ISaR – Inclusive Services and Rehabilitation

Didactic Pool

1 Abacus, the most genius calculation aids in the world By Emmy Csocsán and Solveig Sjöstedt

Technical University of Dortmund Department Rehabilitation Sciences Rehabilitation and Education for the Blind and Visually Impaired Project ISaR D-44221 Dortmund, Germany Tel.: +49-(0)231 / 755 5874 Fax: +49-(0)231 / 755 6219 Email: [email protected] sol[email protected] Internet: http://isar-international.com

© Csocsán, E. & Sjöstedt, S

1. Introduction 2. History 3. Description 4. Principle of calculation 5. Mathematical phenomena and the abacus 6. Implementation in schools 7. Abacus made for the Blind 8. Usage 9. Short manual 10. References 1. Introduction One can find plenty of information about abacus and its usage in different sources (manuals, books or on-line descriptions) which could help but could cause confusion as well. Our intention is now to help students, teachers and any interested person with basic information about that very useful mathematical tool. Abacus is not only a computation aid but a learning assistant to illustrate many mathematical contents and relationships as well. We hope that we can encourage teachers in schools to use that learning aid in their classes not only for students with visual impairment. Abacus has a big advantage for the students because it is working with numbers (structured units) and not only with numerals like a digital calculator. The logic of carrying out the task by abacus provides students that correctness, the precise way of thinking could be transferred into other subjects as well. This field is one of the good examples where the general didactic might learn from the “special” one. 2. History Abacus is a Latin word, means “sand tray”. The word originates with the Arabic "abq", which means dust or fine sand. In Greek this would become abax or abakon which means table or tablet. There are other computation aids which have some of the qualities of abacus. Presumably using marbles, small stones – followed the history of humans in the field of number concepts and calculations – was the first step towards creating the abacus. The abacus is one of the earliest and most effective calculating devices. The first table has been found in Mesopotamia, the cradle of civilization (bordered by rivers Tigris and Euphrates, today Iraq) four thousand years ago. Babylonian Empire is famous for this mathematical development. 3. Principle of calculation Abacus is made for the ten-based system. Addition and subtraction are the basic computations. With the “soroban” like abacus (1:4 beads) you can do your number work in the binary and in the 5 based number systems as well.

© Csocsán, E. & Sjöstedt, S

4. Description Abacus is rectangular in shape. Abacuses with varied columns and number of beads are used in different countries. The common operations for the abacus with 15 columns are the same as those abacuses with fewer columns. A bar is separating the abacus horizontally cutting across all the fifteen columns. Each column has four beads in one part, each bead has the value of one (1). Each column has one bead in the other part. That bead has the value of five (5). The position of the abacus pays no role. There are countries i.e. USA, Poland, etc. in which the user keeps the tool with the bead 5 in the upper part (see below) but there are countries i.e. Sweden, Finland, Hungary in which the users keep the abacus the other way round; the beads with the value “one” are in the upper part. From "Abacus-Online-Museum" (Luetjens)

Basic remarks: In this abacus-online-museum you can see items from different Asian and European countries. Even though there is a diversity in the collection, all abacuses can be reduced into two basic types: the Asian and the Russian abacus.

The Asian type (in three variants)

Criteria: round beads, positioned in a system of (2+5) beads on each rod. This is the traditional design of a Chinese abacus with an upper section and a lower section of beads. Even today it is relatively common.

Criteria: The beads are double cone shaped. They are positioned in a system of (1+5) beads on each rod. This simplified construction was developed in Japan toward the close of the nineteenth century.

Criteria: The beads are also double cone shaped. They are positioned in a system of only (1+4) beads on each rod. This most simplified construction was developed in Japan around 1920. (soroban)

© Csocsán, E. & Sjöstedt, S

The Russian type

A simple system of 10 beads on each rod without any function of "more-unit-beads".

5. Mathematical phenomena and the abacus Using the Abacus to develop awareness in • • • • • • • • •

Cardinal and ordinal aspects of numbers. Grasping simultaneously a number of discrete elements. Experience of parts to the whole in numbers. Structure of numbers. Numbers and signs. Positions in a system (10-(decimal), 2-(Binary), 5-(system to the base five). Number as a result of operations. Connections and relations between operations. Preparation for algebra e.g. 8=5+3 • 8=2+6 • 8=10-2 • 8=50-40-2 etc.

© Csocsán, E. & Sjöstedt, S

6. Implementation in schools In many countries the ten bead abacus is used at a very early age to help children to develop number concept. In Japan students and adults use soroban during their whole life to solve computational tasks. In the Nordic countries and in the United States pupils with VI have the possibility to learn the abacus techniques in schools. In Hungary the curriculum of the school for the blind instructs that kind of calculation methods. It means that it is compulsory for the students to learn and use abacus in the maths class. It is important for the maths teacher to understand that calculating with abacus is a different way of operating than the mental and the written ones. It needs a different algorithm. When calculating with abacus one needs to have elementary competence in mental calculations such as: Decomposing the number in the range of ten; one has to know that 8 as a whole consists of two parts, 5 and 3 and 9 of the parts 5 and 4; Using analogue extending of decomposing of numbers in the range of ten, i.e. 8=10-2. Knowledge of the multiplication table and the number range of 100. Using analogue extending of the multiplication table and the number range beyond 100, i.e. 6 x 6 = 36; 6 x 60 = 360; 60 x 60 = 3600 etc.

7. Abacus made for the Blind The first abacus made for the blind has been used in Japan in the 1920’s. That abacus has a layer under the beads to envoi the easy move of the beads. The produced number can safely be proved/read through touch.

