It arrived! Having ordered a copy of Alex Bellos's second book, Alex Through the Looking Glass right after discovering it existed on 11th February, the gods at Amazon finally decided to deposit it on my doormat this afternoon. I nearly crushed it when I opened the door...

So after hacking into it with an over-large knife (couldn't find the scissors) and discarding the packaging in a very haphazard fashion, I started reading. I've read three chapters tonight, on and off, and am going to finish Chapter 4 before I go to bed (although it does appear to be about conic sections...urrrrgh). I absolutely loved his first book (Adventures in Numberland) and have read it cover to cover at least three or four times, and dipped into it for lesson ideas (it's looking very dog-eared now) and I'm pleased to say I'm enjoying this one just as much. 

So far this evening I've discovered:

• Shakespeare is responsible for turning the mathematical word "odd" into its current meaning of "strange" or "unusual"; prior to his writing, everyone understood it in a mathematical sense only.
• Odd numbers are usually associated with more "exciting" products (Levi 501s, 7UP) while even numbers are more reliable (WD40). And interestingly, they've done studies that show that people will pay more for products depending on the numbers associated with them.
• Why lots of people have the number 7 as their lucky number.
• How Benford's Law is used to catch fraudsters and why the number 1 is the leading digit of about 30% of most data sets.
• A solution to an argument I was having with my tutor group yesterday morning about how everything is related to maths - one of them got quite upset when I showed them this as apparently I'd broken Biology forever.
• Exactly how triangulation works and why it was so useful for navigation before laser methods for measuring distances. Sadly, I discovered that there are no intact trig pillars other than church spires anywhere near my house or school, so that's that field trip idea in the bin. Despite this, I'm definitely going to show this in class the next time I teach trigonometry. You can check your local trig pillars on this lovely little site.

If you like maths books, particularly those which don't give you a headache because the maths requires you to dig out university notes to understand the first paragraph, I'm recommending this. Of course, I'm only three chapters in, and the bit on conic sections may kill me (3D geometry is not my favourite thing ever). Wish me luck!

Thank you, Internet, for wasting my evening. I don't know how I found this on YouTube - one of my random click-a-thons searching for goodies, and I'm a sucker for "magic maths" tricks. This one is how to square any (two-digit) number mentally.

If you don't want to watch the video, here's the basic idea with 33² as an example:

1. Work out the difference between 33 and the nearest 10 (in this case, 30). Difference = 3.
2. Add the difference to 33 to get 36. Subtract the difference from 33 to get 30.
3. Multiply these two numbers together (30 x 36). This is (apparently) the advantage of this method - it's easier to do 30 x 36 in your head than a four-way decomposition. 
4. Finally, square the difference (3² = 9) and add it on to 30 x 36 to get a final answer of 1080 + 9 = 1089.

And here's my workings:

So I did this one...then tried another. 

Cute? Convoluted, really, and not particularly workable in the curriculum (I'd much prefer a sensible approach with proper multiplication), but then I got one of those "oh now how does that work?" moments. I could see how it was kinda linked to quadratics and square areas, so I did some doodling. After many aborted attempts, I came up with this: 

Take a square with side length 33(cm if you insist on units). You can then chop a rectangle off the edge of length 3 (making an easier multiplication of 30 x something) and whack that on the bottom. So then you've got a 30 x 36 rectangle, and the little square left over to add on.

Then, because I'm a mathematician and I like doing things properly, I tried to make it work with algebra:

Typical quadratic expansion, but we think about a as the tens digit and b as the units, because we're trying to change the square into a rectangle with length a (so we have a multiple of 10 to calculate with). Then:

So we've changed the problem from (a + b)² (difficult) to a(a + 2b) + b² (messy but easier mentally). I was quite pleased with this!

Just thought this might be a nice challenge problem to give to pupils when working with quadratics (or square numbers for that matter) - show them the video, then get them to explain why it works - numerical or algebraic approach. I do think this lends itself to an exploration of the link between square numbers, area and quadratics, which I think is often overlooked.

