Mel Lever on a number oblong that helped young children handle the meanings of minus
As maths co-ordinator in a school for dyslexic children I am constantly trying to find ways of helping pupils to acquire mental images on which to hang concepts, pictures to help with problem solving.
The national numeracy strategy emphasises the understanding of number and number processes and the use of mental arithmetic. After giving five workshops for parents, explaining to them the rationale behind the way we work in school and behind the strategy, we now have parents more willing and able to help their children with a variety of homework tasks. Parents understand the importance of allowing children to explore maths before introducing them to mathematical procedures. Parents and teachers are also aware that whatever new ways we find to teach maths, and whatever new philosophies we embrace, children with specific learning difficulties will often take longer to understand and use mathematical concepts than many other children. We know that many dyslexic children do not find it easy to remember facts or to readily recall previous work or concepts covered. We need to be patient and inventive to help these children. To date I have invented "maths monsters", a series of hats to help children learn number bonds, and an alien called the Marlon to encourage children to put their ideas into words. These have all been fun ideas, which have intrigued children through their strangeness.
This year, for a class of Year 6 children who needed a lot of help to be able to manipulate negative numbers, we used horizontal number lines to count on and back (3 - 7= - 4): We did the same with a vertical scale. With each of these it was hard to deal with negative numbers that went into double figures. A way was needed to help the children deal with larger quantities. I could have just said:
"With 78 - 90 you just turn it round to 90 - 78. That gives you 12. Now just put a minus sign in front of it." However, teaching them a procedure would help in the short term but it would be better if they understood the idea first.
We have always done a lot of work with 100-squares, helping the children to understand the logical structure of the number system. These Year 6 children were able to add and take away in steps of 100, 10 or 10s and 1s. Most could now do this in their heads, having acquired fairly good mental images. The next stage was to help them visualise the steps needed to subtract beyond - 10, where they would end up with negative numbers with two digits. I tried using two 100-squares, each going from 1 to 100 with 1 in the top left-hand corner. This clearly would not work. It would not work either if I used two 100-squares which started with 0 at the bottom left-hand corner. What were the relative merits of the various 100-squares available? Would it be better to start at 0 and continue to 99? Should we be using 100-squares where the lowest digit (0 or 1) was at the top? Why was the lowest digit not in the lowest position? Some way was needed of illustrating positive and negative numbers using the 100-square as the basis of my thinking.
After severalattempts at starting with the lowest number in the bottom left-hand square, then in the top left, with numbers increasing by one horizontally or vertically, the solution to the problem turned out to be what you see in Figure A.
First we had to get to know our new number oblong. We discussed the way the numbers were displayed, counting on or back from various numbers. With the positive numbers the children could see that this was one version of the 100 square that they were used to.
Counting back was more interesting. We started by counting back from 20: 20 minus 8 is 12
20 minus 10 requires a jump down a line
20 minus 19 is 1
Using this small step allowed the children to count back in ones from 20 and to begin to see how the negative numbers fitted into the picture. From there we were able to go back in larger steps. This led us to our biggest discovery. While experimenting with counting forward and back in steps of 10, 20, 30 and so on we found some interesting patterns emerging: 24 minus 30 is minus 6
38 minus 40 is minus 2
56 minus 70 is minus 14
We set out our answers and looked for patterns. If we reversed the process and added our result to the number deducted we got back to where we started. Naturally! But look at the final digit of the two numbers being added together: 4 and 6, 8 and 2, 6 and 4. Why did they always come to 10? The children worked in pairs on various sums which they had to solve and write down. One pair worked back from 30, taking away 50, so they started with 30 minus 50, 29 minus 50 and so on, until they got to 20 minus 50. Another pair started at 20 and took away 30. Finally we were able to construct a chart of our findings. Figure B shows part of it.
We worked on our patterns for two lessons. Children were starting to predict what would happen when they started with a positive number and deducted multiples of 10. Through use of this model children began to gain a better understanding of negative numbers and to solve negative number problems more quickly. It was not difficult to take the next step and deduct numbers other than multiples of 10, for example 58 - 74. For this the children would know that 58 - 70 took them to - 12 and by deducting another 4 they reached - 16. Through the use of our number oblong these children were able to acquire a mental map that they found useful when dealing with negative numbers. We could revise and recall it during our mental warm-up exercises.
From this point we could discuss other ways of manipulating the numbers, and some children did indeed come to the conclusion that, "with 78 - 90 you just turn it round to 90 - 78. That gives you 12. Now just put a minus sign in front of it". But for these children it was now their own discovery.
As the children begin to use and manipulate their mental images they will begin to find more efficient ways of calculating but these will be based on a sound understanding of the number system. It is only then that they can build on and extend their mathematical functioning.
For further information contact Mel Lever at Fairley House school, London. E-mail: firstname.lastname@example.org