1. Mastery in Mathematics
Pete Griffin, Assistant Director (Maths Hubs)

2. What do we mean by ‘mastery’?
The essential idea behind ‘mastery’ is that all children need a deep understanding of the mathematics they are learning so that:
future mathematical learning is built on solid foundations which do not need to be re-taught;
there is no need for separate catch-up programmes due to some children falling behind;
children who, under other teaching approaches, can often fall a long way behind, are better able to keep up with their peers, so that gaps in attainment are narrowed whilst the attainment of all is raised.
All children – except for children on an individualised curriculum.
From ‘Assessing Mastery’ - introduction

3. Four ways in which the term mastery is being used :
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.
From ‘Assessing Mastery’ - Introduction

4. 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.
From ‘Assessing Mastery’ - Introduction

5. 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.
Many teachers are asking us, and I guess are asking you “does a mastery curriculum meet the needs of the previously high attainers?”. This depth is just as important and helpful for these pupils and we hope that the assessment materials will help with how to provide this depth.
It is not true that the earlier you learn something the better. Shanghai teachers leave the teaching of fractions until our Y4 and yet by the end of Y6 they are well ahead of us in terms of their mastery of fractions.
From ‘Assessing Mastery’ - Introduction

6. 4. Achieving mastery of particular topics and areas of mathematics
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.
Mastery is a continuum
From ‘Assessing Mastery’ - Introduction

7. A pupil really understands a
mathematical concept, idea or technique if he or she can:
describe it in his or her own words;
represent it in a variety of ways (e.g. using concrete materials, pictures and symbols – the CPA approach)
explain it to someone else;
make up his or her own examples (and non-examples) of it;
see connections between it and other facts or ideas;
recognise it in new situations and contexts;
make use of it in various ways, including in new situations.
Mastery is a continuum
From ‘Assessing Mastery’ – Introduction. Adapted from a list in ‘How Children Fail’, John Holt, 1964.

8. Procedural Fluency / Conceptual Understanding
Debate around the teaching of primary mathematics often opposes procedure (i.e. developing children’s fluency with algorithms) against understanding of mathematical concepts.
However, it is unhelpful to see fluency and understanding in opposition. The 2011 Ofsted survey of good practice in primary mathematics shows that many successful schools successfully teach both fluency in mental and written methods of calculation and understanding of the underlying mathematical concepts.
Mastery is a continuum

9. What is the area of this triangle?
Why?
3 cm
Show the triangle and its dimensions and ask them to calculate its area.
Say “You might have used A = ½b×h, but why is it half the base; why does it have to be the perpendicular height?; what does the formula mean?”
10 cm
Area of triangle = half of the base × height
Area of triangle = ½ b × h

10. What is the area of this triangle?
When you realise that any triangle is half of the rectangle that you draw around it (show other bit of animation) and that the dotted red line shows this then you know why the formula works and can use it with more confidence.
Conceptual understanding and procedural fluency come together.
Area of triangle = half of the (base × height)
Area of triangle = ½ (b × h)

11. What is the area of this triangle?
Why?
3 cm
Show the triangle and its dimensions and ask them to calculate its area.
Say “You might have used A = ½b×h, but why is it half the base; why does it have to be the perpendicular height?; what does the formula mean?”
10 cm

12. Show the triangle and its dimensions and ask them to calculate its area.
Say “You might have used A = ½b×h, but why is it half the base; why does it have to be the perpendicular height?; what does the formula mean?”

13. Show the triangle and its dimensions and ask them to calculate its area.
Say “You might have used A = ½b×h, but why is it half the base; why does it have to be the perpendicular height?; what does the formula mean?”

14. Show the triangle and its dimensions and ask them to calculate its area.
Say “You might have used A = ½b×h, but why is it half the base; why does it have to be the perpendicular height?; what does the formula mean?”

15. Show the triangle and its dimensions and ask them to calculate its area.
Say “You might have used A = ½b×h, but why is it half the base; why does it have to be the perpendicular height?; what does the formula mean?”

16. Show the triangle and its dimensions and ask them to calculate its area.
Say “You might have used A = ½b×h, but why is it half the base; why does it have to be the perpendicular height?; what does the formula mean?”

