3 August 2014

Consistent Mathematics

‘But Miss, that’s not what our last teacher said’

I read an interesting article this week about mavericks in schools. The article talks about the need for consistency in behaviour management - this got me thinking about consistency in mathematics teaching. Us maths teachers all have our own teaching style. We even have preferences for particular mathematical methods. I think it's fine that we all structure our lessons differently - pupils adapt quickly to their new teachers at beginning of each school year. Children are far better at adapting than adults. But it's worth thinking about consistency in our mathematical approaches, terminology and standards, and that's what I want to explore in this post.

Some of the things we teach, or the way we teach them, are confusing. Some of these things we can change, and some we can't. People are generally resistant to change, but now is a good time to make changes because many of us are implementing new schemes of work.

Changing how we teach the order of operations
Whether ‘BODMAS’ is a good teaching method isn’t really up for debate anymore. We all know it doesn’t work. It gives students the impression that division should come before multiplication, and addition before subtraction. There’s been lots of excellent posts about this topic - like this post 'Poorly Executed Mnemonics Definitely Addle Students' by Bill Shillito - so I won’t go through it all again, but let’s think about how to fix it.

As a minimum, we need all teachers in a school to be on board with a new approach. In an ideal situation we’d get all maths teachers in the country – no actually, in the world – on board. Now, in a rather tiny step towards changing the world – I’ll start by discussing this in my department meeting on the first day of term in September.

If our pupils have met BODMAS at primary school, we’ll have to tell them to disregard it, and we’ll have to show them why by demonstrating the misconceptions it may lead to. Like many others, I’m an advocate of teaching GEMA or GEMS instead. Some American maths teachers have already written blog posts about this approach. It’s a pain that most of my lovely resources reference the term BODMAS though. If I get time over summer I’ll make some alternative GEMA resources.

Changing quartile terminology
Now I don’t think I’m going to make any progress with this one because it doesn’t result in any serious misconceptions. Am I the only person bothered by the fact that we use the term ‘lower quartile’ to describe both the bottom 25% of the data and the cut-off value 25% of the way through the data? John Tukey, inventor of the box plot, used the word ‘hinge’ instead ie the lower quartile consists of the values between the minimum value and the lower hinge. I prefer this terminology, and I’ve seen it used in a few places, but I doubt it will be widely adopted.

Maths we want to change but can’t
When we introduce the idea of squaring sinx at A level, we tell pupils to write sin2x instead of sinx2 (to clarify that it’s not the x that’s being squared). But we have to carefully explain that sin-1x doesn't mean the reciprocal of sinx, but actually the inverse ie arcsin. The obvious solution to this confusing notation is to stop using -1 to indicate inverse functions – but then we’d need to redesign calculator buttons! Why didn’t calculator designers write arcsin, arccos and arctan on the inverse trig buttons?

The use of -1 to mean both a reciprocal and an inverse function is one of the many mathematical annoyances discussed in this post on Reddit ‘What Mathematical Conventions would you change?’. You might also be interested in this article ‘The most common errors in undergraduate mathematics’.

Inconsistent mathematical methods
I don’t think it’s a problem if pupils use a variety of methods to answer the same question. Multiplication methods are a good example of this. There’s been some discussion about multiplication methods recently on twitter - most people agree that if we get the right answer then it shouldn't matter what method we use.

As teachers we have our own preferences. And pupils may have their own methods too, learnt from teachers, parents, tutors, Kumon (grrr...) and so on. A good example is the way we find a highest common factor - there’s many different methods, all equally valid (I'm convinced I've finally found the best method though!). Another example is the approach we take to substitution and rearranging. When I use the Cosine Rule to find an angle, I substitute the numbers into the formula a2 = b2 + c2 - 2bcCosA then I rearrange. But I’ve noticed that most of my colleagues tell students to rearrange the algebra before they substitute. It might confuse students when they first see a new approach, but this opens up a useful dialogue about the equivalence of different methods.

