Residential – the final hours

Saturday evening after a peaceful hour spent reading the paper began with dinner at 6. People were a bit more relaxed about what time they came in and the hotel staff were all set so things went more smoothly. Have to say that the food has been brilliant, and so much of it!

The evening session was run by Paul Andrews again and was all about folding paper. We made equilateral triangles out of sheets of A4 paper and then had to prove that they were equilateral triangles. We then turned them into a sort of truncated pyramid and made different size ones to look at ratios.

This was followed by folding strips of paper into thirds by using convergence and then seeing if we could fold our paper into 5ths and 7ths. I was starting to feel a bit daunted by the whole thing at this point, The proportion session had been difficult and although I could follow the proof of the equilateral triangle, I had no idea how to start it on my own. However the idea of finding fractions by convergence was one that I could follow and I managed to find fifths and then sevenths without too much difficulty which made me feel better.

The session ended earlier than planned as Paul was finding it difficult to talk to so many people at once in a large room and a lot of people were having trouble in hearing him and watching what he was doing.

We ended up in the bar (obviously) where we were easily the liveliest (loudest?) group. It was strange how our network group of 10 seemed to have bonded as a group so much better than the other groups. Most others were in 3’s or 4’s whereas we averaged at about 10 plus or minus a few who came to join us at times. It was a fun evening, verging on the hysterical at times which I think was probably the reaction to all the work during the day.

Sunday morning and our final session before going home.

The final session was on Generalisation. I hadn’t realised before how much maths depends on being able to make generalisations about things. We looked at how our generalisations change as we gain more awareness of how complex things are. We tried different generalisations with the ‘Always, Sometimes, Never true’ activity. I have done this with my class before but will definitely use it more often as a way of getting them to explain their ideas.

For our directed tasks we have to develop opportunities for pupils to generalise and note these in our PLL and read and reflect on the article ‘If you can count to ten, you can count to infinity really’.

After coffee we had the plenary session where Debbie brought the ideas of the whole weekend together. We were also taught a clever way of working out answers to tables over 6x by  using your fingers (or in this case rubber gloves). It was clever and I had never seen it before, Definitely one to teach the children

It was a very intense weekend and I veered between feeling confident and enjoying the input to feeling inadequate and struggling to understand. I think that the overall experience was positive, it was just so much to to process in such a short time.

The main problem that I think that I and many maths teachers face is still how to balance the demands of the school and the curriulum against the desire to teach in a way that will foster real understanding and enjoyment. I do try and always teach for understanding but if the children still do not understand then eventually I have to just teach the rule as they need to be able to do the relevant calculations before they leave me. The children are expected to achieve a certain level and I have to make sure that the right percentage reach those levels to meet my targets. This isn’t always compatible with the things that the course is teaching.

So I have lots of new ideas of things to do in the classroom as well as lots of reading to do. My next meeting is at Nottingham Trent University on Tuesdayand is our final session on Mathematical thinking. Before then I have to read the first chapter of Primary Mathematics, Teaching for Understanding byBarmby et al which I have just ordered from Amazon.

More Residential

After coffee and some very nice danish pastries, we moved to our next session all about pattern. A lot of people had commented on how good this session had been so we were looking forward to it. The session was run by Judy Sayers of Northampton university and Dr Paul Andrews from Cambridge university.

The session aimed to consider the nature of pattern and its existence in the maths curriculum (and beyond) and to think about how we might support children in understanding pattern in maths.

We began by considering what pattern was and were going to compare definitions at the end of the session, Sadly we ran out of time so this didn’t happen.

We then looked at the problem where you have to move 2 sets of frogs from one side of the river to another by jumping or sliding. This was acted out. I was one of the frogs which meant that I relaxed a bit and just did as I was told! We considered the patterns of jumps and slides and what the possibilities were and how you could use pattern to predict them.

We then investigated a range of patterns. In each case we moved from a concrete example to a more abstract rule. We looked at patterns such as triangular numbers and seeing squares inside grids. John was very keen to make the point that it is not enough for children just to see a pattern but that they need to explore the maths behind that pattern as well.

This was an enjoyable session but I didn’t find it as thought provoking as the previous session. I think I probably generally am happy if my children can spot a pattern and explain it and don’t go into higher order maths to prove ideas.

For our directed task for this area, we have to read Pattern Power which is on the nrich site and over a period of a week record all the opportunities I provide to draw the children’s attention to pattern.

After lunch we had our 3rd session on proportionality. By now we had already had 4 hours of maths and our brains were beginning to flag a bit. This wasn’t helped by the fact that proportionality and its related areas of fractions, decimals and percentages are some of the hardest ideas to grasp. Some early years teachers found this session challenging as a lot of it was out of their direct teaching experience.

We looked at the part played by proportionality in the maths framework to see how big an area it is for children. As well as the normal FDPRP, we decided that it also plays a big part in measures where children need to see relationships between measures and in money. I’m sure that there other examples out there.

