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.

 

The coming mathematics apocalypse

I am tremendously excited by the current maths curriculum in English primary schools. The expectations are higher by at least a year. It is a no-nonsense raising of standards.

I’m excited because if we can find a way of teaching the curriculum successfully, then our students will have levels of maths previously unheard of in this country. They will be on top of the mathematical world. The average will be above average. They will have the skills and knowledge to found an empire of learning.

Not only is the curriculum at a standard that is a year higher than the previous curriculum, but there is talk of the floor standards, already raised from 60% to 65%, to be raised further to 80%. This means that schools will have to find a way of getting more of their students to a far higher standard. What a fantastic aspiration.

But it’s the ‘if’ I see as being a problem. Let’s face it: we don’t currently have the maths specialists we need in Years 5 and 6 – the upper years of primary school. Secondary school maths departments complain of misteaching, cramming for the end-of-primary-school SATs and students without the knowledge they need. What will it be like with even higher standards, both on the level that children must attain, and also the number of children that must attain it?

I see three options:

  1. Success – we achieve the ambition. We find a way of teaching more of our children to a higher standards than previously attained. A golden age of knowledge is ushered in.
  2. Cheating – we pretend to achieve the ambition by blurring further the grey line between supporting students and telling them what to do in tests. The nation lurches towards a moral crisis.
  3. The Maths Apocalypse – We ramp up the stress. School leaders put the pressure on their teachers with the higher expectations. The teachers crack under the pressure and shout things at their students like: “why don’t you just get it!” A generation is turned off everything to do with mathematics. It’s a maths apocalypse.

The problem with raising standards for students is that we also need to raise standards of teaching. Many primary teachers have a ‘C’ grade GCSE in maths, yet the maths expectations now required would go a long way towards achieving one of those ‘C’ grades.

The clock is ticking. In 2016, the first cohort of students will be tested on the new curriculum. Only eighteen months away and staff rooms across the country should be buzzing with conversations around how we teach maths and the subject knowledge we need. Networks of maths co-ordinators will right now be exemplifying the standards – turning the words into maths that can be taught and practised. Experts from teacher training will be working closely with schools, finding ways of bringing their new trainee teachers up to the required standard and sharing some of their training expertise with teachers who are struggling with their own maths. Teaching schools will be focusing on developing their Specialist Leaders for Education in mathematics so that schools within their networks without maths specialists have a means of accessing their expertise.

We have time to prepare and to succeed. Doing nothing will only prepare us for the mathematics apocalypse.

Key Stage 2 Maths SATs Analysis

Admittedly, not the most exciting post title you’ve ever seen, but let me draw you in with what I found out:

  1. The greatest proportion of maths expectations we need to improve on at my school come from the Year 1 programme of study.
  2. Two of the key questions that we need to get better at are taught through the KS2 computing curriculum, and not the maths curriculum.
  3. At my school we are really good at teaching calculating and number, but we need to improve at teaching problem solving.
  4. Neat, well presented work does not equate to success in maths SATs.

So, I used a useful spreadsheet I found on the TES website to analyse the maths SATs results from 2014 (you’ll need a TES login to access that link). I was particularly concerned about the 6 children who didn’t make 2 levels of progress during Key Stage 2. Six out of thirty is a large percentage for us: it moved us down from well into the top of the half the country (in terms of progress measures) to well into the bottom half. While there was a back story (read: justification) behind each child, I wanted to look more carefully at the results each child had achieved and find out why they hadn’t quite made the grade.

Having analysed the data, I made a presentation for my staff so we could talk through the issues involved. Why not just talk it through with the Key Stage 2 department? Well as I’ve indicated above, many of the statements where we need to get better at are actually taught from Year 1 or 2. I’ve put this presentation into a Movenote here. Please feel free to watch, but don’t expect quality – I was using Movenote to practice my presentation for the staff meeting on Wednesday – it’s a first take, and I’ll be expanding on many of the points during the actual staff meeting.

My two big considerations are the following:

  1. My children need to get better at logical reasoning to achieve well in maths. Logical reasoning is most explicitly described in the computing curriculum – how can I use the computing curriculum to raise standards in maths?
  2. With the foundation for success clearly coming from the teaching in Year 1 and 2, how can I make sure that this teaching is as good as it can be?

