Singapore students have an unparalleled record of achieved world-leading score in international standardized education tests in maths.
Singapore students have consistently ranked as the top in the Trends in International Mathematics and Science Study (TIMSS), which measures the mathematics and science performance of school students for age 13-14 years old (Grade 8) and 9-10 years old (Grade 4) respectively.
Trends in International Mathematics and Science Study (TIMSS)
TIMSS Ranking and Scores for Grade 8 Math Performance
Programme for International Student Assessment (PISA)
Singapore students have also consistently achieved the top position in the Programme for International Student Assessment (PISA), which measures 15-year old school students' scholastic performance in math, science and reading.
PISA Ranking and Scores for Math Performance
The Singapore MOE math program is not about students rote learning math concepts and principles to score in academic tests. It is about equipping students with the confidence and competencies to solve real-world problems. In the first ever global PISA Collaborative Problem Solving Tests in 2015 that involved over 125,000 students across 52 economies, Singaporean students topped the global ranking. This measures students' ability to solve problems collaboratively, negotiate and come to agreements. Singapore had the highest proportion of top performers, with more than 20% of students here achieving the highest level of proficiency (Level 4) in collaborative problem solving, as compared to the OECD average of 8%.
PISA Ranking and Scores for Collaborative Problem Solving
US parents and students often associate the Singapore Ministry of Education (MOE) primary school math program with the Concrete - Pictorial - Abstract (CPA) approach. In this approach, students are introduced to a new math concept via Concrete objects such as everyday items. When they are familiar with the math concept using concrete objects, they then progress to using Pictorial representations such as models, in place of the concrete objects, to solve the problem based on the specific math concept. Finally, when the students have gained mastery of the concept, they progress to using the Abstract of the pictorial, such as via numbers, to apply the concept and solve the problem.
However, the CPA approach is just a very small part of the Singapore MOE’s deep and rigorous approach to its primary school math program, one which has consistently educated Singapore primary school children to become world-class math learners.
Nature of Mathematics
The Singapore Ministry of Education (MOE) considers the nature of mathematics as a study of the properties, relationships, operations, algorithms and applications of numbers and spaces at the most basic level, and abstract concepts and objects at the more advanced levels. It has four recurring themes as below
Four recurring themesSingapore MOE Primary School Math Syllabus
Big Ideas about DiagramsDiagrams are succinct, visual representations of real-world or mathematical objects that serve to communicate properties of the objects and facilitate problem solving. Understanding what different diagrams represent, their features and conventions, and how they are constructed help to facilitate the study and communication of important mathematical results.
Properties and Relationships, Operations and Algorithms
Big Ideas about EquivalenceEquivalence is a relationship that expresses the ‘equality’ of two mathematical objects that may be represented in two different forms. The conversion from one form to another equivalent form is the basis of many manipulations for analysing, comparing, and finding solutions. In every statement about equivalence, there is a mathematical object (e.g. a number, an expression or an equation) and an equivalence criterion (e.g. value(s) or part whole relationships).
Properties and Relationships, Operations and Algorithms
Big Ideas about InvarianceInvariance refers to a property of a mathematical object which remains unchanged when the object undergoes some form of transformation. Many mathematical results are about invariance. These are sometimes expressed as a general property of a class of objects. In each instance, there is a mathematical object (e.g. a sequence of numbers, a geometrical figure or a set of numerical data), there is an action (e.g. re-arrangement or manipulation), and there is a property of the mathematical object that does not change.
Big Ideas about MeasuresNumbers are used as measures to quantify a property of real-world or mathematical objects so that these properties can be analysed, compared and ordered. Many measures have units. Zero means the absence of the property in most cases. Special values such as one unit serve as useful reference. Some measures are governed by a certain formula, e.g. area = length × breadth. Two measures can also be combined to derive new measures, e.g. speed is formed by combining distance and time.
Big Ideas about NotationsNotations represent mathematical objects, their operations and relationships symbolically. They are written in a concise and precise manner that can be understood by users of mathematics. These form a writing system that facilitates the communication of mathematical ideas. Understanding the meaning of mathematical notations and how they are used, including the rules and conventions, helps to facilitate the study and communication of important mathematical results, properties and relationships, reasoning and problem solving.
Big Ideas about ProportionalityProportionality is a relationship between two quantities that allows one quantity to be computed from the other based on multiplicative reasoning. Proportionality is common in many everyday applications of mathematics. Problems involving fractions, ratios, rates and percentages often require the use of proportionality. Underlying the concept of proportionality are two quantities that vary in such a way that the ratio between them remains the same.
The Singapore Ministry of Education (MOE) primary school math curriculum, at its core, seeks to develop the mathematical problem-solving competency of primary school children in the six Big Ideas of Diagram, Equation, Invariance, Measures, Notations and Proportionality.
The Singapore MOE views mathematical problem solving as having five key ingredients of Attitudes, Metacognition, Processes, Concepts and Skills. It is the responsibility of the specially-trained Singapore MOE teachers to impart the understanding of these five key ingredients to the primary school children in Singapore, so as to grow them into confident math learners.
Concepts refers to the understanding of mathematical concepts, their properties and relationships and the related operations and algorithms, which are essential for solving problems. In the Singapore MOE primary school math curriculum, concepts in numbers, algebra, measurement, geometry and statistics are explored.
Skills refers to being proficient in carrying out the mathematical operations and algorithms, visualising space, handling data and using mathematical tools to solve problems. In the Singapore MOE primary school math curriculum, numerical calculation, algebraic manipulation, spatial visualisation, data analysis, measurement, use of mathematical tools and estimation are explicitly taught.
