Unconference my planning

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The danger of following a spiral curriculum (a la Bruner) is that if you always follow the same path, you hit the same bits of learning at the same point on the spiral. Sometimes that means hitting difficult concepts at the end of a term when everyone is tired.


At GTA UK this year I came across the idea of an ‘unconference’ for the first time. This is where you turn up without a specific agenda, but generate it on the day by the people who are there. Google Docs are an ideal tool for this as many people can collaborate in the same online space at the same time. I decided to do an unconference with my Year 6 maths group to generate the plans for the term. I had no pre-conceived idea of how this might work out except for this:


1. We would start with a wallwisher to discuss ‘What is Maths


2. We would use a Google spreadsheet to think about:
  • what we are good at;
  • what we are not so good at;
  • what we would like to learn this year.
As might be predicted, ‘division‘ came up as the concept that most children would like to learn.


So the next day, the first lesson was on division


Bloomfield’s Theorisers

Pavlov began it, thinking he could explain it with dogs.

Thorndike and Skinner experimented further, but it was lost in the Law of Effect and thousands perished in drill and practice.

Then Dewey found it and held it for the desires and motivations of all, while the Gestalt, on the edge of things, encouraged insight and a view of the whole.

Piaget discovered how it worked, but separated it from its core and it was almost lost again.

But then Bruner rescued it and described how it could work, whilst Wood built a tower for it. Then Vygotsky, King of the Tower, opened up the tower for the people to talk and communicate and interact with it. Yet this, his greatest feat, was overlooked by another, the Zone of Proximal Development, which whilst instilled with truth was a distraction from the biggest triumph. And men came and made the most of this distraction, like Von Glasserfeld with his love of the subjective and the internal.

And some, yea, even Bloomfield, were overcome with this distraction and did comment slightly sceptically on the power of social interaction, with words like 'construed' and 'apparent'.

But then Cobb came forward, and Ernest, adding social knowledge to his three worlds, and finally Jaworski with his understanding of story and negotiation

And thus it was that Bloomfield laid aside his slight scepticism and came to declare that knowledge is socially constructed between groups who share meanings.

And so it was that one day all people would understand that knowledge exists neither externally to the individual nor internally; but on the tender wisps of the webs that lie between individuals; on the cusp between the external and the internal; on the expectations and obligations that turn individuals into people.

Mathematical graphics or play? Does it matter?

The focus of HEI day 3 at Edge Hill University was on the Early Years. Although the theme of the morning was developing children’s mathematics, much of the talk was about getting ‘play’ right. The implied assumption then is that if you get play right, children naturally develop mathematical graphics correctly… By the end of the day I had worked out that we had been treated to two of the top experts on Early Years education in the UK and possibly beyond. Much of their practice has informed recent government policy. The two in question were Maulfry Worthington and Elizabeth Curruthers. They have an website that explains much of their work called the Children’s Matehematics Network. Here’s my tweeted journey through their lectures, with a spot of explanation.
Tweet 1: ‘Early years experience should build on what children should know and can do’ #masthei3. What you be the connectivist equivalent?

Aside from the fact that this comment reveals I can’t spell when I’m typing fast, it struck me again that all current ‘best practice’ is based on constructivism. For those who aren’t sure, or who have forgotten what it is, it is pretty much summed up in the above statement. Constructivists would say that all education should be based on what children should know and can do. The seminal writers on constructivism were first Piaget and then Vygotsky, I’ve blogged about them recently when challenged to keep a diary of my daily teaching experiences. You can read those blogs here:

Constructivism has always been rivalled by behaviourism (and a little by cognitivism – but I am really unsure of what that one is about), but there is a new theory on the block – connectivism a concept defined by George Siemens in 2005. It is summed up for me by the statement ‘the pipe is more important than the contents of the pipe’ – it’s all about how we connect with each other as sources of knowledge, skills and attitudes – getting the connections right is more important than actually having the knowledge in your head.

So in my round about way I’m coming back to the question I asked – what would be equivalent statement about early years experience based upon a connectivist view point? ‘Early years experience should build on who the children know and how they relate to each other’? It’s a possibility. The thing is I’m not entirely a connectivist – I believe connectivism is the best theory to be applied for gaining knowledge, but I think constructivism is more appropriate for skills and behaviourism is the best way of describing how people pick up new attitudes. Mainly. I think. But that’s the pub theorist in me again.