It will be interesting to see if my staff agree with me on Wednesday.

A greater stretch in mathematics? If only.

I read the letter from Michael Gove to Tim Oates about how the new National Curriculum Review would affect the Programmes of Study within Primary Schools with a great deal of anticipation, and then a growing feeling of disappointment. There are various areas that disappoint me, but the area of maths teaching is perhaps the greatest. I read:

In mathematics there will be additional stretch, with much more challenging content than in the current National Curriculum. We will expect pupils to be more proficient on arithmetic, including knowing number bonds to 20 by Year 2 and times tables up to 12 x 12 by the end of Year 4. The development of written methods – including long multiplication and division – will be given greater emphasis, and pupils will be taught more challenging content using fractions, decimals and negative numbers so that they have a more secure foundation for secondary school.

Minor Disappointments

Let me break this down then. Here are some of the minor points that disappoint me:

  1. Knowing number bonds to 20 by Year 2 – this is already the case. How can it be more stretching to maintain an expectation – surely the bar should be raised somehow.
  2. Times tables up to 12×12 by the end of year – this is a slight rise in expectation as children are currently expected to know up to 10×10, but in my experience it is not the knowing of extra times tables facts that hold back children in the latter part of primary schools, it is the lack of knowledge of corresponding division facts (which happens to be part of the current national curriculum).
  3. Greater emphasis on written methods (like long multiplication and division) – this sounds good, but it’s already in the current programme of study. Just saying something more doesn’t make it more rigorous, nor does it necessarily raise expectations. In addition, I’m all for children learning skills in school such as the skill of performing long division, but I had been under the impression that the new curriculum would be more knowledge based.
  4. Pupils will be taught more challenging content using fractions, decimals and negative numbers. Again, these are all in the current programme of study for children from the age of about 7. Whether children are taught these are up to their teachers and their schools.

So when I read the phrase “much more challenging content“, and put it next to the above examples of challenging content, I’m struggling to see the giant leaps in standards that Michael Gove would be hoping for when his new National Curriculum is implemented.

A medium-sized disappointment

A greater disappointment is to see the phrase “more proficient in arithmetic” without any reference to algebra. As I have written before, children start learning about algebra from a very young age when they start investigating concepts such as larger and smaller. It is the concepts involved in algebra, often linked with precise language teaching, that I think are under-taught or mis-taught at primary level. Teachers shy away from this vocabulary-driven area because it doesn’t feel like maths to them – there aren’t numbers and operations written in children’s books – it doesn’t look as good as arithmetic. When children leave primary school I think they are often under-prepared for algebra – it is in this area that I was hoping for higher expectations within our new National Curriculum.

If you’re good at arithmetic you can go to the shops. Or maybe become an accountant. If you’re good at algebra you can become a rocket scientist. Not that education should just be about gaining a decent job – Gove himself says he wants “a love of education for its own sake” – but I have a feeling that for many algebra isn’t the elegant thing of beauty that I perceive it as, but is a rather lumpy, ugly thing, ringed with fire and tears.

A major disappointment

Aside from my algebra foibles, there is a further disappointment that I think will have a wider implication on maths teaching: teacher subject knowledge. No matter how detailed a Programme of Study or a National Curriculum might be, neither address this problem: we have many teachers within the primary sector who do not have sufficient subject knowledge to teach mathematics effectively. Many primary teachers only have a grade C at GCSE and have had to repeat their mathematics test required by teaching training in order to pass their course.

I have no problem with teachers teaching mathematics concepts that they’re not to sure about, so long as they know what to do when they’re stuck. There should be an expert teacher within each primary school – the maths co-ordinator or similar – who can share their knowledge and expertise when others don’t know the next steps. Too often less-skilled maths teachers don’t seek help from their more experienced colleagues, but struggle with the text of the National Curriculum and any scheme books that support it. Not knowing what to do, they either miss the tricky areas or teach it badly – characterised by repeating themselves more loudly and more slowly, a bit like the traditional Englishman abroad.

It is not a new Curriculum that is going to improve standards in mathematics. We will stretch primary mathematics only by increasing subject knowledge within our teachers.

