This presentation is on eight features of early grades mathematics in Singapore.

1. NCTM Regional Conference & Exposition Denver Colorado 8 – 9 October 2010 Session 165 This session focuses on eight elements in early grades mathematics that help children develop a strong foundation for a demanding mathematics program in the upper grades. Concrete, visual approaches, coupled with special attention to economically disadvantaged children, systematic variation in representations and content help students learn well.  Colorado Convention Center, 506 – 507 Denver, Colorado

2. Eight Things About Early Grades Mathematics in Singapore Session 165 DrYeap Ban Har Marshall Cavendish Institute Singapore banhar@sg.marshallcavendish.com slides are available at www.banhar.blogspot.com

3. 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 The mathematical problem-solving curriculum framework was introduced in Singapore in 1992 in response to recommendations made in An Agenda for Action (in the US) and the Cockcroft Report (in the UK).

4. NCTM Regional Conference & Exposition Denver Colorado 8 – 9 October 2010 Session 165 This session focuses on eight elements in early grades mathematics that help children develop a strong foundation for a demanding mathematics program in the upper grades. Concrete, visual approaches, coupled with special attention to economically disadvantaged children, systematic variation in representations and content help students learn well.  Colorado Convention Center, 506 – 507 Denver, Colorado

5. Students are expected to solve complex and, often, new problems in the program. The formal assessment such as school examination and national examination contains significant emphasis on problem solving. demanding program

6. A B F E Source: Singapore Examinations and Assessment Board PSLE Mathematics 2004 – 2008 and Specimen Paper 2009 D C

7. Grade 3 Examination Item (Raffles Girls’ Primary School 2009) Challenging Application Devi had some $2 notes and $5 notes. She had a total of $76. She had 4 more $5 notes than $2 notes. How many $2 notes did she have? Source Raffles Girls’ School, Singapore

8. Devi had some $2 notes and $5 notes. She had a total of $76. She had 4 more $5 notes than $2 notes. How many $2 notes did she have? $76 Number of $5 notes 4 $20 $56 8 Number of $2 notes $56 ÷ 7 = 8 8 She had eight $2 notes.

9. Grade 6 Examination Item (PSLE 2009) Challenging Application In the diagram below, ABCD is a square and QM = QP = QN. MN is parallel to AB and it is perpendicular to PQ. Find MPN Answer: 150o Source Singapore Examinations and Assessment Board PSLE Mathematics 2009

10. NCTM Regional Conference & Exposition Denver Colorado 8 – 9 October 2010 Session 165 This session focuses on eight elements in early grades mathematics that help children develop a strong foundation for a demanding mathematics program in the upper grades. Concrete, visual approaches, coupled with special attention to economically disadvantaged children, systematic variation in representations and content help students learn well.  Colorado Convention Center, 506 – 507 Denver, Colorado

11. learning well Students who have learnt well in the early grades should do well in the upper grades. Also, the should find mathematics sufficiently challenging and within their grasp.

12. The majority of students in Singapore opt to study mathematics in Grades 11 and 12 when they are no longer required to. This is confirmed by the TIMSS data on attitude towards mathematics. The attitude index for Singapore students in TIMSS is also relatively high compared to other high-performing countries.

13. Attitude Achievement The figures refer to percentage of students with high PATM (for Attitude) and in the Advanced International Benchmark (for Achievement). Taiwan 37 45 Grade 8 S Korea 33 40 Singapore 60 40 Hong Kong 47 31 Japan 30 26 Hungary 30 10 England 40 8 International 54 2 Source TIMSS 2007 International Mathematics Report, Boston College

14. one: sustained focus on thinking One of the three stated rationales for teaching and learning mathematics is because mathematics is “an excellent vehicle for the development and improvement of a person’s intellectual competence” (Ministry of Education 2006)

15. 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 The problem-solving framework has been in place since 1992. In 1997, Thinking Schools, Learning Nation philosophy was introduced to the entire education system. This is supported by the Teach Less, Learn More initiative in 2004.

16. For example, there is an emphasis on mental strategies. Mental strategies help students become flexible thinkers as they need to make decisions on the best strategy to use for a particular computation. In an earlier presentation (Session 106), I have discussed visualization, number sense, generalization and metacognition. Source Pensar sin Limites 2oBasico

17. Dagoberto Godoy School, Santago Princess Elizabeth Primary School, Singapore Keys Grade School, Manila Springhurst Elementary School, New York

18. There are some apples in a box. There are 5 different ways to put the apples into equal groups. Hint: The number is between 15 and 21. Keys Grade School, Manila

19. sustained focus on thinking Dagoberto Godoy School, Santiago, Chile

20. two: small number of topics This allows in-depth treatment of a few key ideas at the early grade levels. One core idea is number bonds.

