If, like many primary schools, you are considering more of a maths mastery teaching approach, or even if you’ve been doing it for a year or so now, you’ve probably got a few techniques you’re confident in using, but also a few questions about what maths mastery actually means.
With the help of our own in-house Singapore Maths expert, Wendy Liu, we’ve completed an entire remodelling of our own teaching methods for our 1-to-1 maths interventions and during this process, have learnt a lot about the benefits of maths mastery, how we can use it for better pupil outcomes at Key Stage 2, and why mathematics mastery is so key to building pupils’ conceptual understanding.
Here’s what we’ve learnt. We hope it’s useful for you, whatever stage you’re at on your Maths mastery teaching journey.
And if you haven’t got time to ready the entire blog post right now, just download our free Ultimate Guide to Maths Mastery and you’ll have it all at your fingertips when you need it!
Origins of maths mastery
Until recently, teaching children to ‘follow a process’ had been at the core of Maths education. Explaining how or why something happened became merely an afterthought.
As long as a child got the answer correct, their comprehension of the method was less important.
However, by not teaching children conceptually, we left them incapable of making the necessary links required to problem solve (an increasing priority in the new 2014 Maths curriculum).
More importantly, it denied children the opportunity to truly enjoy and engage with mathematics – as it became merely a memory test for many of them.
With the 2014 new curriculum, this began to change. Increasingly children are required to reason, problem solve, and discuss methods. This is where teaching for mastery shines, which is partly why so many schools have begun to make the switch to a more asian-style maths mastery approach.
Looking for a step-by-step guide to mastery...?
Who is the free mastery resource most suitable for?
If you’re unsure about beginning a mastery path or not yet persuaded by the mastery approach to maths, you can see the key aspects of teaching the method and decide for yourself.
If your school has received government funded training and support and have considering implementing mastery, but are unsure where to start, this guide is perfect for you. It takes you through it from the ground up, from mixed ability setting to lesson planning, to using bar models and journals in your classroom.
Finally, if you’re a school that has already implemented the approach, this guide is the perfect health check for your pedagogy!
Why choose a maths mastery approach to learning?
We’ve explored the benefits in our blog on the 6 top reasons that you should adopt maths mastery teaching KS2 or KS1 in your primary school
1. Maths mastery: an investment in your staff
Investing in mastery is an investment in your staff. The training staff receive to implement the approach will improve their teaching practice and, by extent, the quality of teaching in your school.
There is a degree of pedagogical expertise needed to make mastery work in any school. Investing in this means investing in staff CPD, which will inevitably enhance subject knowledge.
There are many organisations that offer training such as NCETM or the Mathematics Association. Providing staff with the necessary training not only ensures teachers have the skillset to teach mastery but, crucially, it shows you trust that they can.
Having a team of Maths teachers you can trust, who are confident in their ability to teach using mastery techniques means you have a team who will doubtless champion a shift towards embedding the approach across your school. These passionate teachers will doubtless help disseminate the teaching techniques throughout your staff.
2. Maths mastery creates conceptual understanding
Teaching concepts alongside procedural algorithms is at the core of mastery. Notably, it arms children with a variety of strategies to tackle problems logically and independently, using what they already know.
Using mastery techniques in this way helps break the cycle of rote learning in mathematics. It provides children with the opportunity to grasp ‘real’ Maths, empowering them with problem solving skills and – crucially – a sense of achievement.
3. Maths mastery – maths in small steps
Small increments of learning are fundamental to this approach. As is “over-teaching” core concepts such as number and place value and the four operations.
Due to this, the initial progress of children taught this way may seem less dramatic when compared to traditional Maths teaching.
However, this is simply because this style of incremental learning ensures children have a thorough conceptual understanding of the Maths, so that no gaps in that area remain.
This avoids instances where the class teacher tries to move on to a new concept, only to discover the class is simply not ready. The “jump” is too large.
By taking smaller steps through learning, maths mastery teachings lead to deeper understanding of mathematical concepts, as children are given ample time to internalise their maths learning.
Grouping and setting in maths mastery lessons
The debate about ‘setting’ children into ability groups is an ongoing one. Many feel that doing so is beneficial: it’s undoubtedly easier to plan for and teach to a smaller spectrum of ability, and it does work well when teaching children to apply a set of rules.