Tweet 2: Maulfry at #masthei3 says there is no place for worksheets in foundation stage or KS1. Contentious? Wise? True also for KS2?

Maulfry (which, incidentally is the best name I’ve heard in a long time) Worthington came up with this statement at the end of her talk on mathematical graphics. It resulted from her explanation that chat children need to learn to represent maths in their own way first, developing from making sense of their own play, before being taught over-prescriptive ways of recording maths. She did go on to say that direct teaching of skills was important, but child-initiated play is to be a significant part of the day for early years children with teachers acting as gentle guides.

Tweet 3: Elizabeth Curruthers at #masthei3 says that her children’s centre has a postmodernist perspective. Prof. G. Lynch says there’s no such thing

Bizarrely, as I write this I’m going to stay with the inestimable Professer Lynch at his small family home in London. Something of an expert on culture, he once told me that post-modernism was a myth. I have no up to date knowledge myself on this front, but I’m sure he will educate me in a few hours time.
Tweet 4: Warnings about superficial play at #masthei3. Not just a prescribed roleplay area but plenty of materials and resources for free-choice.

Elizabeth’s lecture was more about the pedagogy of play, not so much about the theory. This was good for us teachers because we like to see how all these high-falutin educational theories are put into practice. She warned us not to limit play to a roleplay area, but not make it broader and messier. Messier because it requires putting the resources out and clearing them away again. In other words play that’s worth it is hard work (for the teachers).

Tweet 5: Support early maths development by timetabling big chunks of time for play with teachers supporting and guiding.

She actually specified an hour and a half in the morning and more again in the afternoon. That’s a lot of play.

Tweet 6: #masthei3 other good tips: real equipment (spirit level) tech for food (mixers), food room, growing number lines, numbers on trikes

This speaks for itself – make the play real (not plastic), use real things that the children would see at home, do stuff with things the children already know a lot about (like food) and develop it – don’t just stop your numberline at 10, give it space to grow over a few weeks. Number the trikes like racing cars and make the numbers get bigger as the year goes on ans the children get more confident with number. It’s the constructivism principle working out in practice.

Tweet 7: Pedagogy is to build on children’s interests at #masthei3. Could be one child, a group or sustained themes I’ve a period of weeks (should have said ‘over’)

Elizabeth gave an example of a non-English speaking family who ran a restaurant in the city they were in. On a home visit, the teacher was amused by the child treating her like a paying customer, waiting on her and writing down a series of marks and shapes that (to the child) indicated the order that she was going to take. Back in the nursery, the teacher gave that child a chance to carry out that roleplay further and involved other children in the play, so that one child’s graphics soon started influencing others.

Tweet 8: Adults who really listen to children and their play, then co-construct learning are Insiders vs edgers vs outsiders

Elizabeth defined three positions an adult can take to play – as an ‘insider’, an ‘outsider’ or someone on the edge of play. It seems obvious by now what position an adult should ideally take to make play worthwile – that of ‘insider’. I can’t remember much about the distinction between being an outsider and someone on the edge, because I was thinking about my own children and how important it is for me to be an insider in their play. So today we played with the Brio together for the first time in ages and I listened to them as they made up stories about the different trains going round the track. Then we proceeded to make high towers, only for the youngest to knock them down, but it was still fun. For me too!

Tweet 9: Adults interactions: scaffolding, sustained shared thinking, co-construction, participant observer. (bruner, blatchford, jordan)

Elizabeth stated some different ways that adults as insiders have been defined. She talked about the co-constructor being the most powerful. She went on to quote from someone else in tweet 10: ‘Acting and thinking with others drives learning and at the heart of the process is dialogue’ (Stephen 2010).
Tweet 11: Conversation thoughts at #masthei3. Talking more important than reading or listening for developing maths. ‘Talk time’ is a strategy to use.

