Pattern – Second session

This was the first time that most of our network group of 12 had got together since the residential at half term so it was really good to see each other again.

It was also interesting to share our experiences of the university meetings. Most of us had gone to NTU but we had had different tutors. One or two had gone to Northampton instead. The experiences were very different which is interesting and maybe a bit alarming. Those who had gone to Northampton had a very formal lecture session which was mainly note taking. The NTU sessions were a lot more practical and aimed at classroom practice but even these varied from tutor to tutor. We felt as a group that there ought to be more consistency as we are all supposed to be doing the same course. I wonder how much it all varies across the country if there is this much variation on one course?

We then looked at our Professional Learning Logs and the reflections that we are supposed to be putting into section 2 of these.

We need to record examples of things that we have noticed ‘Critical Incidents’. These should refer back to the 5 big ideas or the 8 mathematical powers and should ideally cover a range of these. We should be aiming for one incident to be written about and reflected upon each week.

The session also looked at the work we are expected to do with another colleague in our schools. We have been given references to read on coaching and collaborative working in schools.

Then we looked at practical pattern activities that we can use to improve the children’s understanding of maths.

We looked at trapezium numbers which are exactly the same as triangular numbers but without the 1 on the top. We looked at which totals could be created and what rules and patterns could be found.

We then looked at how a multiplication grid can be used to look for patterns. We looked how the children can look for arrays, especially for prime numbers and square numbers. This could be a good visual reinforcement of work on factors.

The grid is also an excellent resource for seeing equivalent fractions, the top two rows are all equivalent fractions for 1/2 and other fractions can easily been seen.

We finally looked at how the Multiplication Grid ITP can be used to look at pattern in a similar way and also extend this by changing the number of columns.

For our next session which is straight after the Easter holidays, we have to work with a colleague to develop the use of arrays in multiplication or division or the use of a number line , in either case we are to try and extend the use beyond what we usually do. This could be difficult as the next meeting is only 2 days after the start of term so not leaving a lot of time for working with another person.

I am finding the course incredibly stimulating and already feel that it has had an impact on my maths teaching.

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Dicing with fractions

Well, we’ve finally got to the end of unit E2 and it has been a hard slog. They just find fractions so hard in all years and it doesn’t get any easier by the time you get to year 5.

We’ve just about cracked finding fractions and percentages of amounts but I was still concerned by their real understanding of equivalent fractions and the part played by the numerator and denominator.

We had a shortened maths lesson today straight after a very lively assembly so there was no way I was going to deliver my planned lesson. I decided to play an old game and try a new variation of it.

For years I have used making dice numbers as a starter activity. Basically the children draw a 4 square grid in their books and you roll a 0-9 dice. The children have to decide which place to put the digit. The aim is to make the biggest (or smallest) number and is a fun way of reinforcing place value. You can make HTU numbers or 10000 although I think the value is lost after that.

In Primary Maths – Teaching for Understanding by Barmby et al I saw the idea of using the same game to create fractions (pp80-81) and thought that today would be a good day to try it.

We began by playing the original game just to remind them of the point of the exercise. Then I drew the grid for 2 fractions with a greater than symbol between them and told them they had to create 2 fractions, the first being greater than the second.

The 4 dice rolls produced numbers 1, 4, 8, 3 and I was amazed at how many children used the 1 as a denominator which seemed to show a real lack of understanding. Only one child managed to place his 4 digits in positions to generate 2 fractions that made the equation correct and that was by sheer luck.

We then discussed what possibilities there were to create correct statements to lots of cries of “I get it now!”. We played the game again and about a third of the class managed to get a correct statement. Again, we looked at all the possibilities we could have created.

On the third game I stopped after the third dice roll and we discussed what numbers the children were hoping for to make their statements correct. This generated some real discussion about equivalent fractions as the children worked out which numbers would give them a winning equation.

I think that all the children ended the session with a much better understanding of fractions and they all enjoyed the session. I overheard one of my less enthusiastic pupils telling another child that “Maths was really great today” and that was 40 minutes after the lesson had ended.

Representations and Apparatus

At the residential weekend probably the idea that struck me the most was the idea that all children should be supported by the use of concrete apparatus. Having taught the upper ability set of year 5 for several years now, I was definitely of the opinion that apparatus such as cubes, number squares etc were for the younger children or less able. I didn’t really have any basis for this opinion, maybe it was just received wisdom from those around me or even a throw back to my own schooling.

