This is a post about gimmicks; more specifically little tricks to aid pupils remembering “methods”. Its grown from a tweet put out about a month ago by a genuinely curious NQT. This is my campaign for why such tricks should be eradicated from our classrooms.

It’s taken me a long time to get around to writing this post. Mainly because the start of term has been hectic as always but also because I’m well aware I can be opinionated, set in my beliefs and a stubborn old mule and I wanted to completely reflect before I put this post out. I’m still as stuck I my belief as I was then.

The tweet that started this discussion was put out by an NQT and showed the “*butterfly method*” for adding and subtracting fractions:

(Google suggests the source of this image is this post)

This instantly triggered an automatic and physical reaction which quite frankly left me feeling more than a little queasy! Cue a tweet to gauge the response of Maths teachers. I’ll be honest here. I was expecting (and hoping) that most would have a similar reaction.

In some cases my expectations were met as Maths teachers demonstrated their disdain for such trickery:

And thanks to Kevin for making me feel normal by having exactly the same physical response 😉

And then I realised that we weren’t all singing from the same hymn sheet. And it was experienced tweachers who I utterly respect and admire fighting for the place for such methods in our classrooms:

Tweeter Mathsjem linked this to the similar but less cluttered cross-multiplying method which she discusses along with alternatives in this post.

I still just cannot get my head around any justification for using these methods in any classroom (I think I may be willing to make exceptions, as some tweeters argued, for Y12 resit groups who are not likely to be using or studying Maths at a higher level and just need to get over the random Grade C hurdle the government has kindly set them.)

So here’s my argument as succinctly and articulately as I can manage:

**1: Understanding is EVERYTHING**

We need pupils to understand *why* a method works; the actual nuts and bolts. Why? Because then they will recall it. Because then they can extrapolate it to different areas of Mathematics (algebraic fractions for example). Because then they can see and utilise the beautiful links in Mathematics which make us love the subject “that’s why I learnt about LCMs, that’s when I use them”.

**2. We should have high expectations for ALL**

I was shocked by the arguments about low attainers and those who struggle with Maths needing/ appreciating methods like this. Of course they will. You’re teaching them a method they can apply by rote. No thinking. Just doing. They’ll sit there happily doing it until the cows come home. They will never understand even if you show them after because this shouldn’t come as any surprise to you; kids are lazy and they’ll always try to take the shortcut. Here’s a suggestion: why don’t we start by assuming they *can* understand why it works if we teach it well and expect no less of them. Because if there’s one thing I’ve learnt it’s that pupils don’t remember the gimmicks when they walk out of the room, they remember by it making sense to them and being able to work from this understanding to the method and then to the solution.

So there’s my argument: understanding and high expectations. That’s as simple as it boils down for me.

We’re teaching pupils Mathematics. Not teaching pupils how to pass exams. I refuse to be part of an exam factory and it saddens me that some of you don’t. If pupils don’t understand the bottom line is I haven’t taught it well enough not that they need a trick

**Some useful links:**

I sing his praises a lot but Don Steward has some great resources for fractions and teaching for understanding (use the tag index at the side for other specific fraction topics)

This NRich article discusses teaching fractions for understanding

The Guardian – How to teach fractions article links to resources.

The book Nix the Tricks looks at this issue in more detail (thanks to Joe Fraser for sharing).

As one of the people quoted above, I’d like to respond. I fully agree that understanding is the ultimate goal of our teaching. Tricks don’t work and I would hate to see the ‘Butterfly Method’ on a formal assessment.

Many of the pupils I teach are scared of doing Maths, they’ve had so many failures in the past from as early as Infant school. I teach formal methods and proper concepts and they get it – until next lesson when the negative thinking shutters have gone down. The ‘Butterfly Method’ is a friendly face or rather a foot in the door. I’ve found pupils use it once or twice to warm up, but then as they relax they go back to the formal method using the correct terminology (eg common denominator). I’ve never seen a page of ‘butterflies’ in a book.

