Lines, squares or plain? The pages of maths exercise books, I mean. For years I have insisted that pages have to have squares. Without vertical lines as well as horizontal ones how could my students possibly draw all the right-angled triangles and bearing North lines and coordinate axes and so on that I would tell them sternly had to be drawn neatly and with a ruler?
Over the last year or so, I have started to doubt my certainty. I have become concerned that the grid pattern of squared paper is a constraint, a restriction, on my students’ thinking. For example: coordinate axes. It doesn’t ‘die-on-the-barricades’ matter that the x-axis is absolutely horizontal and the y-axis is absolutely vertical, but what is non-negotiable is that they are perpendicular: do I say this, do my students understand this? No, it seems, given my Y11s’ certainty in a recent revision lesson that none of these correctly showed y = x:
But what has convinced me to order plain-paged books is bar modelling. Four years ago I’d never seen or heard of a bar model, and when I first did (in a primary maths lesson in Shanghai, a point I politely mention if I read or hear someone claiming that bar modelling is uniquely Singaporean!) I didn’t grasp the power of it.
Now I’m a passionate convert to bar models. I can’t imagine myself teaching in the future without regularly drawing bar models. Not because they are ‘on trend’ or because they get the thumbs up from certain twitterati, but because they are effective. Bar modelling has dramatically improved the confidence and quality of my students’ reasoning. Four years ago, a trio of questions such as
a) A shop sells red roses and pink roses in the ratio 5:2. One day 420 roses are sold. How many pink roses are sold?
b) A shop sells red roses and pink roses in the ratio 5:2. One day 420 red roses are sold. How many pink roses are sold?
c) A shop sells red roses and pink roses in the ratio 5:2. One day 420 more red roses than pink roses are sold. How many pink roses are sold?
left even my highest attaining students floundering. They realised that there are differences between the language in the questions, but they did not have a systematic, reliable way of unpacking and laying out those differences. Now, though, when given these questions in a recent revision session all my students confidently sketched
then said to themselves “ok, where does the 420 go on this picture?”, then drew
and then reasoned that
a) Each square ‘is worth’ 60 roses, so 120 pink roses are sold
b) Each red square ‘is worth’ 84 roses, so 168 pink roses are sold
c) Each of the three ‘surplus’ red squares ‘is worth’ 140 roses, so 280 pink roses are sold
(It was a much better revision session than that of the y = x debacle!)
Of course, these three ratio problems can be solved abstractly and algebraically:
a) 5x + 2x = 420
b) 5x = 420
c) 5x = 2x + 420
But the pictorial representations are so much clearer in their meaning, and procedurally so much easier for the students to use to solve the problem.
When I train beginning teachers, they often say “Don’t students take a long time to draw the bars? It’s much quicker to solve problems like these algebraically, isn’t it?” Yes it is, for us the teachers who have experience with algebraic manipulation; but for our students who don’t, it isn’t. Success with bar modelling gives students the missing confidence to start to reason abstractly. Once they know for sure the answer from a pictorial argument, then they embark on the algebraic reasoning with expectation of success: they know where they are heading.
But I don’t always want my students to reason abstractly. Why do they need to when the picture conveys all that they are thinking? Now, after lots of practice, and I do mean lots of practice (to paraphrase Michael Jordan, I’m good at bar modelling because of all the models I’ve drawn that didn’t answer the question I was tackling), I turn to algebra increasingly rarely. Give me a question such as
- In my garden, the ratio of red roses to pink roses is 3:2. The ratio of pink roses to yellow roses is 5:2. What is the ratio of red roses to yellow roses?
and I’ll want to draw my answer, not write it:
And this is when I want plain paper. If I draw bars as a whole number of squares, a reader might ‘follow the lines’ and wrongly think there’s a relationship between the sizes of the bars and the values that the bars represent. The size of the two bars I sketched to represent the pink roses (in the top row) was about the same size as the two bars I sketched to represent the yellow roses (in the bottom row), but I was not implying that the number of pink roses was the same as the number of yellow roses.
A bar model is a sketch not a scale drawing. I have found that plain paper has made that much easier for my students (and me, when it’s late and I’m stumped by a ratio problem!) to remember. Now where did I put next year’s stationery order form?
If you’re new to bar modelling and want to practise before you start using these ideas in lessons, there’s lots of guidance on the internet. I’ve enjoyed the games at thinkingblocks.com, for example: fun and free. If you’re ready to start making pictorial reasoning a significant component of your classroom teaching, then you might like to consider the new Discovering Mathematics resources being published by Oxford, which bring Singapore’s long experience with bar modelling to the English KS3 curriculum.
Robert Wilne has over 20 years’ experience teaching mathematics in primary and secondary phases: he has on the same day taught De Moivre’s Theorem and number bonds to 10! He has also been a school governor in both phases.
As the Deputy Director (Maths) for the Atlas Teaching School Partnership, Robert leads the development of teaching and learning in Mathematics in four primaries and three secondaries within the Haberdashers’ Aske’s Federation. He also co-steers Atlas TSA’s maths improvement work across the London borough of Lewisham.
Robert is currently the UK Consultant for Discovering Mathematics: the UK adaptation of the leading Singaporean secondary maths series.