Mathematics can be a complex subject. It is both fundamental and
complex—it is as old as humanity itself and it covers a wide array of topics
and concepts. At present, mathematics is known to branch out to various areas
of study, with the European Mathematical Society and Heidelberg Academy of
Sciences assigning up to 60 different categories to a multitude of
mathematics subjects including general mathematics, algebra, geometry,
statistics, and calculus. It is this—the fact that mathematics has
exponentially grown since it was first studied in the 6th century BC
and has kept abreast with the progress of human civilization ever since—that
makes it a challenging subject not only for students to learn but also for
teachers to teach.

And so it comes as no surprise that Mathematics is the number one
subject wherein Singapore students usually seek tuition. On a national level,
the mathematics performance of students is a measure of the effectiveness of a
country’s education system. On a global scale, a student’s competence in
mathematics is his weapon—in the words of Prime Minister Lee Hsien
Loong—against being “pushed around, shoved about, trampled upon”
as a Singaporean. This statement obviously resonates well among Singaporean
students as they repeatedly place
high on global education rankings. In international assessments such
as the Programme for International Student Assessment (PISA) and Trends in
International Mathematics and Science Study (TIMSS), Singapore consistently
ranks number one—leaving neighbouring Asian countries such as Taiwan, Hong Kong,
and Japan to compete over the remaining top spots, and the
UK and USA lagging far behind.

An assertion can be made then, at least for Singapore: a progressive
math education is fundamental in preparing its youth towards becoming
productive citizens of the 21st century. Yet to go forward
collectively, one must bear in mind the differences of individuals, especially
of learners, who will be making decisions for themselves and for the country in
the future. It should be understood that since students have unique learning
styles and skill sets, they have different starting points in learning, too.
Not everyone will learn at the same pace, not everyone will have the same
interests, and, more importantly, not everyone will have the same inclination
towards math. There will be those who would find its challenging nature
enjoyable, and there will be those who would simply find it difficult.

And so it should be the responsibility of adults—parents, teachers,
tutors, and partners in education—to make the young perceive math as an
engaging and vital subject. However, this task is not as easy as it sounds.
Parents themselves have repeatedly expressed difficulty in helping out their
children with their math homework—even those who have master’s and doctorate degrees.
As for teachers, it would not be as ideal to turn to them for extra help
because as recently reported, Singapore teachers average 46 hours a week in teaching,
and spend an additional 18 hours to take care of administrative tasks and other
academic matters. The number of their work period has slightly lessened by two
hours, yet it’s still a lot compared to the international average of 38 hours.
Given these circumstances, it is only natural that students and parents turn to
tuition. The challenge now falls on tutors as they come up with engaging and
encouraging ways to handhold and inspire confidence in students from different
grade levels who are struggling with math, and to make those
mathematically-inclined ones reach and maximize their full math potential.

As a guide, here is the mathematics syllabus of Primary 1 to Primary 6 students as
per the Ministry of Education. Do take note: Lessons in the P1-P4 syllabus are
common to all students, while lessons in the P5-P6 either continues (P5-P6 Standard
Mathematics) or revisits (P5-P6 Foundation Mathematics) basic concepts and
skills that were learned and developed during the P1-P4 levels. To aid parents
in terms of tutoring, this guide also lists some common challenges that
students face at each level and the learning behavior that they typically
exhibit at that age, as well as the adjustments that tutors usually make to
overcome those obstacles.

