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This plenary lecture on high school mathematics calls for teaching of thinking through mathematics.
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LEAP Educators’ Conference 2011
Developing Thinking Through High School Mathematics
Yeap Ban HarMarshall Cavendish Institute
Singapore
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
