Category Archives: mathematics

Multiplication Tables Check Comparison Data

As ever with such things, it is important to point out that this data is not a scientific sample, has not been verified, and could be completely meaningless. However, in the absence of any comparative data from the DfE, it is an attempt to give some vague indication of the national picture of schools that took part in the MTC sample.

At the time of writing, some 211 sets of data had been submitted to the open spreadsheet online. Because it’s an open spreadsheet, there’s no guarantee that it doesn’t have errors, or that some data hasn’t been damaged, or even completely made up. With that in mind, I have completed some very simple calculations based on the data to give some idea of indicative figures.

Overall Averages

The mean average of all pupils’ results was 18.4

The mean average of all schools’ averages was also 18.4

The following table shows the approximate cut-off points when comparing schools’ averages, to place schools into bands.


Perfect Scores

There was talk at one point of full marks being the expect threshold. It’s no longer clear that this is the case, or even that there will be a pass mark of any sort at all, but within the sample:

Overall proportion scoring 25/25: 17.4%

Bands for proportion scoring 25/25:


Pupil Scores

More pupils did score full marks than any other individual score, with scores clearly more likely to be at the top end of the scale.


School Averages

The majority of schools had an average score of between 16 and 20


Does any of this mean anything? Not really… it’s a tiny sample from a voluntary pilot of a new test with no clear expectations hastily compiled from questionable data. But some of it is at least slightly interesting.


KS2 Maths – Question Level Analysis

As so many schools have evidently used the sample tests to help ascertain their pupils’ progress towards the expected standard (whatever that might be), I’m sure many will welcome the opportunity to analyse the outcomes.mathsqla

Emily Hobson, (@miss_hobson) of Oasis Academies, has kindly agreed to share the template she put together for analysing the KS2 tests.

The spreadsheet can be downloaded below, and then data entered to scrutinise your pupils’ progress in the main areas, and for each question.

Question Level Analysis (Sample Material) – Mathematics

Names need only be entered onto the first page; these will then carry across to later pages.

You can also adjust the % thresholds on the first page, and these will be reflected in the colour bands marked for each pupil.


Free mastery maths resources from White Rose Maths Hub

Carlsberg don’t make teaching schools, but if they did, I’m beginning to suspect that Trinity Academy in Halifax is what they’d come up with. Back in May at a conference I heard vice principal, Tony Staneff, explaining their mastery-led assessment system and I was impressed. Today I have received from the Maths Hub at the school, their excellent resources for organising and planning a primary maths curriculum based on the mastery principles.

I wrote back in April 2014 about how I was using a blocked approach to teaching mathematics, and plenty of people have asked me since then for my resources, or for long-term overviews. I’ve offered what I can, but the White Rose Maths Hub at Trinity have offered far more: a complete long-term scheme of learning for KS1 & 2, supported by excellent additional resources. And what’s more, it’s all free!

Firstly, let me say – as I’ve said before – that mastery has become something of a controversial and confused term. However, in this case they’ve got it spot on: mastery is for everyone, ensuring that all secure the key concepts and skills to allow them to explore things in greater depth.

So what’s available?

For each year group, they have put together an overview document setting out the suggested teaching blocks. This is broadly similar to the approach I took in my previous blog, but with much greater clarity, including the National Curriculum objectives to be covered in each phase. So far, useful indeed, but what really sets this resource apart is the supporting exemplification, which provide examples of the sorts of questions that support fluency, reasoning and problem-solving.

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The resources are provided for every unit in every year group in a manner that helps teachers who feel confident with maths – and maths mastery – but will be a real boon to teachers new to the idea who will value the guidance on pitch and direction.

Opening each resource is a really useful guide to the key ideas behind mastery maths, including the all important use of the concrete-pictorial-abstract model, and some frequently-asked questions. It should help to explain the key messages for teachers new to mastery or who have heard mixed explanations about what it involves. It also pinpoints other useful resources from the NCETM.

And as if that weren’t enough… (I know, I’m beginning to sound like a cheesy advert), the maths hub is also working on assessment resources to accompany the scheme. These are due over the remainder of the term and should be a real bonus in supporting teachers to make accurate assessments that will really support teaching and learning.

***Update – October 2016***
The work has continued. The hub has now produced schemes and assessment resources for every term in every KS1-2 year group, and even schemes for mixed-age classes in primary!

But enough… request the resources for yourself and get started on that journey!