The first session I attended at Celebration of Maths 2015 was titled "Effective Revision" and run by Ryan O'Grady (@tiredoldcliche). My colleague had picked the sessions for us, and I wasn't sure I'd get much out of this - I was pretty convinced that I'd got revision sorted. So far this year with Year 11 I've used:

• Typical past paper practice, with red/amber/green topic sheets to record their results; 
  
• The brilliant Maths Takeaway booklets from Kesh Maths (my pupils seem to really like these - I have had requests for "doing more of those topic booklets, please" from quite a few of them);
• Revision folders to collect all their past papers, posters etc as we've gone through the course;
• Creation of revision posters or cards for key topics, so all the pertinent notes are in the correct place.

However, this session made me think again about how effective that revision actually is. Many of us were using strategies similar to those above, but Ryan raised a few points that made me think twice.

1. Do we take organisation skills for granted?
We assume that pupils know how to revise and organise their time. I'm so guilty of setting the "revise for this mini-test" homework with the implicit assumption that pupils will have the slightest clue how to go about doing this. In some cases, such as those with the permanently lost maths book, they often won't have any meaningful class notes or work to revise from. I've also lost count of the number of times I've had conversations with pupils that go a bit like this:

Me: (Hands back test with less-than-adequate mark) Did you do any revision?
Pupil: Yes I did loads, like four hours!
Me: What did you do to revise?
Pupil: I read through all my notes and then went on Bitesize.
Me: Did you actually do any questions?
Pupil: No...

Despite the fact that I keep telling my pupils that they need to actually do maths to help them revise or learn, I hardly ever give them any direction to where they can find any of these resources. My direction of "use the Internet" is similarly useless - there's so much stuff around now that there's no guarantee that it will be good quality, or that pupils can even find it (MyMaths menus are an excellent example - unless you think in teacher speak, it can be very difficult to find some topics!).

Ryan suggested creating a revision notebook (preprinted with space to write examples) where pupils could record all of the really important stuff from lessons. This has obvious implications for the photocopying budget, but I wonder if it would really help certain pupils. 

2. Why don't pupils revise?
Once we've got round the organisational issues and problems, there's still the issue that some pupils just won't do revision homework (the ones that go "Yessss!" when you say that they need to revise for an assessment, because in their minds, that equals no homework). We know that revision makes a difference to final exam results, but Ryan made the important point that, for some pupils, they don't see the point in revising because they think they are still going to fail. A lot of the ideas behind this next bit come down to a pupil's mindset. A pupil who has experienced success is more likely to want to work hard at something than one who has attempted to work hard in the past but had no success (the "I'm crap at maths" bunch).

3. Working hard makes a massive difference
Combating the "I can't do maths" is one of the most difficult things to do, and I've heard it from all ages and abilities. I'm still on the fence about whether or not there is such a thing as "natural maths ability". If there is, I certainly don't have it - I was much better at French and German at school and breezed through my English Literature exams without lifting a finger, but I had to work hard at Maths. I think this is one of the reasons I came to enjoy it so much, purely because I did actually have to think during Maths lessons and didn't get bored because it was all too easy. Obviously there is some difference between "any old" maths graduate and someone like Andrew Wiles, but my personal opinion (backed up by absolutely no research or concrete evidence at all) is that most people are equally capable of doing maths, it's just that at some point on the mathematical journey, an important building block gets left out and then the cycle of desperation starts.

Sometimes pupils don't see just how important hard work in Maths is, partially because there's this culture of "either you can or you can't", and it's socially acceptable to be part of the "can't" crew. But I've had at least two pupils in the past - "bottom set kids" (with sarcastic quotation marks) achieve well above their targets of Fs and Es to get a grade C at the end of Year 11, and this was purely because they worked really hard every single lesson.