**LEVEL: PRIMARY 1**

**MAIN TOPICS: NUMBER AND
ALGEBRA; MEASUREMENT AND GEOMETRY; STATISTICS**

**CONCEPTS AND SKILLS:**

**A. NUMBER AND ALGEBRA**

**1. Numbers up to 100**

Counting to tell
the number of objects in a given set; Number notation, representations and
place values (tens, ones); Reading and writing numbers in numerals and in
words; Comparing the number of objects in two or more sets; Comparing and
ordering numbers; Patterns in number sequences; Ordinal numbers (first, second,
up to tenth) and symbols (1st, 2nd, 3rd, etc); Number bonds for numbers up to
10

**2. Addition and Subtraction**

Concepts of
addition and subtraction; Use of +, – and =; Relationship between addition and
subtraction; Adding more than two 1-digit numbers; Adding and subtracting
within 100; Adding and subtracting using algorithms; Solving 1-step word
problems involving addition and subtraction within 20; Mental calculation
involving addition and subtraction within 20, of a 2-digit number and ones
without renaming, and of a 2-digit number and tens

**3. Multiplication and Division**

Concepts of
multiplication and division; Use of x; Multiplying within 40; Dividing within
20; Solving 1-step word problems involving multiplication and division with
pictorial representation

**4. Money**

Counting amount of
money—in cents up to $1, in dollars up to $100; Solving 1-step word problems
involving addition and subtraction of money in dollars only (or in cents only)

**B. MEASUREMENT AND GEOMETRY**

**1. Length**

1.1 Measuring and
comparing the length of objects in non-standard units

**2. Time**

2.1 Telling time to
the hour/half hour

**3. 2D Shapes**

Identifying,
naming, describing and classifying 2D shapes: rectangle, square, circle,
triangle; Making/completing patterns with 2D shapes according to one or two of
the following attributes: size, shape, colour, and orientation

**C. STATISTICS**

**1. Picture Graphs**

1.1 Reading and
interpreting data from picture graphs

**Common Challenges:**

At around 6 to 7
years old,** **P1 is the formal
introduction of Singapore children to 10 years worth of education that will
build the foundation of their mathematical knowledge and skills. At this age,
the child is already approaching the end of what is called the Preoperational Stage of cognitive development (by
Swiss psychologist Jean Piaget), which happens from 2 to 7 years old, and is
usually the stage wherein the child learns to use language and develop abstract
thought. What this means for the parent is that the child already has a grasp
of symbols and how to use words and pictures to represent objects. At this
point, the parent should let his child develop a natural love for math by
connecting everyday objects and activities with basic math concepts, such as
counting items in the grocery cart, pointing out display signages with numbers printed
on them, and introducing puzzles and puzzle games. Similarly, parents would see
tutors use these techniques as they try to make mathematics as fun, light, and
relatable as possible for children who are just starting to learn its
fundamentals. At this stage, a parent and/or tutor may typically struggle
getting a child to focus on a single task for more than 15 minutes, as 6- to 7-year-olds normally have the attention span of
10-15 minutes if they find an activity easy and stimulating, and
5-10 minutes if they find it difficult or uninteresting.

**LEVEL: PRIMARY 2**

**MAIN TOPICS: NUMBER AND
ALGEBRA; MEASUREMENT AND GEOMETRY; STATISTICS **

**CONCEPTS AND SKILLS:**

**A. NUMBER AND ALGEBRA**

**1.
Numbers up to 100**

Counting
in tens/hundreds; Number notation, representations and place values (hundreds,
tens, ones); Reading and writing numbers in numerals and in words; Comparing
and ordering numbers; Patterns in number sequences; Odd and even numbers

**2.
Addition and Subtraction**

Addition
and subtraction algorithms (up to 3 digits); Solving up to 2-step word problems
involving addition and subtraction; Mental calculation involving addition and
subtraction of a 3-digit number and ones/tens/ hundreds

**3.
Multiplication and Division**

Multiplication
tables of 2, 3, 4, 5 and 10 use of ÷; Relationship between multiplication and
division multiplying and dividing within the multiplication tables; Solving
1-step word problems involving multiplication and division within the
multiplication tables; Mental calculation involving multiplication and division
within the multiplication tables of 2, 3, 4, 5 and 10

**B.
FRACTIONS**

**1. Fraction of a Whole**

Fraction
as part of a whole; Notation and representations of fractions; Comparing and
ordering fractions with denominators of given fractions not exceeding 12

**2. Addition and Subtraction**

Adding
and subtracting like fractions within one whole with denominators of given
fractions not exceeding 12

