Michaela is different. In many ways. I’ve been asked to explain how a lot recently, as we’ve been recruiting. In terms of the Maths department, I’ve pinned it down to seven principles:
- Cementing the foundations
- Concrete-pictorial-abstract sequences
- Depth over breadth
- Separation of minimal differences
- Learning by heart
- Low stakes quizzing
Cementing the foundations
Let’s get the 80% right first.
Before we plunge into the standard secondary school curriculum, we do whatever it takes, for as long as it takes for children to have a solid grasp of number.
By this I mean the automaticity with basic number facts; fluency with the standard algorithms for the 4 operations; a solid grasp of place value and the base 10 number system; and a strong sense of proportionality.
I like Mark McCourt’s similar priority list:
@daisychristo the pillars of school level maths are: 1. Numerosity 2. Place value 3. Base 10 system 4. Arithmetic 5. Proportional reasoning
— Mark McCourt (@EmathsUK) February 1, 2015
This is the 20% that is responsible for 80% of the success or failure. “Whatever it takes for as long as it takes” may seem a bold claim. I believe it’s the right thing to do.
We find abstract content much harder to understand and remember than something concrete. Mathematics’ beauty comes from its abstraction. But this same beauty can make it tricky for novices to find a way in.
It’s all about the quality of explanations. One thing that can make explanations clearer and easier is the concrete-pictorial-abstract sequence.
Concrete manipulatives like Numicon are the first stage of the CPA sequence. Next comes the visual representation, such as a bar model. Finally we abstract to number and algebra.
Depth over breadth
Progression and challenge is done through interesting problems, rather than speeding through the curriculum. I could easily teach my high attainers some trigonometry right now, but I question the need. We won’t run out of time to cover the curriculum. Instead we go for depth.
My main sources for challenging problems are anything Tony Gardiner has ever written, Don Steward’s median, and UKMT problems.
Automating number facts is a massive priority at Michaela. We expect automaticity in single digit addition facts and the related subtraction facts, and the times tables up to 12×12 and related division facts. That means a fact a second.
We dedicate lesson time to drills every day until pupils are fluent. Pupils carry flashcards to use at school and at home. Our pupils answer tens of thousands of times tables questions a day on ttrockstars.com – the most of any school in the UK. Teachers fire maths questions at pupils around school before assembly, in queues for the toilet and while they’re waiting in the lunch hall. There is no escape!
Separation of minimal differences
We teach concepts with big overlapping ideas as far apart as is reasonable. Perimeter is taught with addition, well before we look at area is touched. Multiples and LCM are introduced with multiplication; factors and HCF with division.
This helps pupils to avoid confusion, but doesn’t wholly prevent it. No matter how much you separate area and perimeter, some pupils still seem to mix them up – especially since they were taught them together at primary. So we do draw the concepts together later, making the differences explicit. We often use a mnemonic to help them tell them apart.
This also gives a lovely opportunity to explore the connections between the concepts. We can look at things like how perimeter changes with area and squash some misconceptions. We can explore how the highest common factor affects whether the lowest common multiple is product of the numbers or not. I love looking at stuff like this, but it flops if it’s done prematurely. Pupils need to understand and remember the concepts properly in isolation before we add the conceptual complexity and cognitive load of drawing links.
Learning by heart
Beyond the number facts, I also want pupils to know by heart important definitions like perimeter is the distance around a two-dimensional shape or to simplify a fraction, divide numerator and denominator by the highest common factor. We often chant definitions. We do a lot of cloze choral response.
It’s not dull at all – the pupils love it. Especially if it rhymes. You can add some pizazz and drama to it all. For instance, when learning powers, pupils stand up, look very solemn, with their hand on their heart and say:
Hand on heart, I promise, I swear
I will never double when I’m meant to square
I guarantee I won’t be a noob
I’ll never triple when I’m meant to cube
I pledge to you and to myself
When I see a power, I’ll times by ITSELF
It brings in a bit of the J-factor, a nice little in joke, and when someone inevitably makes the classic mistake, you can do the whole faux-tearful “but you promised!” that year 7s think is belly-achingly hilarious.
Low stakes quizzing
I can no longer imagine teaching a lesson that did not contain at least 15 minutes of revisiting and testing previous content. Every lesson contains a “nothing new; just review” with a mental maths section, a facts section (testing definitions like perimeter, highest common factor etc.) and a procedures section.
It is the best scientific insight I’ve heard of for making things stick. It’s also a great opportunity to prime pupils for the upcoming lesson by bringing necessary prior knowledge to the forefront.
Over the next few blogs I hope to share some practical ways we’ve made this a reality.
Found yourself nodding along to this? You’d probably be a great fit at Michaela. We’re doubling in size next year, and therefore recruiting in lots of subjects! Take a look at our ads or pop me an email at email@example.com