Part 1 Finger maths
Evidence from research and practice suggests that many children are not getting enough "up-front" teaching, particularly in numeracy. In response, the Government's new numeracy centres are suggesting a model in which whole-class or large group teaching plays an important part. However, not only are most teachers not trained in whole-class teaching, many were not even taught like this themselves. Where then, are the techniques to come from? Just how do we teach 30 or more children effectively, and which numerical topics lend themselves to a more up-front, exposition-type approach? This article, whi ch deals with infants, and its two sequels, offer some practical suggestions.
Fingers are the oldest and the most effective calculators. Right from the moment when children enter the nursery, fingers can become their most useful arithmetic aid. This is because they can serve a triple function. First, they are objects in their own right. They are physical "things" which may be seen, touched and counted. Second, they are also signs - that is, they can stand for something else. Thus, doing 5 take away 3, the child can fold down one finger for every fish that the crocodile eats, and can then see how many are left. The fingers represent the fish; they are signs, carrying a symbolic function. Third, fingers can act as mnemonics, helping children develop their memory. Thus, doing six fives, if we chant "five, ten, fifteen, twenty, twenty-five, thirty..." holding up one finger for each number spoken, they remember the chant through movement. We reinforce how many fives - six fingers as we say "30" - with a movement. Later, all that will be needed is that same movement of the finger-tip to trigger the memory.
Fingers, then, are powerful tools in helping children learn number. Most numeracy work with fingers is done by the teacher with the children sitting around her on the rug. Fingers are used by the teacher to demonstrate the numerical operation, which the children then imitate. The teacher and the children can repeat the process many times. Little and often is the key, with 15 or 20 minutes a day being preferable to an hour once a week.
We start with counting. With children as young as four, chant the numbers daily, matching fingers to the count.
If children have difficulty in matching the spoken number to the numeral, it is a good idea to write the numerals on their finger tips so that as they count and raise each finger, the spoken number matches what they see on their finger.
Start by counting one to five and then extend the count to larger numbers and speed up the finger matching. Children will rapidly become very adept at matching the fingers to the count. Once they have mastered the count to 10, count down as well as up. This time, fold down one finger for each number spoken and don't forget the "blast off!" at the end.
When they are secure in this, they can use their fingers to help them add. First, they can add small numbers, using a very simple counting-on-fingers technique. To perform 4 + 3, we all hold up four fingers - teacher points to the 4 on the number line. Then we add three, counting on from 4 and holding up three fingers, one at a time, five, six, seven. Fingers can also be used to match the doubles, such as two and two, three and three, four and four.
Once the children are secure in counting to 20, using their fingers twice over, they can use them to add by counting on. Start with an addition which is too large for us to do by counting on our fingers, for example, 14 + 4 = . First encourage the children to read the addition, all together, out loud. "14 add 4 is?" Tell them to hold 14 in their heads. "It's a big number, it might get lost! Hold it in your head!" Place the forefinger of one hand on your forehead as you say this. The children do the same. "Count on four. Fifteen, sixteen, seventeen, eighteen." Hold up one finger for each number spoken. Four fingers standing up - we've added on four. "Fourteen and four makes eighteen. " This method can be applied to any addition involving a two-digit number and a single-digit number, for instance, 26 + 3 = ; 42 + 6 = .
Children can use their fingers to count in fives, in twos and in tens. Counting in fives or in twos, we rehearse the chant, again holding up each finger for each number spoken. Two, four, six, eight. How many fingers? Four. Four twos are eight. This not only helps children to memorise the chant, it also helps them think how many? Thus, if we ask for seven fives, they respond by a rapid chant, "five, ten, fifteen, twenty, twenty-five, thirty, thirty-five", holding up one finger with each number. They stop when seven fingers are held up. Regularly chant the two-times and the five-times table like this, using fingers, and rehearse the tables as dismissal activities before playtime or lunch. "What's three twos Sarah?" She responds by chanting, "Two, four, six," until three fingers are standing up. "Three twos are six." "Good, you can go out to play! Amit, what's three fives?" and so on.
Use fingers to help children count in tens. Begin the chant by counting the multiples, "Ten, twenty, thirty..." All together, "throw out" 10 fingers by folding all 10 down and then, on the number spoken, spreading out all 10 fingers as wide as possible. Then fold them down again, and spread them on the next number in the chant. Once the children can chant the multiples, chant a series beginning with any single digit number, for instance, "Six, sixteen, twenty-six, thirty-six," etc. Again, throw out 10 fingers with each number spoken. It helps to reinforce the "adding 10" aspect and it also stops the children from poking each other and keeps them "on task".
Fingers are essential for the learning of the number bonds to 10 - not using a counting-on technique, but using the fact that there are 10 fingers. Say a number to the class, for instance, "Four". The children all hold up 10 fingers and then fold down the number spoken. The number of fingers left standing is the answer ("six"). This is also an excellent dismissal activity. "Tom, three!" Tom folds down three fingers and looks at those still standing. "Seven" "Good, go out to play. Now you, Lina..."
We can take away on our fingers - folding down the requisite number. "Seven take away four?" The whole group holds up seven fingers. Together we fold down four of them. Once again, this is demonstrated by the teacher to all the children, who then imitate. Keep the children as much as possible in unison for these short activities. Once they gain in confidence, then it is possible to start asking them to respond individually. Thus, we start by counting all together in tens, chanting in unison, "Thirty-two, forty-two, fifty-two, sixty-two..." and throwing out our 10 fingers each time. But after a few sessions, the teacher can start the chant and then point at a child to continue it. "Twenty-nine", pointing at Stephen who must say, "Thirty-nine" The teacher then points at Jade, who says, "Forty-nine" and so on. It is remarkable how quiet the class becomes and how hard the children concentrate. It often becomes one of their favourite games, especially when they get to choose the child to say the next number in the series.
There are some very complex and wonderful patterns which can also be demonstrated using fingers. There is space to mention only two. First, the nine times table can be memorised using the finger method illustrated here: 7 x 9 = 63: Turn down 7th finger. Read off answer, 6 tens on left, 3 units on right The children really love this way of working out how many nines, and really will rush home to show their parents. Have fun working out why it works (Hint: The digits of multiples of nine all add up to nine).
Second, we can count to 99 on our fingers using them as an abacus, in the manner ofancient Hindu and Malay cultures. The basic method is simple but brilliant: each finger on the right hand stands forone unit, and the thumb stands for five units; each finger on the left-hand stands for one ten, andthe thumb stands for five tens (50).
This manner of finger counting makes some arithmetic really easy. Try doing 16 - 5, or 23 + 15. Maths on the rug (some call it "whole class teaching") does not have to be intimidating for either children or teachers. The teacher can use the opportunity to demonstrate strategies and rehearse basic numerical skills. The children, in rehearsing these all together, have the opportunity to assist and to learn from each other. They prove remarkably good at this, helping those who are stuck and imitating their faster friends.
Professor Ruth Merttens is the co-author of the ABACUS Maths Scheme published by Ginn.