There’s an easy way of doing this

In the run up to the National tests for eleven year olds called SATs this May, I was practising with some of my pupils what some of the question would look like.

The girl looked at the question and said: “there’s an easy way of doing this.”

The question said 56 ÷ 4 =

It is one of those rare questions in a Key Stage 2 SATs paper that requires a simple answer to a mathematical expression. The girl I’m sure had seen that question every year for the last five years. Yet she was still hesitant – she had no instant response to the question. She had to think of the ‘easy way‘. And unfortunately she went on to choose the wrong easy way.

“My teacher told me you just drop the ‘6’ off the end, add one on to the 5 and that’s the answer.” Unfortunately the girl was remembering the ‘easy way’ for dividing by 9. And she was remembering the answer to the expression 54 ÷ 9 (which of course is 6).

This one of the reasons I dislike teaching children easy ways of doing things. In my experience most children who are taught easy ways have learned the underlying principles behind them. They then can only remember a small number of many easy ways and eventually they forget which way is which and when to use it. The next step is to decide that they can’t do mathematics anymore and they switch off from the subject altogether.

To quote a biblical metaphor, it’s a bit like building your house on the sand. It only takes a single storm of confusion to reveal that there were no foundations and everything is washed away.

Putting it another way, it’s like badly applying Bloom’s taxonomy to teaching. It seems we’re very keen in the teaching world at the moment to find ways of teaching those higher skills of evaluating and creating. But we miss the vital step between remembering things and applying them – that of understanding them.

We teachers often talk about that ‘wow’ moment in lessons – that realisation by the students that they are really ‘getting it’. This most often happens in 1:1 interactions but can also happen with larger groups. When I look at the Bloom’s Taxonomy chart I would say that that ‘wow’ moment comes in the ‘understanding’ phase. It’s not when we’re sure children can remember things by heart, or when we see them diligently applying their knowledge, nor even we see the outcome of a great piece of creativity. It’s when children comprehend, when you can look into their eyes and know they have understand – when they get it.

So, back to the girl with the maths problem.

Striving for that moment of understanding, I asked, “are you sure that’s how to divide by 4?”

She looked at the problem, hesitated for a moment and said. “Oh no. There’s an easy way to divide by 4. Halve it and halve it again.”

I couldn’t argue with that process. She proceeded to halve 56 by writing down 2.5 and 3. Then she wrote 2.8 in the answer box. I almost slapped my forehead in despair.

After a few more minutes of remembering how to halve, she did eventually get to the point where she found that half of 56 was 28 and then half of 28 was 14. She wrote that in the answer box.

Not satisfied, I asked her, “what if it had been 56 ÷ 6? How could you have done that?” She looked at me, blankly. I think that she was a little disappointed that even though she had arrived at the correct answer I hadn’t showered her in praise.

Divide by 6? I don’t have an easy way for that.” OK. She didn’t actually say those words, but I’m sure she was thinking them.

So I showed her the number line in the photo below. I showed her how you could count up in 4s or in groups of 4 to arrive at the answer. I showed her how it would also work for dividing by 6 or dividing by 7.

Trying to teach understanding, not just an easy way

I didn’t really get that ‘wow’ moment I was hoping for. I think she begrudgingly accepted that maybe the number line had some merits. Of course counting up in this way requires good recall of times table facts – facts that she struggled to remember.

It is interesting to me that the first stage on Bloom’s Taxonomy of remembering seems to have been pirated away for this particular student. Where she couldn’t initially remember to halve and her poor recall of times tables facts limited her approach to this question, she could, by contrast, remember quite well that there are some ‘easy ways‘ for doing things in maths. This in turn limited her understanding of the principles of division and stopped her applying any knowledge she had to this problem.

It seems to me then that we need to stop teaching tricks and easy ways that fill up children’s memories. We need to teach children to recall and remember important facts first, such as how to halve and double and times table facts. Then we need to teach children understanding, such as what division is – that it is both grouping and sharing (depending on the context). Then we can give them opportunities to apply their knowledge.

Can you keep the diplodocus safe?

Sometimes the river winds through the country of maths. Many educators see maths as being an entirely different country from the rest of ‘educationland’ and this post will emphasise some of those differences.

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?

Ah! Division by dinosaurs. You can’t beat it!