Project Math Access

Teaching Mathematical Concepts

Basic Number Facts and Operations

Collaborative and Inclusive Strategies

The Personal Perspective of Abraham Nemeth

The abacus, in particular the Cranmer abacus, is certainly one of the most effective calculation tools for blind children, for both low and high achievers, when used either alone or in conjunction with other devices. It allows concrete manipulation, leading to a more meaningful understanding of numbers than does the use of calculators, and it provides an alternative to lengthy and involved calculations done on the braillewriter, although the ability to work with these two tools is also very important and is discussed elsewhere in the Braillewriter as a Calculation Tool and Talking Calculators sections. In fact, the combined use of the braillewriter and the abacus can be very effective; students can use the abacus to check their work on the braillewriter and, if there is a discrepancy, rework the problem using both tools. Generally, it is recommended that the student progress from using the more cumbersome tool to the less cumbersome tool. For some students, picturing the working of problems on the abacus has even increased their ability to carry out calculations mentally.

The abacus is also useful because of its speed, accuracy, portability, and flexibility. It can be used for educational purposes to support a good foundation in addition, subtraction, multiplication, and division. It can also be used to carry out calculations involving fractions and decimals, as well as an aid in completing arithmetic operations included in higher level mathematics. It can also be used for independent living skills such as recording phone numbers, or tabulating costs while shopping.

Skill in the use of the abacus depends on several factors. One of these is readiness. For example, in order to begin working on the abacus, children must understand basic number concepts, be able to count, and know the partners or complements that make up the numbers up to ten. These concepts should be taught concretely with manipulatives, by forming and rearranging sets of objects. Students also need to learn that some beads on the abacus stand for one, some for five, some for ten, and so on, as well as knowing the basic concept of place value. As students work with sets, they can set relevant numbers on the abacus. As they learn about place value, they can reinforce this concept with the abacus. As they become comfortable with all the partners of the numbers up to ten, they can set simple number statements on the abacus. Familiarity with the abacus should start early. Young students should be encouraged to use it in limited ways as they develop number concepts; they can use it as a calculating tool later, as their skills progress.

Students also need the manipulative skills required to operate the abacus itself. Some teachers prefer starting young children on the enlarged abacus because of the larger beads and the greater space between the beads; students can then make the transition to the smaller abacus when appropriate.

A third requirement for the student’s success with the abacus is his or her teacher’s competency and attitude. Teachers must take responsibility for developing their own skill to the level sufficient to teach their students the skills that will benefit them. Teachers must also convey a positive attitude about the use of the abacus, making its use-and the effort required to learn its use-rewarding for the student.

There are several approaches to teaching the use of the abacus. Since one method might not work effectively for all students, teachers should 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 understanding the “what” and “why” of the steps in solving a problem on the abacus. It requires that the student know the partners or compliments of the numbers up to ten (5=2+3, 5=1+4). Verbalizing the steps and the reasons for each movement made on the abacus is an important feature of this approach. At first, the teacher must explain the steps and reasons as the student works through the problem. Then the student should verbalize the process as he or she works the problem. Over time, this “conversation” can be shortened, and finally the process is internalized. This approach would benefit students who can follow the explanation and can understand, and even enjoy, the logical concepts involved.

Example: “The problem is 3+4. What number comes first? The answer is 3. So we set the 3 on the abacus. Now we need to add 4. Do we have 4 more ones to add? No. Since we don’t have enough ones to add, we can add the 5 bead (set 5). But 5 is too many; we only wanted to add 4. So we’ll have to clear the extra bead or beads. What is 4’s partner in 5? The answer is 1. So we’ll clear one extra bead. Now what is our answer? The answer is 7.”

Example: “The problem is 3+4. What number comes first? The answer is 3. Set 3. Can you set 4 directly? No. What is the smallest amount that can be set that is greater than 4? The answer is 5. Set the 5 bead. How many more is 5 than 4? The answer is 1. Clear 1 bead. What is the answer? The answer is 7.”

The secrets method focuses on the process of moving the abacus beads in a particular sequence, following a specific set of rules for different numbers and operations. It does not emphasize the understanding of that process, rather the rote memory of the bead movements. It would be appropriate for students who would benefit from a manipulative process they could rely on, without having to fully understand the principles behind each step of that process.

Example: “The problem is 3+4. What number comes first? The answer is 3. So set 3 (raise three earth counters). Now we want to add 4. In order 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 is added or subtracted, moving from the unit beads to the 5 beads (but counting only 1 for all beads). There are also specific rules regarding certain numbers and operations, but fewer than the full set of secrets. It does not emphasize understanding the concepts behind the bead movements. This approach could also be appropriate for youngsters who would benefit from a manipulative process they could rely on, without having to understand each individual step.

Example: “The problem is 3+4. What number comes first? The answer is 3. So we set 3. Now we want to add 4. To do that, we push up another unit bead (count 1), then another unit bead (count 2), then push down the bead above the counting bar (count 3), and clear all four beads under the counting bar. Finally, push up one more unit bead (count 4). What is the answer? The answer is 7.”

