math – frogphilp.com

The coming mathematics apocalypse

I am tremendously excited by the current maths curriculum in English primary schools. The expectations are higher by at least a year. It is a no-nonsense raising of standards.

I’m excited because if we can find a way of teaching the curriculum successfully, then our students will have levels of maths previously unheard of in this country. They will be on top of the mathematical world. The average will be above average. They will have the skills and knowledge to found an empire of learning.

Not only is the curriculum at a standard that is a year higher than the previous curriculum, but there is talk of the floor standards, already raised from 60% to 65%, to be raised further to 80%. This means that schools will have to find a way of getting more of their students to a far higher standard. What a fantastic aspiration.

But it’s the ‘if’ I see as being a problem. Let’s face it: we don’t currently have the maths specialists we need in Years 5 and 6 – the upper years of primary school. Secondary school maths departments complain of misteaching, cramming for the end-of-primary-school SATs and students without the knowledge they need. What will it be like with even higher standards, both on the level that children must attain, and also the number of children that must attain it?

I see three options:

Success – we achieve the ambition. We find a way of teaching more of our children to a higher standards than previously attained. A golden age of knowledge is ushered in.
Cheating – we pretend to achieve the ambition by blurring further the grey line between supporting students and telling them what to do in tests. The nation lurches towards a moral crisis.
The Maths Apocalypse – We ramp up the stress. School leaders put the pressure on their teachers with the higher expectations. The teachers crack under the pressure and shout things at their students like: “why don’t you just get it!” A generation is turned off everything to do with mathematics. It’s a maths apocalypse.

The problem with raising standards for students is that we also need to raise standards of teaching. Many primary teachers have a ‘C’ grade GCSE in maths, yet the maths expectations now required would go a long way towards achieving one of those ‘C’ grades.

The clock is ticking. In 2016, the first cohort of students will be tested on the new curriculum. Only eighteen months away and staff rooms across the country should be buzzing with conversations around how we teach maths and the subject knowledge we need. Networks of maths co-ordinators will right now be exemplifying the standards – turning the words into maths that can be taught and practised. Experts from teacher training will be working closely with schools, finding ways of bringing their new trainee teachers up to the required standard and sharing some of their training expertise with teachers who are struggling with their own maths. Teaching schools will be focusing on developing their Specialist Leaders for Education in mathematics so that schools within their networks without maths specialists have a means of accessing their expertise.

We have time to prepare and to succeed. Doing nothing will only prepare us for the mathematics apocalypse.

Key Stage 2 Maths SATs Analysis

Admittedly, not the most exciting post title you’ve ever seen, but let me draw you in with what I found out:

The greatest proportion of maths expectations we need to improve on at my school come from the Year 1 programme of study.
Two of the key questions that we need to get better at are taught through the KS2 computing curriculum, and not the maths curriculum.
At my school we are really good at teaching calculating and number, but we need to improve at teaching problem solving.
Neat, well presented work does not equate to success in maths SATs.

So, I used a useful spreadsheet I found on the TES website to analyse the maths SATs results from 2014 (you’ll need a TES login to access that link). I was particularly concerned about the 6 children who didn’t make 2 levels of progress during Key Stage 2. Six out of thirty is a large percentage for us: it moved us down from well into the top of the half the country (in terms of progress measures) to well into the bottom half. While there was a back story (read: justification) behind each child, I wanted to look more carefully at the results each child had achieved and find out why they hadn’t quite made the grade.

Having analysed the data, I made a presentation for my staff so we could talk through the issues involved. Why not just talk it through with the Key Stage 2 department? Well as I’ve indicated above, many of the statements where we need to get better at are actually taught from Year 1 or 2. I’ve put this presentation into a Movenote here. Please feel free to watch, but don’t expect quality – I was using Movenote to practice my presentation for the staff meeting on Wednesday – it’s a first take, and I’ll be expanding on many of the points during the actual staff meeting.

My two big considerations are the following:

My children need to get better at logical reasoning to achieve well in maths. Logical reasoning is most explicitly described in the computing curriculum – how can I use the computing curriculum to raise standards in maths?
With the foundation for success clearly coming from the teaching in Year 1 and 2, how can I make sure that this teaching is as good as it can be?

It will be interesting to see if my staff agree with me on Wednesday.

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.

Ways of dividing by 4 — 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.