This document provides an overview of a seminar on problem-solving heuristics for primary mathematics taught by Dr. Yeap Ban Har. It discusses Dr. Yeap's background and qualifications, as well as the learning outcomes of the seminar, which are for participants to learn about problem-solving heuristics suitable for primary school and how to teach problem-solving lessons. The document also provides several examples of mathematics word problems that could be used to teach problem-solving skills to primary students.

1. singapore international school of bangkok
problem-solving heuristics
for primary mathematics
Yeap Ban Har | yeapbanhar@gmail.com | www.banhar.blogspot.com
Learning Outcomes
Participants are expected to learn about problem-solving heuristics suitable for primary school
and how to teach problem-solving type lessons. The model method will be given some
attention.
Presenter
Dr. Yeap Ban Har is an established name in mathematics education and teacher professional
development. He spent ten years at Singapore's National Institute of Education training preservice and in-service educators. A leading educator, speaker and trainer, Ban Har has authored
dozens of textbooks, math readers and assorted titles for teachers. He has been a keynote
speaker at international conferences, and is currently the Principal of the Marshall Cavendish
Institute. He is also Director of Curriculum and Professional Development at Pathlight School, a
K-10 school for students with autism. Ban Har’s interest is in early numeracy, problem solving
and teacher professional development. In Thailand, he has given mathematics education
seminars at Kasetsart University, Srinakharinwirot University, Chiangmai University, Rajabhat
University Phuket, Rajabhat University Mahasarakham, and Khon Kaen University. He is
teaching graduate courses for masters and doctoral students as a visiting professor this coming
academic year. He will also conduct a seminar in Bangkok jointly organized by Shinglee
Publishers on 26 November 2013.
coursebook
1|Problem Solving

2. introduction
The Singapore approach to teaching and learning mathematics was the
result of trying to find a way to help Singapore students who were
mostly not performing well in the 1970’s.
In 1980, only 58 percent of first-grade students completed secondary school. By 2000, 93 percent of
first-grade students completed secondary school. Achievement, as measured by national examination,
also improved. In 1981, 40 percent of the first-graders would graduate 10th grade with passes in three
subjects. In 1991, the proportion increased to 65 percent. In 2010, the proportion was 88 percent.
Providing students with differentiated curriculum and differentiated examination has arguably resulted
in “a very low attrition rate and a very high average achievement.”
| Yeap 2012
Improving the Education for All: Curriculum Development and Implementation in Singapore
American Institutes for Research
In the first national examination conducted in 1960, a total of 30,615 students sat for the Primary School
Leaving Examination (PSLE) and 45 percent of them passed. In 2011, a total of 45,261 students sat for
the high-stakes examination and 97 percent passed. Only 2.6 percent did not meet the proficiency
required for secondary schooling. Since the 1990s, Singapore has always done well in various
international studies in literacy, science, and mathematics. It is always among the top-performing
countries in Progress in International Reading Literacy Study (PIRLS), Trends in International
Mathematics and Science Study (TIMSS), and Program for International Student Assessment (PISA).
More important than the ranking is that about 12 percent of 15-year-old students in Singapore were
performing at a high level in reading, mathematics, and science in PISA 2009. This compares well with
the Organization of Economic Cooperation and Development (OECD) average, which was 4 percent.
High-performing OECD economies such as New Zealand, Finland, Japan, and Australia had between 8
percent and 10 percent of students reaching the same level. In Shanghai, it was almost 15 percent. The
proportion of students who reach the highest levels was about 15 percent in reading (almost 20 percent
in Shanghai with the OECD average at 8 percent); about 35 percent in mathematics (50 percent in
Shanghai with the OECD average at 13 percent); and 20 percent in science (almost 25 percent in
Shanghai with the OECD at being 9 percent). Similarly, in TIMSS, the proportion of students who reached
the advanced international benchmark has been consistently high. In TIMSS 2007, 36 percent of Grade 4
and 32 percent of Grade 8 students reached this level in science (international median were 7 percent
and 3 percent, respectively), and 41 percent of Grade 4 and 40 percent of Grade 8 students reached this
level in mathematics (international median was 5 percent and 2 percent, respectively). These results
were in great contrast to Singapore’s rank of 16th out of 26 participating countries in the Second
International Science Study in 1982.
| Yeap 2012
Improving the Education for All: Curriculum Development and Implementation in Singapore
American Institutes for Research
2|Problem Solving

