Planning for Reasoning

I’ve known for a while that we need to improve our reasoning and problem solving teaching at my school. Analysis of the SATs papers showed that whilst number skills were really high, skills in ‘using and applying’ (as reasoning and problem solving were called) were less developed. Referring back to the post on why it was good that levels were dropped, it could be said that some children were moving on from primary with level 4 in maths overall, level 4 in number, but only level 3 in using and applying. This use of levels to mask a specific weakness in what a child can do is another reasdon they are bad.

The not-brilliant Ofsted in June, only shed a brighter light on what I already knew – we needed to change our teaching so that children learned to reason more.

This is a hard thing to do, particularly for older children when this expectation hasn’t been there all the way through school. There is a legacy of doing maths in a certain way, which leaves gaps in knowledge and resistance when a new way comes along.

However, 2 key documents from the NCETM have really helped us out. Firstly the Planning for reasoning documents, which have ideas for how to get children reasoning for every objective in the national curriculum. Secondly the Maths Hubs materials for mastery (and mastery with a greater depth) have the expectation that children are reasoning writ large through them.

These documents have been brilliant in supporting me and the teachers to have a greater expectations of what children can do. It would be even better if someone could produce some text books with these expectations equally prevalent…

The Impact of Fluency on Maths Planning

As my week on planning continues, I will now confess the biggest mistake I had made when moving over to the new curriculum in 2014: I only read the programmes of study.

The maths curriculum document is relatively large, and as a deputy headteacher I have a range of responsibilities beyond mathematics, so I thought I had become relatively good at skim-reading important documents and pulling out the key information. In maths I thought it was the programmes of study – the actual content of the curriculum. As it turns out, I was wrong.

In a short section at the start of the document, the aims of the curriculum are spelled out:

  • become fluent in the fundamentals of mathematics, including through varied and frequent practice with increasingly complex problems over time, so that pupils develop conceptual understanding and the ability to recall and apply knowledge rapidly and accurately.

  • reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language

  • can solve problems by applying their mathematics to a variety of routine and non-routine problems with increasing sophistication, including breaking down problems into a series of simpler steps and persevering in seeking solutions.

It would have been easy for me to take that in, but I skipped it and as it turned out, so did everyone else in my school.

This year, as we have striven to correct the mistakes of the year before, fluency, reasoning and problem solving have become the language of maths. Whenever we speak about maths lessons, or books, or planning, or CPD, those words always come out. If you did a Word Cloud of our dialogue around maths, those words would by far be the biggest. But what do they really mean? In this post I’m focused on fluency, but I’ll come to other two in the future.

Fluency is three things:

  • efficiency – instant recall and speed – having the very best method for solving a given problem
  • accuracy – getting things right and knowing you’ve got them right.
  • flexibility – being able to use maths in different contexts and domains.

What’s interesting is that if you just take the programmes of study for maths and deliver the content you find there, you will not make the children fluent. For example, in year 5 it says to teach the pupils this:

add and subtract whole numbers with more than 4 digits, including using formal written methods (columnar addition and subtraction).

If you took that statement at face value without the overarching aim of fluency you might consider it perfectly fine to teach the children how to add and subtract using a number line and if they are always accurate with the number line, you might teach them the formal method, but not really expect them to use it because they already have a method that works. You’ve taught rhe children to become accurate, but they are not yet fluent.

If you want the children to become fluent, you need efficiency too. So you have to teach them the formal method and teach it well, because for most 4-digit calculations, the formal method is the most efficient. You would also teach the children when a mental method is more appropriate, as we’ve all seen children do something like 4000-3998 in a formal method, because they are so conditioned to use it blindly.

But fluency is more than this. It is also flexibility. It’s easy to show that a child can add and subtract 4-digit numbers in a lesson about adding and subtracting 4-digit numbers. But what tended to happen in my school was children only being exposed to 4-digit numbers when they had to. When they were learning something else, say measures or coordinates, the numbers would be much smaller. The implication of fluency is clear – children should be able to add and subtract 4-digit numbers in any context and even in lessons when they have not been directly taught it. The teaching of the add and subtract 4-digit numbers objective should have an impact on the children that lasts into every maths lesson.

