Showing Progress in Fractions

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

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

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

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

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

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


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

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

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

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

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

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

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


Here comes the Dominator

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

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

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

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

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

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

* by this I mean ‘denominator’

** by this I mean ‘numerator’

If only there was a tool like Khan Academy

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

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

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

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

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

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


Trialling Google Classroom

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

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

It’s easy. Really easy.

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

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

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

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

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

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

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

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

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

Computing is maths

I could say: maths is no longer completely maths.

If you compare the statements in the Maths National curriculum (2014) with the questions in the 2016 sample questions (which is when the first children will be assessed on the 204 National Curriculum you find a curious thing: if you only taught children how to do the things in the National Curriculum, they would do badly on the final test.

For example in the old National Curriculum, children were expected to be taught to make decisions about which operations and problem-solving strategies to use. A comparable statement in the new curriculum is that children should be taught to solve problems, including missing number problems, involving multiplication and division, including positive integer scaling problems and correspondence problems in which n objects are connected to m objects. There is currently no statutory guidance in maths for how problem solving should be taught, only that problems should be solved.

But don’t worry, because computing is maths. In fact in the statutory computing curriculum there are statements that describe how to teach problem solving. For example:

  • solve problems by decomposing them into smaller parts
  • use logical reasoning to explain how some simple algorithms work and to detect and correct errors in algorithms and programs.

In those statements for computing there are clear expectations for how problems can be solved, and I think they apply to maths problems as much as any other subject.

Let’s be clear here. I’m not trying to argue which is the better curriculum; what I am trying to argue is that you can no longer see your children make progress in maths by only teaching maths. You have to teach computing too. Computing is maths.

At the end of the last academic year, like most primary maths subjects leaders I did a maths SATs analysis. What I found was that if our children had solved problems as well as they answered questions about number and calculation, then our school would have been well above average, with the majority of our pupils getting level 5s. As it was, the vast majority of the children scored well into a level 4, but I was left wondering, what if we had taught problem solving just that little bit better…

What the analysis of maths SATs 2014 showed me
What the analysis of maths SATs 2014 showed me

And now the text of the maths national curriculum is even less focused on problem solving and more on arithmetic competence. Yet the tests in 2016 will be unforgiving to those children who have only learned to crunch numbers.

That is why believe me when I say maths is computing and teach computing with all your heart.

We should proud of our computing curriculum in England. Inspired by the Next Gen report by Ian Livingstone and Andy Howe back in 2011, the computing curriculum could become the envy of the world. If only 5% more of our students become competent at computing, imagine the world powerhouse it would make us. Ian Livingstone describes the ideal ‘A’ level combination for a student going into hi-tech industries as maths, physics and art. Computing is not only produced by both the sciences and the arts but it supports learning in the sciences and the arts.

And that is why, when Melissa shone at the computing workshop with Miles Berry at Microsoft headquarters back in January (as I posted last time), I was delighted. It was not just because of the great confidence she had gained, nor the insight into being able to write a ‘repetition’ in code.

It was not just the big tick in the box marked ‘self esteem’.

It was because by doing good computing she had also done good maths. She had solved a problem that I knew would stand her in good stead in the next few months and beyond. As a remedial teacher it was a win for me, because when Melissa gets better at computing, she also gets better at maths.

Because computing is maths.


Why Computing?

As I continue to write about the journey of the Mathemateers in their maths learning, I’m going to divert for a post or two into the thorny subject of Computing – a new subject on the National Curriculum. I’m about to argue that computing is just the thing that schools like mine need to raise standards in maths.

As you’ll remember from her pen portrait, Melissa had very low ability in maths a couple of years ago and has made considerable progress to get to where she is, needing only a small boost now to get to national average. Imagine my delight then, when at Microsoft on 7th January for the Quickstart Computing Workshop with Miles Berry, Melissa stood up to explain to the whole room how she had used the ‘For’ function to reduce the lines of code her turtle needed to draw a square from 7 lines of code to 2 lines.

My delight was twofold:

  • Melissa has very low confidence – part of her problem in maths as an inability to try new things out because she doesn’t want to get things wrong.
  • Melissa isn’t very good at maths – using a ‘for’ function shows a level of logic I didn’t know she had.