Multiplication and proportionality are closely linked. Most aspects of proportionality have a multiplication aspect to them.

My head was spinning by this point and although I could do the actual maths, I wasn’t really taking on board the big ideas any more. This is an area that I need to read up on and think about in much greater detail.

For our directed task, we have to do some reading and also to devise a classroom activity involving some element of proportional reasoning.

There should have been a break next but Debbie, the course leader, gave us an extra short session on how to complete our Personal Learning Log which will be  a big part of our assessed work. This did clarify things for us although was quite scary in terms of the volume of work that is expected. She did keep making the point that we should be studying to masters level and therefore need to work at the required depth to achieve that.

5 o clock on Saturday, dinner in an hour and then one more session to go!

Residential part 2

Saturday dawned bright and sunny, well it would be a lovely day wouldn’t it? A very filling breakfast was followed by a brisk walk around Northampton town centre just so I got to go outside at least once during the day. Then it was down to work. We had 4 two hour sessions timetabled with a break between 4.30 and 6.00 so it promised to be a very intensive day. We were divided into groups of approx 20 and did the first 3 sessions on rotation basis. Each session was based on one of the big mathematical ideas, mathematical thinking was Friday eve and we started Sat with a session on Representation.

This was probably the session that got me thinking the most over the weekend. We looked at how all maths is abstract and we need representations to help us to make sense of it. We considered the use of different resources such as number lines and number squares and whether using both could be confusing to children. It was felt that children should be given access to a wide variety of resources and representations to help them make sense of things. A point that struck home strongly was the idea that some teachers and a lot of children feel that they out grow the use of apparatus such as number squares or cubes. Often children who still need the support of apparatus are seen as less able by themselves and others.

The point was made that as the maths gets more complex then children should have more access rather than less to representations and apparatus. KS2 classes in particular often don’t use enough images, representations and apparatus to allow children to fully understand new concepts. Could this be a reason why children find maths harder in KS2 than in KS1? Would they retain their confidence if apparatus were more readily available and more images used?

There should also be a greater emphasis on the value of children’s own representations and jottings. That is certainly an issue at school where some children are reluctant to spoil their nice page with jottings or working out, even asking if they can rub it out! They need to made to see that it is an important part of their work. Maybe this could be helped by doing more jottings and working out myself for them to see as well as getting them to demonstrate their images etc on the IWB.

Lots of ideas about different ways of using representations as well as lots of reading to do around the subject. We also have a directed task to do before our next session on representation. This involves either developing your use of an existing representation such as a number line or arrays and with a colleague developing it further, or to use an alternative resource that represents a mathematical concept and be prepared to explain how you have used it to develop understanding of the concept.

After 2 hours it was definitely time for a cup of coffee!

MaST Residential Part 1

This is the first part of what will probably be several entries about the first ever MaST residential which took place between 19th and 21st February in Northampton.

Everyone was nervous, teachers as well as organisers as no one had ever done this before. Ours was the first residential to take place over the whole country.  However I have to say that I thought the whole weekend went really well so congratulations to all the organisers, tutors and hotel staff for looking after us so well.

The weekend began with registration at 4.30 followed by a period of looking nervously around to see if you recognised anyone from your first network meeting. We were all told that dinner was at 6 o clock but I’m not sure that anyone had told the restaurant staff as they looked slightly startled when we all piled at the same time.

After a good dinner which included sticky toffee pudding, we hesitantly went to begin our first activity. This was a mathematical version of the Crystal Maze. There were a whole range of shape, logic and number puzzles which had to be solved in teams of 6. The puzzles came from all around the world, some were familiar, others totally new. It was fun trying to solve them against the clock and certainly got us all talking.

We followed that by our first actual teaching session which was Mathematical Thinking. We looked at how puzzles and problems can be used to help develop children’s mathematical thinking in a way that is enjoyable. We finished at around 9.30 and then settled in the bar. I was amused to discover the two Leics consultants deep in thought trying to solve the tangram puzzle.

I found it difficult to sleep as my mind was buzzing with lots of ideas going around. I actually replanned my maths lesson for Monday at about midnight and decided to base it on one of the puzzles we had used instead.

Certainly a good start to what promised to be a very challenging weekend.

More Reading

Before our residential tomorrow we were asked to read the following article by Richard Skemp

Relational and instrumental understanding in mathematics

math.coe.uga.edu/Olive/EMAT3500f08/instrumental-relational.pdf

Briefly he discusses how the word ‘understanding’ can have two meanings. Relational understanding is where we understand things in context, the how and why of a thing. Instrumental understanding is where we know a rule for doing something although we may not know why this rule works. This is obviously particularly apparent in maths teaching and there have been several discussions on this particular point recently.

Instrumental maths is often easier in the short term and can lead to success

The rewards are more immediately apparent (pages of ticks) and so can create a feeling of success and confidence.

