Fast, good or cheap? The ‘Triple Constraint’ in Education.

‘Fast, good or cheap? Pick two.’ Originating in software design, this aphorism nicely challenges our ‘have it all’ delusions. Make something quickly and cheaply and quality suffers. High quality products made quickly are expensive. You can have both quality and affordability, but you will have to wait for it. Known in business circles as the ‘triple constraint’, it also lends itself to a great pub game – finding examples in everyday life. Home cooked food can be tasty and easy to make (bacon sandwich) but not terribly nutritious, or easy to make and nutritious (a salad) but not terribly tasty, or tasty and nutritious, but not easy to make (anything in a ‘Anna Jones’ recipe book). I first came across ‘triple constraint’ in this article by Oliver Burkeman and he cites a blog by the entrepreneur Ben Casnocha who has some amusing, if provocative examples. For example, holidays can be exotic and/or cheap and/or relaxing, and our partners can be hot/smart or emotionally stable.

Perhaps teachers can be popular and/or effective and/or emotionally stable whereas with leaders the trade off is between being visionary and/or consultative and/or effective. Pupils’ writing can have good ‘SPaG’ and/or be neat and/or be interesting…although this year’s end of key stage interim assessments are only bothered with the first two. While I hate and detest that dull and boring writing which is neat and has good spelling trumps imaginative, thoughtful work – particularly the perverse incentive to use easy-to-spell vocabulary rather than take a creative risk – I would be willing to trade a little bit of creativity for better spelling and handwriting. Or we could go all out for all three by increasing the curriculum time for English by cutting time for something elsewhere. You might be able to cling onto a broad and balanced curriculum by your fingertips – but depth of coverage in English will have to be paid for by superficiality of study somewhere else. You just might not admit it. At least, not publically.

Project managers describe the triple constraint as defined by choices between time, cost and scope. By scope, performance specification, and/or quality is implied. Project managers use this as a tool to stop kidding themselves that there are no limits upon what can be achieved within a given set of finite resources. It’s a refreshing blast of realism in the face of aspirational, ‘whatever it takes’ woo.   No one gets to have to all. We’ve all made choices along the way – using the model just makes us honestly own the downside of our decisions. If you want something quickly (or frequently) and high quality, it is going to cost. That cost may not be in cash terms, it may be in terms of opportunity cost – you can only spend the time of your teachers once, so make sure you spend that time wisely. So with marking, a set of books can be marked and returned to the class very soon after the initial lesson and the marking can be very effective in that it enable great progress – but this will be at a huge opportunity cost to the teacher. All other calls of their time will have to be rejected – including the calls of their family commitments and personal wellbeing. Here the high quality of the marking and quick turn around is achieved at the expense, or cost, of the teacher’s time. Whereas in days of yore when ‘tick and flick’ was the norm, marking cost relatively little in terms of teachers’ time and could be turned around quickly – but didn’t have much impact of pupils’ learning. Its scope was limited. The Holy Grail of course being finding a system that effectively accelerates pupil progress (scope) whilst still occurring frequently (time) without incurring too large an opportunity cost on the teacher’s time. (Although I suppose theoretically one could reduce opportunity cost by increasing financial costs by employing more teachers to do the marking – not a route likely to catch on in the present funding climate).

The Marking Policy Review Group certainly makes some interesting suggestions and claims that its triplet of ’meaningful, motivating and manageable’ marking is relatively easily achievable – no triple constraint here. Maybe, because meaningful and motivating cover the same ground? Marking’s hardly meaningful if it is not motivating, is it? Clearly, we all want to reduce the opportunity cost to teachers that marking in its present form is extorting. So either we accept that we will have to reduce how frequently work gets marked, or reduce the scope of marking.   So, for example this primary school uses codes and symbols which direct action the next day. A different solution is to only explicitly marking a couple of pieces per class – and sharing these, via a visualiser, with the rest of the class – leaving them to then ‘mark’ their own work by extrapolation. The trail blazer schools are reporting that this approach is working really well; better in fact than the old distance ‘deep’ marking of all pupils’ work ever did.  This of course has its own opportunity cost in terms of curriculum time – curriculum content not covered because lesson time was spent improving and deepening what has already been taught – teaching less but in more depth – in other words, a mastery curriculum. Maybe this is a price well worth paying – for what it is worth I think it probably is – but we should not flinch from owning our choices. There is a shadow side to every decision.

One of the things I really admire about Michaela School is the way it is so up front about its choices. Accepting that it is impossible to do everything – it doesn’t try to. But rather than sweep under the carpet the corners it has cut, it advertises its omissions as a badge of pride. No distance marking here, no siree and no display neither. No computing, or DT or PHSE. Joe Kirby explains here how ideas can be either hornets or butterflies. Hornet ideas are high-effort, low-impact, whereas butterflies are vice versa. Reports and homework are hornets. I don’t think computing and DT are seen as such – just collateral damage in the struggle to teach an exacting and demanding curriculum in the other subjects. You pays your money and you makes your choices. Costs and time being relative fixed within schools – the only give in the system is to reduce scope somewhere along the line. Even when you’ve honed your systems to be as effective as possible – no school can do everything – so choose what you don’t do or what you do less well consciously and not by default.

It would be really useful if schools had to be really honest about the downside of their choices. Particularly in these days of school to school improvement, where we look to schools with amazing results and then try and copy what they do, its really important we are aware of the hidden cost in the choices they’ve made, so we can decide whether the strategies being employed are really replicable, sustainable and ethical. For example, some schools burn through young staff by working them to exhaustion at great cost to the individual teachers concerned. It gets results…but is this sustainable long term? Obviously it’s unethical. (Maybe that should be marked up as an increase cost…to one’s mortal soul!)   But less dramatically, how useful it would be to hear about the things people have decided not to do. School A decides it won’t have a library, thus reducing both financial and curriculum time costs – no more time consuming book-choosing time. The downside is some children who don’t have parents who either buy them books or take them to the library don’t get to read much for pleasure. Maybe their intake means they don’t have many parents like that, or few enough for some different, cheaper strategy to expose those children to a rich selection of books. Or maybe that’s just how it is. Instead, all the children get quality musical instrument teaching. At School B, the priorities are reversed. School C teaches maths in a way that means almost all children make rapid progress. The cost? Children on p levels become more and more isolated, hardly ever working in class with their peers, never taught maths by the class teacher. School D withdraws poorer readers from humanities lessons for extra phonics. The downside is while their phonics improve, their general knowledge suffers, so later on they find it harder to understand what they read.

One problem is often the effect of the downside is displaced a few years, so the school in question does not pay the accountability-price of their choice. The school without a reading-for pleasure strategy doesn’t pick up the tab when that child effectively stops reading fiction. As I’ve written about before, teaching maths with an over-emphasis on the procedural at the expense of the conceptual might engender short term results but at a cost to longer term comprehension of the basics which comes back to bite (some other teacher’s) bum. And the outcome of some choices will make itself felt many years down the line, well into adulthood. An adult drowns; her primary school cut swimming provision to the bone. Another goes to prison; he never received help with his anger management when he was little. Yet another has poor health due to obesity; PE was a Cinderella subject. Obviously the lines of cause and effect aren’t anywhere near as clear-cut as this. But let’s be honest with ourselves. We say we come into teaching to transform children’s lives. Yet the reality of it is, we have to choose which bit of their life it is we are trying to transform. In other words, we have to be clear about the scope of education; what it is we can do well, given the other constraints of limited time and money. What really matters, what will we go to the stake for?

With the coming National Funding Formula, us London schools are bracing ourselves for cuts on an unprecedented scale. The triple constraint reminds us that if there is less money, then either scope will have to be reduced or timescales will increase. In an education context, timescales are fixed. Whether SATs or GCSE’s, those annual results wait for no man – there’s no potential to ask if year 11 can take their Maths GCSE at the end of year 12 as we’ve had to reduce the frequency of Maths lessons due to staff cuts. (Although I love the idea of ‘when –ready’ exams along the piano grades model, I can’t see the government adopting this any time soon). Age-related expectations set tight delivery timescales. Failure to meet them is, well, failure.

So the scope of what we offer will have to take the hit, or several hits, meaning we’re reflecting as rigorously as possible on what is absolutely essential and what is potential cut-able. Not being an academy, I can’t ‘do a Michaela’ and decide we are just not going to teach certain subjects. Obviously English and Maths take centre stage. The time devoted to the rest is already less than ideal, except for music and French – which being taught by subject specialists and taught whilst class teachers have PPA – get an hour per week come what may. Who knows if we will still be able to have subject specialists. Are they the cheapest way of covering PPA? Are they the best use of ‘spending’ precious curriculum time.

We are very proud of our pastoral provision. As well as a (part time) learning mentor and a (part time) home school liaison officer, we also have our own social worker half a day a week. She’s invaluable. They all are. I could reduce this team, or salami-slice their hours. But the inevitable effect would be to reduce the scope of support for the most vulnerable. Maybe, longer term, the scope of the kind of pupil that comes to our school will have to be reduced as we have to cut the resources that enable them to stay in mainstream. They won’t cope. We won’t cope. But hey, there’s always permanent exclusion.

Cutting back on the arts is pretty much inevitable. Swimming provision: how little is too little? We have already cut back on our intervention programmes. Children who are way behind in Maths would have, in previous years, had half an hour a day catch up with a specialist teacher, following a programme proven to be highly effective over the long term. Well, those children won’t get that opportunity anymore. But at least our deputy head teacher is able to run intervention groups – for now.

No wonder oldprimaryhead wrote this heart felt blog recently. That’ll be me next year. I don’t deny that funding needs to be fairer and that in Tower Hamlets we’ve been generously funded compared with everyone else. But it’s not like we’ve been burning fivers; the money’s been used very effectively.

