At the bottom

I've just found out that I'm getting bottom set 11-year-olds next year. It's pretty exciting, because I've never taught that level before - I've taught higher level 11-year-olds, and bottom set 13+. Bottom sets at that age are a bit of a mystery to me, and from the frantic research I've been doing, to a lot of people. Before they enter the world of GCSE target grades (usually 'Get a grade! You're upsetting our A* to G %!') nobody really minds what you do with them. In fact, it's terrifyingly unsupervised and unsupported. And here's the shocker.

Most children in bottom sets in the UK make no progress in maths between the ages of 11 and 14.

None, in over 3 years. That's the sort of stat nobody wants to own up to, and it's the sort that nobody even tries to address, because it looks pretty impossible.

There was an ugly facebook argument last week that I saw, where someone suggested that education isn't good at producing clever, useful people because, well, teachers aren't clever, useful people - if they were, they wouldn't be teaching, would they? They'd be writing software, or working in finance or consultancy, or the civil service, or something a whole lot better than teaching.

It got quite a response, which I shan't bother repeating - if you're reading this, chance are we don't disagree about this anyway. However, it got me thinking of a parallel argument:

Maybe education is so terrible at catering to the needs of the bottom 5% of the (mainstream) school population, because none of the teachers were ever in the bottom 5% of their school populations. In a conversation recently, it turned out that none of the people present had ever been in anything lower than set 2 in school. Bottom sets are a mystery to teachers, and to education advisers in local or national capacities.

A bottom set year 10 kid had me in tears a few weeks ago when she apologised for her shocking behaviour, and explained that she'd just found out she was failing Drama GCSE. As in, she was predicted to get a U, and the teacher couldn't see a way for that to change. She said 'Drama's my best subject! It was the only subject I thought I might get a grade in. Now I'll have nothing at the end of next year - no grades at all. What's the point?'

When she arrived age 11, she met with a host of teachers who had never been in her position, and had little idea how to teach her. She's stuck in a system that really doesn't cater to her, because it was designed by people who had never been in her position. It's probably too late for her to get much out of secondary school, but I want a different experience for my new 11-year-olds, fresh-faced from primary school. How?

Novelty in Equation Solving

I tried a new method for getting kids to rearrange equations this week. Astonishingly, given the amount of stuff out there, I haven't been able to find it anywhere on the web, but I'm sure I can't be the first to think of it.

It was stunningly easy. The majority of the class could already do basic rearranging, such as 3a + 4 = 2a + 6 . I gave them that question then, after they demonstrated the method and answer, asked them to divide themselves into 2 roughly equal groups, based on confidence. Now, I hear what everyone seems to be saying about grouping by ability, but I had a real reason for this. Also, it was the kids' own choice which group they went in. Some of them surprised me. Most got it right. The ones that didn't quickly realised and moved.

Each group then got given a bunch of equations on slips of paper, and each student taped one to the top of their whiteboards (held portrait). The more confident group got fiendish, nasty, horrible equations. The others got ones of the sort above. They sat in a circle. Each student had to write the first step of rearranging the equation under the taped paper, then pass the whiteboard to their left. They would get handed a whiteboard from their other side, read what was on it and write the next step for that equation.

It was fantastic. The amount of brilliant maths talk, productive arguing and explaining was huge. Students were teaching each other and figuring out what worked and what didn't. A sample exchange:

Kid 1 'This step isn't right' *passes board back*
Kid 2 'Yes it is, see, I multiplied by 3p'
Kid 1 'Huh, yeah, I guess you did'

pause

 Kid 1 'But I can't see what to do next, after that step. I don't think we can solve it starting like that. Can you do a different step?'
Kid 2 'Oh, I guess not, ok, sure'*

I sat with the less confident group to start with, and when they'd got on the the second set of equations I moved over. The others were really struggling, but making lots of progress. At the end, I did the litmus test.
'So, who learned something today?'
'Who was challenged today'
A resounding success. So much so, that next lesson I got them to split into 4 smaller circles, and we did some more.


*For interest, this equation had a quadratic on the top of a fraction, which they needed to cancel - if they didn't, they ended up needing to factorise a quadratic with a non-1 value of x squared.

E.g.

At the maths workshop I went to recently, my head of department and I planned an activity around one idea: asking questions of the form 'give me an example of...' There are 3 main ways of using this, and I've been trying them a lot recently.