Ryan showed us one video in the session that illustrates this point beautifully - it's Ed Sheeran on Jonathan Ross's chat show, explaining just how bad he used to be at singing.

Ryan also mentioned Malcolm Gladwell's theory that you need 10000 hours work in a subject to become an expert. A little bit of Googling reveals that actually it seems to be a little less than this, which is comforting news for our pupils, but I think this is also worth sharing with them.

In a similar vein, it's worth sharing the story of Dan McLaughlin, who quit his job in 2010 to take up golfing, despite having never played a complete game in his life. He's not reached the 10000 hour mark yet, but is competing nationally. There are a few great videos on his site.

4. What motivates pupils to revise or work hard?
If pupils do revise, it is likely that they've already bought into some of the points discussed above. One thing that Ryan mentioned (that is apparently backed up by significant research - I'll have to go digging) is that extrinsic rewarding (stickers, stamps, sweets, bribery!) does very little in motivating pupils to work hard. This really stuck with me - many of our reward systems at school are based on this premise, but it kind of makes sense that frequent extrinsic rewards might diminish or mask the intrinsic reward of personal success through hard work.

Something else mentioned (which I think is the most important thing I've learned in teaching) is that pupils are most motivated by a teacher they can trust and build confidence with. I think it's important to start off building small measures of success, particularly with a new class, so when you get onto the stuff they're really going to struggle with, they don't lose complete faith that they'll never get it.

5. Using deliberate practice
Build time into lessons for pupils to deliberately practise the skills they have learned. This is something I shied away from in my first couple of years of teaching (surely every lesson needs to be whizzy with seventeen different card sorts and pupils running round the room), but I've got significantly better at it this year. Working on a mastery curriculum model (and having quite a bit of success with it!) has made me realise how important it is for pupils to practise what they have learned, sometimes over and over again until it's almost automatic. This is why my Year 9s can successfully add fractions a month after I've taught it (that's never happened before). 

Ryan showed us a couple of nice videos to illustrate this point - Odell Beckham making "the greatest catch in the history of football", and then his warmup sessions repeatedly practising exactly the same catch.

As this post seems to be going longer than I'd planned, I'm going to stop there and write Part 2 some other time. In the meantime, Ryan kindly shared the resources from the session on Twitter, so pop over for a mosey!

I have to admit, I was flagging by this point in the afternoon. A combination of an earlier-than-usual start on a Saturday and an inevitable caffeine crash and accompanying headache at about half three meant I was actually (shamefully) considering disappearing early - but I am SO glad I stuck around for Marcus du Sautoy's lecture to finish the day.

I'm a big fan of School of Hard Sums, and I love some of the TV shows that Marcus has done, so I knew this would be good. What surprised me was just how exciting it was being in a room full of enthusiastic mathematicians listening to a lecture from an enthusiastic mathematician; it's this joy that I wish we could bottle and give to students somehow. The hour flew by, and I left with a spring in my step and a reminder of why I love maths so much.

Marcus started the talk by showing us some sequences, and talking about how mathematics was about pattern-spotting. I got the first couple of sequences (I love how I turn into a school child again whenever anyone asks me a maths question and I get excited because I know the answer), but the sequence 1, 2, 4, 8, 16, ... threw me and everyone else in the audience. You'd assume the answer was 32, but it's actually 31, because this is the sequence of circle division numbers. Quite a nice illustration of how mathematical patterns can be deceptive!

Another interesting point from this is the use of the Fibonacci (why can I NEVER spell that) in music, or more accurately, the fact that the so-called "Fibonacci" sequence was being used for beat counts in Indian music way before Leonardo gave his name to it. @Kirstymaths tweeted a pic of this that's worth checking out.

Marcus then talked about prime numbers, and suggested a more intuitive reason that 1 isn't a prime number. If we think of the primes as building blocks for all the other integers (Fundemental Theorem of Arithmetic), then 1 isn't prime because we can't make anything with it. It's interesting that mathematicians have flipped back and forth on 1 for years; I vaguely remember something from my Numbers and Algebra course in the third year of my degree, but Wikipedia is much more accessible than the lecture notes gathering dust in my loft.