Processes refer to the practices of mathematicians and users of mathematics that are important for one to solve problems and build new knowledge. These include abstracting, reasoning, representing and communicating, applying and modelling.
Abstraction is what makes mathematics powerful and applicable. Justifying a result, deriving new results and generalising patterns involve reasoning.
Expressing one’s ideas, solutions and arguments to different audiences involves representing and communicating, and using the notations (symbols and conventions of writing) that are part of the mathematics language.
Applying mathematics to real-world problems often involves modelling , where reasonable assumptions and simplifications are made so that problems can be formulated mathematically, and where mathematical solutions are interpreted and evaluated in the context of the real-world problem.
Metacognition, or thinking about thinking, refers to the awareness of, and the ability to control one's thinking processes, in particular the selection and use of problem-solving strategies. It includes monitoring and regulation of one's own thinking and learning. It also includes the awareness of one’s affective responses towards a problem. When one is engaged in solving a non-routine or open-ended problem, metacognition is required.
Attitudes refers to having positive attitudes towards mathematics which strengthens the inclination towards using mathematics to solve problems. Attitudes include one’s belief and appreciation of the value of mathematics, one’s confidence and motivation in using mathematics, and one’s interests and perseverance to solve problems using mathematics.
Developing problem-solving skills requires attention to all five components of the Mathematics Curriculum Framework i.e. Attitudes, Metacognition, Processes, Concepts and Skills. Even though there are many facts and procedures in mathematics where automaticity and fluency are important, there must be emphasis on conceptual understanding and problem-solving, where reasoning and strategic thinking are important. This means knowing the why, not just the what and how. A focus on relational understanding benefits all students as it helps students apply facts and procedures more skilfully, improve their problem-solving strategies and it deepens their appreciation of the nature of mathematics.
The Singapore Ministry of Education (MOE) primary school math teaching approach seeks to impart in primary school children the understanding of the six big math ideas This requires teachers to teach towards big ideas, where they help students see and make connections among mathematical ideas within a topic, or between topics across levels or strands. An understanding of big ideas can help students develop a deeper and more robust understanding of mathematics and better appreciation of the discipline.
Problem Solving. The central focus of the mathematics curriculum is the development of mathematical problem-solving skills. This means the ability to use mathematics to solve problems. However, not all problems that they will encounter in life will be routine and familiar. Students must have the opportunities to solve non-routine and unfamiliar problems. They must also learn how to approach such problems systematically. Therefore, as part of the teaching and learning of mathematics, students should be introduced to general problem-solving strategies and ways of thinking and approaching a problem. Heuristics such as working with simple cases and working backwards help students to solve problems, and they form part of this framework.
Concepts and Skills. Mathematical concepts are abstract. To develop an understanding of these abstract concepts, it is necessary to start from concrete objects, examples and experiences that students can relate to. The concrete-pictorial-abstract approach is an important consideration in the sequencing of learning. Different concepts require different approaches. It is important to select the instructional strategies and resources based on the nature of the concepts. Mathematical concepts and the related skills are also hierarchical. Some concepts and skills have to be taught before others. Understanding the hierarchical relationships among concepts and skills is important and necessary for sequencing the content and organising learning within a lesson and across lessons. To develop proficiencies in mathematical skills, students should have opportunities to use and practise these skills. There should be repetitions to develop fluency, variations to develop understanding, progression to develop confidence, and frequency to facilitate recall. For example, to develop proficiency in arithmetic algorithms, there should be regular reinforcement of concepts and procedures throughout the learning. It is important to ensure that skills are not taught as procedures only. There must be understanding of the underlying mathematical concepts. For students to understand concepts well, there must be opportunities for them to apply the concepts in a wide variety of problems.
Processes and Metacognition. The teaching of process skills should be deliberate and yet integrated with the learning of concepts and skills. How students experience learning is important. Students will not develop reasoning skills if there is no opportunity for them to do so, or insufficient emphasis in the classrooms. Neither will students develop metacognition if there is no opportunity for thinking about thinking in the classrooms. More specifically, for reasoning and communication, there must be opportunities for students to justify their answers, whether it is during classroom discussion or in written work. Promoting classroom talk is also important. Teachers should encourage students to participate in classroom discourse. For students who need more help, using sentence frames to help them formulate their response or explanation could be used. In general, discourse promotes understanding. For modelling and applications, students should be exposed to a wide range of problems in real-world contexts. At primary level, students can be exposed to real-life problems where they are required to formulate and check on the reasonableness of the answers. For metacognition, there must be opportunities for students to reflect on their problem-solving process and explain this process to one another. Teachers should make a deliberate effort to role model the thinking process by thinking out loud and by encouraging students to constantly monitor their own thinking process and reflect on it. Posing non-routine problems that are within the ability of the students and that students believe they can solve is important to encourage thinking. Without thinking, and thinking about thinking, there will be no opportunity to develop metacognition.
Attitudes. Students will develop confidence in mathematics and feel motivated to learn mathematics if they experience success and feel competent in learning. They will appreciate the value of mathematics and develop an interest in the subject if they can see the relevance of mathematics to their everyday life and in the real world. For teachers to develop positive attitudes towards mathematics in their students, they should scaffold and support the learning so that students can experience success in understanding the concepts. They should ensure that the tasks given are interesting, within the students’ abilities and achievable with effort. Teachers should also bring real-world contexts and applications into the classroom so that students can see the relevance and power of mathematics.Source: Singapore Ministry of Education