Project Math Access

Teaching Mathematical Concepts

Basic Number Facts and Operations

Collaborative and Inclusive Strategies

The Personal Perspective of Abraham Nemeth

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 talking 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 talking 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 at a higher level than that of arithmetic skill (Baggett, 1995), enriching mathematics lessons to include some problems which are at the student’s level of thinking while beyond his or her arithmetic skill level will encourage curiosity and persistence in mathematics. This can be achieved with the use of calculators. Breaking Away from the Mathematics Book (Baggett, 1995) is an excellent resource for ideas about teaching the functions of a calculator, as well as calculator based games and activities.

Talking calculators may be used in educational settings in a variety of ways, including:

- To practice basic facts
- To improve the speed and accuracy of computational skills
- To provide students who have mastered basic operations with a competitive calculation tool
- To provide an alternative to the abacus
- For advanced mathematics and science calculations that are too complex for computation with the braillewriter

For students with additional motor disabilities which impede the use of the abacus, or with cognitive disabilities which hinder comprehension of mathematical concepts or rote learning

- For independent checking of computation, whether using mental math or other aids such as the abacus
- For assisting in other school subjects such as bookkeeping, business, geography, and cooking.

Although a variety of calculators with distinct characteristics are available to aid in a number of unique tasks, the following characteristics are essential:

- Large separated keys for increased accuracy
- Adjustable volume of auditory output
- Adjustable speed of auditory output
- Number keys set apart from the “off’ key to avoid errors.

There are several additional specific questions to consider when selecting the most appropriate calculator for an individual student. These include:

- What type of output is needed?
- What are the applications for this calculator?
- What specific functions or type of calculator is needed (square root, logarithms, standard deviation, adding lists, etc.)?
- What is the range of numbers to be used? It more than eight, it may be necessary to use a calculator with scientific notation.
- What is the level of accuracy (number of decimal places) needed?
- Is there a need for a review function (to review a calculation already entered) or a read key (to reread the number on the visual display)?

- If at all possible, students should learn to understand the concepts of operations first, with manipulatives, hands-on problem-solving, and aids such as the abacus and braillewriter, before relying on the calculator to provide the answers. During these early stages, the calculator can serve as a motivating reinforcer and checking aid for their work.
- It is important to stress development of the ability to estimate and consider the reasonableness of the answers they obtain.
- When considering hand use, the type of task may be relevant. When working from auditory dictation, the dominant hand could be used to calculate, while the other can locate and hold the place. When working from problems in a book, the dominant hand can be used to read the book or to keep place in the book, and the other hand can calculate. Students should develop an effective strategy for themselves.
- The student must devise an efficient finger placement approach. The first three fingers might be placed across the top row of digit keys, moving from this position to depress other keys, and returning to this position before moving on to other keys. Some calculators have a tactile marker on a central key (often the 5 key), and for some students, this base position may be very helpful.
- Several characteristics of calculators can be confusing to younger children. For example, calculators with auditory output often read the decimal point automatically; simply explain this or avoid this type of calculator with younger children. Several calculators read the division sign as “over”; again, simple explanation and comparison to writing methods should avoid confusion.

There are several electronic notetakers, which combine speech synthesis and braille, available. These devices can be used by blind students of any age, as tools for performing mathematics calculations. Each contains a calculator function. We recommend the use of those which have a braille display in order that the braille-reading student has the option of being able to read calculation results in braille. This is particularly advantageous when the factors involved in the calculation or the results contain many digits. Some of the notetakers have models which do not contain braille displays. We do not recommend their use for braille-reading students who use the calculator function.

The calculator function can be used to perform a variety of different mathematical operations. Using these devices, whole numbers and decimals can be added, subtracted, multiplied, and divided; square root and percentages can also be calculated; and the precision can be easily set. Strings of calculations can be performed at one time. The results can be stored in several different memory locations, and can be retrieved later for inclusion in other strings of calculations.

The calculator function also includes a scientific calculator in which trigonometric and logarithmic calculations can be performed. In addition, it contains translation tables in which values in the English system and metric system can be converted to the other system (e.g., kilometers to miles and miles to kilometers).

- Practice entering numbers, storing and retrieving from memory, and basic operations as well as calculator functions by having students enter numbers which are meaningful to them (i.e., their age, number of siblings, telephone number, etc.). Have one child call out a number, while others enter it in their calculator.
- Provide the student with an actual braille menu from a popular fast food establishment. Calculate the total bill for a variety of selections (i.e., sandwich, salad, beverage, and dessert); teach the % function and add tax to the total as an additional challenge.
- Braille a grocery or other retail store receipt in its entirety. Help the student to identify different parts of the receipt, such as the date, the store number, register number, clerk or cashier number, and item prices. Have the student check the total price; this will also serve as a check on keystroke accuracy.
- Use the calculator to compare unit prices. Have the student find out the prices on 4 different sized packages or brands of the same product, e.g., different sized cans of peas. In addition to walking through the steps involved in making a decision about which would be the most prudent purchase for an individual consumer (taking serving size, perishability, and so forth, into account), the student can practice multiple operations and the use of the memory function.
- In a group of three or more students, calculate the individual cost of sharing a pizza equally. Although a cardboard cutout would work, real pizza would be the most effective prop! Use calculators to divide the total cost of the pizza by the number of pieces; and to divide the total number of pieces by the number of students. Again, use calculators to figure the cost for each individual. An additional challenge would be to have an odd number of pieces; the students have to figure out how to distribute the cost.

Baggett, P., & Ehrenfeucht, A. (1995). *Breaking away from the
mathematics book: creative projects for grades K-6*. Lancaster, PA: Technomic Publishing Co., Inc.