As well as my work with Durrington Research School I am also an Assistant Lead for Sussex Maths Hub. At the Maths Hub we have been implementing a Teaching for Mastery curriculum with the support of the National Centre for Excellence in Teaching Mathematics (NCETM) for a number of years now at both Primary and Secondary, and the government’s latest non-statutory guidance for KS3 mathematics also advocates this approach. I thought it would be timely to look at the main principles which underly Teaching for Mastery in maths and how these might be transferred to other subject areas.
The NCETM focuses on “5 Big Ideas” of Teaching for Mastery. Although these are written with maths in mind, they definitely link to the aspects of great teaching that we know work across the curriculum, for example to the “six principles” of challenge, modelling, explanation, deliberate practise, questioning and feedback from Allison and Tharby’s Making Every Lesson Count that we use at Durrington.
The 5 Big Ideas take these principles a step further and focus on subject pedagogy – how to teach maths in the best way possible to elicit deep understanding, enjoyment and ultimately success for our pupils.
Fluency refers to the quick and efficient recall of basic facts so that the working memory is freed up to concentrate on deeper thinking. In maths this might be knowing your times tables or being able to use place value to multiply and divide by powers of ten without using a chart. As students become more experienced mathematicians, they become fluent at more advanced skills such as expanding and factorising expressions.
In the classroom working towards fluency often takes the form of regular recall quizzes on previous topics so that they are revisited often. But the types of question we ask in the lesson can also promote fluency – by embedding one concept within another (for example asking an algebra question which requires knowledge of angle rules) we can continually practise different skills.
I think that this also extends to subject specific vocabulary – by pre-teaching definitions and revisiting them regularly so that they are fully understood, we give our students the tools to discuss and reason at a deeper level. We also use chanting and sentence stems to aid recall and understanding.
Other subjects will also have their “basic facts” which need to be practised to the point of fluency. Considering how fluent our students are in the prerequisite knowledge for a new topic is important and by doing an initial check for understanding we are then able to put measures in place to address any missing aspects.
Representation and Structure
In maths we are dealing with a particularly abstract system of ideas which can be a challenge for students. By using concrete manipulatives, such as counters or tiles to represent negative numbers or algebra, we can help to support their understanding while they grasp a concept by exposing the underlying structure. They may be able to quickly move on to using pictures, sketches or diagrams, and eventually we aim for them to dispense with these altogether. However they may use diagrammatic representations well into their mathematical careers – for example area models for algebra are regularly used at A Level – and this is not a problem. Using these types of representations used to be limited to primary school, but they have many applications in secondary too and have been great for enabling deep understanding from the beginning rather than having students just blindly follow a procedure.
Other subjects will have varying uses for different representations, but where they can be used, they should be used. In all subject areas we are trying to get our students to develop a complex, interlinked schema of ideas which they can draw upon and representing ideas differently provides more ways for students to think about a concept. Maybe using duel coding would help with this, and we have all heard the tenet “a picture is worth a thousand words”.
Variation refers to how we draw attention in maths to the key concept that we are trying to teach, and there are many aspects of this idea that cross over curricular boundaries. Some techniques that we use in maths are:
- Considering “what it is and what it is not” by giving examples and also non-examples
- Carefully structuring our sequences of questions to ramp up the difficulty incrementally, thus keeping all students together and enabling them to access increasingly difficult concepts
- Asking sequences of questions on a topic in which just one small aspect is changed each time, allowing students to make connections and conjectures
- Sticking to one key concept (for example area of a rectangle), but within this revisiting previous topics to promote fluency (for example using fractions, decimals, algebraic expressions as side lengths)
We always try to provide opportunities for reasoning by not only asking “what”, but “why” and “how do you know”. We celebrate mistakes as opportunities for learning. We use “think, pair, share”. We discuss (and pre-empt) misconceptions, we look at improving incorrect work, we use probing questions, and always try to speak, write and talk mathematically to promote disciplinary literacy – we want our students to behave like mathematicians.
Coherence is what ties this all together. We have thought carefully about how to sequence our mathematical curriculum so that the order of topics makes sense, and students have the right foundations to build upon. We have spent time breaking things down into small connected steps that gradually introduce a concept so that students are not overloaded and they can all keep up.
We spend time in our department meetings discussing how we will introduce and teach a topic, and we are careful to be as consistent as we can be with our approaches so that students have the same great deal whichever class they are in, and so that their knowledge makes sense year upon year. We work collaboratively and are continually updating and refining our teaching resources when we can see there are improvements to be made.
Considering these 5 Big Ideas has certainly had a huge impact on my teaching over the past seven years and also on the teaching of many people I have worked with. I think that there are many ways that they can transfer to areas across the school and I hope that these initial thoughts have been helpful. Sussex Maths Hub will be running a Curious about Mastery in Maths taster day later on this term, and also the Talking Maths Conference at Sussex University on June 17th – see the Hub website for details of both of these.
Here at Durrington we are currently looking to recruit a Deputy Leader of Maths. This article gives you a flavour of the way in which we are constantly looking to develop and improve our approach to teaching. If this sounds like a department you think you could make a positive contribution to and you are looking for your first leadership role, please take a look at the job advert.