# Instructions and Tips for Singapore Math Beginners

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Over the past 30 years of using the Singapore math methodology in some American schools, its popularity has only grown. We suggest exploring the instructions and tips below for anyone involved in elementary school math in one way or another: teachers, parents, or students who need to pull up their basic math competency.

## What is Singapore Math?

Let’s start with a short definition. Singapore Math is a particular approach to teaching mathematics, developed in 1982 and gained international recognition and popularity, especially in the United States. The main goal is to explain to students how this science performs, its inner working, before offering them problems. A deeper immersion into the subject can be achieved through visual aids (drawings and diagrams), making it easier to understand for those who have difficulty thinking abstractly.

## Principles of Singapore Math

The Singapore education model is notable for its atypical approach to mathematics. At first glance, you may find it confusing, but in reality, it will help you succeed in learning the offered methods.

Singapore Mathematics involves a special philosophy of the subject. Students should be taught in 5 different directions:

• These are our familiar topics in numbers, algebra, geometry, statistics, analytics, and probability. Students must know not only concepts as individual ideas but also their interrelationships. It is necessary to move to a more advanced mathematical level, where assignments cover a broader range of concepts.
• It includes the ability to calculate, measure, analyze, evaluate, visualize, and master other mathematical manipulations and tools. Many national approaches teach “how” to solve a problem, forgetting to explain “why” it works this way and not that way.
• Processes are the ability to reason, think, connect, discover, apply, and model. Using each of these activities contributes to better understanding and therefore, more effective problem-solving.
• Reasoning helps the student develop the terms of a particular problem to find additional clues.
• Thinking is the student’s ability to think about a problem: classify, compare, order, analyze in whole or in part, visualize, identify connections and patterns, and apply induction and deduction.
• Seeing connections helps unite concepts not only inside but also outside of mathematics. It makes mathematics more useful and multifunctional in the real world.
• Communication refers to the ability to “speak” the language of mathematics. It means understanding proposed tasks and expressing ideas in a mathematically correct way.
• Discovery implies creating new knowledge when solving problems (constructing a graph or diagram, altering the content for better understanding, finding other ways to solve it, etc.)
• Application means using other skills to find a solution, including everyday skills.
• Modeling helps a student to practice several solutions and find the best one.
• Conclude in students’ attitudes and motivation to learn. They should be passionate about the process, confident in their abilities, and persistent in their goals. Mathematics should not be considered a boring subject, and classes should be fun so that students do not lose interest.
• A complex concept of how to think to succeed. It is important not to overwhelm students with complicated and unnecessary information by allowing them to spell out the problem aloud, by offering assignments that focus on the process of finding a solution rather than the solution itself, by organizing students to work collectively on a multi-solution problem.

All this knowledge, skills, methods appear gradually in the process of learning, depending on what needs to apply in a particular concept. Transitions should be smooth and painless. For example, in the beginning, students use numbers to count. Then numbers are replaced by pictures. Eventually, numbers become an abstract concept – a variable.

## Basic Teaching Methods in Singapore Mathematics

• Number bonds (fact families). It means a relationship between some numbers that can be established through manipulations such as addition and subtraction. For example, the numbers 2, 4, and 6 can be related as 2 + 4 = 6 or 6 – 2 = 4. The same goes for multiplication and division: 2 x 3 = 6 or 6 / 3 = 2. It is best to start with numbers whose sum is 10 or less. Once the student has mastered this concept, he can move to numbers divisible by 10.
• Decomposing numbers. Numbers with two or more digits are decomposed into components that are much better to manipulate. Write the number 16 and draw 2 lines down (similar to a family tree). The lines should end with the two numbers 10 and 6, i.e. the number 16 consists of 10 and 6. And the number 23 consists of 20 and 3. Let us explain why it’s necessary. If a problem requires adding 16 and 23, it may seem a bit confusing at first glance. But if we decompose 16 into 10 and 6, and 23 into 20 and 3, and add the components, we have 10 + 6 + 20 + 3. We can easily add 10 + 20, 6 + 3 and get 30 + 9 = 39. The answer is 39.
• Adding from right to left. The method is similar to the previous one with the only difference: the numbers are divided not from top to bottom but from left to right. It is convenient in the case of really multi-digit numbers, such as 6297. We’re used to scoring such numbers from right to left into ones (7), tens (9), hundreds (2), and thousands (6), but try going the other way: 6000 + 200 + 90 + 7. When you add it up to a huge number, like 1358, it looks like this: 6000 + 200 + 90 + 7 + 1000 + 300 + 50 + 8, then 6000 + 1000 + 200 + 300 + 90 + 50 + 7 + 8, then 7000 + 500 + 140 + 15, and we get 7655.
• Multiplication using notions of area. Draw a square and draw 4 equal squares inside it. Choose 2 two two-digit numbers to multiply. Let’s take 37 and 12 and write them on the top and left side of the square. Now you need to multiply all the components with each other and write the resulting products in the squares. 30 x 10 = 300, 30 x 2 = 60, 7 x 10 = 70, 7 x 2 = 14. Then add these numbers: 300 + 60 + 70 + 14 = 444.
• Multiplication using the FOIL method. FOIL is an abbreviation for the order of numbers for multiplication. FOIL stands for: F = multiply FIRST terms, O = multiple OUTSIDE terms, I = multiply INNER terms, and L = multiply LAST terms. Suppose you multiply 14 and 29. You multiply the FIRST terms (10 x 20), then you would multiply the OUTSIDE terms (10 x 9), then you would multiply the INNER terms (4 x 20), and finally, you would multiply the LAST terms (4 x 9). When you add up all the results 200 + 90 + 80 + 36, you get 406.
• Multiplication by division. The most convenient way to divide a large number is to divide it into simple components. To divide 78 by 3, divide 78 by 60 and 18 and divide those numbers by 6. 60 / 6 = 10, 18 / 6 = 3, аnd 10 + 3 = 13.
• Rounding the answer. This method is good for advanced math students who don’t have to find the exact answer. It is good for practicing calculations in your head. For example, to divide 398 by 5, we could round 398 slightly to 400. We would get 400 / 5 = 80. But since we originally had a smaller number, the answer should be less – 79 and something.
• Another way to simplify calculations involves transforming numbers to a convenient form. Add up 98 and 45, increase 98 to 100 and add it to 45, you get 145. Now subtract the compensating part (2) and you get the final answer, 143.
• Drawing to solve problems. It trains students’ ability to visualize the verbal content of a problem. The terms of the problem must be drawn. For example, here is a problem. Mary has 17 apples, and her friend has 12. How many apples in all? To solve this problem, draw Mary’s “basket” and her friend’s “basket” and put a question mark between them. Then “fill” each basket with the appropriate number of apples. By doing this, we will understand that to get the result, we must add the two numbers, 17 and 12. Using addition from right to left, we get 10 + 7 + 10 + 2 or 10 + 10 + 2 + 7 = 29 apples.