10 squares and 1 squares

Earlier today @tomhenzley tweeted about models for teaching addition and subtraction of decimals. I've often found the 10 square and 1 square to be a useful addition to other strategies. In many classes, children are very used to the 100 square to add and subtract 2-digit numbers. It is a natural progression to use the 10 square and the 1 square to move on to decimals.

In many ways the 1 square is easer to pick up because it represents numbers in the way that children are used to see them with money. You could even call it the £ square (or the $ dollar square, or lots of other denominations of currency for that matter). The 10 square is a little bit more difficult to link to real life, although if you produced one with two decimal places you could put it into the context of timing a 100m race.

It is important to remember that number squares are limited as a image for children – number lines are so much more flexible because they are extendable and easier to produce freehand – which is important for jottings – that all-important stage between mental methods and standard algorithms.

If you want to try it for yourself, have a look at my Google Spreadsheet of the 10 square and 1 square.

10_square

10 square.pdf
Download this file

1 square.pdf
Download this file

Notes on maths training

Meter_stick_icon

Alistair Smith, in his book "Accelerated Learning in Primary Schools" told me that one of the secrets to learning new things is re-categorisation. That's the reason that I'm writing this post. It's also the reason I failed as an Electrical Engineer (I took notes in the lectures, but didn't look at them or do anything with them soon enough to properly learn the ideas). You see I was at this maths training day yesterday and I took all my notes on Twitter. So all of the information is out there in the ether, but it's not going to get any deeper into my brain unless I re-categorise it. This post is the first step in the re-categorisation process. There's another step to come when I refine my thoughts and write some focussed posts. Tweets in bold, extra thoughts in normal writing:

