This is part of the professional development for the team that translate My Pals Are Here into Dutch and also people who are going to provide professionald evelopment for teachers using Singapore textbooks in the future.

2. Theories – Bruner & Dienes

3. Teaching for UnderstandingWeten en begrijpen hoe de theorieen van o.aJ.Bruner, R.Skemp, Z.Dieneszijnuitgewerkt in het ontwerp van eeneffectiefrekenprogrammadatalleleerlingen in staatsteltteleren en bovengemiddleldeleerlingen de kansbiedtteexcelleren.

4. Features of Singapore Mathematics Curriculum Yeap Ban Har, Ph.D. Marshall Cavendish Institute Singapore banhar@sg.marshallcavendish.com

5. Introduction This is a brief overview of the development of the Singapore mathematics curriculum, its framework and its rationale and underlying goals.

6. Beliefs Interest Appreciation Confidence Perseverance Monitoring of one’s own thinking Self-regulation of learning Attitudes Metacognition Numerical calculation Algebraic manipulation Spatial visualization Data analysis Measurement Use of mathematical tools Estimation Mathematical Problem Solving Reasoning, communication & connections Thinking skills & heuristics Application & modelling Skills Processes Concepts Numerical Algebraic Geometrical Statistical Probabilistic Analytical Mathematics Curriculum Framework

7. Every Child Counts Huamin Primary School

8. TIMSS 1995 – 2007 Trends in International Mathematics and Science Studies 1995 2003 2007 International Grade 4 5 Advanced 38 41 38 26 High 70 74 73 67 Intermediate 89 92 91 90 Low 96 98 97 North Vista Primary School

9. TIMSS 2007 Trends in International Mathematics and Science Studies Indonesia Thailand Malaysia Singapore International Grade 8 Advanced 2 3 0 40 2 High 15 12 4 70 18 Intermediate 46 44 14 88 50 Low 75 66 48 97 82 Method Used in Singapore Textbooks

10. Why Schools Teach Mathematics We will engage in activities that are used to help teachers understand the rationale for teaching mathematics in school.

11. Problem Arrange the ten cards so that you can do what is shown to you.

12. Method 1 – by drawing

13. Method 2 – by using the cards

14. Scarsdale Teachers’ Institute, New York

15. Some participants tried to solve the Dutch-version of the problem.

16. Problem Rearrange the sticks to show a given number of squares. Wellington Primary School

17. Task Lesson Study Problem Wellington Primary School Move 3 sticks to make 3 squares.

18. Task Move 3 sticks to make 3 squares.

19. Task Move 3 sticks to make 3 squares.

20. Task Move 3 sticks to make 2 squares.

21. Task Move 3 sticks to make 2 squares.

22. Task Move 3 sticks to make 2 squares.

23. Mathematics is “an excellent vehicle for the development and improvement of a person’s intellectual competence”. Ministry of Education (Singapore) 2006

24. Focus on Problem Solving The Singapore curriculum focuses on problem solving. So does the national test. It is no wonder that’s schools place a lot of emphasis on problem solving.

25. National Test

26. Textbooks

27. Textbooks

28. Textbooks

29. Textbooks

30. The CPA Approach & Principle of Variations Yeap Ban Har, Ph.D. Marshall Cavendish Institute Singapore banhar@sg.marshallcavendish.com

31. Number Bonds The focus on number sense right from the start. Number bonds is taught before addition.

32. Number Bonds Number Bonds is emphasized prior to the learning of addition. Children are given, say, 5 unifix cubes and guided to see that 1 and 4 make 5, for example. Others may say that 3 and 2 make 5 or 4 and 1 make 5. Yet others may say that 5 and 0 make 5. Earlybird Kindergarten Mathematics

