The problem with a best-fit approach

Scattergraph
Here’s a handy graph, just ready for a best fit line to be drawn through it

It is not often that I read a 50-page government-commissioned document and get wildly excited. But that is what happened when I read the “Final Report on the Commission on Assessment without levels” (September 2015).

I found pages 12-18 particularly inspiring as the commission explains the rationale behind why levels are bad and ‘not-levels’ are good.

This statement was pertinent:

Levels also used a ‘best fit’ model, which meant that a pupil could have serious gaps in their knowledge and understanding, but still be placed within the level.

And this one:

Levels were used to measure both end of phase achievement and lesson-by-lesson formative progress, but they had not been designed to fulfil the latter purpose, with the result that formative assessment was often distorted.

Just as in yesterday’s post, when I tried to explain a character flaw that had held back my practice, here was a practical example of something I was doing day-to-day which was not actually any good.

As a key stage 2 teacher I had experienced this over many years. Children came up to my phase ostensibly as ‘a level 2 child’, but the best fit approach meant that while they could add a pair of 2-digit numbers with the support of a 1-100 number square, they had no idea of any other strategies for doing the same thing, nor really any sense of the the size of the numbers they were dealing with, nor the purpose of the equals sign.

And likewise I must have been doing the same to my secondary colleagues for years: children from key stage 2 going up to key stage 3 being able to scrape enough marks to get a level 4 in a SATs test, but not with the necessary number or problem solving skills to really go any further. In fact at my school, analysis of SATS shows me that number skills have always been high, but problem solving really low – the children have been able to scrape through with good routine number skills, but lack the fluency to really excel in maths.

What is even more insidious about the ‘best-fit’ approach is what it does to ability groups within a class, particularly low ability children. But that’s tomorrow’s topic.

Image courtesry of: https://upload.wikimedia.org/wikipedia/commons/a/a2/Scatterplot_r%3D.24.png

 

Remember that you are dust, and to dust you shall return.

Remedial

It’s been a while since I had a regular classroom commitment. I’ve always thought that senior leaders suffer from an authenticity failure when they are divorced from teaching. But that’s another story, to be told another time. And to cut that long story short, this term I am teaching a group of seven year 6 children. Maths. For just one hour a day.

I am ‘boosting’.

Here’s what it says in the dictionary about boosting.
boosting: the yearly panic that primary schools go through to ensure they meet their yearly quota of successful students.But this is more than a SATs game. Putting my cynicism aside, each of the seven children in my group have a unique perspective on maths. And it’s a perspective skewed by failure.

Now I know that these days it’s cool to fail. Fail: first attempt in learning, chant the students to me. But not when you’ve failed week after week. Not when you’ve been the last to ‘get it’ lesson after lesson. Not when you’re at the bottom of the achievement rocket year after year.

The children's maths weakness: concentration, telling the time, embarrassment
What the children told me gets them down about maths

My seven Year 6 students are old enough to be embarrassed by their inability to do maths that children four years younger than them can do. They can’t tell the time. They don’t know their times tables. They can’t reliably count on or back. And what’s worse, they can all tell me stories of embarrassment, when their failure to do what their peers find simple has been exposed to the rest of their class.

Embarrassment and repeated failure make a powerful poison that taints the waters of learning. And the antidote to that poison is more than mere boosting. If all I do over the next four months is ‘get these children through their SATs’, I will have failed them. They don’t need my tricks and tips to score the best they can on some 45 minute exam papers in May. They need me to teach them well. They need some core knowledge and some confidence.

So as it turns out, I am not going to be boosting after all.

I’m going to be a ‘remedial teacher.’

I know that sounds awfully old-fashioned, but there are some reasons why I prefer that term. I see their lack of knowledge akin to a sickness and the remedy is good teaching. Each of these children has unique reasons for why they are ‘below national average’ in maths. Whilst I can’t remedy all of the reasons, for some I can do the following:

  1. diagnose the ailment;
  2. identify a treat;
  3. present a cure;
  4. give time for that cure to take hold.

