Mental Abacus is More Than Just "Fast Calculation": Explaining Mathematical Concepts Clearly is True Mental Abacus Teaching

Author: Zhongying Jin (Abacus and Mental Arithmetic Association of Australia) | Vsuccess Education · Mount Waverley

Many parents often have two questions in mind when considering mental abacus for their children: Apart from fast calculation, does mental abacus actually help with my child's mathematics? Will it conflict with the methods taught in school?

Mental abacus (or abacus mental arithmetic) is based on traditional abacus calculation, where the abacus is internalized into the brain—essentially "calculating on an abacus inside the mind." The abacus is both intuitive and functional, making it highly effective for children who are just starting to learn arithmetic. Its computational results are obvious, so why do some teachers and parents still hold reservations about children learning mental abacus?

As a teacher who taught mathematics in China for 20 years and has been teaching the combination of mental abacus and mathematics in Australia for over a decade, I would like to use this article to share a key perspective: Excellent mental abacus teaching is never purely about pursuing calculation speed and accuracy. Instead, it is about clearly explaining the underlying mathematical concepts while the child manipulates the abacus beads. Below, I will discuss how Vsuccess Education integrates mental abacus with mathematical concepts across seven key aspects.

1. Explaining the Concepts and Rules of Addition and Subtraction

When children first start learning mental abacus, they are usually quite young, and some may not even have a concept of addition and subtraction. At this stage, intuitive methods are the easiest to accept. Teachers can establish these concepts through drawings or physical demonstrations.

• ●Addition: For example, if the teacher has 2 books in hand and brings 1 more, how many books are there altogether? Help the child understand that "combining to find the total" uses addition (2 + 1 = 3), while simultaneously demonstrating the bead manipulation for 2 + 1 = 3 on the abacus.
• ●Subtraction: For example, if the teacher has 3 books and gives away 2, how many are left? Help the child understand that "taking away a part from the total to find what remains" uses subtraction (3 - 2 = 1), also demonstrating it on the abacus.

During this process, guide the child to observe the relationships among the four equations: 2 + 1 = 3, 1 + 2 = 3, 3 - 2 = 1, and 3 - 1 = 2, so they can proficiently master the connection between addition and subtraction. Then, elevate it a bit further: 2 + 1 + 1 = 4, 4 - 2 - 1 = 1, 4 - 2 + 1 = 3, letting the child understand that when an equation contains only addition and subtraction, it should be calculated from left to right.

Once the concepts are clear, proceed to mental abacus addition and subtraction exercises. Throughout the process, the teacher must continuously reinforce this by linking it to real life, letting the child understand that both mathematics and mental abacus originate from life and ultimately serve life.

In the mid-stage of learning, after the concepts of addition and subtraction have been mastered, students can upgrade to "comparing more and less" problems. Here, I believe the most critical point is to let children truly understand what "the same amount" means.

For example: I have 7 apples and you have 5. How many more do I have than you? Children can quickly write down 7 - 5 = 2. But if you ask them to explain the meaning of the equation, they often say, "Subtract your 5 apples from my 7 apples." At this point, they must be explicitly told: This understanding is incorrect—my apples are mine, so how can I subtract "yours" from mine?

The correct understanding is: From my 7 apples, subtract the 5 apples that are "the same amount" as yours, and what is left over is the amount I have more than you. Paired with a diagram, children will understand this logic much more intuitively.

It can be seen even more clearly on the abacus: My apples are divided into two parts—one part is the same amount as yours, and the other part is what I have extra. Therefore, 7 - 5 = 2 means exactly "my apples minus the same amount as yours equals my extra apples."

In the later stages of learning addition and subtraction, operations with parentheses can be introduced. For example: I have 10 pens. I give 4 pens to female students and 2 pens to male students. How many pens are left? This can be written as 10 - 4 - 2.

Can we first add up the 4 and 2 pens that were given away, and then subtract the total of 6 given pens from 10? That is, 10 - 6, which works. However, if we directly replace 6 with 4 + 2 and write it as 10 - 4 + 2, the result will conflict with the actual remaining amount—proving that writing it this way is wrong.