Elizabeth defined what a good conversation looks like and played us a video of her nursery angaging in ‘talk time’ a freeflow activity where everybody starts together with some artifacts and stuff to talk about. Children can wander off or back as they pleas, but the teacher is there to guide and be a part of the conversations. The children know that that particular time and space is for talking. I’ve been wonbdering how I can structure that into my Year 6 class, but I’m not quite sure how to do it.
Tweet 12: Discussion at #masthei3. How do we encourage child-initiated learning beyond the early years?

One of the things that was becoming obvious to me was that adults were vital for child-initiated learning because they have to guide it. However as children get older, generally staffing levels are reduced, so there is often less space for child-initiated learning. I personally believe this is where technology comes into it’s own. If you look at the work of @deputymitchell, @oliverquinlan or even my own work with my CATsEYES film-makers (to name but a few), then it is clear that child-initiated learning can still exist for older children through the use of technology. Not that any of this is maths based per se. Not yet.
Tweet 13: Maulfry says she has never seen any evidence that worksheets help develop mathematical thinking.

OK I seem to be repeating old-ground here – but it is quite a contentious statement. Worksheets are such a time-saving device for teachers that is it really fair to say there is no place for them? Ideally I would like to see my school without them, but practically… It might take a few years.
Retweet 13: @timstirrup: who agrees? RT @frogphilp: Maulfry says she has never seen any evidence that worksheets help develop mathematical thinking.

Tim Stirrup retweeted this onto #mathchat and into his own network on Twitter. I’m yet to see any response from this, but it would be interesting to see it come up on the #mathchat forum sometime.
Tweet 14: Maulfry talks about language for thinking and language for communication at #masthei3. Reminds me of @ewanmcintosh’s talk to CCE back in Feb

Ewan McIntosh at the CCE conference in February had talked about a new language that children had to learn – the language of technology. He explained how it overlapped with the language for thinking and the language for communication. Myself, I’ve not really though too much about these areas, but just recognising that there is more than one seems important.
Tweet 15: Maulfry says that recording maths is not the emphasis at #masthei3. Mental methods and thinking are mor important

This backs up some of the earlier points. Just learning how to record maths without thinking is not successful.
Tweet 16: #masthei3 only 36% attain FSP point 8 in mathematical developing. Do some children ever get it?

This made me think back to the previous days lecture by Nick Dowrick when he said that 6% of children do not achieve a level 3 at KS2 and 40% of children do not get a C grade at GCSE. FSP point 8 reads ‘The child solves or attempts to solve problems and challenges by applying mathematical ideas and methods. The child explores problems such as missing numbers, grouping, sharing and estimation, and responds to questions such as ‘What could we try next?’ or ‘How shall we do it?’’. Do some children ever get this? Do all adults have it? I’m not sure.

Tweet 17: @timstirrup: @frogphilp the day sounds interesting, but where/what is #masthei3? I have tried many google searches with no luck.

I’m really gald that Tim stirrup was getting interested enogh in my tweets to ask where I was coming from. This programme is part of a two years Masters level study course that, if passed, should gain the participants 60 points towards a Masters degree. The programme came out of the Williams Review of maths in Primary Schools, which is a good read, I reckon. There are 10 HEI days with lectures given at Edge Hill University and the day I’m writing about was the third of them.

Tweet 18: I need a subject and a title to do with primary maths for my master level assignment (2500 words). Any ideas? #mathchat

It suddenly struck me that there’s a whole network out there who could give me some good ideas for an assignment – thanks if you’ve already made a suggestion – any more suggestions gratefully received.

Tweet 19: #masthei3 is over. On way back home with 2 Birmingham maths consultants. Next stop #gtauk…

Just a word for my consultants, Muriel and Ian -they have been marvellous. Not only did they gave me a lift back to Birmingham from Ormskirk, but they have also fully engaged with the course, had a go at some of the modules and been really positive about it. I’m sure everyone in the Birmingham group would agree they’ve done a cracking job.

Mathematical masters study for grumpy teachers

Perhaps spending the first days of your summer holiday at a 2 day conference on maths is not the perfect thing for many teachers. I have certainly heard some teachers expressing opinions other than perfect happiness today. In fact you could say that some of them are downright grumpy. I’ll proceed to explain why towards the end of the post, but first for the important bit – my learning.