 

There’s an easy way of doing this

In the run up to the National tests for eleven year olds called SATs this May, I was practising with some of my pupils what some of the question would look like.

The girl looked at the question and said: “there’s an easy way of doing this.”

The question said 56 ÷ 4 =

It is one of those rare questions in a Key Stage 2 SATs paper that requires a simple answer to a mathematical expression. The girl I’m sure had seen that question every year for the last five years. Yet she was still hesitant – she had no instant response to the question. She had to think of the ‘easy way‘. And unfortunately she went on to choose the wrong easy way.

“My teacher told me you just drop the ‘6’ off the end, add one on to the 5 and that’s the answer.” Unfortunately the girl was remembering the ‘easy way’ for dividing by 9. And she was remembering the answer to the expression 54 ÷ 9 (which of course is 6).

This one of the reasons I dislike teaching children easy ways of doing things. In my experience most children who are taught easy ways have learned the underlying principles behind them. They then can only remember a small number of many easy ways and eventually they forget which way is which and when to use it. The next step is to decide that they can’t do mathematics anymore and they switch off from the subject altogether.

To quote a biblical metaphor, it’s a bit like building your house on the sand. It only takes a single storm of confusion to reveal that there were no foundations and everything is washed away.

Putting it another way, it’s like badly applying Bloom’s taxonomy to teaching. It seems we’re very keen in the teaching world at the moment to find ways of teaching those higher skills of evaluating and creating. But we miss the vital step between remembering things and applying them – that of understanding them.

We teachers often talk about that ‘wow’ moment in lessons – that realisation by the students that they are really ‘getting it’. This most often happens in 1:1 interactions but can also happen with larger groups. When I look at the Bloom’s Taxonomy chart I would say that that ‘wow’ moment comes in the ‘understanding’ phase. It’s not when we’re sure children can remember things by heart, or when we see them diligently applying their knowledge, nor even we see the outcome of a great piece of creativity. It’s when children comprehend, when you can look into their eyes and know they have understand – when they get it.

So, back to the girl with the maths problem.

Striving for that moment of understanding, I asked, “are you sure that’s how to divide by 4?”

She looked at the problem, hesitated for a moment and said. “Oh no. There’s an easy way to divide by 4. Halve it and halve it again.”

I couldn’t argue with that process. She proceeded to halve 56 by writing down 2.5 and 3. Then she wrote 2.8 in the answer box. I almost slapped my forehead in despair.

After a few more minutes of remembering how to halve, she did eventually get to the point where she found that half of 56 was 28 and then half of 28 was 14. She wrote that in the answer box.

Not satisfied, I asked her, “what if it had been 56 ÷ 6? How could you have done that?” She looked at me, blankly. I think that she was a little disappointed that even though she had arrived at the correct answer I hadn’t showered her in praise.

Divide by 6? I don’t have an easy way for that.” OK. She didn’t actually say those words, but I’m sure she was thinking them.

So I showed her the number line in the photo below. I showed her how you could count up in 4s or in groups of 4 to arrive at the answer. I showed her how it would also work for dividing by 6 or dividing by 7.

Trying to teach understanding, not just an easy way

I didn’t really get that ‘wow’ moment I was hoping for. I think she begrudgingly accepted that maybe the number line had some merits. Of course counting up in this way requires good recall of times table facts – facts that she struggled to remember.

It is interesting to me that the first stage on Bloom’s Taxonomy of remembering seems to have been pirated away for this particular student. Where she couldn’t initially remember to halve and her poor recall of times tables facts limited her approach to this question, she could, by contrast, remember quite well that there are some ‘easy ways‘ for doing things in maths. This in turn limited her understanding of the principles of division and stopped her applying any knowledge she had to this problem.

It seems to me then that we need to stop teaching tricks and easy ways that fill up children’s memories. We need to teach children to recall and remember important facts first, such as how to halve and double and times table facts. Then we need to teach children understanding, such as what division is – that it is both grouping and sharing (depending on the context). Then we can give them opportunities to apply their knowledge.

Solution to Einstein’s Problem

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Quite a few of my class know the answer to 'Who Keeps the Fish?' know.