  1. Am off to #mastHEI4 armed with a metre stick. Getting rather dubious looks already on the train – Yes. 100 primary maths teachers all walking through Birmingham City Centre. Slightly reminiscent of some kind of re-inactment society meet, except with maths teachers. Still it showed the dedication to the task in hand – you've got be into the subject to be willing to face that kind of embarrassment.
  2. Mathematical talk by Phil Butcher at #mastHEI4 effective classroms include high quality dialogue. Dialogue = AfL – (explanation: AfL stands for Assessment for Learning) that's really important. As a senior manager in a school, I'm really concerned that my teachers have the right skills and opportunities to spend their time as effectively as possible for the benefit of the children, without spending all their time doing it. It's why I'm so disappointed by the recent APP framework. (APP stands for Assessing Pupil Progress). The APP focuses on the correct skills to raise standards, but it is paper-based and cumbersome. It takes too much time. And what's worse some schools have begun using it as a summative assessment tool, when it was designed as a tool for increasing teacher subject knowledge. Teachers with high subject knowledge have the confidence to assess their children with dialogue, to implement interventions on a minute by minute basis and not to waste their time on meaningless paperwork. They assess as children learn. Children learn whilst they assess. We should be concentrating our efforts on developing our dialogue skills and increasing our subject knowledge. The two go hand in hand.
  3. RT @frogphilp: Mathematical talk by Phil Butcher at #mastHEI4 effective classroms include high quality dialogue. Dialogue = AfL – thanks to @TLTP for re-tweeting this line.
  4. #mastHEI4 misunderstandings shape dialogue. – Skilled teachers know the progression needed to develop concepts and can pick up what the misunderstandings of the children are through dialogue. It means we need to encourage a culture where misunderstandings are seen as good – a step towards success. The problem about doing this for students is that this culture is not in place in any other part of the society: teachers can't be seen to fail by their senior managers; senior managers can't be seen to fail by their governors or local authority; local authorities can't be seen to fail by government. It takes a very brave teacher or indeed headteacher to break this cycle and create a culture where misunderstandings are valued and where process is more important than product.
  5. Teachers.tv problem solving in maths – children on their knees with real equipment talking to each other #mastHEI4 – I haven't found the link on teachers.tv yet for the lesson that Phil Butcher showed. I will do before my summary most.
  6. I'm taking my notes at #mastHEI4 by Twitter. I will collect onto a blog later: http://philpmaths.posterous.comthat's what I'm doing now. Although this is only an intermediary post.
  7. Teachers.tv video shows children struggling to have confidence put data into tables. Is this a skills problem or attitude? #mastHEI4 – I tweeted this because I have a theory that social connectivism is a better pedagogical approach that constructivism for inculcating positive attitudes to learning in maths.
  8. Effective dialogue: teacher acting as chair of discussions, encouraging all to contribute + scaffolder of ideas. #mastHEI4 – It's really good to see the different kinds of dialogue that teachers engage with – chairing, encouraging, scaffolding.
  9. Teacher uses dialogue to guide children to an elegant solution. Remember maths is elegant. It is graceful and swift. #mastHEI4 – despite declaring earlier that process is more important than product (it is!), product is still important. I think of them in a kind of 60/40 ratio. The product in maths should be an elegant solution – it's important to remember that maths IS the most elegant and efficient way of describing the world. And that is why maths is graceful and swift. The teacher in the video didn't just stop at a tabulated answer with an oral explanation from the students. He guided them to a more elegant solution. I love that.
  10. Following elegant solution, teacher encourages children to explain. This helps embed their learning. Learning becomes development. #mastHEI4 – One of the things that Vygotsky demonstrated was that learning is different from development. Students can forget stuff they have learned, but they can't undevelop. Good teachers hand the learning over to the children, letting them verbalise and picture it in different ways; letting them re-categorise it in their minds to give the students every chance of making that learning become development.
  11. Recording dialogue provided evidence for MA1. (note to all APP users!) #mastHEI4 – If you are doing the paper-based APP approach, then MA1, which is maths attainment target 1: using and applying maths, must be a nightmare. So much of how children have really developed in using and applying is evidenced orally. Get your video cameras or audio recorders out – it's the only way to record this sort of stuff.
  12. Teacher dialogue strategies in maths: eavesdropping, chairing, prompting, rich questions (see Bloom's taxonomy) #mastHEI4
  13. Why is dialogue an important part of learning?validation, identify misconceptions, time for rehearsal (thinking time). #mastHEI4
  14. Software to support dialogue in unpicking misconceptions: 'the number crunch bunch' tool for stimulating dialogue – Fiery ideas #mastHEI4
  15. Vygotsky says that talk can take people beyond the 'edge of their thinking' ZPD #mastHEI4
  16. RT @frogphilp: Vygotsky says that talk can take people beyond the 'edge of their thinking' ZPD #mastHEI4 – These last five tweets have been referred to earlier.
  17. IRE (Initiation Response Evaluation) many teachers are locked into this – worried that too much talk = poor classroom management #mastHEI4 – Possibly from the pressure of school culture, it was interesting to hear Phil Butcher (the lecturer) talk about how many teachers don't give the children the opportunity to talk in the classroom. Ask yourself the question: 'what does a good classroom sound like?'
  18. Phil Butcher's experience – takes at least half a term to develop good talk in the classroom. Helps if kids develop the rules. #mastHEI4 – Some practical experience. If children haven't been practised at doing constructive talk, you won't be able to change their habits in a couple of days. It took Phil a half term. Sobering and yet somehow encouraging at the same time.
  19. Alexander: 'written work tends to be seen as the only 'real' work and oral activity is the prelude to written work.' #mastHEI4 – I think this comes back to what kind of evidence we collect as senior managers. If we only ever judge students on what we see in books, then we are not encouraging teachers to take the shackles off on talk.
  20. Longitudinal study (Fiona Walls 2007) shows children drawing their maths lessons over time. Depressing to see boredom as get older #mastHEI4 – Fiona Walls had done a study following students through school and tracking their attitudes to maths. It was characterised by younger children drawing themselves in maths lessons having fun, playing with bricks and then building things. As they got older they draw themselves sitting at desks and wrote how boring maths was. That's just sobering without any of the encouragement.
  21. Games encourage talk. Snakes and Ladders is not a maths game because it relies completely on chance. #mastHEI4
  22. Games make maths learning purposeful. Puts pressure on children to work mentally. Can create discussion of all kinds. #mastHEI4 – some comments on games playing in maths lessons.
  23. Bruner: maths representions Enactive – Iconic – Abstract. Don't throw away equipment in Y6 – kids still need it to affirm learning #mastHEI4 – nice to hear Phil Butcher getting Vygostky and Bruner in the same lecture.
