Showing Progress in Fractions

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

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

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

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

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

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


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

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

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

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

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

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

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


Here comes the Dominator

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

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

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

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

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

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

* by this I mean ‘denominator’

** by this I mean ‘numerator’

Welcome to Fraction County in the United States of Maths. Population: ⅗

twinpeakspsykopaintI’ve often heard it argued that learning maths is like learning another language. There is a whole vocabulary and a way of speaking that is alien to people who don’t live in the land of maths. Abstract concepts are understood by saying things that only maths people understand. The conjunction is the equals sign; verbs are operators; a degree-level literature essay is a second-order differential equation.

If that’s true, then Fraction County is the kind of place where the banjo stops playing when you walk into a bar. The talking stops. The locals all put down their home-made moonshine and all that can be heard is the faint rustle of tumbleweed blowing along the street outside. And you realise that the language they were talking is a completely different dialect from one that you’ve understood before.

It is no wonder that many children panic when they hear the word “fraction”.

Think about this.

The children walk into a room and see ¼ written on the board. The teacher asks “how do you say this?”


A brave child sticks their hand up and says “one line four”. Another child, emboldened by the first contribution, suggests “one point four”. Then someone asks “is it a fourth?”


“That’s not exactly how we say it,” corrects the teacher, obliquely referring to some shadowy group of people the children have never heard of. A group of people that obviously can already speak ‘Fraction’. “We say ‘quarter’” The teacher smiles reassuringly, but inside is concerned. She knows that the children should already be able to read and say a quarter and she utters a silent curse at the children’s previous teacher.


The lesson continues. The children learn that fractions are something to do with pizza (or if you listen to Sal Khan, pie). Then, after seeing that ¼ of a pizza is one piece out of 4, the teacher holds up 4 multilink cubes that are all joined together in a small tower. She asks the children how many cubes are in the tower. The children say “four”. The teacher breaks off a cube. She asks how many cubes she broke off. The children say “one”. “Ah, but what fraction did I break off?” asks the teacher, with an air of mystery. “Half?” asks a child. “A third?” asks another.


Ever patient, the teacher persists. “How many cubes were in the tower?”




“So what is the ‘out of’ number?”




“So this cube is one out of four,” declares the teacher triumphantly, writing ¼ on the board again. “How do we say that?”


“One line four” says a child.


“One four” says another.


“Quarter” says a third.


“Yes,” says the teacher, pouncing on the learning. She vigorously shakes the child in sheer joy that someone has got it. “And we write a quarter, one over four.”

The problem is in the language. The children have already learned that division is one word that means two different things – sharing and grouping. Now there’s the whole same thing going on with fractions. They’re sharing pizzas and calling each piece a fraction. Then they’re grouping sets of objects into equal subsets and calling each subset a fraction. Then despite the fraction being called “a quarter”, the teacher describes it as being “one out of four” whilst explaining that you write it “one over four.” The concepts behind these aren’t impossible to grasp, but the language we use to describe them is just so inefficient.

This is one of the reasons that my favourite thing to come out of the old National Numeracy Strategy was the book on maths vocabulary – describing the kind of words that children should be taught in each subsequent year.

But knowing the words is only part of the problem. I know some French words and some Spanish words but (to my shame) I find it hard to put them in the right order. The language of ‘Fraction’ is similar. It takes practice and good teaching to put them in order. If your teacher is woolly in their teaching and you don’t practice enough, you won’t learn the language. Worse, I know plenty of people whose maths teacher lost patience with them during some maths lesson or the other and shouted at them for not getting it quickly enough. This is often a reflection on that teacher’s subject knowledge, not the maths ability of the student. It is a reason why I recommend Derek Haylock’s excellent book on teaching maths.

So next time you’re on the road to Fraction County, make sure you’ve rehearsed some of your lines – you may just teach your child to know their denominators from their numerators.

Good for the fractions learning; bad for the coffee mug


Sometimes children hear the word 'fractions' and they turn off.

