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This is Session 106 on mathematical problem solving.
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NCTM Regional Conference & Exposition Denver Colorado
8 – 9 October 2010Session 106
This session includes a selection of problems used in grades K – 3 classrooms in Singapore. Look at how these problems are used to achieve different instructional goals. See how these problems play a role in helping children develop a strong foundation for subsequent learning.
Colorado Convention Center,
Korbel 3B
Problem Solving in the Singapore Classrooms
Dr Yeap Ban HarMarshall Cavendish
InstituteSingapore
banhar@sg.marshallcavendish.com
www.banhar.blogspot.com
Session 106
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).
Mathematical Problem
Solving
Attitudes
Metacognition
Proc
esse
s
Concepts
Skills
NumericalAlgebraic
GeometricalStatistical
ProbabilisticAnalytical
Reasoning, communication & connectionsThinking skills & heuristicsApplication & modelling
Numerical calculationAlgebraic
manipulationSpatial visualization
Data analysisMeasurement
Use of mathematical tools
Estimation
Monitoring of one’s own thinkingSelf-regulation of learning
BeliefsInterest
AppreciationConfidence
Perseverance
intellectual competence
The Singapore mathematics curriculum states that mathematics is an “excellent vehicle for the development and improvement of a person’s intellectual competence” (Ministry of Education 2006). The next few examples show how problem solving can be used to develop visualization, number sense and ability to see patterns as well as to facilitate language development.
example one
SourceEarlybird Kindergarten Mathematics – Kindergarten Textbook in Singapore
PCF Kindergarten Telok Blangah, Singapore
example two
SourcePrimary Three Examination Item from Henry Park Primary School, Singapore
Angela, Bala, and Cheng Hao have 872 marbles altogether. Angela has 4 times as many marbles as Bala. Cheng Hao has 50 more marbles than Bala. How many marbles does Bala have?
example two
SourcePrimary Three Examination Item from Henry Park Primary School, Singapore
Angela, Bala, and Cheng Hao have 872 marbles altogether. Angela has 4 times as many marbles as Bala. Cheng Hao has 50 more marbles than Bala. How many marbles does Bala have?
Angela
Bala
Cheng Hao
50
872
example two
SourcePrimary Three Examination Item from Henry Park Primary School, Singapore
Angela, Bala, and Cheng Hao have 872 marbles altogether. Angela has 4 times as many marbles as Bala. Cheng Hao has 50 more marbles than Bala. How many marbles does Bala have?
Angela
Bala
Cheng Hao
822
1 unit = 822 ÷ 61 unit = 137Bala has 137 marbles.
600180
42
example two
SourcePrimary Three Examination Item from Henry Park Primary School, Singapore
Angela, Bala, and Cheng Hao have 872 marbles altogether. Angela has 4 times as many marbles as Bala. Cheng Hao has 50 more marbles than Bala. How many marbles does Bala have?
Angela
Bala
Cheng Hao
50
872y
y y y y
y
4y + y + (y + 50) = 872
6y + 50 = 872
6y = 872 – 50
y = 822 ÷ 6
y = 137
Angela
4y + y + (y + 50) = 872
Using a box to represent
known quantities.
Using a box to represent unknown
quantities.
Using a letter to represent
unknown quantities.
example three
SourceGrade 4 Lesson on Long Division in Keys Grade School, Manila
SourceGrade 4 Lesson on Long Division in Keys Grade School, Manila
481644816
4000 800
16
4
81
12
8
16
0
16
0
4
example four
Source2o Basico Lesson on Model Drawing in Escuela Republica de Guatemala, Chile
Hugo has ? toy cars.His friend has ? toy cars.They have ? toy cars altogether.
Source2o Basico Lesson on Model Drawing in Escuela Republica de Guatemala, Chile
Hugo has 10 toy cars.His friend has 12 toy cars.They have ? toy cars altogether.
Source2o Basico Lesson on Model Drawing in Escuela Republica de Guatemala, Chile
Hugo has 10 toy cars.His friend has 12 toy cars.They have ? toy cars altogether.
Source2o Basico Lesson on Model Drawing in Escuela Republica de Guatemala, Chile
Hugo has 121 toy cars.His friend has 74 toy cars.They have ? toy cars altogether.
121 74
?
Source2o Basico Lesson on Model Drawing in Escuela Republica de Guatemala, Chile
Hugo has 20 toy cars.His friend has 74 toy cars.They have ? toy cars altogether.
Source2o Basico Lesson on Model Drawing in Escuela Republica de Guatemala, Chile
Hugo has 100 toy cars.His friend has 74 toy cars.They have ? toy cars altogether.
Source2o Basico Lesson on Model Drawing in Escuela Republica de Guatemala, Chile
Hugo has ? toy cars.His friend has 74 toy cars.They have 100 toy cars altogether.
example five
SourceMy Pals Are Here! Maths – Grade 1 Textbook in Singapore
example six
Use five consecutive whole numbers so that the vertical sum is equal to the horizontal sum. Start with 1 to 5.
SourceResearch Lesson at a Professional Development Course in Singapore
Use five consecutive whole numbers so that the vertical sum is equal to the horizontal sum. Start with 1 to 5.
15
4
3 2 2 1 53
4
31 52
4
32
6
4 5 3 4 52
6
62 53
4
Princess Elizabeth Primary School, Singapore
24
6
3 5
Princess Elizabeth Primary School, Singapore
Use the digits 0 to 9 not more than once to make a correct multiplication sentence.
xexample seven
SourceTeacher Professional Development Course in Singapore
1 2x 56 0
4 3x 28 6
2 9x 38 7
1 5x 46 0
1 2x 78 4
1 5x 23 0
1 8x 23 6
1 8x 35 4
1 8x 47 2
1 4x 79 8
1 2x 89 6
responses
problem solving for
various instructional
goals
We have seen the use of problem solving to teach a basic concept or skill, to consolidate learning as well as to provide opportunities for students to apply what they know.
problem solving to
teach a new concept or
skill
PK – K
G 1
G 2
G 3G 4
PK – K
G 1
G 2
G 3G 4
visualization
visualization
metacognition
number sense
problem solving to
consolidate a skill or
concept
xreasoning
generalization
problem solving for
students to apply their knowledge
Angela, Bala, and Cheng Hao have 872 marbles altogether. Angela has 4 times as many marbles as Bala. Cheng Hao has 50 more marbles than Bala. How many marbles does Bala have?
Problem Solving in the Singapore Classrooms
Slides are available at
www.banhar.blogspot.com
Da Qiao Primary School, Singapore
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