What should we do? Add parentheses to 4 + 2, letting the child understand that the items inside the parentheses "are all given away and form a single whole," so the operations inside the parentheses must be calculated first: 10 - (4 + 2). In this way, the child naturally understands: If there are parentheses in an expression, calculate what is inside them first.

2. Mental Abacus and Column Method Are Not Contradictory, But Interconnected

Many people claim that the calculation methods of mental abacus and the column method (vertical calculation) taught in schools are "contradictory." Is this really the case? Let's sort it out.

• ●High-to-Low Calculation: Mental abacus calculates from the highest place value. Why? Because human memory habits naturally go from high to low, from left to right—this is how we remember phone numbers, and how we read and write numbers. Calculating from the highest place value aligns perfectly with the natural laws of reading, writing, and memory. For instance, when a teacher calls out "25," an image of this number immediately appears in the child's mind. When the teacher adds "plus 54," by the time "plus 50" is spoken, the number 75 is already formed in the mind. Just as the voice pronounces the "4" in the ones place, the child has already calculated 79. This is why the answer comes out the exact moment the dictation ends. Of course, this requires the child to be highly focused: listening to the numbers, calculating on the abacus in their mind, and remembering the results.
• ●Understanding Place Value: The abacus also helps children understand the "place value system" very well. After learning mental abacus addition and subtraction, when you teach them the school's column method, you will find that it often takes less than a minute for them to fully master it. When handling large-number calculations in the future, many children will combine both methods—using mental abacus to mentally calculate a certain place value and then calculating another, which is both fast and accurate.

Some people argue that "school column methods start from the lowest place value," but this statement is not entirely accurate. Let's look at it: For situations like 23 + 25 or 29 + 30 where there is no carrying, or for straightforward integer addition and subtraction, you can actually calculate directly from the highest place value. It is only in cases like 37 + 68 where carrying is involved that the rule states we must start from the ones place—mind you, this is just a "rule" designed to avoid dropping the carried digits from lower place values. Multiplication can be viewed as a special form of addition that involves carrying, so it also starts from the lowest place value; whereas division starts from the highest place value.

Think about it carefully: In column methods, it is actually "addition/subtraction with carrying and multiplication that start from the lowest place value, while straightforward integer addition/subtraction and division start from the highest place value."

Evidently, the two methods—mental abacus and column calculation—can be used in parallel or interconnected with each other. There is no contradiction at all.

3. Explaining the Meaning of Multiplication, Division, and Their Relationship

When teaching mental abacus multiplication, whether using multiplication rhymes (Times Tables) or the "Yi Kou Qing" (instant calculation) method, it is essential to first explain the meaning of multiplication.

For example, when teaching "multiplication by 2," let the child manipulate the abacus while the teacher draws a diagram. As the child adds a 2 on the abacus, the teacher draws 2 apples in a dish—this is 1 group of 2, 2 × 1 = 2. Then add another 2 and draw another group, which makes 2 groups of 2, 2 × 2 = 4, and so on.

This process emphasizes the child's understanding of "how many groups of a number." They should know how to write the addition equation, how to rewrite it into a multiplication equation, and be able to articulate the meaning represented by the multiplication. In this way, the child clearly knows that multiplication is just addition with identical addends, and that there is a definitive connection between addition and multiplication. After understanding the meaning, they can then perform mental abacus multiplication.

The same applies to division; explain the meaning of division before diving into the actual calculations. To give children a deeper understanding, I usually illustrate with two scenarios:

• ●First Scenario (Equal Sharing): The teacher takes 6 pens and distributes them equally to 2 students, giving them out one by one in turns. After finishing, both have the same amount, which is 3 pens each. The equation is written as 6 ÷ 2 = 3.
• ●Second Scenario (Grouping / Measurement Division): There are 10 pieces of chocolate, and every 2 pieces are packed into a bag. How many bags can be packed? The equation is written as 10 ÷ 2 = 5. Here, children can experience the relationship between division and subtraction: subtracting a 2 from 10 each time, a total of 5 times until nothing is left—which means "how many 2s are subtracted from 10," and "how many groups of what number" happens to be the focus of multiplication.

