Pensar sin Limites Seminars This seminar was held at various places including Colegio Inmaculada in Conception, Universidad Andres Bello in Santiago & San Benito School in Santiago. The seminar explains the pedagogy behind the Spanish edition of My Pals Are Here! Mathematics.

1. Dr Yeap Ban Har Marshall Cavendish Institute Singapore banhar@sg.marshallcavendish.com Seminar on Singapore Math in Chile SANTIAGO  CONCEPTION This seminar is about how textbooks can help average students learn mathematics well. Presentation is available at www.banhar.com

2. 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 In Singapore, a curriculum that focuses on mathematical problem solving was introduced in 1992.

3. thinking schools learning nation The emphasis on thinking continues in the late 1990s with the introduction of Thinking School, Learning Nation philosophy to the entire education system in 1997.

4. Textbooks that place an emphasis on problem solving and thinking naturally will be able to help many students becoming good in problem solving. Singapore textbooks do not focus on tedious computations or learning procedures without the corresponding conceptual understanding.

5. Textbooks place an emphasis on problem solving and thinking.

6. first day first day first day

7. It focuses on the development of visualization. The bar model is used extensively to help students represent problems well as well as to develop visualization.

9. 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.

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

11. It focuses on the development of number sense.

12. 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.

13. A strong foundation is necessary for the students to do well in mathematics. In the Singapore textbooks, such a strong foundation is achieved through the application of a few learning principles.

14. bruner’s theory concrete Singapore Math in Malaysia

15. concrete experiences Singapore Math in Malaysia

16. 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.

17. For example, students learn the idea of division by sharing 12 cookies among 4 persons as well as by putting 12 eggs in groups of 4 before progressing to using drawings to solve division problems. Later they learn to use division sentence 12 ÷ 4 = 3.

20. The Spiral Approach – students get to revisit core ideas as they deepen their understanding of those ideas. This is also one of Jerome Bruner’s ideas. Singapore Math in the Netherlands

21. 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. Singapore Math in USA

22. 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. Singapore Math in The Philippines

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

24. Students in Singapore have demonstrated high achievement and positive attitude towards mathematics. In Trends in Mathematics and Science Study, more than 40% of Singapore’s 4th and 8th graders are in the Advanced International Benchmark (the international average is 5% and 2% respectively). 1995 2003 2007 International 5 Advanced 38 41 38 Grade 4 26 High 70 74 73 67 Intermediate 89 92 91 90 Low 96 98 97

25. Average Hong Kong Singapore S. Korea Taiwan Grade 8 Advanced 2 40 31 45 40 High 15 70 64 71 71 Intermediate 46 88 85 86 90 Low 75 97 94 95 98 Singapore Math in Indonesia

26. TIMSS Trends in International Mathematics and Science Studies Grade 4 2003 Grade 8 2007 Grade 4 1995 Grade 8 1999 38 40 Advanced 38 42 73 70 High 70 77 91 88 Intermediate 89 94 97 97 Low 96 99

27. Mrs Hoon made some cookies to sell. 3/4 of them were chocolate cookies and the rest were almond cookies. After selling 210 almond cookies and 5/6 of the chocolate cookies, she had 1/5 of the cookies left. How many cookies did Mrs Hoon sell?

28. Mrs Hoon made some cookies to sell. 3/4 of them were chocolate cookies and the rest were almond cookies. After selling 210 almond cookies and 5/6 of the chocolate cookies, she had 1/5 of the cookies left. How many cookies did Mrs Hoon sell? 210

29. Mrs Hoon made some cookies to sell. 3/4 of them were chocolate cookies and the rest were almond cookies. After selling 210 almond cookies and 5/6 of the chocolate cookies, she had 1/5 of the cookies left. How many cookies did Mrs Hoon sell? 210 MrsHoon sold 960 cookies.

30. In the PSLE, the national average for students who are capable of completing the most challenging items in the examination is between 40% and 45%.

31. Schools make problem solving a priority in their mathematics programme because of the demands of the PSLE. Grade 3 Examination Item (Raffles Girls’ Primary School 2009) 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?

32. Schools make problem solving a priority in their mathematics programme because of the demands of the PSLE. Grade 3 Examination Item (Raffles Girls’ Primary School 2009) 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? $78 Number of $5 notes 4 $20 $58 8 Number of $2 notes $58 ÷ 7 = 8 8 She had eight $2 notes.

33. The attitude index for Singapore students in TIMSS is also relatively high compared to other high-performing countries. Also, the majority of students in Singapore opt to study mathematics in Grades 11 and 12 when they are no longer required to.

34. Attitude Achievement Singapore 71 41 Grade 4 Hong Kong 67 40 Taiwan 50 24 Japan 62 23 Kazakhstan 89 19 England 62 16 Russia 80 16 International 72 5

35. Attitude 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

36. Other than the curriculum, assessment and textbooks, another important factor is the teachers.

37. Teachers received about 100 hours of courses in mathematics teaching and learning during pre-service training.

38. Teachers are expected to engage in professional development throughout their career.

39. Some study for certification while others just for improvement. It is also an expectation.

40. Textbooks are designed for teachers to learn the mathematics that they teach.

41. Dr Yeap Ban Har Marshall Cavendish Institute Singapore banhar@sg.marshallcavendish.com Seminar on Singapore Math in Chile SANTIAGO  CONCEPTION Singapore Math in Chile Presentation is available at www.banhar.com