. When we ask an 11-year-old to add 10 to 27 and he counts on in ones, or when we ask for an approximate answer to 19 x 32 and he cannot choose between 60, 600 and 6,000, we realise that something more than a few more pages of written sums is required.
Children need to build up mental models of how numbers work. They need to develop both a familiarity with numbers and a suitable vocabulary. The language which teachers give children is crucial. Quite literally, children need to articulate their mathematics; both to speak it and to make the connections between one part and another.
Teachers can help by encouraging oral repetition and counting. When chanting numbers as a class, or responding as a large group to teacher prompts, the children gain both a memory of, and a confidence with, some of the routine aspects of number. They no longer have to think about what they are doing when performing simple number operations such as adding 10. It becomes automatic. Teachers can also support children's growing familiarity with numbers by providing the vocabulary they need. They teach the terms and phrases which enable children to "talk through" what they are doing as they add and subtract, multiply and divide.
There are a number of key skills that make all the difference to children's fluency in number (see right). If these are not in place by the time children are 10 or 11, they may become further entrenched in a pattern of underachievement and lack of confidence in maths.
There are strategies that help in the teaching of these skills. The most obvious is oral repetition. Chanting in tens begins in Year 1 and continues through Y2, Y3 and Y4. First chant the multiples of ten, and then progress to counting in tens, starting with a single-digit number: "Six, sixteen, twenty-six, thirty-six . . ." The best method is to start by counting in unison as a large group or whole class.
Little and often is the key. Once the children are confident, start the count by saying a number, for instance "Nine!", and then point to a child. They say the next number in the chant, "Nineteen!", then choose a child who says the next number, and so on.
Once the children are confident chanting in tens, adding a multiple of 10 is simple. Faced with 45 + 30, the children count on in tens, "Fifty-five, sixty-five, seventy-five", holding up a finger with each number spoken to help them keep track of how many tens they are adding. The same methods can be applied at a later stage (Y4Y5) to counting in hundreds and adding multiples of 100.
Similarly, counting in unison forms the basis of rounding up or down. To round numbers to the nearest ten, we need a number line from 1-100 made of large colourful numbers going round the classroom.
This is then used to help children find the nearest ten. The teacher says a number, such as 63. She chooses a child to point at that number. Then she encourages the children to choose the nearest ten and count down to it. Sixty-two, sixty-one, sixty. Sixty is the nearest ten. This supports children's understanding of the size of numbers. They develop a mental picture of numbers as relative positions in a line or series, and this helps them to approximate later on.
Clearly, chanting the multiples plays an important role in helping children to memorise their multiplication facts, particularly in relation to the 2x, 5x and 10x tables. Counting in threes or in fours along the number line reinforces these multiples as numbers to remember. Stick a coloured ring, or a sticker, on every third or fourth number and spend a week focusing on that set of multiples.
The key to learning the number bonds is little and often. And most of the important work is the oral maths, which takes place on the rug, not the written reinforcement.
In the first article of this series, we stressed how children build up a kinesthetic memory of number bonds to 10 by using their fingers, and develop instant recall. In Years 1 to 3, these bonds need to be rehearsed once a week at least. Dismiss or welcome the children using a "number-response" pattern. Call the register with "Seven, Alice", and the child responds "Three, Miss!" In the upper juniors, introduce number bonds to 100. Hang six balloons from the ceiling. On one side, written in felt-tip, are the bonds to 10 (4 + 6, 8 + 2, etc). On the other side of each balloon are the bonds to 100 (40 + 60, 80 + 20, etc). Show the children that if you know the number bonds to 10, you know the multiples of ten which add to 100. Once the children are secure with these facts, move on to the harder bonds to 100. Teach these using zig-zags.
First demonstrate these on the board, and then invite some of the children to come to the front to talk one through for the whole group. Then give each child a card with a number between 50 and 90 on it, and ask children to work in threes or pairs to find the number which matches theirs to make 100.
When the children are confident about their zig-zags, start encouraging them to perform this operation mentally. Talk it through. Card 67, 3 more to 60. Forty more to 100. That's 43. To practise, play a form of number-bond football. Put the children in teams of four or five. Each team get a card, and have to agree the matching number to make 100. If they are correct, they score a goal. If they are wrong, they score an "own goal" and the other teams get a point. Or play bingo, where pairs of children work together. They write 10 numbers on a piece of paper. The teacher has cards 10-90, shuffled. Turn over one card at a time and call out the number. If a pair of children have drawn a number which matches it to make 100, they can cross that number out. The first pair to cross out four numbers are the winners.
Doubling and halving can be broken down in a similar fashion for teaching purposes. Make or draw a "doubling machine". Feed in a number, say 13. The machine doubles the tens first (20), then the units (6). Answer: 26. Try a harder one, say 27. Double the tens (40), double the units (14). Answer: 54.
Once children become confident with certain routine aspects of number, they will start to use this knowledge in their more general number work. Thus, the child who is confident adding and taking away tens will address the calculation 45 - 29 by pointing at the 29 and saying, "Call it 30!" This is what is meant by transforming a calculation to make it easier to do. Similarly, the child who knows how to multiply by 10, when faced with 9 x 24 will say, "Call it 10 times 24!" and can then subtract the extra 24.
We have been talking about teaching on the rug, about addressing fairly large groups of children and encouraging them to talk through their maths. This process has three huge advantages. First, the maths is oral and mental. We speak it first and then think it. Second, it provides an opportunity for the teacher to demonstrate how to do something, to talk children through a method (such as the zig-zags) or to provide a strategy (such as fingers). Third, it enables weaker children to be helped by those who are stronger, partly by being able to imitate, and partly by being able to respond. Stories and rhymes have long been part of teachers' "rug" repertoire. It is good that maths is now seen that way as well.
1. Knowing the number bonds to 10 and, by extension, to 100
2. Counting in tens, or in hundreds; adding multiples of 10 or 100
3. Rounding up or down to the nearest 10, 50 or 100
4. Doubling and halving
5. Multiplying and dividing by ten
6. Having a sense of the size of numbers and being able to estimate an approximate answer
7. Knowing key multiplication and division facts, and being able to give a quick answer to the others
*There are also more general skills
Choosing an appropriate method to perform a particular calculation with reference to the actual numbers involved
Transforming a calculation to make it easier to do
* Starting with 67, ask how many to make the next 10 (Answer: 3)
* What is the next 10? (Answer: 70)
* How many to get to 100? (Answer: 30)
* Read the number along the bottom line?(Answer: 33)
Ruth Merttens is co-author of The Abacus Maths Scheme, published by Ginn