The comment that as the maths gets more complex then the children should have more access to apparatus and other representations rather than less seemed to be a revelation that really made sense. There has to be a reason why children’s confidence and enjoyment of maths reduces rapidly during KS1 and I wonder if our insistence that the children cope with abstract concepts without the support of concrete apparatus could be partly to blame.

Today I was introducing my class to the ideas of ratio and proportion. I have always found in the past that this is a difficult area for children to grasp and understand fully. It seems to be an area where they appear to have grasped it initially but further investigation reveals that this is very superficial.

I had always used representations in my teaching of this area, I have a lovely powerpoint of dragons to illustrate how the numbers can be scaled up but the children have always struggled to understand this properly.

I decided today that I would give the children a lot more concrete support so I sorted out cubes and counters that they could use if they wished. I also started by getting children out to the front in a ratio of 2 girls:3 boys. We discussed the idea of ratio and then I added another set of 5 children. There was a lot of discussion between the children who felt that the ratio had changed and those who saw that it hadn’t. By getting the two groups of children to stand separately, they were able to see that the ratio was still the same although the numbers had changed. 

I then used the IWB to demonstrate this with purple and turquoise ladybirds and the children seemed to have fully grasped the idea. I then gave them sample ratios for them to show as a picture in the same way that I had done on the IWB. They had either cubes or counters which they could use if they wished. They then had some simple problems to solve. Probably about half of the children chose to use the apparatus but there was no obvious split between more or less able. It was also useful for me to have the apparatus to use when I was helping the children who needed a bit of extra reinforcement

At the end of the lesson I asked the children to evaluate what they had done and how they felt about their own level of understanding. All but two of the children felt that they had succeeded with the lesson and were confident that they could move on further tomorrow.

I will definitely make more of a point of having apparatus for the children to use if they wish in future lessons as it certainly seemed to make the children a lot more confident.

Caleb Gattegno

Reading up on Caleb Gattegno is one of the areas we were pointed to as part of our study of ‘Pattern’ as a big idea

http://en.wikipedia.org/wiki/Caleb_Gattegno

This article gives a brief description of his work and ideas. I’m not sure that I totally understand his idea of awarenesses but it seems to relate quite closely to the idea of relational rather than instrumental learning. The idea that just committing something to memory does not mean that it has been truly learned is common to both Skempe and Gattegno.

Gattegno’s ideas seem to be to do with the idea of creating links with what is already known so that new information is not just arbitrary but links with what is already in the learner’s brain. There is also a strong emphasis on images which he sees as low energy learning so there are obviously links between his ideas and representations in maths.

I probably haven’t understood this properly and need to read up on ‘the Silent Way’ and other ideas but these are just my initial thoughts.

Mathematical Thinking 3

This was the final session connected with our first ‘big idea’ Mathematical Thinking’. Had to drive all the way to Nottm Trent Uni after being at school all day. Not the ideal way to put yourself in the right frame of mind for mathematical thinking!

The session was a bit strange as it didn’t really build on the previous one. This was because whereas some of us had completed our residential and done session 2, others hadn’t and had only had their initial network session.

The aims of the session were to explore the role of mathematical thinking in teaching and learning, to consider how teachers can develop their own mathematical thinking skills and support their development in other teachers and to consider how to develop children’s mathematical thinking skills.

A huge set of aims for a 2 hour session!

We looked at a whole range of activities that could be used in classrooms to develop children’s mathematical thinking. One of the lovely things about this course is how many practical ideas I get that I can use in the classroom

I loved the idea of making estimation into a kind of competition between groups that really would make them think mathematically. The idea of using an unlabelled graph to promote discussion of ideas is one that I have used before but not at such a high level. Not sure that my year 5’s would be able to identify that particular sport!

Although the practical activities were enjoyable and useful as a resource, it was a shame that we skipped rapidly through the remainder of the slides which were more focused on how to develop children’s thinking through problem solving.

Our latest directed task (to add to the ones we already have from the residential) is to plan and carry out a problem solving activity in class. We are to try and involve as many as the pedagogic ideas as possible and record their impact. I think that is definitely something that I need to leave until the Easter holidays. Planning a problem solving activity isn’t a problem but involving all the pedagogies will take a lot of thought.

There was some discussion at the end of the session about the Personal Learning Log which will form a large part of our assessed material. People (myself included) are nervous about how much work is expected to be in this and what level of reflection is expected of us. It’s a long while since I did anything like this and I have no idea what the expectations are.