I suppose I could label the same 4 point sequence like this:

  1. identify misconceptions
  2. plan some good lessons
  3. teach well
  4. allow children time to practice so that their confidence grows.

The majority of my posting this term is going to be about the journey with these children.

 

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:

  1. 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.
  2. 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.
  3. 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:

  1. The greatest proportion of maths expectations we need to improve on at my school come from the Year 1 programme of study.
  2. Two of the key questions that we need to get better at are taught through the KS2 computing curriculum, and not the maths curriculum.
  3. At my school we are really good at teaching calculating and number, but we need to improve at teaching problem solving.
  4. 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:

  1. 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?
  2. 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.

On the teaching of handwriting

I’m writing this both as a parent and as a follow up to a previous post concerning roleplay in schools. You may ask what is the relationship between handwriting and roleplay? Well, I suppose I’m asking a wider question – what are schools really for?

When my son was in Reception, my wife and I were concerned about his letter formation. We told the teacher about it. She assured us it would be alright – he’s a boy after all – he just needed some time. She gave us some photocopied sheets of letters to practice at home .

When my son was in Year 1 we were concerned about his handwriting. We told the teacher about it. She told us that it was early days and he just needed some work on his hand strength and motor control.

When my son was in Year 2 we were concerned about his handwriting. We told his teachers. They told us there was still time. The important thing was getting him to write in sentences for his SATs. He was assessed at level 2B in writing, despite being level 3 in everything else. His handwriting had held him back apparently.

When my son was in Year 3, the teacher told us she was concerned about his handwriting. She implied that we hadn’t done enough as parents and we should be supporting him more at home. We too were concerned about his handwriting, but now we learnt that it was our fault, we gave him writing exercises and paid for piano lessons to build up his hand strength some more.

When my son was in Year 4, the teacher told us she was concerned about his handwriting. She put him in a small group so that during times when the rest of the class were doing something he was already good at (like reading, or maths), he could practice his handwriting. By the end of of Year 4 we were less concerned about his handwriting.

When my son was in Year 5, the teacher told us that our son was the best writer in the year group, but he did sometimes forget his capital letters and full stops. She didn’t mention his handwriting.

As parents we made the mistaken assumption that during his first years at school he would be taught handwriting.

As a teacher I know it is easy to assume that things like handwriting will just sort themselves out.

As a child educated in the 70s, I went to a school that, whilst it hadn’t completely sold itself out to ‘discovery learning’, it didn’t teach things like pencil grip or handwriting. My handwriting is pretty dire as a consequence.

In following up that post I mentioned on roleplay, my question is what are schools for? Should they ensure that children are brilliant at handwriting? Or should they make up for the lack of open-ended play that takes place in the modern home by focusing more on roleplay? Do I sound too much like Michael Gove if I suggest that schools should focus on academic skills such as handwriting, reading, mathematics, to the detriment of play?

In fact, I don’t think this is what has happened in the case of my son. I don’t think it was a war between play and academia. I think the school focused heavily on their needs to have my son perform well in various assessments – maths, reading or writing, and unfortunately handwriting is an insignificant part of the writing assessment.

A third of children at level 4 aren’t

It is Monday morning. Today, primary schools across the country are beginning their week of judgement. Should their children in Year 6 do well in their SATs, then the primary schools will meet their targets – maybe be beating the floor targets; perhaps by demonstrating that all their children have made the required amount of progress; or possibly by beating their previous best and thereby showing year-on-year improvement.

Doing well could mean a higher place in the league table. It could mean families from more well-off backgrounds choose that school in the future. It could mean that the school scores well enough to avoid an Ofsted inspection (Ofsted undertake an annual risk assessment to establish which schools need inspecting). In short it can mean less stress and anxiety for teachers, and who wouldn’t want that?