I saw it on Wednesday when I started my lesson on comparing and ordering fractions. I had barely uttered the words when I saw a few heads drop. A few children joined in when I asked them what they knew about fractions – one knew the word 'third'; someone else knew 'part'; yet another one knew they have something to do with division. But quite a few heads with dropped.

So while the keen had their hands up, and others were looking to avoid eye contact, I slid an empty coffee mug into an empty plastic bag. Then, for security, whilst the conversation continued, I placed the first plastic bag into a second one.

Then I smacked it against the wall. Really hard.

All the children looked – some jumped.

I proceeded to pull pieces out of the bag and estimate how much of the mug each piece had been, from the large chunks (1/3 or 1/5) to the tiny chips that were only 1/1000 or maybe even smaller.

The children were engaged and by the end of the lesson all of them had made some progress about ordering and comparing fractions. Even the special needs group children who, according to their data, struggle to order numbers 1-100.

As a bonus, we even specified that the bottom of the fraction was called the denominator and the top number the numerator – I love it when children learn proper maths words, although it was amusing to hear one child call the top number the nominator and the bottom number the dominator.

So, if you're stuck with teaching fractions – break something. At least you'll stop the heads from dropping…

Valuing misconceptions on the way to explaining fractions

explaining fractions_0001.wmv
Watch on Posterous

I filmed this about 6 months ago, following an excellent session about fractions on the Mathematics Specialist Teacher Programme. The challenge that we were given, and then I in turn gave to the children, was given a 4-pint bottle of milk that gets 3/5 of a pint drunk each day, how many days does the milk bottle last for? Those of us with a formal background in maths would say:

÷ 3/5
= 4 ÷ 3 x 5
= 4 x 5 ÷ 3
= 20 ÷ 3
= 6 r 2.
So the milk lasts for 6 and a bit days. If we wanted to be really fancy we would say the milk lasts for 6 and 2/3 days. And isn't it more practical to say the milk lasts for 6 days and there's 2/5 of a pint left over? Does our understanding of the algorithms let us say that?

Also can children, who are without the drilled-in knowledge that when you divide a divisor you actually multiply, do this question?

That's what the video explores – and there's some interesting misconceptions on the way.

Fractions: learning something new

Yesterday was a complete surprise to me. I learnt something new about maths. And I enjoyed it.

Without trying to show off, I do know a lot of maths. I won’t bore you with too many of the details, but I am both interested in maths and quite good at it. I recognise that there are a lot of people who are much better than me – without some of those people I would never have got through my ‘A’ level maths (thanks Greg, thanks Yao) nor my Engineering degree (thanks, Jim, thanks Dan). However, in primary teaching I haven’t met too many of those people. Most of my colleagues are good at teaching maths, but would say that it is not their main interest. Some would demonstrate an enthusiasm for a particular branch of maths, whilst a few would express some negativity about areas of maths, particularly at the higher levels of the primary age range.

Yesterday’s topic at the local area meeting of the MAST programme was ‘fractions’ – an area of maths which usually generates the word ‘Hmph’ from children, parents and teachers alike. I was so excited by some of the fractions problems we attempted I took them straight back to school the next day and filmed my Year 6 children trying to solve them. Here’s the video:

Hopefully you can see how the children progressed in the lesson. Many of the children, despite being the most able in the school, had quite a negative attitude to solving problems involving fractions. Through using models and images the children now have a better conceptual understanding of fractions – they have linked the visual to the concrete – and are now ready to move on to using the abstract: numerical fractions themselves.

It struck me that as teachers we often move too quickly from the concrete to the abstract. If the highest ability children needed this level of input to begin to ‘get it’, then younger children and less able will need far more input at the concrete and visual stage before they move on to the abstract. This makes complete common sense, but in our overly prescriptive curriculum, how often do we rush children on to using and failing with the numbers when they don’t get the concept?

So if two and half men take two and half days to dig two and half trenches, how many trenches can one man dig in one day?

My answer was one trench and I was completely wrong. The feeling was exquisite – some maths that I didn’t get. My table group had to work hard to try and solve the problem and we still didn’t get it. Finally when someone provided a solution and the concept started to sink in it was marvellous to realise that I had been challenged with something and learnt something new as a consequence.


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