I’ve experimented a little today. as I have attended lectures, rather than taking notes I have tweeted what I think have been the key points under the hashtag #mastHEI2 (that stands for Mathematical Specialist teacher Programme Higher Education Input Day 2). Now I’m going to go back over those tweets and see if I can explain the learning.

Tweet1: Ian Sugarman lectures on subitising. Introduces Mayan numbers.

In Ian’s lecture he introduced the concept of ‘subitising’, which is defined as: ‘Instantly recognizing the number of objects in a small group, without counting,‘ according to mathsisfun.com It is important because it is the step between counting and recalling number facts that leads to really confident calculation skills. He showed us how Mayan numbers use this concept by having up to 4 dots in their number system (apparently it’s hard to subitise more than 5). Mayan numbers are logically – a bit like Roman Numerals or indeed the Arabic number shapes that founded our own number signs: 1, 2, 3, 4, etc.

It’s useful when you see groups of things. Take a group of 7 dots for example. Do you actually count each dot? Children do, when they are first starting out, but then they learn to do something else. At least most of them do. Some children seem to miss that bus and need a chance to catch it again.

Tweet 2: Ian Sugarman says we are aiming for automacity of recall at #mastHEI2

Of course what I meant to say was ‘automaticity’ but my fingers didn’t quite go fast enough. so we don’t just teach counting. We teach subitising too. Then children have a chance to have really quick recall of number facts – they can just see them in their minds eye, and eventually even turn that visual concept into an abstract one. Ian has written some software called ‘Numbergym‘ that helps develop this concept.

Tweet 3: Andy Tomkins talks about search strategies at #mastHEI2.

A little bit dry for me and stuff that I already knew, this short lecture covered ground such as Boolean operators and the * for making searches. He went into detail around academic searches for online journals and the like. However the journal that he experimented on, I subsequently found on Google Scholar a few moments later, so my question is – do universities still hold information that you can’t get hold off anywhere else? Or is everything accesible via the web? That is why I tweeted this: Tweet 4: excited about amount of academic info available through Edge Hill’s online databases – but are they rivalled by Google scholar?

Tweet 5: Sharon Pieroni speaks about Harvard Referencing at #mastHEI2 Got to get this right – don’t want to be accused of plagiarism..

So in a few months I have to write an assignment. I haven’t done that since 1996. This was a short lecture that reminded us of the basics of how to reference something and then include that reference in your bibliography.

Tweet 6: Nick Dowrick speaks about Every Child Counts at #mastHEI2 6% of KS2 children don’t achieve level 2. 40% don’t get C at GCSE. Hence ECC

It seems that some of our children just aren’t getting it in maths. Every Child Counts is a 1:1 intervention over the course of 3 months that tries to help 5, 6 and 7 year olds catch up. It is led by a well-trained specialist teacher and seems to be doing an excellent job. It’s ironic that with the first lecture being about the importance of subitising, this one referred to counting so much. I wonder if there will be an intervention programme called Every Child Subitises? It doesn’t have quite the same ring to it.

Tweet 7: Numbers count programme focuses on lowest attaining children rather than targeting the children who are just behind.

‘Numbers count’ and ‘Every Child Counts’ are both terms that seem to be used interchangeably by the lecturer, Nick Dowrick, although there is probably some subtle difference that I didn’t quite get.

Tweet 8: Numbers count programme focuses on lowest attaining children rather than targeting the children who are just behind.

I felt this was important. So many of our interventions and foci in school are on those children that are just behind the average. If we could get them to achieve, then we would be making a significant difference to our performance figures with minimum effort. It was refreshing to hear someone talking about a meaningful intervention for the lowest achievers, and one that actually works too.

Tweet 9: Nick says ‘the floor is a natural place to do mathematics’ referring to the fact that many of our lowest achievers are kinesthetic learners and just need to get on with doing maths in their space and at their level. This leads on to Tweet 8: ECC isn’t filling the gaps in the wall, it’s knocking down the wall, re-laying the foundations and rebuilding the wall fr scratch I’ve met so many older children in primary schools who have holes in their conceptual understanding and they’re desperately trying to plug them or they have just given up. We need to recognise that for some children we need to start some of their concepts again from scratch. An example is my friend (whom I won’t name) who at the age of 24 was getting many subtractions wrong. For example trying to work out what he was doing 6 years ago, he would go 24-6 and then get 19. He did that because he counted the 24 when he counted backwards. I taught him to count the jumps (the gaps) not the numbers themselves and he got it. At what age should he have learnt this concept?