Of course, we are not allowed to admit that scope – quality and breadth of provision – have taken a knock. Since we are held accountable for standards in English and maths, we will move heaven and earth to maintain quality there, while wondering what we can pay lip service to, while maintaining a veneer of quality? What can we get away with? What can we live with and still sleep at night?

I didn’t start this blog to moan about education.   There’s plenty of people doing that already. I’m not decrying moaning. It’s necessary. Done well, it galvanizes us to change things. But I wanted to mull things over, suggest solutions, share what I’ve read. This post seems to be a bit scarce on the sharing suggestions front. Sorry about that.





Fast, good or cheap? The ‘Triple Constraint’ in Education.

‘If they sit with us, they become lazy.’ TA’s, growth mindset and the MITA Project

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Back in the day when the DfE had a rainbow in its logo, Peter Blatchford and his colleagues were commissioned to start their longitudinal study into the impact of teaching assistants on pupil outcomes.[1] The Deployment and Impact of Support Staff Project (DISS Project) lasted 5 years, from 2003-2008 and researched in primary, secondary and special schools. It remains the most comprehensive piece of research into teaching assistants[2] in the world to date.

The key finding took everybody by surprise. While TA’s had a positive effect on teachers’ work load, job satisfaction and levels of stress, their deployment made pupil outcomes worse. Researchers were stunned to find out that even after adjusting for SEN, fsm etc. there was a very strong negative correlation between the amount of TA support a pupil received and the progress they made. The more support, the less progress. Clearly, things had to change.

The researchers were clear that the reasons for the poor outcomes originated in managerial decisions about their deployment and preparedness – rather than in some deficiency innate in teaching assistants.   TA’s were overwhelmingly deployed to work either with children who had some kind of special educational need or with children with lower prior attainment. Very often, they acted as a kind of replacement teacher rather than an additional, enhanced provision on top of excellent teaching from the class teacher. What is more, very few TA’s had any meaningful liaison time with the class teacher and generally learnt what was being covered that day along with the children by listening to the teacher. It is not surprising then, that under these circumstances and lacking sustained professional development, TA’s tended to focus upon task completion rather than the actual learning.

Following the report, the Education Endowment Foundation released guidance for schools to alert them to the pitfalls of poor TA deployment and make seven recommendations to maximize the impact of teaching assistants. Three of these (V, V1 and V11) are to do with interventions – the one area where TA’s could be shown to be making a difference. Only last week, the EEF released further evidence that TA’s can and do make a real positive difference when deployed and trained properly to deliver high quality, well researched interventions.

Back in Bethnal Green we already knew that our interventions had positive outcomes for the very reasons given. For example we used several maths interventions from the Every Child counts stable, such as 1stClass@Number[3].

As a staff we had been looking at growth mindset research and that had already challenged us to move away from grouping by ‘ability’ to letting children choose their own level of challenge from a range of differentiated tasks.   There was no point in us promoting a growth mindset message if we then went on to undermined it by corralling children in tacitly fixed-ability groups. We were amazed on what a difference it made and how a sizable proportion of children made stunning progress once liberated to work at higher levels of challenge.

However, we still had teaching assistants sitting next to the statemented children and their lower achieving acolytes. Did the very presence of a teaching assistant give out the sign ‘abandon hope, all ye who sit on this table’ – whatever other positive messages we were promulgating? And were our TA’s focused on task completion? Did our statemented childrem – like their DISS counterparts – mainly interact with TA’s rather than other children? And what could we do to change?

What was surprising was that our teaching assistants agreed with the research. ‘If they sit with us, they become lazy;’ said one TA. ‘They rely on us to do the thinking for them’. What transformed the situation was discovering the follow up to the DISS report – the Effective Deployment of Teaching Assistants Project (EDTA). This action research project looked at ways of enabling schools to use TA’s effectively. As a direct result, the researchers published Maximising the Impact of Teaching Assistants,   a book so good and so accessible we bought a copy for every TA and class teacher and which I highly recommend. It is highly readable and set out to be used directly for professional development purposes. There is also a MITA website to accompany the book – that’s definitely worth a look too.

We did, of course, reflect on deployment and preparation of TA’s, but the key thing we have been concentrating on has been how TA’s can ensure that their presence promotes resilience, self reliance and autonomy in the pupils they support – the entire ‘growth mindset’ table d’hôte as it were.

One of the first things we did was conduct a survey among a subset of our pupils. We surveyed all the children in ks2 with statements, a selection of lower prior attainment children, and a selection of higher prior attainment children. When asked, who helps you learn in school – the ‘higher’ children almost all named their teacher whereas the ‘lower’ and SEN children were more likely to name a TA and not mention the class teacher at all. But the most illuminating finding was when asked to answer ‘true’ or false’ to the question ‘I learn better when a TA is there to help me’, a third responded no (including the SEN and ‘lower children), and explained that it often interrupted their thought processes and was an unwanted distraction. Another third said that TA presence was sometimes helpful and the final third were positive.

We shared these findings, along with highlights from both the DISS and the EDTA research, with TA’s and teachers during an INSET day this January. Prior to the training, we also videoed 3 out of 10 TA’s as a benchmark tool to help us evaluate the impact of our initiative. This video was only seen by the TA themselves and the 3 teachers leading the MITA project group in school. The best thing about the MITA project is what we now call ‘the MITA triangle’. This is a simple visual reminder of a hierarchy of when to intervene – or not intervene – when supporting students. I say ‘not intervene’ advisedly, because a key message from the project is to intervene as little as possible when supporting students. Assume they can do it, observe carefully, and only intervene after sustained careful observation shows you that the child needs some support.

Our teaching assistants found this really challenging. The idea that it was ok to just sit next to a pupil and effectively do nothing but watch for a few minutes was difficult to take on board. Fortunately, I had videoed myself ‘being a TA’ and supporting a year 6 child with his maths. The sight of the headteacher sitting alongside a pupil and just watching him work for a good few minutes was liberating for them. I was, of course, watching very carefully, so when I did intervene, it was because I could see a misconception getting in the way of learning. We talked a lot about the difference between this and spoon feeding a pupil. ‘I definitely don’t spoonfeed’ said one TA. Imagine her surprise when she saw her video…

The triangle outlines 5 possible ways of intervening. At the top of the triangle the pupil is autonomous and self scaffolds their own learning. Nothing to see here – move along.

The next rung down is prompting (or I prefer to call it nudging) a pupil. This might be as gentle as a meaningful look or strategically focused cough. It might be as simple as saying ‘so, what have you got to do now’ and that will be enough to get the ball rolling. If that isn’t sufficient we might then suggest a very generic strategy, leaving as much of the leg work as possible to the pupil. ‘So, is there anything on the board that might remind you of what you are meant to do now?’

The next step down the triangle is to give clues to the pupil. These will be more specific to the learning at hand than prompts. So we might say ‘ would it help you remember if you looked at the success criteria/spelling bank/100 square/periodic table/working wall? Remember this is happening after some quality input from the class teacher. This is not the child encountering information for the first time. Or being re-presented with it after it has become obvious that the first time didn’t work. This is not an intervention situation where the TA is acting as a teacher and imparting information. This is when the pupil is working with information or a process they have just had explained to them during whole class teaching.

If that doesn’t still work then the TA has to do some explicit modeling. This means the original teaching has failed in some way – it might mean it was too hard for the pupil in the first place. Modeling is basically re-teaching the concept or information. Modeling is fine – but the idea is that the pupil gets it first time along with the rest of the class and doesn’t get to have their own private mini lesson with a TA. That way, dependency lies. I am sure we have all come across pupils who realise that they don’t have to listen the first time around because they will be rewarded with special 1:1 time if they switch off when the teacher is talking.

At the very bottom of the triangle is just telling the pupil what to do. Not because telling pupils things is bad per se – but because we tell them instead of expecting them to work for the answer. Obviously this means that the answer has to be something they should be able to work out for themselves. It isn’t something they don’t, in theory, already know. This isn’t about championing discovery learning – this is about expecting a child to use strategies to find information we have already shared. If, for example, we have shown pupils how they can find the atomic number of an element by looking at the periodic table – it is not ok for them to shrug and look hopeless when asked to find the atomic number of sodium when there is a copy of the periodic table readily available to them.

If they’ve got counters in front of them, and worked examples, and – if all else fails – someone to re model the concept to them again, they should be able to re-create a 3×2 array. What shouldn’t happen is for them to be told,’ put three counters in the first line and another three counters in the second line – there – that’s three times two, draw that in your book.’   In this case, the TA might as well have done the work in the book herself and cut out the middle man!

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Following the INSET day, we shared the videos with the original 3 ‘benchmark’ TA’s, to see what they thought about how they performed, now that they had received the training. Each TA watched their video alongside one of the 3 teachers leading the project. The reaction was amazing. The very TA who had been adamant she didn’t spoonfeed was running around the staff room afterwards exclaiming ,’oh my God – I do everything – I turn the pages, I give him a pen, I jump straight in before he’s even had a chance to think, I interrupt him when he is talking, I practically tell him the answer – I’m spoon feeding all the time.’ The child she was supporting is in year 5.

Another TA was similarly horrified. ‘I don’t give him any space, I hold the book like his a baby, I turn the pages – my hand are in the way.’ Indeed, the child she supports being a feisty character – there are veritable tussles for who gets to turn the page and hold the book. The child she was supporting was in year 4.

