There is, clearly, a difference between “doing” and “understanding” maths, but in the secondary context there are often the circumstances that turn this difference into tension: difficulties recruiting specialist maths teachers, and high-stakes exams. Some pupils arrive in KS3 and are taught maths by teachers who are not trained or experienced maths teachers, and then they move into KS4 with insecurities of core knowledge and skills that put them at risk of not reaching grade C at GCSE. The school’s response to this is teaching that, given the limited time available and the importance for the pupil (as well as the school) of achieving a pass or better grade, prioritises procedural fluency above conceptual understanding – “how” not “why?”.
If this is to change, then the maths teachers’ subject knowledge must be deepened through professional development – and PD that does this well is as important for existing maths specialists as it is for out-of-field teachers who want to, or have been asked to, become maths specialists. It is as necessary for KS3 teachers as it is for those assigned to KS4 and 5 classes; it is as important in departments fully staffed with well-qualified maths specialists as it is in departments where that is not the case.
Consider this question, used by Kath Hart et al. in the CSMS research:
Mr Short is as tall as 6 paperclips. He has a friend Mr Tall. When they measure their height with matchsticks, Mr Short’s height is 4 matchsticks and Mr Tall’s height is 6 matchsticks. What is Mr Tall’s height measured in paperclips?
In the research, most pupils answered this wrongly, and said “8”. “Deep subject knowledge” means knowing why they said that, and knowing what should have been in the scheme of work in the weeks, terms and years prior to this so that the right answer was given quickly and confidently, and with a coherent explanation.
Teachers with deep subject knowledge think critically, and successfully, about
- the choice of the representation / model with which they introduce a (new) concept
- the reasoning they cultivate and sharpen through the in-class discussions they foster and steer
- the misconceptions they predict and confront as part of the sequence of questions they plan and ask
- the conceptual understanding they embed and deepen through the intelligent practice they design and prepare for the pupils to engage in and with.
Expanding the first point, this means that they consider carefully if their chosen explanation or model
- can at first be explored “hands on” by all pupils, irrespective of prior attainment
- arise naturally in the given scenario, so that they are salient and hence “sticky”
- can be implemented efficiently, and increase all pupils’ procedural fluency
- expose, and focus all pupils’ attention on, the underlying mathematics
- are extensible, flexible, adaptable and long-lived, from simple to more complex problems
- encourage, enable and support all pupils’ thinking and reasoning about the concrete to develop into thinking and reasoning with increasing abstraction.
For example, in issue 116 of the NCETM secondary magazine there is a discussion of a model for division of and by fractions that seems to fit well the above criteria, and in issue 123 there is proposed a sequence for teaching mechanics at KS5 that exemplifies this. In contrast, in issue 118 there is an analysis of why the acronym “BODMAS” is a poor model – indeed, the recent NCETM Director of Secondary would, at any and every opportunity, call “BODMAS” a “waste of oxygen”!