21. 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. Source Earlybird Kindergarten Mathematics Book B

22. PCF Kindergarten TelokBlangah, Singapore

23. One duck is big. Six ducklings are small. PCF Kindergarten TelokBlangah, Singapore

24. PCF Kindergarten TelokBlangah, Singapore

25. PCF Kindergarten TelokBlangah, Singapore

26. PCF Kindergarten TelokBlangah, Singapore

27. PCF Kindergarten TelokBlangah, Singapore

28. Source My Pals Are Here! Mathematics Primary 1

29. sustained focus on thinking focus on fewer topics Princess Elizabeth Primary School, Singapore

30. Circle Graphs are taught in Grade 6 in Singapore. Source TIMSS 2007 International Mathematics Report, Boston College

31. Based on Jerome Bruner’s ideas of enactive, iconic and symbolic representations, Singapore curriculum, textbooks and teacher education emphasized the importance of concrete materials and the progression to visual and symbolic representations. three: CPA approach Source Earlybird Kindergarten Mathematics – Kindergarten Textbook in Singapore

32. Is the rectangle cut into equal parts? illustration equal parts Source Professional Development Course for Singapore Teachers

35. bruner’s theory concrete Source mathz4kidz Learning Centre, Penang, Malaysia A Lesson from Earlybird Kindergarten Mathematics

36. concrete experiences Source mathz4kidz Learning Centre, Penang, Malaysia A Lesson from Earlybird Kindergarten Mathematics

37. The CPA Approach – the progression from concrete to pictures to abstract symbols is recommended for concept development. This is based on the work of Jerome Bruner. Source Fuchun Primary School, Singapore Research Lesson to Develop Formula for Circumference of Circles

38. use of visuals Source Math in Focus Grade 4

39. Source Math in Focus Grade 4

40. Visuals are used extensively to develop conceptual understanding. In learning how to write mixed numbers as improper fractions in Grade 4, the visual approach is a common approach in the textbooks.

41. This visual shows why It is clear that there are 5 halves in

42. The use of visuals in learning multiplication facts. Source Primary Mathematics Grade 3

43. Source Pensar sin Limites 2oBasico

44. sustained focus on thinking focus on fewer topics the CPA approach Da Qiao Primary School, Singapore

45. four: spiral curriculum Students have many opportunities to learn a core idea. For example the concept of division takes five years to develop – in a systematic manner, with the goal of advancing students’ notion of the concept.

46. Source My Pals Are Here! Mathematics Primary 1

47. Princess Elizabeth Primary School, Singapore Source Pensar sin Limites 1oBasico

48. For example, students learn to do division of discrete quantities without the need to write division sentence in Grade 1. In Grade 2, the revisit this idea and use division sentences to represent the situations. In Grade 3, the idea is extended to include the idea of a remainder. They also learn to regroup before dividing for 2-digit and 3-digit numbers. The quantities used are limited to discrete quantities. Princess Elizabeth Primary School, Singapore My Pals Are Here! Mathematics Primary 1

49. In Grade 4, 4-digit numbers are used. In Grade 5, division of continuous quantities are dealt with where 13 ÷ 4 = 3.25 rather than 3 remainder 1. Keys Grade School, Manila

50. Up till 1992, there was no college graduates teaching in primary schools in Singapore. Today, half are college graduates. In pre-primary levels, many teachers hold only a Grade 10 certificate (with a teaching certificate), although increasingly more teachers completing the bachelor and graduate degrees. Compulsory education is Grade 1 to Grade 6. five: non-specialist teachers

51. textbooks

52. Source My Pals Are Here! Mathematics Primary 1

53. Source Pensar sin Limites 1oBasico

54. pre-service teacher education

55. Teachers received about 100 hours of courses in mathematics teaching and learning during pre-service training. National Institute of Education, Singapore

56. in-service professional development

57. Teachers are expected to engage in professional development throughout their career. Princess Elizabeth Primary School, Singapore Marshall Cavendish Institute, Singapore

58. Some study for certification while others just for improvement. It is also an expectation. SEED Institute, Singapore

59. Tasks are presented in a systematic sequence to provide students with both mathematical as well as perceptual variability. six: systematic variation of tasks

60. Source Math in Focus Grade 2

61. Source Math in Focus Grade 2

62. Source Math in Focus Grade 2

63. Systematic Variation – students are presented with a variety of tasks in a systematic way. This is based on the idea of ZoltanDienes.

64. Source Primary Mathematics Grade 1 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 Source Professional Development Materials Marshall Cavendish Institute, Singapore Words Numerals Tens and Ones Notation

65. 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

66. non-specialist teachers sustained focus on thinking focus on fewer topics the CPA approach spiral curriculum Da Qiao Primary School, Singapore systematic variation

67. Through education, education level and consequently, social-economic status of families increase over one generation. Young parents have high aspirations for their children. Most send their young children to pre-schools and there number of school-age teenagers who are not in school is negligible although there is no compulsory education beyond Grades 1 – 6. seven: high aspirations among parents

69. TelokKurau Primary School, Singapore “Sometimes my teacher gives me remedial classes. Some people go for Maths Olympiad training. My maths is not good. I don’t go for that.” Fifth Grader school support “In my school, we have a fixed day for remedial lessons. I get a group of students to attend this regularly.” Teacher

70. community support TelokKurau Primary School, Singapore

71. In grades one and two, students who did not attend pre-school, received a pull-out learning support for English and mathematics (LSP and LSM). They are taught in smaller groups of not more than ten while the regular class size is 30. Formal school hours end around 1:30 to free up time for a range of informal co-curricular and curricular activities. eight: learning support and remedial

72. safety nets and interventions non-specialist teachers Slides are available at www.banhar.blogspot.com sustained focus on thinking focus on fewer topics the CPA approach spiral curriculum systematic variation high parental expectations

73. Eight Things about Early Grades Mathematics in Singapore Slides are available at www.banhar.blogspot.com Da Qiao Primary School, Singapore