Could ‘overlearning’ in mixed-age classes provide a blueprint for others to follow?

About a year ago, I wrote a blog entitled ‘How can the mastery approach work in a mixed-age class?’ In this blog, I put forward the approach that I have now been using for over 3 years with my own mixed age Year 3-4 class and talked about the huge improvements many of my children were displaying when they got to Year 4.

To summarise the approach briefly, all Year 3 and 4 children are taught together in a mixed ability class, moving through the same curriculum at broadly the same pace. They are exposed to both the Year 3 and Year 4 maths objectives throughout the year and the same objectives are then repeated the following year. There’s more to it than that and further details can be found by reading the blog here.

Each year that I have followed this approach, I have been really taken back by the progress shown by so many of the Year 4 children as soon as they have started repeating the content that they covered the previous year. In a significant number of cases, children who weren’t seen as being secure on the majority of their Year 3 objectives the previous year, were suddenly showing that they were secure on most of the Year 4 objectives, while other children were given the chance to really deepen their understanding of each concept taught.

Having first penned the blog, I remember asking Clare Sealy for her opinion as to why she thought I was seeing such gains being made. I wanted to know how the approach being adopted affected children’s memory when having to repeat the same maths concepts they worked on a year earlier. Clare explained that the new Year 4 children were doing well because they had the opportunity to ‘overlearn’. She went on to explain that overlearning is something which is ‘rarely done in a mistaken rush to ‘stretch’ higher prior attainers.’

Since the blog was written, another year has passed and yet again, I am seeing some similar results. Over the weekend, I asked on Twitter whether teachers of Year 5-6 classes were able to follow a similar approach, bearing in mind that they have the extra distraction of SATs to consider. I was really heartened by the response as it appears that many teachers are organising their maths curriculum in similar ways and, from the feedback I received, the surprising improvements I have seen when children have repeated content have been witnessed by others too.

Only a few weeks ago, I had the pleasure of talking to a couple of local head teachers and their maths leads about the way that maths is organised in my school. When we got onto the subject of teaching mixed-age classes and repeating the content each year, I was told that, at one of the schools, they had a mixed Year 4-5 class as well as straight Year 4 and Year 5 classes. The mixed-age class repeated the same content for 2 years whereas the single age classes obviously just focused on the objectives for that particular year group. They had noticed that the children going through the mixed-age class were, on the whole, performing better than those going through the single year groups by the time they had all finished Year 5

Of course, all of this evidence is merely anecdotal, but I am convinced that it is more than just a coincidence. So with this being the case, I would like to know whether any educational researchers out there would be interested into looking into this in more detail. What kind of impact does teaching a mixed age curriculum and repeating the same mathematical content the following year really have?

I would never want to second guess what the outcome of such research might be, but let’s just suppose for a minute that clear evidence did emerge that this approach has a positive impact on children’s learning in maths. Would it need to be restricted to mixed-age classes? Would there be anything to stop single age classes from following a 2-year cycle where content is repeated, giving the children the chance to overlearn? I don’t believe so. It might be fanciful to think that this could have the potential to really change the way that maths is taught in this country, but I think that it is definitely worth exploring further.

Number Bonds – The Other Essential Building Blocks to Success in Maths

In my last blog, I focused on the need to devote regular time to learning multiplication facts and expressed my belief that knowledge of key number facts such as times tables and number bonds is absolutely fundamental to children becoming confident and competent at mathematics.

Times tables are often a hot topic amongst teachers on social media, especially when it comes to how and when we teach them to children and their profile is only likely to further increase with the first multiplication tables checks now less than 2 years away. And yet, the subject of number bonds, which to my mind are equally important doesn’t seem to receive quite the same attention (not that for one second I am suggesting a further national test is required!).

The beautiful thing about maths is that patterns and links can be found everywhere: if you know one fact, you can use it to work out countless related facts. This is especially true with number bonds: if you know all number bonds that make any total up to 20, and understand how to apply these facts, the opportunities to use this knowledge are huge.  As an example, I constantly remind my Year 3-4 class that if they know their number bonds that total 10, they can use this to work out what to add to any 2-digit number to make 100. E.g. for 53 + ? = 100, I know that 7 goes with 3 to make 10, so I know that 7 also goes with 53 to make the next multiple of 10. I know that 6 goes with 4 to make 10, so I know that 6 tens and 4 tens will add to make 10 tens. Likewise, knowing that 7 + 4 = 11 should help children to calculate 70 + 40,  700 + 400 or 0.7 + 0.4 and of course, with related subtraction facts being taught at the same time, that 11 – 4 = 7, 110 – 70 = 40 and so on.

A few years ago, knowing how important number bonds are, I decided to ‘help’ my colleagues by coming up with some guidelines of which number bond facts should be known by the end of each year group from Reception up to Year 6. On the face of it, there was a clear and logical progression from one year group to the next, but the statements themselves were actually of very little use. For example, the expectation for the end of Year 2 was for children to know all addition facts that total any number up to 20 and all related subtraction facts. That’s fine but it’s not until you sit down and go through all the possible permutations that you realise when just looking at all of the addition facts alone, you are dealing with 121 facts to learn. This is a huge amount and so a much more structured approach is clearly required.

