The document discusses Singapore Math and its spiral curriculum approach. It provides examples of how fractions are taught over multiple grades, with concepts being revisited and built upon each year. It also discusses enrichment lessons, and gives an example of a lesson where students explore different methods for dividing fractions by whole numbers.

2. Enrichment LessonsDe structuur van het curriculum onderzoeken en begrijpen hoe de rekenconceptenzorgvuldig op elkaaraansluiten. Weten hoe verrijking is onderbracht in het programma en hoe betrokkenheid van leerlingenwordtvergroot door gebruiktemaken van activiteitengericht op exploratie, onderzoek, leren door tedoen, interactie, reflectie en meer.

3. Singapore Math in Rotterdam 3 Opleiding Singapore rekenspecialist Review of Day 2 and Going Ahead Let’s solve another problem using the model method. Today our focus is on fractions and area and use these to study the idea of the spiral curriculum. We will also look at some enrichment lessons.

4. Materials developed by Poon Yain Ping

5. Materials developed by Poon Yain Ping

6. MrsHoon made some cookies to sell. of them were chocolate cookies and the rest were almond cookies. After selling 210 almond cookies and of the chocolate cookies, she has of the cookies left. How many cookies did MrsHoon sell? 210

7. MrsHoon made some cookies to sell. of them were chocolate cookies and the rest were almond cookies. After selling 210 almond cookies and of the chocolate cookies, she has of the cookies left. How many cookies did MrsHoon sell? 210 MrsHoon sold 960 cookies.

9. The Spiral Approach Yeap Ban Har, Ph.D. Marshall Cavendish Institute Singapore banhar@sg.marshallcavendish.com

10. The Process of Education “A curriculum as it develops should revisit this basic ideas repeatedly, building upon them until the student had grasped the full formal apparatus that goes with them.” Bruner 1960

11. Adding & Subtracting Fractions This is learned over a period of four years – from grade two to grade five. This is an example of the spiral approach adopted in the Singapore curriculum. It is not up to textbook authors. It is based on the national curriculum.

25. Key Ideas in Fractions How are these taught? Let’s take a look.

28. Fraction is introduced in Grade 2. Why are the terms numerator and denominator introduced only in Grade 3? Can you explain this using one of the theoretical underpinnings of Singapore Math?

33. Please prepare a set of digits 0 to 9 for the activities.

34. x

38. Scarsdale Middle School New York

39. The Anchor Problem – to make a correct division sentence using one set of digit tiles 0 to 9. Why do you think the teacher provided the restriction of not repeating any of the digits?

40. Students were given time to make some sentences on their own. It was evident that the students were already familiar with the algorithm. The purpose of this lesson was for students to make sense of the algorithm. Three approaches were used by the students.

41. First, the teacher wanted to discuss an incorrect use of notation. One pair has thought that 2/0 = 2 and made the sentence 8/4 ÷ 1 = 2/0 This example is interesting because it involves the idea of division by 1 which results in the two fractions being equal. It was agreed that 8/4 = 2 and that the right-hand side is equal to 2. Other students offered that 2 can also be written as 2/1 and, incorrectly 2 (2/0). The teacher explained that 2/0 is not 1. 2/1 is. And that division by zero is not defined. Subsequently, students offered the right-hand side can be 4/2 and 6/3. 4/2 was rejected because they realized the condition of the problem – no repetition of digits.

42. The first explanation a student gave for the algorithm is the fact that 2 fourths shared equally by 3 is equal to each getting 1/3 of 2 fourths. The second explanation was based on the idea of division of whole numbers. If 2 divided by 2 is 1 then 2 fourths divided by 2 is 1 fourth. If 4 divided by 2 is 2 then 4 fourths divided by 2 is 2 fourths. The rest of the lesson was focused on the whole class grasping these two ideas.

43. In discussing the other responses forwarded by the students, the teacher challenged the students to use the second method to explained cases such as 8 fourths divided by 6 where 8 is not divisible by 6. In the case of ¾ divided by 8, students were able to suggest changing the numerator to 24, writing ¾ as 24/32 before dividing by 8 since 24 is divisible by 8.

44. The third approach used was the bar model that the students have become familiar with. The teacher was pleased that a student who seemed unsure of himself found his voice to explain to the class why 1/5 ÷ 4 = 1/20. The lesson ended with students reflecting on the methods used to explain division of a fraction by a whole number.

48. Think of two digits. Use the digits to make two numbers. First, the first digit be the tens and second digit be the ones. So if you think of 4 and 5, the number is 45 (not 54). Then, add the digits to make a second number. So 4 and 5 make 9. Finally find the difference between the two numbers (45 and 9).

49. Reference www.banhar.blogspot.com TIMSS 2007 Europe

50. TIMSS 2007 Europe Trends in International Mathematics and Science Studies Latvia England Russia Kazakhstan Lithuania Hungary Armenia Denmark Netherlands Germany Italy Grade 4 Advanced 11 16 16 19 10 9 8 7 7 6 6 High 44 48 48 52 42 35 28 36 42 37 29 Intermediate 81 81 79 81 77 67 60 76 84 78 67 Low 97 95 94 95 94 88 84 95 98 96 91

51. TIMSS 2007 Europe Trends in International Mathematics and Science Studies Slovenia Slovak Rep Scotland International Austria Sweden Ukraine Czech Rep Norway Georgia Singapore Grade 4 Advanced 3 4 5 5 3 3 2 2 2 1 41 High 25 25 26 26 26 24 17 19 15 10 74 Intermediate 67 62 63 67 69 68 50 59 52 35 92 Low 92 88 88 90 93 93 79 88 83 67 98

52. TIMSS 2007 Europe Trends in International Mathematics and Science Studies Latvia England Russia Kazakhstan Lithuania Hungary Armenia Denmark Netherlands Germany Italy Grade 4 Advanced 11 16 16 19 10 9 8 7 7 6 6 High 44 48 48 52 42 35 28 36 42 37 29 Intermediate 81 81 79 81 77 67 60 76 84 78 67 Low 97 95 94 95 94 88 84 95 98 96 91

53. TIMSS 2007 Europe Trends in International Mathematics and Science Studies Latvia England Russia Kazakhstan Lithuania Hungary Armenia Denmark Netherlands Germany Italy Grade 4 Advanced 11 16 16 19 10 9 8 7 7 6 6 High 44 48 48 52 42 35 28 36 42 37 29 Intermediate 81 81 79 81 77 67 60 76 84 78 67 Low 97 95 94 95 94 88 84 95 98 96 91