  24. Techniques to come away from think-pair-share: expert groups, snowballing, envoys. #mastHEI4 – Phil discussed some techniques from coming away from the standard think – pair – share that many teachers use. I'll describe all these more in a later post.
  25. Ideas for dialogue: 'What maths can you see in the picture?' use snowballing. (Note to self: must share @tombarrett maths maps) #mastHEI4 – after the lecture I asked Phil Butcher if I could demonstrate one of the maths maps that @tombarrett had shared at #GTAUK. I had particularly wanted to show the 55 shape activites in Paris. But shock! There was no internet connection. An educational event in the 21st Century without an internet connection? Is that possible?
  26. #mastHEI4 ideas for pictures: golden ratio, square numbers, shapes, etc.
  27. #mastHEI4 try googling '10 ideas for energising classroom discussions' – I did and I got to this: http://web.grcc.edu/CTL/faculty%20resources/ten_techniques_for_energizing.htm It's pretty good actually.
  28. Cambridge review: many children sit in groups but work individually. Why? #mastHEI4 – I'm really passionate that children should sit in positions appropriate for them. Practically it can be a pain to move desks around, but the benefits can be worth it. When I do extended writing sessions I encourage children to sit, stand or lie as the mood takes them. I don't think desks should be a barrier to learning for children. It's interesting that the Cambridge Review noticed the same thing.
  29. What about vocabulary: In maths what is the different between technical and specialist vocabulary? #mastHEI4
  30. #mastHEI4 Answer: technical words are specific to maths (eg triangle); specialist words are general words used in a maths context (eg table)
  31. #mastHEI4 more ideas for talk: talking tins, start with end – children design problem, use huge variety in language for subtract and divide – later on in the day I used this information to tweet @tucksoon about talking tins.
  32. #mastHEI4 final thought from Phil Butcher: be precise. If we can't define prone numbers accuratly how can we expect children to? – Ah! At last a mis-tweet! I meant prime numbers. Now it makes more sense doesn't it. Prone numbers indeed.
  33. RT @frogphilp: Cambridge review: many children sit in groups but work individually. Why? #mastHEI4 – thanks to @dan_bowen for re-tweeting this.
  34. Problem-Cube lowered into water half submerged with one vertex at lowest point. What shape does cross-section make at water level? #mastHEI4
  35. #mastHEI4 what did I FEEL about answer? Theoretical but nervous that I couldn't see it and do it. Other responses…?
  36. Possible responses to problems include: I don't care, what did you get, not enough information, j just knew it #mastHEI4
  37. Insurance salesman question is a cracker. I will share on my blog later. #mastHEI4
  38. Challenge: with difficult problems how do we stop being smug and give the answer away? How do w stop children being unstuck? #mastHEI4
  39. At #mastHEI4, Brian Robinson talks about 'Big questions… Small steps' – NRICh masterclass – what can you see? – The problems that Brian Robinson had been sharing in the second lecture of the day were so exciting that I didn't actually share the title of his talk until now (the first tweet was number 35). Brian was advocating a problem solving based approach to all maths. Start with an interesting problem. Decide the skills needed to solve it. Teach the skills, then solve the problem. The small steps are the guidance the teacher gives to the students to take them through the problem – to scaffold their thinking. It's very much a 'Brunerian' approach, although it seems to me that much of Bruner's model has been dismissed practically because of the failure of 'discovery' teaching in the 1970s, my personal feeling is that was down to poor execution of the model, rather than the model itself being wrong.
  40. Ideas for Say What you see: target words, talk ping pong, use shapes. Also: http://nrich.maths.org #mastHEI4
  41. RT @KnikiDavies @frogphilp have you noticed… have you tried… why not try… #mastHEI4 (to add to other tweets!) – thanks to @KnikiDavies for your interest in my tweets. I hope this blog is a useful first step in explaining what I was up to on Saturday. I will blog about this particular lecture in more detail at a later date.
  42. Idea: use SATs questions to ask more interesting questions. E.g table – data handling – ask who gets most pocket money #mastHEI4
  43. Brian says we need to encourage children to feel good about being stuck. #mastHEI4 – again this theme about being stuck, not fearing failure, encouraging children to demonstrate their misunderstandings so we can work on them. It's all part of a good maths education. It's all part of good education.
  44. When children are stuck – don't just sit there: mime it; talk it; model it; draw it; act it… Do something! #mastHEI4 – Brian finished by talking about some practical things to do when you are stuck. This was really useful. It's really important to recognise the full range of feelings that are evoked by being stuck so that when it happens some children aren't excluded from the next steps.
  45. Mary McAteer speaks about approaches to Masters level assignment at #mastHEI4 – May not tweet about this one too much
  46. Assigment tips from #mastHEI4 – grid evidence against learning outcomes, think audience, understand 'critically', write analytic narrative.
  47. Assignment tips at #mastHEI4 – fluent English, keep focus tight, evidence claims, be questioning, conclusions be properly supported
  48. Assignment tips from #mastHEI4 – look at grid that examiner uses to assess you, don't 'go large', articulate the intuitive
  49. What is 'critical'? Evidence of self-awareness, discussion, descriptivity + #mastHEI4
  50. Assignments: is mine like chips and custard or like Blackpool? #mastHEI4 – OK – this section was really just about writing assignments. I probably won't blog about it.
  51. Graham Smart talks at #mastHEI4 about ratio and proportion – Graham Smart finished the day with a highly practical session on ratio and proportion. I will probably blog about it after I've taught it as a lesson and won't say anything further now.
  52. Ratio and proportion are hard concepts. Hard to separate, then link to division, fractions, decimals and percentages. #mastHEI4
  53. Proportion = in every; Ratio = for every. Ratio breaks with pattern of fraction – decimal – percentage – proportion. #mastHEI4
  54. Ratio question: "3 times round my head = my height: is this true?" #mastHEI4
  55. @tucksoon that's the one. It's been recommended to me for developing math talk in 3-11 year olds. #mastHEI4
  56. Graham Smart teaches about unitary ratios. Height to head ratio = 1:0.33 #mastHEI4
  57. RT @frogphilp Graham Smart teaches about unitary ratios. Height to head ratio = 1:0.33 #mastHEI4 or 3:1
  58. The Giant of Biblical Proportions talked about by Graham Smart (a way of engaging children with ratio) #mastHEI4
  59. Graham sums up by giving ideas for FDPRP: filling up petrol, mixing paint, number lines, recipes, stories. #mastHEI4
  60. FDPRP are really the same thing. Model on number lines. Line of people #mastHEI4
  61. people pie chart at #mastHEI4
  62. Smart quotes Mike Askew who said the best teachers of ratio and proportion are those who are good at making links. #mastHEI4
  63. Mary McAteer concludes #mastHEI4 by quoting from 'Dead Poet's Society': "why are we standing here? To see things differently."
And that was the end of my tweeting. I then walked to Birmingham New Street carrying my metre stick. Today I used Twapperkeeper to collect my tweets, sent them to an Excel spreadsheet (note to Tapperkeeper – it would be great if you could output straight to Google Spreadsheets) and put them into this e-mail to posterous. Do you think Tweeting lecture notes is an effective way of learning something? In my next maths post I will go into more depth on the whole dialogue in maths topic.

