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- 1. NCTM Regional Conference & Exposition Denver Colorado<br />8 – 9 October 2010<br />Session 165<br />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. <br />Colorado Convention Center, 506 – 507 <br />Denver, Colorado<br />
- 2. Eight Things About Early Grades Mathematics in Singapore<br />Session 165<br />DrYeap Ban Har<br />Marshall Cavendish Institute<br />Singapore<br />banhar@sg.marshallcavendish.com<br />slides are available at<br />www.banhar.blogspot.com<br />
- 3. Beliefs<br />Interest<br />Appreciation<br />Confidence<br />Perseverance<br />Monitoring of one’s own thinking<br />Self-regulation of learning<br />Attitudes<br />Metacognition<br />Numerical calculation<br />Algebraic manipulation<br />Spatial visualization<br />Data analysis<br />Measurement<br />Use of mathematical tools<br />Estimation<br />Mathematical Problem Solving<br />Reasoning, communication & connections<br />Thinking skills & heuristics<br />Application & modelling<br />Skills<br />Processes<br />Concepts<br />Numerical<br />Algebraic<br />Geometrical<br />Statistical<br />Probabilistic<br />Analytical<br />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).<br />
- 4. NCTM Regional Conference & Exposition Denver Colorado<br />8 – 9 October 2010<br />Session 165<br />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. <br />Colorado Convention Center, 506 – 507 <br />Denver, Colorado<br />
- 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.<br />demanding program<br />
- 6. A<br />B<br />F<br />E<br />Source:<br />Singapore Examinations and Assessment Board PSLE Mathematics 2004 – 2008 and Specimen Paper 2009<br />D<br />C<br />
- 7. Grade 3 Examination Item (Raffles Girls’ Primary School 2009)<br />Challenging Application<br />Devi had some $2 notes and $5 notes.<br />She had a total of $76.<br />She had 4 more $5 notes than $2 notes.<br />How many $2 notes did she have?<br />Source<br />Raffles Girls’ School, Singapore<br />
- 8. Devi had some $2 notes and $5 notes.<br />She had a total of $76.<br />She had 4 more $5 notes than $2 notes.<br />How many $2 notes did she have?<br />$76<br />Number of $5 notes<br />4<br />$20<br />$56<br />8<br />Number of $2 notes<br />$56 ÷ 7 = 8<br />8<br />She had eight $2 notes.<br />
- 9. Grade 6 Examination Item (PSLE 2009)<br />Challenging Application<br />In the diagram below, ABCD is a square and QM = QP = QN. MN is parallel to AB and it is perpendicular to PQ.<br />Find MPN<br />Answer: 150o<br />Source<br />Singapore Examinations and Assessment Board PSLE Mathematics 2009<br />
- 10. NCTM Regional Conference & Exposition Denver Colorado<br />8 – 9 October 2010<br />Session 165<br />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. <br />Colorado Convention Center, 506 – 507 <br />Denver, Colorado<br />
- 11. learning <br />well<br />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.<br />
- 12. The majority of students in Singapore opt to study mathematics in Grades 11 and 12 when they are no longer required to.<br />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. <br />
- 13. Attitude<br />Achievement<br />The figures refer to percentage of students with high PATM (for Attitude) and in the Advanced International Benchmark (for Achievement).<br />Taiwan<br />37<br />45<br />Grade 8<br />S Korea<br />33<br />40<br />Singapore<br />60<br />40<br />Hong Kong<br />47<br />31<br />Japan<br />30<br />26<br />Hungary<br />30<br />10<br />England<br />40<br />8<br />International<br />54<br />2<br />Source<br />TIMSS 2007 International Mathematics Report, Boston College<br />
- 14. one: sustained focus on thinking<br />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)<br />
