1. Primary PD Lead Support Programme Introduction to: The focus and demands of the new curriculum

2. The Aims of The New Curriculum The National Curriculum for mathematics aims to ensure all pupils:  become fluent in the fundamentals of mathematics so that they are efficient in using and selecting the appropriate written algorithms and mental methods, underpinned by mathematical concepts  can solve problems by applying their mathematics to a variety of problems with increasing sophistication, including in unfamiliar contexts and to model real-life scenarios  can reason mathematically by following a line of enquiry and develop and present a justification, argument or proof using mathematical language. 2

3. The focus and demands of the new curriculum The new National Curriculum is likely to have An increased focus on arithmetic proficiency A greater emphasis on learning number facts Higher expectations in fractions, decimals and percentages Less emphasis on data handling 3

4. Mathematical Proficiency Mathematical proficiency requires a focus on core knowledge and procedural fluency so that pupils can carry out mathematical procedures flexibly, accurately, consistently, efficiently, and appropriately. Procedures and understanding are developed in tandem. 4

5. Arithmetic Proficiency: achieving fluency in calculating with understanding... an appreciation of number and number operations, which enables mental calculations and written procedures to be performed efficiently, fluently and accurately. 5

6. BALANCE 6 Procedural Fluency Conceptual Understanding INTEGRATION

7. Is this image preferable? Procedural Fluency Conceptual Understanding

8. Developing conceptual understanding The teacher’s job is to organise and provide the sorts of experiences which enable pupils to construct and develop their own understanding of mathematics, rather than simply communicate the ways in which they themselves understand the subject. (1989 National Curriculum, Non- Statutory Guidance, page C2) Is this how teachers see their role?

9. Ofsted: Made to Measure “Schools should: tackle in-school inconsistency of teaching increase the emphasis on problem solving across the mathematics curriculum develop the expertise of staff: in choosing teaching approaches and activities that foster pupils’ deeper understanding, in checking and probing pupils’ understanding during the lesson, in understanding the progression in strands of mathematics over time, so that they know the key knowledge and skills that underpin each stage of learning ensuring policies and guidance are backed up by professional development for staff “ (Source Ofsted 2012 - Mathematics: made to measure) 9

10. Flexibility ●25 + 42 ●25 + 27 ●25 + 49 ●1462 + 2567 ●24  50 ●24  4 ●24  17 ●1424 x 17 ●123  3 ●325  25 ●408  17 ●67 - 45 ●67 - 59 ●3241 - 2167 ●672 - 364 What aspects of conceptual understanding and procedural fluency are required for each of these calculations?

11. Improving children’s Arithmetic Proficiency Findings from Ofsted 2011: Practical, hands-on experiences of using, comparing and calculating with numbers and quantities … are of crucial importance in establishing the best mathematical start … Understanding of place value, fluency in mental methods, and good recall of number facts … are considered by the schools to be essential precursors for learning traditional vertical algorithms (methods) Subtraction is generally introduced alongside its inverse operation, addition, and division alongside its inverse, multiplication High-quality teaching secures pupils’ understanding of structure and relationships in number … 11

12. Other findings: Skills in calculation are strengthened through solving a wide range of problems. Schools recognised the importance of moving towards more efficient methods Schools in the good practice survey had clear, coherent calculation policies Schools recognise the importance of good subject knowledge and subject-specific teaching skills and seek to enhance these aspects of subject expertise. 12 Improving children’s Arithmetic Proficiency (Source Ofsted 2011 - Good Practice in Primary Mathematics)Source Ofsted 2011 - Good Practice in Primary Mathematics

13. Primary PD Lead Support Programme What can we learn from other countries?

14. Problem solving, reasoning and procedural fluency’ “...there is a wider consensus amongst mathematics educators that conceptual understanding, procedural and factual fluency and the ability to apply knowledge to solve problems are all important and mutually reinforce each other.” 14 DfE RR178, 2011

15. 15 Singapore mathematics framework

16. Procedural fluency and conceptual understanding Singapore Maths does not rely on rote learning of facts and procedures without the underlying understanding required to use them effectively. The Ministry of Education (2006) explicitly states: “Although students should become competent in the various mathematical skills, over- emphasising procedural skills without understanding the underlying mathematical principles should be avoided”; Improving pupils’ attitudes is likely to raise achievement – links between success, confidence, enjoyment; High expectations; Concrete, pictorial, abstract (CPA); Visual problem solving strategies; Focus on number and calculation from the beginning. 16