However, maths mastery teaching is different. Here, the teacher no longer tells the pupils what to do; instead they facilitate learning by guiding the pupils to think for themselves. Communication is a vital part of this process.
Increased metacognition and peer discussion within mixed ability classes
It is through peer discussion, and the proving and disproving of ideas, that metacognition happens (being aware of one’s thought processes). Given that peer discussion in mastery teaching and learning is so important you should consider the advantages of the mixed ability classroom:
• There will be more fruitful discussions for lower ability children to participate in.
• The higher ability children have the opportunity to explain their methods to lower achievers which ensures they have ascertained a depth of understanding.
For more on the debate on setting in classrooms, read Mixed ability vs ability grouping: Where do you stand?
How to manage mixed ability grouping within maths mastery
If a school is introducing a mastery initiative for the first time, then mixed ability groups can work well up to year 4. However, by year 5 and 6 the range of ability may be too wide to implement a totally mixed group at this stage. In this case, it might be possible to separate the highest or lowest achievers from the rest, however this will depend on the cohort.
Planning interventions such as Third Space Learning’s 1-to-1 maths boosters to plug gaps in core knowledge is an effective way to ensure that the whole class can progress together. These could be based on an audit of each child’s needs as opposed to one regular intervention group.
Having a ‘Maths meeting’ is an effective teaching model adapted from Shanghai schools. Here, the teacher is released for 30 minutes in the afternoon to support those pupils who didn’t understand the concepts taught that day.
Read more in our blog post: What every teacher should know about teaching for mastery in a mixed ability maths class
Challenge rapid learners
Mastery learning happens in small steps which enables all children to move through key points of understanding together. Advanced learners are not left without challenge, as this can be achieved through higher–level questioning and extension tasks.
How Third Space Learning uses a mastery approach in its 1-to-1 tutoring sessions
At Third Space, we utilises a mastery approach to our 1-to-1 tutoring sessions as we are of the belief that each topic within maths should be understood inside and out before moving on to the next.
To see this in action, book your no commitment 10 minute demo to discover the impact our 1-to-1 interventions can have for your pupils – it’s never been easier or more affordable. Call us on 0203 771 0095 or contact us here to learn more about how we can help turbocharge maths in your school, even for low prior attainers!
How to plan a maths mastery lesson
Historically, government schemes (Numeracy Strategy, Numeracy Hour) limited the amount of flexibility teachers had in terms of when they taught each topic.
The new curriculum instead permits a freer approach, as long as a certain amount is covered by the end of each Key Stage. As such, teachers may spend a long time planning and creating resources with a wide range of differentiation.
However, with mastery it is possible to be flexible without the need for excessive planning.
So what should you teach, and when?
Having more autonomy to decide the direction of your lesson is a good thing, but it can leave teachers with questions such as:
• Which topic should be planned first?
• Is there a hierarchy of topics?
• How long should be spent on a topic?
• What are the non–negotiables for each lesson?
• Is there a sensible order of topics to maximise learning?
In answer to these questions there is no definitive order, but there are some concepts that must be taught prior to teaching others. Online tools can aid planning by mapping out lesson objectives.
Maths No Problem is based on Singapore Maths and designed by experts.This is a subscription–based website, however it provides a breakdown of the
small steps within each topic. As the whole class works together, tasks should be presented in a format that all students can relate to, for example pictures or visual will support EAL and SEN children.
The problems should be ‘low threshold, high ceiling’ (LTHC) activities that enable all pupils to access the learning, but have depth to cater for more advanced learners.
Example Low Threshold High Ceiling activity
This Year 2 question looks at subtraction.
For many children, 31 – 5 can be meaningless and confusing. Renaming (or exchanging) the numbers in a context that is familiar to children gives the Maths meaning. Exploring the problem through discussion, and coming up with the number sentence as a class, means children are more likely to access the Maths.
Allowing children to do this independent of the teacher also has merit. The teacher can circulate the room, assessing learning and asking questions to support learners or challenge their thinking.
Differentiation and assessment within maths mastery
For more advanced learners, challenge can be achieved by proving their answers or explaining a method that they have used, or thinking of an alternative method to solve the above problem.
For struggling learners, they may approach the above problem by counting the number of remaining balloons or using maths manipulatives like base ten.