Many months later, I was sat at home one night and I decided to write out all 121 facts then started to cross out those that could be grouped together in certain ways. I crossed out number bonds to 10, doubles, near doubles, calculations that involved adding 0, 1, 2 or 10. I also looked at facts that required bridging through 10. Before I knew it, I was left with just 4 facts that all involved adding 3. This had to be a better approach: to group the facts into smaller families of facts that could be learnt discretely in much more manageable amounts. Why hadn’t I thought of this before? Still, I was feeling quite pleased with myself after this lightbulb moment. All I needed now was a way of displaying all the facts in their groups, so they would be easy to follow.

Only a few weeks later, a colleague from another school visited and I discussed what I had come up with. Her response was ‘The NCETM have already put together a really good graphic that shows all the key addition facts grouped like this.’ Needless to say, the wind was taken straight out of my sails! Enough of my rambling though. The important thing is that there is a great document out there which I certainly found to be really useful, as did my colleagues when I shared it with them. I hadn’t seen it before and haven’t stumbled across it in any other context since so I suspect that there are others out there who might benefit from using it too, which is why the NCETM have kindly allowed me to share it via this blog. As you will see, the facts are split into those to be learnt in Year 1 and those that are for Year 2.

Addition facts for blog

 

The next step for me was to come up with a timetable for learning each set of facts which could be a focus for our daily maths skills sessions. As we have mixed Year 1-2 classes, we had to include all the groups of facts together. There is no exact science to this timetable and I must stress that this is an example which I would never expect the KS1 staff to stick rigidly to, but the feedback I got from them was very positive.

timetable for blog

As you can see, each group of facts is practised and then returned to a number of weeks later and with the classes being mixed age, the whole timetable then gets repeated for another year (You will notice that I prefer to use the term Making 10 rather than Bridging / Compensating). With single Year 1 and 2 classes, there are less groups of facts to learn so there would be the opportunity to return to each set more frequently. I’m sure others more qualified than me might have tips on how the timetable could be improved to achieve maximum benefit but it’s a starting point. As with times tables, there is no magic wand. The key thing is regular practice and this, of course, can take many different forms, but with so many facts to learn, I am hoping that this approach will pay dividends in the long run.

Making Time for Tables

Over the last few days, there has been a huge amount of debate about the rights and wrongs of having a times table check and it has clearly split opinion amongst teachers. Aside from concerns about the tests being used as an accountability measure, there have also been lively discussions on social media about the importance of times table knowledge in the maths curriculum.

For what it’s worth, my opinion is that knowledge of key number facts such as times tables and number bonds is absolutely fundamental to children becoming confident and competent at mathematics. In isolation, of course, knowing the 9 times table is pretty useless but an understanding of how to apply this knowledge to a range of situations involving the likes of multiplication, division, fractions, percentages, area and ratio, to name but a few, is only going to be extremely beneficial.

Helping children to learn their times tables clearly isn’t going to happen by chance and, with so many facts to learn, the key for me is exposing them to regular practice and giving them the opportunity to truly learn a times table before moving on to a new one. There are many ways to teach times tables and I like to vary the activities I give to my children, but one particular resource which has made a bit of a comeback in my classroom over the last few years is the counting stick.

The ideal end scenario for me is that children know times table facts off by heart, but getting to this point takes time and the way they get there can be as important as the end result. When introducing a new times table, I usually start with the counting stick by getting my class to count up and down in steps of that particular number. Obviously, this is a long way from knowing multiplication facts – and I certainly don’t want them to rely on counting on their fingers – but it begins to expose children to the numbers which appear in a given table. When counting down, I also regularly get the children to carry on below zero as this is an excuse to work with negative numbers whilst continuing to count in steps of the same amount.

Now there’s nothing new about using a counting stick to teach times tables and I’m sure that many people have seen videos of Jill Mansergh teaching adults how to learn their 17 times table in a matter of minutes. For me, this isn’t about being able to learn an impressive times table in double quick timer. Rather, it’s about promoting the message that if you know one fact, you can quickly work out another one and this is a message that I regularly preach to my class. Instead of thinking ‘I don’t know that multiplication fact,’ I want them to think ‘I don’t know it yet, but what do I know that can quickly help me to work it out?’

The 6 times table is perfect for this approach. As long as you know that 2 x 6 = 12, then it is easy to double this amount to work out that 4 x 6 = 24. Likewise, doubling 24 to work out 8 x 6 is straightforward as in both cases, no tens boundary is crossed. We now know that 8 x 6 = 48 and can use 8 – 6 to help work out 48 – 6, thus finding 7 x 6. Knowing that 10 x 6 = 60 makes finding 5 x 6 much easier, especially as children find it far less challenging to halve an even number of tens. If we know that 5 x 6 = 30, adding another lot of 6 to calculate 6 x 6 doesn’t require a multiple of ten to be crossed, so this shouldn’t be too tricky. Knowledge of 10 x 6 also helps when finding 9 x 6 as it is only 1 lot of 6 less. You can carry on finding links for 11 x and 12 x and, of course, the same links apply, regardless of which times table is being learnt. As I have blogged about before (Tell Them What They Know), maths is full of patterns and links, but I feel that this needs to be made explicit to children.