- 15. Beliefs<br />Interest<br />Appreciation<br />Confidence<br />Perseverance<br />Monitoring of one’s own thinking<br />Self-regulation of learning<br />Attitudes<br />Metacognition<br />Numerical calculation<br />Algebraic manipulation<br />Spatial visualization<br />Data analysis<br />Measurement<br />Use of mathematical tools<br />Estimation<br />Mathematical Problem Solving<br />Reasoning, communication & connections<br />Thinking skills & heuristics<br />Application & modelling<br />Skills<br />Processes<br />Concepts<br />Numerical<br />Algebraic<br />Geometrical<br />Statistical<br />Probabilistic<br />Analytical<br />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.<br />
- 16. For example, there is an emphasis on mental strategies.<br />Mental strategies help students become flexible thinkers as they need to make decisions on the best strategy to use for a particular computation.<br />In an earlier presentation (Session 106), I have discussed visualization, number sense, generalization and metacognition.<br />Source<br />Pensar sin Limites 2oBasico<br />
- 17. Dagoberto Godoy School, Santago<br />Princess Elizabeth Primary School, Singapore<br />Keys Grade School, Manila<br />Springhurst Elementary School, New York<br />
- 18. There are some apples in a box. There are 5 different ways to put the apples into equal groups. <br />Hint: The number is between 15 and 21.<br />Keys Grade School, Manila<br />
- 19. sustained focus <br />on thinking<br />Dagoberto Godoy School, Santiago, Chile<br />
- 20. two: small number of topics<br />This allows in-depth treatment of a few key ideas at the early grade levels. One core idea is number bonds.<br />
- 21. Number Bonds is emphasized prior to the learning of addition.<br />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. <br />Source<br />Earlybird Kindergarten Mathematics Book B <br />
- 22. PCF Kindergarten TelokBlangah, Singapore<br />
- 23. One duck is big. <br />Six ducklings are small.<br />PCF Kindergarten TelokBlangah, Singapore<br />
- 24. PCF Kindergarten TelokBlangah, Singapore<br />
- 25. PCF Kindergarten TelokBlangah, Singapore<br />
- 26. PCF Kindergarten TelokBlangah, Singapore<br />
- 27. PCF Kindergarten TelokBlangah, Singapore<br />
- 28. Source<br />My Pals Are Here! Mathematics Primary 1<br />
- 29. sustained focus <br />on thinking<br />focus <br />on fewer topics<br />Princess Elizabeth Primary School, Singapore<br />
- 30. Circle Graphs are taught in Grade 6 in Singapore.<br />Source<br />TIMSS 2007 International Mathematics Report, Boston College<br />
- 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.<br />three: CPA approach<br />Source<br />Earlybird Kindergarten Mathematics – Kindergarten Textbook in Singapore<br />
- 32. Is the rectangle cut into equal parts?<br />illustration<br />equal parts<br />Source<br />Professional Development Course for Singapore Teachers <br />
- 33.
- 34.
- 35. bruner’s theory<br />concrete<br />Source<br />mathz4kidz Learning Centre, Penang, Malaysia<br />A Lesson from Earlybird Kindergarten Mathematics <br />
- 36. concrete<br />experiences<br />Source<br />mathz4kidz Learning Centre, Penang, Malaysia<br />A Lesson from Earlybird Kindergarten Mathematics <br />
- 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.<br />Source<br />Fuchun Primary School, Singapore<br />Research Lesson to Develop Formula for Circumference of Circles<br />
- 38. use of visuals<br />Source<br />Math in Focus Grade 4 <br />
- 39. Source<br />Math in Focus Grade 4 <br />
- 40. Visuals are used extensively to develop conceptual understanding.<br /> <br />In learning how to write mixed numbers as improper fractions in Grade 4, the visual approach is a common approach in the textbooks. <br />
- 41. This visual shows why <br />It is clear that there are 5 halves in <br />
- 42. The use of visuals in learning multiplication facts.<br />Source<br />Primary Mathematics Grade 3<br />
- 43. Source<br />Pensar sin Limites 2oBasico<br />
- 44. sustained focus <br />on thinking<br />focus <br />on fewer topics<br />the CPA approach<br />Da Qiao Primary School, Singapore<br />
- 45. four:<br />spiral curriculum<br />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.<br />
- 46. Source<br />My Pals Are Here! Mathematics Primary 1<br />
- 47. Princess Elizabeth Primary School, Singapore<br />Source<br />Pensar sin Limites 1oBasico<br />