Relational maths is more adaptable to new tasks and situations.

Once grasped, it is easier to remember as it makes sense as a whole. It can also be effective as a goal in itself and may lead to further exploration and growth for its own sake.

The author was heavily in favour of relational maths teaching but looked at why teachers so often use instrumental teaching. These reasons basically boiled down to curriculum and test demands on teachers and schools.

I am as guilty as anyone of teaching children how to do something because they need to be able to perform a particular type of calculation or operation and either there isn’t time to fully develop their understanding or they haven’t grasped the teaching. I hope that as time goes on, then they will develop the understanding behind the rule. That was certainly true in my own case. I remember being taught decompostion at primary because that was how subtraction was done. A few years later, the penny dropped as to why the method worked.

Curriculum demands are a very valid point. The maths curriculum that ‘must’ be covered seems to allow no time for reflection and development of understanding by pupils. The new framework has tried to address the problems of the NNS by making links between areas and using the ‘little and often’ method. However this just seems to mean having to move on before anything is properly understood as there is so much to cover in each unit.

These are exactly the same problems that came up with the first reading Teaching children to think mathematically. As teachers we have a duty to our schools to get children to the required standard as well as teaching to the best of our ability. Sometimes it seems that it is not possible to do both things at once. Teaching children to think mathematically and using methods to ensure that they have relational understanding have no time allocation in the curriculum. It is much quicker just to teach the rule and leave the understanding for another time.

It is a sort of tightrope that has to be walked. Balancing the demands of the curriculum with the requirements of the subject and the needs of the children and trying not to fall off.

Ncetm Audit

So half term is here and I am looking forward to spending the weekend doing maths instead of at home relaxing. Must be mad.

Before we go to the residential on Friday, we are expected to fill in the self evaluation audit at www.ncetm.org.uk. It covers all primary age groups including early years and is divided into subject knowledge and pedagogy. I began the processs this evening although I didn’t follow the rules exactly. The instructions were to begin with early years and move through the age ranges. Now as I know very little about early years that didn’t seem very helpful as a place to start so I began with Key Stage which is all I have ever taught.

Generally I scored highly on mathematical subject knowledge. This is probably down to the fact that I teach top set in year 5 and so I need to cover pretty much everything in the KS2 pos. If I had still been teaching in year 3 then I suspect that my results would have been different. Maths is one of those subjects that you need to use or you forget.

It has made me think a bit about what this course may demand of me in widening my experience. I can’t really expect to become a specialist maths teacher with little or no knowledge of anything outside KS2. To be fair, I do try to keep up with what is happening in KS1 and foundation but I own up to not really being knowledgeable about pedagogy in the early years. I suspect that I need to widen my knowledge base and experience.

I will have a go at the younger areas on the audit tomorrow and see how my results vary

Network Meeting 1

This was our first proper session. We have been split into 3 groups of 10, each one led by a maths consultant from the local authority.  Today’s meeting looked at the rationale behind the course and to begin to consider mathematical thinking as a big idea in maths.

We began with an introductory activity and then introduced ourselves with some background. I was surprised by how few of the 10 were maths co ordinators, there were more RE co ordinators than maths ones! I think I was reassured by the range of backgrounds that people had, from being a real maths specialist with a pure maths degree to having no more than a GCSE pass but a real interest in how to teach the subject better.

We have been given 5 key themes which will run through the course

                     Mathematical Thinking

                    Proportionality

                    Pattern

                    Generality

                   Representation

As well as that, we are being introduced to the ideas of Learner’s Mathematical Powers

                     Imagining and Expressing

                     Specialising and Generalising

                     Conjecturing and Convincing

                     Organising and Classifying

We had a paper to read based on the work of Professor John Mason in 2005. I was struck by the idea of conjecturing as a learning power. My most able children at the moment appear to be unable or unwilling to conjecture. They are reluctant to suggest ideas for solving problems which seems to suggest an underlying lack of confidence and an unwillingness to suggest ideas that may turn out to be incorrect.

Somehow I need to change the atmosphere in my classroom so that it encourages conjecture. I wondered whether a way of doing this may to give the children easier problems to have a go at, things that are well within their comfort zone. If  I do that then perhaps the children will gain confidence in their ideas and also gain familiarity with the idea of suggesting ideas and finding a solution together.

We then looked at a range of mathematical puzzles and looked at how we used the learning powers to help us solve them and how this would impact on our work in school. There are definite things that I need to work on such as giving the children scaffolding to develop their thinking rather than just expecting them to develop it on their own. Some probably will but others will need teaching or having possible patterns modelled to them.

A lot of ideas and food for thought in this morning’s session.

We are off to Northampton on 19th Feb for our weekend residential. Before then I need to read a paper by Richard Skempe on Relational and Instrumental understanding in mathematics. I also need to look at the use of questions such as ‘What do you notice?’ and ‘What is the same and what is different?’