I regularly come back to the counting stick but also like to make use of games such as Top Marks’ Hit The Button (https://www.topmarks.co.uk/maths-games/hit-the-button) and Transum’s Tablesmaster (http://www.transum.org/Software/Tablesmaster/) to build up sharpness and accuracy. Throwing in weekly timed tests amongst other activities, I tend to spend at least 3 weeks on a times table before moving on and it is important to point out that, during this time, related division facts are given just as much prominence. I also see it as vital to return to tables that have already been worked on as, at this stage, I believe it is very much a case of ‘use it or lose it!’

I’m certainly not claiming that this approach is perfect, but there is a positive attitude towards times tables in my class and improvements in knowledge of multiplication facts are clearly being made.

All of this is pretty time consuming and I would have plenty of sympathy for any teacher reading this who is thinking ‘Where on earth would I get the time to fit in all of this practice?’ I’m lucky because, at my school, we have a dedicated maths skills session every day where we have 25 minutes that can be devoted to times tables, number bonds and recapping on previously covered work.

At the end of the day, no matter which approach is followed or how much time is devoted to it, I remain convinced that teaching children times tables remains a vital stepping stone in helping them to become confident and competent at mathematics, regardless of whether knowledge is checked at the end of Year 4 or not.

How can the mastery approach work in a mixed age class?

A few months ago, I followed a thread on Twitter which was started by a teacher who was concerned about his daughter going to a school where classes were made up of mixed age groups. There were many reassuring replies about the benefits of being in a mixed age class, but a large question mark still appeared to hang over one particular subject: maths. The great thing about many of the curriculum objectives covered in almost every subject is that they don’t rely upon a particular topic having to be taught first. So if a child enters, say, a Year 3-4 class, it doesn’t matter whether they learn about the Ancient Egyptians in Year 3 or in Year 4, as long as the appropriate curriculum is covered during that two year period.

Sadly, maths doesn’t work like that. As an obvious example, it wouldn’t make sense for a child to work on place value in 4-digit numbers in their first year in a mixed age class then look at place value in 3-digit numbers the following year. Maths is all about building on previous knowledge and certain concepts need to be taught before others. So how can maths work in a mixed age setting, especially when trying to follow a mastery approach where classes are expected to work their way through the curriculum at broadly the same pace?

I know that in some schools, children from one year group work with a Teaching Assistant while the other year group receive their teaching input, then halfway through the lesson, both year groups swap. Having never taught in this way, I can’t really pass judgment on how effective this approach is, but for those teachers who don’t have an extra adult in their class for each maths lesson, this isn’t really an option anyway. This was the case for me when I began teaching mixed age classes for the first time just over 2 years ago, so I needed a different approach. My class was one of three Year 3-4 classes and, in the past, the children had been ‘ability grouped’ for maths, but as I have alluded to in a previous blog, this isn’t something that sits well with me. We decided to go for a different approach and while I certainly don’t claim that it is perfect, I wanted to share it because I feel that there are real merits to it.

An alternative approach to teaching maths in a mixed age setting

The main features of the way we teach for mastery in mixed aged classes can be summarised as follows:

  • All Year 3 and 4 children are taught together in their own mixed ability classes, moving through the same curriculum at broadly the same pace
  • All the children are exposed to both the Year 3 and Year 4 maths objectives throughout the year and the same objectives are then repeated the following year

As you read this, you can be forgiven for having several questions in your mind at this point and I will attempt to answer as many as I can.

How can 2 years of maths objectives be covered in just one year?

The straight answer is ‘It can’t.’ Well not quite anyway. We decided to strip out all of the times table objectives and ensure that these were covered in the daily maths skills sessions that we do, while money was to be covered through work on addition, subtraction and decimals. It does mean that, for example, in a 4 week block on fractions, a lot more objectives need to be covered than would be the case for a single year group, but when this 4 week block is repeated the following year, children get the chance to really consolidate their understanding.

How do you plan for teaching both Year 3 and 4 objectives?

Fortunately, many of the objectives from consecutive year groups dovetail each other extremely well. As an example, in Year 3, children are expected to add and subtract 3-digit numbers using formal written methods and in Year 4, they need to do the same thing with 4-digit numbers. The key to planning each unit of work is to look at the objectives for both year groups and plan a path to take the children from the starting point for the Year 3s in that particular area of maths to where the Year 4 children need to get to. The mixed age mastery overviews produced by White Rose Maths group together objectives from the different year groups in a logical manner which really helps with this process.

How do the Year 3 children cope with working on Year 4 objectives?

When Year 4 objectives naturally follow on from Year 3 ones, the children are usually in a good position to build on what they have just learnt. As an example, having worked on ordering and comparing numbers up to 1000, the obvious next step is to then order and compare numbers beyond 1000. Of course, there are some Year 4 objectives which introduce a completely new concept, such as Roman numerals, but my hope is that those Year 3 children will have securely understood the concept by the following year, when they will have been exposed to the topic for a second time.