Throughout the video was an endless litany of ‘Good boy! Good boy!’ It was suggested to her that pointing out what successful strategy the child had just used was likely to be much more helpful. In discussion, each TA came up with 3 or 4 things they were going to try to do differently and the teacher summarized the discussion and these targets on one side of A4. After this, we videoed the other 7 TA’s for the first time and went through the same process. Unsurprisingly, because these TA’s had already had the training, they were already using the techniques in their practice. The discussions with the teacher were really useful. Everyone found it really hard to back off and wait at the start of the session. Everyone wanted reassurance they were doing it right. People were a bit confused if they were clueing or modeling at various points. Looking at the footage, it becomes obvious that we all flit up and down the triangle at different points as the child encounters and then overcomes difficulties. It didn’t really matter if we couldn’t quite decide if a particular intervention was a prompt of a clue – what mattered was that we were all thinking hard about how to let the child – or children in the case of a group – do the thinking.

After a couple of weeks, we re-videoed the original three. The results were astonishing. I actually cried watching the footage of one child. Here was a boy who used to do anything to avoid reading, and now here he was, blending those sounds like a pro, going back again and again if it didn’t make sense, showing amazing levels of self motivation and ploughing through the book like an Olympian. You know I never thought we’d ever teach this child to read and he is miles and miles behind – a 1C probably in old money. But he now has the determination and drive to go for it. It’s two and a half years until he hits secondary school and we are going to move heaven and earth to get him as far as we possibly can so he transfers a fluent (ish) reader. I had better make it clear that his failure to read previously was not down to the kind of support he had received previously – there’s a long and complicated story and I’ve named my school so I can’t say more – nor was the new style of support the only factor in his resurrection (no other word will really cover the extent of the transformation).   However, now that his TA was supporting him in a way that took the stabilisers off – he was flying. No more holding the book for him or being the one pointing at the words on the page, no more vacuous praise, lots more praise for using specific strategies of for showing perseverance, lots and lots more waiting and observing. Really insightful intervention just at the right time, building on strategies. It was like was like watching a masterclass in supporting reading. I watched the footage alone on the weekend before I was due to meet with the TA to look at it with her. It was so brilliant I texted her there and then to thank her for making such a difference to this child’s life.

But is wasn’t just her. Both the other TA’s have really changed how they do things. And all three children they work with have taken off. The gap between them and the rest of the class is closing rapidly. By chance, both classes having been doing fractions – notoriously challenging and made even more so by the hike in expectations from the new curriculum. Both children – and they both have statements for learning difficulties – can how add and subtract fractions with ease, and in the case of the older child, including when they have different denominators.   Before they come and take the money away, I had better add that their difficulties, particularly with processing language, remain as severe as ever. But as for maths – extraordinary!

And the main thing the TA’s are doing differently? Waiting. Taking time before jumping in. in fact, not jumping in…gently putting a toe in the water if really necessary.   Reminding pupils of strategies they could use, giving students space to struggle and become a little frustrated without rushing in to soothe and calm.

With the other TA’s we have also seen a sea change –waiting, gentle prompting – expecting independence as the new normal. There’s been a real buzz about this project with TA’s chatting about it a lot on the staff room. When your TA’s are reflecting about ways to improve their teaching and students learning on their tea breaks, you know you are on to a winner.

Particularly when TA’s work with groups, I can see how strong the temptation to go for task completion is and how we need to reinforce that sometimes you just need to model the input again – and then try the other strategies. One group of young children were meant to use objects such as toy dinosaurs to tell a ‘real story’ based on an equation such 3+2 and then draw a pictorial representation of this. The word ‘story’ was a distractor to some in the group who started on elaborate tales – as if it were a literacy lesson – without reference to the maths. Despite the TA reminding the children about how the class teacher had told a real story based on the maths story, the link between the two – obviously the whole point of the exercise, had not been grasped by a couple of children. Instead of modeling this again – or – even better – sharing the work of other children who had been successful – the TA spent most of her energy trying to get the children to draw first 3, then 2. When actually the drawing was an afterthought – a recording of the thinking and not the thinking itself. But the MITA model allowed us to have a great conversation around this and for her to self identify her own desire to please the teacher by getting the task done – whether or not this actually helps the child learn. So a powerful learning experience for her.

The project’s not over yet. The idea is that a second round of videoing is shared in TA triads – sort of lesson study style – but less threatening. That was meant to be this week – but we have fallen behind. The main problem was that videoing in class made it really difficult to capture the sound properly – even when the rest of the class were really trying to work as quietly as possible to facilitate the video, the sounds of chairs scraping and background talking just made the soundtrack too hard to analyse. So we re-videoed with TA’s taking their group or child out just for this lesson. We could invest in a directional mike, I suppose – but for our purposes videoing lessons in a different room worked fine. We had to make sure that the prompts available in class were still present, of course.

The project has also helped class teachers reflect on how they promote independence for all children.  In one notoriously dependent class, the teacher has shared the triangle with all pupils – suggesting that some of them rather like life at the bottom of the triangle. She now gives them feedback about how well they are able to self-scaffold – the expectation being that children who actively listen during whole class time should usually be able to get down to solid independent work without needing further reassurance. Her TA and herself will circulate, reinforcing where things are going well and spotting where children are making mistakes and then prompting, clueing and possibly re-modeling as necessary. But the children need to risk doing something wrong first.

At a recent open day day review – when teachers from 7 different schools came to observe classes and look in books – all present noted the high levels of resilience and autonomy. I am sure we have further to go with this and it will be interesting to see how this impacts on end of year data in the summer. I wholly heartedly recommend the MITA project to you – it’s been instructive, it’s been transformative and it’s been a blast.

[1] The rainbow doesn’t have anything to do with the rest of this post – I just included it because it looks hopeful and I was feeling wistful.

[2] I’m going to use the term TA – teaching assistant for all classed based support staff. I know a variety of other terms are used and some school differentiate between learning support assistants for SEN pupils and TA’s for more general support – but for brevity and clarity I am using the one term – in the same way the researchers themselves do.

[3] Full disclosure: husband works for ECC. Also, to be quite clear, ECC’s teacher led intervention has been subject to published independent study which found to made a very positive impact but its suite of TA led interventions do not yet have published independently verified research –although EFF trails are taking place. Their own databank is substantial and shows positive impact, as this report from Learning Wales shows.

‘If they sit with us, they become lazy.’ TA’s, growth mindset and the MITA Project

Efficacy, efficiency and fancy counting (Or why primary schools will ruin fractions forever.)

My husband and I once had an epic row one Saturday morning about whether all of maths could be reduced to fancy counting. He is a maths consultant specializing in teaching children who struggle with early maths. I am a primary school headteacher. He’s the one who has read the books. I’m the one who felt like they were doing a vicarious Masters when asked to read essay after essay. His knowledge is deep and focused – mine is more broad brush but shallower. He’s probably a bit more on the progressive side of the continuum. An innocent comment over a supposedly relaxed weekend breakfast sometimes degenerates into maths wars. My sons, if they haven’t already left the table bored rigid by our endless pedagogy discussions, depart exclaiming ‘they are arguing about bloody maths again!’

Anyway, my side of the argument was that once you realized that counting in groups was much more efficient than counting in ones, everything else in maths was really just a set of footnotes to that basic principle.

Footnote 1: counting in tens (or multiples thereof) is a fancy way of adding or subtracting something efficiently

Footnote 2: multiplication and division are just fancy ways of counting something lots of times

Footnotes 3: fractions, decimals and percentages are just fancy ways of counting bits of things.

Footnote 4: graphs are just fancy pictures of counting

Footnote 5: everything else in maths is just combining notes 1-4 in some way or other.

Footnote 6: by maths I of course mean mainly number. Geometry is some alien cuckoo-in-the nest, cruelly inserted into the maths curriculum by nasty people who can read maps and parallel-park easily. That it is not fancy counting, just proves my point. It should be a weird option at GCSE and not inflicted upon the rest of us.

For some reason he thought this was some gross over simplification.

I suppose what I was trying to express was that much (maybe ‘all’ was stretching it a bit) of learning maths is learning ever increasingly efficient ways of applying our ability to count in different situations. Maybe that would be better phrased as, ‘learning ever increasingly efficient ways of manipulating the number system.’

Hin-Tai Ting’s blog post about efficacy, efficieny and mastery brought it all back. What was really helpful about Hin-Tai’s post was how it contrasted effective strategies with efficient ones. Effective ones work, but may be laborious. Efficient ones work better, but you really have to ‘get’ the maths to be in any position to be able to decide which strategy is more efficient. Procedural understanding gives you effective strategies; conceptual understanding enables you to choose between procedures to find – or even invent for yourself – the most efficient for the task in hand. And when teaching and wanting to reduce the cognitive load on students we tend to go for the method that they are most likely to get and teach this as a procedure. This is fine up to a point – but with only one ‘tool’ in their mathematical toolkit, the student has no alternatives.

I had a friend once who passed her driving test without being confident at turning right at junctions. Left was easy because you don’t have to cross the traffic – turning right freaked her out. If she gave you a lift you could end up going some really strange routes to avoid right turns. And some places you just couldn’t reach! She hadn’t mastered turning – that’s for sure. What we want is to give students flexibility of thinking. We want them to have a range of tools in the kit, and – crucially – to be able to work out on the hoof, which to use when. This is what mastery understanding is. It’s not some special level clever kids get to, it’s what proper maths teaching is all about. But there is a real tension between only having one effective tool and having so many tools you’ve know idea which one does what – let alone make a judicious choice about which is most efficient for the job in hand.

This was one downside of the otherwise groundbreaking National Numeracy Strategy. Primary old lags like myself will recall how we were expected to plough through pages and pages on different strategies. See for example, the range of mental calculation strategies we were meant to teach year 3 for addition:

Mental calculation strategies (+ and –)

  • Use knowledge that addition can be done in any order to do mental calculations more efficiently. For example:
    put the larger number first and count on in tens or ones;
    add three small numbers by putting the largest number first and/or find a pair totalling 10;

partition into ‘5 and a bit’ when adding 6, 7, 8 or 9, then recombine (e.g. 16 + 8 = 15 + 1 + 5 + 3 = 20 + 4 = 24); partition additions into tens and units, then recombine.