1. Ask students for an example of something, and keep adding criteria until there is one correct answer (attributed to lots of different people, as far as I can tell). Cute, but I don't like the idea of their first answers being 'wrong', so I want to be careful about how I phrase this. Also, it's less focussed on creativity.
For example: 'Give me an example of an even number.' 'Now see if you can find an example that's a multiple of 3' 'Now see if you can find an example that's a square number' etc. until everyone has, say, 36.

2. Make it really open ended: 'Give me an example of a hard question for this topic. Why is it hard? What would be an easy question?' I like this in really small groups or one-to-one, but in a class of 30+ I find it tricky to have everyone involved in the discussion.

3. Ask for an example of something, then compare. My favourite way of doing this involves everyone thinking of the first thing that comes into their heads. Get it down on paper, get it out of the way. Now the pressure's off, everyone's got someting, think of another example that you don't think anyone else will have got. I did this with 2 classes last week, asking each of them for a shape with an area of 5. After the initial shapes (rectangles, mostly, and a few right triangles), it got interesting. I asked them all to stand, then asked the people with rectangles to sit down. Then the people with triangles, then parallelograms, then trapeziums. Then we looked at what was left. The second time I did this, I talked more about the different categories as the students sat down, or rather, got them to talk about 'how to draw an x with area 5', which allowed them to share expertise and made those ideas seem more valued.

We extended this into another favourite tactic, question swapping, by asking each student to pick an integer between 6 and 20, then draw a shape with that area on a card. Then I paired them up and swapped them, and they had to find the area of the shape drawn on the card they were given, and check their answer with the person who drew it. Paired by ability, this meant I managed to challenge most of them. Incidentally, it also meant I won a bet with my head of department: I said that I was sure at least one of my top set would draw a circle. 'Surely not! What would you even write on it? The radius? Like....root of (5 over pi)?' In fact, I got a circle with a decimal radius, but said I wouldn't accept inaccurate answers, could they write it accurately please? Then they had it! Never underestimate.

The creativity in maths is so often lost, especially when the teacher does all the creative work! What would make a good question here? How did they get that answer? Handing the reins over for a bit was far less work for me in terms of preparation, though harder in terms of generating ideas and think-time. It got some kids involved who rarely volunteer answers in maths, either through anxiety about their ability or general shyness. It was so nice to see the kids who usually think their strengths lie in what they see as 'creative' subjects have a chance to shine in maths, and it gave the 'good-at-maths' rule followers pause for thought. I just wish more lessons could be more like this.

What's the point?

'The focus should really be on progress. People complain that the children don't enjoy my subject, and not many of them carry it on to GCSE or A Level, but that's not the point. That's not what OFSTED are looking for. Progress is what really matters.'

That's a direct quote from an 'expert' on an INSET day this week. It terrifies me. I'm all for not needing to be liked as a teacher, and not needing every lesson to be fun, and not sacrificing subject content for the sake of sheer entertainment, but if you're making students dislike your subject, to the extent that they don't want to carry it on, something is wrong. Statements like these, from professionals, make me think that all the Standards and OFSTED stuff about 'teachers should inspire a love of learning' is not being taken seriously. Of course it's harder to measure than academic progress, but we shouldn't chuck it out because of that.

I think, as a maths teacher, that it's massively important that students feel confident working with numbers, and not put off by them for the rest of their lives. Moreover, we may lose our best students from the subject if we take the approach that we don't care if they enjoy it as long as they're learning - the brightest will learn, but they'll drop it as soon as possible. Two kids who now have level 8 in year 9 told me at the start of the year that maths was their worst subject. We need more bright kids doing maths A level.

What about the gender divide? Stats suggest that girls are even more likely to be put off a subject than boys by a 'shut up and learn' approach, and their talents are more likely to be left undiscovered. We can't afford the gender gap in maths to grow any larger.

So much for the top end, what about the other side? If the weakest students don't enjoy maths, they will feel less confident and be even more likely to chuck in the towel and give up. We don't need their motivation to decrease any more - for these students, it will interfere with their progress, and that (so say these experts) is all-important.

Some kids are never going to like maths, say its detractors. Maybe. But we're losing kids who could like maths, at the top end where it's affecting the people we have in the subject, and at the bottom end, where it's affecting progress. Besides, I'm not so sure that some kids will never like maths. Let's try a bit harder before we say that, no?

