nctm 2010 regional conferences & expositions denver 2

NCTM Regional Conference & Exposition Denver Colorado

8 – 9 October 2010Session 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

Dr Yeap Ban HarMarshall Cavendish

InstituteSingapore

[email protected]

slides are available atwww.banhar.blogspot.com

Session 165

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





507

Denver, Colorado

demanding program

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.

Example of Basic Skill Item

Example of Direct Application

Which of the following numbers is the largest?(1)6.59(2)6.95(3)6.509(4)6.905

Mr Tan rented a car for 3 days. He was charged $155 per day and 60 cents for every km that he travelled. He paid $767.40. What was the total distance that he travelled for the 3 days?

A movie started at 11.45 a.m. The movie was 2 hours 25 minutes long. What time did the movie end?

Prawns are sold at $1.35 per 100 g at a market. What is the price of 1.5 kg of prawns?

In the figure below, ABCD is a square, AED is an equilateral triangle and BFD is a straight line. Find <AFD

A B

CD

EF

Source:Singapore Examinations and Assessment Board PSLE Mathematics 2004 – 2008 and Specimen Paper 2009

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?

SourceRaffles 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?

Number of $5 notes

Number of $2 notes

4 $20

$76

$56

$56 ÷ 7 = 8

8

8She 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 MPNAnswer: 150o

SourceSingapore Examinations and Assessment Board PSLE Mathematics 2009





507

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.

Taiwan

Singapore

Hong Kong

S Korea

Att

itude

Ach

ievem

en

t

37 45

33 40

60 40

47 31

Gra

de 8

Japan

England

International

Hungary

30 26

30 10

40 8

54 2

SourceTIMSS 2007 International Mathematics Report, Boston College

The figures refer to percentage of students with high PATM (for Attitude) and in the Advanced International Benchmark (for Achievement).

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)

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.

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

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.

SourcePensar sin Limites 2o Basico

Keys Grade School, Manila

Springhurst Elementary School, New York

Dagoberto Godoy School, Santago Princess Elizabeth Primary School,

Singapore

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.

16 17 18 19 20


sustained focus

on thinking

Dagoberto Godoy School, Santiago, Chile

two: small number of

topicsThis 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.

SourceEarlybird Kindergarten Mathematics Book B

PCF Kindergarten Telok Blangah, Singapore

One duck is big. Six ducklings are small.


SourceMy Pals Are Here! Mathematics Primary

1

sustained focus

on thinking

Princess Elizabeth Primary School, Singapore

focus on fewer

topics

SourceTIMSS 2007 International Mathematics Report, Boston College

Circle Graphs are taught in Grade 6 in Singapore.

three: CPA approach

SourceEarlybird Kindergarten Mathematics – Kindergarten Textbook in Singapore

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.

illustrationequal parts

SourceProfessional Development Course for Singapore Teachers

Is the rectangle cut into equal parts?

bruner’s theoryconcrete

Sourcemathz4kidz Learning Centre, Penang, MalaysiaA Lesson from Earlybird Kindergarten Mathematics

concreteexperiences

Sourcemathz4kidz Learning Centre, Penang, MalaysiaA 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.

SourceFuchun Primary School, SingaporeResearch Lesson to Develop Formula for Circumference of Circles

use of visuals

SourceMath in Focus Grade 4

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 2

12

2

5

2

12

The use of visuals in learning multiplication facts.

SourcePrimary Mathematics Grade 3

sustained focus

on thinking

Da Qiao Primary School, Singapore

focus on fewer

topicsthe CPA approach

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.

SourceMy Pals Are Here! Mathematics Primary 1

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.


My Pals Are Here! Mathematics Primary 1

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.


five: non-specialist teachers

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.

textbooks

SourceMy Pals Are Here! Mathematics Primary 1

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

six: systematic variation of

tasks

Tasks are presented in a systematic sequence to provide students with both mathematical as well as perceptual variability.

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

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

Numerals WordsTens and Ones Notation

SourceProfessional Development Materials

Marshall Cavendish Institute, Singapore

SourcePrimary 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

Numerals WordsTens and Ones Notation Primary Mathematics

sustained focus

on thinking


focus on fewer

topicsthe CPA approachspiral

curriculumsystematic variation

non-specialist teachers

seven: high aspirations

among parents

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.

homesupport

Seminar for Parents on Problem Solving

“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

schoolsupport

Telok Kurau 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

“In my school, we have a fixed day for remedial lessons. I get a group of students to attend this regularly.”

Teacher

communitysupport

Telok Kurau Primary School, Singapore

eight: learning

support and remedial

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.

sustained focus

on thinkingfocus on fewer

topicsthe CPA approachspiral

curriculumsystematic variation

non-specialist teachers

safety nets and

interventions

high parental expectations

Slides are available at

www.banhar.blogspot.com

Eight Things about Early Grades

Mathematics in Singapore

Slides are available at

www.banhar.blogspot.com