Here’s a few more examples of where teachers take different approaches:
  • When teaching trigonometry in right-angled triangles, not everyone says ‘SOHCAHTOA’. At school I learnt a mnemonic instead. I’ve recently discovered trigonometry triangles too.
  • I’ve already written about differing methods for finding the equation of a straight line in my linear graphs post.
  • I once had a Year 11 class who’d been taught by their previous teacher to find the nth term of a sequence using the a + (n-1)d formula that we normally teach at A level. It worked well.
  • I don’t use the ‘FOIL’ order when I expand brackets on the board and this confuses some of my pupils who seem to think that FOIL is the only correct method.

Inconsistent vocabulary and pronunciation
I say parab-ola, my colleague says para-bola. I’m pretty sure I say it right (in fact, I just checked here to make sure!). But I had a moment of doubt when my pupils all laughed at my pronunciation because they’d heard it pronounced differently the previous year. Now this really isn’t the end of the world, but it made me wonder about my pronunciation of other words, such as cyclic (as in cyclic quadrilateral), alternate (as in alternate angles) and ln (as in natural log).

As for vocabulary, I switch between the terms 'index' and 'power' all the time. This is just one example of my inconsistencies. I don't think it's a problem to use a variety of words, in fact it might be a good thing, as long as all new vocabulary is explained to students.

Inconsistent standards
I taught a Year 12 last year whose work was set out so badly it was almost impossible to follow. She scattered her workings all over the page, sometimes showing me where to look next with flowery arrows. She put multiple equals signs on the same line of working, and her final answer was always hidden away somewhere. Her work was also mathematically incorrect. Although I commented - many times - on the state of her workings, I decided it was more important to focus our limited time together on fixing her mathematical misconceptions. I felt frustrated that her appalling approach to setting out her work hadn’t been dealt with at an earlier age.

I remember when I was a student in the Upper Sixth Form, my maths teacher wrote on a homework that I’d used the symbols for 'equals' and 'therefore' interchangeably. I’d seen her use a double arrow symbol on the board and misunderstood its meaning. I was utterly confused (and embarrassed!) because I’d been doing it wrong for over a year and she’d never mentioned it before!

I also think we could do a lot more to encourage clearer mathematical handwriting (both numbers and algebra). We all have our own styles though. I’m guilty of writing diagonal lines in fractions on the board even though I know I shouldn’t.

A case study
It’s worth reading this good practice report from Ofsted: “In many schools, students’ long-term progress in mathematics is slowed by inconsistent teaching. At Archbishop Temple School, very effective leadership of a strong team of teachers ensures that students experience teaching that is consistently good or better. Students in all groups benefit because different teachers use common approaches and teaching styles”. There are lots of good ideas in this case study, such as the formal sharing of resources between teachers and ‘book swapping’ – not in the critical ‘how much marking do you do?’ sort of way, but with a view to encouraging consistent practice.

I think that new products and technologies like Complete Mathematics by La Salle Education will do a lot to support teachers, improve consistency and raise teaching standards. The flipped classroom approach, which has been gaining popularity recently, may also have an impact on consistency.

I’d be interested to hear your views on whether we need to improve consistency in maths teaching and how we could go about doing so.

Image: mentalfloss.com


  1. Thanks Jo - yet more to help with planning for the new year. The language of negative numbers is something that students and teachers can struggle with too. I always try and say 'negative 4' instead of 'minus 4' when using a number, although it often helps to ensure that students understand what is meant by this, otherwise they'll get confused.

  2. That's a really good point. As I'm still relatively new to teaching, it helps me to think about the way I communicate. I probably still sometimes say minus when I'm talking about a negative number.

    I've trained myself to say zero instead of nought but I often hear pupils say the letter O when they mean zero, and I'm not very consistent about correcting them.

  3. Excellent post, I agree with a lot if it. I have a few previous posts you might find interesting on these topics, will tweet you.

    I am equally irked by some of these points. I Don't like trig triangles though!

  4. Also, I'm totally with you on the inverse functions, it's a real pet hate of mine.

  5. Thanks, glad you like it. And thanks for the links, I was particularly interested in your blog-off about straight line graphs because I've always wondered whether there was a 'better' method.

    I'm not sure about the trigonometry triangles either - a student told me about them and I used them in teaching for the first time this year. I figured it's no different to me using speed/distance/time triangles, which I've never questioned.