Go forth multiply
Ruth Merttens turns the tables in the second of her series on teaching numeracy.
Some teachers have observed that multiplication (like most numerical operations) begins with counting. Here the image of the number line is crucial (see below). The washing line with numbers pegged along is a part of every good infant classroom - and is an essential piece of equipment in the juniors as well. Children can be asked to unpeg numbers and describe which come "in-between" two other numbers. They can use it to look for the nearest 10, to round up or round down, and for a host of other things.
We can vary the numbers which are pegged along the line, using it to illustrate any set of numbers we need. Thus we can have decimal numbers, 0.1, 0.2, 0.3, 0.4, or four-digit numbers, 3,489, 3,490, 3,491, 3,492, or negative numbers, -1, -2, -3, -4, -5. In the context of learning tables, the number line can be used for multiples. It will quickly become apparent just how useful this is.
Children start by counting in ones, using their fingers (see last week's TES). They progress rapidly to counting in twos and in fives. Two, four, six, eight, holding up one finger for each number spoken. The number of fingers indicates how many twos. Six fingers - six-twos are 12. Peg these numbers, 2, 4, 6, 8, 10, 12 I along the line to reinforce the chant. In this way, children can become confident with their two-times, five-times and ten-times tables by the time they are seven. Although they will not necessarily have instant recall of every one of these 30 number facts, they will have a fast, efficient means of finding the answer.
In the infants, children also need to learn to read number sentences involving multiplication. The vocabularies which teachers provide are crucial here; children need a way of reading number sentences which will help them to make sense of what it says andalso enable them to perform the necessary numerical operation. Thus, we teach 3 x 4 as three lots of four, three sets of four, and then, later, as three rows of four.
Reading 3 x 4 as "three rows of four buns" provides a way of making sense of what is being said and helps the child to work out the answer by counting the rows. Thus, they count in fours - four, eight, twelve. We can see here how this connects with our counting in twos, in fives and in tens which the children have perfected through their finger-chanting with help from the number line. Counting is the primary skill; we teach a reading of multiplication sentences which enables them to use and build on it. They then come to count in rows of three, or rows of four, or rows of six, and so on. This links directly to their developing knowledge of these multiplication tables.
We can now begin to envisage a staged process of teaching (see table below).
The order in which the tables are taught is important here. Building on their earlier counting, we encourage counting in threes, using finger joints in the Indian style: Hang the multiples of three on the number line to reinforce their recognition. The nine-times table can be taught using the finger method described last week, where one finger is turned down, and the answer read off the remaining digits as tens and units.
At this stage, it is important to stress to children that they already know more than three-quarters of their tables facts. They know the one-, two-, three-, five-, nine- and ten-times tables - and many facts from other tables are included in these. Thus, they know 8 x 3 so they also know 3 x 8. This point can be emphasised by using the tables square and colouring in the known facts.
It is vital that children from seven onwards perceive division as the inverse of multiplication. A good teaching strategy is to call divisions "multiplications with holes in". Thus 12 V 3 is read as "how many threes in 12?", or "what times three is 12?", which is the same as n x 3 = 12 (a multiplication with a "hole"). The same vocabulary can then be used for both. 42V6 is read as "how many sixes in 42?" and to find the answer, the children count in sixes until they reach 42. Six, twelve, eighteen, twenty-four. ..they hold up one finger for each number spoken. Seven fingers. There are seven sixes in 42.
Later on, when we need to teach more complex divisions requiring remainders, the washing lines of multiples become indispensible. The formulation 4 )37 is read "how many fours in 37?" and is addressed by counting in fours or by finding the nearest lower multiple on the line - 4, 8, 12, 16, 20, 24, 28, 32, 36, 40 - the children count along the line to 36. Nine fours are 36. There are nine fours in 37 and one left over. The lists of multiples can be provided on snakes hanging down from the ceiling so that all the relevant tables are available.
Personally, I never give the children the twos, fives, tens or nines (which they should already know). But providing the other tables helps them to engage with the process of performing a division with a remainder. Once they have the hang of this process, we can gradually require that they do without the lists of multiples.
Teachers can and should chant the tables with the children. We can count in multiples - in threes, fours and so on. We can chant the tables as multiplications - one three is three, two threes are six. We can consistently read division as multiplications with holes in - how many fours in 20? - and relate these to their tables knowledge. Tests and little quizzes all help with this memorisation. Give a quick "l0-question" test first thing in the morning and have the children mark their own. Dismiss them at break time by giving them a tables question each. "Annie - seven fives?" "Thirty-five." "Good, you can go out to play." "Jimmy, six eights?" Vary the difficulty to suit the child. The washing line numbers and lists of multiples help them to see possible answers and reinforce rather than hinder their memorisation.
A final but consoling point. Even when teachers use all of these techniques and others, and are as enthusiastic about number as it is possible to be, some children simply do not achieve instant recall, while other numerically confident children find their own ways of using key facts and arithmetical strategies to get fast answers.
Professor Ruth Merttens is the co-author of theABACUS Maths Scheme, published by Ginn