Project Math Access

Grasping the fundamental concepts underlying the addition, subtraction, multiplication, and division of fractions is difficult for many students. Not being able to visualize the concepts as they may be depicted in pictures makes the understanding of these fundamental operations even more difficult. One can train students to readily calculate the correct answers to problems containing fractions, but being able to complete the operations correctly does not guarantee that the student has a fundamental understanding of the meaning of the four operations as they pertain to fractions. What does it mean to add 3/4 and 1/2? What does it mean to subtract 1/2 from 3/4? What is the basic meaning of the answer to the problem, 3/4 times 1/2? What does it really mean to divide 3/4 by 1/2? The use of manipulatives to illustrate the basic operations in which fractions are involved will enhance understanding by the blind learner. The following illustrate that point.

In the problem, 3/4 added to 1/2, the teacher might use two cardboard circles divided into fourths. Three pieces from one circle represent 3/4. Two pieces from the other circle represent 1/2. Bringing the three pieces (3/4) together with the two other pieces (1/2) results in a total of five pieces (5/4). The pieces can be re-assembled into one whole circle comprised of four pieces, with an additional single piece representing 1/4. The answer, thus, is 1 1/4. The manipulation of the pieces provides concrete evidence of the concept of adding fractions.

In subtracting 2 1/2 from 3/4, a similar activity would be effective. In this case, three pieces would be displayed. To subtract 1/2 from 3/4, one would simply remove two of the pieces, representing 2/4 (1/2). The remaining piece, then, is the answer: 1/4.

Multiplication of fractions is a somewhat more difficult operation to understand. If 3/4 and 1/2 are multiplied, the result is 3/8. Multiplying 1/2 times 3/4 actually is asking the question, what is 1/2 of 3/4? To illustrate this point, a cardboard circle is divided into eighths. To convert the fraction 3/4 to 6/8, 6 pieces of the circle are used, each of which constitutes 1/8 of the circle. One-half of 6 is 3. Thus, multiplying 1/2 times 3/4 is 3/8.

The basic meaning of division of fractions is even more difficult to comprehend. If one divides 3/4 by 1/2, the result is 1 1/2. This is the way one would determine how many 1/2’s are in 3/4. Once again, the teacher can display a partial circle with three 1/4 pieces. Two of these 1/4 pieces are removed, representing one 1/2. The remaining piece represents 1/4. The fraction 1/4 is one-half of 1/2; thus, there are one and one-half 1/2’s in 3/4.

The following strategies can facilitate the student’s working with fractions:

- The student should be instructed that what is between the opening fraction indicator (dots 1-4-5-6) and the fraction line (dots 3-4) is the entire numerator, and what is between the fraction line and the closing fraction indicator (dots 3-4-5-6) is the entire denominator.
- When teaching order of operations, students need to learn that all computations above the horizontal fraction line are to be completed, all computations below the horizontal fraction line are to be completed; then division of the numerator by the denominator can be carried out.

The following is attributed to Professor Abraham Nemeth (reference unknown). He recommends use of the following language in order to communicate with clarity when reading mathematics problems containing fractions to a blind student:

A simple fraction (which has no subsidiary fractions) is said to be of order zero... A fraction of order 1 is frequently referred to as a complex fraction, and one of order 2 as a hypercomplex fraction. Complex fractions are fairly common, hypercomplex fractions are rare, and fractions of higher order are practically non-existent. The order of a fraction is readily determined by a simple visual inspection, so that the sighted reader forms an immediate mental orientation to the nature of the notation with which he is dealing. It is important for a braille reader to have this same information at the same time that it is available to the sighted reader. Without this information, the braille reader may discover that he is dealing with a fraction whose order is higher than he expected, and may have to reformulate his thinking accordingly long after he has become aware of the outer fractions.

The reader is referred to the section on spoken math for detailed information regarding how fractions should be spoken.