1. The New Primary Curriculum for mathematics: what does it mean to you?
Caroline Clissold:

2. The National Curriculum for Mathematics aims to ensure that 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

3. 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. They should also apply their mathematical knowledge to science and other subjects.

4. The expectation is that the majority of pupils will move through the programmes of study at broadly the same pace. However, decisions about when to progress should always be based on the security of pupils’ understanding and their readiness to progress to the next stage. Pupils who grasp concepts rapidly should be challenged through being offered rich and sophisticated problems before any acceleration through new content. Those who are not sufficiently fluent with earlier material should consolidate their understanding, including through additional practice, before moving on.
Stage 1: emerging
Stage 2: expected/achieving: it is at this stage that teachers often accelerate through new content
Stage 3: exceeding – great depth and complexity of problem solving

5. Calculators should not be used as a substitute for good written and mental arithmetic. They should therefore only be introduced near the end of key stage 2 to support pupils’ conceptual understanding and exploration of more complex number problems, if written
and mental arithmetic are secure. In both primary and secondary schools, teachers should use their judgement about when ICT tools should be used.

6. EYFS handbook:
Key aspects of effective learning characteristics include children:
• being willing to have a go;
• being involved and concentrating;
• having their own ideas;
• choosing ways to do things;
• finding new ways;
• enjoying achieving what they set out to do.
These develop the children’s ability to reason and problem solve – these characteristics must be developed throughout a child’s education.

7. Something to consider……
Divide the requirements into these sections to promote reasoning:
Number sense
Additive reasoning
Multiplicative reasoning
Geometric reasoning
Build two or three week units that are based on these. Include measures and statistics within them and not separately – they are real life applications of the mathematics we teach.
This will allow more time for teaching, practising, consolidating and deepening children’s understanding of the fundamentals of mathematics.

8. Mastery curriculum Some points to consider:
Differentiation is achieved by emphasising deep knowledge and through individual support and intervention - not in the content taught. Questioning and scaffolding vary, different problems to solve, higher attainers given complex problems which deepen their knowledge of the same content. Misconceptions dealt with immediately.
Fluency comes from deep knowledge and practice. Ability to recall facts and manipulate them to work out other facts is important.
Concrete and pictorial representations help build procedural and conceptual understanding.
Practice and consolidation within different contexts, e.g. time, money, length.
There is a focus on the development of deep structural knowledge and the ability to make connections.

9. Where it all begins…… Counting: Stable order principle
One to one principle
Cardinal principle
Order irrelevance principle
Abstraction principle
Gelman and Gallistel

10. = = The equals sign the same as the same as equivalent equivalent
Not the answer
to a calculation!
Not the answer
to a calculation!
equal
equal
balance
balance

11. What is Place Value? 2 3 4 5 (Ross 1989) Positional Base 10
Multiplicative
Additive
(Ross 1989)

12. Does this represent understanding of place value?
JULIA ANGHILERI1, MEINDERT BEISHUIZEN2 and KEES VAN PUTTEN (2001)
FROM INFORMAL STRATEGIES TO STRUCTURED
PROCEDURES: MIND THE GAP! Educational Studies in Mathematics 49: 149–170, 2002.

13. 1 2 3
1 0
2 0
2 0 3

14. 1000
100 10 1 .
10th
100th

15. Using Place Value Charts1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90
100
200
300
400
500
600
700
800
900
Activity: Give out place value charts and counters Demo activities involving partitioning and combining. 1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90
100
200
300
400
500
600
700
800
900
1 000
2 000
3 000
4 000
5 000
6 000
7 000
8 000
9 000

16. Place Value and Decimals
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9 1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90
100
200
300
400
500
600
700
800
900
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
20000
30000
40000
50000
60000
70000
80000
90000
Each row is 10 times bigger/smaller than the row above/below. Move up and down each row multiplying and dividing by 10 and multiples of 10.

17. Calculators for place value

18. A sledgehammer to crack a nut 3 1 9 97
x 100 00
000
9700 16
- 9 7 1
0 8

19. Well known mental calculation strategies
Partition and recombine
Doubles and near doubles
Use number pairs to 10 and 100
Adding near multiples of ten and adjusting
Using patterns of similar calculations
Using known number facts
Bridging though ten, hundred, tenth
Use relationships between operations
Counting on
x4 by doubling and doubling again
x5 by x10 and halving
x20 by x10 and doubling

20. Resources to support the memorisation of number facts
This resource is based on the slavonic abacus and represents quantity through a more concrete and visual representation, rather than just abstract number symbols
Establishing 10 as a visual image of two groups of 5
These cards can be helpful for children developing mental images of number bonds

21. Resources to support the memorisation of number facts
How many are there?
How many more to make 10?
Seeing number and quantity without counting.

22. Resources to support the memorisation of number facts
Could make reference to work on the Slavonic Abacus by Tandi Clausen-May (MT 168 – “Spatial Arithmetic”
Also look at flip flops as representing an equation
How many are there?
How many more to make 10?
Seeing number and quantity without counting.

23. Tens Frames
Notice how the numbers are arranged in Numicon shapes, drawing out the odd and even properties.
Flip flops!