The lecture then progressed to talking about cicadas, insects whose life-cycle lasts a prime number of years. I remember finding out about this last year when a friend in America sent me a video very similar to this one of cicadas in his local area; I was amazed at the volume the insects create, and even more amazed when I read some of the linked news articles that explained that these particular cicadas emerge once every 17 years. There are some theories that this is to do with the cycles of now-extinct predators; choosing a prime numbered life cycle would mean that the cicada has less chance of meeting a surge of predators whose life cycle works on multiples of 2 or 4 (for example). I've also just found a nice video from the BBC's Life in the Undergrowth narrated by David Attenborough - I'm feeling inspiration for a lesson on primes and LCM here!

We then did a lottery activity, and Marcus predicted (pretty accurately) how many of us would have 1, 2 3 or 4 numbers correct. He talked about the ideas that people don't select consecutive numbers because they think these are less likely, but pointed out that half of all possible choices contain consecutive numbers. He also talked about the 1, 2, 3, 4, 5, 6 selection - I love using this when I do combinations with Year 12, and having a discussion about how you think you're clever because (if you're a mathematician) you know that this is equally as likely as any other possible combination, but how you'd be kicking yourself if you won, as there are (apparently) about ten thousand people in the country who pick this per week, and you'd be splitting the prize more than if you went for 22, 23, 24, 25, 26, 27.

Marcus finished with a lovely demonstration of patterns in populations, looking at one model for lemmings to explain the four-year "suicide". Thanks to QI (about 20 minutes in, profanity warning), I was already pretty clued up on the lemming myth (another proud schoolkid moment!), but Marcus demonstrated (using quite a simple mathematical model) how a population could stabilise, then changed this model to show how the lemming population could vary wildly to explain the four-year dip. Again, tempted to try this in the classroom!

I'm sure there's lots of stuff I've missed, but this was a brilliant way to end the day. Like I said at the start, there's something infectious about being in a room full of people who love maths and are enjoying themselves doing or thinking about mathematics, and it's a shame that this doesn't translate to our students sometimes. 

I've been using bar modelling quite extensively in my teaching since being involved with the NCETM's multiplicative reasoning project last year. Part of the project was to emphasise the importance of diagrammatic representations of problems in teaching maths for understanding; we were given materials to deliver to Key Stage 3 classes, some of which included use of Singapore bar modelling for topics such as fractions, percentages and ratio. I found the work we did really altered my teaching; I think that I managed to teach addition of fractions successfully for the first time since I started teaching, and I was amazed at just how well my students retained efficiency and accuracy with "traditional" written methods.

It seemed like a bit of a no-brainer to pick the bar modelling workshop at the Celebration of Maths, so I and my colleagues trotted along to the session, sat down with our mini-whiteboards and got ready to draw some bars. One thing I was really keen to get out of this session was to iron out some issues I still had with using the bar to solve problems with negative amounts, and I was still struggling to see how to apply bar modelling to exam technique (see my attempt with the Edexcel SAMs here).

First of all we looked at some simple problems, like fractions of amounts. I'm already pretty happy with this - there are plenty of examples of stuff like this in my Year 7's books at the moment. I was so proud of myself that I thought I'd add a really constructive "what went well" to my work too.

Just a quick post to say I've just found an excellent site for quick starters and practice - Flash Maths is chock full of Flash applications for generating questions on loads of different topics. A five-minute poke-around unearthed these little gems:

Substitution Grids

Having taught this to nearly all of my KS3 groups at some point in the last few months, I think I'm becoming something of an expert. This year, we've started the Mastery Pathway at KS3, and nearly all of our students in Years 7, 8 and 9 (unsurprisingly) had gaps in their understanding of fractions. So I've now taught this about five or six times; it's true what they say about practice making perfect (or pretty close).

