I’m at the very beginning of my teaching career. Amongst the day-to-day business of teaching, whilst on my PGCE I spent quite a lot of time thinking about how to break down mathematical ideas into key concepts that the kids could understand and thought about how best to communicate them. I found this time valuable and illuminating as it challenged my own deep understanding of concepts that I’d taken for granted. The area of maths that I have so far found most interesting to think through in this way has been proportion and the link with fractions.
Why do so many kids really struggle to understand the idea of proportion and also how you can represent it as a fraction? I’m not sure for certain but I think one explanation is that they don’t understand division.
I remember sitting in a dingy, damp mobile classroom about 15 years ago, when I was in Year 8.This building did not inspire learning with it’s bouncy, rotten, squeaky floor boards and noisy, single electric heater that occasionally emitted a puff of luke warm air that kept the frost bite at bay during those period 6, dark lessons at the start of the spring term. Fortunately the inadequacies of the learning environment were counterbalanced by the inspirational, eccentric maths teacher, Mr M. He rode the boundary between challenging us and making us feel totally stupid but had a sense of humour sharper than iced lemon juice that endeared us all to him. Mr M flat out refused to teach procedures. We spent many lessons focussing on concepts alone, talking them through as a class endlessly with Mr M steering the conversation like a helmsman in a yacht race, keeping it moving towards the goal despite us zig-zagging off on conversational tangents.
Despite the time we locked Mr M in the store cupboard, the most vivid lesson I remember was about proportion. Mr M asked us “somebody explain to us what division is”. What seemed like a minute later, but what was probably 20 seconds, he asked us again. After another deathly silence he finally relented and went into an explanation of what we would today call chunking. He explained the idea of separating things up into groups of a certain size and then counting how many groups you have. In hindsight I feel ashamed to not have understood this in Year 8 but in my defence would say I remember learning division procedurally at primary school rather than by chunking.
Division is a double-edged sword. There are two ways of looking at it and I personally wonder whether we need to show kids both ways, rather the one way most teachers convey it.
The first way of looking at division is through the chunking method. If I want to do 10 divided by 5 I imagine I have 10 of something, split them into groups of 5 and the answer is how many groups I have. This is the method now commonly taught in primary schools and the kids seem to really understand it (on the most part). 20 divided by 5 becomes the question “how many 5s in 20?” and the kids are happy answering it. It’s a nice visual method that link to the times tables, showing how division is the opposite of multiplication. The idea of remainders is also tangible and consistent.
The second way of thinking about division is precisely the opposite of the first way. When doing 10 divided by 5 we said split into groups of 5, how many groups do we have? We could equally have said split 10 into 5 equally sized groups, how many in each group? Think hard about this last sentence and really get it clear in your mind. Division is a double-edged sword because you get the same answer whichever conceptual model you use for x divided by y:
- Split x into groups of size y, how many groups do you have? or,
- Split x into y number of equally sized groups, how many in each group?
The first conceptual model, which for ease I’m going to call the chunking model, is useful when you use division in arithmetic. I put forward that we should consider actively teaching the second conceptual model for division, which I’m going to call the proportion model when teaching fractions, pie charts and any other topics that are based on the idea of proportion. It may not be immediately obvious why I have named this the proportion model but I hope to convince you the name is appropriate in the following paragraphs.
If we were teaching fractions and wanted to show 2/6, many of us would draw a shape, lets say a circle, split it into six equal sized sectors and colour in two of them. This conceptual model of showing fractions is simple to understand and shows the idea of proportion very clearly but is, I would put forward for your consideration, a poor model. It communicates the idea of proportion well but that is it. In another lesson we would then teach the kids that to turn 2/6 into a decimal they do the division sum 2 divided by 6. We tear our hair out trying to enlighten them with the fact that 2/6 and 0.3 recurring are the same thing but many of them don’t see it.
Is there any wonder? Think back to the conceptual model we used to explain 2/6 as a fraction. We ‘divided’ a shape into six equal pieces and coloured in two of them. If you are going to look at fractions as division sums, which I think we have to if we are going to try to explain that they are equivalent to decimal numbers, then to explain 2/6, why are we taking one of something, splitting it into six pieces and colouring in two of them? The conceptual model is neither the chunking model or the proportional model. The problem is that we are thinking of the ‘whole’ as being unity rather than the numerator. If it were the chunking model we would say we start with two circles and split them into ‘groups of six’, how many groups do we have? We obviously don’t even have one whole group but we do have part of a group, two out of six of one group of six. This gets across the idea of proportion as well as providing a conceptual model for representing fractions that is consistent with thinking of them as division sums, which should make the link between fractions and decimals clearer. If we use the chunking model the “whole” is taken as the number of things in one group (the denominator).
Alternatively we could teach 2/6 using the proportional model which would take two circles and split them into six equal sized pieces and asking the question “how much of a circle is in each piece?” which also conveys the same fractional part which is less than unity, the idea of proportion and a model that is consistent with division. If we use the proportional model the “whole” is taken as unity.
To sum this up, when we are explaining fractions and the idea of proportion, shouldn’t we use a conceptual model that incorporates the idea of division which we then teach them is necessary to convert them into decimal numbers? The link between fractions and decimals is much clearer and they still learn proportion. If students learn fractions are actually just division sums then this will dispel the common misconception that fractions are pictures showing proportion rather than actually also being numbers themselves.
This is all quite higher-order thinking and something that should definitely not be taught to your average year 7. It would blow their mind and they would never want to set foot into your classroom again! What about discussing the different approaches with your level 8-10 pupils though? Rather than teaching them the procedural routines those kinds of kids absorb and replicate so freely, why not also show them (if time allowed) the different conceptual models and see what links they can make to other parts of maths. Let them see why a fraction is equivalent to a decimal number rather than them just accepting it because you said so. If they understand why 12 divided by 4 is the same as 6 divided by 2 using the chunking model and they looked at fractions as division sums, wouldn’t they get a whole new perspective and understanding of equivalent fractions? Wouldn’t this make their understanding of algebraic fractions where they need to change the denominators to match by using equivalent fractions so much better?
No one model is perfect for all topics. If you think about it the chunking model is totally inappropriate for explaining how to calculate sector angles in a pie chart, the proportional model is much better. I don’t intend to stray far from the traditional path early on in my teaching career. Much smarter people than me have been teaching the traditional conceptual models for a long time. Nonetheless, I will keep thinking about when it might be appropriate to dip into alternative conceptual models if they have the potential to boost understanding.
Your thoughts on any of my mumblings above would be appreciated. Please add to them in the comments section below and fuel the debate!
If Mr M is still around he should feel proud for asking that question “what is division?”. It is about 15 years too late but I think I might now just about have a reasonable answer. Sorry Sir…