Thus, the relationships among division, subtraction, and multiplication/division are neatly tied together. This segment can also be conducted through open-ended group work: give each group 10 chocolates and let them figure out how to "make the shares equal" on their own, and the teacher can guide them based on the situation.

Regarding mixed multiplication and division: If there are 2 boxes of pencils, with 6 pencils in each box, and they are distributed equally among 3 students, how many pencils does each person get? First, calculate that there are two groups of 6, which is 6 × 2 = 12, and then divide it equally into 3 parts, 12 ÷ 3 = 4. The combined expression is 6 × 2 ÷ 3. From this, the child understands that when an expression contains only multiplication and division, it should be calculated from left to right. The reasoning behind parentheses in multiplication and division expressions is exactly the same as mentioned earlier for addition and subtraction.

4. Explaining the Order of Operations in Mixed Arithmetic Operations

Once children have learned addition, subtraction, multiplication, and division, and have mastered operations of the same level (calculating from left to right when there is only addition/subtraction or only multiplication/division; calculating inside parentheses first if they exist), they can move on to mixed four arithmetic operations.

For example, ask: How many apples are there in total? Children can easily list out 3 + 2 + 2 + 2 + 2. Guide them to observe that the latter part consists of 4 groups of 2, which can be written as 2 × 4, leading to the expression 3 + 2 × 4.

Then, let the children explore how to calculate it themselves: since it means "3 plus four 2s," they cannot calculate 3 + 2 first and then multiply by 4 (as that would become four 5s, which contradicts reality). Through verification, they conclude: calculate multiplication first, then addition.

From this, the rule is summarized: If an expression contains addition, subtraction, multiplication, and division, multiplication and division must be calculated before addition and subtraction; as long as there are parentheses, always calculate what is inside them first.

5. Explaining the Meaning of Decimals Before Calculating Decimal Operations

In reality, some teachers simply tell children "where to place the decimal point." As a result, children only know how to calculate but do not understand why. Therefore, before calculating decimals, the concept of decimals must be explained clearly, so that children "know not only the 'what' but also the 'why'."

For example: Let the children measure how long their desks are, or how tall they are. How do we represent the parts that exceed whole integers? Children will realize that decimals are ubiquitous in life. This introduces the meaning of decimals: dividing the unit "1" equally into 10 parts, 100 parts, etc. One or several of these parts represent tenths, hundredths, and so on. Writing them without denominators creates decimals. A decimal consists of an integer part, a decimal part, and a decimal point.

• ●Decimal Addition and Subtraction: You can use a pizza divided into 10 slices to let children realize during "pizza sharing" that the key is to align the same place values and align the decimal points, then add or subtract as usual.
• ●Decimal Multiplication and Division: Before learning these, children need to understand the concepts of "magnifying by 10 or 100 times" and "reducing by 10 or 100 times" (by this time, children are already clear about unit conversions). After that, decimal multiplication and division become very easy to understand.

Example problems can be used to help understand decimal multiplication. For example: A box of chocolates costs $4.2. How much does it cost to buy 3 boxes? $4.2 is 42 dimes (or 42 ten-cents), and 3 groups of 42 dimes is 126 dimes, which is $12.6. Using both "unit conversion" and "magnifying 10 times then reducing 10 times" allows children to understand that they can calculate it as integers first, and then place the decimal point on the product.

As for how to place the decimal point, let's look at three examples:

• ●Example 1: 0.83 × 4. Magnify by 100 times first, then reduce by 100 times, which equals 3.32. Let the child observe the relationship between the number of decimal places in the factors and the product.
• ●Example 2: 4.8 × 0.3. Both numbers are magnified by 10 times, and the result is then reduced by 100 times, which equals 1.44. Children will notice that there are 2 decimal places in total across the factors, so they count 2 places from the right of the product to place the decimal point.
• ●Example 3: 0.83 × 0.04. The factors have 4 decimal places in total. When the product does not have enough digits, pad it with 0s in front before placing the decimal point, which equals 0.0332.

