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teaching approaches

there are several approaches to teaching the use of theabacus. since one method might not work effectively for all students, teachersshould be familiar with several methods. the most commonly used approaches are:

• the partners or logic approach,
• the secrets approach,
• the counting method, and
• adaptations or combinations of these approaches.

each is briefly described below, with an example (3+4=7)worked out according to that approach.

the logic method or partner method focuses on understandingthe “what” and “why” of the steps in solving a problem on the abacus. itrequires that the student know the partners or compliments of the numbers up toten (5=2+3, 5=1+4). verbalizing the steps and the reasons for each movementmade on the abacus is an important feature of this approach. at first, theteacher must explain the steps and reasons as the student works through theproblem. then the student should

verbalize the process as he or she works the problem. overtime, this “conversation” can be shortened, and finally the process isinternalized. this approach would benefit students who can follow theexplanation and can understand, and even enjoy, the logical concepts involved.

example: “the problem is 3+4. what number comes first? theanswer is 3. so we set the 3 on the abacus. now we need to add 4. do we have 4more ones to add? no. since we don’t have enough ones to add, we can add the 5bead (set 5). but 5 is too many; we only wanted to add 4. so we’ll have toclear the extra bead or beads. what is 4’s partner in 5? the answer is 1. sowe’ll clear one extra bead. now what is our answer? the answer is 7.”

example: “the problem is 3+4. what number comes first? theanswer is 3. set 3. can you set 4 directly? no. what is the smallest amountthat can be set that is greater than 4? the answer is 5. set the 5 bead. howmany more is 5 than 4? the answer is 1. clear 1 bead. what is the answer? theanswer is 7.”

the secrets method focuses on the process of moving theabacus beads in a particular sequence, following a specific set of rules fordifferent numbers and operations. it does not emphasize the understanding ofthat process, rather the rote memory of the bead movements. it would beappropriate for students who would benefit from a manipulative process theycould rely on, without having to fully understand the principles behind eachstep of that process.

example: “the problem is 3+4. what number comes first? theanswer is 3. so set 3 (raise three earth counters). now we want to add 4. inorder to do that, we must set 5 (bring down a heaven counter) and clear 1(clear one earth counter). what is our answer? the answer is 7.”

the counting method has the student count each bead as it isadded or subtracted, moving from the unit beads to the 5 beads (but countingonly 1 for all beads). there are also specific rules regarding certain numbersand operations, but fewer than the full set of secrets. it does not emphasizeunderstanding the concepts behind the bead movements. this approach could alsobe appropriate for youngsters who would benefit from a manipulative processthey could rely on, without having to understand each individual step.

example: “the problem is 3+4. what number comes first? theanswer is 3. so we set 3. now we want to add 4. to do that, we push up anotherunit bead (count 1), then another unit bead (count 2), then push down the beadabove the counting bar (count 3), and clear all four beads under the countingbar. finally, push up one more unit bead (count 4). what is the answer? theanswer is 7.”

there are several resources available which demonstrate howto teach the use of the abacus employing the above approaches. abacus made easy(davidow, 1975) utilizes the logic approach. the japanese abacus: its use andtheory (kojima, 1954) describes the secrets approach. abacus basic competency(millaway, 1994) employs the counting approach. use of the cranmer abacus(livingston, 1997) explains both logic and counting approaches.

teachers of blind children have made a variety ofmodifications to all of these approaches in order to meet the individuallearning styles of their students. for example, students included in theregular classroom for much of the time can work their addition and subtractionproblems from right to left to coincide with the way the teacher works throughthe problem with the class.

an example of a more specific modification relates todivision, and is sometimes referred to as the “subtraction method” of division.the divisor is placed to the far left on the abacus, then 2 columns are leftblank, followed by the dividend. the quotient is the sum of partial answersobtained as the student works through the problem, and is placed to the farright.

another specific example of a modification of the logicmethod involves multiplication of one, two or three digit multipliers and oneor two digit multiplicands. for example, in the problem 93x25, the first factor(93) is set in the billions place, the second factor (25) in the millionsplace, and the answer in the thousands and hundreds places. instead of workingfrom the outside in, the entire multiplicand is multiplied by the first digitof the multiplier; then the entire multiplicand is multiplied by the seconddigit of the multiplier.

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