This has made a huge impact on our planning, some good and some bad:

  • Good in that expectations are much higher – when an objective is taught, the expectation is that it will not be dumbed down in future lessons throughout the term. This means when I where my monitoring hat, I am deliberately looking for occasions when the child can use something they have been taught out of the original context. It also means that teachers are now always on the look out for ways to use a taught skill in a different context and are more flexible in their own planning, adjusting lessons to to recap and consolidate key skills when children have not quite learned them.
  • Bad because all of our text books and resources don’t work like that, so teachers have had to spend a lot more time preparing activities then they used to.

There’s a another aspect to the impact of fluency on planning. The mental maths skills that underpin the ability to fluently add and subtract 4-digit numbers are not explicit in the programmes of study, but they are explicit when you read each statement with the words in the ‘fluency aim’ bullet point echoing round your brain. For example, when should children be able to double 36? At what point should a child be able to say that 3.4➗2 is 1.7 because half of 34 is 17? When do they get the sense of number that means they don’t blindly use a formal method to do 4000-3998? The reason I’ve put this at the end of the post is that we’ve not solved the mental maths side of things yet. That’s next half term’s work.


Medium Term plans in Primary Mathematics

How does your school organise their mathematics curriculum? There are broadly two ways of doing this. You could, in a three term year, divide the content equally between each term and teach a third in the Autumn term, a third in the Spring term and the final third in the Summer term. Or you could teach each domain of maths in more depth, sequentially going through them during the year and not revisiting them.

Here are three different models that all have their strengths and weaknesses.

1. The Scattergun Coverage Model by Hamilton Trust.

I’m not entirely sure how this model was planned, except that it seems to have covered all the objectives throughout the year.

Hamilton Trust cover objectives in a seeming scattergun approach
Hamilton Trust cover objectives in a seeming scattergun approach

This little extract informs teachers in which weeks they will be visiting different objectives. In my school, with the amount of work required to adopt the new curriculum, we thought this would do us fine, but actually what happened (as I’ve written) was that teachers did not take ownership of their planning and instead merely delivered the lessons they found in the order they were given. The outcome was that, whilst individual lessons were often good, the units (a week or fortnight of lessons) were often aimless with no sense of the children being on a learning journey. Maths only happened in the maths lesson and was very much isolated from the rest of the curriculum.

Much of this was my fault. As the maths leader I hadn’t taken responsibility for the planning that teachers had adopted. I’m sure many schools can make the Hamilton Trust plans work for them, but for us it had led to failure and so we needed a change.

2. The year-long blocked approach by White Rose Maths Hub

It’s too late to adopt this now, but you could plan to take it on for September.

The Overview for the medium term plans created by the White Rose Maths Hub
The Overview for the medium term plans created by the White Rose Maths Hub

As you’ll see from the picture, all the number and calculation objectives are taught by the end of Week 9 Spring term, and in much longer blocks than we have used in the past, enabling children to learn concepts with a much greater depth. I think this model will work brilliantly, and I’m currently discussing with the staff whether to adopt it for the next academic year. This maths hub have created tests for their medium term plans so you can assess how well the children are getting on. You can sign up for their free resources here.

3. The Term-by-Term approach (by me)

This approach was recommended to me by a headteacher who was part of the first peer review we had after our not-brilliant Ofsted. She said that I should base my plans on the 1999 schemes of work. So I went back to them and found them still hosted on a website in Dudley. Then I used a document that I only have a paper copy of called the ‘Steps Up’. I believe this was created by a group of Birmingham mathematics SLEs and provides some useful guidance as to what are the steps up to achieving the age-related expectations in each year group.

The model I went for was for the teachers to visit each domain during a term. They would cover all the content by the end of the Spring term (half in the Autumn term, the other half in the Spring term) and then use the Summer term to consolidate the age-related expectations.

My medium term plans are here:

Medium Term Plans

You’ll see that they are not quite finished yet. I haven’t written the Summer Term, nor have I completed the mental maths part of the plans – what you see in the mental maths part are still the old expectations from 1999.

Finally, I made some tests for the plans. These tests are half termly and I used Testbase to make them – a paid for service that hosts all the test questions asked of children in the England’s testing system for the past 20 years. You can find the tests here in their own folder.

You may consider it quite an effort to write a whole load of medium term plans. But it’s given me a much better insight into the new curriculum and a much greater sense of ownership over what we are teaching at my school. The problem with using other people’s plans, is that you end up believing the lie that you shouldn’t have to reinvent the wheel, when actually the process of reinvention is healthy all by itself.