Free Learning Schemes


Stop teaching ‘thousands’

If you haven’t already read my rant Stop teaching simile! then I’d suggest starting with that first. However, having started something, now things keep cropping up that I think the same sort of thing about, so here’s an addition to what seems to be turning into a series of “Stop teaching….” posts.

Stop teaching thousands

This will seem silly at first. Of course we need to teach thousands. But I’m coming to the conclusion that we tackle it in the wrong way in some ways. We expect children to work with increasingly large numbers as they go through primary education, and so once they seem to have grasped hundreds, it seems to make sense to move them onto thousands. Except, there’s a difference in the way the numbers work here, and it’s not always obvious when we teach it in that way.

The problem is not so much the thousands, and the next ‘column’ in our place value system. We often call it the ‘ten-thousands’ column, but like with the ‘tens’ column, we don’t often use that language for numbers that include a digit in that place.

Consider the number 54,321.

We don’t treat the 5 as a digit in its own right here; rather it becomes the tens digit of the section of the number that we describe as 54 thousand. It works just like the tens digit.

The same is true as we move over one more column. Consider 654,321

Here the 6 is merely part of the 654 thousands that are needed. It’s why we use commas after every third digit starting from the right. It’s not just a handy number, it actually helps us to read them.

So when we teach thousands, we should teach them as a block. It makes dealing with larger numbers much simpler. Recognising that each section of up to 3 digits is read as a single ‘chunk’ of a number makes it easier to read large numbers, and to avoid the common errors with placeholder zeroes. When a child needs to write four hundred and six thousand, and seventy four, it’s much easier to think of the blocks of 406 in the thousands block, and then 74 in the units section. It even invites the ‘punctuation’ of the number:

406, 074

(I grant you that the last section is lacking a name. I’m tending to prefer to call the very right-hand column “ones” and then refer to the last three digits as units, but there may be a better term. Suggestions welcome!)

The wondrous thing about so much of maths is that patterns are often scalable. The same system now allows us to consider millions, billions (so long as you’ve come to terms with the US billion) and to extend the system in groups of three, rather than one place at a time. Children are then very quickly able to read


as 123 million, 456 thousand, 789.

It also allows them to see the structure of the system so that they can identify any point in the place value structure. So, in the case above the digit 5 is clearly in the tens position within the thousands block: it shows us how many tens of thousands there are.

So, in truth, the argument is not for teachers to stop teaching thousands, but rather to consider thousands as a block of three digits in the numbering system following the HTO pattern.


Perhaps most importantly, before even thinking about numbers larger than 999, we should ensure that children have a secure understanding the relationships between the first three digits and their positions, fully grasping the nature of powers of 10. Once that is secure, rather than simply extending to thousands, it should be easier to teach the whole thousands block, right up to 999,999 with ease – and then to further extend.

Since posting this, Alex Weatherall has raised some perfectly reasonable points about the use of this structure. As with so many things, there is a risk that it becomes a prop which fails to secure clear understanding. So let me stress, I’m not proposing that we extend place value grids in this way. I present the image merely to make the point. My personal view is that if a child still needs a place value chart to organise numbers, then they shouldn’t be dealing with anything greater than a three-digit number.

Sins of the classroom revisited…

Personalisation and differentiation is the order of the day isn’t it?

Imagine a year five classroom, where the teacher has identified the progress every child has made, and tailored the lesson for each of them. When the main input begins, some are straight off to a task, while others stay on for extra support after the teaching. In the main session some are working in small groups, others work independently but on a common task, while some have a specific task just for them to meet their needs, whether more or less able. Some form of self-assessment lets the children identify their own success, and when children struggle they can call on professional support for near-immediate personalised feedback; when work is progressing well, a new challenge is set.

Sounds almost too good to be true doesn’t it? An outstanding lesson, maybe?

Peak MathsIt describes some of my year five maths lessons pretty well. But I don’t mean my, naturally, excellent teaching… I’m talking about me, as a second-year in middle school under the tutelage of Miss Ashworth. With the aid of Peak Maths. The form of self-assessment we used was checking the answer book. And the personalised feedback? We could line up at the desk for help when we were going wrong.

We wouldn’t countenance it, would we? It’d be lucky to scrape RI, let alone outstanding. Can you imagine offering such an experience up for your appraisal observation? Although I can’t quite put my finger on why.