Children need ample time to explore and demonstrate ‘exchange’
with physical equipment. Once this has happened, learning can be linked to the column method. At each stage, pupils should be able to explain what is happening by referring back to the balloon context. This is only possible if children have a solid understanding of place value and the base ten system – without this, the ‘crossing out’ and ‘carrying over’ just becomes a learnt
Differentiation through teacher expectation and challenge
Differentiation in mastery sessions is not presented on different worksheets, but through teacher expectation and challenge. The positives of this are twofold. Less time-consuming planning and boosted confidence for lower ability children (as the work isn’t tiered).
For quick finishers, in depth effective questioning techniques such as these can be used to further their learning:
• “Can you come up with a ‘brand new’ technique to solve the problem?”
• “Can you write a leaflet to explain what you were working on today for an absent classmate?”
By assessing what they have written, teachers gain insight into pupils’ depth of understanding and highlight any possible misconceptions. Activities from the NCETM National Curriculum Assessment Materials are useful for assessing children in their mastery learning.
Know the lesson inside out in maths mastery
To teach for mastery means being prepared, and teachers need to take into account what the non-negotiables are for that lesson:
• Subject knowledge. How best to teach the concepts with sound pedagogy
• Knowing the common misconceptions in the topic.
• Technical terms. Introducing the correct terminology
• Expectations. Having an expectation that children answer in whole sentences, using mmathematical language
• Feedback. If possible, encourage staff to plan together and give feedback of practices that were or were not successful.
Spend ample time on concepts
‘Over–teach’ concepts, especially the core areas of:
• Number and place value
• The four operations
• Times table and division facts
Seeing patterns and structures in Maths is the key to being able to generalise thinking and enables children to make sense of topics such as fractions and decimals. Children will be unable to see, for example, the relationship between equivalent fractions if they don’t know their times tables.
Questioning techniques within maths mastery
A hugely important part of a mastery classroom is questioning. More traditional ‘chalk and talk’ lessons can be transformed into an excited discussion by incorporating good questions.
In successful mastery classrooms, it is common to see hands thrown in the air in order to prove or disprove what has been asked. A cleverly asked question will see children ‘struggling’ between their own preconceived ideas and the logic that may be in front of them.
The peer discussions that take place as a result of questions posed are invaluable as a way for teachers to assess levels of understanding, and as a platform for pupils to learn from one another
True mastery in maths is demonstrated when two elements are combined:
• The reasoning and justifying of a concept.
• The correct use of technical language needed to convey a message.
What does this look like in the primary classroom?
Noise level: This will be good noise – pupils on task trying to prove their point during peer or group discussions.
Teacher circulating: Asking appropriate questions will facilitate learning and deeper thinking.
Unbiased attitude: It is important that the teacher neither validates nor rejects an answer or a theory for two reasons:
If the answer is confirmed correct, the rest of the class no longer need to think for themselves. An alternative approach is to ask several children what they think the answer is before asking, ‘How do you know?’
It is up to the whole class to justify and reason the theory correct or incorrect. Teachers avoid ‘telling’ the students the answers, and the pupils have to work out why it must be correct – a more powerful type of learning that helps link concepts together.
What questions to ask in a maths mastery lesson?
Some generic questions to ask that are perfect to help assess learning include:
• How do you know?
• Can you prove it?
• Can you come up with a different method?
• What do you notice?
• Will it always do that, and why?
• What happens if?
• Does your answer seem reasonable? Why/why not?
Thought must be given to the types of questions teachers ask. These need to be tailored to support or challenge learners.
One of the most important parts of challenging higher ability children is questioning. The power of a simple question should not be underestimated.
For example, ask questions such as:
• ‘What do you notice?’
• ‘Will it always happen?’
Teacher Research Groups
Teachers require in-depth knowledge of the topic – something that may not come naturally but can be learnt. To aid this, it is good practice for teachers to share things that went well and things that could be improved. Learning from each other and not being afraid of making mistakes are significant parts of mastery.
‘Teacher Research Groups’ are very common in Asian countries – however, they are given dedicated time for these meetings to happen. Nonetheless, it is possible to promote such feedback informally.
Lack of knowledge shouldn’t deter a teacher from asking questions beyond their own understanding. In fact, allowing advanced learners to discover and explore, can lead to the teacher learning something. This is beneficial as children enjoy seeing teachers who don’t know all the answers.
