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.
This post was written by Louise Adrianne Lopez.
