## Archive for the ‘Action Research’ Category

Over the course of the next couple of weeks in school, I intend to begin my work with the children. I haven’t really finished my reading yet, but I have a solid foundation to begin from – also I have the time available to me in school over these two weeks and after that it will be increasingly difficult to be able to do these sorts of things.

**Session 1**: Show a pattern made from multilink cubes. 1^{st}, 2^{nd} and 3^{rd} stage of the pattern. Following this concrete modelling, introduce the use of an input-output table to organise data about the number of blocks used for each stage of the pattern. The table helps quantify the pattern so that children see both the growing pictures and the growing numbers in the table. They can note the change from stage to stage and work to explain how the change in the table matches the change in the picture of the growing pattern. Finally, children can try to write a general rule that will work for any stage of the pattern without having to build it or know how many blocks were used in the stage before it. This is an important abstraction of the pattern and the rule must make sense to children and be in their own words or in their own mathematical notation that reflects the level of their current understanding. These may not be accurate at this stage – but that’s ok (and the whole point of the sessions…)

**Fir Tree investigation**: uses pattern block triangles to create growing fir trees. Children must extend the pattern, complete the table of values and describe the 10th tree.

**Session 2**: use ideas from algebraic infants. Modelling how a family of dogs can be constructed. Legs, shoulders, bum, body, head – with only legs and body growing, Children need to show each stage in a table for the dog. They then create their own animal, growing it in the same way. Multilink cubes needed.

**Session 3**: **Tables & Chairs investigation** challenges students to find a rule to describe the relationship between the number of small square restaurant tables placed together in a line and the number of diners that can be seated at the larger table if only one person sits on each side. Can model with shapes if needed (squares and circles). As before, children make a table to shapes used and differences to aid thinking.

**Session 4**: Part 1: **Ships Ahoy**. Children look at a simple pattern of horn blows and predict future patterns.

Part 2: **Rockets**. Children look at a pattern for building rockets. They are challenged to see how many parts will be needed for the 50^{th} stage. Whiteboard/paper drawings probably best – stickers might be helpful though?!

**Session 5**: **Task Cards**. Children choose a task card then construct it using squares or triangles. Encourage them to look at the patterns and decide how it is growing. Create and complete a table to show the growth pattern. Use the tiles to make the next two shapes in the pattern. Be prepared to explain how the pattern is growing. If there is time, choose another task card. Cards attached at the end.

Session 5 – Challenge Cards (PDF)

**Session 6**: Same as session 1. Can the children identify the growing pattern and express the rules algebraically?

Show a pattern made from multilink cubes. 1^{st}, 2^{nd} and 3^{rd} stage of the pattern. This pattern must be similar but not the same as session 1. Following this concrete modelling, introduce the use of an input-output table to organise data about the number of blocks used for each stage of the pattern. The table helps quantify the pattern so that children see both the growing pictures and the growing numbers in the table. They can note the change from stage to stage and work to explain how the change in the table matches the change in the picture of the growing pattern. Finally, children can try to write a general rule that will work for any stage of the pattern without having to build it or know how many blocks were used in the stage before it. This is an important abstraction of the pattern and the rule must make sense to children and be in their own words or in their own mathematical notation that reflects the level of their current understanding.

**Hexagon dragon investigation**: dragons made from equal numbers of hexagons and triangles, each new term adding on more. Requires children to extend the pattern, create an input/output table to describe the growing pattern, then draw and/or describe the 10th dragon in words. An extra challenge asks students to generate a rule for this pattern so that Miguel can figure out how many blocks he will need to build a dragon of any size.

Each of these sessions will be completed during the children’s Numeracy time. I am likely to choose 4 or 5 children to work with from across ability groups and a mix of genders. I will take copies of their written work and record conversations to help me analyse their progress and thinking over the course of the sessions – I feel that this will give me the data I need to go further. Session 6 is a repeat of session 1 in order to set a baseline and see the progress, if any, the children have made in between.

Clearly, this is such a small scale research project that results can’t be read into too much. However, it is a beginning for my school to look at how we can use algebra in wider contexts. I have been careful to choose similar tasks throughout the sessions as I will only have at most half an hour to complete the tasks with them.

I tried this with my current Year 6 group of 24 children to disappointing results. The children break the stick of 18 Multilink cubes, describe what they have done and put something on the sheet of paper that will show what their actions to the rest of the group.

