In my job my favourite times are in the classroom where the action is. I learn so much from the interactions between teachers and children about learning. I also indulge myself a little and teach when I can.
When reflecting on interesting lessons I taught or observed last year, I fondly remembered my time supporting Year 3 staff in Shackleton Primary, Bedford (amongst many others of course). It was a couple of sessions focussed on fractions – a tricky area for Year 3 learners.
The first session was focussed on ordering and comparing unit fractions. There were several adults in the room. The teacher had already taught a session and was reasonably confident the children were able to order them on a number line. It is a very tricky area however and I expected that children would
b. still be bringing in prior knowledge about whole numbers, generalising that 3 will always be smaller than 8.
Forgetting is important as it allows us to remember. Remembering strengthens neural pathways helping us to imprint knowledge into the long term memory.
So I began with a hingepoint question that would:
a. re-ignite learning
b. allow me to assess whether the over generalisation about whole numbers was impacting on ordering unit fractions.
Where would one eighth and one third live on this number line?
We soon saw that the vast majority of the children placed one third before one eighth believing one third to be smaller than one eighth.
None of the children divided the lines into 3 or 8 approximately equal parts while we observed either, demonstrating that children had not necessarily grasped the relative size of each of the parts compared to the whole. We needed to allow children to focus on the critical aspects of ordering unit fractions and the first task was practice in the form of a game.
In groups of three or four, children were each provided with different coloured markers (counters in this case) and a 1-6 dice and 6 fraction strips per team – strips of the same length divided equally into 1, 2, 3, 4, 5 and 6 equal parts. Equal parts were labelled to help make links to the size of the part, number of parts to the whole and the abstract notation. A start and finish line was taped to the tables using masking tape.
The object of the game was to be first to move their marker from the start line to the finish line. Each child took turns to throw the dice. The number thrown related to the denominator on a fraction strip e.g. throwing a 3 means the player picks up fraction strip with three thirds and moves their counter the length of one of the thirds. Player 2 does the same but throws a 6 for example and moves the length of one sixth.
It’s at this point the learning occurred, visibly. Usually in a game you want to throw a 6, good things often happen! But now disaster you don’t move as far as the player who moved one third. This polar difference to normal game playing situations caused surprise and therefore thinking. Soon all of the children were hoping to throw ones or twos. The game focused winning on the critical aspect that the larger the denominator the smaller the part and vice versa. Not why just yet and they hadn’t been exposed to non-unit fractions, but the children were able to generalise something mathematical from the game playing (credit for this game goes to @MathThatCounts Rachael Watt, one of my lovely fellow advisers).
Once all the cheering and groaning died down, we finished the game task and asked children what they noticed about the numbers on the dice and the distance they could move. We ensured that the generalisation was shared by everyone.
The next task involved dividing a whole and sharing it ‘fairly’ between more and more children, causing the more groups = smaller parts relationship to be blatantly obvious - ‘My bit is getting smaller,’ and ‘oh no we don’t have to share it equally again, I’ll have even less’. We wrote this next to the fractional part to show the relationship between the denominators and the number/size of parts the whole was divided into.
Subsequently, we returned to the number line example and observed as the majority of children now thought more carefully about the number line and successfully reasoned that an eighth would come before one third. Adults noted children who were still insecure with this aspect of ordering.
The children were desperate to play the game again! But wanting to move the learning on slightly I told the children that they still had to move one sixth when they threw a 6 for example but now they had to choose which three out of the six strips I could take away from them. Children, with further support from adults, reasoned that they could still move one third even if they didn’t have the thirds strip because they could move two of the sixths instead. Some used reverse reasoning for the same relationship noticing, I could get rid of the sixths strip because I can fold one of the thirds in half and move half of a third.
All of the groups realised they didn’t need the 1 whole strip because they could use any other strip to make it.
Through the tasks we had moved children into doing tasks and seeing aspects necessary to partially understand the learning – comparing and ordering fractions. This is not to say they wouldn’t forget or revert back to previous whole number generalisations but by playing the game again in a couple of weeks and then a couple of months to help them remember and revisiting the concept again in future years they will eventually secure this tricky area.
The following week I went to visit the parallel Year 3 class, whose teacher was able to observe the first session and had taught the tasks as I had and then moved the children on. I’m always grateful to teachers who throw their rooms open and are reflective practitioners, and this teacher allowed the subject lead and I to come and see how the children were getting on. And they were getting on marvellously too. The children had a great part whole fractional understanding developing and were several sessions into adding and subtracting fractions with the same denominator. In fact the majority (a small groups was still rehearsing adding then subtracting before making links) were at the stage where they were assimilating the critical aspects of part whole thinking and applying additive relationships from whole number to fractions when adding and subtracting fractions.
Firstly, the teacher showed the following examples of solutions for the same calculation* to the whole class. I must confess I became over excited and went to join the teacher to probe what the children were thinking. The teacher supported the children’s communication with a model (cubes) to help them reason about the top left example. Many children identified two of the others were incorrect and with communication support could say why – in fact they scoffed at the bottom left example forgetting that this was exactly what they themselves were doing at the beginning of this learning!
Interestingly, to me anyway, was the fact that the children found the top right example the most difficult, at first they felt it was right. Mainly because the picture showed 5 parts coloured and 4 parts subtracted leaving 1 part. This matched their expectations for an answer. It wasn’t until one child (prompted) counted the ‘altogether’ parts that the children twigged that the model represented eighths. The teacher wrote that calculation with the children.
With the exception of a targeted group who were still getting to grips with the inverse relationship evident in the example top left, the rest of the children began completing the following examples. The teacher modelled a worked example and left it on the board. But by and large it was left to the children to reason about the operation needed depending on the example.
For some, success with this task meant modelling using the cubes to find solutions but once this happened they independently completed further examples. By the end of the lesson the majority of children had completed the examples and many moved on to a still more complex grid example.
The lessons above were interesting to me because they exhibited purposefully planned tasks, designed to help children notice what was critical about the object of learning in each case. This is in contrast to where examples appear more scattergun too early in sequence of learning and therefore do not draw attention necessarily to what it is important for children to notice. Where the children had not yet discerned those features, the teacher in the second lesson was working in a different way to allow the learners more time to do so.
Obviously no lesson is perfect, and external factors play their part too. But there reader I’ll leave you to perhaps some musings, if not, hopefully some interesting tasks to teach fractions with.
* Lesson two was taught using materials from ESSENTIALmaths