#MathsConf15 Part 1
#mathsconf15liftshare was a brilliant hashtag that led to Rob, Pete, Jo and I travelling together up to Manchester. Rob was designated driver and went above and beyond to secure us a comfortable journey with sweets, Coke and water – what a star! We sand along to some great playlists, played the number plate word game (bonus points for maths words) and found prime factorisations of house numbers. (This was definitely more fun than it sounds!)
Friday night drinks was brilliant as ever, with a great turnout. It’s so lovely to see the same faces and meet new people too. We even went out dancing after drinks at the bar!
At breakfast, my day started wonderfully when Claire presented me with a gift to say thank you for my website – a lovely notebook! She’d even draw beautiful calendar pages in it for me. I think I still see PixiMaths as just a form of online storage for all my stuff, and I often forget how widely used it is. Thank you all, for your continuous support.
Bleary-eyed and rather sleepy after dancing the night away until 2am, Rob, Pete, Jo and I arrived at Manchester Enterprise Academy for another MathsConf. We had offered up our help with setting up the MA stall with Rob, then manning the welcome desk. It was great to say hello to everyone as they came through the doors.
Mark gave a warm welcome, as always, then passed over to Andrew Taylor from AQA who spoke about the beauty of maths. He used Euler’s identity as an example of how #mathsisbeautiful. As Craig Barton had described it, it has “e bombing about, pi bombing about and i bombing about”! Andrew also applied this beauty to accessible maths for our students in the form of magic squares and exam questions, for example on simplifying algebraic fractions.
Mark then showed us the awesomeness that is Complete Mathematics – complete with schemes of work, lesson resources, homework, quizzes, full student profiles, times tables app… and the UK’s largest network of maths teachers. They offer 65 days of CPD across the year that is entirely evidence-based.
My first workshop was with Jo Morgan on indices, including possible resources, misconceptions and approaches. Jo gave us some great questions to look at before her session started – I recommend having a go at these! She has set us the challenge of planning and teaching topics in depth. Often, Jo said, we “skim the surface of every topic we teach”, but if instead we take the time to research every topic properly students would have a much better understanding.
Jo talked us through the curriculum and what students cover on indices from year 5 upwards. They meet squares and cube numbers for the first time in primary school, but indices are new to students at secondary school. At GCSE level, students need to incorporate indices into other topics too, such as standard form and prime factorisation. Jo described all maths as “a journey to calculus” and discussed how this definitely applies to indices.
She then talked through how the topic of indices would be delivered, starting with going back to basics in year 7, starting with 2 x 2, then a x a. Jo talked about correct vocabulary: who knew the number at the top is not a power, but instead the whole thing. And the way we say 2^4, for example, varies from classroom to classroom and across countries too.
Jo showed us an extract from a Raynor textbook with loads of deliberate practice and raised the question of if we giving our students enough fluency practice? She showed us the explicit instructions in a Victorian textbook too, for example, “when the index is unity it is omitted. Thus we do not write a^1 but simply a”. Jo suggested using binary as an enrichment lesson at end of year 7 to further extend understanding of indices.
I’ve always taught the laws of indices (all of them!) in one or two lessons. It made sense to do it this way. “The whole of index laws in one hour?! What am I doing?!” Jo introduced us to the term SLOP – shed loads of practice – and emphasised the importance of this with deliberate practice and clear instructions: “the bases are the same, we’re multiplying the terms, so we add the indices”. Jo now teaches the power law after multiplication law then the division law after that. She said with all of these we should be varying practice including fractions, negatives and algebra. The Victorian textbook that Jo showed us described the zero index as “a curious conclusion”. It’s clear that beginners to indices were acknowledge and supported.
Jo wrapped up her session with a list of possible resources including minimally different questions, OUP textbooks and brilliant.org. She asked us to think about how many lessons should we spend on indices. We hadn’t even looked at fractional and negative indices in this mammoth idea-filled workshop! I’ll probably be covering indices in at least 10 lessons from now on, to ensure students are more fluent with this amazing topic. This session has really got me thinking about my teaching. I would love to spend the time thinking this deeply about every topic, but I don’t have enough time, and we definitely don’t have the time to cover topics to this level of depth with our classes. However, my pledge to myself is to try my best.
The second session I attended was with Andy Elwell (you may not have heard of him but he’s responsible for the awesomeness that is MethodMaths). The session was called ‘Procedures are not the enemy’ and part of went along to play devil’s advocate! However, I took a lot from this session. Enough, in fact for me to change my lesson resources for my Pythagoras’ Theorem lesson (I’ll be uploading this later this week). Andy was keen to encourage debate around teaching maths and his workshop definitely did just that.
Andy shared with us the journey that students take with long multiplication, including the calculations policy from AET, the MAT within which Andy works. He explained how the teaching moves from concrete to pictorial to abstract, including Deines and Numicon, the grid method and then long multiplication which is a requirement for SATs questions in year 6. Andy then compared long multiplication to the lattice method. Now I love the lattice method and used to teach it to students, but I was always concerned that I was missing out the importance of conceptual understanding. The lattice method has far fewer procedural steps than long multiplication, and the multiplication steps are all completed in one go, rather than being mixed in with addition and carrying numbers here and there.
After watching this segment of the workshop however (see video clip below), I am definitely going to go back to the lattice method. I have no doubts now that I’ll be able to answer students when they ask why this works.
Pythagoras’ Theorem is generally a little abstract with students needing to use algebra and substitution, but just becomes the processing of a formula. Instead, as a “procedural bridge”, Andy’s students draw squares on the sides of right-angled triangle instead – the areas of the two small squares give the area of the big square, which requires very little working out. Once students understand this, it is possible to draw it back to the abstract.
The next procedure Andy shared with us was proportion grids. First he gave us some questions to try ourselves and asked us to think about the steps we took to solve them. There were a variety of methods used in the room, from algebraic manipulation to use of scale factors. Next we saw the DM (divide/multiply procedure) and I still can’t decide whether I like this or not. Andy explained that proportion grids can organise thinking and minimise overload to help students build success before going back to why it’s working. He also showed us how these can be used for the sine rule in trigonometry, and I definitely see the benefits, but I’m undecided about how I will teach trigonometry next time I meet it with a class.