They were sat in two circles of 12 each with a large sheet of flip chart paper in the middle, three pens to record their work and a stick of 18 multilink cubes. I chose 18 for its number of factors: 1, 2, 3, 6, 9 and 18. This gives many possibilities for different sums being created. 12 is also a good number (with factors of 1, 2, 3, 4, 6 and 12), as demonstrated in the taught session. Of course outcomes could be expressed as aspects of addition, subtraction, multiplication, division, fractions, proportions, ratio and percentages (there may be others, arrays could be used for example too).

I wanted them to keep discussion to a minimum and they worked silently for the most part. I had to keep reminding them to be mathematical as the activity progressed, but that was all I said. I didn’t say that anything they had written was right or wrong, remaining neutral and fairly detached throughout. I also made a point of not mentioning any potential things they could record and stated that if the stick had been around the circle once to carry on – especially after seeing the responses… this was in the hope that they might actually use some mathematical ways of recording!

One group in particular, took this as an opportunity to create a silly theme. Their recordings were written sentences of what they had done, such as, “I dropped it on the ground and 9 fell off. ^{9}/_{18 }= half ½” or “I dropped it and 14 came off” and even “I headbutted it and 13 came off.” Needless to say, I wasn’t particularly happy with that group as their work showed little thought or care for the maths they were doing nor did they relate this task to the fractions work we had done previously.

The other group’s responses were more considered. Although the first four responses were variations of the sum 18-5=13: “-13 = 5 left” and “5=13left” being two of their recordings. The rest of their responses try to take the form of ratio statements, which is a great piece of thinking from them – although they don’t quite get it right as they write “6:18 *[which is followed by]* Took off 6 cubes which makes 12 cubes” or “8:18”.

In all, this tells me that my class need a little more guidance and structure when it comes to tasks. Although I had a feeling that this may not be a useful activity with these children, having taught them for a year and a half now and knowing how they can be, I didn’t expect the outcome to be this far removed from my expectations. I suppose it shows that not all good activities work with all classes.

Yesterday, I detailed how I was planning to use Cuisenaire Rods with half of my Year 6s to investigate and consolidate their knowledge of fractions.

What I love about the photo above is that despite the general age of the sets we had in school, the rods looked like new… also, the lovely glimpse of working out on the whiteboard in the top corner… (click to enlarge)

My aims for working with the Cuisenaire Rods were:

- to allow them some hands on work with little recording needed;
- to give them lots of thinking time – time to play with the ideas behind the rods and questions;
- to explore fractions in a different way – I doubt they will have used this equipment before (it always surprises me how, in this computer filled age, the simplest thing like a wooden block can fascinate children in the ways they do);
- to have some interesting discussions about their mathematical thinking. I want to hear their reasoning throughout this task.

My class relished working with these. I gave them a brief introduction to the rods – none of them recalled using them before – and a quick run down of the task *(shown in full at the bottom of the full post, click read the rest of this entry to display it)*, to investigate relationships between the rods when given a statement about them. I emphasised that it was OK not to give an answer as a definite decimal number, but they could leave it as a fraction, or even give their answer as a colour in certain cases. Nevertheless, a few children asked for calculators and I allowed this as I didn’t want to dampen their enthusiasm for finding an answer!

They had around half an hour to explore the task and the rods, during which I took the photos you see dotted around this post and kept a careful ear open for the language and discussions the children were having. They had whiteboards to make jottings on and were told that I wouldn’t be marking them but I wanted them to explain to me how they knew things.

The majority of the 12 were sensibly exploring the relationships and questions associated with the rods. A couple were unsure about how to approach the task – in particular the questions that didn’t have an easily definable answer. One child found it hard to see that with each question the rules were reset. For example, question 1 states that red is equal to one, and in question 2 that association is broken as brown is equal to 10, equivalent to 4 red rods, making red equal two and a half. This child is one who is particularly talented at number work and is adept at working mentally. I expected him to really enjoy and take to this task with minimal effort but it didn’t click with him at all. He wanted to join the rest of the class in their task claiming that he, “didn’t get fractions”. I think his issue was down to a different structure to this lesson than normal and the safety net of a right or wrong answer not being there – he loves being right, and struggles to cope with being wrong.