Does anything sound wrong in what I’ve just said? That’s right – it’s all about the performance of schools and not about the performance of students. You would hope that the former is the same as the latter, but teachers, and especially school leaders, suffer a considerable role conflict in this. Is it possible that school leaders and teachers can behave in such a way that their school performance is enhanced whilst the learning of their students actually suffers?

No-one would do this intentionally, neither do I want to get into the how or why this happens, as I have blogged extensively on it in the past. However I have observed that the education secretary, Michael Gove has noticed something of a discrepancy between the performance of schools and the performance of students. In his recent speech he tells us:

Nearly a third of children who get at least a good level 4 in English and maths fail to go on to secure five A*- C passes including GCSE English and maths – the minimum level of literacy and numeracy required for future employability.

I find that statistic truly shocking. It means either that those third of children achieving level 4 are badly let down by the secondary schools, or that they weren’t actually performing at Level 4 in the first place. Sorry, that is a bit sensationalist. It could also mean that the whole system for leveling a child is wrong. Or that after primary school, a third of children lose all motivation and fail to perform for reasons entirely independent of their secondary school. Let’s be honest – education is complex and it’s probably some mix of all those reasons.

I was pleased to read therefore from Michael Gove “We’ve taken action to deal with this scandal.” The action seems to be twofold:

  1. introduce a phonics screening check for 6 year olds.
  2. introduce a grammar and punctuation test for 11 year olds.

In his speech Michael Gove then spends several sentences criticising unions and Michael Rosen for opposing these tests. Now I do not oppose those tests, but they don’t seem to me to be the kind of action to ‘fix the scandal’ of seemingly good performers at primary school under-performing at secondary school. The whole problem with SATs is that they are internally invigilated and then linked to whole school performance. The way to fix this is not to introduce more tests that are also internally invigilated and linked to whole school performance.

I would suggest that the way to fix this is to do some decent analysis of that third of students. Which of those four possibilities are true for them? Is it:

  1. They weren’t actually at level 4 in the first place (indicating too much support during primary schools SATs – you might say this a kind way of saying ‘cheating’.)
  2. The secondary schools didn’t do a good enough job.
  3. The system for leveling children in English and maths is all wrong.
  4. The students lost motivation completely independently of the education system.

Once this has been determined, some change of policy and practice could be put into place that would sort this scandal out, rather than just trying out another test in primary schools that might actually make things worse.

But we’re all cheating, remember.

So it’s great to see that more primary schools are meeting their targets. I love the way the BBC headline is phrased: “Fewer primary schools fail to meet their target.” Hooray: we have less failure. It shows what a glass-half-empty culture we’re in at the moment in education. We could so easily have said – Hooray: more success!

So I could go on about this over-negativity I suppose. Or I could make the ironic link between this and what Michael Gove said yesterday. Today the DoE tells us that: “Heads, teachers and pupils deserve credit for meeting the challenge head on”. Yesterday Michael Gove wrote to schools criticising an admittedly small minority of teachers for having a detrimental effect on their schools through working to rule.

However what I want to draw attention to is that today’s headlines come at a time when I believe we have an over-supportive assessment system in this country. Some people have pointed the finger at how the system has been geared to judge the school and not the pupil. Others have expressed their fears that our system is being ‘gamed’ – teachers and school leaders are ‘playing the game’ to make their schools look good, without actually doing the job of raising standards through good teaching. A helpful way of looking at this may be David Weston’s hastily but well-drawn graph of the continuum between teaching and cheating. Or you just might say that actually we’re all cheating.

You see, the BBC say: “The results are based on the national tests children in all state schools take in their final year of primary school.” But they don’t say that these tests are entirely self-invigilated. Unfortunately the phrase ‘national test’ distracts us from the fact that schools can sit their students with as many support staff as they want, in whatever group size they want, and nobody but the staff in the school watch them do it.

Yes there are one or two ‘invigilators’ that Local Authorities send out to schools on random monitoring visits, but they are so few and far between that school can pretty much guarantee that they won’t be there on the crucial day.