**C. MONEY**

**1. Money**

Counting
money in dollars and cents; Reading and writing money in decimal notation;
Comparing two or three amounts of money; Converting an amount of money in
decimal notation to cents only, and vice versa; Solving word problems involving
money in dollars only (or in cents only)

**D. MEASUREMENT**

**1. Length, Mass and Volume**

Measuring—length
in metres/centimetres, mass in kilograms/grams, volume of liquid in litres;
Measuring and drawing a line segment to the nearest cm; Using appropriate units
of measurement and their abbreviations cm, m, g, kg; Comparing and
ordering—lengths, masses, volumes; Solving word problems involving length/mass/
volume

**2. Time**

Telling time to 5 minutes; Use of ‘a.m.’ and ‘p.m.’; Use of
abbreviations h and min; Drawing hands on the clock face to show time; Duration
of 1 hour/half hour

**E.
GEOMETRY**

**1. 2D Shapes**

Identifying,
naming and describing 2D shapes—semicircle, quarter circle; Identifying the
basic shapes that make up a given figure; Forming different 2D figures with
rectangle, square, triangle, semicircle, quarter circle; Copying figures on dot
grid or square grid

**2. 3D Shapes**

Identifying, naming, describing and classifying 3D
shapes—cube, cuboid, cone, cylinder, sphere

**F. STATISTICS**

**1. Picture Graphs with Scales**

Reading and interpreting data from picture graphs with
scales; solving 1-step problems using data from picture graphs

**COMMON CHALLENGES:**

At P2, when the child is aged 7-8, he enters what Piaget
called the concrete operational period. At this stage, the child starts to
learn logical reasoning and thought organization. These steadily develop until
the child reaches the age of 12, or until he is in P6, by which he is supposed
to have gained an understanding of how mental operations work (how to solve a
problem and think of possible outcomes and similar mental operations). Parents
and tutors at this point may start to introduce problem solving and predicting
outcomes, although at a slow pace or depending on the child’s pace, as learning
styles and paces vary from child to child. From this stage until P5, the
child’s development in terms of mathematical knowledge and skills is supposed
to progress as each academic year passes until he is ready for his Primary
School Leaving Exam or PSLE at the end of P6.

**LEVEL: PRIMARY 3**

**MAIN TOPICS: NUMBER AND
ALGEBRA; MEASUREMENT AND GEOMETRY; STATISTICS**

**CONCEPTS AND SKILLS:**

**A. NUMBER AND ALGEBRA**

**1. Numbers up to 10 000**

Counting in hundreds/thousands; Number notation,
representations and place values (thousands, hundreds, tens, ones): Reading and
writing numbers in numerals and in words; Comparing and ordering numbers;
Patterns in number sequences

**2. Addition and Subtraction**

Addition and subtraction algorithms (up to 4 digits); Solving
up to 2-step word problems involving addition and subtraction; Mental
calculation involving addition and subtraction of two 2-digit number

**3. Multiplication and
Division**

Multiplication tables of 6, 7, 8 and 9; Multiplying and
dividing within the multiplication tables; Division with remainder;
Multiplication and division algorithms (up to 3 digits by 1 digit); Solving up
to 2-step word problems involving the 4 operations; Mental calculation
involving multiplication and division within the multiplication tables

**B. FRACTION**

**1. Equivalent Fractions**

Equivalent fractions; Expressing a fraction in its simplest
form; Comparing and ordering unlike fractions with denominators of given fractions
not exceeding 12; Writing the equivalent fraction of a fraction given the
denominator or the numerator

**2. Addition and Subtraction**

Adding and subtracting two related fractions within one whole
with denominators of given fractions not exceeding 12

**C. MONEY**

**1. Money**

Adding and subtracting money in decimal notation; Solving
word problems involving addition and subtraction of money in decimal notation

