## Archive for February, 2011

This reading discusses how an average student can benefit from a structured and ordered approach to the teaching of fractions. It could probably be applied to any mathematics topic.

The approach taken was one of understanding the needs of the child first and having an in depth idea of where her weaknesses were. There was a strong focus on the language of fractions – this relates to some of the findings I wrote about yesterday. Fractions are a relative part of maths and their outcome depends entirely on whatever the whole is.

However, we initially observed Audrey having problems in comparing the equivalent fractions 1/3 and 2/6. We therefore used the fraction-lift to clarify equivalent fractions, by making them fractions living at the same floor of the fraction-building and by introducing the metaphor of “roommates” for fractions at the same position on the number line. We now observed how Audrey used the strategy of doubling both the numerator and the denominator to generate equivalent fractions, for example by replacing 2/3 by 4/6 to compare the latter fraction with 5/6.

This gives Audrey a hook to hang her ideas on, something concrete that she can build her learning on. I’m convinced that children need to have the basics of any topic before moving on. I also think that the current approach we have to teaching in this country, based on the current Mathematics Framework, is far too fleeting and jumpy. As practitioners, I feel we need to consider the needs of our children – group parts of topics together so that they have more time to practise and consolidate their learning.

**Link (the full article)**: Ronald Keijzer and Jan Terwel: Audrey’s Acquisition Of Fractions: A Case Study Into The Learning Of Formal Mathematics.

Below are the general responses to the questions posed – I recap the questions and the views are generalised notes from talking to a range of teachers from nursery, through Key Stage 1 and 2. I provide the detailed breakdown of two colleagues from Years 3 and 4.

Throughout these you can see that fractions is a huge, varied and tricky concept to think about, teach and learn. I have found that, throughout my teaching career, children have always found fractions hard, although there are a core of children who grasp it quickly, these are the exception rather than the rule.

It is clear that a hands-on, physical approach is needed at the beginning of fractions work – indeed practical maths was a key talking point of all the teachers throughout the Primary phase. But also, I feel that language is a huge barrier to children’s learning. The language of fractions is often misused throughout life and if they don’t have a solid understanding in the first place, their will only serve to cloud the issue further. Another difficult aspect is the link with division – children who don’t know their multiplication and division facts can’t begin to develop their ideas of fractions

So, how can we bring all these parts together to make up one cohesive whole?!

I think it’s a case of reviewing the way it is taught throughout the Primary Phase. Children have to encounter teachers who are confident in approaching fractions and the subject needs to be taught consistently, removing areas of conflict and making sure that each part of their learning isn’t conflicting with another area. If it is to be taught alone, then it needs to be done until that child grasps that particular stage of learning. My belief however, is that is must feature in each part of mathematics – there can always be a question relating to fractions in whatever is taught. This may help to break some of the barriers to learning that exist – I currently feel that children are very negative towards them.

One key to learning is children’s difficulty with fraction language. Maybe teachers are trying to make too many jumps at the same time, moving too quickly. Maybe this is down to pressures from the curriculum. It is often better to avoid comparisons. For instance, when focussing on halves, it would be better to focus on and describe objects or models that are either halves or not halves, rather than giving objects other labels *(much in the same way that children find it easier to learn that bricks are heavy and feathers are not heavy rather than comparing them as heavy and light. Floating and sinking is another example, it is easier to get children to understand things that float and things that don’t float BEFORE investigating things that sink – it’s too confusing)*.

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.