Tweet 10: ECC gain is over a year in just 3 months

The data from Every Child Counts is very encouraging. In just 3 months each child had made over a year’s progress, however: Tweet 11: ECC problem in Year 3 for lower attaining children who make only 3 months progress in 6 months. Is this consistent for all in Y3? indicates that all is not a complete bed of roses. Those children who had ECC intervention late (at the end of Year 2, when they were 7) slipped back, making only 3 months progress in 6 months when they got to year 3. Why? No-one yet knows. But the lecturer finished by saying that it could be down to an attitude change – first of all changing attitude to one of confidence and positivity by being part of the ECC program, then having to change attitude again when met by barriers of the broader curriculum experienced in the new Key Stage at Year 3. Tweet 12: Nick Dowrick at #mastHEI2 says a successful intervention should be indicated by a complete sea-change in the attitude of a child.

I finished my lecture-tweet extravaganza by attending the course rep meeting. I hadn’t intended to be a course rep, but the person who was doing it went on holiday and asked me to step in. This is where I experienced some of the grumpiness. Tweet 13: Now acting as Birmingham course rep for #mastHEI2. People disgruntled about workload, but hey, that’s Masters study. The thing is people were complaining that the course has demanded too much workload, but I think this is down to different levels of communication between the course, the LAs, the government who set the course up (who are no longer in power), the headteachers and the teachers – lots of different groups. I’ve got no personal frame of reference for how much study a third of a masters should be, and I suspect neither do many other teachers on the course.

After the meeting I bumped into a couple of teachers, who were beyond disgruntlement, or even grumpiness, one could even be said to have been angry, or even mildly furious. He said he had got nothing out of the day of any consequence at all.

“Subitising,” I could have said, “Strategies for engaging lower achievers”, even “Mayan numbers.” But no I just listened. Then he confessed to having got very drunk on the previous night and was worried about making his ferry to France the next day – both factors that may have affected his concentration and enjoyment of the day I suspect.

Unsticking the stuck

“Twitter is too hard for me to use.” That statement stopped me in my tracks earlier today when I was trying to persuade some colleagues about the benefits of networking via Twitter.


I’m on a maths training programme called MAST. It’s a big 2 year academic thing that can contribute to a Masters degree (if you want it too). Myself and the other participants are at different stages of study and some of us are stuck.


Some of us are stuck because the university’s VLE is old and outdated, without the functionality and flexibility of the kind of interfaces we’re used to like facebook.


Some of us are stuck because the material is hard. There are hard words to understand and apply, like ‘didactic’ and ‘quotitive’ (a word I used in my last post).


Some of us are stuck because in the local area network meetings we still haven’t actually formed a network – we come to them, we learn some stuff but then we don’t talk about it in between.


Now it seems to me that Twitter would be a good solution to this. We could have a tag for our group and communicate when we’re stuck to each other, post useful links, help each other with the tricky parts of the course. It might even make the rest of the face-to-face meetings more meaningful.


The problem is that out of the 28 participants and 2 course tutors, only I use Twitter. A small group of them use Facebook and have set up a Facebook group, but say that ‘Twitter would be too hard for them to use.’


That was such an interesting statement and I have to say that 6 months ago I would have said the same thing. Since then however I’ve decided that it is my responsibility to learn about new technologies so that I can help the children I teach and their parents understand them better and use them more safely. On the way I have discovered that they have really helped me plug in to learning networks and be generally more effective.


So how do I persuade the others that Twitter is the thing to do?


Firstly, am I wrong – is there a better solution? For example, setting up a wiki page buckling down and setting up a discussion on the university’s oh-so-clunky VLE? Something else.


Secondly, maybe I should just join the Facebook group and convince people to come to Twitter from there?


Any other solutions? Does anyone know how to create a Twitter epiphany amongst thirty sceptical maths specialists?

Can you keep the diplodocus safe?

Sometimes the river winds through the country of maths. Many educators see maths as being an entirely different country from the rest of ‘educationland’ and this post will emphasise some of those differences.

