## What is Singapore Math?

### Introduction

The Singapore math method is a highly effective teaching approach originally developed by Singapore’s Ministry of Education for Singapore public schools. The method has been widely adopted in various forms around the world over the past twenty years following our introduction of the curriculum to the U.S. in 1998. Read our story here.

### Why Us?

We started Singapore Math Inc. because we thought everyone should have access to this effective teaching approach. All of our programs preserve the techniques, sequencing, and rigor that define Singapore math. We continue to provide the original curriculum that put Singapore math on the map, while evolving the Singapore math approach through other series to better serve today’s students and educators.

Proven Results

Singapore consistently ranks at the top in international math testing. The intentional progression of concepts in the Singapore math approach instills a deep understanding of mathematics.

Two international tests, the TIMSS (Trends in International Mathematics and Science Study) and the PISA (Programme for International Student Assessment), assess math and science competency in countries around the world. Singapore students consistently rank among the top on both tests.

Our Singapore math programs raise U.S. student performance internationally and at home on standardized and state assessments. With the use of our programs, more students rank “At or Above NAEP Proficient” on U.S. national math assessments.*

The Method

The Singapore math method is focused on mastery, which is achieved through intentional sequencing of concepts. Some of the key features of the approach include the CPA (Concrete, Pictorial, Abstract) progression, number bonds, bar modeling, and mental math. Instead of pushing through rote memorization, students learn to think mathematically and rely on the depth of knowledge gained in previous lessons.

An attitude that math is important and approachable is also essential. Students perform at a higher level when their potential for understanding and success is assumed.

So how is this different from the way math is widely taught in the U.S.?

In typical U.S. math programs, students get a worked example, then solve problems that very closely follow that example, repeating all the same steps with different numbers. In Singapore math, students must think through concepts and apply them in new ways from the very start. Since they can’t rely on simple replication, students are pushed to greater engagement and broader thinking. In U.S. math programs, concepts and skills are more compartmentalized within and across grade levels than in Singapore math, where a strong sense of connectivity to past learning is woven throughout.

Singapore math not only helps students become more successful problem solvers, it helps them gain a sense of confidence and resourcefulness because it insists on conceptual depth. This naturally prepares students to excel in more advanced math.

The Components

**Concrete, Pictorial, Abstract (CPA) Approach**

The Concrete, Pictorial, Abstract (CPA) approach develops a deep understanding of math through building on existing knowledge. This highly effective framework introduces concepts in a tangible way and progresses to increasing levels of abstraction. In the concrete phase, students interact with physical objects to model problems. In the pictorial phase, they make a mental connection between the objects they just handled and visual representations of those objects. For example, real oranges (or counters standing in for oranges) are now represented as drawings of oranges. In the abstract phase, students use symbolic modeling of problems using numbers and math symbols (+, −, ×, ÷).

By varying the methods and phases of CPA fluidly, educators help reinforce important connections. Students work towards math mastery when they view concepts with increasing levels of abstraction over time. Not all lessons include all three CPA stages as application of this approach varies by topic. Instead, CPA principles are woven throughout the curriculum, and support other important strategies such as number bonds, bar modeling, and mental math.

**Number Bonds**

Number bonds are a pictorial technique that show the part-whole relationship between numbers. Initially, the whole number is written in one circle, and the parts of the number are written in adjoining circles connected by lines to the first circle. This method helps early elementary students work towards addition and subtraction, and illustrates strategies to solve expressions mentally. Using number bonds fosters a solid number sense that serves students throughout their math education.

This problem from Dimensions Math Textbook 1A shows the stages of the CPA approach, incorporating the use of number bonds to illustrate addition.

**Concrete Stage**

In this stage, teachers lead a classroom activity.

Students represent birds. Have 5 students go to the front of the class and ask the rest of the class how many students there are. Send 2 more students up to the front and ask, “How many students are there now?” Ask students to explain what happened. The interaction introduces students to the problem in a tangible way. (Adapted from Dimensions Math Teacher’s Guide 1A).

**Pictorial Stage**

Students are then shown a visual representation of the problem.

**Abstract Stage**

Students are then shown an equation of the problem.

**Bar Modeling**

Bar models are a versatile and transferable tool that students can use to visualize a range of math concepts, such as fractions, ratios, percentages, and more. Drawing bar models for word problems allows students to determine the knowns and unknowns in a given situation. It extends the CPA approach, especially the pictorial phase, as it allows students to illustrate the mathematical information given in problems. It prepares them to understand more complex math on a conceptual level. This method is most effective when used frequently throughout the program.

The problems below demonstrate various examples of bar modeling.

Click on the image to expand.

**Mental Math**

The Singapore Math approach teaches techniques and skills to easily and accurately perform mental math. These strategies help students develop number sense and flexibility in thinking about numbers. Many mental math strategies involve factoring numbers into parts, then performing operations on them in a different order from the original expression. The thought processes involved in mental math are often illustrated by number bonds.

Some mental math strategies are taught as early as grade 1. As students progress, they learn to apply new mental math strategies to specific types of problems and adapt ones they already know. Students are encouraged to develop their own strategies, and to use their discernment in deciding when and where to use them.

Here are some examples of mental math, click on the image to expand.