As far as collaborative work went, the children mostly worked well together, discussing the implications of the criteria set down in the rules and finding relationships within the rods that they could use to guide them to the answer. Some of the questions I have likened to solving a Sudoku puzzle in that you only have a finite knowledge of the whole set of 10 coloured rods and you need to slot the values of the rest in carefully around that knowledge.

One such example of this is question4:** “If orange subtract pink is thirty, what is: a) orange plus red? b) orange plus yellow? c) half of orange?”**. Above is one child’s working out for this question – they insisted on neatening it up for the second photo… From this, they have worked out:

- Orange subtract pink is dark green;
- Therefore, dark green is worth 30;
- 3 red rods have the same value as dark green (30);
- 1 red rod is worth 10 (30 ÷ 3);
- 5 red rods are the same as orange, so orange is worth 50 (5 x 10);
- Orange plus red is 60 (top picture);
- Yellow is worth 25 (half of orange, so 50 ÷ 2);
- Orange plus yellow is worth 75;
- Brown plus red = orange;
- Dark blue plus white = orange.

Quite a lot of maths for such a short question – the last two not having been asked for, but they wanted to find equivalent lengths.

The great thing is, the quality of the maths here was good. It showed their ability to think in a logical way and discuss their thinking well. The discussions spread further than their pairs, the two tables became one mass talking point in the class, trying to explain to each other how they had worked out their different results. Another positive is that, generally, they agreed on a solution – no matter how they each worked it out.

In conclusion then, this has been a worthwhile experiment. It has given the children a new experience, it has given a new lease of life to dusty Cuisenaire Rods, it has allowed the children to stretch their mathematical thinking and developed their explanation skills. In all, something that ticks many boxes in quite a simple way.

Tomorrow I intend to work with half of my Year 6 group, 12 children, on the included Cuisenaire Rods task from the handbook (an example of the type question we will look at is below).

Georges Cuisenaire was teaching at his school in Thuin in Belgium when he invented these now famous rods as a means of helping his pupils with their study of arithmetic. He made then a discovery now established as a vital component in mathematics teaching today. He found that by making use of children’s natural inclination to play, and giving them an appealing material which demonstrated the relationships on which mathematics is based, it was possible to provide understanding for them all. [Cuisenaire.co.uk]

I know these children well having taught them both last year and this. I know that they enjoy working with equipment and that my teaching over the past couple of weeks has been pretty hands on as we have reviewed our shape transformation work (translation, rotation, reflection etc.). I’m also acutely aware that they haven’t quite managed a solid grasp of fractions yet. They are good at shading fractions of shapes, if those shapes are split into equal segments, and are beginning to apply their ideas to numbers, but some of them find it difficult. However, one bit of teaching that has stuck in their heads is the phrase, “You divide by the bottom and times by the top!” They often repeat this chunk of learning while applying the actions they were taught in Year 4 – hitting their bottoms and heads. I’m never quite sure that they understand why they do this, but it has clung to their brains like chewing gum to a carpet and I may as well take advantage of that.

So, my aims for working with the Cuisenaire Rods are:

- to allow them some hands on work with little recording needed;
- to give them lots of thinking time – time to play with the ideas behind the rods and questions;
- to explore fractions in a different way – I doubt they will have used this equipment before (it always surprises me how, in this computer filled age, the simplest thing like a wooden block can fascinate children in the ways they do);
- to have some interesting discussions about their mathematical thinking. I want to hear their reasoning throughout this task.

Looking at the third bullet point above, Cuisenaire Rods were around when I was at school in the 80s/90s (I left primary school in 1994), but I can’t remember ever seeing them used in our classroom – apart from the single cubes which often appeared alongside the plastic Dienes equipment. It seems that many sets of Cuisenaire have found their way into bins over the years from the discussions at our last meeting. My school clearly either clings onto things or we have an insightful Maths co-ordinator who knows the value of equipment – both in a monetary sense and for their use in teaching. Either way, I discovered 7 sets in school when I looked hard enough… As well as this, there are online versions of the rods available. I’ve included some links at the end of this post.

This makes exploring the task we looked at in the session with my Year 6s an exciting possibility, and one I’m looking forward to. Their job will be to explore the relationships between rods, using their logic and reasoning skills to explain their ideas. One typical question is: **“If red is one, what is: a) pink? b) light green? c) blue?”**

I fully expect them to compare the rods side by side, possibly using more than one red to work out the ratios between the sizes. I can think of one child who will be able to see the links fairly quickly. He is very visual in his learning but also is able to play with numbers and ideas with ease in his head. He will want a definitive answer to each sum and I may have to work on this as, in later questions, fractional answers are the only realistic sensible answer!