Three ways to Improve Planning in Primary Mathematics

Since June (and the bad Ofsted and bad SATs results) I have been working on improving planning in maths and it’s really worked. Here’s how.

1. Impose a planning format

If you know anything about union and indeed Ofsted guidance you’ll know that is the *wrong* thing to do. But it’s exactly what I did at my school. I imposed a planning format that everyone had to use.

But planning is for the teacher, not the senior leader, I hear many cry.

No. I disagree.

Planning is for the child.

For too long we had written plans that support what the teacher might say in front of the children, how they might model or an activity or what learning resources they should prepare before the lesson. What we had not done is really think about what the child needs to do during a sequence of lessons to achieve the goals set out in the unit of work.

So in fact, while it was an official planning proforma that I imposed, it was actually the principle behind it that was important. And the principle was this: start the unit plan by writing the final lesson’s learning objective. Then work backwards through the learning objectives that build up to the final one to create a learning journey for the children. It’s a bit like that picture I shared 2 days ago. Then, for each day, plan what children who were likely to exceed the set learning objective might accomplish, and do the same for children who would struggle each lesson.

When this was done, each lesson would have 3 learning objectives around a similar aim, and each lesson would also built towards the next lesson.

What was great was that the teachers who got this, swiftly diverged from the planning proforma I had set up. They got the principle, so who cares if the format looks different? So much for the planning format imposition! What I found is that the teachers who needed most support where those who thought they were doing it right by merely filling in the boxes of the planning proforma without really thinking about the principle. And for these the proforma was a great starting point for explaining the principle in greater depth, thereby developing their mathematics subject knowledge.

2. Supply some medium term plans

There’s a lot to think about in mathematics, especially when you’re a non-specialist dealing with a brand new curriculum that you’ve had little or no training for. While I’m a big fan of ‘ownership’, as I wrote about yesterday, there is a limit to how much new stuff any teacher can take on in one year.

So I gave everyone medium term plans. I had some help. Various folk from different parts of the country have made maths plans and overviews and the like, so I used a few different documents to create medium term plans for mathematics for Years 1-6. I’m going to save the post for the ideal mathematics medium term plan model for another day (probably tomorrow), but suffice it to say that the medium term plans I’ve written have been a useful framework for teachers creating their unit plans mentioned above.

3. Link the medium term plans with assessment

The big problem we have at the moment in Primary is assessment. Nobody is quite sure what the age-related expected standard will look like. To make that more confusing, if a child is ‘age-related’ by the end of the academic year in mathematics, what sort of maths should they be able to do by December? or Easter?.

I looked around for some tests, and found none that suited what I wanted, so I made my own. Well that’s not quite true. I am making my own. So far I’ve made tests for Autumn 1, Autumn 2 and Spring 1. Each of these tests is linked with the medium term plan that should have been taught during that half term.

Of course, I don’t actually agree with testing a curriculum that is not best-fit. I’ve already written about the problems of the previous level-based assessment system being best-fit and how it disadvantages children. And of course any test is a best-fit measure. It seems ludicrous to me to create an ‘expected standard’ curriculum and then use a ‘best-fit’ tool (like a test) to measure how well each child has done, but that, again, is another post, for another time.

The point is that by analysing the tests, each teacher has been able to find out which bits of that unit the child did less well on and use that knowledge to follow up and plan further interventions during the next half term.

In conclusion

I’m biased obviously, but I think mathematics is in a much stronger place at my school than it was six months ago. We’ve had more conversations around mathematics knowledge in the last few months than in the previous five years, and I can really see teachers taking ownership of their maths teaching, rather than relying on third party solutions and merely delivering lessons.

In my next post I’m going to attempt to compare some different models for mathematics medium term planning that I’ve seen.

My first original idea

I’m being observed tomorrow. I’m teaching maths.

I’ve come up with an idea – the children need to learn how to use Pie Charts and I’m going to teach them how to do it using perfect numbers as the vehicle. What’s great is that it’s an original idea. I Googled it and found no results. Look:

Evidence that my idea is indeed original
Evidence that my idea is indeed original

The lesson may not work, but at least it’s original. And if it does work, who knows, maybe it will sell. Maybe I could publish it to the TES website and make – I don’t know – pounds.

