Should maths be learned by rote?
Some of the most egregious pedagogy is born when the answer to that question is ‘100% yes’ or ‘100% no’.
“100% yes” conjures up – perhaps rightly – an image of maths as a joyless subject whereby pupils are learning algorithms without meaning. Although it can feel like an easy way to teach, pupils are unlikely to succeed with equations such as (4a+4)/3a = 17 if the approach to linear equations has simply been ‘change side, change sign’ and practise only the simplest problem types (e.g. 4a + 3 = 23). Automaticity with times tables, simple written calculation and being able to regurgitate the order of operations is of limited help if the pupils aren’t taught how to think flexibly (i.e. if they can’t see the deep structure of a question).
“100% no” is also problematic. Typecast as the progressive approach to maths, it is founded on exploring maths as a way to develop deep understanding (and an assumption that fluency and confidence arise from there). It is championed by academics such as Jo Boaler and many teachers (and maths consultants…), and the heart of much debate. This approach argues that relational facts needn’t – and shouldn’t – be taught as such and certainly don’t need to be explicitly memorised.
Relational facts are those that can be derived from a smaller field of arbitrary conventions (such as ‘angles in a straight line sum to 180o’ is derived from the convention that angles around a point sum to 360o) or easily understood and recalled relationships (e.g. I can calculate 3 x 8 by doubling a relationship I do recall – 3 x 4 = 12 – to get 3 x 8 = 24).
There is clear merit in an approach that builds relational understanding1. It is an important part of building the storage strength of concepts2 (how well a concept or fact connects to other memories and concepts) but, used alone, it ignores what is happening in pupils’ brains as they work.
Simplistically put: as pupils work on a new problem or idea, their working memory is gradually being ‘used up’ until there is little capacity for additional processing. Take this problem:
0.8 + 0.4 x 52 ÷ 0.01
A pupil has to think about all of the following:
- The order of operations (that they should complete the multiplication and division first AND that, within that, that they should work from left to right)
- What the notation 2 means
- The value of 52
- A strategy to multiply an integer by 0.4
- A strategy to divide by 0.01
- How to add 0.8 to the answer
That is a lot to think about! If trying to think about each idea from scratch, their working memory will soon overload, making the calculation seem more complex than it is.
In comparison, the problem is much simpler for a pupil who confidently knows the following facts by heart:
- 52 = 25
- ÷0.01 = x100
- To multiply an integer by a decimal, I can ignore the place value at first and adjust afterwards
- 4 x 25 = 100
- 4 x 25 = 4 x 2.5
- 4 x 2.5 = 10
…they will see this instead:
0.8 + 0.4 x 52 ÷ 0.01 = 0.8 + (0.4 x 25 x 100) = 0.8 + (4 x 2.5 x 100)
A much less daunting calculation, and one where much less tricky processing or self-doubting thought has taken place.
What does a knowledge grid have to do with it?
In the Michaela maths department, we aim to identify all the facts and relationships that can be codified as a single nugget of knowledge (or set of clear steps) that will reduce pressure on pupils’ working memories. This frees them up to tackle more complex and interesting problems and allows them to feel confident in their reasoning and solutions.
This does NOT mean teaching without understanding. It is the opposite: we aim for pupils to understand why something works, or is the way it is, and then to be so confident of that fact or relationship that they can recall and use it with minimal effort and worry.
The purpose of a knowledge grid – explained in detail by Joe Kirby – is to set out what these facts and relationships are, and to support pupils in learning them by heart.
Take indices, which the Y7 pupils have just learned about:
This sets out what we expect pupils to know by heart if they are going to be able to tackle more complex or interesting problems involving indices (e.g. What is the final digit of 10100+999+598?). Knowing by heart that ab x ac = ab+c doesn’t replace knowing why this relationship is true. But, knowing it by heart – and practising explaining why it is true – frees pupils up to tackle problems like ‘evaluate 23 x 52 x 22 x 53′.
Here is the grid for Y8 pupils at the outset of learning to solve linear equations:
Here is an example for Y8s learning to substitute and use formulae:
Sometimes it is solely a collection of relationships, such as the grid Y7 are about to work from:
(shading in grey typically indicates ‘optional’ knowledge, in that it is possible to be successful in maths without knowing those facts by heart…at least not at their stage!).
A useful rule of thumb is: if we, as maths teachers, know these facts by heart because they help us work more efficiently and confidently, then the pupils should know it by heart too.
How is it used?
In lessons, the knowledge grid lays out the agreed definition and procedures that we want to share with pupils. The constraint of the definition means we teach to a higher technical standard, ensuring that we stick to language like ‘eliminate this operation’ (instead of saying ‘get rid of the 4’ in a bid to make the maths feel more accessible). Knowing that the pupils must understand and use a phrase like ‘isolate the unknown’ forces us to explain it with greater clarity, check they understand it precisely, and then use it constantly.
In most lessons, pupils are quizzed on the terms and facts in the knowledge grids. This can be cold calling (asking questions and picking students), checking everyone’s answer on mini-whiteboards, or giving a 1-minute quiz in books (e.g. “write the formula for the area of each of these shapes” or “rewrite each of these as a multiplication: ÷0.5, ÷0.1, ÷0.25, ÷0.125, ÷0.01, ÷0.2”).
Once a week, pupils ‘self-quiz’ at home on the definitions and facts the teacher has set for that week. Typically, this is 10-15 facts/definitions. Pupils first practise saying the facts to themselves, then cover the right-hand side and write the definitions based on the prompts on the left-hand side, and then correct their errors in green. They continue this until a page is filled. It is possible to game it by mindlessly copying, but it becomes obvious if they’re doing so because…
Once a week, pupils take a formal, but low-stakes, written quiz, of which half will be a knowledge grid test (the other half tests their ability to apply procedures and try unfamiliar problems).
The levels of scaffolding vary; these are the knowledge grid sections Y8 took recently:
Pitfalls We Fell Into
An easy temptation is to produce a ‘revision mat’ full of facts, examples, diagrams and mnemonics. Although this is close to a knowledge grid, it isn’t as useful. It must be REALLY EASY to test yourself from a knowledge grid without ‘accidentally’ seeing the answer, or having prompts. It must be really clear what they should know by heart (the definitions and terms and facts) and what is just useful for jogging their memories (examples, where appropriate).
Another easy error is to go overboard with how much you try to codify and write down. If you, as teachers, struggle to articulate the definition or steps for something, it probably isn’t useful or suitable. Make steps for a strategy (e.g. solving equations) as generalised as possible so that pupils aren’t learning multiple minimally different steps and becoming muddled and frustrated. The more generalised the steps, the more they can be used to illuminate the common features of varied problems (and thus help pupils see the underlying structure).
Pitfalls We’re Still Trying to Avoid
We are still struggling to decide which aspects of algebraic simplification can be listed as facts: here is the start of a debate I was having in my head this morning for updating the facts in the ‘expressions and simplification’ grid:
Any that are included are there because pupils had become faster by recalling them as facts (as opposed to working them out) or their work was slowed because they weren’t confident when simplifying a fundamentally identical expression.
I hope it goes without saying that we would love to know what you think and if you have tried anything similar. Do you have facts and rules, besides those set out in examination specifications, that make a big difference to your pupils when learned by heart?
Whether this fascinates or enrages you, get in touch and come see the pupils (and grids…!) in action. You’ll have a great time 🙂
1: See Skemp, R.R (1977) Relational Understanding and Instrumental Understanding, Mathematics Teaching, 77: 20-6
2: See https://www.youtube.com/watch?v=1FQoGUCgb5w for Bjork discussing research in this field.