Mathematics is an excellent vehicle for the development and improvement of a person’s intellectual competence in logical reasoning, spatial visualisation, analysis and abstract thought. Students develop numeracy, reasoning, thinking skills, and problem solving skills through the learning and application of mathematics. These are valued not only in science and technology, but also in everyday living and in the workplace. The development of a highly skilled scientifically- and technologically-based manpower requires a strong grounding in mathematics. An emphasis on mathematics education will ensure that we have an increasingly competitive workforce to meet the challenges of the 21st century.
Mathematics is also a subject of enjoyment and excitement, which offers students opportunities for creative work and moments of enlightenment and joy. When ideas are discovered and insights gained, students are spurred to pursue mathematics beyond the classroom walls.
AIMS OF MATHEMATICS EDUCATION IN SCHOOLS
Mathematics education aims to enable students to:
(1) Acquire the necessary mathematical concepts and skills for everyday life, and for continuous learning in mathematics and related
(2) Develop the necessary process skills for the acquisition and application of mathematical concepts and skills.
(3) Develop the mathematical thinking and problem solving skills and apply these skills to formulate and solve problems.
(4) Recognise and use connections among mathematical ideas, and between mathematics and other disciplines.
(5) Develop positive attitudes towards mathematics.
(6) Make effective use of a variety of mathematical tools (including information and communication technology tools) in the learning and application of mathematics.
(7) Produce imaginative and creative work arising from mathematical ideas.
(8) Develop the abilities to reason logically, communicate mathematically, and learn cooperatively and independently.
This framework shows the underlying principles of an effective mathematics programme that is applicable to all levels, from the primary to A-levels. It sets the direction for the teaching, learning, and assessment of mathematics.
Mathematical problem solving is central to mathematics learning. It involves the acquisition and application of mathematics concepts and skills in a wide range of situations, including non-routine, open-ended and real-world problems.
The development of mathematical problem solving ability is dependent on five inter-related components, namely, Concepts, Skills, Processes, Attitudes and Metacognition.
Mathematical concepts cover numerical, algebraic, geometrical, statistical, probabilistic, and analytical concepts.
Students should develop and explore the mathematics ideas in depth, and see that mathematics is an integrated whole, not merely isolated piece of knowledge.
They should be given a variety of learning experiences to help them develop a deep understanding of mathematical concepts, and to make sense of various mathematical ideas, as well as their connections and applications, in order to participate actively in learning mathematics and to become more confident in exploring and applying mathematics. The use of manipulatives (concrete materials), practical work, and use of technological aids should be part of the learning experiences of the students.
Mathematical skills include procedural skills for numerical calculation, algebraic manipulation, spatial visualisation, data analysis, measurement, use of mathematical tools, and estimation.
The development of skill proficiencies in students is essential in the learning and application of mathematics. Although students should become competent in the various mathematical skills, over-emphasising procedural skills without understanding the underlying mathematical principles should be avoided.
Skill proficiencies include the ability to use technology confidently, where appropriate, for exploration and problem solving. It is important also to incorporate the use of thinking skills and heuristics in the process of developing skill proficiencies.
Mathematical processes refer to the knowledge skills (or process skills) involved in the process of acquiring and applying mathematical knowledge. This includes reasoning, communication and connections, thinking skills and heuristics, and application and modelling.
Reasoning, communication and connections Mathematical reasoning refers to the ability to analyse mathematical situations and construct logical arguments. It is a habit of mind that can be developed through the applications of mathematics in different contexts.
Communication refers to the ability to use mathematical language to express mathematical ideas and arguments precisely, concisely and logically. It helps students develop their own understanding of mathematics and sharpen their mathematical thinking.
Connections refer to the ability to see and make linkages among mathematical ideas, between mathematics and other subjects, and
between mathematics and everyday life. This helps students make sense of what they learn in mathematics.
Mathematical reasoning, communication and connections should pervade all levels of mathematics learning, from the primary to A-levels.