24. Addition How would you solve these? 25 + 42 25 + 27 25 + 49 127 + 113

25. 25 25

26. 26 26

27.
4 5
+ 7 7
1 2 2 1 27 27

28. Subtraction How would you solve these? 67 - 45 67 - 59 178 - 99

29. . 29

30. 30

31.
1 812
3 5 31

32. Multiplication How would you solve these? 24 x 50 52 x 4 26 x 15

33. Models for multiplication
Arrays lead to the grid method and then the short written method 30 3 90 8 24 10 1
See Primary Magazine Essentials maths to share link – click on ‘calculation’ Issues 23 and 24 cover structures for addition and subtraction 38
x 3
1 1 4 2
What’s different about all these models?
What is the same?

34. Division How would you solve these? 123÷ 3 165÷ 10 325÷ 25 408÷ 17
728÷ 5
623÷ 24

35. Models for division
Using manipulatives to develop a conceptual understanding
of the short written method
Develop a conceptual understanding of the short written method
135 ÷ 3
We can’t make any groups of 3 hundred with the 1 hundred we have. Exchange the 1 hundred for 10 tens.
100 10 1
1 3 5 3 10 1 10 1
See Primary Magazine Essentials maths to share link – click on ‘calculation’ Issues 23 and 24 cover structures for addition and subtraction
What’s different about these models?
What is the same? 1 1

36. Models for division
Using manipulatives to develop a conceptual understanding
of the short written method
We now have 13 tens 10 10 10 1 3 10 10 10 1 10 10 10 1
See Primary Magazine Essentials maths to share link – click on ‘calculation’ Issues 23 and 24 cover structures for addition and subtraction 10 10 1 10 10 1
What’s different about these models?
What is the same?

37. Models for division
Using manipulatives to develop a conceptual understanding
of the short written method 4
We can make 4 groups of three 10s, leaving one 10 10 1 3 1 10 10 10 10 1
See Primary Magazine Essentials maths to share link – click on ‘calculation’ Issues 23 and 24 cover structures for addition and subtraction 10 10 10 10 1 10
What’s different about these models?
What is the same? 10 10 10 1

38. Models for division
Using manipulatives to develop a conceptual understanding
of the short written method
We need to exchange the one 10 for ten 1s 1 1 1 4 1 1 1 3 1 1 1 10 10 10 10 1 1 1
See Primary Magazine Essentials maths to share link – click on ‘calculation’ Issues 23 and 24 cover structures for addition and subtraction 10 10 10 10 1 1 1
What’s different about all these models?
What is the same? 10 10 10 10

39. Models for division
Using manipulatives to develop a conceptual understanding
of the short written method 4 5
We can make 5 groups of three 1s,
giving an answer of 45 3 10
10Z 10 10 1 1 1 1 1
See Primary Magazine Essentials maths to share link – click on ‘calculation’ Issues 23 and 24 cover structures for addition and subtraction 10 10 10 10 1 1 1 1 1 10 10 10 10 1 1 1 1 1
What’s different about all these models?
What is the same?

40. The bar model (Singapore Bar)
This has been extremely successful in helping children to make sense of problems in Singapore and Japan. It is increasingly being used in the UK.
‘It helps me see the story of the problem’
‘I can have a go at any problem now’
David spent 2/5 of his money on a book.
The book cost £10.
How much money did he start off with?
This also illustrates a part whole relationship
Concrete to abstract
In Singapore there would be a progression from concrete objects, in this case footballs, to pictures of objects to spaces on the bar(s) to represents objects
Refer back to the structures of addition explored in residential 1
In this model aggregation can be seen in the two sets of balls coming together and being combined in the one strip
There is also the potential to sow the seeds and link to proportional reason in seeing 3 balls as a proportion of the whole
The importance of making connections in mathematics!
What if the book cost…..
£20?
£6?
£5?
£10

41. Findings from Ofsted
Good Practice in Primary Mathematics, 2011

42. Peter has 4 books. Harry has five times as many books as Peter
Peter has 4 books. Harry has five times as many books as Peter. How many more books has Harry?
Peter’s books
Harry’s books

43. Sam had 5 times as many marbles as Tom
Sam had 5 times as many marbles as Tom. If Sam gives 26 marbles to Tom, the two friends will have exactly the same amount. How many marbles do they have altogether?
A computer game was reduced in a sale by
20% and it now costs £48. What was the original price?
A gardener plants tulip bulbs in a flower bed.
She plants 3 red bulbs for every 4 white bulbs.
She plants 60 red bulbs.
How many white bulbs does she plant?

44. Generalisation
This can then help the children solve, for example, missing
number problems:
45 + ? = 93, ? – 62 = 13, ? = 79, ? + 82 = 147
Generalising the structure of the mathematics to any context

45. KS2 2012
Ask participants to solve this problem and ask how hard Y6 children found this?
The majority of children found this difficult – even those at L5!
Explain that we will explore representations that might help children to more easily access the structure of the mathematics and solve the problem.

46. Led by St Paul’s Catholic College and St Richard’s Catholic College