 Do the Year 4 children repeat the same work they did the previous year?

I explained in a previous blog how children are allowed to choose their own level of challenge depending on how confident they are feeling and the real test is to ensure that there are tasks available that will deepen conceptual understanding and allow children to progress from where they were the previous year. A year on, very few of the tasks themselves are remembered by the children but in the vast majority of cases, they end up choosing a more challenging task than they did the previous year. Some tasks are changed from one year to the next although it is possible for a child to repeat a task, but if this means they consolidate to a point where they are more secure than they previously were, then I don’t view this as being a negative.

 How are Year 4 children challenged during teacher input when working on Year 3 objectives?

One of my biggest concerns when repeating the maths curriculum for the first time was ensuring that those children who had clearly mastered Year 3 objectives didn’t switch off and instead remained motivated when working on those same objectives again in Year 4. To compound my worries, the cohort which were moving from Year 3 to Year 4 were one of the strongest ones I had ever taught. I decided that these children would have a key role to play in supporting a Year 3 partner by checking their understanding during whole class and partner work. All of a sudden, they had to think about questions to ask which would guide their partner without giving them the answers and in doing so, their own levels of conceptual understanding needed to be sound. At the same time, it was important to offer more challenging examples for these children to work on during whole class work, if they felt confident enough to do so.

 What has the impact of this approach been?

Just over 2 years have passed since we began this approach to teaching mixed age classes. In that time, one thing in particular has been absolutely striking: the progress made by the Year 4 children. Taking my current Year 4s as an example, I know we are only half way through the first term but I have been left amazed by how many children, who weren’t assessed as secure on the majority of their Year 3 objectives last year, are now secure on most of the Year 4 objectives covered so far. These children generally chose the least demanding work in Year 3 and now they are choosing much more challenging work. The Year 4 class from the previous year, albeit a much stronger cohort, displayed similar levels of progress. I don’t know a great deal about cognitive science although I have read blogs and articles from the likes of Clare Sealy and Dylan Wiliam with great interest and I would be fascinated to find out how the approach we have adopted affects children’s memory having to repeat the same maths concepts they worked on a year earlier. The anecdotal evidence I have, certainly suggests that it has had a very positive effect.

It is still a work in progress, but I believe it is a genuine option for teachers to explore if they are wrestling with the complicated challenge of following a mastery approach with a mixed age class.

Place Value: It’s not just about columns!

As another new school year begins, thousands of children up and down the country will no doubt begin their maths work for the year by looking at place value. To start here is understandable. After all, if we are hoping for children to carry out any of the 4 number operations, it makes sense for them to first be able to read and understand the value of the numbers they are going to be dealing with.

As children progress through the primary year groups, they are expected to recognise the value of each digit in numbers of ever increasing sizes. Time is rightly spent ensuring that they can identify the ‘column’ that a digit is in and therefore, then explain what that digit is actually worth. This is a really important step in being able to recognise numbers but if we’re not careful, the idea of columns can actually lead to misconceptions. As an example, I would be confident that almost all the children in Key Stage 2 at my school would easily be able to recognise that there are 6 tens in 62 but, in my experience, there would be a significant number (including some Year 6s) who would incorrectly say that there are 4 tens in 341 and it is clear to see where this misconception has come from. After all, when looking at a number, it is easy to fall into the trap of asking ‘How many tens/hundreds have we got?’ when we actually want to ask ‘Which digit is in the tens/hundreds column?’

I actually think it is important to ask both of these questions when teaching place value and to draw a distinction between the two, spending time focussing on how many tens/hundreds etc. there are altogether in a particular number. Once children are able to recognise how many tens there are in a number, it can be a huge help when it comes to mental addition and subtraction. Take, for example, finding 10 more or less than a given number. I’m sure I am not alone in teaching children who are confidently able to add or subtract 10 from numbers like 53 or 326 but when asked to do the same with a number where they need to cross a multiple of 100, then it all falls apart. Seeing a mistake such as 203 – 10 = 293 isn’t unusual and yet once children are able to recognise that there are 20 tens in 203, they can see that subtracting 1 ten from the 20 tens leaves us with 19 tens and an answer of 193. It is equally effective, of course, when it comes to multiples of 10 e.g. for 288 + 40, we add 4 tens to the 28 tens to give us 32 tens and an answer of 328. It tends to be a real lightbulb moment when I demonstrate this method to children and, in my opinion, it helps them to improve their understanding of number and how our number system works. The added bonus, of course, is that the mental calculations involved are usually much simpler than having to bridge through a multiple of hundred.

So as I begin to teach place value next week, I’ll keep reminding myself: it’s not just about columns!

Tell them what they know

 

A few years ago, I came across a great term to describe how the most effective mathematicians operate: lazy intelligence. It might sound derogatory but the idea is that they use the least amount of effort to get to the answer that they need. In order for anybody to be able to do this, they would need to look at a calculation, a problem or puzzle and assess all of the strategies available to them, before choosing the method which involves the least work. By doing so, it would also usually get them to the answer in the quickest time. They may seemingly take short cuts or see solutions that aren’t always obvious to everyone.