17. Show the triangle and its dimensions and ask them to calculate its area.
Say “You might have used A = ½b×h, but why is it half the base; why does it have to be the perpendicular height?; what does the formula mean?”

18. Show the triangle and its dimensions and ask them to calculate its area.
Say “You might have used A = ½b×h, but why is it half the base; why does it have to be the perpendicular height?; what does the formula mean?”

19. Show the triangle and its dimensions and ask them to calculate its area.
Say “You might have used A = ½b×h, but why is it half the base; why does it have to be the perpendicular height?; what does the formula mean?”

20. Show the triangle and its dimensions and ask them to calculate its area.
Say “You might have used A = ½b×h, but why is it half the base; why does it have to be the perpendicular height?; what does the formula mean?”

21. Show the triangle and its dimensions and ask them to calculate its area.
Say “You might have used A = ½b×h, but why is it half the base; why does it have to be the perpendicular height?; what does the formula mean?”

22. Show the triangle and its dimensions and ask them to calculate its area.
Say “You might have used A = ½b×h, but why is it half the base; why does it have to be the perpendicular height?; what does the formula mean?”

23. Show the triangle and its dimensions and ask them to calculate its area.
Say “You might have used A = ½b×h, but why is it half the base; why does it have to be the perpendicular height?; what does the formula mean?”

24. Show the triangle and its dimensions and ask them to calculate its area.
Say “You might have used A = ½b×h, but why is it half the base; why does it have to be the perpendicular height?; what does the formula mean?”

25. Show the triangle and its dimensions and ask them to calculate its area.
Say “You might have used A = ½b×h, but why is it half the base; why does it have to be the perpendicular height?; what does the formula mean?”

26. Show the triangle and its dimensions and ask them to calculate its area.
Say “You might have used A = ½b×h, but why is it half the base; why does it have to be the perpendicular height?; what does the formula mean?”

27. Show the triangle and its dimensions and ask them to calculate its area.
Say “You might have used A = ½b×h, but why is it half the base; why does it have to be the perpendicular height?; what does the formula mean?”

28. Show the triangle and its dimensions and ask them to calculate its area.
Say “You might have used A = ½b×h, but why is it half the base; why does it have to be the perpendicular height?; what does the formula mean?”

29. Show the triangle and its dimensions and ask them to calculate its area.
Say “You might have used A = ½b×h, but why is it half the base; why does it have to be the perpendicular height?; what does the formula mean?”

30. Show the triangle and its dimensions and ask them to calculate its area.
Say “You might have used A = ½b×h, but why is it half the base; why does it have to be the perpendicular height?; what does the formula mean?”

31. Show the triangle and its dimensions and ask them to calculate its area.
Say “You might have used A = ½b×h, but why is it half the base; why does it have to be the perpendicular height?; what does the formula mean?”

32. Show the triangle and its dimensions and ask them to calculate its area.
Say “You might have used A = ½b×h, but why is it half the base; why does it have to be the perpendicular height?; what does the formula mean?”

33. Show the triangle and its dimensions and ask them to calculate its area.
Say “You might have used A = ½b×h, but why is it half the base; why does it have to be the perpendicular height?; what does the formula mean?”

34. Show the triangle and its dimensions and ask them to calculate its area.
Say “You might have used A = ½b×h, but why is it half the base; why does it have to be the perpendicular height?; what does the formula mean?”

35. Show the triangle and its dimensions and ask them to calculate its area.
Say “You might have used A = ½b×h, but why is it half the base; why does it have to be the perpendicular height?; what does the formula mean?”

36. Show the triangle and its dimensions and ask them to calculate its area.
Say “You might have used A = ½b×h, but why is it half the base; why does it have to be the perpendicular height?; what does the formula mean?”