And who benefits from these tests being, shall we say, over-supportive? Schools do. When we read from the DfE:

“Schools with a long history of underperformance, and who are not stepping up to the mark, face being taken over by an Academy sponsor…” 

we begin to get get a hint of the intense pressure that schools are under. Headteachers and Governors are replaced in this process. Poor SATs results can trigger an Ofsted inspection bringing more pressure onto teachers. Surely if we just give one or two suggestions to those border-line children just to tip them over onto the right side of the grade boundary it won’t matter? Will it?

The problem with this approach is that it sucks the independence out of children. No matter what their ability these children expect the same level of support as they go into secondary school. Then to achieve their own progress measures, the secondary schools are under the same pressure as primary schools to ‘tweak’ their GCSE results just that little bit. And all along our students are losing their independence. We are creating a generation of exam zombies – education consumers who expect education to be done to them, not to be active participants.

Ever wondered why they need so many plagiarism checkers at universities these days? It’s not just the ease of access to online material – it is also the fact that our students expect to be supported. They do not expect to think for themselves. It is not a ‘thinking skills curriculum’ that fixes this. It is a rigorous, well-invigilated exam system that demands students think for themselves.

So unfortunately, even though I’d like to be glass-half-full about today’s announcement, celebrating with the many primary schools that, like mine, are now above the floor targets, all I can think of is the increasing amount of learners who will need support to hit their inflated targets throughout the rest of their education. It weakens us as an education system and it weakens us as a society, even though it might look good in the short term.

Over-supporting is a nice way of saying cheating

 This helpfully drawn graph by  David Weston on today’s blog post at the Teacher Development Trust has helped visualise a problem that has been growing in my mind for at least ten years. I referred to it obliquely in a previous post entitled “Moral Purpose” when I mentioned the notion of Game Theory in assessments. Game Theory is one of those things that I haven’t really understood until now, but a really helpful post from maths teacher Owen Elton this morning has helped me understand it better, and in particular how it applies to the particular situation in the English education system.

It started in a Year 6 class I had a decade or so ago. I’d been in the school in a deprived part of Birmingham for a couple of years and we had had a new headteacher that year. The head, who hadn’t shown much interest in my class for most of the year, took a sudden and very intense interest in the class during SATs week, including offering to help read the questions to the children in the maths SATs papers – as everyone knows, maths is not a test of reading and children who can’t read so well shouldn’t be disadvantaged in their maths just because odd their poor reading skills. So in that first test, I saw the headteacher position himself next to a girl who was very border line between level 3 and 4. He read the questions, yes, but then went on to talk her through the steps to solving the problem. He didn’t actually tell her the answer. Then in the next test, which was a mental mathematics test read by a lady on a cassette tape (a voice that all teachers in England will know very well), he stopped the tape. After the first 5 questions the children were allowed some ‘extra thinking time’ – and likewise after the next 10 questions.

I felt really uncomfortable about what had happened, so the next day, before schools started and before the final test of the week I went to the headteacher to express my discomfort. “Did we go to far?” he asked, smiling. I felt that we had and we agreed not to support the children so much. Relieved, I went away to prepare my children for the final test of their SATs week. And after it, I remember he called back into his office, I thought to ask me how I thought the week had gone. But no, he told me that he had seen me playing too roughly with the Year 6 boys at playtime during football. I should be careful not to do that he asserted. I knew what he was saying – don’t grass me up and I won’t grass you up.

It was a difficult time for me – I spent the next year or so really struggling with my practice and it was only when I moved schools that I started to regain my confidence.

But aside from that, it was my first experience of the grey area of over-support. The facts are:

  • There are no entirely externally assessed national tests in England.
  • Primary schools are held accountable by national government using the results of their tests.
  • Teachers are held accountable by their schools using the outcomes from tests.