  • Find a small difference by counting up from the smaller to the larger number (e.g. 42 – 39).
  • Identify near doubles, using doubles already known (e.g. 8 + 9, 40 + 41).
  • Add/subtract 9 or 11: add/subtract 10 and adjust by 1.
    Begin to add/subtract 19 or 21: add/subtract 20 and adjust by 1.
  • Use patterns of similar calculations.
    State the subtraction corresponding to a given addition, and

vice versa.

  • Use known number facts and place value to add/subtract mentally.
  • Bridge through 10 or 20, then adjust.

(From the National Numeracy Strategy 1999, p2).

What the authors wanted was for children to have such mastery over the number system that they would appreciate that you can partition any number in a myriad of different ways; the trick being to partition the numbers you wish to add in a clever, efficient (or as I would say, ‘fancy’) way that made life easier for you. Instead, what actually often happened was that children were taught a range of seemingly unconnected procedures that all got muddled up, leaving them not really knowing which method to use at all.   So schools quickly chose one method that seemed best at getting the right answer – the most effective method – and let considerations of efficacy go hang. I remember one school where it was absolutely forbidden to teach children that you could partition both numbers into tens and ones and then recombine the tens first. You were only allowed to teach partitioning the second number; anything else was wrong.

Accountability pressures are our enemy here. When the reputation of the entire school is at stake, better teach one tool that will always work – possibly inefficiently, – than two that they muddle. Especially when the consequences for inculcating inflexibility won’t be felt until a few years hence, when some other educational institution gets to pick up the tab. In theory we are all mastery teachers now; taking our time teaching less in more depth; spending quality time with topics so that procedural and conceptual understanding develop in tandem. In reality schools are tearing through the curriculum at breakneck speed in order to cover the new curriculum before accountability exams kick in.

Which brings me to fractions. Secondary school teachers; I’m so sorry. On behalf of my primary colleagues I apologise for ****ing up the understanding of fractions for the children you will shortly inherit. Here’s the backstory.

The higher expectations of the new curriculum now have us teach year 6 to multiply and divide fractions. Previously, children could easily obtain a level four without even knowing how to add or subtract fractions with different denominators. That was for level five children. But not now. The sample paper is chock full of fractions being manipulated ever which way. What we used to teach in year 6 is now meant to be done in year 4. But we didn’t do it  when the present year 6 children were in year 4 ‘cos the curriculum was only finalized in July of that year. We didn’t do it in year 5 much either because we were too busy teaching the column methods and formal long division, which also weren’t previously necessary. So finally here we are with three years worth of fractions curriculum to teach (on top of prime factorization, cube numbers, area of a circle, algebra…) by early May. Added to that, to be honest, our own conceptual understanding of multiplying and especially dividing fractions is a bit hazy. Added to that, the actual procedure for these two operations is ridiculously easy. So why waste time explaining why they work, eh?

That was the general consensus at a conference I was at last week. I’ve begged my year 6 teacher, please…humour me, at least show them, albeit briefly, a quick area model of why multiplying a fraction by a fraction works. And then teach them the trick. Which incidentally seems to break all the rules about denominators. Previously, for addition and subtraction,  we’ve stressed that when numbers have different denominators you absolutely can’t muck about with them with them at all. That would be like saying that 3 litres and £4 was 7 metres.   Different denominators were evil and dangerous and had to be rendered safe by finding a common denominator.  But now children are positively encouraged to play fast and loose with denominators and multiply them even when they are different. The illicit is, for reasons best known to the maths police, made licit.

This really bothers me. If we teach children to do things that make no sense and seem arbitrary, we run the risk of children assuming maths is not meant to make sense, that it just a case of complying with whatever apparently random routine has been served up to us today. Don’t touch the denominators: touch the denominators. Whatever. No wonder they are not bothered by answers that are clearly wrong. No one ever said it was meant to make sense.

Confession time: I’m, 53 and until last week, I didn’t realize that the quick, ‘effective’ methods we use for multiplying and dividing fractions, are actually underpinned by first of all finding a common denominator. That just makes so much sense. This video has really helped me understand this. In fact I felt a palpable sense of relief. All these years I’ve been defrauding maths, performing algorithms like they were magic spells. Made me feel dirty – and not in a good way. At last I am an honest woman. No wonder when at secondary school I encountered algebraic fractions I didn’t understand the various rules for manipulating them. In fact, it was only last year helping my child with GCSE maths I realized that algebraic fractions are just…fractions! Same rules and everything! Who knew? Seriously it was a light bulb moment for me.

However, probably year 6 teachers will just be drilling ‘flip and multiply, flip and multiply.’ Anyway squeamishness they may feel overwhelmed by noxious accountability radiation.

“Dividing fractions, as easy as pie,
 Flip the second fraction, then multiply.
And don’t forget to simplify, 
Before it’s time to say goodbye”

 Which is fine as an aide-memoire alongside work to develop conceptual understanding, but not as the main course. Which is what it will be. Sorry about that.

So when they are in year 9 and trying to manipulate algebraic fractions and it’s all going horribly wrong, curse your primary colleagues. (I presume it’s year 9, maybe it’s year 7 these days?)

Actually don’t curse your primary colleagues. Like you, we have to operate in a strange land where the government imposes a shiny new curriculum on all of us all at once. It’s not put into place one year at a time so that we can build children’s understanding of the new maths landscape one step at a time. Which is how they did it in Singapore. Like you, we operate within the accountability force-field, which distorts everything in its path. On top of this the accountability system is getting even more punitive with schools still pressured to get kids looking like they can do the stuff they are as yet not quite ready for. There is a fundamental disconnect here between the desire to promote deep and sustainable learning and the desire to meet fairly arbitrarily specified targets. Rock and a hard place anyone?

Hin-Tai’s post was much more positive and solution focused, so I will endeavor to end on a positive note. He talks about the need to design curricula that build-in mastery approaches by identifying the deep maths and then devoting quality curriculum time to do just that. This is clearly beyond the headspace of individual teachers and madness that we should all try to do this in isolation. Fortunately, our school had adopted mathsmastery , which does precisely that. I was a bit suspicious, initially, of its highly detailed –almost scripted – lesson plans and had all the usual objections (as outlined, but not championed, by David Didau here.)   The name is seriously naff too. Can’t we just call it ‘deep maths’. However, two years down the road I have to say the curriculum is a work of pure genius. I fancy myself as able to plan great maths lessons – but as a day-to-day practitioner I simply don’t have the time to read the books necessary to really get to the heart of the matter. This curriculum does. And it goes s-l-o-w-l-y at first. The first half term of Reception covers numbers 1 to 3. I thought some kids would be so bored they’d be sticking pins in their eyes. But no. With these three numbers, children work on – among other things ­– getting a firm grasp of equivalence, by randomly generating these numbers twice, and describing the two numbers as the ‘same’ or different’. There are, after all, 9 possible outcomes, if we say that 1 followed by 2 is a different outcome from 2 followed by a 1. What concept could be more fundamental than understanding what ‘same’ and different’ means? If you think about it, ‘same’ is quite a tricky concept, since it does not mean ‘identical’. We mean ‘same’ in a specified way. Variation theory for 4-year-olds.

In another great lesson, the children play a variation of Nim’s game. Using bricks, build a wall in a 3-2-3-2 formation. They can choose to remove 1,2 or 3 bricks at a time, the person removing the last brick being the winner. I can’t begin to overstate how clever this is. Some children just do this practically and don’t realize until they remove the last brick(s) that they have won. Others begin to plan ahead – to visualize the maths in their head. Some of them begin to theorize about a way that works every time.

And it turns out, I was dead wrong about fancy counting. In fact, in the UK we are far too wedded to counting as a means of calculating. In Singapore, close on the heels of ‘same and different’ comes ‘part, part, whole’. These two form an essential part of the deepest of deep maths children that need to master. Really, once you have got 1 to 1 correspondence, you shouldn’t really count anything much beyond 5. [1] 7? That’s the same as 5 and 2.  I might concede to counting to 10 at a stretch. Adding is much more about applying really simple number facts than it is about counting. 8+5? Don’t teach the children to count on 5. Instead, partition the 5 into 2 and 3, pop the 2 with the 8 to make 10 and voilà.

Occasionally, I think mathsmastery miss a trick. For example, in year 1 they leap straight from circling pictures into tens and ones to dienes, without the all important physical bundling and unbundling of tens using straws. Lots of children in upper key stage 2 with poor place value haven’t actually grasped that the ten stick is made up of 10 ones as it seems so obvious we brush over it. Printing with dienes would also help in the tricky move in transferring what we’ve done concretely to what we record pictorially. And of course, I think fractions should start with inculcating a secure grasp of the denominator before introducing numerators. As I’ve said before.   And I read something I should have bookmarked about teaching Time by removing the minute hand from clocks and getting children to approximate telling the time by the position of the hour hand. ‘Oh look, its gone past the 6 and is close to the 7 – it must be a few minutes to 7,’ which I also think would really work much more effectively.

But these are mere quibbles. I would really recommend it. Or something similar. Something with a better name, perhaps.

[1] Actually him indoors says counting is vital for development of place value and mentions Carpenter and Moser, or possibly Nunes. But can’t be arsed to pull himself away from the rugby and actually find a proper citation. Ok. But we should rely less on teaching calculation through counting. Looks like we’ve done a volte-face on our previous positions.

Efficacy, efficiency and fancy counting (Or why primary schools will ruin fractions forever.)


The old bear roared in frustration and waved at the empty air with his huge paws, then reared up on his hind legs .’