In Blantant Disregard for the Healthy Eating Policy

To wrap up a topic on surface area and volume, I wanted to do something really exciting, and investigatory. It's not a topic that lends itself to that so much, as most of it is learning to correctly follow formulae (at the early KS3 level). I'd read Fawn Nguyen's post on doughnuts as great vehicles for this, and it seemed too fantastic an opportunity to miss. I messed with the lesson a bit, talking about models with the kids first (a cylinder inside a cylinder, or one long, bent cylinder?) and helping the weakest ones out with some of the rules they'd forgotten, but otherwise it was mostly the same.

The one thing the kids did differently to Fawn's, though, was that almost all of them wrote their calculations on the large sheet of paper I'd put the doughnut on. The drew lines by the doughnut and labelled them with measurements, and wrote their workings around them. I had a 'no touching the doughnut' rule, and they were really good at that, so the doughnuts stayed fixed until their estimates had been approved. They had to justify their estimates as well - I don't know it Fawn's kids had to do that.

The sheets looked really awesome by the end and they all took photos. I wish I'd got some, they'd be great for my display boards though I'm sure they'd make the other kids jealous! We did this 2 weeks ago, and already it is a legend around the school: 'The maths lesson that had doughnuts in!' I think that says something sad about maths lessons, mine and everyone else's. Clearly we can't reach doughnut-eating heights of kid enjoyment every lesson, but we should at least be trying to compete with that on a regular basis. It shouldn't be an event so massively out of the ordinary that everyone in school has now heard of it. So, what's as awesome as doughnuts?

Making Mistakes

Last week, I went to a dance class. I'm a pretty dreadful dancer, and it's no surprise that I was a complete beginner. It went really well, and I enjoyed the successes of getting to the end of each dance in tact - until the last dance. As soon as it started, someone called across to tell me I should be doing something, although I thought I should be standing still. No one had said that I should be doing this thing, but it was obvious to everyone else, as they were more experienced. It was really tough after that to catch up with the dance, and to figure out what was going on, and I got more and more confused until I gave up and ended up being pulled and jostled through to the end. I was feeling pretty cross with myself and with the instructors for not making it clear to a beginner. I felt that I was dreadful at dancing, and that it was pointless to try to improve.

As I was heading home, the parallels jumped out at me. The times when a child panics in class because they don't 'get it', and I tell them to wait - I'll help them later - but they feel the class is running along without them. The times when a student declares a test 'stupid' before they even try, because of a previous bad experience. The many students who give up when they start to find something hard, and look for someone else to blame.

So what can we do about this? How can I make failure less damaging to kids self-belief and attitude, and to my own? Someone once told me never to start by saying that something was easy. It puts the students in a lose-lose situation: if they can do it, that's no big deal, because it's easy. If they can't, well, they must be really dumb, because it's easy. The converse can help here, I think. If the teacher starts off by saying 'this is really tricky', it instantly becomes more ok to make mistakes. If the dancing instructor had said 'this is a tough dance', I'd have felt better when I went wrong.

We can get even more explicit than that though, and I've been trialling it this week with my youngest students. 'This topic is really tough. You guys are probably going to make loads of mistakes, but that's ok, because that's how we learn best.' When mistakes are made, I've been quietly observing them and then writing them on the board in an altered form for everyone to discuss, pointing out how useful the mistakes are. Most kids didn't recognise their own mistakes on the board, because I'd changed the numbers, but the ones who did didn't seem to mind.

I think this can only get us some of the way. I don't want to get into discussing the 'learning objective' culture in detail, but it seems to me that if you have a fixed objective, failure is always going to be more problematic. If it's open ended, not being able to do something is ok. Obviously we need kids to be focussed and have goals, but I think much of their learning should still be open-ended. They don't need to be driven to a specific goal every lesson, and when they instead set off and see where they end up, without fear of failing to meet a criteria, they allow themselves to make mistakes.

Maths in the Library

Last week, there was a maths class in the library for the first time ever at my school. It was amazing. I was trying to figure out why that 'data' topic - a quarter of the course - is so damn dull. I found it dull and, as I discovered at uni, I love stats! I figured that it might be partially to do with the fact that you never get to deal with data you care about in school, so of course the results are boring. And frankly, the colours-of-cars / survey pseudo experiments that are often said to be the solution are almost worse than the textbooks in terms of providing data the kids might actually find interesting.

So, the library. Every kid got a data set that they were actually interested in - world records for the 50m front crawl, for the boy who swims for the county. CD sales per month for the girl in love with Bieber. And one kid - that one, and he's top of the class - comparing sales of cheese to GDP.

Now, every lesson they all draw a particular type of graph - Thursday was scatter graphs - using their own data, and to their own level of challenge. Pick the scales for your axes. Find equations of lines of best fit and predict more data, do some research, do other people agree with you?

