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
- 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.  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.
 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.