33. PCF TelokBlangah Kindergarten

34. One duck is big. Six ducklings are small. PCF TelokBlangah Kindergarten

35. PCF TelokBlangah Kindergarten

36. PCF TelokBlangah Kindergarten

37. PCF TelokBlangah Kindergarten

38. PCF TelokBlangah Kindergarten

39. My Pals Are Here! Mathematics Number Bonds continues to receive attention in Grade 1.

40. Addition Facts Addition Facts are given emphasis in the first six months of grade one. The children learn it in stages as the textbooks distinguished between Numbers to 10 and Numbers to 20. Count On and Count All are used in Numbers to 10. Earlybird Kindergarten Mathematics

41. Addition Facts &Number Sense While Count On and Count All are used in Numbers to 10, Make Ten is given emphasis in Numbers to 20. Princess Elizabeth Primary School The topics are arranged in a way that makes learning progressive and systematic – this is part of the idea of a spiral curriculum.

42. Jerome Bruner Ideas are introduced using concrete materials. Pictorial representations follow. Finally, children engage with the abstract symbols. This is called the CPA Approach in Singapore.

43. mathz4kidz Learning Centre, Penang, Malaysia bruner’s theory concrete A lesson from Earlybird Kindergarten Mathematics

44. mathz4kidz Learning Centre, Penang, Malaysia concrete experiences

45. mathz4kidz Learning Centre, Penang, Malaysia from concrete to pictorial

46. from pictorial to abstract All Kids Are Intelligent Series

47. mathz4kidz Learning Centre, Penang, Malaysia symbols

48. Variations Tasks are varied in a systematic way to ensure that average and struggling learners can learn well.

49. Math in Focus 2A

50. Math in Focus 2A

51. Math in Focus 2A

52. ZoltanDienes The three lessons include mathematical variations within the same grade. This is referred to as a spiral approach.

53. It is likely that a teacher will start this unit using the sticks. This is followed by the use of base ten blocks. Finally, non-proportionate materials such as coins are used. In each of these lessons, the teacher is likely to introduce the following five notations in turn – place value chart, expanded notation, number in numerals, number in words and the tens and ones notation. The question is what is an appropriate sequence? Should the place value chart be used first? Or the expanded notation? Give your reasons. Place Value Chart Expanded Notation Words Numerals Tens and Ones Notation Primary Mathematics

54. ZoltanDienes The one lesson include perceptual variations. This is Dienes’ idea of multiple embodiment. The mathematical concept is constant while the materials used to embody it are varied.

55. Teaching for Understanding Ideas from Richard Skemp Yeap Ban Har, Ph.D. Marshall Cavendish Institute Singapore banhar@sg.marshallcavendish.com

56. Example 1 Grade 1 Division

58. The concrete  pictorial  abstract approach is used to help the majority of learners to develop strong foundation in mathematics. Grade 1 Division National Institute of Education

60. Curriculum Document Division of a quantity (not greater than 20) into equal sets Given the number of objects in the set Given the number of sets Solving 1-step word problems with pictorial representation

61. Princess Elizabeth Primary School Division

62. Links between concrete and pictorial representation must be carefully constructed.

63. skemp’s theory conceptual understanding BinaBangsa School, Semarang, Indonesia

64. Example 2 Division in Other Grade Levels

65. My Pals Are Here! Mathematics 3A

66. My Pals Are Here! Mathematics 3A

67. My Pals Are Here! Mathematics 3A

68. My Pals Are Here! Mathematics 3A

69. My Pals Are Here! Mathematics 3A

70. My Pals Are Here! Mathematics 3A

71. My Pals Are Here! Mathematics 3A

72. My Pals Are Here! Mathematics 3A

73. My Pals Are Here! Mathematics 3A

75. Keys Grade School, Manila, The Philippines

76. Keys Grade School, Manila, The Philippines

77. Example 3 Division Involving Fractions

78. Skemp Understanding in mathematics relational (conceptual) instrumental (procedural) conventional Teaching for conceptual understanding is given emphasis in Singapore Math. Primary Mathematics Standards Edition Grade 6

79. Primary Mathematics Standards Edition Grade 6