1. What is Teaching for Mastery? MA2M+ 102 Mark Moody
Secondary Mastery Lead
What is
Teaching for Mastery?
MA2M+ 102
Mark Moody, Head of Maths, Caldew School
Secondary Mastery Lead, North North West MathsHub

2. What is ‘Mastery’? A set of principles and beliefs.
A mastery curriculum.
A set of pedagogic practices.
Achieving mastery
The essential idea behind mastery is that all children need a deep understanding of the mathematics they are learning.
A mastery approach; a set of principles and beliefs.
This includes a belief that all pupils are capable of understanding and doing mathematics, given sufficient time. Pupils are neither ‘born with the maths gene’ nor ‘just no good at maths.’ With good teaching, appropriate resources, effort and a ‘can do’ attitude all children can achieve in and enjoy mathematics.
2. A mastery curriculum
One set of mathematical concepts and big ideas for all. All pupils need access to these concepts and ideas and to the rich connections between them. There is no such thing as ‘special needs mathematics’ or ‘gifted and talented mathematics’. Mathematics is mathematics and the key ideas and building blocks are important for everyone.
3. Teaching for mastery: a set of pedagogic practices that keep the class working together on the same topic, whilst at the same time addressing the need for all pupils to master the curriculum. Challenge is provided through depth rather than acceleration into new content. More time is spent on teaching topics to allow for the development of depth. Carefully crafted lesson design provides a scaffolded, conceptual journey through the mathematics, engaging pupils in reasoning and the development of mathematical thinking.
4. Achieving mastery of particular topics and areas of mathematics.
Mastery is not just being able to memorise key facts and procedures and answer test questions accurately and quickly. It involves knowing ‘why’ as well as knowing ‘that’ and knowing ‘how.’ It means being able to use one’s knowledge appropriately, flexibly and creatively and to apply it in new and unfamiliar situations.

3. What is Mastery? Mastery means that learning is sufficiently: Embedded
Deep
Connected
Fluent
In order for it to be:
Sustained
Built upon
Connected to
NCETM MathsHubs definition of Mastery

4. Procedural understanding
To know how to ...
Facts, skills, procedures, algorithms
No understanding of underlying meaning
Conceptual understanding
Procedural understanding and conceptual understanding need to be taught and learned in tandem
Use tasks that allow students to develop procedural fluency by working on an example space of carefully selected questions which can then be discussed and analysed leading to conceptual understanding
To know why ...
Understanding concepts and relationships between concepts
Having a “sense” of something

5. 21 x 53 22 x 53 21 x 54 Which is greater? Problem from ICCAMS
Have you mastered multiplication?
Changing the product by increasing one of the numbers by 1, which gives the larger amount (or are they the same)?
How would pupils tackle this problem. Can they find the answer by reasoning rather than performing the calculations?
Possible extension: what is the difference between the two answers? Try it with other numbers – what’s the rule?
Which is greater?

6. 21 x 53
22 x 53
21 x 54 53 1 21
Using a representation draws attention to the mathematical structure and leads to conceptual understanding (why the products are different – and by how much) rather than carrying out the calculation, where the focus is on carrying out a procedure.
Can the representation enable students to calculate the product mentally leading to greater fluency? (what is 20 x 53 …) 1

7. What are these questions assessing?
How should pupils answer these questions?
Questions taken from the 2016 KS2 arithmetic tests
What are the questions assessing – concepts or processes?
How would we want our pupils to approach these questions – using an algorithm or using conceptual knowledge and reasoning?
How would your pupils answer these questions? Lower prior attainers, higher prior attainers? Are different approaches what define the “more able” or “less able” mathematicians?
How would you teach these concepts – separate to the algorithms, before or after algorithms have been mastered?

8. What are these questions assessing?
How should pupils answer these questions?
Conceptual understanding or procedural fluency?

9. What are these questions assessing?
How should pupils answer these questions?
What type of pupils (typically) would attempt the question this way?
How would you raise awareness of the reasoning without it becoming another trick to learn?