It may well seem fanciful aiming to develop a class of ‘lazy’ mathematicians at primary school, when for many children, having one method that reliably gets them to the correct answer could justifiably be seen as a real success, but we can start to sow the seeds. If we see mathematics merely as a set of procedures that we want children to learn and use in isolation then it is unlikely that this will happen. If, however, we are able to get across to children, even at a young age, that mathematics is a discipline full of patterns and relationships, they can start to make links between what they know and what they are trying to find out; they can spot patterns and relationships and begin to use lazy intelligence.

There will always be some children who appear to have a natural aptitude for spotting patterns and making links for themselves but they are often in the minority and for those who don’t seem to be able to do this yet, how will they ever begin to do so, or even realise it is possible unless patterns and relationships are, initially at least, pointed out to them?

Children in my class are used to me regularly making comments such as:

‘You know what to add to 3 to make 10, so what do you add to 173 to make the next multiple of 10?’

‘You know 2 + 6 so use it to work out 42 + 6.’

‘Use your knowledge of 7 x 9 to work out what 7 x 90 is.’

They still have to do the work, but I am signposting for them and I feel that, at this stage, I very much need to. I know that if I asked my class to calculate 763 – 499, some of my most confident mathematicians would still plump for a compact written method and although they would almost certainly get the right answer (which shouldn’t be undervalued), in the long run, I would love them instead to be seeing the relationship between 763 – 499 and  764-500 as much less effort is then required.

I do feel that we are moving in the right direction though and this has been demonstrated recently when working on times tables during maths skills sessions. I gave the children the following 4 questions:

4 x 8

8 x 8

12 x 8

16 x 8

Not all of my children know their 8 times table off by heart (I’ve still got 5 weeks to get the year 3s there!) but they are learning how to calculate any facts that they may not be able to quickly recall. With that in mind, it was great to hear the explanations of how some added their answers for 4 x 8 and 12 x 8 to work out 16 x 8, while others doubled their answer for 8 x 8 and they could explain why this worked. The thing is that I didn’t give them this set of questions by chance and it wasn’t the first time a set of questions like this had been given to them. I had to start showing them how to use what they know and many are now starting to do this independently. We’ve still got a long way to go as a class, but using more procedural variation will surely only help to foster that all important idea of looking for relationships. At the end of the day, I’m not content with teaching children how to ‘do maths’, I want to teach them how to become mathematicians.

So the next time a friend or colleague tells you they have a lazy class, don’t dismiss it straight away as it may not necessarily be a bad thing! In the meantime, I’m going to carry on telling my children what they already know.

A Hidden Gem

Over the last 18 months, my school has made many changes as we have gone on a journey towards following a mastery approach to mathematics. With more and more changes taking place, it soon became apparent to me that the school’s old calculation policy was no longer fit for purpose as it didn’t align with the way we were now teaching children to add, subtract, multiply and divide. The old policy was pretty much entirely based around the abstract and that alone wasn’t how we wanted the children to understand methods of calculation. One of the central pillars of the mastery approach, of course, is the idea of first beginning to understand a concept through using concrete materials, then moving on to the pictorial before finally being able to work effectively with the abstract. If this was now going to be the way that we wanted to teach the four number operations, then, for me, this idea of CPA would need to feature prominently in any new calculation policy.

Ordinarily, I would see one of the main purposes of a calculation policy as ensuring a consistent approach to teaching calculation across a school. While this is still the case, at a time when many colleagues might understandably be feeling unsure about mastery and particularly using concrete materials to teach calculation, I believe that a well thought out policy can provide real support in following a CPA approach.

 

Like all the good mathematicians, I’m a bit lazy!

With this in mind, just over a year ago, I did what I’ve always done when looking to write a new policy: I had a good look around at what other people had already created to see if there was something I liked which I could use as a starting point. I found many policies out there that had been designed with mastery in mind, some of which were being very highly spoken of at maths network meetings that I attended. When I looked at most of them, however, I couldn’t help feeling that they were a little too cluttered for my liking and if I was finding them hard to follow, what might colleagues and even more so, parents think of them? Not only that, in almost every case, that vital concrete step appeared to have been somewhat overlooked which wasn’t what I wanted.

I was searching for a calculation policy that would deliver a real clarity as it took readers through an easy to follow path from concrete to pictorial, then abstract but alas, it looked as though I would have to do it myself from scratch. That is until I stumbled across a tweet from the White Rose Maths Hub, explaining that they had created their own calculation policy.

 

The Gift That Keeps On Giving

Now, I make no apology if I sound effusive in my praise for the White Rose Maths Hub. On more than one occasion, I have referred to them as ‘the gift that keeps on giving,’ and they are quite rightly held in high regard up and down the country for very good reason. The detailed schemes of learning that they have created are a fantastic resource and I had the pleasure of working with them when they designed schemes of learning specifically for mixed age classes. This year, they have received great acclaim for the work that they have done on bar modelling while they have created extremely useful daily revision questions for the Key Stage One and Two SATs.

