What do we mean when we talk about teachers’ subject knowledge in the context of maths? Do we mean teachers have to have at least a GCSE Grade C in maths? Or an A level maybe? Whatever the measure, we want proof that they can ‘do’ maths themselves, don’t we?
That doesn’t get anywhere near what an effective maths teacher must know. A maths teacher must know how maths is learnt; what journey the mind needs to go on to acquire deep and lasting understanding. In primary schools he/she must know what goes on in a child’s mind when the concepts of number and counting are first introduced, what misconceptions most commonly occur in those minds; how to present concepts in a way which minimises the chance of those misconceptions occurring, and what questions to ask to check that they haven’t occurred. This domain of knowledge was referred to by Shulman (1986) as pedagogical content knowledge.
Let’s take something that most primary teachers would rate themselves as feeling fairly confident in doing: carrying out column subtraction. After all, the algorithm most commonly taught is the same one which most teachers learnt at school themselves. But effective teaching of this goes far beyond ‘do as I do’. A teacher with good pedagogical content knowledge would know that a common difficulty with column subtraction is that pupils may (as in the mistaken version on the left) subtract the smaller digit from the larger digit, regardless of which is in the minuend and which in the subtrahend (*):
This teacher would have thought about a good model to represent the need to regroup in this situation: both Dienes and place value counters will demonstrate nicely the need to break up a ten to provide additional ones. They will have carefully planned the accompanying language: when working with the remaining tens do we say “sixty subtract fifty” or “six tens subtract five tens” or just “six subtract five”? Or does it depend on where we are in the learning process? The implications of each will have been carefully considered.
This teacher will also know that overuse of column subtraction is a possibility, and a fluent mathematician will use a ‘mental first’ approach. They may encourage children to look at a range of subtraction calculations and group then into ones it’s probably efficient to use column subtraction for (maybe 726 – 468) and ones it is less efficient for (maybe 351 – 298).
They will get the children to attend to common situations where column subtraction is actively unhelpful: finding change from notes for example is unwieldy with column subtraction. Confident mathematicians will work this (£20 minus £6.99) out mentally, and so the confident teacher will teach a representation which will lead to this mental approach.
And of course the teacher will have carefully attended to the building blocks needed to perform column subtraction fluently (getting a class of Year 3 children fluent in the subtraction facts which bridge 10 such as 13 – 4, 15 - 8 is a pedagogical challenge in itself), so that at the introduction of the algorithm children can focus their full attention on that.
In primary schools the mechanisms for developing this kind of pedagogical content knowledge are not always planned for but, as noted in a 2014 Sutton Trust Report What Makes Great Teaching, time spent on it will pay dividends not just for that year but for future years too and is an investment with great payback.
(*) The minuend is the number from which another number (the subtrahend) is to be subtracted