Here’s what I got from the day.
David always has something new to say at conferences. No rehashed powerpoint slides and tired jokes to be seen. This time, he demonstrated three powerful techniques: bar modelling, algebra tiles and double number lines.
I was fairly au fait with bar models already, but the talk did prompt me to track down some My Pals are Here books once and for all. These are the books used in Singapore primary schools. He had sourced some lovely problems from them, like the following (rephrased because I didn’t get the exact wording):
I have a mixture of 20c, 50c and $1 coins. I have 20c and 50c coins in the ratio 2:5. I have 3 times as many $1 coins as 20c coins. If I have 28 more $1 coins than 20c coins, how many more 50c coins do I have than 20c coins?
Lovely with a bar model; horrid without.
Next up were algebra tiles. This was of much interest to me: I have to make the call on whether to buy class sets for the department pretty sharpish before we teach negative number to our pupils.
I could see how they would be excellent for wiping out misconceptions when collecting like terms and expanding brackets.
I was a little more wary about them for solving equations. Take an equation like 4x + 1 = x + 4
It starts off really well. The idea of taking the same thing from both sides works nicely.
Dividing both sides by three seems OK as well, though there’s something slightly less intuitive about it – maybe because I can already see that the sides don’t look equal?
This is my real bugbear. Look at that picture. The length x is clearly not the same as the square of unit 1. They are different!
I wonder whether the algebra tiles don’t just reinforce the misconception that equals means “the answer is…” rather than “the same as”.
The jury’s still out. No manipulative is perfect, and its advantages may outweigh the disadvantages I currently see. It’s one I’m pondering.
No such misgivings about David’s third technique, however: double number lines. What a fantastic way to introduce proportionality! I was mentally replanning my unit of work on proportional reasoning as he spoke.
The idea of starting with simple times tables is genius:
David then suggested giving number lines that get harder and harder, getting pupils to fill in numbers using a variety of techniques.
Finally, you can extend to word problems: currency, converting units, percentages, you name it.
I really think this technique could be game changing for schools, especially with the heavier focus on proportional reasoning in the new GCSE.
Tweet up and do some maths
I’m convinced there’s a circle of hell called “speed networking”. I get what conferences are trying to do: talking to other delegates is often the best part of a conference. But you need to let it happen more organically for it to work. Best catalyst for a room of mathematicians? Some maths problems like this on the walls:
From the wonderful @solvemymaths
I thought this worked an absolute treat. I had a boatload of fun, talked to some new people, and did some maths.
Kris said many sensible things on assessment. He started off with his definition of mastery: a term which has come to mean whatever people want it to mean of late. Is mastery about refusing to move on until everyone’s got it? Is it about kids showing some sort of “understanding” over and above procedural knowledge? Kris stuck to something more tangible: depth before breadth and spending longer on topics before moving on.
Kris gave a crystal clear explanation of validity and reliability, and how that affected how we choose to assess in mathematics. I second his recommendation to read Measuring Up by Koretz. It will give you the tools you need to think clearly about assessment.
I appreciated Kris’ bold principles for assessment design: one mark a question, with no marks for working, making it possible to mark and record results in under one hour. We’ve all spent weekends scrutinising messy exam papers for whether Jimmy can pick up that M1 mark and filling in complex QLA grids – the fact I saw multiple people marking GCSE papers in corners of the conference centre during coffee breaks is evidence of that. It’s one of those workload issues I now have to think about a lot more carefully as head of department than I did as a lone ranger. I want my teachers to have an excellent work-life balance.
A further principle that Kris espoused was that the mastery assessments should be endlessly resit-able. The ideal would be endless assessments with the same structure questions but different numbers. I was left pondering how small tweaks might make that a reality with our assessment system. We’re already ticking many of the boxes with no marking for teachers, QLA automatically generated, and limitless resits (but with the precise same questions each time). With the flexibility of Excel imports and the =randbetween and =concatenate functions, it’s possible to generate many subtly different assessments with minimal workload. One I’ll be working on.
Jo (@mathsjem) is a veritable powerhouse of ideas. @danicquinn and I were sat together, nattering like excitable schoolgirls about the pros and cons of the exciting methods we were exposed to. I loved the format: we tried some questions, compared methods and Jo then showed us four or five different ways it could be done.
Jo made some important points about “nixing the tricks”. I have a lot of sympathy with Nix the Tricks. I hate awful things like the adding fractions butterfly. I do, however, think it goes a bit far sometimes. I won’t be simplifying surds like this anytime soon:
Jo said we should “keep pupils brains free of unnecessary memorisation”. This gives me pause. I think memorisation is often more necessary than we think – including the type of rote, instrumental, procedural memorisation that is so slated. It’s important to remember that it’s not a choice between “memorising something” and “learning something”. Unless something has changed in long term memory, nothing has been learned. So memorisation is learning. You can’t learn without memorisation. It’s what you memorise that we should be discussing.
All in all, a really enjoyable, useful and thought-provoking day. Thanks to all involved in making it happen!