As soon as I realised that the White Rose Maths Hub had created a calculation policy, I was excited and confident that it would be well thought out. Being such a fan of them, the only thing that surprised me was that I didn’t already know about it and even now, I believe it is still a bit of a hidden gem, which is why I wanted to write this blog. When I downloaded it for the first time, I certainly wasn’t disappointed. I was looking for CPA and a layout that was easy to follow and that is exactly what I found as the short example below shows:

blog 5 picture

(N.B. The White Rose Maths Hub’s calculation policy has been revised in the last couple of months from the image shown above but it still follows a very similar approach)

Of course, having found what I wanted, I still had a considerable task in turning it into the calculation policy that I wanted. I added parts, removed and changed many others as well as creating an appendix containing a range of recommendations and teaching ideas aimed at further supporting staff, some of which came from the NCETM’s Calculation Guidance document. Not for the first time though, as far as I was concerned, the White Rose Maths Hub had delivered.

If you’re looking at writing a new calculation policy or haven’t yet seen the White Rose Maths Hub’s other resources, in my opinion, it’s definitely worth visiting http://whiterosemathshub.co.uk/free-learning-schemes/.

I have a confession to make…

I am recovering from an affliction that has affected me for most of my teaching career and I know that I’m not alone: for many years, I significantly over-marked. While I was doing it, I was fully aware that the time I spent marking was hugely disproportionate to the impact it was actually having on the children I taught, yet I still did it. In fact, let’s be honest, I didn’t even believe I was marking for the sake of the children. I was doing it for others because that is what was expected of me.

At its worst, there were times in the past where I would take home upwards of 50 books in an evening and I have been reliably informed by my colleague, that he has talked about the box I carry to and from school, at conferences up and down the country when speaking about ‘Doing Less, But Better’ (the shame of it!).

It might seem a bit rich, therefore, that I should write a blog about marking in maths, but I like to think that I am now something of a reformed character and yet I know that there are teachers up and down the country who are bound by policies requiring marking to be carried out in maths which is both onerous and ineffective. When I hear stories about schools that ask their teachers to ‘bubble and block’ every piece of maths work, it troubles me greatly.

Marking is, of course, just one of a number of ways of giving feedback. Maths lessons provide countless opportunities for teachers to give immediate verbal feedback to a child, which research suggests can have a far greater impact than written comments, especially when you consider the latter often aren’t seen by the child until the next day. Looking at the maths work a child produces is still an essential part of the feedback process, yet I believe that this is much more because of the feedback that a teacher can gain from the child rather than any feedback that can be given to them in the form of a written comment. It can give a great insight into whether children have understood the mathematical concept that has been taught or not and, as a result, I believe that books should be looked through at the earliest possible opportunity but this doesn’t mean that marking should be a lengthy process.

It’s time to burst the bubble and remove the block to marking maths work efficiently

Whether it is called ‘bubble and block’, ‘a star and a wish’ or anything else, the idea of spending so much time producing written comments that let children know what they have done well and what their next step should be in maths really is counterproductive. Certainly, for those of us following a mastery approach, there is a simple message: the next step should be the next lesson. And herein lies the problem. The more time that is spent writing down next steps in books, the less time there is available for teachers to do what really matters: adapting and preparing for the next lesson.

Of course, for some children, looking at their work may reveal a lack of understanding. If this is the case, some form of intervention (e.g. a post or pre-teach) before the next lesson would certainly seem much more favourable than trying to address any misconception through marking. The likelihood is that a couple of written sentences won’t be able to significantly improve understanding in a way that an hour long lesson failed to do.

Don’t just take my word for it

Last year, the NCETM produced a document called Marking and Evidence Guidance for Primary Mathematics Teaching (https://www.ncetm.org.uk/files/33333022/NCETM+Primary+Marking+Guidance+April+2016.pdf) where they stated the following:

‘Marking and evidence-recording strategies should be efficient, so that they do not steal time that would be better spent on lesson design and preparation. Neither should they result in an excessive workload for teachers.’

When it comes to the process of marking, the NCETM also offer this interesting piece of advice which, as well as being beneficial to children, could also help to reduce a teacher’s workload:

‘Evidence shows (Black and Wiliam 1998) that pupils benefit from marking their own work. Part of this responsibility is to identify for themselves the facts, strategies and concepts they know well and those which they find harder and need to continue to work on.’

 

Ofsted’s School Inspection Handbook clearly states its own position on the matter of marking and feedback:

‘Ofsted does not expect to see any specific frequency, type or volume of marking and feedback; these are for the school to decide through its assessment policy. Marking and feedback should be consistent with that policy, which may cater for different subjects and different age groups of pupils in different ways, in order to be effective and efficient in promoting learning.’

 

In theory, this is great news from Ofsted but if a school’s policy expects teachers to mark maths books in detail on a regular basis, then there is very little that can be done about it. At a time when the profession is seemingly facing the prospect of a retention crisis with workload seen as a major factor, surely it is time for senior leaders to adopt a common sense approach to marking in maths. And who knows, having a bit more time to spend on planning maths lessons may just prove to be a good thing for the children too.

