Order and Disorder

Happy New Year everyone. One of my resolutions for 2016 is to get less worked up about my pet hates and try to learn more about my own bad habits.

I, like many mathematics teachers, have some pet hates about some shortcuts (while of course being blind to my own bad habits). One that bothers me is when my PGCE students tell me that they are going to teach ‘BIDMAS’ (or other versions such as BODMAS or BEDMAS). I have cultivated an equally irritating reaction to this – I look blank and then say, with a puzzled expression, ‘I thought you were going to teach some mathematics’. Most are then able to say ‘I mean order of operations’, but some cannot and this is deeply bothering. We know very well that if mathematics appears to be a collection of meaningless rules, young adolescents are at best prevented from enjoying the subject and at worst thoroughly turned off. Furthermore, the implied order of BIDMAS does not always give the correct answer, and here is the clue to its purpose.

The reason why it is important for students to learn the conventional order of operations is because mathematical notation has to be interpreted to avoid ambiguity. So rather than presenting ‘order of operations’ as an authoritarian rule, why not immerse students in the ambiguities so that they understand the need for conventions?

A quick look through various new textbooks shows a variety of approaches to this issue. Most textbooks avoid the ambiguities and some even offer the rules before they are necessary, i.e. when a left-to-right approach will give the mathematically correct interpretation. Why would I need a rule to calculate 6 x 5 + 4? Why do I need a mnemonic to decide how to calculate it?

Fortunately, it is a natural human activity to try to sort, match and classify a variety of similar looking objects, such as these nine very similar calculations:

6 × 5 + 4 6 × 4 + 5 5 × 6 + 4
4 + 6 × 5 5 + 4 × 6 4 × 6 + 5
6 + 4 × 5 6 + 5 × 4 4 + 5 × 6

Try printing these nine calculations on cards and asking ‘same and difference’ questions:

Do any of these give the same answers? What has to be assumed to make some of them equal to others? Do students get the same answers as each other for the same calculations? Do all calculators give the same answers to the same calculations?

It is interesting that some textbooks do not introduce brackets until later, although these may have been introduced in primary school in years 4, 5 or 6 to express distributivity. It is crucial therefore to find out what your students already know about brackets. When thinking about multiplication many children find the discovery that ‘multiplying two numbers by five and adding them gives you the same answer as adding them and multiplying the total by five’ is quite exciting, and they may have been shown brackets then as a way to express their discovery.

How would you extend the set of sorting cards to include their existing knowledge of brackets?

Can the cards be arranged in ascending or descending order of value? What is the largest/smallest value that can be made using digits 4, 5, 6 and the operations of addition and multiplication?

Many students will be used to this kind of activity from primary school.

Back to ‘BIDMAS’ – I would also ask if it carries with it too many warnings to be useful. For instance, some textbooks interpret it to mean ‘brackets first, then indices, then division, then multiplication, then addition, then subtraction’ but it is quite easy to make up examples where doing addition before subtraction gives a wrong answer e.g. 6 – 5 + 4. Other textbooks clarify this by interpreting BIDMAS to mean ‘brackets first, then indices, then division or multiplication, then addition or subtraction’. How are students to choose whether to do division or multiplication first? How are students to choose whether to do addition or subtraction first? The apparent simplicity of the BIDMAS ‘rule’ begins to become more and more complicated in order to work correctly.

The new curriculum for primary Year 6 offers strong guidance that algebra should be introduced to express what children already know about number operations and relations; the new curriculum for Key Stage 3 includes a similar statement. If students know that ambiguities need to be sorted out, they are likely to be more willing to learn how mathematicians sort them out through precise notation. The notation follows from their mathematical needs, rather than their learning needs following from the notation.

Why not let us all know how you get on with the card sorting approach, and what other interpretations you had to deal with when you did it? Have you created and used other card sorts yourself?

All the best,

Anne Watson

Anne WatsonAnne Watson has two maths degrees and a DPhil in Mathematics Education, and is a Fellow of the Institute for Mathematics and its Applications.  Before this, she taught maths in challenging schools for thirteen years. She has published numerous books and articles for teachers, and has led seminars and run workshops on every continent. 

One thought on “Order and Disorder

  1. Nice and important article. One thought: “algebra should be introduced to express what children already know about number operations and relations;” If that one simple guideline was followed, I think maths teaching would be transformed and massively improved.

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