NCTM 2010 Regional Conferences & Expositions Denver 2

NCTM Regional Conference & Exposition Denver Colorado
8 – 9 October 2010
Session 165
This session focuses on eight elements in early grades mathematics that help children develop a strong foundation for a demanding mathematics program in the upper grades. Concrete, visual approaches, coupled with special attention to economically disadvantaged children, systematic variation in representations and content help students learn well. 
Colorado Convention Center, 506 – 507
Denver, Colorado

Eight Things About Early Grades Mathematics in Singapore
Session 165
DrYeap Ban Har
Marshall Cavendish Institute
Singapore
banhar@sg.marshallcavendish.com
slides are available at
www.banhar.blogspot.com

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
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).

8 – 9 October 2010
Session 165
Denver, Colorado

Students are expected to solve complex and, often, new problems in the program. The formal assessment such as school examination and national examination contains significant emphasis on problem solving.
demanding program

A
B
F
E
Source:
Singapore Examinations and Assessment Board PSLE Mathematics 2004 – 2008 and Specimen Paper 2009
D
C

Grade 3 Examination Item (Raffles Girls’ Primary School 2009)
Challenging Application
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?
Source
Raffles Girls’ School, Singapore

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?
$76
Number of $5 notes
4
$20
$56
8
Number of $2 notes
$56 ÷ 7 = 8
8
She had eight $2 notes.

Grade 6 Examination Item (PSLE 2009)
Challenging Application
In the diagram below, ABCD is a square and QM = QP = QN. MN is parallel to AB and it is perpendicular to PQ.
Find MPN
Answer: 150o
Source
Singapore Examinations and Assessment Board PSLE Mathematics 2009

8 – 9 October 2010
Session 165
Denver, Colorado

learning
well
Students who have learnt well in the early grades should do well in the upper grades. Also, the should find mathematics sufficiently challenging and within their grasp.

The majority of students in Singapore opt to study mathematics in Grades 11 and 12 when they are no longer required to.
This is confirmed by the TIMSS data on attitude towards mathematics. The attitude index for Singapore students in TIMSS is also relatively high compared to other high-performing countries.

Attitude
Achievement
The figures refer to percentage of students with high PATM (for Attitude) and in the Advanced International Benchmark (for 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
Source
TIMSS 2007 International Mathematics Report, Boston College

one: sustained focus on thinking
One of the three stated rationales for teaching and learning mathematics is because mathematics is “an excellent vehicle for the development and improvement of a person’s intellectual competence” (Ministry of Education 2006)

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
The problem-solving framework has been in place since 1992. In 1997, Thinking Schools, Learning Nation philosophy was introduced to the entire education system. This is supported by the Teach Less, Learn More initiative in 2004.

For example, 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.
In an earlier presentation (Session 106), I have discussed visualization, number sense, generalization and metacognition.
Source
Pensar sin Limites 2oBasico

Dagoberto Godoy School, Santago
Princess Elizabeth Primary School, Singapore
Keys Grade School, Manila
Springhurst Elementary School, New York

There are some apples in a box. There are 5 different ways to put the apples into equal groups.
Hint: The number is between 15 and 21.

sustained focus
on thinking
Dagoberto Godoy School, Santiago, Chile

two: small number of topics
This allows in-depth treatment of a few key ideas at the early grade levels. One core idea is number bonds.

Number Bonds is emphasized prior to the learning of addition.
Children are given, say, 5 unifix cubes and guided to see that 1 and 4 make 5, for example. Others may say that 3 and 2 make 5 or 4 and 1 make 5. Yet others may say that 5 and 0 make 5.
Source
Earlybird Kindergarten Mathematics Book B

PCF Kindergarten TelokBlangah, Singapore

One duck is big.
Six ducklings are small.

Source
My Pals Are Here! Mathematics Primary 1

sustained focus
on thinking
focus
on fewer topics

Circle Graphs are taught in Grade 6 in Singapore.
Source
TIMSS 2007 International Mathematics Report, Boston College

Based on Jerome Bruner’s ideas of enactive, iconic and symbolic representations, Singapore curriculum, textbooks and teacher education emphasized the importance of concrete materials and the progression to visual and symbolic representations.
three: CPA approach
Source
Earlybird Kindergarten Mathematics – Kindergarten Textbook in Singapore

Is the rectangle cut into equal parts?
illustration
equal parts
Source
Professional Development Course for Singapore Teachers

bruner’s theory
concrete
Source
mathz4kidz Learning Centre, Penang, Malaysia
A Lesson from Earlybird Kindergarten Mathematics

concrete
experiences
Source
mathz4kidz Learning Centre, Penang, Malaysia
A Lesson from Earlybird Kindergarten Mathematics

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.
Source
Fuchun Primary School, Singapore
Research Lesson to Develop Formula for Circumference of Circles

use of visuals
Source
Math in Focus Grade 4

Source

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.

