Three weeks into teaching my Year 10 GCSE class, I’ve come to the conclusion that a large part of my task is going to be psychological. In a class of 28, there are so many different responses to the process of learning maths as well as to the maths itself. Some of these are probably common to all forms of teaching but others seem more specific to learning maths. The last time I wrote about maths it was based on observing lessons – ( Meaning and Magic amid the Muddle of Mental Mathematical Models ). There are so many mental models to wrestle with. All very interesting as an observer but now I’m teaching maths myself it’s a constant challenge – and a fascination.
For context, this is Set 2 out of four; all expected to get Grade 4/5 as a minimum; all aiming for more. We’re following the MyMaths Edexcel Higher scheme with what I think is a great textbook. The first unit is a recap unit called Calculations focusing on ordering decimals, multiplying and dividing by decimals, long multiplication, significant figures, decimal places and rounding. Everything in the chapter is something they would have encountered before, pitched slightly higher.
Here’s a sample of the mindsets I’ve encountered:
I don’t know.
M says this every time I ask her a mental maths question. What’s 45 x 9? I don’t know. I repeat the mantra: “I’m not expecting you to know; I’m asking you to work it out”. She struggles with that idea. She perceives other students to ‘know’ answers rather than realising that they’ve also had to work them out.
Guess: you might get lucky.
A starter activity ordering fractions and decimals. eg 0.1, -0.4, 2/7, 0.22, 3/8, 2/5, 1/4, 0.3…. etc. The boys in the front row just banged them out in some order making lots of errors. They hadn’t worked out 2/7 as a decimal and often put it above 0.3; similarly with 3/8. They were surprised to learn that even I don’t ‘know’ 2/7 as decimal; it’s not a fact I carry around. In my model answer I showed them that I have go through the drill: 7s into 2..(bus stop method – as they call it) to produce 0.29 (2dp). Guessing isn’t going to work. This comes up a lot. Answers need to worked out, not just plucked out of the air by having a punt.
I shouldn’t be in this class.
We’re pitching high so some students are up against their limits fairly often. D’s early conclusion was that he must be in the wrong set. ‘I don’t belong in this class’. He’d been to see his Director of Studies. I had a quiet word to encourage him…he’s bang in the middle of the class but this is what it’s like when you’re learning and being challenged. You struggle but then you get better at it. His attitude since has been exemplary. But he’ll probably never find it easy and I can see that being a challenge – to keep him motivated. Do I soften? Give him some easy wins? Not yet.. I think he needs to toughen up a bit first.
It’s too easy.
Cocky and Glib on the front bench. Scoffing at the need to review the basics. But wait… what are those errors I see? C and G thought that writing out the method in full was rather beneath them. Long multiplication? Easy? Well.. yes, if you follow the method! A lesson learned. This is a frequent barrier/masking technique – the polar opposite of ‘I don’t know’. Rather than accept that things need to be worked through systematically, the bravado of ‘too easy’ leads to all kinds of surface learning. I’ve tackled this head on with them.
My method is better.
Another masking mindset. Column addition and subtraction with decimals works every time. ‘But my method also works and it’s not like that Sir’ said J. Show me….. Oh! Not quite so easy after all. This was just a way to avoid the need to follow a method in a disciplined manner all the way to the end. There must be short-cut? Nope. This is how you do it.
Distorted method memory
B had the idea that when you’ve got numbers in columns you always carry to the left. But we’re doing subtraction. Here you borrow from the left. He kept on carrying, every number borrowed was added back on. NO. ‘But don’t you have to carry?’…..He had a half-learned method that was misplaced. He couldn’t shake it off. We had to go back to basics; the reason for borrowing and carrying. He’d completely lost sight of that. Method learnt; model forgotten. It’s a common issue.
A minus x a minus is plus. Yes? Why? No idea. Dividing by numbers smaller than 1… complicated! These are the maths model issues that stem from shaky foundations. 200 divided by 4. No problem. That means breaking up 200 into 4 pieces. But 20 divided by 0.4? CRISIS! The idea that division also means ‘how many times does 0.4 go into 20’ is so much harder. This seems to stem from insecure associations between multiplication and division. 50x 0.4= 20; that’s ok, but not the reciprocal calculation. Times tables need to learnt as divisions as well – and some students haven’t got this embedded.
Proximity to others in the classroom is a constant issue. Look left, look right; have a sneaky peek at what the others are doing. Grrrr. Y has a huge dependency on this. He needs to sit by himself because otherwise he spends the lesson like a tennis spectator. He’s too insecure to rely on his own problem solving methods and continually looks for clues and tips from his neighbours. This explains his maths test performances. Not good. We need to do more on meta-cognition and self-help; where to start; what to do next. By ourselves.
Cruise now, work later.
The age-old resistance to hard work; lazyitis; procrastination. I’ve preached; I’ve lectured; I’ve given the inspirational talk. Every lesson counts; work hard and you will succeed. For some this is all too unreal. It’s only Y10. The exam that matters is May 2017. That’s like AGES. Relax Sir, said S. I’m doing my work… He’s in for a shock. Quite soon in fact because he’ll need to complete the Assignment by next week. The gap between the students with drive and determination and those without it is quite stark – every lesson getting through fewer problems, getting less practice and generally being sluggish in their thinking. No quick wins here… but it’ll be key to getting them where they need to be.
Banter…negative peer influence
Cocky got an answer wrong. No big deal. We discussed this as I was moving around the room. K was thrilled. ‘C got it wrong… C got it wrong, Shame-up!!!’ K is someone who gets lots of things wrong and needs to work hard to keep up. Stop the class! Let’s talk about this. We examined the whole peer pressure issue. Boys with fragile self-esteem needing to act cocky one minute, knock someone down the next and then panic at being exposed for making errors – this is not a recipe for success. I’m calling it out every time so we can clear mental space for the actual maths problems in hand. It’ll need continual reinforcement because these attitudes run deep.
Positive peer influence.
The greatest joy of this week was hearing M explaining a rounding/decimal places problem to E. E hadn’t been receptive to my interference in her flow of work…’I’m Ok Sir..I get this’. But M noticed what I had – that E made a mistake. E was prepared to listen to M, but not to me! 273.3 to 1dp means that it could have been 273.25 to 2dp .. so 273.3 to 1dp is not ‘definitely bigger’ than 273.28. Harnessing positive peer influence is something I need to do more of.
The journey continues…..