The thing is, most teachers don’t get maths. Not only do they not get it, but they do not want to get it. That is why the numeracy strategy in the UK has been a moderate success – it provided such a tight framework for teaching maths that teachers didn’t have to get it, they just had to deliver the lessons put in front of them. As Sir Peter Williams put it:

“The United Kingdom is still one of the few advanced nations where it is socially acceptable to profess an inability to cope with mathematics. We need to urgently reverse this trend so every pupil leaves primary school without a fear of maths..”

This attitude if prevalant in the nation must also be prevalant amongst teachers.

So, how many 4s are there in 20?

At what age should a child be able to work this out? It is a quotitive division – you could express ot as 20 ÷ 4 = 5, but it is quite a different question from saying ‘share 20 sweets between 4’. In the latter question, each of the ‘4’ would receive 5 sweets. In the former there are 5 groups of 4 in 20.

It is grouping vs sharing. It is quotitive division vs partitive division.

Children are exposed early to partitive division. They are sharing from reception and before. They share sweets, teddy bears, small plastic dinosaurs and even an 8 chunk bar of chocolate if they’re lucky (incidentally this latter example is actually a fraction problem – but don’t tell the reception teacher that).

They are not on the whole exposed to quotitive division. In fact grouping isn’t really referred to until Year 2.

So here we have 2 equally valid meanings to the word division, with one being taught from reception and one being taught from Year 2. I wonder which one will be better understood? Forgive my sarcasm, but it seems obvious to me that our framework has let us down here. Younger children can group. They could make a group of the chewy sweets and the hard sweets. They can group their teddies by size, colour or even in groups to go off to their teddy bear’s picnic. They can group dinosaurs by how angry they are.

So when it comes to the question ‘how many 4s are there in 20?’ We may at the moment say that we can’t begin to talk about that with children until they are 7. We certainly can’t express it in symbols unto they are 9. But actually, if they understand the numbers, there is no reason why they can’t be asked the question much earlier:

  • The camels cross the desert (sand pit) in herds of 4 at a time. How many herd can you see in the sand pit?
  • The teddy bears only ever have their picnic in groups of 4. Look at those twenty bears… how many picnics do you think there will be.
  • Those diplodocus are only safe from the tyrannosaurus rex when they are in groups of 4. Can you make those 20 safe? Please?

Ah! Division by dinosaurs. You can’t beat it!

Exploring Children’s Attitudes towards Mathematics

The river winds on its course and today has taken an unexpected detour from Scalextric into maths.

I’ve just read a paper by Ben Ashby with the above title and for my MAST study I’m to reflect on what the key factors are that inhibit maths learning within my setting. I think the two main ones are confidence (particularly in girls) and self-belief.

I think Mr Ashby is spot on when he writes that girls ‘frequently attributed success and failure to external factors, such as luck and the perceived difficulty of a question.’ I’m often frustrated that talented bright girls don’t take any credit for their own skills – it’s almost as if it’s not cool for them to do so – they have no role models who are good at maths – no-one to aspire to – so why should they be. If only every up and coming female celeb was as forthright about maths as Carol Vorderman…

I disagree, however, with the the author of the paper when he writes: ‘The reason for this is currently unclear and warrants further research.’ From reading the ALPS book, which draws from a range of well-known brain-based learning research, including Howard Gardner, it is clear that high achieving girls in particular have a problem with their concept of intelligence. They think they can’t learn more past a certain point – that they have reached the limit of their intelligence. I spend much of my time with higher achieving girls teaching them the attitude of resilience rather than discrete knowledge or skills. Not that I’ve cracked it yet…

This is a big problem at our school. So many of our children have convinced themselves that they are no good at maths. Some parents tell them that they were no good at maths either. It’s also not cool to be good at maths.

We have tried some things that have addressed the balance. Maths happens first now each day, so that children can do it when they are most alert. We also use sets in Key Stage 2 so that the range of ability is not so vast as it once was. This helps both the teachers, who have less differentiation to sort out, and the children who can see that everyone in the group suffers from the same amount of struggle.

We have also tried maths classes for parents, but so fare only a small number (10 or so) have taken it up – but I think changing parental attitudes is key.

I’ll be back on Scalextric tomorrow.

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