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

The use of visuals in learning multiplication facts.
Source
Primary Mathematics Grade 3

Source

sustained focus
on thinking
focus
on fewer topics
the CPA approach
Da Qiao Primary School, Singapore

four:
spiral curriculum
Students have many opportunities to learn a core idea. For example the concept of division takes five years to develop – in a systematic manner, with the goal of advancing students’ notion of the concept.

Source

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. The quantities used are limited to discrete quantities.

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.

Up till 1992, there was no college graduates teaching in primary schools in Singapore. Today, half are college graduates. In pre-primary levels, many teachers hold only a Grade 10 certificate (with a teaching certificate), although increasingly more teachers completing the bachelor and graduate degrees. Compulsory education is Grade 1 to Grade 6.
five: non-specialist teachers

textbooks

Source

pre-service teacher education

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

National Institute of Education, Singapore

in-service professional development

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

Marshall Cavendish Institute, Singapore

Some study for certification while others just for improvement. It is also an expectation.
SEED Institute, Singapore

Tasks are presented in a systematic sequence to provide students with both mathematical as well as perceptual variability.
six: systematic variation of tasks

Source

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

Source
Primary Mathematics Grade 1
It is likely that a teacher will start this unit using the sticks. This is followed by the use of base ten blocks. Finally, non-proportionate materials such as coins are used. In each of these lessons, the teacher is likely to introduce the following five notations in turn – place value chart, expanded notation, number in numerals, number in words and the tens and ones notation.
The question is what is an appropriate sequence? Should the place value chart be used first? Or the expanded notation? Give your reasons.
Place Value Chart
Expanded Notation
Source
Professional Development Materials
Marshall Cavendish Institute, Singapore
Words
Numerals
Tens and Ones Notation

It is likely that a teacher will start this unit using the sticks. This is followed by the use of base ten blocks. Finally, non-proportionate materials such as coins are used. In each of these lessons, the teacher is likely to introduce the following five notations in turn – place value chart, expanded notation, number in numerals, number in words and the tens and ones notation.
The question is what is an appropriate sequence? Should the place value chart be used first? Or the expanded notation? Give your reasons.
Place Value Chart
Expanded Notation
Words
Numerals
Tens and Ones Notation
Primary Mathematics

non-specialist
teachers
sustained focus
on thinking
focus
on fewer topics
the CPA approach
spiral curriculum
systematic variation

Through education, education level and consequently, social-economic status of families increase over one generation. Young parents have high aspirations for their children. Most send their young children to pre-schools and there number of school-age teenagers who are not in school is negligible although there is no compulsory education beyond Grades 1 – 6.
seven: high aspirations among parents

home
support
“Last Saturday, Mum and Dad went to school – to learn the model method so they can help me with schoolwork. But Mum said she did not get it.
In the end I had to teach her.”
Third Grader
Seminar for Parents on Problem Solving

TelokKurau Primary School, Singapore
“Sometimes my teacher gives me remedial classes. Some people go for Maths Olympiad training. My maths is not good.
I don’t go for that.”
Fifth Grader
school
support
“In my school, we have a fixed day for remedial lessons. I get a group of students to attend this regularly.”
Teacher

community
support
TelokKurau Primary School, Singapore

In grades one and two, students who did not attend pre-school, received a pull-out learning support for English and mathematics (LSP and LSM). They are taught in smaller groups of not more than ten while the regular class size is 30. Formal school hours end around 1:30 to free up time for a range of informal co-curricular and curricular activities.
eight: learning support and remedial

safety nets and interventions
non-specialist
teachers
Slides are available at
sustained focus
on thinking
focus
on fewer topics
the CPA approach
spiral curriculum
systematic variation
high parental expectations

Eight Things about Early Grades Mathematics in Singapore
Slides are available at

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NCTM 2010 Regional Conferences & Expositions Denver 2

by Jimmy Keng on Oct 08, 2010

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