Posts Tagged ‘Reasoning’
The article gives a history of the mathematics landscape in the UK since 1837. Algebra has been part of this throughout.
We see algebra as a key tool to help solve problems now, but historically “problem solving was seen as a specialized skill, only for mathematicians in opening up new fields.”
Key points:
 “A problem solver needs a rich, connected understanding of mathematics and the abilty to see patterns of similarity and association, as well as the skills to carry out the planned attack, and to check that the results make sense in the context of the problem.”
 “A Royal Commision, reporting on the state of mathematics teaching in nine leading Public (i.e., private) Schools in 1837, noted that the typical two weekly hours of mathematics consisted of Arithmetic, a little manipulative Algebra, and “Euclid“, learned by rote.”
Reference:

Burkhardt, H., Bell, A. (2007) ‘Problem solving in the United Kingdom’ ZDM, Vol. 39, no. 5, pp. 395403
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 coordinator 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.