3. The CPA Approach as well as the Spiral Approach are fundamental to
teaching mathematics in Singapore schools. The national standards,
called syllabus in Singapore, is designed based on Bruner’s idea of spiral
curriculum. Textbooks are written based on and teachers are trained to
use the CPA Approach, based on Bruner’s ideas of representations.
“A curriculum as it develops should revisit this basic ideas repeatedly, building upon them until
the student has grasped the full formal apparatus that goes with them”.
| Bruner 1960
“I was struck by the fact that successful efforts to teach highly structured bodies of knowledge
like mathematics, physical sciences, and even the field of history often took the form of
metaphoric spiral in which at some simple level a set of ideas or operations were introduced in
a rather intuitive way and, once mastered in that spirit, were then revisited and reconstrued in
a more formal or operational way, then being connected with other knowledge, the mastery at
this stage then being carried one step higher to a new level of formal or operational rigour and
to a broader level of abstraction and comprehensiveness. The end stage of this process was
eventual mastery of the connexity and structure of a large body of knowledge.”
| Bruner 1975
spiral approach
concrete-pictorial-abstract
Bruner's constructivist theory suggests it is effective when faced with new material to follow a
progression from enactive to iconic to symbolic representation; this holds true even for adult
learners.
| Bruner 1966
3|Problem Solving

4. problem-solving
approach
| Singapore 2013
…able to pose and solve problems …
| Australia 2012
Make sense of problems and preserve in solving them.
| Common Core State Standards USA 2010
.. can solve problems by applying their mathematics to a
variety of problems with increasing sophistication,
including in unfamiliar contexts and to model real-life
scenarios …
| United Kingdom 2012
4|Problem Solving

5. |Example 1
Find the value of 6 + 5 + 7.
5|Problem Solving

6. |Example 2
Make 4 equal parts.
6|Problem Solving

7. |Example 3
Use one set of digit tiles 1 to 9 to make two 3-digit numbers.
Make the difference the least possible.
7|Problem Solving

8. |Example 4
There are 1111 ducks in a farm.
There are 299 fewer ducks than chickens.
How many chickens are there?
How many chickens will be left if 418 chickens are sold?
Source | New Syllabus Primary Mathematics
|Example 5
At first, Mario had 8 coins more than Nina.
Then, Nina gave Mario 3 coins.
Who had more coins after Nina gave Mario the coins? How many more?
Source | Kong Hwa School (Singapore) Lesson Study
8|Problem Solving

9. |What if …?
At first, Nina had 8 coins more than Mario.
Then, Nina gave Mario 3 coins.
Who had more coins after Nina gave Mario the coins? How many more?
|Example 6
There were 94 children in a group.
2
3
of the boys and of the girls in a group chose swimming as their after-school activity.
5
7
In all, 39 children chose swimming as their after-school activity.
9|Problem Solving

10. |Example 7
10 coins are either 2 baht or 5 baht.
Their total value is 41 baht.
How many of the coins are 2-baht coins?
|Example 8
Draw a polygon with four dots on its sides.
Investigate its area.
10 | P r o b l e m S o l v i n g

11. 11 | P r o b l e m S o l v i n g

12. Additional Examples
|Example 9
37 + 29
|Example 10
1 3
3 
2 4
|Example 11
Multiplication Facts




18 + 6 = 
Use 3 × 6 = 18 to get 4 × 6.







12 | P r o b l e m S o l v i n g

13. |Example 12
Cindy has 6 more candies than David.
Cindy has 15 candies.
How many candies do they both have altogether?
|Example 13
Feliz had twice as much money as Ginny.
After Feliz gave Ginny $12, they both had the same amount of money.
How much did they both have altogether?
|Example 14
Elvi made some paper cranes on Monday.
On Tuesday she made 3 more paper cranes than on Monday.
Subsequently, she made 3 more paper cranes than the day before.
Elvi said that she had made 50 paper cranes by Saturday the same week.
Is that possible?
|Example 15
Han used ¼ of his savings to buy a gift for his father and ½ of the remainder on a book
which cost $12. What did the gift cost?
bar model method
visualization
|Example 16
The number of boys to the number of girls in a group was in the ratio 2 : 3.
After ¼ of the boys and 69 girls left the group, there were 51 more girls than boys in the
group.
13 | P r o b l e m S o l v i n g

14. |Example 17
The figure, not drawn to scale, shows an isosceles triangle XYZ, and a parallelogram UVWX.
Given that UY and WZ are straight lines, XY = XZ and the sum of  YZX and  XUV is 162o, find
 UVW.
Source|Keming Primary School, Singapore
|What if …?
The figure, not drawn to scale, shows an equilateral triangle XYZ, and a parallelogram UVWX.
Given that UY and WZ are straight lines, find  UVW.
14 | P r o b l e m S o l v i n g

15. |Example 18
A figure is formed using a length of wire.
The figure, shown below, consists of 4 squares.
X
Y
The length of the straight line XY is 15 cm.
Find the length of the wire used.
|Example 19
A group of boys share some sweets in a box.
When they tried to take 11 sweets each, one boy got 6 sweets.
When they took 8 sweets each, there were 25 sweets leftover.
Find the number of sweets in the box.
singapore textbooks
Singapore textbooks are designed for a problem-solving approach. They
are also designed for differentiated instruction. A typical mathematics
lesson is in a three-part format – anchor task, guided practice and
independent practice.
15 | P r o b l e m S o l v i n g