**D. MEASUREMENT**

**1. Length, Mass and Volume**

Measuring length in kilometres (km), volume of liquid in millilitres
(ml); Measuring length/mass/volume (of liquid) in compound units; Converting a
measurement in compound units to the smaller unit, and vice versa--kilometres
and metres, metres and centimetres, kilograms and grams, litres and milliliters
(numbers involved should be within easy manipulation); Solving word problems
involving length/mass/ volume/capacity excluding fractions and compound units

**2. Time**

Telling time to the minute; Use of ‘past’ and ‘to’ to tell
time; Measuring time in hours and minutes; Converting time in hours and minutes
to minutes only, and vice versa; Finding the starting time, finishing time or
duration given the other two quantities; Solving problems involving time in
hours and minutes

**3. Area and Perimeter**

Concepts of area and perimeter of a plane figure; Measuring
area in square units, cm2 and m2, excluding conversion between cm2 and m2;
Perimeter of rectilinear figure, rectangle, square; Area of rectangle/square

**E. GEOMETRY**

**1. Angles**

Concepts of angle; Right angles, angles greater than/smaller
than a right angle

**2. Perpendicular and Parallel
Lines**

Perpendicular and parallel lines; Draw perpendicular and
parallel lines on square grid

**F. DATA REPRESENTATION AND
INTERPRETATION**

**1. Bar Graphs**

Reading and interpreting data from bar graphs; Using
different scales on axis; Solving 1-step problems using data from bar graphs

**LEVEL: PRIMARY 4**

**MAIN TOPICS: NUMBER AND
ALGEBRA; MEASUREMENT AND GEOMETRY; STATISTICS**

**CONCEPTS AND SKILLS:**

**A. NUMBER AND ALGEBRA**

**1. Numbers up to 100 000**

Number notation, representations and place values (ten
thousands, thousands, hundreds, tens, ones); Reading and writing numbers in
numerals and in words; Comparing and ordering numbers; Patterns in number
sequences; Rounding numbers to the nearest 10, 100 or 1000

**2. Factors and Multiples**

Factors, multiples and their relationship; Determining if a
1-digit number is a factor of a given number within 100; Finding the common
factors of two given numbers; Determining if a number is a multiple of a given
1-digit number; Finding the common multiples of two given 1-digit numbers

**3. Four Operations**

Multiplication algorithm up to 4 digits by 1 digit and up to
3 digits by 2 digits; Division algorithm (up to 4 digits by 1 digit); Solving
up to 3-step word problems involving the 4 operations

**B. FRACTIONS**

**1. Mixed Numbers and Improper
Fractions**

Mixed numbers, improper fractions and their relationships

**2. Fraction of a Set of
Objects**

Fraction as part of a set of objects

**3. Addition and Subtraction**

Adding and subtracting fractions with denominators of given
fractions not exceeding 12 and not more than two different denominators;
Solving up to 2-step word problems involving addition and subtraction

**C. DECIMALS**

**1. Decimals up to 3 decimal
places**

Notation, representations and place values (tenths,
hundredths, thousandths); Comparing and ordering decimals; Dividing a whole
number by a whole number with quotient as a decimal; Converting decimals to
fractions; Converting fractions to decimals when the denominator is a factor of
10 or 100; Rounding decimals to the nearest whole number, 1 decimal place, 2
decimal places

**2. Addition and Subtraction**

Adding and subtracting decimals (up to 2 decimal places)

**3. Multiplication and
Division**

Multiplying and dividing decimals (up to 2 decimal places) by
a 1-digit whole number; Solving up to 2-step word problems involving the 4
operations; Rounding off answers to a specified degree of accuracy

**D. MEASUREMENT**

**1. Time**

Measuring time in seconds; 24-hour clock; Solving problems
involving time in 24-hour clock

**2. Area and Perimeter**

Finding one dimension of a rectangle given the other
dimension and its area/perimeter; Finding the length of one side of a square
given its area/perimeter; Finding the area of figures made up of rectangles and
squares

**3. Angles**

Using notation such as Ð ABC and Ð* **a* to name angles; Measuring angles in degrees; Drawing an
angle of given size; Relating quarter, half and complete turns to angles in
degrees; 8-point compass