It’s the Easter holidays. Reading booster revision class for year 6.  I have a group of 4 children. They need to get a level 4. It’s not going well. The bear in question has recently changed his behaviour – we are medddant to be describing how. And why.  I’m settling for how in the first instance.

‘So,’  I enquire, with that  strained upbeat  tone adopted by teachers when the odds are against us, ‘can anybody tell me what it means when it said the bear  ‘reared up?

Blank looks.    Someone says that rear means your bottom – which is sort of encouraging – though not particularly helpful in this context. Mental note to self:-must make sure we teach them the words ‘rear up’.   Time for a spot of drama. I rear up out of my chair and we all practice rearing up and think about what sort of occasions might make us want to rear up. Not taking a bottom group booster class three weeks before sats, that’s for sure. All though I am – like the bear – frustrated.  Quick check that we know ‘frustration’. Nominalisation is a problem – they know frustrated but don’t automatically make the link. Mental note to self:-must make sure we teach them more about nominalisation, root words, looking for links.

‘Ok, are we all clear now about the meaning of all the other words in the sentence?’

A pupil enquires, ‘what does hind mean?’  How come they don’t  know they word ‘hind?’ We do lots of stuff about animals. Surely it came up somewhere.  I check if they know the word ‘fore’.   Nope. Don’t know that either.

I reread our sentence, one more time, confident now that every word is now understood by my four pupils and that we can begin to answer our 2 mark question. So if we were, by some sort of miracle, to get a very similar sort of passage in the real test, where someone reared up in frustration – possibly on his hind legs – we might even be able to glean a mark or two.

‘Miss…? The undulating tone indicates a question is coming. ‘What’s a paw?’ Some of the group are excited now because they know this.  They rear up in excitement. ‘I know, I know, it’s what you call an animal’s foot!’ Mental note to self:-must make sure we teach them the word paw. Surely we teach them the word ‘paw’. Surely in the early years during ‘people who help us’ the vet makes paws better? I’m sure we learn ‘paw’ in phonics.  Despondently I turn and  reread the sentence yet again. My face a mask of fake brightness as I strain to erase any hint of sarcasm from my voice I enquire, ‘we are all sure what a bear is – right?’

By the way, I ought to point out, only some of this group are EAL. Others speak English at home. Allegedly. It’s a different kind of English though. Standard English is definitely a foreign tongue.

Look, I’ve got as  big a ‘growth mindset’ as the next ‘ResearchEd’ nerd, but there is something about trying to increase children’s vocabulary that drains every last scrap of optimism out of me. With just about everything else in education, it is simply a case of finding the right technique and then practicing the living daylights out of it. Actually that’s the grossest of gross generalisations. It is not simple for a start. And there’s infinitely more to education than just mastering basic literacy and numeracy. There is creativity and wonder and stuffing your brain with glorious morsels of knowledge. There’ s critical thinking and problem solving  and reasoning and immersing yourself in the imaginary worlds of brilliant stories.  But to get to all that properly you need to read, write and be numerate at least a little bit. And there are techniques like phonics in reading and using the concrete-pictorial- abstract approach in maths that have a proven track record of being pretty efficient.  I like the way the universe has been even handed here: favouring traditional phonics on the one hand and trendy ‘cpa’ on the other.

When it comes to increasing a child’s vocabulary, the way ahead is so much more mysterious. Something like phonics is, for the most part, generalisable.  Once you know the 44 grapheme: phoneme correspondences and how to blend and segment them – you’ve got that skill for life. You can transfer that skill onto any other phonetic language you encounter. Sure the gpc’s may vary a bit or even a lot, so you may need to upgrade you knowledge of those, or learn to decode from right to left if you learn Arabic, but the basic skill is there: it doesn’t need to go on being taught over and over once securely grasped.

Maths is pretty much the same. Once you understand the base 10 place value system, you don’t need to have it explained afresh each time we increase or decrease a factor of 10. It’s quite a complicated and abstract principle so may take some time to really understand, but once it is done, it is done for ever.  Same with the four operations. Yes we apply them in ever more complicated situations, but addition is always addition, whether it’s 2+3, 233+32.3  or  2a +(-3b).  Obviously there is a lot more to both, but you get the point. In both there’s an element of the really hard work being done in the early years and ks1.  After that, it’s mainly applying what you already know in new, more demanding situations. But words are so damn specific.  If today I teach you ‘paw’ that  doesn’t help you know ‘hoof’ tomorrow. If today’s character ‘rears up in frustration’, tomorrow’s will ‘slump in deep despondency’. Am I meant to have some sort of list- starting perhaps with aardvark and ending with zygote?

It has always struck me as miraculous that any child ever learns to speak at all. It’s creation ex nihilo par excellence. One minute you have a babbling baby who can’t say or understand anything and the next they start issuing orders like ‘more’ and ‘milk’. Before you know it they’ve moved beyond giving orders and labelling objects to generating their own, unique sentences- beyond just copying and into innovation. Watching a young child learn to speak is exciting and enthralling; no wonder parents bore everyone else senseless with news of the latest cute thing their genius  two year old has just said.

I am sure those who have EAL new arrivals turn up in February into year 4 /8/11 ( insert numeral of choice) have often witnessed the same miracle. One minute you are announcing to the class ‘here is a new boy,Yan Ye. He doesn’t speak any English yet’ and arranging a buddy system  to get him through geography and the next he gets a level five in his English reading sats. So- and here is the million dollar question, how come Yan Ye can learn English in little more than two years when he only really gets to speak it at school whereas some of this classmates have been surrounded by English since birth, been to the same school for 8 years and don’t yet know what ‘frustration’ means? Unlike their teachers. Why does this everyday miracle fail to ‘take’ for some children and what can we do about it?

Of course it is linked to disadvantage. For a variety of reasons, there is a much higher risk of growing up within a family with a limited language environment if that family is also poor. It’s absolutely by no means inevitable, but statistically, if you are poor, the odds of you being raised in a language rich environment are just lower.  When comparing  the exposure to language of the richest and poorest children, a seminal American study that makes me want to weep describes the grim reality that a child from a high-income family will experience 30 million more words within the first four years of life than a child from a low-income family. But it is also true that socioeconomic status is not destiny and that there are a variety of providers out there working with parents of babies and toddlers to empower them to break the inter- generational cycle of deprivation.  But what about the children already in the system? How can we help them?

Well it turns out that there is some sort of list. Or at least there should be, although there is a lot of argument about what should be on it. Unfortunately, Mr Gove was rather in favour of having a list. I say unfortunately because I think there is quite a lot in this list business, but Mr Gove tended to taint everything he advocated with the venom of his invective against teachers. A bit like the new maths curriculum. It’s really good- but you have to get past one’s stomach churning aversion to anything Govian to appreciate it. E.D Hirsch, an American academic and chair of the Core Knowledge Foundation, argues persuasively that in order to read, one needs to know not only certain words but something about the cultural context in which those words are set. It’s hard for a British reader to make much sense of an account of baseball or an American an account of cricket even though we understand the individual words perfectly well. It’s just together they don’t make sense.  Socioeconomic disadvantage in education is best overcome, he argues by the explicit teaching of knowledge and its accompanying vocabulary. Instead of wasting time teaching generic reading skills, precious curriculum time should be spent on the humanities and science, building up knowledge.  As his colleague Daniel Willingham argues in this video, teaching content is teaching reading.  The trouble is that Hirsch’s list, as well as being American and rather dated, just seems so arbitrary. Back here in the UK, Civitas tried to translate it into  our kind of English. Apparently, every year 1 child should know about Machu Picchu- something I confess I know precious little about (despite my knowledge based private secondary education). I think it’s a sort of templey thing somewhere in Latin America – Peru maybe- and was built by the Inkas or possibly Aztecs. The same list says every  year 2 child must know about rabies. Why? To give them nightmares, perhaps.

Much more usable is the work of Isabel Beck  . She divides words up into three tiers. Tier one words are everyday words that do not need teaching- table, house, book etc.  Tier three words are specific to certain domains and are fairly low frequency, words like isotope. Words like this are taught fairly easily as they occur in the curriculum. But it’s tier two words where the action is.  These are words that don’t usually come up in spoken language but are high frequency in written texts and give the learner a more mature, nuanced way of expression a concept they already understand. A child who knows ‘sad’ can be taught the word ‘forlorn’, for example.  Rather than providing a definitive list, she teaches teachers how to identify and choose suitable tier two words for explicit word work. she explains this well here.  There are two books by her that are definitely worth reading; Bringing Words to Life and Creating Robust Vocabulary. (Although the phrase ‘robust vocabulary’ brings a smile to my face: no need to put any effort into teaching that!)

Beck (and colleagues) explain that children don’t easily pick up new vocabulary just from the context and from their reading. Rather they need explicit instruction and creative ways to practice what they have been taught. Katie Ashcroft from the fascinating Michaela school explains how they are explicitly teaching vocabulary there.    How does this contrast with current practice in primary schools?

In my experience (almost entirely based in schools in Tower Hamlets), schools work very hard to provide language rich environments. We have to, since our population is predominantly EAL and those who are not EAL tend to number among the word poor.  We all know our BICS from our CALP.  Thinking of my own school, starting with the early years promoting high quality communication is our top priority.  We’ve done ‘Every Child A Talker‘ and refresh our memories its key messages every couple of years. We have a speech therapist on the payroll who regularly observes practitioners to develop their practice. We run interventions such as TalkBoost and Speech Bubble. We value speech as integral to learning. Talk partners are a completely routine part of every lesson from Nursery onwards. We expect children to answer in full sentences. We use sentence frames and dictogloss. We orally rehearse before we write and use drama across the curriculum. We talk for writing and internalise key texts. We spend a lot on our library which has high quality non fictions well as fiction.  Each year a group of children who have read widely get to visit the book wholesalers and spent a budget of thousands electing new books for our library. Authors come and visit.  We insist children read daily for homework.  We read to them daily.  Key vocabulary is highlighted and explained. Yet still, frustratingly, vocabulary problems rear up.