The challenge went like this:

  1. Miles Berry asked the children to define a square.
  2. The children struggled to define a square. Apparently this knowledge has been removed from the national consciousness sometime in the last few years.
  3. Mile Berry showed the children how to make the turtle draw a line and turn using Microsoft’s online programming teaching tool: Touch Develop.
  4. The children used Miles Berry’s start to program the turtle to draw a square. Most of the code looked like this:
    Code for how to draw a square on TouchDevelop
    Code for how to draw a square on TouchDevelop


  5. At this point the children near me started noticing that the code repeated itself rather a lot. I wondered out loud whether there was a of making the code repeat and eyed the screen meaningfully.
  6. Melissa immediately started looking beyond the ‘right turn’ and the ‘forward’ button and noticed that there was another button called ‘For’ with the words ‘repeat code’ under it.
  7. She started dragging clicking and dragging things around and soon came up with this:
    Better code for drawing a square.
    Better code for drawing a square.

    We were all delighted and Miles got Melissa up the front to explain what she had done. Since then she has become a bit of a celebrity back in year 6 – her class teacher has been pleased to get her to do the same demonstration to the rest of the class when he introduced Touch Develop. Then Melissa had to go to Year 5 where she again demonstrated her computing prowess.

So. Here I have Melissa, self esteem going through the roof and she has associated this computing success with maths. Over the last couple of weeks, she has solved problem after problem, met target after target – she is truly flying. Maths is going great because of a positive experience in a computing workshop in London.

So what’s the lesson here – give children a chance to shine and they will?

No, there’s more than that. Computing is maths. And I’ll explain that statement in my next post.


The trouble with three

I noticed a really interesting thing when I was working with Ebony Rose in the Colouring In lesson.

She has a real problem with ‘three’.

As you’ll have worked out if you’ve read the post and considered the problem I set, the children have to colour in different amounts of squares. There are only two choices – when you come to a new colour you’ll either colour one of each square next to the old squares, or you’ll colour three squares. That’s the choice: one square, or three squares.

But Ebs has a real problem with three.

She could cope with all the ones. She could see that each old square would generate one new one, but every time we came to one that would make three squares she froze. If I wasn’t watching here, she would just blithely colour in a single square, even though three were required. And then when I was talking her through it, and getting her to talk it back to me it was like she didn’t want to even say the word three – whenever we came to it, it was like she was trying to out-wait me – to see who would crack first before saying the word three.

It became really hard work. The fact is that during the activity you have to colour in three squares lots of times – she wanted to progress and do well, but she didn’t want anything to do with the number three. I begun to believe that she had some kind of ‘three’ phobia – as if she thought it was cursed or out to get her or something.

Over the weeks since then I’ve begun to understand why: Ebony-Rose often confuses the ‘3’ digit with the ‘5’ digit. Show her ’50’ and she may say “thirty”. Show her ’13’ and she may say “fifteen”.

I remember last year when I was working with a child who, at the age of seven, couldn’t reliably count to ten (or for that matter in tens to 100). By ‘reliably’ I mean she could do it, but 4 out of 10 times, she would get it wrong. When I listened to her count I noticed that she didn’t like pronouncing the ‘f’ in five or fifty: she found it awkward to say. So instead of saying it she would just move swiftly on to six or sixty. We worked for  a few weeks on this reliability and (this academic year) her year 4 teacher tells me she is flying in maths.

Imagine being ten and trying to do maths when you can’t reliably differentiate between a ‘3’ and a ‘5’.

Now I know there may be lots of excuses I could throw at Ebony-Rose at this time  – dyslexia, dyscalculia and all of that stuff. But the excuses aren’t going to help me this week or next week. I need to teach her know how to diferentiate between those two digits. I need to get her confident at knowing ‘the threeness of three’ and give her practice at using threes.

Who knows, she may even master the 3 times table by half term…

The Quiet Coach

As it happened, my second day with the Mathemateers was a trip to London. I’m going to tell you the story of that another time.

Today’s story is about the train journey.

No. Today’s story is about the inflexibility of booking systems and the lengths we British will go to to avoid being embarrassed.

There I was, a couple of weeks before Christmas about to click the ‘submit order’ button on the Virgin Trains website. One click and the booking was made. Myself, a teaching assistant and my not-even-nascent group of ‘low-ability’ maths children would go to London, by train, visit Microsoft headquarters, attend a workshop led by Miles Berry, and maybe even see some of the sights.

And then the ‘e-ticket’ came back through to me and it said ‘quiet coach’.