Or maybe I should wait and see if the children actually learn something first. And if they do, I could always find contentment in that.

Mathemateers and their Chromebooks

Something I’ve not mentioned too often in my posts about my remedial group: the Mathemateers, is that each of them have a Chromebook.

A Chromebook is a complete non-event as a device. All it does is provide seamless access to the online materials you need to use to educate your children.

So I’ve previously written about using Khan Academy and Google Classroom to give my children meaningful homework and challenging practice. Seamless. Khan Academy and Google Classroom just work.

And here’s the thing: my school owns the devices, yet I let the children take them home. How can that be? Where is the safety in that? The management console in Google Apps allows me to enforce safe search in both Google and Youtube. I’m pretty confident that the Chromebooks are the safest device the children have at home.

But it must be an effort managing that sort of thing? No. Not really. It’s less time than marking a set of books and moreover my technician in school spends no time managing Chromebooks. He spends some time distributing apps to iPads and considerable time managing our Windows network, but no time managing Chromebooks.

I’m going to be speaking in more detail about how ace I think Chromebooks are at the Google Education on Air conference at the start of May. Here’s the details of my session. Even better, the Mathemateers will be there in person, through the power of the Google Doc. Might see you then.

Showing Progress in Fractions

One of the great things about teaching fractions to my Mathemateers group is that they knew pretty much nothing about them. This meant that whatever they learned would show oodles of progress – always good for impressing line managers.

(Not that it matters in this circumstance. I am the booster (remedial) teacher for the Year 6 group, therefore reporting to the Year 6 teacher who in turn reports to the Key Stage 2 Phase Leader. However I line manage both these people, so it’s less of a line and more of a circle…)

The assessment system we use showed that all the children in the group struggled with fractions. For example, Sarah‘s profile in ‘number’ looked like this:

Sarah's lowest 'ticks' were in Fractions
Sarah’s lowest ‘ticks’ were in Fractions. My school used the Incerts assessment system.

Of course when I asked Sarah some questions, it transpired that her prior assessment in fractions was, shall we say, over-ambitious, in that the assessment system said “she is developing the ability to use simple fractions that are several parts of a whole”, when actually she couldn’t do this question from ‘Recognizing Fractions 1‘ in the Khan Academy (which I have written about previously).

Most of the children couldn't do this question when we started.
Most of the children couldn’t do this question when we started.


Of course there’s the whole issue about performance and learning here. Sometimes children really do know something, but for whatever reason they don’t show it. This is performance. Performance variation is one of the main reasons for the difficulty in carrying out accurate assessment in education.

But for me as a teacher, this is great. I can now teach some stuff to the children and show great progress. And that’s what I did. Pretty soon the children had motored on to ‘Recognizing Fractions 2’ and even managed to do questions like this by the end of the first week.

By end of week 1 children could do this
By end of week 1 children could do this

No I’m not saying this is world-record teaching, but it does show progress. And what’s great is there’s an image, you can talk about it with the child and then the child has to write down the answer in fraction notation. It’s the perfect move from the Pictorial to the abstract. The downside, if you only use the Khan Academy is that children don’t write down what they did in their books and so their progress isn’t there for external visitors. And that’s not good if you’re a very book-scrutiny focused kind of school.

What would be great would be if we had already moved on to the New National Curriculum. However, as you well know, Year 6 are still working to the old curriculum. You see Incerts have just released their tracking system for the new curriculum and it looks fantastic. Here’s a picture of the ticks I could make about Sarah’s fraction learning:

What the new assessment for fractions looks like in Incerts.
What the new assessment for fractions looks like in Incerts.

However I can’t use that for my current group because they’re in Year 6. Nope. I’m going to have to cope with the learning that’s actually happened in the children’s brains and their SATs results in a few weeks time. Speaking of that, the final tool I’ve used to show progress is the Testbase tool that is a store of all the previous SATs questions. Sounds boring, but it’s really, really handy at the stage of the school year when teaching in Year 6.


Here comes the Dominator

So of course, it was Melissa who came up with the classic line. It’s a line that I’m sure is heard in many Key Stage 2 classrooms whenever fractions are being taught.

To the question “and what do you call the number at the bottom of the fraction?”