Answers are overrated

Last weekend, I read an excellent blog by Nicola Clements (https://firestartingblog.wordpress.com/) in which she reminded me of some great advice: spend less time focusing on the final outcome produced by the children and more on the journey they go on in order to reach that outcome, as this is the point where learning really takes place.

It got me thinking about how this approach relates specifically to maths. Unlike a lot of other subjects, children usually produce several ‘outcomes’ in every lesson, with each answer they generate being an end product in its own right. Taking Nicola’s advice into account, it made me wonder whether too much importance is placed upon the answers that children give in maths. After all, the answer to question 3 will no doubt be instantly forgotten by little Freddie as soon as the lesson has finished, but the skills that were honed or the links that were made on the way to getting that answer could stay with him for a lifetime.

With this in mind, I decided to try something a little bit different with my class this week. As well as giving them a question, I also gave them the answer. They seemed a little confused for a few seconds, but I explained that I wanted them to find as many ways as they could to get to that answer and to discuss with their peers the methods they preferred and why. I suppose it’s a little bit like giving them all a map, telling them the final destination they need to get to and asking them to find the best route by looking at all the options available to them. The fact that I have told them where they need to get to makes the task no less challenging while the opportunities for learning are doubtless much greater than if I had given them all the same set of directions to follow and asked them to find out where it leads them to.

A number of things struck me as the class worked enthusiastically on the task they had been given:

  1. Those children who are usually in a race to be the first to get to the answer realised that the rules of the game had now changed. I like to think that this was a good thing because, while I do want to promote mathematical fluency, children can miss so many opportunities to develop their mathematical thinking due to being in a rush to get to the answer. Children who fall into this group are often very capable procedurally but is their conceptual understanding as secure as I would want it to be?
  1. From the feedback they gave me, it was clear that less confident children had just had a real weight taken off their shoulders. That pressure of needing to get the correct answer had been removed which can only have helped to increase their confidence and motivated them to get stuck into the task.
  1. By already knowing the answer, they had some instant feedback available to them. If they had made a mistake in their calculations, they knew about it straight away and wanted to identify and correct any errors. Far too many children, having generated an answer for a question, see it as a case of ‘job done’ and they move on. Knowing the answer actually encouraged children to grapple with the question more and this surely has to be a good thing.
  1. The different methods that were shared generated some excellent discussions and doubtless helped a number of children to make links that they hadn’t picked up on before.

Overall, I feel that it was a really worthwhile exercise and something which I will definitely look to repeat. Having said that, it is important to acknowledge that children need to generate their own answers on a regular basis. Producing accurate answers with good speed and efficiency is an extremely important skill that needs to be encouraged while a set of answers in a child’s maths book can provide a teacher with valuable feedback. Therefore, I am certainly not advocating giving out the answers all of the time but let’s not give too much status to them. We all want our children to be able to ‘do maths’ but the aim should surely be to turn them into mathematical thinkers who value what they can learn from the process much more than the final outcome.

Transforming fixed mindsets towards maths

For a long time, I have held the belief that one of the biggest inhibiting factors when it comes to children making progress in maths is their own lack of confidence. Let’s be honest, almost every child can have a go at writing a story or drawing a picture but when faced with a maths question which involves a concept that they really don’t understand, how do they even make a start? So it is no surprise that many children have a fixed mindset when it comes to maths.

I have always seen one of my biggest roles in teaching maths, therefore, as that of a confidence builder and yet until the last couple of years, without realising it, I was doing certain things in my practice that I believe encouraged a fixed mindset. When I moved to my current school, I made two significant changes to my practice which I had previously only briefly experimented with and the difference it has made to the mindset of the pupils I have taught has been quite striking. Now I can imagine some of you reading this are waiting to hear something radical but I’m afraid there is nothing new about what I am about to reveal. Both approaches have been around for some time but I know that there are many teachers out there who haven’t taken the plunge so I thought I would share my experiences.

What did I do?

I abandoned seating the children in ability based tables and I started letting them choose their own level of challenge in their work. It might not be revolutionary but it was a highly significant change for me and for a number of reasons, it initially took me out of my comfort zone, so much so that after a few weeks, I thought I had made a big mistake. But I stuck with it and I am so happy that I did.

Why had I always sat children in ability groups?

When I look back, the only reasons I can come up with are hardly compelling ones: It was what everyone else did and it made it easier for me to organise resources or to work with a particular ‘ability group’. I had never told the children which ability group they were in but they always knew (not only that, so did their parents!). The children that I had decided were less able tended to have a particularly fixed mindset and why wouldn’t they have? Year after year, they had been sat with other children who also held the view that they couldn’t do maths either. They had lower expectations placed upon them, and because they all sat together, either I or a Teaching assistant would work with them and rarely a day went by where they were truly working independently.