Now, I’m not proposing an “it never did me any harm” line here. I was only 9 so maybe I missed some aspect that would make it unacceptable. But doesn’t it show something about where we’ve ended up? There are plenty of schools who would happily demand 5 different activities for different abilities, why not 6… Or 12? You could do that with Peak. The kids who got it could get on, those who needed more support got it. It sounds almost idyllic.

Of course, reality is that times change, and every system has its flaws. But is it just possible that there were some positives here?

Would a couple of minutes waiting for personalised help in a queue (a sight you’d never see in today’s Ofsted-ready classrooms) be a better use of time than children working on a low-level task to ensure that enough levels of differentiation were evident?

Would a bit of time working from a text book be more constructive than rolling dice to make questions, the answers to which your teacher won’t check until the next day?

Might the opportunity for immediate feedback for all help to reduce the marking workload of teachers while supporting the learning of children?

I’m not suggesting we all run every lesson like this – Miss Ashworth certainly didn’t. But might we do well to stop and question whether or not our “improvements” are quite as great as we imagine? Maybe the occasional queue at the teacher’s desk might not be such a bad thing?

Key Instant Recall Facts for mathematics

I am a massive fan of drilling and practice for children who need to learn number facts. And the reality is that that’s all children. Whether it’s the earliest number bonds, or the prime numbers, the new curriculum is very clear that fluency in these areas underpins much of what else is done in mathematics – and it’s right to do so, in my opinion.

Key Instant Recall Facts (Y2 example)

Key Instant Recall Facts (Y2 example)

I was, consequently, thrilled when the documents below were sent to me by Jo Harbour (@joharbour) of Mayfield Primary School. As a maths subject leader she has taken the time to set out a programme of teaching and learning to secure those essential number facts that runs from Year 1 through to Year 6. Beginning with the basic number bonds to 6, and developing to the knowledge of equivalent fractions and decimals by the end of KS2 they set out a useful progression for schools, and an excellent support for parents wishing to help at home.

Jo has kindly said that I can share these here, and so I am delighted to do so. They are created in Powerpoint format, which means that most schools can edit them. Note that some of the core elements are saved in the master slide, so to change the logo, for example, you’ll need to edit the slide master (accessed via the View menu on recent editions of Publisher). I have also uploaded a PDF version for those schools who cannot access the originals, but might want to follow the model.

Thanks must go to Jo Harbour for both creating and sharing these excellent resources (here contact details are contained within the files).

Key Instant Recall Facts (editable PowerPoint)

Key Instant Recall Facts (PDF)

What should the primary curriculum really look like?

Or: What is the point of teaching them all this stuff anyway?

I’m firmly of the belief that a majority (perhaps the large majority) of primary teachers share the same view: that we force-feed the kids in our classes a diet of breadth over depth because the curriculum, or the tests, or Ofsted, or SLT’s demand it. I think most primary teachers – particularly in infants and lower juniors – find themselves teaching things that they think are being delivered ‘too soon’ for the children in their care.

This is not an argument for the molly-coddling of children, or the lowering of standards. Rather it is an argument for a rationalisation of what we try to teach.

Coming from a middle school background, I have long wished that the 9-13 Middle Schools of the 70s had really taken off. I wish that the National Curriculum from its first inception had been built around the three main phases of first, middle and upper schools. Then, we might perhaps have had a different approach. Perhaps not in 1988, but maybe by now we might have recognised that very little really matters in the curriculum for children under 9 unless they are already confident with number and language.

I raise this point because of a brief discussion I had with Heather () on Twitter this evening. She quite rightly pointed out that starting to teaching persuasive writing in Year 1 didn’t seem to be contributing to a significant growth in the transferability of such skill at GCSE level. And if the skills aren’t transferable after 10 or 11 years’ teaching, then what’s the point? My response was both complete agreement and disagreement.

I disagreed because I think the point of teaching persuasive writing at KS1 is not to enhance the persuasive writing skills of 16-year-olds. In fact, I think the only purpose for any form of writing at KS1 is the practice of the basic skills of writing itself: the building of sentences; the use of capital letters; the simple formation of the symbols. However, I agree that expecting the teaching of varied genres at KS1 to have much impact on the ability of children to write for different purposes is frankly erroneous.

So, what then, is the point of any such work?

Looking back at the three-tier model, I’d be quite happy to see a curriculum substantially different to the one we have in place at the moment. This links in with Michael Fordham’s (@mfordhamhistorypost on an altered Secondary curriculum (which is well worth a read). In it, Fordham argues that English as a separate subject (as distinct from Literature) ought to be removed from the curriculum and its various aspects be properly addressed in domain-specific subject lessons. A genuine approach to Literacy across the curriculum. I’d be happy with that model, and what’s more, I think that it should be balanced by the inverse approach at first school age.

