1. Standards & Learning Effectiveness Service Jon R. Brown Standards & Learning Consultant (0 - 19) Calculations: policy and practice

2. Session aims: Highlight some key elements that support progression through calculations What is an ‘effective’ calculation policy? How does clarity and consistency support AfL and vice versa? Calculations

3. Key elements for progression: 1 A and C

4. Key elements for progression: 1 The language of place value including ‘zero as a place holder’

5. 465 378 700 130 13 + 843 465 378 700 130 3 + 843 465 378 + 1 13 4 1 8 What’s missing from this picture?

6. 465 378 700 130 13 + 843 465 378 700 130 3 + 843 465 378 + 1 13 4 1 8 Non-negotiable htuhtuhtu The language and notation of Place Value Understanding carrying “…thirteen, that’s one ten and 3 units….”

7. The story of decimals: 1000100110 1 / 10 1 / 1000 1 / 100 272

8. 272 £ How do pupils ‘see’ the 7? What are their mental images?

9. Key elements for progression: 2

10. 12 ÷ 3 “How many threes are in twelve”

11. Key elements for progression: 2 Calculation structures Aggregation/Combining Augmentation/Counting on +++++ Dino Maths

12. Calculation structures with Year 1

13. Small step with Year 1 to introduce another key image. Adding or subtracting?

14. Key elements for progression: 3 Practical equipment, manipulatives, models and images.

16. +

17. Division: Grouping on a number line. Do we add or subtract? Adding groups Subtracting groups Model or image relates to number line approach - the underlying structure. Bigger issues…. Compact formal division methods generally rely on subtractive approaches. When will pupil be expected to make the transition between adding and subtracting?

18. Beware!

19. Key elements for progression: 4 Side by side: linking methods and images Key elements for progression: Teacher and (where appropriate) pupil awareness of underlying structures. Use of the language of place value including ‘zero as a place holder’. Use of practical equipment, manipulatives, models and images.

20. 1 5 4 UT 1 6 - 4

21. 86 THU 57 + 86 + 57: Adding the units (least significant figure) first 806 + TUU 5 07 +

22. 86 THU 57 + 806 + TUU 5 07 +

23. 86 THU 57 + 3 1 806 + TUU 5 07 + 13

24. 86 THU 57 + 3 1 806 + TUU 5 07 +

25. 86 THU 57 + 3 1 806 + TUU 5 07 + 130

26. 86 THU 57 + 3 1 806 + TUU 5 07 + 41 130

27. 86 THU 57 + 3 1 806 + TUU 5 07 + 41 130 +

28. 86 THU 57 + 3 1 806 + TUU 5 07 + 41 130 +=

29. 86 THU 57 + 3 1 806 + TUU 5 07 + 41 130 += 143

30. 14 x 6 TU U Grid method side-by-side with a formal vertical column method

31. 14 x 6 TU U 6 10 4 x 14 TU 6 x TUU

32. 14 x 6 TU U 6 10 4 x 14 TU 6 x TUU

33. 14 x 6 TU U 6 10 4 x 14 TU 6 x TUU 24 24

34. 14 x 6 TU U 6 10 4 x 14 TU 6 x 24 24 TUU

35. 14 x 6 TU U 6 10 4 x 14 TU 6 x 24 24 TUU

36. 14 x 6 TU U 6 10 4 x 14 TU 6 x 24 24 TUU 60 60

37. 14 x 6 TU U 6 10 4 x 14 TU 6 x 24 60 2460 TUU

38. 14 x 6 TU U 6 10 4 x 14 TU 6 x 24 60 + 2460 TUU Total

39. 14 x 6 TU U 6 10 4 x 14 TU 6 x 24 60 84 + 246084 TUU Total

40. 23 x 47 TUTU

41. TUTU 20 40 7 3 x 47 THU 23 x TUU

42. 23 x 47 TUTU 20 40 7 3 x 47 THU 23 x TUU

43. 23 x 47 TUTU 20 40 7 3 x 47 THU 23 x TUU 1 2 21

44. 75 3 45 3×10 - 3 0 4 5 3×10 15 3×10 - 3 0 1 5 3×10 0 3×5 - 1 5 0 3×5 The number of 3’s in each chunk/group. The number of 3’s in each chunk/group. 75 ÷ 3 = 25

