Posted 1st October 2019
Have you ever wondered why your brain defaults to carrying out a mathematical calculation a certain way?
Over the years whilst teaching mathematics, I’ve always been passionate about the use of multiple methods, both as a learner and a teacher of maths. The idea of being able to work out a question in a variety of ways and then reason behind why we choose one over the other is something that fascinates me, and I continue to explore this now.
Helen Drury, author of Mastering Mathematics, says ’A mathematical concept or skill has been mastered when, through exploration, clarification, practice and application over time, a person can represent it in multiple ways'. It is vital that as teachers, we expose students to concepts presented in a range of formats. It’s equally as important that they also have a wide range of tools in their toolbox to tackle each situation in the most efficient and effective way.
A teacher once said to me “I have some great lessons on fractions using pizzas. The kids love it, so I always teach fractions using pizzas now”. I’m sure this teacher’s lessons are great, and I would encourage them to continue to use it, but I am concerned that if the children only learn about fractions using pizzas, how would they tackle a question like this one?
There are a number of ways students could tackle answering this question. For example, they could split the shape into smaller triangles:
Cutting the shape up and placing each triangle on top of each other will show that each triangle is the same size. They could then discuss whether this was a proof or a demonstration.
Students could also find the area of the shaded part to show it is one quarter, either by choosing numbers for the length and width or using algebra.
Again, they could discuss whether one numerical example is enough to prove the result, or whether the algebraic approach calling the sides a and b would be better…or could the idea be generalised using words rather than algebra? Or was the visual approach better after all? What interests me is the huge amount of mathematics that can be garnered from the problem, hence this discussion of alternatives, rather than the ‘You do it like this, now do lots of the same’ approach that can sometimes mask how creative a subject mathematics can be.
Should we let students choose their own method?
‘Which method do we teach?’ ‘Do we have to teach every possible method for every mathematical concept?’ ‘How can we encourage students to see things in different ways?’
These are the sorts of questions I often hear when supporting teacher colleagues across schools in the country and within the team at White Rose Maths.
Multiple ways can certainly be introduced through mathematical talk, exploring a problem in the classroom. For example, how would you work out 4 + 3 + 18 + 6 + 7? Many students (and indeed adults) would start working from left to right and possibly start to struggle mid-way through due to cognitive overload. But, if we exposed students to thinking differently and looking for links in calculations, they would quickly spot that 4 and 6 and 3 and 7 both make 10, therefore the answer is 38. 10 + 10 + 18 is a much simpler calculation to hold and perform in your head. Through talking strategies and allowing students to share ideas in the classroom, it can give greater exposure to how we calculate more efficiently.
I have found effective teaching exposes students to multiple methods for solving a mathematical problem – through discussion and/or teacher/peer modelling – then pull these ideas together and decide on the most efficient method to collectively move forward with. Students need to be empowered to make appropriate decisions depending on the problem they face. Too often we see students doing things like this:
They are mindlessly using a procedure rather than thinking about the most efficient and appropriate method. The fraction question in particular would be so simple if students spotted that 26 is a multiple of 13 giving an answer of .
How do we get students thinking differently to support this way of working?
In hindsight, now, as an adult, the idea of multiple methods is evident even more so in the workplace. I often get shown efficient ways of using basic functions on an IT package and now wonder, if I had been taught multiple ways at the start when using Microsoft Office, would this ‘gap in knowledge’ exist 15 years later? The saying ‘you don’t know what you don’t know’ certainly springs to mind in my case!
Conversely, what is the rationale behind the need to teach or even learn multiple methods, is efficiency & time the goal or accuracy? Certainly, in today’s 24-hour society, I would argue both. So where does flexibility sit in all this? That’s my point, how can you be flexible if you don’t know any other way….is it up to me as a learner to work this out and create my own efficiency or should I have been taught it?
Secondary Mathematics Specialist