Given the choice, I’d happily see a three-tier curriculum (as in first, middle and upper stages) that broadly followed this pattern:

First School (age 5-9): Only English, Maths and Modern Languages would be statutorily prescribed programmes of study. All other subjects currently in the National Curriculum would become part of required areas of study (Arts, Humanities, Sciences, etc.) which were intended to provide breadth of experience and support the core subjects. Physical Education would also remain statutory, with no programme of study.

English and Maths programmes of study would be re-shaped to focus on Literacy and Numeracy. That is, all children would be expected to focus on developing oracy, and reading and writing basics (comprehension, building sentences, vocabulary, paragraphs, etc.), without concern for genres or required areas of study.That’s not to say that children wouldn’t meet other genres, or contexts, but that these would merely be to support the core teaching aims, rather than becoming additional goals in their own right.

Similarly, in Maths the requirements would focus largely on number work with relatively brief forays into shape as appropriate. To be fair, the new Maths curriculum has moved a good way towards this. I have often heard many secondary maths teachers say they’d be happy to teach Y7s who came to secondary secure with number bonds and tables and relatively little else. I’d agree, but think we could move to that sooner. Let’s have all 9-year-olds ready for the next level.

By removing the requirements to study particular programmes of study in all areas, it ought to be possible to move towards a system where the current Level 4 expectations could be met by the majority of 9-year-olds, rather than 11-year-olds. As Mark McCourt (@EmathsUK) said this weekend at the maths conference: Maths is like Jenga – pupils don’t fail because of weaknesses in the blocks at the top!

Middle School (age 9-13): The current subjects of the National Curriculum would remain, although English and Maths would be radically re-shaped to reflect the changes in the first school range. English could now begin to focus more on literature, although as Michael Fordham suggests, ought not to need as much curriculum time as at present (often 7.5+ hours a week in primary schools) as literacy should be mastered by age 9. There would still be study of language and some genre-linked ideas, but the shift towards domain-specific writing should be reflected in a shift in timetabled hours. I would argue that Middle Schools used to do this, until the KS2 SATs demanded that they narrow their timetables to focus on meeting the odd demands of the tests.

This model should leave more time in this phase for the study of subject knowledge. It would be far more sensible, for example, to begin a study of chronological history at age 9 and maintain it until at least age 16, rather than the current 7-14, and would be far more successful if students had already mastered the required literacy skill. Of course, this also would be combined with the middle school approach to specialism. We should expect all teachers of first school-age children to be expert in the teaching of early reading, writing and mathematics. We simply cannot expect that to apply right up to the age of 11 any more. It isn’t working.

Upper School (13+): The model that Michael Fordham suggests seems to make a good deal of sense to me here. By this stage children should have a broad experience of all the subjects, underpinned by their ability to access and use texts and a secure knowledge of number work. Ideally I’d argue for greater breadth until the age of 18 as well


Of course, none of this is rocket science. Indeed, most of it fits with what many primary teachers already think: if we spent less time ploughing through genres, or tackling history concepts with 8-year-olds, we could focus more on the things that really matter, and give those kids the freedom to access all matter of higher level material as they got older. Surely that’s got to be better than the current system which tries to build all curriculum areas from age 5… and too often leaves interventions at 16 to try to plug the gaps the system leaves?

Addendum: I ought to note that it wouldn’t necessarily be a requirement to change the whole system to a three-tier model. But I would argue quite strongly that expecting any primary teacher to be an expert in all areas of the curriculum up to Y6 level is never going to provide us with the best system; middle schools present a good solution to this; specialisation in small primaries is much harder.


Reading, Writing & Maths Key Objectives (new curriculum)

The Key Objectives for Reading, Writing, Maths and Science for KS1/2 can now be found in the Free Resources section.

Primary Progression Documents for English & Maths

Example progression documentThe nature of the new curriculum documentation is such that the primary section alone lasts for some 200 pages. It makes sense that it is organised in year group order for the core subjects, but it also makes it harder to visualise the progression of concepts and skills. That’s particularly problematic if you’re trying to identify key thresholds for assessment or planning.

