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Singapore Math in Rotterdam 3Opleiding Singapore rekenspecialistThis set of slides cover two presentations made on the final day:The Spiral Curriculum
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.
Singapore Math in Rotterdam 3Opleiding Singapore rekenspecialistReview of Day 2 and Going AheadLet’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.
Materials developed by Poon Yain Ping
Materials developed by Poon Yain Ping
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 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?  210MrsHoon sold 960 cookies.
Singapore Math in the Netherlands Day 3
The Spiral ApproachYeap Ban Har, Ph.D.Marshall Cavendish InstituteSingaporebanhar@sg.marshallcavendish.com
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
Adding & Subtracting FractionsThis 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.
Singapore Math in the Netherlands Day 3
Singapore Math in the Netherlands Day 3
Singapore Math in the Netherlands Day 3
Singapore Math in the Netherlands Day 3
Singapore Math in the Netherlands Day 3
Singapore Math in the Netherlands Day 3
Singapore Math in the Netherlands Day 3
Singapore Math in the Netherlands Day 3
Singapore Math in the Netherlands Day 3
Singapore Math in the Netherlands Day 3
Singapore Math in the Netherlands Day 3
Singapore Math in the Netherlands Day 3
Singapore Math in the Netherlands Day 3
Key Ideas in FractionsHow are these taught? Let’s take a look.
Singapore Math in the Netherlands Day 3
Singapore Math in the Netherlands Day 3
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?
Singapore Math in the Netherlands Day 3
Singapore Math in the Netherlands Day 3
Singapore Math in the Netherlands Day 3
Singapore Math in the Netherlands Day 3
Please prepare a set of digits 0 to 9 for the activities.
x
Singapore Math in the Netherlands Day 3
Singapore Math in the Netherlands Day 3
Singapore Math in the Netherlands Day 3
Scarsdale Middle School New York
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?
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.
First, the teacher wanted to discuss an incorrect use of notation. One pair has thought that 2/0 = 2 and made the sentence8/4 ÷ 1 = 2/0This 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.
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.
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.
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.
Singapore Math in the Netherlands Day 3
Singapore Math in the Netherlands Day 3
Singapore Math in the Netherlands Day 3
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).
Referencewww.banhar.blogspot.comTIMSS 2007 Europe
TIMSS 2007 EuropeTrends in International Mathematics and Science StudiesLatviaEnglandRussiaKazakhstanLithuaniaHungaryArmeniaDenmarkNetherlandsGermanyItalyGrade 4Advanced1116161910987766High4448485242352836423729Intermediate8181798177676076847867Low9795949594888495989691
TIMSS 2007 EuropeTrends in International Mathematics and Science StudiesSloveniaSlovak RepScotlandInternationalAustriaSwedenUkraineCzech RepNorwayGeorgiaSingaporeGrade 4Advanced345533222141High2525262626241719151074Intermediate6762636769685059523592Low9288889093937988836798
TIMSS 2007 EuropeTrends in International Mathematics and Science StudiesLatviaEnglandRussiaKazakhstanLithuaniaHungaryArmeniaDenmarkNetherlandsGermanyItalyGrade 4Advanced1116161910987766High4448485242352836423729Intermediate8181798177676076847867Low9795949594888495989691

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Singapore Math in the Netherlands Day 3

  • 1. Singapore Math in Rotterdam 3Opleiding Singapore rekenspecialistThis set of slides cover two presentations made on the final day:The Spiral Curriculum
  • 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 3Opleiding Singapore rekenspecialistReview of Day 2 and Going AheadLet’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? 210MrsHoon sold 960 cookies.
  • 9. The Spiral ApproachYeap Ban Har, Ph.D.Marshall Cavendish InstituteSingaporebanhar@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 FractionsThis 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 FractionsHow 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
  • 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 sentence8/4 ÷ 1 = 2/0This 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).
  • 50. TIMSS 2007 EuropeTrends in International Mathematics and Science StudiesLatviaEnglandRussiaKazakhstanLithuaniaHungaryArmeniaDenmarkNetherlandsGermanyItalyGrade 4Advanced1116161910987766High4448485242352836423729Intermediate8181798177676076847867Low9795949594888495989691
  • 51. TIMSS 2007 EuropeTrends in International Mathematics and Science StudiesSloveniaSlovak RepScotlandInternationalAustriaSwedenUkraineCzech RepNorwayGeorgiaSingaporeGrade 4Advanced345533222141High2525262626241719151074Intermediate6762636769685059523592Low9288889093937988836798
  • 52. TIMSS 2007 EuropeTrends in International Mathematics and Science StudiesLatviaEnglandRussiaKazakhstanLithuaniaHungaryArmeniaDenmarkNetherlandsGermanyItalyGrade 4Advanced1116161910987766High4448485242352836423729Intermediate8181798177676076847867Low9795949594888495989691
  • 53. TIMSS 2007 EuropeTrends in International Mathematics and Science StudiesLatviaEnglandRussiaKazakhstanLithuaniaHungaryArmeniaDenmarkNetherlandsGermanyItalyGrade 4Advanced1116161910987766High4448485242352836423729Intermediate8181798177676076847867Low9795949594888495989691