It reminded me of the multiplication tables, and the way children have to start at the beginning and recite their way through in order to answer a multiplication question. It happened with my elder son when he was about seven. He told me that they said their tables every morning at school before they went into assembly. I forced a smile, and asked him what seven sevens were. He started at the beginning and chanted his way through to "five sevens are 35", but then pleased me by adding on 14.
I remembered about a year before, when for some reason or other he had wanted to know what seven sevens were, and when it was obvious that I was not going to help him, said: "Well, three sevens are 21, so six sevens are 42, and another seven is 49."
When I was in infant school we had a chart on the wall with our names down the side and the tables from 2 to 12 across the top. We could report to the teacher and chant a table, and have a star put in that column opposite our name. The two times table was all filled in; some of the three times was; the five times was mostly done, and so was the 10 times.
Then word got around that the 11 times was easy if you learned the last three lines, and suddenly everyone got stars for that. Maybe some of us noticed something about the nine times too. It seemed an effective method for the Thirties, but I don't know if we in suburban Surrey were typical pupils.
I went on to 10 other primary schools during the Second World War. At the next one, instead of saying once two is two, two twos are four, three twos are six, we said two ones are two, two twos are four, two threes are six. This was, perhaps, my first introduction to commutativity of multiplication, but I certainly had to learn a new set of words to the tune.
At another school we said each line three times, and at yet another we learned up to our 16 times table. It helped our calculations in weight (16 ounces in a pound, 14 pounds in a stone), but I could never see the point of the 13s or the 15s.
Schonell, years ago apparently, found that one interesting result of this method of learning the multiplication facts was that the "squares", like 7 x 7 or 8 x 8, were less reliable than the other facts because they only appeared in the tables once, whereas all the others appeared twice. For example, 7 x 8 appeared in both the 7s and the 8s. Schonell had obviously not met those children I had, who when asked eight sevens would answer, "Eight sevens? Seven eights. 56!" It depended on which way round they said their tables, but one was easier than the other because they had learned it earlier. (If Schonell had done his research some years later, when chanting tables became less common and an interest in square numbers was developed in more concrete ways, he might have found the opposite to be true.) One problem about starting with the tables themselves is that they are a sophisticated organisation of some 121 number facts. In what other situation do we ask children to start to learn the organisation before they have anything to organise?
That may have been the situation in 1938 or so when I started my rote learning, but later on we did not begin the formalisation so early. We ask children to build up their tables in more practical ways, perhaps related to whatever topics they were investigating. An infant class working on transport were, among other things, using counters or cubes to build up patterns like: 1 bicycle, 2 wheels 2 bicycles, 4 wheels 3 bicycles , 6 wheels.
Tricycles produced a pattern of threes; cars produced a pattern of fours; and so on.
The numbers were represented by objects, and the results could be counted each time, or, if the children had learned to be more sophisticated, the six "wheels" could be derived from adding two on to the previous result, rather than counting all six.
You were certainly encouraged to work this way when you got up to numbers like 20, but this relied on the children having the concept of "counting on" - the idea that you did not have to go back to the beginning each time. This is satisfactory in a differentiated classroom: some children will count on, some will not, and others will learn to count on during these activities.
However, this activity does pose the question: are the children learning about multiplication, or about addition here?
I must emphasise what I am querying. It is true that the children are learning something worthwhile, and that this type of activity is useful. But I would question whether they are learning to multiply. The only person who thinks of this as multiplication is the teacher. And they do so because they already have an idea of what multiplication looks like, and it looks like the multiplication tables, that sophisticated organisation with which they are familiar. Indeed, I have often heard teachers, even when they are talking about multiplication facts per se, happily refer to them as "the times tables".
Suppose we start from the other end. We forget any sophisticated organisation, and we look at multiplication rather than addition.
A friend years ago was told by his son that he knew his seven times table, and asked to be tested. My friend said: "OK. What's 56?" He was told that was not fair.
Maybe 56 is not a good starting point, but it depends on the children. Say we start with 20. Better still, we start with 20 counters, or plastic squares, or wooden cubes, and we arrange them in a rectangle. Children will perhaps know that they can make a rectangle 2 by l0. Some insight or previous experience leads them to break this rectangle into two halves and rearrange them. They try now to arrange the 20 cubes in rows of three instead of two, and find they are missing a corner of their rectangle. This can be recorded as (3 x 6) + 2 or as ( 3 x 7) - 1.
They have already put the 20 cubes in rows of four, so now they try rows of five, which turns out to be the same as rows of four, but the other way round; so they find that 4 x 5 is the same as 5 x 4, and the sequence continues backwards.
The continuation of this idea is obvious. Different numbers of cubes are arranged as far as possible into rectangles.
Children thus start from the numbers and investigate them in terms of their factors. In this way they build up an experience of numbers in a more analytical way, developing experience of their divisors. Numbers like 4, 9, 16 and 25 can be arranged into rectangles that are also squares: these are the square numbers. Numbers like 2, 3, 5, 7, and so on cannot be arranged into rectangles, unless you are using squares or cubes and consider, say, 1 x 5 to be a rectangle: these are the prime numbers.
The first really interesting number is 12, because it can be arranged into more different rectangles than any lower numbers; the next such number is 24, then 36, but not 48. Martin Gardner called these "abundant" numbers.
In this way the children learn about squares and primes; but they also learn to become familiar with numbers, and to see them in terms of their factors, so that 36, for instance, has a different feel about it from 37. Later on they will appreciate the Babylonians' choice of 360 degrees in a complete revolution: 360 is an abundant number.
Furthermore, they are learning about relationships like 4 x 5 = 2 x 10, or 2 x 20 = 4 x 10 = 8 x 5, and thus developing a flexibility with multiplication that will enable them, for example, to see 5 x 18 as l0 x 9 and so be able to calculate it immediately in their heads.
Above all, they are seeing multiplication facts as multiplication rather than addition. They are also seeing the relation between multiplication and division, because a 3 by 5 array can be recorded as3 x 5 = 15, or 5 x 3 = 15, or 15 V 3 = 5 or 15 V 5 = 3.
Some other time they may wish to organise all these facts they now know into a standard format like the multiplication tables, but by then it may not be necessary.
David Fielker is a freelance lecturer and writer, and is teaching part time at the American Community School, Egham.