At the other end of the spectrum were those children that I had decided were more able. They also had a fixed mindset: they saw themselves as being good at maths and knew that other children in the class held the same view about them. Many weren’t prepared to take risks because they didn’t want to be seen to get things wrong or to be struggling. I did occasionally move children ‘up’ and ‘down’ and that always ran the risk of parents coming in wanting to know why their child was no longer ‘on the top table’.

When I read that the Sutton Trust went as far as to say that ability grouping can have a negative impact on pupil attainment, I knew it was time to sit up and take notice. I needed to change my practice but would sitting the children in mixed ability tables be enough to really change these fixed mindsets? After all, I would still need to come up with names for the different ability groupings that I had so that they would know which work I had chosen for them to do. It wouldn’t take long for the children to work out which group were ‘the clever ones’ and who were ‘the strugglers’ regardless of the names I came up with.

What more could be done?

I felt that, if I was to truly start tackling these fixed mindsets, more needed to change. Traditionally, I had always set the work for each ‘ability group’. There were times when I didn’t pitch the level of work as appropriately as I had hoped but overall, I liked to think that I had got it right more often than not. But had I really? How many times had I unwittingly prevented a child from going on to achieve at a higher level than I thought they were actually capable of doing? If you set the bar low, of course many children will be ‘successful’ at reaching the standard expected of them but it is amazing what can happen when children are given the opportunity to aim higher.

How many times did I knock a child’s confidence because I made them do a piece of work that they really didn’t understand? I know that my confidence in my ability as a teacher can fluctuate from one day to another (it’s certainly not unusual for me to question my ability as a teacher altogether) so it is only reasonable to expect that children can and do feel the same. What if I allowed children to choose their own level of challenge based upon how they felt on any particular day? Could I trust them to make the most appropriate decisions? I would never know unless I gave it a go.

Starting at a new school gave me an ideal opportunity to confine ability grouping to the past and to allow children to start choosing their own level of challenge in their work. My colleagues in the two other Year 3-4 classes were really open to trialling the same approach so we boldly took the plunge together.

The early days

After 2 or 3 weeks, I was beginning to question the wisdom of my decision. What had I done? I had 5 children who seemingly couldn’t do any work at all without me being there to support them and I had just spread them far and wide around the room. All of a sudden, I was having to chase around the classroom like a lunatic in a bid to get any work out of these children while their classmates might have been lucky enough to get a few seconds of my time if they happened to be on my route.

What changed?

Just as I was getting to the point where I was considering giving it up as a bad job and reverting to my old ways, I started to see some real chinks of light. I noticed that there were some really good relationships blossoming around the room. More confident children were getting the opportunity to verbalise their understanding of different concepts to their partners while less confident children were listening, asking questions and learning from them. The quality of interactions appeared to be much greater than what I had been used to seeing in ability grouped tables. I consciously started planning more opportunities for ‘Maths Talk’ in lessons and collaborating in this way has had a really positive effect on the way that the children participate in lessons.

The idea of allowing the children to choose their own independent tasks was warmly received from the beginning. Of course there were occasions (and there still are) when children made choices where the tasks weren’t challenging them enough or were a step too far and I have to admit that at times, I might encourage a child towards a different level of challenge although I always leave the final decision down to them.

As the class got used to this new way of working, I began to make some interesting observations. I remember a child putting his hand up to tell me “I started the middle challenge but I was finding it too easy so I have moved on to the harder one. Is that ok?” Ok? I thought that it was absolutely wonderful. A great example of self-assessment had just taken place and instead of happily carrying on and getting all of his questions right without breaking sweat, the boy decided that he wanted to challenge himself more. I stopped the class and explained what had happened. Before long, children across the class were feeling the freedom to make similar decisions. Sometimes children dropped down to a less challenging task if it really was a step too far for them and the great thing is that they clearly felt comfortable in doing this. There was no stigma attached to it.

What followed next really took me by surprise: occasionally, some of those children who craved my support so much during lessons and seemed incapable of working independently began to feel confident enough to try the middle level of challenge. Sometimes it was a step too far but that wasn’t really the point. They were showing the first signs of developing a growth mindset and that felt like a mighty triumph.

Now this is, of course, purely anecdotal evidence but what reassured me about this approach was that discussions with my Year 3-4 colleagues revealed that they were seeing very similar things happening in their classrooms. Surely that wasn’t a coincidence?

What do the children make of it all?

Part-way through last year, I carried out some pupil interviews with children from all three Year 3-4 classes. When I asked which subjects they preferred, maths kept cropping up. I was intrigued to know why it was so popular and so I dug a little deeper. The children loved to choose their own level of challenge because they were in the best position to know on any particular day just how confident they were feeling about the learning that had taken place. They liked the freedom they were given and the fact that they were taking ownership of their learning.

A couple of months ago, a teaching assistant was carrying out her own pupil interview with a Year 4. The child was asked what his favourite subject was. I can still see his reply now: I love maths. Even if I get it wrong, I still love it. What a marvellous attitude, especially considering that it came from one of the 5 children who only a year earlier didn’t believe that he could do maths without an adult there to help him.

Looking back, I’m so glad that I didn’t give up. It looks like I’m developing my own growth mindset!