Upon reflection, I reckon we explain too much vocabulary too fast.  If I am reading Beck et al correctly, less is more. Fewer words, carefully identified and revisited over a series of activities.  This is something we can change.  We magpie literature, ‘stealing’ good words. We list ‘wow’ words and ‘star words’ and ‘powerful verbs’.  Beck’s tier framework provides real scope here to revisit our practice.  Those ‘powerful verbs’ are bound to be tier two words. Time for them to come down from our working walls and be rehearsed in the ways Beck recommends.

Beck’s recipe is as follows:

First introduce a new word in context. I think this is pretty standard primary practice. I don’t think primary schools tend to just dole out list of edifying words to learn. I hope not anyway.

Secondly, provide a friendly explanation. Not a dictionary definition unless it is from something like the Collins COBUILD Dictionary that defines words in full sentences. Beck suggests explaining words with sentences that use words such as someone, something, if and you. For example,’ if someone is jaded,me has or has seen so much of something that he begins to dislike it.’ (p24 Creating Robust Vocabulary). I think, given time, this is the sort of explanation most of us would tend to give. The trouble is, because our vocabulary teaching tends to be so ad hoc as we encounter words in the middle of stories, we try to articulate a coherent definition on the hoof and fail miserably, coming up with something garbled.

But even in programmes such as Success For All that go out of their way to highlight and define high value vocabulary, this goes nowhere unless the students have opportunities to encounter this word several more times over a series of days. So step three is that old favourite spaced repetition.  Beck suggests spending  a few minutes over 5 days on a few words. After introducing the word and briefly defining it, on subsequent days, children can be given opportunities to process the new words mentally in a variety of ways. One technique is called example/ nonexample. Here students are asked to decide if various sentences illustrate the target word or not.

If any of the things I say might be sleek, say ‘smooth man’. If not, don’t say anything.

a porcupine 

a duck

a leaf

a car

Another technique is word associations. This is basically matching. A card sort really- though maybe without the cards.

The next activity she suggests involves  generating situations, contexts and examples.

How might a …..cook……a musician…..a basketball player……..a teacher show they are:





And so on.  Then at the end of the week,mother words go in a word back. Each week, three ‘old words’ are drawn from the bank to jog those memories. That way, they stick and might actually get used.

I’m not sure why it is secondary school teachers who are leading the way here (or maybe they are just the ones blogging about it) but there is some inspiring practice being developed.   As well as Katie Ashcroft, l have also found this by Josie Mingay on using root words as our bread and butter teaching. We already do this is spelling – so why not in vocabulary instruction too? Joey Bagstock  – a colleague of Josie’s – is worth reading too.  Which brings me back to nominalisation.  This term foxed loads of primary teachers when it first appeared in the writing APP grid for level 6. It’s only now after reading this brilliant blog ( nominalisation comes right at the end) I realise it is a great way of turning an ordinary sounding verb into a much posher sounding noun and making your writing seem so much more intelligent. Contrast ‘when the Romans arrived’ with ‘The arrival of the Romans’.

Of course, I can bet my bottom dollar that however many words I successfully teach my charges, when it comes to test time, those words won’t be in the test.  ‘If only I had taught them frugal instead of exuberant’,’  we will exclaim.  And when it’s a test we are talking about, that arbitrariness remains. But we are teaching these words for life and not for the test. Be nice if they coincided though- just this once.


The light switchers solved- with actual photos!

In my previous post (silly carpets and the light switchers) I mentioned an interesting problem and promised to post the solution – I’m sure it was easy enough the work out especially since the title also mentioned prime and square numbers. I’ve never actually done it practically before. But for you, dear reader, I’ll go that extra mile. Really glad I did because I’ve just appreciated that that this is a really great problem for upper ks2 ( possible ks3. – what do I know?) and a fun way to rehearse those tables yet again. I wanted to video it but lack of something to fix my phone/ iPad into eluded me. Too many shots of me in my dressing gown (hey it’s the weekend, you don’t expect me to actually get dressed?) So just what is probably one of the most boring sets of photos ever- but what was strangely satisfying to do in practice. I needed an object I could turn over to signify the light being on or off.  Having no plastic cups I resorted to nespresso coffee capsules which makes this the most middle class poncy maths investigation ever….in class maybe use those mini ketchup cups you get at McDonalds or biscuits like Jaffa cakes which have very different sides and which we now all know are zero rated for vat, thanks to MP Stella Creasy.

So first of all all the lights are off so my capsules are all the right way up.


Then number one switches all her multiples on – i.e turns all of the capsules over.


As every number is a multiple of one, all the capsules are turned over- all the lights are now on.

Then along comes number two and turns over all,of her multiples, making this obvious pattern.


Then number three turns over all of his, making this interesting pattern.


Number four turns over all his multiples…


And I’m losing the will to live inserting all these photos, so I will cut to the chase. Here is what it looks like after number 25 has done her switching. Drum roll please….


So the lights left on are 1,4,9,16 and 25 ( and following- I decided modelling it to 25 was quite enough) because, of course, square number have an odd number of factors whereas all the other numbers have an even number so return to their original ‘off’ state.

The light switchers solved- with actual photos!

Silly carpets and the light switchers- investigating prime and square numbers

This is just a short post about a very simple idea I did with my yr2 class a long time ago when I was still a class teacher. It sort of follows on from my previous ‘ hooray for an array’ post. It was a way of children investigating prime and square numbers. Our class carpet was made from carpet tiles.  We were learning area by counting squares. I had a few spare carpet tiles….not quite sure why. So I showed them that with some numbers like  8 you could make a sensible (rectangular) carpet, and some numbers like 9 you could even make square shaped carpets. But some numbers, like 11, no matter how you tried,  you could only make a silly carpet that was very long and very thin. There was something about the notion of the class trying to fit onto a long thin carpet, one behind the other, that the class found hilarious- and with squared paper quickly found all the squares and primes to 50. Made a great maths display. Hardly an original investigation, but I hope an interesting context that your children will enjoy doing.

For older children (upper ks2, secondary, adults) here’s an interesting problem. With primary children I’d use something like two coloured counters or paper cups for them to model this- basically anything that could be used to indicate if a switch is on or off. The problem starts like this…..

There is a long corridor in school with fifty light switches- numbered 1-50.  There are 50 pupils also numbered 1-50. All the lights are off, so pupil number one helpfully goes along the corridor and flicks every switch on.

Pupil number two,comes along and thinks- what a waste of electricity, I’m going to flick all the switches for lights that are multiples of two. So she turns off lights number 2,4,6,8 etc.

Pupil number three comes along and wonders why some lights are on and some off. She  flicks every third switch ( so if it were on she flicks it off and if it was off she flicks it off).

Pupil four flicks every fourth switch, pupil 5 every fifth and so on, until finally pupil number 50 just flicks switch 50.

At the end of this, how many lights are left switched on? Which ones? Why these?

I will post the solution tonight, but the title should help a lot!

Silly carpets and the light switchers- investigating prime and square numbers

Hooray for an array!

Now I maybe wrong, but I was taught that strictly speaking mathematically 3×2 means 3 taken 2 times or in other words 3+3 rather than 3 lots of 2 (2+2+2).  Which means that 3y means y lots of 3 rather than the easier to understand 3 lots of y…. But that’s what I understood the notation to officially mean. Not that it matters with the commutative law and all- but a teacher in America marked some poor kid down for saying 3×5 was 5+5+5 instead of 3+3+3+3+3 (i.e disagreeing with me). The teacher even marked the array the child had drawn for 4×6 wrong because it was orientated the’wrong’ way. The whole point of arrays is that they can be ‘read’ either way and hence are very useful for teaching the commutative law. I love an array! Do you know they are also brilliant for showing the associative and distributive laws, link to the bar model, the grid method, prime factorisation and multiplying by 10!  You just need some spotty wrapping paper……


Here is 2+2+2 or 2, three times- what I’m going to call 2×3.  Like this it is also a bar model. But like this


it is an array that could be seen as 2, 3 times or


3, 2 times.  Or back to the bar model



Now I’m sure we’ve all done similar with our children in yr2 and yr3- I used to do it with pegboards and exploring 24. Dotty paper works brilliantly though- get the children to do the cutting and experimenting.

But I’d never thought about using it to teach multiplying by 10, until the new curriculum. The new curriculum stresses multiplicative reasoning. In yr3 they are supposed to understand that 4x12x5=4x5x12=20×12=240. (Associative law and all that).  Which got me thinking. Obviously we NEVER tell children ‘ just add a zero’ when multiplying by 10 and tell then about moving digits one space to the left. My colleague gets the whole class to stand up, jump to the left while chanting ‘multiply, go large, left’ ( for division it’s ‘division, reduce, go right). The local secondary teacher came to see a lesson and finally understood why a quarter of her class would jump up out of their seats and do his routine everytimemshe asked about multiplying or dividing by10! But even though this helps children remember what to do – does it help them understand?  But what if we taught it ( alongside the dance) with arrays? Make an array of 6, then make 10 of these arrays to make an array of arrays eg 2x3x10=6×10=3×20


Which they can play about with and establish that this also equals


5×12 and even perhaps after a lot of cutting 2x3x2x5.  Hence prime factorisation. Remembering of course to make the link with factory… The factors are what the 60 factory uses to make 60.

