Blog: Ian Davies, Director of Secondary
Forget-me-not: how to embed mathematical principles and not let memory derail your teaching
Posted: 16/10/15Picture the scene. Your lesson goes really well – really really well. The students are engaged, they’re learning – and you can almost see the progress striding across the room.
But inside you can’t help feeling a tiny bit sad that, today of all days, no-one is observing you. No one of any significant standing – not OfSTED, the Headteacher or the Head of Department – were there to witness the magic of today’s lesson.
You’re trying your best to look on the bright side. The fact is, the lesson was a corker and WOW these kids can handle the rules of indices! In the absence of external validation, you bask instead in the solitary knowledge that today’s lesson was a true success.
Scene change. Six months on, you’re teaching the same class in a hot double period and the students are getting a little bogged down with the rules of congruency (admit it, you’ve never liked teaching geometry).
You decide to mix it up a bit and put on your excited voice: “OK – quick quiz!! Five questions”. As the mixture of mock groans and excitement floods the room, you decide to revisit THAT lesson – the holy grail of teaching, if you will – and you quickly make up five indices questions.
Suddenly the atmosphere darkens, it’s the nightmare of nightmares…they’ve all forgotten how to do them! HOW CAN THIS BE?!
I’m sure we’ve all experienced similar scenarios. Our students have forgotten some maths that we teachers were sure they had really understood – or even “mastered” in some sense of the word.
Some degree of forgetting is perhaps inevitable. Our students learn so many new things in maths over the year (and most unfairly they have to study other subjects too!) The question is, how can we minimise the forgetting and maximise the remembering?
One of the key strategies we use in our 5 year secondary plan is the structure of the curriculum itself. The approach here is two-fold. Firstly, we spend longer on new topics when they are first introduced, to allow for varied practice in different contexts which helps deepen understanding.
Secondly, we use the cumulative structure of mathematics itself to organise the curriculum. This means we keep revisiting concepts again and again, in different areas.
Here’s our Year 7 map of key ideas as an illustration:
The dark grey bars show where we focus on a particular concept and the light grey sections show where these concepts are revisited, but within a different area of focus.
Take fractions, decimals and percentages for example – a key area where students often get mixed up and confused because there are so many “rules”. We study decimals and tenths in some detail in the first half term of the year as we’re looking at place value. There is an obvious link in the second half term as we reinforce this understanding in increasingly complex multiplication and division problems. A key focus here is area, so it also makes sense to include questions where the given lengths/areas are decimal values and to link them to money problems.
Similarly when we move on to fractions, we link back to the previous angles and lines work with tasks like this:
We then focus on fractions, decimals and percentages together – making the links and encouraging students to see and use the connections.
During algebraic notation in half term 5, the notation of a/b is linked to division and fractions and so the key area is seen again – including substitution of decimals into algebraic expressions to reinforce multiplication and everything else of course.
Hopefully you are now seeing the strategy and won’t be surprised that in half term 6, when we look at pie charts (revisiting angle rules, amongst other things) and percentages of amounts – we again, explicitly, do fraction and decimal work. Here’s another example activity:
This of course continues in Year 8, where we extend understanding to include percentage increase and decrease. For example, we include equations of the form 2.1x + 3.45 = 10.8 (why ever do 2x + 3 = 11 anyway? You can just spot it!) and we throw in fractions when dealing with areas of parallelograms. And so it goes on.
The cynics among you might think fractions, decimals and percentages are easy examples which happen to fit naturally into lots of other mathematical areas. You might argue that other topics don’t lend themselves as well to this approach.
There is a little bit of truth in this – but the broader principle holds true. By regularly revisiting and applying previous mathematical learning throughout the curriculum, all pupils are in the best possible position to truly, truly master the fundamentals of mathematics (and are fluent in applying their skills, even in unfamiliar contexts) by the end of secondary school.
So going back to the rules of indices, how do we keep that fresh? Well, keep watching because I’ll be covering this topic in another blog soon…