17. TIMSS 17 HIGHER performance compared with England Participants performing at a significantly higher level than in England SIMILAR performance compared with England Participants performing at a similar level to England (not statistically different) LOWER performance compared with England Participants performing at a significantly lower level than England 6 countries [and 1 benchmarking participant], with their scale scores 6 other countries [and 1 benchmarking participant], with their scale scores 37 countries [and 5 benchmarking participants] including… With their scale scores Singapore Korea Hong Kong Chinese Taipei Japan Northern Ireland [North Carolina, US] 606 605 602 591 585 562 [554] Belgium (Flemish) Finland [Florida, US] ENGLAND Russian Federation United States Netherlands Denmark 549 545 [545] 542 541 540 537 [Quebec, Canada] Portugal Germany Ireland, Rep of [Ontario, Canada] Australia Austria Italy [Alberta, Canada] Sweden Kazakhstan Norway New Zealand Spain 533 532 528 527 [518] 516 508 [507] 504 501 495 486 482 Table 1.1 TIMSS 2011 performance groups: mathematics at ages 9-10 Source: Exhibit 1.3 international mathematics report

18. Primary PD Lead Support Programme Towards written algorithms: Addition and subtraction

19. “Dependence on facts and procedures alone cannot ensure competence with arithmetic at the required level; competence also depends on identifying, understanding and acting on the underlying relations, equating and estimating quantities.” ACME Report on Primary Arithmetic, Dec. 2010

20. Now what was the next step? 3 5 + 4 7 9 1 2 4 5 - 3 7 1 2 3 5  3 915 248  25 1240 496 Pupil forgot to ‘put a nought’ here because there already was one here. 1 8 216 2

21. The need for algorithms “Civilization advances by extending the number of important operations we can perform without thinking.” Alfred North Whitehead

22. A sledgehammer to crack a nut 1 0 0 0 - 7 9 9 3 10 1 1 9 9 16 - 9 7 0 1 97 x 100 00 000 9700 7 5 6 5 0 8 0

23. See NCETM Mathemapedia – for example Counting and calculating in the Early YearsCounting and calculating in the Early Years 23 Addition

24. 57 Models for addition Combining two sets of objects (aggregation) 24 12 Context: Combined price of items Adding on to a set (augmentation) 12 Context: Increase in price Increase in temperature Age in x years

25. More than single digits? 25 25 + 47

26. 26

27. 27

28. Compacted leading to 28

29. Compacted 29 5 7 2 4 leading to

30. 30 Subtraction

31. 12 Models for subtraction Removing items from a set (reduction or take-away) - 1- 2- 3- 4- 5= 7 Issue: Relies on ‘counting all’, again. Comparing two sets (comparison or difference) Issue: Useful when two numbers are ‘close together’, where ‘take-away’ image can be cumbersome Seeing one set as partitioned Seeing 12 as made up of 5 and 7 Issue: Helps to see the related calculations; 5+7=12, 7+5=12, 12-7 = 5 and 12-5=7 as all in the same diagram N.B. When this is done on a bead bar, there are links with both counting back and difference on a number line 31

32. Models for subtraction Counting back on a number line Finding the difference on a number line 12 0 - 5 7 Issue: Number line helps to stop ‘counting all’. Also: Knowledge of place value and number bonds can support more efficient calculating 12 0 5 7 Issue: Useful when two numbers are ‘close together’, use of number bonds and place value can help. 52 10 -2-3 32

33. 72 - 47 More than single digits? 33 This is now “Sixty- twelve” 7 1 2 6

34. Compacted 34 2 7 7 4 7 2 47 leading to

35. Task Explore some subtraction calculations using the different manipulatives. How well do the manipulatives help you to solve the calculation problems? How well do the manipulatives help to move pupils towards written methods? 35

36. Primary PD Lead Support Programme Towards written algorithms: Multiplication and division

37. 37 Multiplication

38. Models for multiplication Bead Bar Number Line Fingers “6”“6”“9”“9” “12” “3”“3” 0 3 6 9 12 38 Lots of the ‘same thing’

39. Models for multiplication 39 Scaling 3 times as tall

40. Models for multiplication 40 3  44  3

41. An image for 7  8 = 56 41

42. An image for 7  8 = 56 42

43. More than single digits? 18 x13 18 13 1810 8 13 3 10 10080 2430 43

44. Progressing towards the standard algorithm 1 0 8 3 1 0 08 0 3 02 4 44

45. 108 3 10080 30 24 18  13 54 180 234 ? When? Whether? How?

46. Division

47. An image for 56  7 Either: How many 7s can I see? (grouping) Or: If I put these into 7 groups how many in each group? (sharing) The array is an image for division too 5 67 8

48. An image for 56  7 How many 7s can I see? 5 67 8

49. How many 7s can I see? 5 67 8

50. How many 7s can I see? 5 6 7 8

51. How many 7s can I see? 5 67 8

52. 120  3 52 40 1203 The power of the place value: counters for larger numbers

53. 1200  3.. and for 100s 53 400 12003

54. 54 Now try one yourself 138 divided by 6 Use the counters

55. 2 0 1 3 8 HundredsTensOnes 6 1 0 55 3 1 138 6 23

56. Task Explore some division calculations using the different manipulatives. How well do the manipulatives help you to solve the calculation problems? How well do the manipulatives help to move pupils towards written methods? Reflect on your own practice about how a written method for division can be taught. 56