The document discusses the theoretical underpinnings of the Singapore Math approach. It describes three key theories that influenced the development of Singapore Math: 1. Bruner's theory of the concrete-pictorial-abstract approach to help students develop a strong mathematical foundation. 2. Skemp's theory of conceptual understanding versus procedural understanding, with Singapore Math emphasizing the former. 3. Dienes' theory of variation in mathematics education through perceptual and mathematical variability in tasks. The document provides examples of how these theories are applied in Singapore Math teaching materials, lessons, and instructional models.

1. Theoretical Underpinnings of Singapore Math Sheraton San Diego Mission Valley Hotel, San Diego CA SingaporeMath.com Professional Development Please download from www.mathz4kidz.com Yeap Ban-Har, Ph.D. National Institute of Education Nanyang Technological University Singapore banhar.yeap@nie.edu.sg DaQiao Primary School

2. Catholic High School (Primary)

3. video of dice problem solving problems instructional models overview Bruner’s theory Skemp’s theory Dienes’ theory

4. introduction Wellington Primary School

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

6. Task Move 3 sticks to make 3 squares.

7. Task Move 3 sticks to make 3 squares.

8. Task Move 3 sticks to make 2 squares.

9. Task Move 3 sticks to make 2 squares.

10. Task Move 3 sticks to make 2 squares.

11. A Problem from Singapore Grade 6 National Test Jim bought some chocolates and gave half of them to Ken. Ken bought some sweets and gave half of them to Jim. Jim ate 12 sweets and Ken ate 18 chocolates. After that, the number of sweets and chocolates Jim had were in the ratio 1 : 7 and the number of sweets and chocolates Ken had were in the ratio 1 : 4. How many sweets did Ken buy?

12. Jim bought some chocolates and gave half of them to Ken. Ken bought some sweets and gave half of them to Jim. Jim ate 12 sweets and Ken ate 18 chocolates. After that, the number of sweets and chocolates Jim had were in the ratio 1 : 7 and the number of sweets and chocolates Ken had were in the ratio 1 : 4. How many sweets did Ken buy? chocolates sweets Assuming that both boys did not have any sweet or chocolate before they bought the chocolates and sweets. 12 Jim 12 18 12 12 12 12 Ken 3 parts  12 + 12 + 12 + 12 + 18 = 66 1 part  22 Half of the sweets Ken bought = 22 + 12 = 34 So Ken bought 68 sweets.`

13. 88 children took part in a swimming competition. 1/3 of the boys and 3/7 of the girls wore swimming goggles. Altogether 34 children wore swimming goggles. How many girls wore swimming goggles on that day? A Problem from a Singapore Classroom Fairfield Methodist Primary School

16. 34 – 20 = 14 54 – 34 = 20 34 54

17. 3 x 7 = 21 21 girls wear goggles. A Curriculum That Helps Average Students Reach High Achievement

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

19. TIMSS 2007 Trends in International Mathematics and Science Studies Average Indonesia Thailand Malaysia Singapore 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

20. 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

21. Every Child Counts

22. effective mathematics teaching BinaBangsa School, Indonesia

23. Primary Mathematics 1A Pedagogical Principle: Bruner

24. Number Bonds PCF Kindergarten TelokBlangah

25. Number Bonds PCF Kindergarten TelokBlangah

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

27. Division Princess Elizabeth Primary School

28. Catholic High School (Primary)

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

30. mathz4kidz Learning Centre, Penang, Malaysia concrete experiences

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

32. from pictorial to abstract All Kids Are Intelligent Series

33. mathz4kidz Learning Centre, Penang, Malaysia symbols

34. using concrete materials Professional Development in AteneoGrade School, Manila, The Philippines Lesson Study in a Ministry of Education Seminar on Singapore Mathematics Teaching Methods in Chile

35. Primary Mathematics (Standards Edition) 2A Pictorial Before Abstract

36. bruner Lesson Study in a Ministry of Education Seminar on Singapore Mathematics Teaching Methods in Chile

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

38. Keys Grade School, Manila, The Philippines

39. Keys Grade School, Manila, The Philippines

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

41. Fraction Division Primary Mathematics Standards Edition Grade 6

42. skemp Scarsdale Middle School New York

43. Dienes Dienes encouraged the use of variation in mathematics education – perceptual variability and mathematical variability. Pedagogical Principle: Dienes Primary Mathematics Standards Edition Primary Mathematics Standards Edition Grade 1

44. Pedagogical Principle: Dienes Primary Mathematics Standards Edition Grade 1

45. Pedagogical Principle: Dienes Primary Mathematics Standards Edition Grade 2

46. homework Are you able to see how these tasks are varied according to Dienes’ idea of mathematical variability?

47. How is Task 4 different from Task 5? 16 Primary Mathematics Standards Edition Grade 5

48. 16 What is the given in Task 5? What is the given in Task 6? Are these different? 30 Primary Mathematics Standards Edition Grade 5

49. Earlybird Kindergarten Mathematics Standards Edition Can you see how Dienes’ idea is used in designing these tasks? diene’s theory of variation

50. dienes Princess Elizabeth Primary School, Singapore

51. Emphasis on pictorial representation and systematic variation to enhance conceptual understanding

52. conclusion PCF Kindergarten PasirRis

54. Modeling

55. Providing

56. ExplaningDaQiao Primary School

57. “Children are trulythe future of our nation. “ Irving Harris

58. This presentation is based on part of one of Singapore pre-service mathematics method courses. 50% of Singapore elementary teachers are not college graduate and they are not trained to be specialists. The TEDS-M findings provide some evidence into the effectiveness of this form of professional development.

59. TEDS-M Elementary Teachers Content Knowledge TEDS-M Elementary Teachers Pedagogical Content Knowledge

60. TEDS-M Middle School Teachers Content Knowledge TEDS-M Middle School Teachers Pedagogical Content Knowledge