Unconference my planning

planning3.mp4
Watch on Posterous

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

 

Scattered and Superficial Thinker

A few days ago I finally turned to some academic work that I had been putting off for a while. I turned off all my distractions – Tweetdeck, Googlemail, my phone, the tv. Then I sat in a quiet room and did the work using only a PDF of the arthcle I was studying and notepad on my laptop.

 

I had spent the week leading up to that being in charge of childcare, but nevertheless had grabbed a few minutes here and there to get some work done – planning, preparation, admin and the like. I had also held some really interesting conversations on Twitter, read some interesting blogs and responded to the odd e-mail. You may be wondering exactly how I care for my children, but it’s amazing what you can fo with CBeebies on in the room…

 

Somehow I’d never felt able to focus on the academic stuff with the kids about, and when I came to the study itself, I had also felt the necessity of turning off the online distractions.

 

I hadn’t thought conciously about that decision until today when I read a really good article in the Telegraph called ‘How the Internet is making us stupid’ by Nicholas Carr.

 

He has pulled together various bits of research that show how all the distractions we engage reduce the depth at which we think. We are becoming shallow thinkers.

 

He writes things like: ‘people who juggle many tasks are often less creative and less productive than those who do one thing at a time.’

 

And: ‘People who read text studded with links, the studies show, comprehend less than those who read words printed on pages.’

 

He also quotes developmental psychologist, Patricia Greenfeld who says that while ‘every medium develops some cognitive skills at the expense of others’ there are ‘new weaknesses in higher-order cognitive processes.’

 

And Roman philosopher Seneca who said: ‘To be everywhere is to be nowhere.’

 

He goes on to quote neuroscientist Michael Merzenich who said that as our brains adapt to this shallow way of thinking, ‘the long term effect on the quality of our intellectual lives could be deadly.’

 

Now I’m not to sure about that. I think we need to be able adapt to different ways of thinking for different purposes, which is what I found the other evening when I successfully engaged in some study. But I do agree with him when he says that ‘skimming is becoming our dominant mode of thought’. I’ve been guilty of spending too long in skimming mode recently and that whole way of thinking has stopped me from even being ready to attempt any academic study.

 

My conclusion

 

I must be determined not to let ‘skimming’ be my default mode and schedule myself time to engage in different types of thinking.

 

Do you agree with Carr’s article? Have you read any research that indicates the positive impacts on thinking of using social media?

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!

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