I came across a very strange situation the other day…a student who can apply all four rules of number on paper with fluency. She can complete long multiplication and forget the place holder and still get the answer right. She can do long division, using a written method I don’t quite recognise yet (but I will come back to this later). But, she doesn’t have any number sense…
Number sense, that understanding of how numbers work and knowing whether the answer you give is realistic.
When I asked her to draw me a square number, she drew me a picture of a house and a garden with the number 4 embedded in the diagram. Drawing maths was a whole new concept. Luckily we had some cookies and we were able to think about how 4 or 9 cookies could be arranged to look like a square.
It became evident the thought processes that link numbers wasn’t there especially when we started to look at problem solving questions.
For example, she knows that there are 1000m in a km, an absolute result in my mind. But, didn’t really know what 250m means when you try to connect it to km because she couldn’t connect 25 x 4 = 100 or 250 x 4 = 1000. This made the question “how many 250m in 30km” almost impossible.
I came home and asked my own sons the question how many 250m in 30km? (Yes, I’m a maths teacher, yes they’re a few years older, but their number sense was evident in their explanations)
“120!” was my 12 year olds reply and the 14 year old looked surprised “Why?” is what I asked next.
“Well 4 x 250m = 1000m which is 1km and 30 x 4 =120”
Everything linked together all of those facts visualised in his mind and then explained clearly (not bad for a child with verbal dyspraxia)
I asked my student “Without writing anything down what is 18 x 5?”
90 was her answer and when I asked her to explain how she got the answer she told me she had visualised the column method in her head. She isn’t the first student I have encountered who visualises the column method but we then discovered it doesn’t always work.
So I asked what about 18 x 15?
This threw up lots of problems because how do you do long multiplication in your head? I think we ended up with 140, because 10 x 10 = 100 and 8 x 5 = 40. No checking, no thinking about whether that could be right or wrong. NO thought about 15 x 10 being 150.
What have we done to help with number sense?
The girl is 11 years old and over the last few weeks we have worked extensively on explaining her thinking and it has been delightful to see her beginning to make sense of the questions she is being asked. Considering whether 240 could be the correct solution 46 x 22 is something she wouldn’t have thought about before but now she knows it can’t be right even if she’s not sure why…but she does know that 46 x 10 =460
This week I asked her 467 + 44 and for the first time she had partitioned the numbers and added them – no column method in her head.
Here was her thinking:
7 + 4 = 11
60 + 40 = 100
100 + 400 = 500
500 + 11 = 511
It was great to hear her put the hundreds together before she added the 11. She then told me she had thought more about these methods in lessons at school and was beginning to answer questions with more confidence.
Overcoming long division
Earlier, I said she was able to complete long division and then I discovered she couldn’t! It’s not so much as she can’t follow a process or method but without a sense of number she struggled to understand the process. She can’t easily do her times table of “big numbers” and this is what stopped her. She only knows as far as 12 x 12.
I’m a big fan of diagrams to represent maths and so with the help of a rectangle we have developed her number sense to complete long division problems. It was great to see the student finally recognising the importance of the 2, 5 and 10 times table and then using these to find out 608 ÷ 32.
So what did we do?
We would begin with a rectangle and we know the width is 32 and knew the inside is worth 608.
Then the student would madly write down known number facts:
32 x 10 = 320
32 x 5 = 160 (because we know the half of 10 is 5)
32 x 20 = 640 (because we know that 32 x 2 = 64 and 20 is 2 x 10)
She looks at her numbers and thinks…
608 – 320 = 288
288 – 160 = 128
32 x 2 = 64 (because 32 x 20 = 640)
128-64 = 64
and then a smile as she realises she has 64 left. But, this looks a jumbled mess, how does she know how many 32’s she has used?
Well she uses the rectangle and fills in the areas she has..
It became an absolute joy to watch her as she made sense of number and used the facts that she knew. She would draw the rectangle and begin her calculations. The calculations aren’t formal and I am going to have to move her to something more formal looking but she can do long division. She was beginning to make sense of her learning and explain what she was doing and why.
Slowly as we make sense of number I am sure we will also be able to develop her reasoning and problem solving – two things that her teacher says are a weakness. Of course this is a weakness she isn’t confident with how numbers really work.
We still have a long way to go but I am sure that as soon as she begins to make sense of numbers then maths will begin to make sense too.