Thinking skills and heuristics Students should use various thinking skills and heuristics to help them solve mathematical problems. Thinking skills are skills that can be used in a thinking process, such as classifying, comparing, sequencing, analysing parts and wholes, identifying patterns and relationships, induction, deduction and spatial visualisation. Some examples of heuristics are listed
below and grouped in four categories according to how they are used:
• To give a representation, e.g. draw a diagram, make a list, use equations
• To make a calculated guess, e.g. guess and check, look for patterns, make suppositions
• To go through the process, e.g. act it out, work backwards, before-after
• To change the problem, e.g. restate the problem, simplify the problem, solve part of the problem
Application and modelling Application and modelling play a vital role in the development of mathematical understanding and competencies. It is important that students apply mathematical problem-solving skills and reasoning skills to tackle a variety of problems, including real-world problems.
Mathematical modelling is the process of formulating and improving a mathematical model to represent and solve real-world problems. Through mathematical modelling, students learn to use a variety of representations of data, and to select and apply appropriate mathematical methods and tools in solving real-world problems. The opportunity to deal with empirical data and use mathematical tools for data analysis should be part of the learning at all levels.
Attitudes refer to the affective aspects of mathematics learning such as:
• Beliefs about mathematics and its usefulness
• Interest and enjoyment in learning mathematics
• Appreciation of the beauty and power of mathematics
• Confidence in using mathematics
• Perseverance in solving a problem
Students’ attitudes towards mathematics are shaped by their learning experiences. Making the learning of mathematics fun, meaningful and relevant goes a long way to inculcating positive attitudes towards the subject. Care and attention should be given to the design of the learning activities, to build confidence in and develop appreciation for the subject.
Metacognition, or “thinking about thinking”, refers to the awareness of, and the ability to control one’s thinking processes, in articular the selection and use of problem-solving strategies. It includes monitoring of one’s own thinking, and self-regulation of learning.
The provision of metacognitive experience is necessary to help students develop their problem solving abilities. The following activities may be used to develop the metacognitive awareness of students and to enrich their metacognitive experience:
• Expose students to general problem solving skills, thinking skills and heuristics, and how these skills can be applied to solve problems.
• Encourage students to think aloud the strategies and methods they use to solve particular problems.
• Provide students with problems that require planning (before solving) and evaluation (after solving).
• Encourage students to seek alternative ways of solving the same problem and to check the appropriateness and reasonableness of the answer.
• Allow students to discuss how to solve a particular problem and to explain the different methods that they use for solving the problem.
PRIMARY MATHEMATICS CURRICULUM
Objectives of the primary mathematics curriculum primary 1 to primary 4, primary 5 and primary 6
(including Foundation Mathematics)
The objectives of the primary mathematics programme are to enable pupils to:
• Develop understanding of mathematical concepts:
• Recognise spatial relationships in two and three dimensions
• Recognise patterns and relationships in mathematics
• Use common systems of units
• Use mathematical language, symbols and diagrams to represent and communicate mathematical ideas
• Perform operations with
• Whole numbers
• Use geometrical instruments
• Perform simple algebraic manipulation
• Use calculators
• Develop ability to perform mental calculation
• Develop ability to perform estimation
• Develop ability to check reasonableness of results
• Present and interpret information in written, graphical, diagrammatic and tabular forms
• Use mathematical concepts learnt to solve problems
• Use appropriate heuristics to solve problems
• Apply mathematics to everyday life problems
• Think logically and derive conclusions deductively
• Develop an inquiring mind through investigative activities
• Enjoy learning mathematics through a variety of activities
USE OF CALCULATOR AND TECHNOLOGY
Calculators and other technology tools are tools for learning and doing mathematics.
The introduction of calculators at P5 and P6 reflects a shift to give more focus to processes such as problem solving skills. The rationale for introducing calculators at the upper primary levels is to:
(1) Achieve a better balance between the emphasis on computational skills and problem solving skills in teaching and learning and in assessment
(2) Widen the repertoire of teaching and learning approaches to include investigations and problems in authentic situations
(3) Help students, particularly those with difficulty learning mathematics, develop greater confidence in doing mathematics
The introduction of calculators would not take away the importance of mental and manual computations. These skills are still emphasised as students need to have good number sense and estimation skills to check the reasonableness of answers obtained using the calculator.
The use of scientific calculators is encouraged, although only the basic keys are required. Students can continue to use the same calculator when they are in secondary schools.
There are many softwares and open tools that teachers can use to enhance the teaching and learning of mathematics. Basic word processing and spreadsheets skills would support the different strategies and activities that students could engage in. Technology also has an inherent motivating and empowering effect on students. Teachers and students should harness technology in the teaching and learning of mathematics.