It's amazing, producing nearly 100% engagement and raising interesting questions about how to present difficult data that doesn't fit neatly into categories. As a side benefit, I chummed up with the librarian. She just emailed me with a question.

The library 'maths books' budget hasn't been spent for the last 8 years. The ordering deadline for this year is a week away. Is there anything you'd like me to buy?

So, everyone, is there anything I should buy - of course there is! What?

Invention or Discovery?

With snow flurries outside the windows and the prospect of an imminent school closure, you'd think nobody would be feeling particularly focused in year 7 maths, especially given the topic. Order of operations (or BIDMAS/BODMAS) must be one of the dullest topics in the spec, especially on a snow day. However, it was one of the best lessons this week, and I'm trying to work out why.

We always start with a question, such as 5 + 2 x 3 and ask what answers we could give. What would a calculator say? (Interesting, because it varies depending on the calculator.) With such a bright group, we raced through to 'because you do multiplication before addition' and 'because of the rule'. Then, because it's almost reflex by now, I asked 'Why?' It was a particularly bright girl, let's call her Sparky, answering, and she was flummoxed. Why do we have a rule? Why is it THAT rule?

S: 'Because that gets you the right answer'
Me: 'What do you mean, Sparky?'
'Huh?'
'What do you mean, 'the right answer'?
'...'

What makes one answer right? It stopped me in my tracks as well, as the whole class paused to get philosophical.

S; 'The rule says what's right'
Me: 'Why?'
S: 'Well, you've got to have a rule, or you won't know what's right'
Other student: 'Yeah - and people would do it differently'

We'd quickly worked out that the rule gave consistency, so we knew what people meant when they wrote that notation. But we couldn't stop there...

Me: 'So this is just a rule we picked?'
Students: 'I guess...'
Me: 'Is everything in maths just rules?'
Students: 'No, some of it's obvious, it's got to be true'
Me: 'What else is a rule that people decided?'
Students: 'Numbers?'

Then we spent 20 minutes talking about different number systems, and how the way we write numbers is an invention, like BIDMAS, but other things are discoveries. They'd done Roman numerals in history, but never made the link. The point that the rules can also be different was nicely made when a student explained that whenever you have a smaller number before a bigger number, you subtract, giving the example of IV. I wrote 15 on the board, and said 'So that means 4, right?'

They really love the ideas of having some rules, that varied between cultures (we got onto Egyptian, Chinese, Babylonian and Arabic numerals, all through them), and also having discoveries that didn't vary. It is so strange to me that this is the first time these students have thought about this - the idea of discovery and invention in maths. I think it might help a lot for those students who think maths is just 'a load of random rules'. It wasn't on the lesson plan, but I think they enjoyed themselves nearly as much as I did, and I think it was far more valuable than a worksheet.

Finding Compromises

With bottom sets in years 10 and 11 (leading up to GCSEs), there's a big conflict surrounding what we actually teach them. Thanks to 'No Child Left Behind', we are mandated to teach them the GCSE syllabus, or at least all the content up to C grade level, so that every child has a chance to reach that magic threshold. It's a laudable idea, but in practice, this means that our schemes of work are full to bursting with topics like percentage increase and decrease, area of compound shapes and finding nth terms of sequences - all for students who cannot reliably count to 20.

If I was to try to cover all that content, I would have to go at a pace which means that almost none of it, not even the more basic areas (such as what percentages represent), would be comprehensible to the students. It would also totally prevent me from addressing issues such as: they don't know their 3 times table, they don't have any understanding of decimals, and they cannot subtract any number larger than 5. Don't get me started on division.

As most teachers do, I cobble together the best I can, trying to keep myself accountable but also addressing what I feel there children need to understand to allow them to function well as adults. It's a tricky compromise, and even more so when mock exams and tests are thrown at them ('We haven't done this, Miss!'). Even when I settle the time allocations, it's tricky to find ways of helping them with their basic concepts that are effective, not too boring, and don't feel babyish. Enter the 99 club.

The 99 club is fantastic. It's not my invention at all, and it's used in primary schools across the country, but I can't find any company or organisation claiming credit. This is the way I do it. The students start off trying to do 11 times tables questions in an allotted time (say, 3 minutes). When they can get them all right, they try to do 22. Then 33, 44 and so until they can do 99. To differentiate it further, the '11 club' (where they try to do 11 questions) only involved the 2 and 10 times tables. One times table is added each club.