45. The rapid recall of multiplication and division facts is extremely important and underpins mental and written calculations. Of equal importance is a pupil’s ability to find new facts from their existing bank (or box) of known facts and familiar processes. Key elements for progression: 5 Balancing rapid recall and the use of known facts/processes

46. Instrumental understanding: “Rules without reasons”. ‘Memorising’ a procedure that works for a specific problem and learning a different procedure for each new ‘class’ of problems. Relational understanding: “Knowing what to do and why”. Building up a ‘conceptual structure’ where packages of knowledge are interconnected. Skemp (1976) ‘The fundamental issue for teachers is how better to develop pupils’ mathematical understanding. Too often, pupils are expected to remember methods, rules and facts without grasping the underpinning concepts, making connections with earlier learning and other topics, and making sense of the mathematics so that they can use it independently.’ (UNDERSTANDING THE SCORE: OFSTED Sept 2008, p.5)

47. Briefly summarise (in under a minute) the ‘bigger picture’ when describing progression through calculations. How would your colleagues respond?

48. Key elements for progression: 5 Balancing rapid recall and the use of known facts/processes Counting Stick

49. An informal ‘ad hoc’ multiplication approach Find 19 ‘lots of’ 24 1 ‘lot of 24 is’ 24 2 ‘lot of 24 is’ 48 4 ‘lot of 24 is’ 96 8 ‘lot of 24 is’ 192 16 ‘lot of 24 is’ 384

50. Find 19 ‘lots of’ 24 1 ‘lot of 24 is’ 24 2 ‘lot of 24 is’ 48 4 ‘lot of 24 is’ 96 8 ‘lot of 24 is’ 192 16 ‘lot of 24 is’ 384 An informal ‘ad hoc’ multiplication approach

51. Find 19 ‘lots of’ 24 1 ‘lot of 24 is’ 24 2 ‘lot of 24 is’ 48 4 ‘lot of 24 is’ 96 8 ‘lot of 24 is’ 192 16 ‘lot of 24 is’ 384 I have 16 ‘lots of’ 24. How many more do I need? An informal ‘ad hoc’ multiplication approach

52. Find 19 ‘lots of’ 24 1 ‘lot of 24 is’ 24 2 ‘lot of 24 is’ 48 4 ‘lot of 24 is’ 96 8 ‘lot of 24 is’ 192 16 ‘lot of 24 is’ 384 Final addition: Mental, Horizontal, Vertical, Ad hoc, Formal? I have 18 ‘lots of’ 24. How many more do I need? An informal ‘ad hoc’ multiplication approach

53. Find 15 ‘lots of’ 24 1 ‘lot of 24 is’ 24 2 ‘lot of 24 is’ 48 4 ‘lot of 24 is’ 96 8 ‘lot of 24 is’ 192 16 ‘lot of 24 is’ 384 Progression is shown by the efficiency of the calculation. An informal ‘ad hoc’ multiplication approach

54. Find 15 ‘lots of’ 24 1 ‘lot of 24 is’ 24 2 ‘lot of 24 is’ 48 4 ‘lot of 24 is’ 96 5 ‘lot of 24 is’ 120 10 ‘lot of 24 is’ 240 An informal ‘ad hoc’ multiplication approach Progression is shown by the efficiency of the calculation.

55. Find 102 ‘lots of’ 24 1 ‘lot of 24 is’ 24 10 ‘lot of 24 is’ 240 100 ‘lot of 24 is’ 2400 2 ‘lot of 24 is’ 48 An informal ‘ad hoc’ multiplication approach Progression is shown by the efficiency of the calculation.