10. Teaching for Mastery: Some themes and key principles
All pupils can succeed.
Procedural fluency and conceptual understanding are developed in tandem.
Whole-class interactive teaching, where all pupils work together on the same lesson content.
Pupils gain secure and deep understanding before moving to the next part of the curriculum sequence.
Early intervention ensures pupils keep up (not catch up).
Challenge is provided not by acceleration into new content but by deeper thinking tasks within the same learning aim.
NCETM document released in June 2016
First bullet point links to Growth Mindset – Jo Boaler / Carol Dweck
Whole-class interactive teaching – what are the implications for classroom practice?

11. Some themes and key principles
Lesson design focuses on small steps through a carefully sequenced learning journey.
Teachers encourage demonstration, explanation, exploration, analysis and generalisation.
Practice is intelligent practice, developing conceptual understanding and reasoning whilst reinforcing procedural fluency.
Examples use variation to draw attention to ‘what it is’ and ‘what it is not’.
Significant time is spent developing deep knowledge of the key ideas needed to underpin future learning.
Key facts are learnt to avoid cognitive overload and to enable pupils to focus on new concepts.
Small steps, moving on when pupils are secure. “Rapid graspers” deepen their understanding of the same content rather than being moved on with a superficial understanding.

12. 5 big ideas
5 big ideas – the big picture of Teaching for Mastery

13. In mathematics, new ideas, skills and concepts build on earlier ones.
Coherence
In mathematics, new ideas, skills and concepts build on earlier ones.
If you want build higher, you need strong foundations.
Every stage of learning has key conceptual pre-cursors which need to be understood deeply in order to progress successfully.
When something has been deeply understood and mastered, it can and should be used in the next steps of learning.
A key element of mastery is that ideas are deeply understood and don’t need to be re-taught.
However, important that these ideas are used and built on as this provides the necessary consolidation.

14. 97 - 48 = 100 - 51 97 48 100 51 Representation and Structure
Bar models can be used to highlight underlying mathematical structures.
Playing with these ideas may reveal easier calculations for example – 2.37 could become 9.99 – 2.36 and subsequently no need for decomposition/partitioning

15. Representation and Structure
A representation is used to pull out the concept being taught, exposing the underlying structure.
In the end, pupils need to be able to do the maths without the representation.
A stem sentence can be used to describe the representation and helps pupils move to working in the abstract.

16. 1 2 1 7 1 3 1 9 1 12 1 100 4 1 2 3 3 8 4 15 triangle unit fraction
Variation
How is a triangle typically drawn? What (false) generalisations might pupils draw from this?
If we give simple examples, to try and help our pupils, do we risk causing problems later on? 4 1 2 3 3 8 4 15

17. 290
2900
5290
52900
5690
56900
150
1500
Variation
455
4550
Careful choice of examples can provide practice while still highlighting the mathematical structure. Pupils can then generalise and deepen their conceptual understanding at the same time as developing procedural fluency.
200 20

18. Highlight the essential features of a concept or idea through varying the non-essential features.
When giving examples of a mathematical concept, emphasise what it is and what it is not.
Variation
Variation is not the same as variety – careful attention needs to be paid to what aspects are being varied (and what is not being varied)

19. Mathematical Thinking
Mathematical thinking is central to deep and sustainable learning of mathematics:
looking for pattern in order to discern structure;
looking for relationships and connecting ideas;
reasoning logically, explaining, conjecturing and proving.
Shift the focus from “doing maths” to pupils thinking mathematically. Use variance and intelligent practice to develop an example space for pupils to explore – spotting patterns, conjecturing, generalising, proving, …

20. move between different contexts and representations
Quick and efficient recall of facts and procedures is important in order for learners to keep track of sub-problems, think strategically and solve problems.
The flexibility to:
move between different contexts and representations
recognise relationships and make connections
make appropriate choices from a whole toolkit of methods, strategies and approaches.
Fluency
The aim remains for students to be able to achieve fluency supported by, not at the expense of, conceptual understanding.

21. 5 big ideas

22. What is Teaching for Mastery? MA2M+ 102 Mark Moody
Secondary Mastery Lead
What is
Teaching for Mastery?
MA2M+ 102
Mark Moody, Head of Maths, Caldew School
Secondary Mastery Lead, North North West MathsHub