This problem is ascribed to Einstein who reckoned that only the top 2% of the population could solve it. That might have been true back then, but I reckon we're a lot more adept as problem solving these days. Anyway, I was really impressed with Lauren's neat, tabulated solution to the problem. See if you can work out how she did it.

Ray Maher at Athena

Today, UK Maths expert, Ray Maher visited the Athena schools in Birmingham. This is the record of the tweets I made of the event.

5 schools meet for #Athena day at the Beeches for #RayMaher on Raising Standards in Mathematics

#RayMaher still does teaching in school – says best teachers should be in Year 3 – it's the engine room of the school
#RayMaher- best resource in schools is human
#RayMaher – if get foundations right, you get the rest right – times tables by rote – make sure 7 year olds know them off by heart.
#RayMaher wonders why some Year 6 don't have methods for + – x and ÷
#RayMaher says UK is 4th in World in maths (behind, Korea, China, India)
#RayMaher -think of a number, double it, add 5, multiply by 50, +1760, – year you were born. My answer = 738
#RayMaher talks about weighting of maths. 20% U&A, 50% N@C, 20% SSM, 10% DH
#RayMaher questions how secure children are at end of Year 3. (should be level 3)
#RayMaher says if you get the calculation right, you're going to get standards up
#RayMaher – child methodology: in my head? Using drawings / jottings? can I use expanded / compact written method? need calculator?
Teaching one method means less able improve, but more able stay the same… #RayMaher
#RayMaher says models and images in Y1-Y3 are really important
#RayMaher – sort out misconceptions in Year 3
#RayMaher Use number lines to develop mental imagery – then move quickly on to efficient written methods when understanding is secure
#RayMaher advocates using Grid Method – Level 3: 2 dig by 1 dig / Level 4: 2 dig byt 2 dig. Can't do grid method unless timetables secure
#RayMaher gives example of using bus stop for 47 ÷ 8. #fail!!!!
#RayMaher Chunking is repeated subtraction. Egof Y5 non-divider who caught on with chunking 10 lots of 17 on a number line (for 191 ÷ 17)
#RayMaher indicates that children need to OWNtheir one method for +, -, x, ÷. It's not one method per teacher, but one method per child.
#RayMaher – we need progression of calculations policy from foundation stage to Y6. Gives example of very simple policy.
#RayMaher says use calculations policy as displayed curricular targets
#RayMaher use easy-language display of calculations policy to go home for parents
#RayMaher has marvellous counting stick with velco on and 'Mortimer' a puppet who counts up and down.
#RayMaher shows place value mats for Lower KS2. Place value strips for addition. Number squares – 0-99 better than 1-100.
#RayMaher shows viewfinders for number squares. – Algorithmic Ls – great resource for revitalising our number squares
#RayMaher Know 1, get 3 free sheets. Show that one addition fact gives 3 more (1 +, 2 -); 1 multiplication fact gives 3 more facts (1 x, 2÷)
#RayMaher – round tables – look ace. TAs – make up packs of all these FAB resources
#RayMaher – Language pyramids for maths language
#RayMaher – shows kinaesthetic resource for counting
Useful numbers game like countdown http://bit.ly/1HqVtn #RayMaher
Good place for maths games (although the cost…) http://bit.ly/cAghdW #RayMaher
#RayMaher shows Busy Bees good SODA (Start Of the Day Activities) for maths http://bit.ly/dqDGOZ (it costs £20)
#RayMaher – can you gain 5 minutes during register? Research shows that if you do the children will get cleverer…
#RayMaher advocates buying Nintendo DS to support mental recall – brain traing / maths training
#RayMaher uses numeracy passport to support progression of key skills. http://bit.ly/8ZEHNl
RT @frogphilp: #RayMaher uses numeracy passport to support progression of key skills. http://bit.ly/8ZEHNl
#RayMaher says Wirral Authority give children a 'travel bag' for pre-passport activities where children get stuff when they achieve learning
#RayMaher demonstrates resources that include left-handed children
#RayMaher shows Fiery Ideas Passport kit. Costs £55 but is comprehensive for developing mental maths and instant recall http://bit.ly/b0FX3h
#RayMaher points out that doing bonds to 6, 7, 8, 9 are as important as bonds to 10.
#RayMaher – organisation of passport instant recall means that you can focus on skills for different groups during main activity of lesson