Decimal division is similar to multiplication and can also be calculated using unit conversion or "magnifying first then reducing." Summarize the rules through examples: the decimal point of the quotient must align with the decimal point of the dividend; if the integer part is not enough to divide, write a 0 for the quotient and place the decimal point; if there is a remainder, append a 0 and continue dividing.

6. Using the Abacus to Explain Unit Conversion Clearly

Unit conversion is an area where children often make mistakes when they first learn it, but the rods on an abacus (where adjacent units are base-10) can precisely help children complete conversions quickly and accurately.

• ●2 dollar 65 cents = 2.65 dollar = 265 Fen. (Note the position of the decimal point relative to the rods).

Let's try another one:

• ●6 km 501 m 6 dm 2 cm 7 mm = 6501.627 m.

The rule is clear: whichever rod you place the decimal point to the right of, just add that rod's unit name, and the conversion is complete directly—making it simple, fast, and accurate.

7. Explaining Positive and Negative Numbers, and Calculating with the Abacus's "Dual Representation”

Let's first look at the concept of positive and negative numbers. For instance, during winter in the Northern Hemisphere, the temperature in the north might be 4 degrees below zero, written as -4°C. Numbers smaller than 0 like this are called negative numbers, which are opposite to positive numbers; positive numbers are numbers greater than 0. Negative and positive numbers represent quantities with opposite meanings.

Next, let's understand a concept unique to abacus calculation—Dual Representation.

The so-called dual representation means that when beads are moved toward the beam to represent a number, the beads toward the frame simultaneously represent another number (i.e., its complement; the number of beam beads and frame beads are complementary; on a "1-upper, 4-lower" abacus, the last rod needs +1 to complement to ten).

Dual representation can use the beam bead count and frame bead count to represent the absolute values of positive and negative numbers, respectively. This perfectly embodies the unity of opposites between positive and negative numbers, allowing them to be taught synchronously:

• ●The absolute value of positive numbers is read from the beam beads.
• ●The absolute value of negative numbers is read from the frame beads.
• ●Suspending a lower bead represents subtracting 1 (1) on the current rod, or represents the negative sign of the frame bead count on the subsequent rod.

For example: 23 - 48 = -25 (read from the frame beads, with +1 on the last rod).

Another example: 23 - 48 + 87 = 62. When adding 87, a 1 is carried from the tens place to the hundreds place, which offsets the original -1, and the calculation proceeds normally. In other words—regardless of whether there are enough beads to subtract, calculate using the normal method without needing to change the order of operations.

In this way, teaching positive and negative numbers comes naturally: add when you should add, subtract when you should subtract, with consistent calculation rules that are simple and clear.

Conclusion: Why We Insist on Teaching Mental Abacus This Way

Having explained these seven points, I believe everyone can understand why there are sometimes discrepancies between school math teachers and mental abacus teachers. Many school teachers must follow a rigid curriculum syllabus and rarely have the opportunity to truly understand mental abacus deeply. What they often see is the emphasis on calculation speed and accuracy, rather than the underlying mathematical thinking and concepts. To be honest, if I hadn't deeply researched mental abacus and spent over twenty years teaching the integration of mental abacus and mathematics, I might not have agreed to treat mental abacus as a part of mathematics education myself.

But precisely because I have done it deeply, I am more convinced than ever: the true value of this world cultural heritage, the mental abacus, lies not in "a different way of playing" or in blind, mechanical training for speed, but in tightly integrating mental abacus with mathematics—allowing children to truly understand mathematics while manipulating the abacus. This is exactly what Vsuccess Education has always been doing.

“Reference: The Optimization Factor Project of Mathematics Teaching (by Qishu Guo)”

Vsuccess Education, located in Mount Waverley, is a professional mathematics and mental abacus educational institution. We provide mental abacus, mental arithmetic, mathematics enrichment, NAPLAN, and VCE mathematics courses for children from Prep to Year 12, serving families throughout the Monash region including Glen Waverley, Wheelers Hill, and Clayton.

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