Singapore Math in the Netherlands Day 3

Singapore Math in Rotterdam 3
Opleiding Singapore rekenspecialist
This set of slides cover two presentations made on the final day:
The Spiral Curriculum
Enrichment Lessons
De 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 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.

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?
210
MrsHoon sold 960 cookies.

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

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

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?

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

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 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.

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.

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).

Reference
www.banhar.blogspot.com
TIMSS 2007 Europe

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

TIMSS 2007 Europe
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

TIMSS 2007 Europe
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

http://banhar.blogspot.com	50
http://singaporemathz4kidz.blogspot.com	13
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Singapore Math in the Netherlands Day 3

by Jimmy Keng on Sep 18, 2010

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Singapore Math in the Netherlands Day 3 — Presentation Transcript