56. Aims of the NC: Fluency … through varied and frequent practice with increasingly complex problems …, so that pupils develop conceptual understanding and the ability to recall and apply knowledge rapidly and accurately. Mathematical reasoning: following a line of enquiry, conjecturing, generalising, justifying and proving. Problem solving: Applying their mathematics. Breaking down problems and persevering.

57. Key elements for progression: Teacher and (where appropriate) pupil awareness of underlying structures. Use of the language of place value including ‘zero as a place holder’. Use of practical equipment, manipulatives, models and images. Side by side: linking methods and images Balancing rapid recall and use of known facts and processes Some key changes in Year 3 children are now expected to count in multiples of 4, 8, 50 and 100 they are expected to mentally calculate with three-digit numbers learning the eight-times table has been included tenths are new to Year 3 children now need to add and subtract fractions of the same denominator measuring perimeters of simple shapes was in Year 4 but now it is in Year 3 children are expected to be able to tell 24-hour time - this previously appeared in Year 5 they are also expected to be able to read the time on clocks with Roman numerals children need to be able to identify perpendicular and parallel lines (Dynamic geometry) NCETM

58. Creating an ‘effective’ calculation policy: School Action Plan priority? Subject leader action plan? o Initial audit

59. Creating an ‘effective’ calculation policy: School Action Plan priority? Subject leader action plan? o Initial audit

61. Creating an ‘effective’ calculation policy: School Action Plan priority? Subject leader action plan? o Initial audit including staff and pupil voice

62. Creating an ‘effective’ calculation policy: School Action Plan priority? Subject leader action plan? o Initial audit including staff and pupil voice o Calculation guidance. To be used in conjunction with an existing policy or drawn upon to develop/enhance a policy based on pupil needs and preferred approaches.

63. Creating an ‘effective’ calculation policy: School Action Plan priority? Subject leader action plan? o Initial audit including staff and pupil voice o Calculation guidance. To be used in conjunction with an existing policy or drawn upon to develop/enhance a policy based on pupil needs and preferred approaches. o Calculation policy. Part 1: WhatPart 2: How Calculation GuidanceExpectations

68. Creating an ‘effective’ calculation policy: School Action Plan priority? Subject leader action plan? o Initial audit including staff and pupil voice o Calculation guidance. To be used in conjunction with an existing policy or drawn upon to develop/enhance a policy based on pupil needs and preferred approaches. o Calculation policy linked in to long and medium term plans.

69. Creating an ‘effective’ calculation policy: School Action Plan priority? Subject leader action plan? o Initial audit including staff and pupil voice o Calculation guidance. To be used in conjunction with an existing policy or drawn upon to develop/enhance a policy based on pupil needs and preferred approaches. o Calculation policy linked in to long and medium term plans. o Future proof – Informed flexible updates (Age or stage??)

71. Yr 4 should focus on additive approaches to grouping on the number line. Suggested activities: Dino division Grouping carousel Side-by-side with number rods Etc.

72. Creating an ‘effective’ calculation policy: School Action Plan priority? Subject leader action plan? o Initial audit including staff and pupil voice o Calculation guidance. To be used in conjunction with an existing policy or drawn upon to develop/enhance a policy based on pupil needs and preferred approaches. o Calculation policy linked in to long and medium term plans. o Future proof – Informed flexible updates (Age or stage??) SLT and Maths SL working together strategically. Regular joint meetings. Review, track impact and pupil progress. Planning updates, interventions and QFT approaches.

73. Supporting AfL: Success criteria

74. Formative assessment and feedback.

76. Aims of the NC: …….. and persevering.

77. Session aims: Highlight some key elements that support progression through calculations What is an ‘effective’ calculation policy? How does clarity and consistency support AfL and vice versa? Calculations Aims of the NC: Fluency … through varied and frequent practice with increasingly complex problems …, so that pupils develop conceptual understanding and the ability to recall and apply knowledge rapidly and accurately.

78. Stepping off the level ladder and having a look around.