Therefore, I have created these simple documents to support schools. They are not revolutionary, but simply present the objectives from the National Curriculum in a sequence of progression strands from Year 1 to Year 6 across Reading, Writing and Mathematics. Hopefully they might help schools in organising their curricula, and also in identifying progression across these very large subjects.

As with all my materials, they are also available at, and I recommend looking at the other resources available there to support schools’ journeys in implementing the new curriculum.

The documents are all shared here for ease, including easily printable versions:

English progression document (editable Excel file)

Reading progression document (printable A3 PDF)

Writing progression document (printable A3 PDF – 2 pages)

Maths progression document (editable Excel file)

Maths progression document (printable A3 PDF  – 3 pages)


Mastery Maths in KS2

Around this time last year I started reading about the work of the Ark group and Mathematics Mastery. So it was that as I moved to KS2 in September, I set about leading my year team – and an adjoining one – on a mastery maths journey. We’ve not reached the end-point yet, but it seems to be a hot topic at the moment, and following on from Bruno Reddy’s great blog about how he’d tackled mastery maths at Secondary, I thought it would be worth sharing what we’d done in KS2.

My initial thinking was led by what I’d read about the Ark scheme, and then built on by what I read in Dan Willingham’s excellent “Why don’t students like school?” about how children learn. It was soon put into context by my early experience in KS2. Having moved from KS3 I had previously taught maths sets; now I would be teaching a mixed ability group at a very different level. In KS3 I had previously moved towards what I’d considered to be longer blocks of two or sometimes three weeks on a unit. That had worked quite well for the high ability groups, but it had become clear for others it had still been too much too fast. I hadn’t previously been dealing with the need to teach and learn tables, or introduce area, but it felt like this was a good way of getting it right!

As the new curriculum was on the horizon, it was a useful starting point, and seemed to fit rather well with the mastery approach anyway. I began by mapping out broad units, using a model based very loosely on the Mathematics Mastery secondary curriculum map. It has’t held fast all year, but it provided a perfectly good starting point.

Year 5 Mastery Overview draft

Year 5 Mastery Overview draft

It meant that the first half term of the academic year was spent almost exclusively on place value and addition/subtraction. Within that we drew in elements which related to those skills. So, it seemed a sensible time to tackle in aspects like calculating perimeter, or finding missing angles on a straight line. Interestingly, there are plenty of similarities between our plan and that of KSA, particularly in that first term. See also what it says about separating minimally-different concepts (such as area & perimeter!)

In the Spring term, we took the step of spending a whole half-term on fractions. I’ll be honest, I was nervous about it. It’s never been my favourite area to teach, and rarely is it students favourite area to study. However, the system seems to have paid off. Knowing that we had weeks to spend on it meant that we weren’t afraid to take the time to secure the basics before launching into the higher level skills suitable to their age. And we weren’t abandoning it for another topic just as they were getting into things.

thinkingblocksWhat’s more, I drew on the things I’d seen of the Singapore bar method to really secure understanding of fractional calculations. We’d been using in school as a general problem-solving tool, but it seems that for fractions this approach really comes into its own. It allowed the children clearly to visualise the problems we were tackling, and to secure a much clearer understanding of why mathematical approaches worked. I cannot speak highly enough of the bar model in the context of mastery!

We haven’t been working on this approach for anything like as long as Bruno Reddy’s school, but initial results look positive. We’ve trialled the approach in Years 4 and 5 and seen a substantial improvement in ability to master the key methods, as well as spending more time to drive a focus on number bonds and tables. It seems that the approach will likely be even more successful in data terms once the new KS2 tests begin with the additional arithmetic paper!

Although it’s early days for us, some of the most significant evidence of success has come from the teams teaching the curriculum. Not all were sold on the idea at the beginning, but it has garnered the support and enthusiasm of those involved because it’s working! You can see it in the progress made by groups who traditionally do well, but perhaps more importantly in the successes of those learners who might traditionally have found making progress more challenging!

There’s still plenty to iron out and tweaks to be made over the coming years as different cohorts come up with different experiences. I still don’t think I’ve spent enough time and effort on securing number bond and tables knowledge – despite finding myself in every week’s work saying at some point “Now, can you see why it helps to know your tables?”. I still think we can do more to incorporate the important stages of concrete and representational development before the abstract. It’s not perfect yet.

But I can no longer imagine teaching any other way. Five years ago I was arguing that we needed to move away from week-long planning for maths; now I’d argue that anything less than six weeks is probably doing our students a disservice!

Ask me next summer how it’s paying off in terms of KS2 results!