QUIET COACH? Why would an automated booking system put a group of 5-15 year old children on a quiet coach? I could feel the embarrassment of the situation and it was still a whole month away. Mid-December and I was worried about saving face in January. Immediately I rang up the Virgin Trains helpline. A very nice man from India answered the phone and told me he could do nothing about it, but he did give me the phone number of the customer care team. I rang them up and a very nice lady from India told me that as it was an ‘e-ticket’ nothing could be done.

I was stuck with a quiet coach. My mathemateers (as I now know them are lovely), but not guaranteed to be quiet. Sarah, in particular, has a hearing impairment and an enthusiasm that knows no bounds. By herself I knew she would ruin the ambiance of any so-called quiet coach.

So I fretted quietly to myself, wondering what I would do in that first week back. On the first day back in January, it was teacher training – I had some time in the afternoon to plan a solution, but I did nothing about it. Then the next day, I met the maths group for the first time and we came up with that name and I decided to blog about what happens as I teach them. I still had nothing. All I could do was plan a lesson. So that’s what I did. Yes, the fear of embarrassment in a public place led me to plan a lesson for a train journey. I know.

And when the day came. After the taxi and wait at Birmingham New Street and the inevitable finding of seats (I always find asking adults to leave their seats due a reservation awkward) I pulled my lesson sheets out…

The lesson didn’t start immediately. I had to let the awe and wonder die down first. You see three of the seven children had never been on a train before. 2 of them had never been out of Birmingham. Only one had been to London.

So after about twenty minutes during which the classic question “Do they have different money in London?” was asked, I was getting increasingly conscious of the looks from various commuters on the quiet coach. So I started the lesson.

Do you know what? – it went really well. But that’s yet another story for another time.


Starting at zero

In my very first session with the Mathemateers last week I spotted a significant thing that Jules was doing wrong.

When he counted he did not start at zero.

It seems bizarre doesn’t it, that someone can get to ten and still be unsure how to count.

However I remember a friend I was at school with, who after finishing his degree in politics, confessed to me that whenever he subtracted in his head he was always one out. When I listened to him subtract I could hear that he counted the first number as ‘one’. So if he was doing, say 25- 9 he would count back 25 as one, 24 as two, 23 as three and so on, rather than starting at zero. He didn’t make this error adding and strangely had got all the way through GCSEs, ‘A’ levels and a university degree with this slight impediment to his maths.

My friend never looked back from the revelation I presented to him. A couple of years later he did a masters in Computer Science and now project manages big software design projects in Australia.

I’m not sure, but Jules seems to have had a similar epiphany. He has stood still in maths for a year or so – he should be working at level four, but is struggling to maintain level 3. On Tuesday last week I noticed that when trying to read the time, he counted round in chunks of 5 to what should have been 30, but read out 35. The minute hand you see was on the ‘half-hour’, Jules knew that each digit on the clock face meant 5 minutes, but somehow managed to construe half past 5 as 5:35. He started by counting the ’12’ as 5, the ‘1’ as 10 and so on, rather than starting at zero minutes.

In my pen-portrait of Jules I mentioned this misconception, thinking that I would be spending the next couple of weeks teaching him how to get out of it. But not so. It seems he is out of it…

From then, he hasn’t  looked back. At the weekend he practised 33 different Khan Academy activities and almost doubled his points (he’s been using Khan Academy for nearly 2 years). When he came in this morning he was bursting with confidence and pride and whizzed through the activities I had for him – I’m going to have to pitch things harder tomorrow!

Was this going to happen anyway? Did he just need a small group to express himself in and a trigger for his confidence? Was it just that he was at the end of an achievement plateau and ready to climb another hill? Or was it just a quick pointer that he should start counting at zero?

And I wonder how many children across the country are struggling right now in maths just because they don’t have this basic skill?

Melissa the Mathemateer

Melissa is amazing. Two years ago, she could barely count to ten, but now she knows written methods for the 4 operations and can carry them out reasonably well. Her progress has been exceptional. Unfortunately she is one of those children who scraped a 2C in Year 2 (more of this later), so while I know her progress is exceptional, it doesn’t look very exceptional to external visitors.

Melissa is cared for by her Nan, after her natural mother was deemed incapable of caring for her. Nan is pretty close to illiterate, Mum was educated in special school, so I always think it’s a vindication of our education system when generations of families achieve progressively higher academic standards. I have high hopes for Melissa.

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