- 48. <br />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.<br />Princess Elizabeth Primary School, Singapore<br />My Pals Are Here! Mathematics Primary 1<br />
- 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. <br />Keys Grade School, Manila<br />
- 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.<br />five: non-specialist teachers<br />
- 51. textbooks<br />
- 52. Source<br />My Pals Are Here! Mathematics Primary 1<br />
- 53. Source<br />Pensar sin Limites 1oBasico<br />
- 54. pre-service teacher education<br />
- 55. Teachers received about 100 hours of courses in mathematics teaching and learning during pre-service training.<br /> <br />National Institute of Education, Singapore<br />
- 56. in-service professional development<br />
- 57. <br />Teachers are expected to engage in professional development throughout their career.<br /> <br />Princess Elizabeth Primary School, Singapore<br />Marshall Cavendish Institute, Singapore<br />
- 58. <br />Some study for certification while others just for improvement. It is also an expectation.<br />SEED Institute, Singapore<br />
- 59. Tasks are presented in a systematic sequence to provide students with both mathematical as well as perceptual variability.<br />six: systematic variation of tasks<br />
- 60. Source<br />Math in Focus Grade 2<br />
- 61. Source<br />Math in Focus Grade 2<br />
- 62. Source<br />Math in Focus Grade 2<br />
- 63. Systematic Variation – students are presented with a variety of tasks in a systematic way. This is based on the idea of ZoltanDienes.<br />
- 64. Source<br />Primary Mathematics Grade 1<br />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.<br />The question is what is an appropriate sequence? Should the place value chart be used first? Or the expanded notation? Give your reasons.<br />Place Value Chart<br />Expanded Notation<br />Source<br />Professional Development Materials<br />Marshall Cavendish Institute, Singapore<br />Words<br />Numerals<br />Tens and Ones Notation<br />
- 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.<br />The question is what is an appropriate sequence? Should the place value chart be used first? Or the expanded notation? Give your reasons.<br />Place Value Chart<br />Expanded Notation<br />Words<br />Numerals<br />Tens and Ones Notation<br />Primary Mathematics<br />
- 66. non-specialist <br />teachers<br />sustained focus <br />on thinking<br />focus <br />on fewer topics<br />the CPA approach<br />spiral curriculum<br />Da Qiao Primary School, Singapore<br />systematic variation<br />
- 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. <br />seven: high aspirations among parents<br />
- 68. home<br />support<br />“Last Saturday, Mum and Dad went to school – to learn the model method so they can help me with schoolwork. But Mum said she did not get it. <br /><ul><li>In the end I had to teach her.”</li></ul>Third Grader<br />Seminar for Parents on Problem Solving<br />
- 69. TelokKurau Primary School, Singapore<br />“Sometimes my teacher gives me remedial classes. Some people go for Maths Olympiad training. My maths is not good. <br />I don’t go for that.”<br />Fifth Grader<br />school<br />support<br />“In my school, we have a fixed day for remedial lessons. I get a group of students to attend this regularly.”<br />Teacher<br />
- 70. community<br />support<br />TelokKurau Primary School, Singapore<br />
- 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. <br />eight: learning support and remedial<br />
- 72. safety nets and interventions<br />non-specialist <br />teachers<br />Slides are available at<br />www.banhar.blogspot.com<br />sustained focus <br />on thinking<br />focus <br />on fewer topics<br />the CPA approach<br />spiral curriculum<br />systematic variation<br />high parental expectations<br />
- 73. Eight Things about Early Grades Mathematics in Singapore<br />Slides are available at<br />www.banhar.blogspot.com<br />Da Qiao Primary School, Singapore<br />

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