LEAP Educators’ Conference 2011

Developing Thinking Through High School Mathematics

Yeap Ban HarMarshall Cavendish Institute

Singapore

banhar@sg.marshallcavendish.com

Presentation is available on www.banhar.blogspot.com

Link to Keynote Lectures

First lecture raised the question: “Why is the achievement level eroded when students progress

from elementary to high school?”. The second lecture calls for changing mindset.

This lecture attempts to answer the question and suggest an alternate way of looking at school

mathematics.

Introduction

Mathematics is an excellent vehicle for the development and improvement of a person’s

intellectual competence…

Ministry of Education 2006

Problem Solving

National Test

National Test Grade 10 (Secondary 4)

A fly, F, starts at a point with position vector (i + 12j) cm and crawls across the surface with a velocity of (3i + 2j) cm s-1. At the instant the fly starts crawling, a spider, S, at the point with position vector (85i + 5j) cm, sets off across the surface with a velocity of (-5i + kj) cm s-1, where k is a constant. Given that the spider catches the fly, calculate the value of k.

Problem Solving

Textbook

Draw a 4-sided polygon.Cut it into triangles.

How many triangles can you get?

Example: Algebra

Factoring a quadratic expression.

Algebra Tiles

122 xx

122 xx

1x

2)1( x

12 2 xx

132 2 xx

132 2 xx)1)(12( xx

Example: Algebra

Completing the square.

162 xx

88162 xx

8)3( 2 x

232 xx

232 xx

232 xx

232 xx

2)2

3()

2

3(

23

22

2

x

xx

Example: Algebra

32

311

xSolve .

32

311

xSolve .

22

31

x

431 x

53 x

3

5x

Example: Coordinate Geometry

Find the distance between two points.

Problem: Find the distance between points A (2, 3) and B (10, 9).

Students working collaboratively on a problem to learn the coordinate geometry topic on distance between points. Traditionally, students were given a formulae to do this. Instead students had to conduct a mini-inquiry to use what they had previously learned.

Bina Bangsa School, Indonesia

Every group was able to use their previous learning (Pythagoras Theorem) to solve the main problem.

Bina Bangsa School, Indonesia

Later they were also asked to find points where the distance is a whole number.

Bina Bangsa School, Indonesia

Example: Circles

Learning definition and making sense of formulae.

ExampleTo engage pupils in the learning of mathematics.

Fuchun Primary School

Task 1 Find the longest distance between two points on a circle.

Fuchun Primary School, Singapore

Fuchun Primary School, Singapore

Fuchun Primary School, Singapore

Task 2 Measure circumference and diameter of four circles.

Fuchun Primary School, Singapore

Task 3 Draw a circle given that its circumference is 6 metres.

Fuchun Primary School, Singapore

Fuchun Primary School, Singapore

Fuchun Primary School, Singapore

Ratio 2.8 3.0 3.1 3.2 3.3 3.5 3.8Frequency 1 3 5 7 18 4 1

Example: Circles

Learning definition and making sense of formulae.

Area > 2r2

Area < 4r2

2r2 < Area < 4r2

Conclusion

Mathematics is not just for its content but also for the opportunities to develop

generic thinking skills.

Example: Trapezium

a

b c

d

h

Example: Trapezium

a

b c

d a

d

h

2

)( hdaA

International Benchmarks

Photo: Catholic High School (Primary)

Advanced

Intermediate

Low

High

Sing

apor

e

Inte

rnati

onal

USA

TIMSS 2007 Achievement

40 62

70 3115

88 6746

97 9255

Grade 8

Photo: Catholic High School (Primary)

High

Low

Medium

PATM

SVM

SCM

TIMSS 2007 Affect

60 4177

20 3419

20 254

Grade 8