Building Bridges
In Issue 118 and Issue 119 I’ve written about the flexibility and longevity of the area model for multiplication, and I’ve tried to explain that I think it’s a “good” mathematical model because it can be used to help pupils think about multiplication in a range of different contexts over a number of Key Stages. More generally, I believe that when we choose and use models and representations in our teaching to help our pupils understand the mathematical concepts that they are exploring, we need to ensure that the models / representations
 can at first be explored “hands on” by all pupils irrespective of prior attainment
 arise naturally in the given scenario, so that they are salient and hence “sticky”
 can be implemented efficiently, and increase all pupils’ procedural fluency
 expose, and focus all pupils’ attention on, the underlying mathematics
 are extensible, flexible, adaptable and longlived, from simple to more complex problems
 encourage, enable and support all pupils’ thinking and reasoning about the concrete to develop into thinking and reasoning with increasing abstraction.
I would argue that the area model for multiplication meets these six criteria, as does the model for division as fitting sticks into gaps (discussed in Issue 116): please do let me know (email or Twitter) if you disagree. Currently there is a lot of discussion about – and advocacy of – bar modelling (sometimes called Singapore bar modelling because it is regularly used in both primary and secondary schools there): does it fit the criteria? Is it the wonderpanacea its most zealous proponents claim?
If you’re unfamiliar with bar modelling, there are many online demonstrations and explanations you can easily look up. It’s often taught in the context of ratio and fraction problems, for example “I share some sultanas between Alice and Bob in the ratio 3:5. Alice gets 28g fewer sultanas than Bob. How many grams of sultanas does Bob get?”
To solve this problem, a pupil might first draw
to represent the 3 portion / handfuls / scoops / boxes that Alice gets, and the 5 that Bob gets. The pupil would then add to the picture that the discrepancy is 28g:
and then see that two of Bob’s portions weigh 28g and so one must weigh 14g. Therefore his five portions must weigh 70g. The diagram would end up looking like:
There are two immediately clear and powerful benefits to using this representation to model and answer this question: the pupil has got the right answer, and the diagrams capture precisely the pupil’s reasoning. This argument – pictorial though it is – is as incontrovertible as a formal symbolic proof, and it is accessible to pupils with a wide span of prior attainment. I certainly think that the bar model meets the first five of the six criteria I proposed earlier.
To test whether it meets the sixth, let’s imagine giving the pupil these subsequent questions:
 I share some sultanas between Alice and Bob in the ratio 3:5. Alice gets 28g fewer sultanas than Bob. How many grams of sultanas does Bob get?
 I share some sultanas between Alice and Bob in the ratio 6:10. Alice gets 28g fewer sultanas than Bob. How many grams of sultanas does Bob get?
 I share some sultanas between Alice and Bob, so that Bob gets ^{5}⁄_{8} of all the sultanas. Alice gets 28g fewer sultanas than Bob. How many grams of sultanas does Bob get?
 I share some sultanas between Alice and Bob, so that Alice gets 60% of what Bob gets. Alice gets 28g fewer sultanas than Bob. How many grams of sultanas does Bob get?
A pupil who doesn’t see that, or see why, these are all the same question hasn’t yet got a deep conceptual understanding of the question; the pupil who answers the question a further four times is still reasoning about the concrete. My caution about the bar model is that it doesn’t naturally or automatically develop the abstract reasoning that enables the pupil to see that these are all flip sides of the same (five sided!) coin, because drawing boxes/bars focuses pupils’ attention on the additive structure of the Alice and Bob problem (Alice has two boxes/bars fewer than Bob, Bob has two boxes/bars more) and not on the multiplicative structure of the problem, that
 Alice’s share is ^{3}⁄_{5} of Bob’s share
 Bob’s share is ^{5}⁄_{8} of the total
 Alice’s share is Bob’s share reduced by 40%
 the scale factor from Alice’s share to Bob’s is 1^{2}⁄_{3}.
This is not a fatal flaw in bar modelling, as long as we recognise it and in our teaching address it and overcome it: when we draw bar representations we must ask pupils for multiplicative as well as additive comparisons between the two (or more) bars drawn, so that they get into the habit of looking for multiplicative relationships between numbers and variables.
Whatever the mathematical concept and the associated model, if pupils’ thinking and reasoning about the concrete is going to develop into thinking and reasoning with increasing abstraction, they need us their teachers to help make this happen. Crucial to this happening successfully is our choice of representation / model, but equally important are
 the reasoning we cultivate and sharpen through the discussions we foster and steer;
 the misconceptions we predict and confront as part of the sequence of questions we plan and ask;
 the conceptual understanding we embed and deepen through the intelligent practice we design and prepare for the pupils to engage in and with.
Bars, fields and sticks’n’gaps are all contexts in which this can happen easily, naturally and powerfully, hence my advocacy (though not uncritical) of them. As for “BODMAS” … well, the less said about that the better!
Agree or challenge? Let me know robert.wilne@ncetm.org.uk or @NCETMsecondary.
Image credit
Page header by PauliCarmody (adapted), some rights reserved