© Csocsán, E. & Sjöstedt, S

8. Usage Some methodical principles Pre-requisite skills for the efficient learning of abacus: • Demonstrating the correct finger movement in using abacus i.e. setting, clearing • Demonstrating correct hand positions in using abacus • Understanding the concept of complement of addition and subtraction • Memory of Multiplication tables • Memory of squares of numbers and squares of numbers 1 to 25 in higher classes • Understanding of the relation between multiplication and division. 9. Short manual The fundamental idea to good technique is the rule to always work from left to right. It's one of the abacus’ biggest advantages, the numbers are added and subtracted in exactly the same way we read and hear them. Calculations begin with clearing the abacus.

© Csocsán, E. & Sjöstedt, S

Addition 26 + 13 Set the 26 in the right place (on the second and first column on the right hand side) Add one (bead) on the column of the tens Add 3 on the column of the ones The answer is 39. 32 + 46 Set the 32 on the right end of the abacus. Add the forty on the column of the tens: plus fifty, minus ten Add 6 on the column of the ones. The answer is 78. 134 + 252 Set the 134 Add two hundred, plus fifty. Add two so: plus 5, minus 3. The answer is 386.

Subtraction Subtraction is the complement to addition. The rules followed in abacus addition are applicable in subtraction too. The highest digit must be subtracted first. 79 – 18 Set the 79 in the right place. Start to subtract the highest digit of the 18: minus ten on the column of the tens. Then minus 8 on the column of the ones. The answer is 61. 132 – 41 Set the 132. The 40 you subtract as follows: Minus 100, plus 60. Then you subtract the 1 on the column of the ones. The answer is 91. 765 – 327 Set the 765. The 300 you subtract as follows: minus 500, plus 200. The 20 you subtract as follows: minus 50, plus 30. Minus 7 you subtract: minus 10, plus 3. The answer is 438.

© Csocsán, E. & Sjöstedt, S

Multiplication For setting the multiplicand at the right side of the abacus, count the number of digits of the multiplier. If the multiplier has one digit, you set the multiplicand so, that two columns on the right hand side of the abacus are free. If you have a two digit multiplier, place the multiplicand so that you have three free columns on the right. The multiplier is always set to the extreme left. The multiplier and the multiplicand together are called factors. 23 x 6 Set the 23 on the fourth and on the third columns. Set the 6 on the left end of the abacus. 6 times 3 equal 18. Set the 18 in the right place. Clear the 3 of the multiplier. 6 times twenty equals 120. Add this to the 18. Clear the twenty. The answer is 138. 312 x 8 Set the 312 on the fifth, the fourth and the third columns. Set the 8 to the left. Follow the steps: 8 x 2 = 16, set in the right place. Clear the 2 8 x 10 = 80. Add it. Clear the 10. 8 x 300 = 2400. Add it. The answer is 2496. 43 x 21 = Set the 43 in the fifth and in the fourth columns. (Three columns on the right side are free because of the two digit multiplier.) Set the 21 on the left. The steps are 20 x 3 = 60, set in the right place 1 x 3 = 3, add it to the 60 Clear the 3 20 x 40 = 800, add to the former sum 1 x 40 = 40, add to the former sum Clear the 40. The answer is 903.

© Csocsán, E. & Sjöstedt, S

Division Division is complement to the multiplication. You place the dividend on the right hand side of the abacus and set the divisor on the left. You will set the quotient on the columns before the dividend. First make an estimation concerning the number of the digits of the quotient. See the example: 78÷6= We set the 78 on the second and on the last columns on the right hand side. We estimate the number of the digits of the quotient. We have now one digit divisor. So we will get a two digit quotient. We start with the tens. 7 divided by 6 is 1. Set the 1 on the fourth column from the right end. 1 x 6 equal 6. We subtract this from the 7. We get 1 (the rest of the first step) put the ones together with that. 18: 6 equal 3. Set the 3 on the third column. 3x6 equal 18. We subtract this from the 18. The rest is 0. The answer (quotient) is 13. 156 ÷7= Set the 156 to the right. We should start with the hundreds but 1 hundred cannot be divided by 7. We take the tens. 15: 7 = 2. We will get two digit quotient again, so we set the 2 on the fourth column. 2 x 7 = 14. 15 – 14 = 1. Put together with the ones. 16: 7= 2. Set the 2 in the third column. 2x7= 14. 16 – 14 = 2. The rest is 2. The answer is 22 with the rest 2. 144÷12= Set the 144 to the right and the 12 to the left. We are going to get two digit quotient, so, we start on the fourth column. 14:12= 1. Set it on the fourth column. 1x10 = 10. Subtract it from the 14. 1x2 = 2. Subtract from the 4. The rest is 2. Put together with the ones. 24:12 = 2. 2x10= 20. 24-20 =4. 2x2=4. 4-4=0. The rest is O. The answer is 12.

Examples to computation with decimals and fractions you find in the book Mathematics Made Easy by Dr. Mani, edited by L. Campbell. Publication of ONNET/ICEVI 2005 Philadelphia ISBN: 1-9305526-02-4

10. References Csocsán, Emmy – Klingenberg, Oliv – Koskinen, Kajsa-Lena – Sjöstedt, Solveig: Maths „seen“ with other eyes – A blind child in the classroom – teacher`s guide in mathematics. Schildts Förlags. 2002. Mathematics Made Easy by Dr. Mani, edited by L. Campbell. Publication of ONNET/ICEVI 2005 Philadelphia ISBN: 1-9305526-02-4 http://www.joernluetjens.de/start-eng.html www.joernluetjens.de