37. What is the area of this triangle?
3 cm
Show the triangle and its dimensions and ask them to calculate its area.
Say “You might have used A = ½b×h, but why is it half the base; why does it have to be the perpendicular height?; what does the formula mean?”
10 cm
Area of triangle = half of the base × perpendicular height
Area of triangle = ½ b × h

38. Write down the number that is 1000 times bigger than …
143
It is important to know that:
our place value system is based on each successive digit representing 10 times the value of the preceding one to its right;
that each time you multiply by 10 each digit moves along into a place that has a higher value;
that the places that become empty as a result of this multiplying up have to be filled with a place holder ‘zero’.
All these awarenesses are important but to quickly multiply by 1000, the procedure of adding the same number of zeros as the number you are multiplying by is a neat and efficient procedure.
Again conceptual understanding and procedural fluency come together.
(Without the conceptual understanding it may be that students will still apply this rule to decimal numbers; without procedural fluency, the amount of processing required to keep track of each of each digit, its value and its new value after moving, etc., etc. would slow down the process considerably).

40. Is there evidence of conceptual understanding?
Is there fluency and efficiency?

41. Sally knows all her tables up to 12 x 12 When asked what is 13 x 4 she looks blank. Does she have fluency and understanding?

42. The aims of the mathematics curriculum
The National Curriculum for mathematics aims to ensure all pupils:
become fluent in the fundamentals of mathematics, including through varied and frequent practice with increasingly complex problems over time, so that pupils have conceptual understanding and are able to recall and apply their knowledge rapidly and accurately to problems
reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language
can solve problems by applying their mathematics to a variety of routine and non-routine problems with increasing sophistication, including breaking down problems into a series of simpler steps and persevering in seeking solutions.
These three aims should not be lost in the detail of the programmes of study. The new draft seeks to strengthen these aims.
These aims come from research carried out by the DfE into high performing jurisdictions

43. The whole is not the sum of the parts
Mathematics is an interconnected subject in which pupils need to be able to move fluently between representations of mathematical ideas. The programmes of study are, by necessity, organised into apparently distinct domains, but pupils should make rich connections across mathematical ideas to develop fluency, mathematical reasoning and competence in solving increasingly sophisticated problems.

44. Conceptual Understanding
Keeping the balance
Procedural Fluency
Conceptual Understanding
Talk about how there needs to be a balance of the two, but more so than this, the two need to be integrated.
Many schools are doing either one or the other well but few are doing both, this is the challenge
Is this the right image – discuss and see the next slide. We want to emphasise the integration 44

45. Knowing when you are focussing on different aspects
Procedural fluency
Conceptual understanding
May discuss, are the two fully integrated, how large is the intersection?

46. Integrating the two so that one strengthens the other
Procedural fluency
Conceptual understanding

47. Teach, learn, confuse Think: Which line is longer?
We are going to look at some activities from the materials in a little bit but I have just picked out a couple of examples we saw from Shanghai colleagues which illustrate this feature of teach – learn – confuse.

48. 1 2 1 3 Teach, learn, confuse Think: Which line is longer?
We are going to look at some activities from the materials in a little bit but I have just picked out a couple of examples we saw from Shanghai colleagues which illustrate this feature of teach – learn – confuse. 1 3

49. Teach, learn, confuse

50. Solve the following
 + 17 = –  = 90 – × 2.5 =  × 10 3 ÷ 4 = 15 ÷ 
This illustrates how a conceptual method rather than a procedural method can lead to a quicker answers. Whilst procedures are important, they do not stand alone but need to be underpinned by conceptual understanding.

51. Structural Arithmetic
These examples are from the recent ‘Teaching for Mastery’ assessment materials materials.
Here’s the question. Read, internalise…
How can we represent that with coins?
4 x 10p
8 x 5p
2 x 20p
From Multiplication and Division Year 2

52. Structural Arithmetic
18 ÷ 3 = 6
Where is the 18?
Where is the 3?
Where is the 6?
“I see the six. Two groups of three make six”
(Higher attaining Y3)

53. Structural Arithmetic
Examples of practice development

54. What do we mean by ‘mastery’?
The essential idea behind ‘mastery’ is that all children need a deep understanding of the mathematics they are learning so that:
future mathematical learning is built on solid foundations which do not need to be re-taught;
there is no need for separate catch-up programmes due to some children falling behind;
children who, under other teaching approaches, can often fall a long way behind, are better able to keep up with their peers, so that gaps in attainment are narrowed whilst the attainment of all is raised.
All children – except for children on an individualised curriculum.
From ‘Assessing Mastery’ - introduction