There are several resources available which demonstrate how to teach the use of the abacus employing the above approaches. Abacus Made Easy (Davidow, 1975) utilizes the logic approach. The Japanese Abacus: Its Use and Theory (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 of modifications to all of these approaches in order to meet the individual learning styles of their students. For example, students included in the regular classroom for much of the time can work their addition and subtraction problems from right to left to coincide with the way the teacher works through the problem with the class.

An example of a more specific modification relates to division, 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 left blank, followed by the dividend. The quotient is the sum of partial answers obtained as the student works through the problem, and is placed to the far right.

Another specific example of a modification of the logic method involves multiplication of one, two or three digit multipliers and one or 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 millions place, and the answer in the thousands and hundreds places. Instead of working from the outside in, the entire multiplicand is multiplied by the first digit of the multiplier; then the entire multiplicand is multiplied by the second digit of the multiplier.

In addition to modifying general approaches to teaching the abacus, teachers have found several strategies that can help to facilitate students’ learning of this skill. Some of these are included here:

- Familiarity with the abacus should be started at an early age as the child begins working on number concepts; in the elementary grades it can be used to support the learning and understanding of operations and calculations, as well as the use of fractions and decimals. It can easily be used in conjunction with the braillewriter; by middle school, students should be proficient at using the abacus, making less writing with the braillewriter necessary. Guidelines for combining the use of these two tools are presented in the Calculation Tools and Aids section.
- It is important that the student develop a positive attitude about using his or her abacus. One teacher told her student that he won it, making him feel special and eager to use it. Providing simple but relevant tasks in which the student uses the abacus would also be helpful in motivating the student.
- The abacus can be used for a variety of functional and motivating classroom activities, such as keeping scores for games, tabulating scores on daily quizzes, using in simple money games (1=pennies, 5=nickels, 10=dimes).
- Games such as an abacus “bee” can add to the fun of learning the abacus. Teams can be selected in a variety of ways. Individuals can compete against individuals in a predetermined order that could rotate. Scores or winning teams can be determined in a variety of ways. Each student should have his or her own abacus in this or similar games.

Since blind students cannot see the gestalt of where their beads are placed on the abacus, it is extremely important to teach them place-keeping habits. This will be especially critical when they are dealing with problems involving multiplication, division, decimals, fractions, and any problems involving zeros.

Two abaci, placed either one above the other or side by side joined by a coupler, can be used effectively to record and sum partial products or answers on one while working additional steps of the problem on the other.

Some addition and subtraction can be carried out directly on the Cranmer abacus, that is, appropriate beads can be moved without having to make any exchanges. For example, 3+1, or 5+2 or 8-5 can be done directly because moving the beads with the real value is sufficient to solve the problem. However, this is not always the case. In a problem such as 3+2, 3 beads can be set, but 2 more beads cannot be set directly. Instead, the five bead must be set and the four previously set beads must be cleared. The logic method would have the student verbalize this process. The secret method would offer steps for moving beads in a specific manner to arrive at the right answer. A list of these secrets for addition and subtraction is presented below.

The number to add followed by the secret for that number:

- Set 5, clear 4
- Clear 9, set 1 left
- Set 5, clear 3
- Clear 8, set 1 left
- Set 5, clear 2
- Clear 7, set 1 left
- Set 5, clear 1
- Clear 6, set 1 left
- Clear 5, set 1 left
- Set 1, clear 5, set 1 left
- Clear 4, set 1 left
- Set 2, clear 5, set 1 left
- Clear3, set 1 left
- Set 3, clear 5, set 1 left
- Clear 2, set 1 left
- Set 4, clear 5, set 1 left
- Clear 1, set 1 left

The number to subtract followed by the secret for that number:

- Clear 1 left, set 9
- Set 4, clear 5
- Clear 1 left, set 8
- Set 3, clear 5
- Clear 1 left, set 7
- Set 2, clear 5
- Clear 1 left, set 6
- Set 1, clear 5
- Clear 1 left, set 5
- Clear 1 left, set 4
- Clear 1 left, set 5, clear 1
- Clear 1 left, set 3
- Clear 1 left, set 5, clear 2
- Clear 1 left, set 2
- Clear 1 left, set 5. clear 3
- Clear 1 left, set 1
- Clear 1 left, set 5, clear 4

Davidow, M. E. (1977). *The abacus made easy*. Louisville, KY: American Printing House for the Blind.

Kojima, T. (1954). *The Japanese abacus: its use and theory*. Rutland, Vermont: Charles E. Tuttle Company, Inc.

Livingston, R. (1997). *Use of the Cranmer abacus, 2nd edition*. Available at the following site: www.tsbvi.edu/publications/abacus.htm

Millaway, S. M. (1994). *Abacus basic competency*. Norristown, PA: Eye-Deal Materials for the Visually Handicapped.