Then photocopy or scan the 10×6 array and by cutting and pasting make another array ten times bigger by putting 10 copies together and establish you know have 600 because 60 can be written as 6×10, so 6x10x10= 600. Then photocopy again to make 6000 and- well maybe leave it at that. When I showed my yr6 that 60×10 could be factored into 6x10x10 they were enthralled and wanted to know if it worked for multiplying by 100 – which they quickly proved it could. This kind of multiplicative reasoning not having been common until recently, they were entranced by it. Last year I showed the level 6 group how you can easily work out eg18x6 by halving the 18 and doubling the 6 and they thought I was teaching them some kind of dark magic- arrays of course to the rescue. Indeed the bottom two photos above show precisely this.  Much experimentation later they got the general rule that you can multiply one factor by any number as long as you divide the other factor by the same eg 96×15=12×120=3×480. By now, they can ditch the arrays and experiment with factorising numbers and then recombining in ways that make multiplying easier. Multiplying even numbers by multiples of 5 being an easy win eg 24x 15= 4x6x5x3=2x2x2x3x5x3=2x5x2x2x9=10x90x4=360. Well,maybe it’s not easier but  I find it kind of satisfying. And they are going to have to understand this if they are going to be able to reason about questions like these from NCETM’s brilliant assessment document. To be honest it did take my yr6 teacher and I a few moments, quite long moments, to get our head around these at first. The last one still gets me every time.


Have you ever used the dotty wrapping paper method as a prelude to the grid method? It works a dream. The photo below was taken in bright light so it’s hard to see the pen annotations dividing it into 10×10, 10×9, 10×7 and 9×7- try zooming in perhaps.


Again, get the kids to do the dividing up and cutting, finding the arrays that will make the calculation easiest (the multiples of ten).

This method can form a bridge to the column method. In the class I’d do this with actual paper we had cut up and made into a grid first but I’ve drawn it ( very badly). So put the actual arrays where I’ve put the drawings.


Finally the distributive law. Easy Peasy when you just split you array into two pieces


10×6 distributed to make 6×6 and 4×6. Cue investigation- how many ways can we distribute this array. I had to model this to a primary teacher who didn’t believe you were allowed to split numbers up like this in the middle of a multiplication when I did a staff meeting on multiplicative reasoning using the post on Shanghai maths from resourceaholic. This is what we were trying to work out mentally


He got the bit where you divide 45 by ten and adjust by multiplying 1.58 by 10, but didn’t agree that 4.5x 15.8 + 5.5x 15.8 was the same as 10×15.8. Until I drew this.image

Then the penny dropped. So hooray for arrays and remember CPA all the way . (That’s concrete pictorial abstract, in case you haven’t encountered this yet). If I haven’t converted you yet to the joys of arrays, read this from NRich.

Hooray for an array!

Teaching Biff and Chip to read.

When I was at home on maternity leave with my 19 yer old son- I mean- with my son who is now 19 years old- my husband came back from a school he had been visiting as part of his PGCE where everybody in year 2  could read- where everybody passed their reading sats and many got level 3. They did this by teaching them phonics.  It is difficult to convey quite how shocking this was. At that point (1996), the idea that you could teach something out of context- without some meaningful frame of reference that made sense to the child- was seen as almost tantamount to child abuse. Children were said to learn to read by being immersed in a rich world of high quality children’s literature, by bathing in wonderful stories and glorious poetry. Not by dissecting language into individual atoms and labelling them with differently voiced grunts. It was like being told they taught witchcraft. Surely, I said, the children had sprouted horns as a result, or hair on the palms of their hands or something?  But husband- usually the sort of person to champion wonderful stories and glorious poetry- was adamant that  this atomistic sound grunting business had something to it. So I went to visit.

Within 5 minutes it was obvious these kids were streets ahead of mine. About a year ahead, I reckon. So although if I were God I would have created things so that children learnt to read best through exposure to rich language as that appeals to my liberal tendencies- I conceded that unfortunately the seemingly less inspiring, dry and dull phonics method was annoyingly superior. Witchcraft was now in. We adopted it, found that children actually really enjoyed it because they loved the success that went with learning to read. Our results soared.  Every child except some of those with statements for thorough going global delay learnt to read. The headteacher of that school was Ruth Miskin and we’ve been using her products through their many iterations ever since.

At this point, the phonics champions are whooping, poised to retweet. But fear not, oh phonics denier- there is more to this tale. Actually SPOILER ALERT, both purist phonics advocates and deniers are going to be disappointed as I will come to a ‘sit on the fence- we need both phonics and other strategies’ moderate conclusion.

Anyway, time passed and we realised that while we were really good at getting almost everyone a 2c or higher at the end of year 2 ( if you remember the old days when we used to have these levels), the ones who got 2c – and some who got 2b, limped through ks2 and might just scrape a 4c on a good day, with a fair wind behind them. They might be able to read, but they couldn’t always pass that damn test.  Pot holing at Dingley Dell in 2011 was a particular low point- with an able reader scoring N, yes N, in the reading test. Reading for meaning was a real issue for some children. In fact for some, their tolerance of non meaning was truly shocking. They just didn’t expect reading to make sense- it was  just an exercise in barking at print. The most proficient print barkers would get quite cross when you tried to slow them down and enquire what a sentence actually meant. They know they were good at decoding and saw reading as nothing more than that. Yet here we were, calling their hard earned prowess into question. For some children, this may have been exacerbated by the way they were taught to read Arabic at Quran school, where as far as I am aware ( do correct me if I am wrong) learning to read the Quran means learning to decode it fluently- sometimes beautifully- but not actually understanding what the text says.

We’d been using ‘accelerated reader’ for years to keep tabs on whether our children were actually reading the books they took home from our library and this clearly showed that a stubbon minority of children – let’s call them Biff and Chip- just don’t read at home ever. And if we force feed them reading at school- they still don’t really understand what they’ve read.

Anyway- enter ‘ project meaning’.  We kept doing all the phonics stuff but added in extra ‘reading for meaning and pleasure’ stuff too. We revamped guided reading in ks2 by introducing reciprocal reading.  This was fabulous. Our proportion of level 5’s shot up- 85% one year and we are in a very deprived area  (school deprivation index 0.58). We also did Power of Reading and made fancy reading corners and bought lots of great children’s fiction and rewrote our curriculum centred on high quality children’s literature. We also realised our phonics programme had moved on and put a lot more emphasis on reading for meaning and enjoyment itself  so got retrained in that.  We introduced guided reading into ks1 so on top of  a hour’s rwinc ever day and 45 minutes of text based literacy, kids also got 30 mins guided reading daily. Biff and Chip got additional phonics teaching in years 3 & 4, until they were scoring a level 3 in their end of year assessments. Throughout ks2 all children did a phonics based spelling programme. And guess what? Biff and Chip  ( and really unexpectedly Wilma) still missed a 4c by 1 miserable mark. Which means we now have red blobs on our raiseonline saying the proportion of our disadvantaged pupils making expected progress in reading  is ‘well below’ that of other schools. (Although we get a yellow blob for our proportion of disadvantaged children making more than expected progress). So we are simultaneously really good and really bad at teaching disadvantaged children to read.

Deep breaths. Yes the red blobs are annoying- but actually- I sort of get it.  Say Biff and Chip had scraped a 4c? (I’m ignoring Wilma and assuming that was down to exam nerves- she’d got solid 4’s in all practice tests.) I would have patted myself on the back, smug in the knowledge that all was well with reading. But seriously – 4c…! You can ( or rather could)  be pretty poor at reading and scrape a 4c. Hardly setting you up for success at secondary though- is it?  Not good enough. When I looked back at our results over past few years realised most years there was at least one Biff……what was going on? Why weren’t we picking up that these children were so weak and doing something about it?  I could hear David Didau tutting.

‘But Biff and Chip don’t read at home’, decried my SLT. ‘Mrs Biff took Biff on holiday in term time… year 6!  And neither Biff nor Chip came to February or Easter booster classes! ‘ they continued. True, but we need to be good enough to enable our children to succeed even without parental support. With support- well just look at those other disadvantaged but well supported children with all those level 5’s. But we can’t rely on it for all children. What were we doing wrong? Or what weren’t we doing? Tracking showed that the children were 3c by the end of yr4, 3a by the end of yr5- so 4b here we come. 4a should be possible.

Sure when they actually entered yr6 the teacher ranted about them not actually being able to read properly…..but then she always does this about every pupil at the beginning of the year. ‘ ‘Woe is me…there is a mountain to climb….they are all barely level 2….what am I supposed to do.’ That’s just what year 6 teachers say ever year- it’s tradition. I don’t take any notice of that!  Especially because look, the tracker is all green- they are on track to make good progress. And usually- almost all of them do. Except Biff. And Chip.

Then I read Daisy Christodoulou’s piece about reading ages versus national curriculum levels and I felt sick. Daisy shares the evidence that shows that national curriculum level  2 reading assessment can span a seven year range of ability when the same children were assessed using reading ages. Now we’d always poo  pooed reading ages as when we used the bit of accelerated reader that assesses reading age- the results were so out of kilter with our national curriculum judgements that we just simply didn’t believe them. Sats held such sway that those results just had to be infallible- didn’t they?

And to be fair- the accelerated reader test has its flaws. It’s a multiple choice test done on a computer.  Without sitting on top of  a child and forcing them to take it seriously, some children find the thrill of having a laptop just too exciting and get distracted- clicking merrily at any answer without actually you know, er reading. I know sharing grades is not great practice but it does work wonders on the bright 10 year old who hasn’t concentrated on his  ‘star reader’ test to be told- ‘apparently you read like a six year old- how about you do the test again and actually concentrate?’  Who then achieves a 13 year old’s reading age. Earlier this term a teacher left the test to do with a supply and got a dire set of results. Only weeks later did the children mention the fire practice that had interrupted the test. Since it’s done on a computer and time limited, it’s a wonder that the results were as good as they were. And then there is a legendary tale of a struggling year 2 reader who presumably by random chance managed to score a reading age of 12 once ( and only once).  And then with young children there is the fatigue effect. As soon as you get a certain  number of questions wrong, your test ends- so the better you are the longer your test lasts. With lunchtime looming, Wilf sees Biff and Chip heading to the playground after a mere 20 questions and wonders why he is on number 34. He quickly clicks at any old answer to get the stupid test over and done with and lo and behold- the test ends. How do you make a test high stakes enough for children to try their hardest but low stakes enough that they don’t panic?