#RayMaher – passport objectives: know all recall by Year 5; more confident at doing sums in heads; key skills for ECM; kids love it!
#RayMaher argues that APP should be standardised – 5 schools on this day – we could meet to standardise our practice
#RayMaher Shows a grid of standardisation
#RayMaher warns that great resources like Pitch and Expectations http://bit.ly/bn0sia may be removed from website as funding is withdrawn
#RayMaher says learning objectives should be broken down – different buzzwords 'steps to success', 'learning ladders'
#RayMaher example of steps to success for Grid method: partitioning, multiply by 10, recall of times tables, spatial awareness, addition…
#RayMaher shows resource linking AFs – Objectives – Steps to Success – Resources
Robin at #RayMaher day points out that steps to success can become a barrier for teachers if they don't use common sense.
Robin at #RayMaher says that a teacher's job is to make planning exciting and relevant; not to pull planning from the ether.
At #RayMaher day, http://bit.ly/OuI2Y (Mathletics) is demonstrated. Looks great – Education City rival – Athena have paid for 2 years
#RayMaher endorses Mathletics
"If there's a problem, there's a solution" says #RayMaher about to introduce section on problem solving…
#RayMaher says put maths into context – 'Deal or No Deal' with special offers from supermarkets
#RayMaher Shows problem solving frames – RUCSAC (Read, Understand, Choose an operation, Solve, Answer, Check)
#RayMaher Problem with RUCSAC – no estimation
#RayMaher 6 schools meet #Athena My tweetdeck is keeping me fully informed with @frogphilp chirps. Sounds great. Chirp…
#RayMaher Other Problem solving Frame – RACECAR (Like RUCSAC but with estimation)
#RayMaher Problem Solving – Use maths in a real context – real money, use roleplay, prices, 5-a-day, special offers, price vs mass, data.
#RayMaher – lesson idea work out how much Cola is replaced by ice at Macdonalds (Tip: answer = about 67%)
#RayMaher shows free game to show supermarket maths http://bit.ly/bTKk8H
#RayMaher shows Number Crunch Bunch http://bit.ly/cHgOzt (£25) to encourage dialogue
#RayMaher – Could use Flips to film children speaking about conceptions / misconceptions to promote class discussion
#RayMaher If there are 196 legs and 126 eyes at a dog show, how many people and dogs are present?
#RayMaher shows Maths Talk TV http://bit.ly/bqfsA9 (£60) (NB Teacher.tv has stuff on dialogue http://bit.ly/ckYhd9)
#RayMaher shows beginning of Charlie and the Chocolate Factory and asks: What maths can you see?
#RayMaher Logic and Reasoning is possible area of weakness in UK primary schools
#RayMaher shows logic problem: http://bit.ly/bTzAXT – Can you solve it?
#RayMaher shows progression for making decisions – build up complexity from simple decisions
#RayMaher points out you can get free education resources about London Olympics from http://bit.ly/bCLD21
#RayMaher shows Professor Problemo – interactive software and resource sheets (£25) http://bit.ly/9atnwC
#RayMaher recommends free resource: Mathematical Challenges for able pupils http://bit.ly/apfZxn #gtchat #mathchat
#RayMaher recommends Gecko Maths http://bit.ly/aIFBqj Korean problem solving that has contributed to SK being number 1 in maths

Does ‘mean’ mean ‘mean’, ‘mean’ or are you being mean?

Ambiguities

In researching for my assignment (about whether social media can enhance maths learning), I've come across lots of interesting stuff about language. I'm bemused with myself that I'm saddened at the fact that the mere 2500 words I've got to write my assignment won't be enough to include many of the interesting things that I've found out. Interesting questions more than anything – because it's a whole world of learning. But it's also another essay to be written at a different time.

Suffice to say, at this point: maths teachers – you are amazing.

Why are we amazing?

Because we can explain to our children when leaves means equals.

And when vulgar isn't rude.

We know that multiply can make things smaller; that even isn't always smooth; divide can mean share and group; differentiation is more than just good practice for including all children in learning.

And we can explain that mean means mean (as in average), as opposed to nasty (although it might feel like it), or intend.

Maths. It's all about the words, don't you know.