Spatial arithmetic

1st October 2004 at 01:00
A Japanese abacus enables pupils to use visual skills when doing calculations that involve addition and subtraction. Tandi Clausen-May explains how it works

The soroban is a traditional Japanese abacus. It offers teachers a powerful tool to enable pupils who think more easily in pictures than in words and numbers to carry out numerical calculations visually and kinaesthetically, making the process of computation much more meaningful.

The human eye can scan up to four objects, and see at a glance how many there are, without counting. Children can learn to see a column of four beads, with no need to count.

This ability to see up to four beads at a glance is the key to the great power of the soroban. On the soroban very large (and very small) numbers - much greater (or smaller) than four - are represented by patterns of beads that can be seen at a glance. As calculations are carried out, the beads are moved to represent each arithmetical operation as it occurs. The soroban turns numerical problems, such as 3582 + 406, into spatial problems. All the "working out" is done by manipulating the beads, not by applying numerical operations. The result is interpreted back into a number only after the calculation is complete.

How does it work? Numbers up to four are represented on the soroban by a simple column of up to four beads below a horizontal bar on a rectangular frame. In addition, the number five is represented by one special bead above the bar. A soroban therefore consists of a set of columns of beads, each with one "five" bead above and four "unit" beads below the bar.

When all the beads are positioned so that they are not touching the bar, the number zero is represented. To represent a greater number, the beads must be moved so that they touch the bar. In this way any number up to nine may be represented on a single column. For example, Numbers from one to nine are always positioned in one of the columns with a dot on the bar - the units column. Columns to the left of the units represent 10s, 100s, 1000s and so on, while columns to the right represent decimal numbers. So, for example, 3582 is represented using four columns, with two in the units column.

For the number 35.82, the bead in the units column represents five, with 30 in the column on the left and eight-tenths and two-hundredths on the right.

The positioning of powers of 10 in relation to one another is one of the soroban's great conceptual strengths. When using written numbers, pupils may develop the misconception that a whole number can be multiplied by 10 or 100 by "adding noughts". The function of the "noughts" as place holders in the place-value system is not obvious in a written calculation, especially when the numbers are presented horizontally. But "adding noughts" does not work when pupils try to multiply a decimal number, such as 2.4, by a power of 10. On the soroban, however, the eight is quite clearly in a different position when representing eight 10s (80), eight units (8), eight 100s (800) or eight-tenths (810). The calculations 3582 + 406 358200 + 40,600 35.82 + 4.06 are obviously similar, as they all involve the same movements of the same patterns of beads, but they each take place in a different position on the soroban. Soroban calculations always start from the left - so big numbers are added or subtracted first.

This is conceptually more powerful than the standard written approach to the addition or subtraction of multi-digit numbers, where the units are dealt with before the 10s, which come before the 100s, and so on. For example, to carry out the calculation 3582 + 406, the number 3582 is first set on the Soroban: Then the 400 in 406 is added, by moving four "one" beads that are below the bar in the 100s column up: There are no 10s to add in 406, so the 10s column is not changed, but the six units must be added. This is accomplished by moving a "five" bead that is above the bar in the units column down, and a "one" bead that is below the bar up.

Now the answer, 3988, may be read off the Soroban. The process of the calculation was done with a series of movements, rather than by adding figures. It is only at the final stage that the position of the beads is translated back into a number.

This simple example illustrates how the Soroban works. More complex calculations, such as those that cross the 10s boundary and would involve "carrying" with a standard written algorithm, may require more complicated movements. Adding five (in 6 + 5, for example), has to be done as subtract five, then add 10, as there is no "five" bead available in the units column.

But the basic principle remains the same. Each step in the calculation - all the "thinking" - is carried out in the beads. The interpretation back into numbers comes only at the end.

Soroban arithmetic is spatial arithmetic, so it makes sense to pupils who struggle with numbers on the printed page.

Dr Tandi Clausen-May is principal research officer, Department of Assessment and Measurement, National Foundation for Educational Research

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