Project Math Access

Teaching Mathematical Concepts

Basic Number Facts and Operations

Collaborative and Inclusive Strategies

The Personal Perspective of Abraham Nemeth

According to the National Council of Teachers of Mathematics, all students need meaningful mathematics with appropriate tools. It is particularly important for students with visual impairments to be exposed to the use of multiple mathematics strategies and tools, including use of the abacus, braillewriter, mental mathematics, talking calculators, and the calculator function of electronic notetaking devices. These strategies and tools should be introduced early in a child’s educational program, and continually reinforced throughout the school years.

This section deals with the recommended sequence of instruction of the braillewriter, the abacus, and the talking calculator as tools for arithmetic calculation. Each of the three tools has specific advantages and disadvantages.

Of the three tools, use of the braillewriter is the most time-consuming and cumbersome. The braillewriter is extremely slow and awkward to use as a calculation tool. Using it for this purpose is analogous to doing mathematical calculations with a typewriter by a sighted individual. Under these circumstances, the reader may dispute the advisability of expending the time and effort needed to teach the use of the braillewriter as a calculation tool. Nevertheless, blind students benefit from knowing the steps which are required to carry out arithmetic calculations in this manner. In general, while it should not receive heavy emphasis and should be combined with the use of other tools, the braillewriter as a calculation tool should be introduced at the beginning of mathematics instruction. For details on how to use the braillewriter as a calculation tool, the reader is referred to the Braillewriter as a Calculation Tool section.

Use of the braillewriter emulates how sighted individuals perform arithmetic calculations. Although it is not absolutely necessary for blind persons to perform calculations in the same manner as sighted individuals in order to be successful, it is beneficial to the blind learner to know the processes which are necessary to perform general arithmetic calculations. For true understanding of the procedures for carrying out arithmetic calculations, and thorough comprehension of basic mathematics, the braillewriter is the only device available to blind students. It is the only way for a blind student to write the steps involved in carrying out an arithmetic operation. In the past, arithmetic slates such as the cubarithm and the Taylor slate were advocated for this purpose. These are no longer used in up-to-date programs for blind students because experience has shown that these tools are very cumbersome and inefficient when compared to the use of the braillewriter.

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 section. 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. For more information regarding the use of the abacus, the reader is directed to the Abacus section, which is devoted to that topic

As described above, during initial instruction in arithmetic operations, the braillewriter and abacus should be the major tools used in calculations. The talking calculator should only be used as a reinforcer for skills learned with the braillewriter and abacus until a student masters the fundamental concepts involved in computation. As the student becomes more proficient with the braillewriter and abacus, and demonstrates understanding of basic mathematics concepts, progressively less emphasis should be placed on the braillewriter and more emphasis should be placed on the use of the calculator.

Eventually, the calculator is likely to become the student’s major tool for performing calculations. This is especially recommended in the advanced study of mathematics, for example, algebra, where the emphasis is upon learning content far advanced from the simple performance of arithmetic calculations. Steps should be taken to give the student the most efficient tool to use so he or she is not expending inordinate amounts of time in the performance of arithmetic calculations, but rather devoting study time to mastering the subject matter content of a course.

While the calculator is the most efficient method for a blind student to perform arithmetic calculations, it has two major disadvantages. First of all, reliance on the calculator does not afford the student the advantage of practice in the underlying steps needed to perform the calculation. One can use a calculator without actually understanding the underlying mathematical principles. Secondly, heavy reliance upon use of the calculator results in the loss of instant recall of the basic addition, subtraction, multiplication, and division facts. If a calculator is used too early, while the youngster is learning the basic facts, he or she will not have immediate recall of mathematics facts in his or her repertoire of skills. On the other hand, the calculator enables students to solve problems which are challenging and interesting, since intellectual development is often more advanced than the ability to perform the actual calculations. For additional information regarding calculators, the reader is directed to the Talking Calculators section.

The ability to calculate mentally with efficiency is a very important skill for all students, but especially for visually impaired and blind students. Using the braillewriter, and the abacus can be very labor intensive and time consuming, and calculators have their own limitations, as described above. The more efficiently students can estimate, calculate, and check the reasonableness of answers using mental math techniques, the more facile they will be at using numbers, in both schoolwork and independent living skills. These strategies should be taught to students as soon as they begin to count and work with simple numbers. For more detailed information regarding strategies for teaching mental math techniques, the reader is directed to the Mental Math section