**4. Rectangle and Square**

Properties of rectangle and square, excluding diagonal
properties; Drawing rectangles and squares on square grid

**5. Line Symmetry**

Identifying symmetric figures; Determining whether a straight
line is a line of symmetry of a symmetric figure; Completing a symmetric figure
with respect to a given line of symmetry on square grid

**E. STATISTICS**

**1. Tables and Line Graphs**

Completing a table from given data; Reading and interpreting
data from tables/line graphs; Solve 1-step problems using data from
tables/graphs

**LEVEL: PRIMARY 5**

**MAIN TOPICS: NUMBER AND
ALGEBRA; MEASUREMENT AND GEOMETRY; STATISTICS**

**CONCEPTS AND SKILLS:**

**A. NUMBER AND ALGEBRA**

**1. Numbers up to 10 million**

Reading and writing numbers in numerals and in words

**2. Four Operations**

Multiplying and
dividing by 10, 100, 1000 and their multiples without calculator; Order of
operations without calculator; Use of brackets without calculator; Solving word
problems involving the 4 operations

**B. FRACTIONS**

**1. Fraction and Division**

Dividing a whole number by a whole number with quotient as a
fraction; Converting fractions to decimals

**2. Four Operations**

Adding and subtracting mixed numbers; Multiplying a
proper/improper fraction and a whole number without calculator; Multiplying a
proper fraction and a proper/ improper fractions without calculator;
Multiplying two improper fractions; Multiplying a mixed number and a whole
number; Solving word problems involving addition, subtraction and
multiplication

**C. DECIMALS**

**1. Four Operations**

Multiplying and dividing decimals (up to 3 decimal places) by
10, 100, 1000 and their multiples without calculator; Converting a measurement
from a smaller unit to a larger unit in decimal form, and vice versa—kilometres
and metres, metres and centimetres, kilograms and grams, litres and
millilitres; Solving word problems involving the 4 operations

**D. PERCENTAGE**

**1. Percentage**

Expressing a part of a whole as a percentage; Use of %;
Finding a percentage part of a whole; Finding discount, GST and annual
interest; Solving up to 2-step word problems involving percentage

**E. RATIO**

**1. Ratio**

Notation, representations and interpretation of a:b and
a:b:c, where a, b and c are whole numbers, excluding ratios involving fractions
and decimals; Equivalent ratios; Dividing a quantity in a given ratio;
Expressing a ratio in its simplest form; Finding the ratio of two or three
given quantities; Finding the missing term in a pair of equivalent ratios;
Solving up to 2-step word problems involving ratio

**F. RATE AND SPEED**

**1. Rate**

Rate as the amount of a quantity per unit of another
quantity; Finding rate, total amount or number of units given the other two
quantities; Solving word problems involving rate

**G. AREA AND VOLUME**

**1. Area of Triangle**

Concepts of base and height of a triangle; Area of triangle;
Finding the area of figures made up of rectangles, squares and triangles

**2. Volume of Cube and Cuboid**

Building solids with unit cubes; Measuring volume in cubic
units, cm 3 and m 3 , excluding conversion between cm 3 and m 3; Drawing cubes
and cuboids on isometric grid; Volume of a cube/cuboid; Finding the volume of
liquid in a rectangular tank; Relationship between l (or ml) and cm 3

**H. GEOMETRY**

**1. Angles**

Angles on a straight line; Angles at a point; Vertically
opposite angles; Finding unknown angles

**2. Triangle**

Properties of isosceles triangle, equilateral triangle,
right-angled triangle; Angle sum of a triangle; Finding unknown angles in
geometric figures without additional construction of lines

**3. Parallelogram, Rhombus and Trapezium**

Properties of parallelogram, rhombus, trapezium; Finding
unknown angles without additional construction of lines

**I. STATISTICS**

**1. Average of a Set of Data**

Average as ‘total value ¸ number of data’; Relationship
between average, total value and number of data