Anyway the team needed convincing- so I read the  boring small print bit in accelerated reader about how they ensure their test is reliable and it may not be perfect but it’s got heaps of trials behind it and is way, way more reliable than the old nc test.  So we are using that now. Alongside some other stuff.  And each time it throws up a couple of kids in each class that are doing less well than we had thought. Not bad enough for us to already have them flagged for phonics intervention but a good year behind where they should be.

I think it’s a reading fluency problem. Rwinc teaches them phonics and reciprocal reading nails the meaning but somewhere in between the two some kids don’t quite take off.  I see now that when we ended Biff and Chip’s phonics based intervention – because they’d got a 3c- that 3c was probably cobbled together by getting easy marks here and there and masked the fact they weren’t really fluent refers yet. Because they got their intervention  from a TA during guided reading- the class teacher didn’t actually know much about what they were like as readers. So went on the test result alone.  BIG MISTAKE. Although it’s tricky- what should Biff miss to catch up on his reading. Mum won’t bring him before or after school so should he miss guided reading or history? If he misses history- he’ll miss out on acquiring knowledge- and that’s vital for reading too, we now understand. I’m sure if we’d been pot holing in 2011 our ‘Dingley Dell’ Sats results would have been much better!   Assembly time? But what about developing his SMSC, hey and all those Britsh values.  Because he never did his reading homework, he already had to stay in every lunchtime to catch up. You’d see him, staring at his open book but not actually reading it. Or, when forced to, reading it word by word without enjoyment or real understanding. Every lunchtime and it did him no good- put him off reading more like. What he needed was someone to read with him. If we had realised how bad it was we might have found someone to do just that.

So that’s where we are at now. Screening all ks2 using reading ages and trying to catch all the Biffs and Chips for whom 3 years of high quality phonics in ks1 still hasn’t been  quite enough to enable really fluent reading or, unsurprisingly, a passion for reading.  A literature rich curriculum, a well stocked library, author visits and teachers who are passionate about passing on a love of reading still hasn’t inspired them to really want to and so to voluntarily put the work in.  Not even the lure of being able to read ‘ The Recruit’ ( parental permission required because a character says ‘shit’ or something) has been sufficiently tempting- because it’s just a bit too hard for them to read by themselves.   So now we are overhauling our lower ks2  for children who are not yet fluent.  We’ve bought in ‘ Project X Code‘ for them- rwinc is great but they need a change after 3 years on the programme. And they like the stories. Then in upper ks2 trying to  find enough staff to give up their lunch break  to help children become fluent and eager readers even when we’ve deprived them of football, using real books  via the Chatterbooks project.  As they remind us, being someone who reads for pleasure is a better indicator of long term success than  education or social class.  But will try and supplement with yet more phonics (rwinc’s fresh start) at some other point in the day. Just don’t have enough staff to do this at the moment. Am in school tomorrow trying to squeeze budget for solutions.  And we are really well funded compared to other local authorities. No idea how the rest of the world copes!

So that’s my solution. Don’t choose between phonics and other approaches. Do it all!

Teaching Biff and Chip to read.

Fractious fractions

Imagine that you’ve got two pizzas…mmm…but you’ve got to share them between three people (boo), and share them fairly, how much does each person get?  With this apparently simple question started the ‘big maths cpd ruckus of 2015.  Teacher A said of course you got two thirds of a pizza whereas teacher B said you got a third of the pizza and apparently it all got quite heated and they ended up having to get some paper plates and cut them up in order to prove who was correct. Which of course they both were.  It’s a determiner problem ( Oh no, not grammar and fractions…is this the new curriculum gone mad?)  Teacher A said two thirds of a (i.e one) pizza, whereas teacher B said one third of the pizza ( i.e one third of two pizzas). Which just goes to prove that fractions are tricky. They are tricky because they aren’t really numbers at all ( ok ok maths peeps, I’m playing fast and loose with definitions here, but I’m talking from the perspective of primary kids ok).  Up until we introduce fractions, when we talk about numbers, we mean natural numbers……the counting numbers. If I show a child a 2 I can get 2 pencils or 2 chairs to illustrate what I mean. Whereas if I show a child 1/2…well I could show them half a pizza or half a triangle but 1/2 is so much more loaded than that. Back to determiners. If I show them half of a pizza, I am actually saying – look- here is one pizza. Now let me divide it into two pieces. Each of those two pieces can now be called ‘ a half’. The difficulty being that if I were to have two pizzas to start with, then a half would be one pizza. So  sometimes a half is half a pizza and sometimes it is one pizza or 8 pizzas or…..well pretty much any number could be half of the pizzas.depends on how hungry you were in the first place. Fractions are not naturally numbers at all- quite often they are an instruction to do something: veritable bossy verbs. 1/2 really means ‘divide 1 by 2.’  It also means, ‘ how much you end up with when you divide 1 by 2.’ Which is an adjective apparently ( although it feels like a noun to me).  The answer and the operation are the same, which seems a little tautologous.  And when you cut three things into four equal pieces then the answer  is 3/4 of one thing- even though you started with three things.

The technical term for this is that 1/2 and other fractions are not natural numbers but rational numbers (ratio-nal numbers; they describe ratios). You have to know what the whole is for them to make sense. The whole is not always one. But the final answer is always expressed with reference to one, regardless of the size of the original whole. But we don’t make that clear. Or appreciate that clearly ourselves. Hey, I’m confusing myself writing this!

My better half (little fractions joke there) was recently at some training where they said that children learn about fractions more effectively if taught it in the context of division. Now a reference would be nice here, but husband can’t remember. Did I say he was my better half? It was probably Nunes, it always seems to be Nunes. Anyway – that got me thinking.

A while back resourceaholic posted something by some other secondary teacher (which of course I can’t find) about teaching  addition of fractions with different denominators by using Playdoh . Each pot of Playdoh was divided into different amounts and rolled into little balls of equal size. Because the different denominators are obviously different sizes, the students are less likely to make the error of adding different denominators together. So that made me think that at primary level maybe we should start teaching fractions by just investigating different denominators – and not mention numerators AT ALL until children clearly get that larger denominators means smaller pieces. In terms of writing, we would just write the / line ( that’s probably got a special name…) and the denominator underneath.   I imagined telling the children that the vinculum ( thanks google) aka the fraction bar or slash was a knife that cut the Playdoh ( or whatever) into a certain number of pieces and that we could record how many pieces by writing that number underneath the knife slash. Writing nothing at all above the line.  Then, after many happy hours slashing Playdoh into different amounts and becoming really fluent at writing these (sans numerator) and at naming these,  I finally pose the question- as if it has just occurred to me- ‘ but what if I wanted two of these things called fifths, how would I write that?’ And hopefully someone would suggest we could write that number above the knife/slash/fractions bar/vinculum. And the entire class would henceforth have a rock solid understanding of what a denominator is and how it is logically prior to the numerator- which is a mere adjective to the denominator’s  verb.

Because playdo is sticky, it also makes it easy to divide say  3 tubs of playdo into 4, by firstly amalgamating the 3 tubs worth into one great big ball and the dividing it into four, and then also by dividing each tub into four and then amalgamating one piece from each- and realising this is the same.  And of course it allows us to see that half of one tub is half but half of 4 tubs is 2 and finally that 1/3 of 2 tubs is 2/3 of 1 tub.

Edit: March 2016. My colleague Keeley Warren was much taken with this idea and spent a couple of days with her year 3 class and some play dough dividing it into different denominators….or ‘doughnominators’ and recording these saying the vinculum was the knife and not putting any numerator above it.  As a result, the entire class quickly grasped that the bigger the denominator the smaller each piece is.  Then when she returned a few weeks later to do fractions again- they had no problems whatsoever and grasped everything really quickly. She was amazed at how well it worked and it is definitely going to be standard procedure here from now on.

But that was before I heard about the ‘learn fractions as division’ thing. Which made me want to adapt this lesson further and start with bigger numbers. In fact, to teach division as sharing whole numbers using the vinculum thingy from the off- again saying it’s the knife that divides thing up – or pushes them into piles…..two for you and two for me. Sticking to even numbers initially but them moving onto dividing one thing by two, four etc.  Meanwhile division as grouping could be represented using the other sign ( the obelus, apparently) which keyboards  don’t tend to let you do- certainly not iPads on WordPress- but you know the one I mean. Not quite sure how and when you’d tie the two together yet. Via arrays probably. Love a good array.

Haven’t actually taught it like this- but I think it might work well. One problem though is that the numerator means slightly different things in each case and teachers would need to be very clear in their explanations. In traditional fractions teaching, the denominator indicates the number of equal parts into which a whole was cut and the numerator indicates the number of parts taken.  In division, the numerator refers to the number of items being shared and the denominator refers to number of recipients: if 2 chocolates are shared among 4 children, the number 2 refers to the number of chocolates being shared and the number 4 refers to the number of recipients; the fraction 2/4 indicates both the division – 2 divided by 4 – and the portion that each one receives.  It’s two quarters of one chocolate. Not of two chocolates. Our fraction is now an adjective, or arguably a noun, with its own special space on a number line. Hey it could probably grace the back of a football shirt.

Ps. Here’s that reference. It was Nunes! 

Fractious fractions