The child responds: “Is it the D-d-d-dominator?”

Fractions are counter-intuitive to many people. They get smaller as they get bigger. When you multiply them they get smaller, sometimes. And when you divide them you make them go upside-down. They are just weird. And then you add new words like the dominator* and the nominator** and the children get even more confused…

I spent quite a bit of time teaching my Mathemateers about fractions in the last term and I’m hoping that my next few posts will detail some of my failures and successes as I attempted to teach them something they had previously known very little about.

I have written some time ago about the importance of accurate vocabulary when teaching mathematics, particularly with fractions.

* by this I mean ‘denominator’

** by this I mean ‘numerator’

If only there was a tool like Khan Academy

So I was speaking to an inspector a few months ago who was trying to look a bit more deeply into my schools maths data. She asked out loud, “couldn’t you make a system that finds out how well children are doing in each individual area of maths, rather than these overall numbers?

Broadly speaking, that is the problem with data in schools. There’s always the danger of there being so many interpretations and approximations between the numbers that come out of the system and what’s actually in a child’s brain that the data becomes meaningless. Here’s how assessment works:

  • we decide what children should be able to do by particularly ages or stages and write it down in sentences.
  • we assess how well children can do the things we wrote down.
  • we turn those assessments into numbers.

Sometimes those assessments are called tests, at other times they are called observations. Either way it’s more or less the same process. However, quite often as teachers we get distracted and over-focused on the last stage of the process – on the numbers and less on the ‘what the children can actually do’ part of it.

This is where Khan Academy is brilliant. I’ve been using it this term with my Mathemateers group, and even though it doesn’t entirely match with the UK National Curriculum, it does help spot the gaps that children can’t do and provide the children with ways to practice skills that they are still shaky on. I also like the way I can focus the children on a particular skills at a time so that I’m not having to teach each child individually. For example for a few weeks I was focusing on fractions, so I directed children to activities that helped them visualise and practice fractions. I used Google Classroom quite often this – I would post a link in the Google Classroom assignments that would take the children directly to the Khan Academy challenge I wanted them to do.

Why Khan Academy fits in to the inspector’s question is that it gives a brilliant assessment of how children are doing in each area. For example, when setting my fractions challenge I mentioned earlier, I could see that one child had already mastered it, another was struggling at it and the rest had never tried it – it meant I could focus the challenge precisely on what I wanted the children to learn, support the child who was struggling and set a harder challenge for the child who had already mastered it. Ace.


Trialling Google Classroom

Google Classroom: a streamlined easy experience
Google Classroom: a streamlined, easy experience

I mentioned early on in my Mathemateers posts that I would be using Google Classroom to help me ‘deliver content’. So a few words about Google Classroom.

It’s easy. Really easy.

As the teacher, I choose my students from the Google Apps for Education users (we have Years 2-6 set up as individual users). The children receive an email to ‘accept’ the invite, or they can enter a code to join the new class that has set up. From there I can do one of two things:

  1. Make an announcement.
  2. Set an assignment.

The only difference in functionality between the two is that the children don’t have to respond to announcements. With assignments I write a title, write a sentence or two of description, set a due date and then I can attach ‘content’ in various ways:

  • as an uploaded attachment,
  • as a Google Drive file (docs, slides, sheets or drawings),
  • as a Youtube video,
  • as a URL.
The assignment screen on Google Classroom
The assignment screen on Google Classroom

It’s over to the students then. Each of my students has a touchscreen Chromebook – this may seem extravagant, but at less than £170 per device I think it is well worth the investment.

I’ve added Google Classroom to the screen of their Chromebooks via the Google Apps admin console, so it’s right there whenever they log on to their device. They can open it and quickly see which assignments they have done, or are yet to do, or (occasionally) are late at handing in.

Like the teacher, they can attach work to their ‘turn in’ comment. So far this has range from Google Drawings to screenshots of other work they have done online. This takes a bit of training, but once they’ve been through the routine a couple of times they soon have the hang of what to do when they have finished their assignment.

So far I’ve mainly used it for homework – it’s so satisfying to know that students are doing meaningful work without sending them home with polypockets full of photocopied worksheets.

It’s early days so far – I’ve only been using it with children for four weeks, but I can’t wait to get it going with the whole school. It may just revolutionise the way we do homework…