The thing is, most teachers don't get maths. Not only do they not get it, but they do not want to get it. That is why the numeracy strategy in the UK has been a moderate success - it provided such a tight framework for teaching maths that teachers didn't have to get it, they just had to deliver the lessons put in front of them. As Sir Peter Williams put it:
"The United Kingdom is still one of the few advanced nations where it is socially acceptable to profess an inability to cope with mathematics. We need to urgently reverse this trend so every pupil leaves primary school without a fear of maths.."
This attitude if prevalant in the nation must also be prevalant amongst teachers.
So, how many 4s are there in 20?
At what age should a child be able to work this out? It is a quotitive division - you could express ot as 20 ÷ 4 = 5, but it is quite a different question from saying 'share 20 sweets between 4'. In the latter question, each of the '4' would receive 5 sweets. In the former there are 5 groups of 4 in 20.
It is grouping vs sharing. It is quotitive division vs partitive division.
Children are exposed early to partitive division. They are sharing from reception and before. They share sweets, teddy bears, small plastic dinosaurs and even an 8 chunk bar of chocolate if they're lucky (incidentally this latter example is actually a fraction problem - but don't tell the reception teacher that).
They are not on the whole exposed to quotitive division. In fact grouping isn't really referred to until Year 2.
So here we have 2 equally valid meanings to the word division, with one being taught from reception and one being taught from Year 2. I wonder which one will be better understood? Forgive my sarcasm, but it seems obvious to me that our framework has let us down here. Younger children can group. They could make a group of the chewy sweets and the hard sweets. They can group their teddies by size, colour or even in groups to go off to their teddy bear's picnic. They can group dinosaurs by how angry they are.
So when it comes to the question 'how many 4s are there in 20?' We may at the moment say that we can't begin to talk about that with children until they are 7. We certainly can't express it in symbols unto they are 9. But actually, if they understand the numbers, there is no reason why they can't be asked the question much earlier:
- The camels cross the desert (sand pit) in herds of 4 at a time. How many herd can you see in the sand pit?
- The teddy bears only ever have their picnic in groups of 4. Look at those twenty bears... how many picnics do you think there will be.
- Those diplodocus are only safe from the tyrannosaurus rex when they are in groups of 4. Can you make those 20 safe? Please?