As Owen Elton says when analysing the ‘teacher’s dilemma’ in the blog post I referred to above, teachers “should mark generously”. He was referring to coursework, but translating that to SATs tests in primary schools, teachers should give whatever support they can to their children.

Now, I would maintain that stopping a mental mathematics test to give children more thinking time is cheating, but there are grey areas of support, which as David Weston says would be called cheating by some, but not by others. Here are some:

  • test papers can be opened up to an hour before the test by their teachers – but should they be?
  • children can be placed in different rooms and group sizes around the school to give them different levels of emotional support – but should they be?
  • teachers can read test questions and instructions to their children – but should they do so?
  • Children are allowed to sit in the same room where they have done their learning – but should they?

For each of those examples above I can think of examples where the regime has been abused. For example:

the deputy headteacher who opened the writing tests an hour earlier, saw that the test was on a certain form of writing and spent the next 45 minutes reminding children about features of that kind of writing…

Or

the school where a friend’s daughter was placed in a room with only two other students and found that the atmosphere was a lot more conversational than previous tests had been…

Or

my son reported his teacher had raised her eyebrows and pointed at an answer in a maths test…

Or

the headteacher who accidentally left the science display up in the classroom where the science test was being sat…

And of course, when a teacher has stepped into the grey area one year, is it easy to step back from it? Doesn’t the grey area get bigger?

My second clue that a grey area of over-support existed was in speaking to a colleague from a university a few years ago. She had noticed that the year that they had had to introduce plagiarism checkers as standard was the year that corresponded with the students who had sat the very first SATs back in 1996. She also noticed that it was that same year group of students who suddenly demanded far more from their tutors. It was like they had lost their independence and almost needed their studies doing for them.

My third clue was looking over the shoulder of my mother who is still a marker for one of the exam boards. She was marking a set of maths coursework, and each one was virtually identical. Some of the names and places and numbers were slightly different from each other, but the format and the nature of the maths represented was identical – no child seemed to demonstrate any independent thinking or mathematics. But I suppose is there any wonder that secondary school teachers need to over-support their students if their students are assessed in an over-supportive way at primary school?

The final pointer towards over-support has been the GCSE English fiasco that has been debated long and heard over the last few months. Who is the wrong? The schools? The exam boards? The politicians? I think we all have to take some of the blame. And hears why:

When I hear students say things like “the system has let me down”, which is a quote I’ve heard from the news in recent weeks, that’s when I have to think that actually we all share some of the blame. We have created an assessment culture that is at best over-supportive and at worst is cheating. Michael Merrick foretold this in his post back in March, when he asked the question: “When primary responsibility for success or failure is taken away from the student and placed instead on the shoulders of teachers, what effect might this have on the education system? ”

To me it is inevitable that we have come to this point – we are so lost in ‘playing the game’ or ‘keeping the Ofsted wolf from the door’ that we have forgotten that an education system should be built to let individual students triumph. Is ours? Is it really?

I don’t do a good job by moving my school to outstanding, or moving my school up the league table – I do a good job by moving my students from consumers to contributors, by educating them so they can achieve for themselves. Yes, hopefully those 2 things are synonymous. In an ideal world they would be. But in the real world (to quote Michael Merrick once more) “that murky landscape of educational ethics comes into view, with exhausted and anxious teachers straying over the once clear demarcation lines, in the process creating a culture that absolves students from real responsibility (and even, sometimes, effort) in their own learning and achievement.”

And to quote Frank Zappa, which is becoming a bit of a habit at the moment: “go to the library and educate yourself if you’ve got any guts.” And he said that in 1966.

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.

The Mental Maths SATs Lady – reflections by Year 6 children.


At Teachmeet BETT 2011 I tweeted that my year 6 had created various renditions of the SATs Lady using Audacity. I also realised that I had not at any point shared their musical productions. The year 6 children that made these recordings have left now (the project happened in Summer 2010), but maybe their creativity can serve to inspire future generations of Year 6 SATs musicians.

(Apologies – these files haven’t copied across properly from Posterous).