I don’t just teach a ‘trick’ – don’t get me started on the division of fractions taught as change to multiple and flip upside down. Is it so hard to explain reciprocals?!

I think a method that opens up pupils to trying something in conjunction with proper understanding is useful. Unfortunately the way that some people try to build success is to ‘dumb down’ the questions so pupils see lots of ticks on work. This doesn’t support the development of mathematical thinking and reinforces the negative self esteem through preventing growth.

So in summary:

Tricks – Bad

Tools to open up closed mindsets – Good

Proper understanding of underpinning Maths – Essential

Keep up the great blog – open discussions and opinions keep our teaching fresh!

Interesting article by Richard Skemp (1976) on this issue, ‘Relational Understanding and Instructional Understanding.’ He argues whether there are in fact two types of mathematics, not simply two types of understanding.

Thanks for that. Will have a read. Found some links here if anyone else is interested. http://www.skemp.org.uk

Meant to change name to @EJmaths

If you really believed that people should only use techniques that they fully understand then I suspect all of our mathematical repertoires would become somewhat limited. I haven’t met a maths teacher yet who hasn’t confessed to learning _how_ to do something only to acquire the understanding of what it is and how it works at a later date. Examples of this persist through mathematics where techniques are learnt only before the learner is competent to understand the proofs.One of the most obvious cases of this is the quadratic formula which most of us learnt to use long before we actually understood where it came from – this is not bad.

Perhaps this is a naive point of view but I think that it is legitimate to learn a technique/algorithm without understanding the details. Obviously understanding the meaning and use of the algorithm is important but the ‘how it works’ details are less so. This approach is common in many areas of life; computer programmers do it all the time and scientists are often good at looking up standard results which they probably couldn’t derive.

Going back to your fractions example – a person who can add fractions and knows what adding means has a useful skill. If they do not understand the exact details of the process then it doesn’t affect the use of the skill.

– obviously it is educational nirvana to achieve competence in the use of techniques AND full understanding of them.

My point was more about the fact that “methods” often lead to misunderstanding which leads to errors. As an aside I would never and have never taught the quadratic formula without showing pupils how to derive it or scaffolding it for them to do so in the case of top sets.

Thank you for responding to my post – please forgive me for disagreeing further. Your original point above was not a pragmatic one about methods leading to misunderstanding leading to errors. Under your ‘understanding’ section you say that understanding (vs methods) leads to recall and to extrapolation.This is not the case – the fashion in maths teaching for at least as long as I have been teaching (12 years) has been about teaching for understanding. In practice this has been at the expense of teaching methods and performing rote practice. Both of which are traditional elements of maths teaching. I am not arguing against teaching students to understand (no sensible person would) but methods are important and they do not always need to be understood. I would be very surprised if you couldn’t identify some areas which you teach without fully derivation and understanding (obviously not the quadratic formula). Perhaps one of:

– the formula for cones, spheres etc… at GCSE

– the reason for the n-1 in the standard deviation formula (I would be interested to see this explained to a GCSE statistics class from first principles.

– the working of the polynomial long division algorithm (particularly to students who probably have never been taught the traditional long division because it is a ‘method’)

– why we flip fractions and multiply rather than divide (even if you do explain this I would dispute that it aides understanding and or recall for most people – especially compared with KFC etc…)

– standard results all over the place – for example the location of centers of mass for sectors or arcs (mechanics) – students will not have been taught the integration necessary to derive them.

– the derivation of spearmans rank for GCSE statistics (another easy to use and understand formula – but good luck understanding why it works without further study)

Personally I think that teaching for understanding goes alongside teaching methods and giving opportunities for practice (ie/rote) as part of the skillset of an outstanding teacher. A lot of damage has been done by the insistence that ‘methods’ are wrong and shouldn’t be taught unless they are fully understood. ‘Understanding’ as you seem to envision it often hinders recall whereas a good method is easy to remember and apply. Finally, to address your second point – teaching methods is not about having low expectations, it is about expecting students to be able to do things in an efficient manner so they can then use those skills to solve more complex problems.

Regards

Rupert