In each question, the rules are reset so that there is a given idea to start with. This will also need to be made clear to some of my children. Another example investigation is, ** “If brown is 10, what is: a) pink? b) red? c) blue?”** Now, this involves colours we’ve previously given values to in pink, blue and red and I know some will fall into the trap of using the previous examples results here.

I am looking forward to seeing how my class get on with this. I will be taking photos of them working and reporting on thei use of language later this week.

**Links**:

- Online Cuisenaire Rods– a flash file. Click on ‘Rods’, to choose a Cuisenaire rod and then drag it onto the squared background. More rods can be added in a similar way and aligned as you wish. A rod can be rotated by 90° by clicking any key whilst dragging. The background squares can be altered (for example increasing/decreasing their size) using the ‘View’ menu.
- NRICH activity ideas – their search results for Cuisenaire Rods.
- Numicon Number Rods – the current makers of the rods.

Having just finished my work with a bunch of Year 3 – who are far fussier than I remember – I feel quite pleased.

They were able to build and complete the secret construction quicker and more accurately than I first hoped, they used a good range of language – next to, on top of, it looks like a house (!) – and generally they worked well together.

I used magnetic polydron to build the shape for the secret construction again. In fact, I decided I would use the same shape entirely for the job, with the same colours too! It was clear that they had done this activity before. They knew what was expected of them – although that didn’t stop me from telling them – and how to go about getting to a satisfactory end. Certainly the secret construction is an activity that works lower down school than I am used to and I can see it being of value to them from a vocabulary, communication and shape point of view. The only problem with magnetic polydron is that currently we only have squares and equilateral triangles, which limits the number of shapes we can make. I have included some of their creativity with the equipment from the very end of the session when I let them have a little play…

During the string activity, it was clear that they had limited knowledge of shape names. They struggled to predict what shapes would be made – although a couple did correctly identify a hexagon. When a star shape was created with a heptagon inside, I already knew they wouldn’t have a clue what it was, so focussed on the outside shapes instead.

Now, because of troubles when lowering the string to the floor, we ended up with some unusual patterns around the edge. We had trapeziums and triangles… the trapezium intruiged me. So I asked them what they thought it was called. Instantly, we had the name ‘square‘, which lead me to ask why they considered it a square. The response I got showed their knowledge, but also immediately reminded them they were wrong. They said, “It has four sides… but they’re not the same size, so it can’t be a square.” Intelligent thinking! This lead another child to say, “Well it must be a rectangle then!” Prompting another to say, “But it doesn’t have four right angles.”

While I wasn’t expecting this at all, it showed that a simple thing can generate such a wonderful discussion. To me it doesn’t matter that they didn’t know what a trapezium was, it was valuable enough for me to go back to their teacher and tell him that those children knew the properties, roughly anyway, of a rectangle and a square. And thinking about it, that’s all they should know. After all, it’s the fourth week of their first half term in Key Stage 2 – their knowledge of shape hasn’t been touched since the back end of Year 2 anyway, and that would consist of looking at the names of basic shapes.

If it taught me one thing, it’s that I haven’t been around Year 3 enough lately, that I’ve become used to the language and abilities of Years 5 and 6 so much that I’ve desensitised myself from children further down the school.

I certainly need to make time to work with them more throughout this course.

Tomorrow, I work with Year 3, completing some of the activities that were so successful with Year 6. I plan to introduce the session with a secret construction, along similar lines to the one made at the start of last Wednesday’s lesson with Year 6 – a simple house like structure using only a couple of colours. Hopefully this will be at a reasonable level for them to work securely.

After this, I’m moving outside with them to work in circles. I will be having 13 children, more than I would like – so I will have a group of 6 and a group of 7. Hopefully these groups won’t be too small for a successful string challenge – I can always combine them into a group of 10 with a few onlookers and rotate the children as and when so everyone gets an opportunity to be in the circle. I am hoping for dry weather – standing up and lowering the string was harder than I thought for the 6s and so if they were sat on the floor, it would help a lot.

My main concern is that I haven’t worked with children of this age for a long while now…