Growth mindset within maths mastery
The power of having an open mind is a teaching and learning asset. A growth mindset from a learner’s point of view is the self belief that you can improve at anything if you work at it, and that the brain gets ‘stronger’ as you do.
Pupils with a growth mindset:
• Thrive on challenge
• Are resilient to setbacks
• Show a great deal of perseverance
A fixed mindset is the opposite:
• Those that give up easily
• Say things like, “I can’t do it!”
• Won’t even attempt something new
• Are afraid of making mistakes
• Never answers in class unless they are 100% sure
• Rub out mistakes in their books
Research shows that pupils with a fixed mindset are less likely to reach their full academic potential.
How can teaching for mastery help with creating a growth mindset?
A mastery class adopts a ‘can do’ attitude. There is an underlying message that all children can learn maths and that no one is born being ‘good’ or ‘bad’ at maths.
Mastery aims to promote a growth mindset, providing a safe environment where mistakes are celebrated as a key to unlocking new learning. Through class discussions, misconceptions are challenged by concrete proof. This provides evidence that the ‘understanding’ a child had before must have been incorrect.
It is vital to approach this in a supportive way, so no child is made to feel uncomfortable or stupid.
As mastery is about conceptual learning, children begin to make links across Maths. As their understanding improves, self-belief and confidence grow. Therefore, over time children begin to look forward to the challenge of Maths as opposed to be daunted by the prospect.
Further resources for growth mindset
Classdojo have some great videos to use to help encourage a more positive mindset –these work well during PSHE.
The story of the butterfly is a good metaphor for mastery: If the butterfly does not struggle as it emerges from its chrysalis, it will never fly. This demonstrates how struggling with learning helps us to be better. Encouraging a positive attitude towards learning helps the school to be a much happier place.
For more guidance on how to create a resilient and learner-friendly culture in your primary school or classroom download our free Guide to Getting Growth Mindset Right in Primary Schools.
Manipulatives for maths mastery
In mastery, it is widely accepted that children learn best through a Concrete Pictorial Abstract approach (CPA).
The initial concrete phase brings concepts to life by using
physical objects. Manipulatives play a key role in this as they are the concrete resources used to support pupils thinking as they explore abstract ideas. Using something ‘real’ to make sense of the Maths takes away the need to imagine or visualise at the early stage of learning a new concept.
Read this blog post on how maths manipulatives transform primary Maths at KS2 together with a free guide to our top recommended hands on resources for your maths classroom.
Best practice use of manipulatives
• Investing, not only in resources, but also on training will ensure that manipulatives are
• It is important that manipulatives are not solely relied upon – there is a danger that
over–reliance on equipment will hinder progress.
• With experience, teachers will gauge when is most appropriate to use materials, which
materials to use and when they should be removed. This can also be personalised to individual children depending on their needs.
Choosing the concrete resources you need for maths mastery
Overall, you should allow autonomy when choosing equipment. When children are familiar with the manipulatives and how to use them, give them a chance to select their own choice of equipment to represent the problem they are to solve. This is helpful because:
• The teacher can see how well the student has understood the concept by assessing the relevance of the resources chosen.
• It promotes high–level thinking – pupils have to ensure that the resources chosen are made relevant to the problem and represent it adequately
What to know about anchor charts for maths mastery
Anchor charts are posters based on key concepts in a lesson and help children to ‘anchor’ their learning to these displays. They can be created during the lesson as learning unfolds or when children share their methods and ideas.
Anchor charts can also be created post lesson and used as a learning aid in subsequent lessons. They should be constructed with the pupils and can be made personal by adding the name of the author: e.g. ‘David’s method’. They then serve as a memoire to learning and should be displayed prominently in the classroom and referred back to when appropriate.
Anchor charts should be utilised whenever there is a purpose for having one displayed in the classroom. This may differ year on year as it is somewhat dependent on the cohort of children.
What to put on your anchor charts
Anchor charts are easily created by sticking a piece of sugar paper onto the whiteboard. The class teacher and a well–prepped TA can write and draw out key ideas. Children can help by writing their own methods and explanations.
There are no rules to dictate what an anchor chart must look like, however:
• It should be clear and precise.
• It should be colourful and neat.
