One very clear memory I have of my time at school is the first term of work I did in Year 7. My teacher was properly "old-school" - he'd demonstrate for 15 minutes, then we'd work in silence for the remaining 45 minutes on similar problems. He was the deputy head, and we were all pretty terrified of him; while I don't want to debate if that's a good teaching model or not, one thing that really stuck with me was that first term's work - we did nothing but solving equations, pretty much from September until Christmas. It wasn't just simple linear though; we went all the way to solving quadratics using the formula, although I confess I didn't really know what I was doing or why I was doing it. As a result, I've always loved algebra and been pretty good at it (conversely, my geometrical reasoning skills are pretty dire).
Because it stuck with me so much, I tried it when I started teaching. Didn't work in the slightest; here's a prime example of having to find your own ways of explaining things - what worked for a group of top set kids in a pretty high-attaining middle school in Staffordshire just didn't seem to translate to my teaching with kids who had never understood algebra in the slightest and were just bemused by the pictures of animals appearing everywhere. I realised that, although this method had clicked with me, it was clouding the issue for many of my pupils, and I stopped trying to teach animal algebra soon after that.
During the second week of my NQT year, we had a department Ofsted - as I discovered later, the lady who observed me was Jane Jones, their mathematics leader, and our school was one of those used to compile their "Mathematics: Made to Measure" report published in 2012. As I was absolutely petrified at the thought of big ol' scary Ofsted turning up when I'd not even got settled in, I was pleased that my lesson went OK. However, in my feedback, Jane mentioned one thing that I'd not even thought about before - the way that my presentation was confusing some of the pupils:
At this stage, I was in the habit of writing the inverse operation below the equation before doing the next line of working.
Although this worked fine for most pupils, some were getting confused about exactly what the "divide by 2" referred to, and were simply dividing the x coefficient by 2, rather than the whole of the left-hand side of the equation.
Some were also mixing up their working with the equation, ending up with lines of working and solution blending into each other and creating even more confusion.
She consistently used this presentation in her working; she explained that she wrote down how she was transforming both sides of the equation on the right-hand side of the line, with her solution on the left-hand side.
I went away and trialled this with my Year 10s; they loved it as a method for organising their working, so I now present all my algebraic working like this.
Once I'd got my notation sorted, I started thinking more critically about the conceptual issues pupils were having with solving equations. My teaching at this point was still mostly procedural; do one example, then get the pupils to practise loads more. I think there's still an argument for doing this, as fluidity in solving equations is so important for higher-level topics, but I think I was teaching the process and hoping that understanding would miraculously appear later (it didn't).
Following on from my experience on the NCETM course last year, I started to try and incorporate more ideas about modelling situations into work across the board. I also took a different approach to the whole topic; before even going near solving equations, I got pupils to build their own. I used Fred and George (yes, the Harry Potter twins - they pop up a lot in my random examples), who have to have equal money spent on them for presents, and got pupils to model this, using p as the cost of a present.
We spent a couple of lessons creating our own equations before even thinking about solving them, and I started to encourage the line notation for pupils to show their working out.
This seemed to work quite well as a conceptual model for them, and was infinitely preferable to using animal algebra, as a present can cost an amount of money, whereas having "a cat is worth 5" is a little meaningless.
Something that was embedding misconceptions rather than illuminating them was my presentation of examples and problems to pupils. Making up an equation on the spot, I'm more likely to come up with ax + b = c than b + ax = c or even c = b + ax. Now, I'm consciously trying to include examples of equations written every way round and getting pupils to think carefully about what effect (if any) this has.
Where possible, I think it's important that pupils are presented with relevant reasons to solve equations. It's a great opportunity to link in prior knowledge, such as angles on a line or in a triangle, and get them to construct their own equations pretty much straight away.
I also now teach solving inequalities right after (or sometimes simultaneously to) solving equations. I was initially mystified that pupils who could solve complicated equations well somehow got in a complete muddle with inequalities, and I think some of that is our fault; it's frequently presented as a completely disparate topic, not touched until Year 10 or 11, rather than essentially the same process with a different symbol in the middle.
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