**LEVEL: PRIMARY 6**

**MAIN TOPICS: NUMBER AND
ALGEBRA; MEASUREMENT AND GEOMETRY; STATISTICS**

**CONCEPTS AND SKILLS:**

**A. FRACTIONS**

**1. Four Operations**

Division of a whole number/proper fraction by a proper
fraction without using calculators excluding division of an improper
fraction/mixed number by a proper fraction and division by an improper
fraction/mixed number

**B.** **PERCENTAGE **

**1. Percentage**

Finding the whole given a part and the percentage; Finding
percentage increase/decrease; Solving word problems involving percentage

**C. RATIO**

**1. Ratio**

Expressing one quantity as a fraction of another, given their
ratio, and vice versa; Finding how many times one quantity is as large as
another, given their ratio, and vice versa; Expressing one quantity as a
fraction of another given the two quantities; Finding the whole/ one part when
a whole is divided into parts in a given ratio; Solving word problems involving
2 pairs of ratios

**D. SPEED**

**1. Distance, time and speed**

Concepts of speed and average speed; Relationship between
distance, time and speed; Calculation of speed, distance or time given the
other two quantities; Writing speed in different units such as km/h, m/min, m/s
and cm/s; Solving up to 3-step word problems involving speed and average speed

**E. MEASUREMENT**

**1. Area and circumference of
circle**

Use of formulae to calculate the area and circumference of a
circle; Finding the area and perimeter of semicircle (half circle) and quarter
circle; Solving word problems involving area and perimeter

**2. Area and perimeter of
composite figure**

Finding the area and perimeter of a figure made up of some of
the following shapes: square, rectangle, triangle, semicircle and quarter
circle

**3. Volume of cube and cuboid**

Finding one
dimension of a cuboid given its volume and the other dimensions; Finding the
length of one edge of a cube given its volume; Finding the height of a cuboid
given its volume and base area; Finding the area of a face of a cuboid given
its volume and one dimension; Use of the symbols √ and ^{3}√;
Solving word problems involving volume of a cube/ cuboid

**F. GEOMETRY**

**1. Geometrical Figures**

Finding unknown angles in geometrical figures involving
square, rectangle, parallelogram, rhombus, trapezium and triangle

**2. Nets**

2-D representation of prism and pyramid; Identifying nets of
cube, cuboid, prism, pyramid; Identifying the solid which can be formed by a
given net; Making 3-D solids from given nets

**G. DATA ANALYSIS**

**1. Pie Charts**

Reading and interpreting pie charts; Solving 1-step problems
using information presented in pie charts

**H. ALGEBRA**

**1. Algebraic expressions in
one variable**

Representation of an unknown number using a letter; Simple
algebraic expressions; Simplification of algebraic expressions; Evaluation of
simple algebraic expressions by substitution; Solving word problems involving
algebraic expressions

**COMMON CHALLENGES:**

At this stage, when the child is about 11-12 years old, he is about to face the dreaded PSLE which is known for its difficult exam items. However, the child at this point has been equipped with all that he needs for the exam given six years of formative mathematics education from P1 to P6. The child has also entered what is called the Formal Operational Stage according to Piaget, which means that cognitively, the child, or the adolescent, has already learned to think abstractly and reason about hypothetical problems. He also begins to think deductively—a top-down reasoning that utilizes general ideas down to specific information, which is usually utilized in solving math problems. At this point, the parent and the tutor have equal challenges of helping the child along as he copes with all the physiological and psychological changes of puberty as he tries to succeed not only in mathematics but in his PSLE, which will ultimately dictate which secondary school he goes to or what academic plans he should pursue next.

Given the outlined challenges, it is every parent’s
responsibility to support the child mentally and emotionally in his scholastic
pursuit. With all the available options out there—from tuition centres to
one-to-one private tutors to online tutorials—parents must make smart decisions
that would help their children along, especially in terms of math. And that is
what iMath is here for: to help parents assist their children towards math
mastery to ensure academic success through the aid of a fun and supportive
community of math experts and enthusiasts—right at the comfort of their own
home.