All the fun of the square;Mathematics
It started with colouring in every fourth square on a numbered ten by ten grid. The class of eight-year-olds needed some reinforcement of their table patterns, as well as some work on calculations with whole numbers.
As they neared the end, I stopped them and discussed the possibility that the rows of ten on the hundred square could be continued. If they carried on colouring in every fourth number, would 192 be coloured in?
The children had various ways of working this out, the most popular being that the pattern on the "next" hundred square would be the same as the first, and as 92 was coloured in on the first square, 192 would be coloured on the second.
This was not the answer I had in mind, although it was perfectly acceptable, of course. I rescued the situation by asking how many 4s it would take to get to 192.
Jason had learned how to do division, he thought, and set out the numbers in the usual way, but then forgot how to proceed. This was just as well, because like the rest of the class, who had not learned the algorithm, he had to find another way of solving the problem.
Some counted up the 4s on the hundred square, most finding quick ways of doing it, like noticing there were five 4s in every two rows. The consensus was 48. I asked the class if they could think of a way of checking this.
Several added up four lots of 48, although Thomas added eight lots of 24 instead because he found it easier.
Kristina, new to the class, showed us how to multiply 48 by 4, but was unable to explain the algorithm very well. Four eights made 32, but the "four times four" that followed was a problem. Was it really 4 x 4? What was the "4" in 48? Yes, it was four 10s, so it was really 40, and 4 x 40 was . . . ? We discussed this, until they could tell me what was 4 x 400, or 4 x 4,000, or 4 x 4 million, even 4 x 4 squillion.
What happened if you coloured in every third square, or fifth, or any other pattern? Would you hit 192 or not? The class started work on this more general problem using a variety of methods.
Jason started adding three lots of numbers, such as 61 + 61 + 61. When I asked him how many 8s were in 192 he said 24, because there were 48 4s.
Kristina used a similar trial and error method to investigate 5s, using her multiplication algorithm to calculate 32 x 5 = 160, 33 x 5 = 165, up to 38 x 5 = 190, and realising that 192 would be missed.
Matt and Devon were also working on 5s, but they were building the numbers up by noting that ten 5s were 50, twenty were 100, thirty were 150, and another eight of them made 190.
Sindre and Dani were looking again at 4s, but by counting in 20s. It took nine 20s to get to 180, and because there were five 4s in 20, there were 45 of them in 180.
Garrett began by counting single 4s, but then he used trial and error to add quadruples of numbers to try to reach 192.
For those who had investigated various ways of getting to 192 I suggested they find how many 4s it took to get to 1,000. We discussed this together. It was possible to subtract 4s, but it took a long time. How could we speed it up? They suggested subtracting 12s, 20s, 80s, 40s. Sindre subtracted 400s until he had had 200 left, then subtracted 40s.
I had always known children could invent their own algorithms, but I had never had a chance to explore the idea myself with them. The children used a variety of methods. They demonstrated a knowledge of relationships between the four operations, and some useful relationships between numbers. Above all nobody protested that they had not been taught how to do division, and they got on with different ideas about how to solve the problem.
Furthermore, I was able to use the ideas they had produced to discuss the development of a more systematic method of subtracting appropriate multiples of 4 from 1,000, say - a method that could eventually lead to something like the standard algorithm. Incidentally, most of them do not yet know an algorithm for multiplication, but we can deal with that soon. Who said it had to come before division?
David Fielker is a freelance lecturer and writer and teaches part time at the American Community School, Egham, Surrey