Two major changes this year have been use of the bar model (read about my discovery here) and linking cubes. I'm trying to do a lot more in concrete situations before moving to rules, and using manipulatives and models seems to get the ideas to stick.

1.  Start with a chocolate bar
Think Cadbury's Dairy Milk (single bar), Kinder Bueno, Hershey Bars - anything which comes in a single row of squares or pieces of chocolate. The first thing I did was show the students some pictures of these chocolate bars, and get them to tell me each piece size as a fraction - e.g. the Bueno has four pieces, so if I eat one piece, I've eaten 1/4, two pieces is 2/4 and so on. We spent some time discussing what the numerator and denominator tell us about the different chocolate bars, and emphasising that the denominator told us about the size of one whole bar.

2.  Move to a physical model
I then gave students different amounts of link cubes, explaining that they were pretending these were squares of chocolate. I asked them to make me as many whole chocolate bars as they could - we used the Bueno 4 square model to start with. There were a few comments about how I'd not given them enough cubes in some cases (they were expecting multiples of four); I kept quiet and just told them to build what they could. Once they'd built for a bit, a couple of the students realised that this was the point of the task.

We then collected ideas on the board and looked at different students' examples. I deliberately started using a bar model at this point to represent the cube pictures they were explaining. This is also a good point to get students to draw their models on the board themselves and explain what they've done.

3.  Introduce improper fraction notation
I took one example and modeled the use of fraction notation, drawing on our original chocolate examples. So 13/4 means 13 pieces of chocolate, and one whole bar has 4 pieces. We then discussed how this related to mixed number form:

• How many whole chocolate bars? This is the big number.
  
• How many pieces that don't make a whole bar? This is the numerator of the fraction part.
• How many pieces in a whole bar? This is the denominator.

4.  Practise problems using cubes and diagrams
I then gave students some problems to try for themselves. They started with more examples using four-square chocolate bars, then progressed to other sizes. I kept denominators fairly small to start with (2, 3, 4 and 5), just because modelling and drawing gets a little impractical with larger denominators. I encouraged students to use the cubes where necessary, and insisted they drew diagrams for the first few they attempted.

Some students quickly spotted the "rule" and then began working without diagrams. Towards the end of the activity, we discussed what they had discovered as a shortcut method and why it worked. However, I haven't been pushing rules too much this year - I'd rather students thought about what they were doing, and if necessary got the right answer through drawing a diagram, rather than learn a rule which is quickly forgotten.

5.  Work the opposite way

Some of the classes I've tried this with were ready to move on to converting back the other way, either with diagrams or by applying understanding gained from working from improper to mixed within the same lesson. 

Some of them needed a little more consolidation work one way before we moved on. One lesson I'm learning this year is how important it is not to push too quickly; a new idea takes time to cement thoroughly, and I think it's worth working on one thing properly at a time, rather than charging ahead and progressively losing students along the way.

If you're looking for questions quickly, Math Aids has a great worksheet generator, differentiable by easy, medium and hard denominators.

Last year, I attended the best CPD of my teaching career. It was a year-long course run by the NCETM; a project on multiplicative reasoning. Now don't get me wrong, training days can be useful, but I can honestly say that this course changed how I teach fractions and proportion on a fundamental level. We had six days of training, spread out over the year, and had to deliver certain materials to our KS3 classes, then evaluate their performance at the end of the year to assess the effects of the project materials. I'm going to blog more about this in later posts, but in this post I wanted to focus on one of the first questions they asked us on Day 1, because it really altered my thinking about teaching fractions. It's a simple question:

Just to give a bit of context to this, this was one of four diagnostic questions we had to give our classes prior to delivering the project materials. The students had to answer the questions in as much detail as they could, and explain their reasoning. 

We also had to try the questions. I suggest you give it a go now, too! Go on, before you peek at mine...