• Use of diagrams and pictures ensures that it is easier to read and understand.
• Include a challenge or an in–depth question can promote thinking in learners.
• Avoid making the posters text heavy as pupils will not want to read it.
If there is limited space on the walls, a ‘washing line’ display can be used. Alternatively, arrange the charts so that only the headings can be seen and the relevant pages can been ‘flipped’ to when necessary.
Download the Ultimate Guide to Maths Mastery for some examples of what anchor charts look like.
Journalling for maths mastery
A huge part of the National Curriculum 2014 required students to develop reasoning and fluency with number facts in Maths, but most teachers had difficulty evidencing this – particular if work that was set was just a page of calculations.
A way to combat this was to introduce the ‘Maths Journal’ – a book dedicated to the thoughts and justifications of the individual pupil.
Here, children can log and articulate the different methods and ideas for solving a problem, enabling teachers to assess what is going on inside the child’s head.
Journalling could be considered a ‘window into the child’s thinking’ and where misconceptions arise, they can be dealt with in a meaningful way.
5 things your pupils might journal
1. At the start of a ‘new’ topic children can journal what they already know and what they would like to find out.
2. As the topic progresses, children may journal about their thought processes and practices that they have used to come to a solution.
3. They may also journal about the different methods, techniques and strategies that they have come across, explaining their understanding.
4. When looking at various methods, children may want to comment on the most efficient and explain why the efficacy is dependent on the numbers used.
5. Towards the end of the topic, children could log what they have learnt in the unit of work and use examples to demonstrate their learning.
What they log is entirely up to the teacher, however their journals should provide teachers with a clear path of learning and progress that has taken place for each individual child.
Aside from the ongoing assessment that takes place during a lesson, journals are a fantastic source of formative assessment. However, this can only work effectively if books are marked and kept up–to–date.
When, why and how to journal
Daily journalling – All students should keep a journal, but when to write in it depends on what learning the teacher wants to log. Daily journalling is recommended if different concepts are being introduced each day.
Assess understanding – Aside from being an assessment vehicle, journalising also encourages metacognition as the pupil is being forced to think again about the learning and how best to record it.
Have high standards – One of the fundamentals of journalling is that ideas and
explanations are very clear. Precise language and diagrams should be used too. With advance learners, the expectations and the quality of work produced may be higher than that of a struggling learner, however ideally teachers shouldn’t accept mediocrity – even lower ability children are able to use representations to explain what they are thinking.
Help with vocabulary – Producing a list of keywords with the class will help children focus their work further.
A typical journalling session should only last 10–15 minutes, so there shouldn’t be copious amounts of marking. However, teacher feedback should help the pupil to identify an underlying misconception, whilst next step marking can promote depth of understanding.
Use of pictures and diagrams
Pictorial representations are just as vital as concrete materials. These are particularly helpful for engaging EAL pupils, SEN pupils and visual learners, as well as being a platform to express their understanding. Initially, the teacher may need to model different visual representations but as the children get used to using them, they may well go on to create their own.
There is no right or wrong model as long as it expresses accurately an explanation.
Download the Ultimate Guide to Maths Mastery for some examples of what journalling can look like.
Bar modelling- the cornerstone of maths mastery
Bar models were developed and used in Singapore and are an effective visual tool to help students break down, understand and solve a problem. Children need plenty of exposure to the different types of bar models before they can use them productively.
Through maths mastery learning, children should first learn simple bar models to solve addition and subtraction maths word problems, then multiplication and division. Gradually they progress to being able to solve more complex maths word problems for KS2 SATs with bar models.
What are bar models?
Bar models are rectangular bars drawn to represent a scenario posed in a worded problem. These rectangles provide a way to ‘see’ the steps and the appropriate operations needed to solve the problem.
They model the worded problem in its most simplistic state, removing all words except for the key information. This allows the student to deal with one section of the worded problem at a time, and gives space
to decide whether the piece of information is important in the context.
For EAL and SEN learners, this visual image is powerful, as they no longer need to worry about long sentences which they may find difficult to comprehend – they can now see what they have to do.
By allowing children real time to reflect and provide ample feedback, the approach is revolutionising Maths learning across the UK. So, to take the next step for your school and download the Ultimate Guide to Maths Mastery. We hope you find it useful!