The students love the routine, and the fact that they're all working at their own level of challenge. Every student in my tiny bottom set can be on a different club. They can mark it themselves, and it ensures that I can start every lesson with 5 minutes of silence, followed by lots and lots of praise. And it motivated them to learn their times tables like nothing else I've found. It could be competitive, but I try to make it about individual progress, giving equal praise for any student who makes progress, whatever club they are on. Students who would argue about doing 5 for a starter will happily do 33 now, because the graded work means they start off being successful. As I collect the sheets in every lesson, looking at the hard work on them, it invariably puts me in a good mood.

Argh! Algebra!

Every year, with every class, in every school, it's there on the scheme of work: Into to Algebra. A lesson of review or true introduction, depending on the class, and how much they've forgotten from last year. There's rarely any guidance or ideas provided for this lesson, even though it's crucial to how the students perceive algebra. I haven't taught it the same way twice yet, but this is where I'm at so far.

We talk about how sometimes, either we don't know what a number is (how many pencils in this box), or the number can change (how many siblings someone has, will change depending on the person). This is when we use letters. We go straight from there into simplifying - one jar of sweets, plus another jar of sweets, is 2 jars of sweets. j + j = 2j

I haven't seen other teachers do this, but I think it's important to get straight into manipulating the letters, rather than using numbers straight away (substituting or solving), so the students get used to using the letters. We use the simplifying to explain the hidden multiplication signs in expressions like 2j, and then think about why we write 2n not  n2. I ask the students to tell me a thing that begins with n - the best so far is ninjas. If, I ask, Sophie and Charlie suddenly revealed that they were ninjas, what would we say? We'd say 'Look, there's....' I repeat this short pantomime until the whole class are shouting at me 'Look, there's 2 ninjas!' Would we say 'Look, there's ninjas 2!'? Clearly not!

That's enough abstract algebra for most 11-13 year olds to handle, so back to the numbers. Everyone pick their favourite number. Yes, go on, do it! You can call that number by the first letter of your name. Alex's favourite number is 7, so he's going to use A for 7. Then, using their favourite numbers, the students have to make a list of numbers that I give them, writing the rules using the letter. So if Alex was making 20, he could write A + 13. Students like to feel clever, so soon they start coming up with more complicated ways, as I encourage them to write multiple methods for the same answer. 2A + 6, for example, using their simplifying knowledge. It's a good lesson in equivalence too.

We review this exercise by asking people for what they wrote, and I write one 'answer' next to every target number on the board. For 20, I might write Alex's answer: 2A + 6. Then we try to guess Alex's number.

The great thing about this lesson is that, after some explanation of simplifying, the kids can figure everything else out by themselves. I go slowly on the simplifying, taking maybe half an hour, so that everyone can follow. Then they're off! They can make the problems as complicated as they like. I'd like to make the simplifying half of the lesson less teacher-led, and mind-mapping might work for older students, but when they haven't seen algebra before, there needs to be some instruction. Possibly a card sort could replace some of it?

Games that are actually useful

My year 8s are a noisy bunch. They're not the most able, and not the most inclined to sitting still or doing worksheets, or questions from the board, or anything much that involved writing. If you can market it as a game, however, their enthusiasm is astonishing. The problem is that most maths games only involve one student at a time - taboo, round-the-world, even fizz-buzz. Bingo is a good solution but doesn't quite have the excitement they're looking for. So I've developed a new way of playing blockbusters so that everyone does every question.

In blockbusters, I display a hexagon grid on the smartboard (from TES), and split the class into two teams. Each team has to make a path across the grid, one from top to bottom and the other from left to right, by picking hexagons and correctly answering the revealed question, which turns the hexagon to their team's colour. Traditionally this is done with one student answering each question. The year 8s love it, but it gets a bit loud and 31/32ths of the class are doing nothing at any one point.

So, one student picks a hexagon. I click on it and when the question appears, there are ten seconds of silence. Any communication forfeits the question. After ten seconds, I pick someone from the team (no hands up allowed - a random name generator would work well here but I just pick). They have to answer the question correctly within 3 seconds. If they don't, I pick someone from the other team, who has 3 seconds, and repeat. If the questions are hard enough, this happens often, so every student in the room is working out the question in the 10 second silence, ready to be called upon.

They love it even more. The pressure is exciting, and there's a huge feeling of not wanting to let down their team, so everyone works it out. It keeps the pace up, and the number of questions each student does in the 5 minutes we play for is huge compared to normal lesson time!