Advanced Mathematics

The study of algebra, geometry and other advanced mathematics topics by blind individuals presents several challenges. One of the most serious of these is the fact that blind students cannot visualize the graphical representations of complex mathematical concepts which aid sighted students in their understanding of those concepts. An obvious example of this problem is the difficulty in representing three dimensional objects which may be the focus in the study of certain advanced topics such as geometry. However, even basic concepts often taken for granted can present difficulty for a student who is blind as he or she pursues more advanced study. An individual who is blind and who is employed as a programmer-analyst at a major university illustrates this in her comment:

I have been totally blind since birth and have studied algebra, geometry, and calculus. I found geometry especially difficult because I lacked the understanding of many spatial concepts ... I found that I had difficulty understanding such concepts as how four walls meet the ceiling, and I actually stood on a chair to study this (Dick & Kubiak, 1997, p.344).

Specific issues and guidelines related to the facilitation of the student’s use of tactile displays and graphics to aid in such study is described in the section on tactile graphics.

A second problem relates to the fact that a student who is blind cannot visualize complex mathematical expressions in their entireties. He or she must view these expressions in small portions rather than all at once as their sighted peers do. This inability to view mathematical expressions in their entirety requires the student to retain large portions of the expression in his or her memory as it is manipulated. This can make the study of algebra and other advanced topics more difficult for a less capable student who is blind than it is for his or her sighted peer with equal cognitive ability.

Techniques which facilitate students’ abilities to calculate and remember mathematics facts mentally, and to use tools for keeping track of partial sums and products, can be very helpful in this regard, although they will not eliminate the problem.

As with the development of basic concepts, the need for concrete experiences is critical to the development of many advanced mathematical concepts for a student who is blind. Acting out story problems and applying these problems to everyday situations is just as important at the advanced levels of mathematics as it is at basic levels, if students are to become capable of using their mathematics skills in functional situations or as a foundation in their pursuit of even more advanced mathematical and scientific learning.

The use of models, manipulatives, and real life items found in everyday classrooms and living environments also plays an important role in providing support to the development of mathematical skills and concepts at all levels. These items can be used for functional measuring, comparing, deduction and induction, as well as for motivating students to solve relevant problems. The following strategies are intended as illustrative examples of the use of concrete experiences to support higher mathematics concepts. They are included here as representative samples only; teachers are encouraged to develop an array of concrete experiences and problem-solving situations to help students understand the concepts involved in higher mathematics.

Strategies for teaching concepts in higher mathematics

• Many students are confused by the signs of inequality (< and >), and inappropriately apply one of them when the other is needed. Since the symbols for each of these is a two-celled symbol, he or she needs only to remember that the cell containing only one braille dot is always pointing to the value which is less than the other. For example, in “nine is greater than four” (9>4), in which the Nemeth Code greater than sign is composed of dots 4-6, dot 2, the single dot 2 is pointing toward the lesser value, four. Likewise, in “four is less than nine” (4<9) in which the Nemeth code less than sign is composed of dot 5, dots 1-3, the single dot 5 is still pointing toward the lesser value, four.
• When using a braille keypad, each sign of comparison begins with a configuration formed with the right hand.
• Each of these symbols in braille retains one cell of the two cells of the braille symbol for equality. This can be used to demonstrate that the compared relationship is skewed in favor of one direction. The braille symbol for equality uses two braille cells that are symmetrical (dots 4-6, 1-3), just as the left and right side of an equation are symmetrically balanced.
• When presenting the braille symbols and teaching the concepts of intersection (dots 4-6, 1-4-6) and union (dots 4-6, 3-4-6), note that the braille “plus” sign (dots 3-4-6) is part of the symbol for union. This is because the new set which results from combining two original sets is similar to the sum in an addition problem.
• Print users often compare the symbol for union to an uppercase letter “U”. A comparison for a braille student would be to notice that the dot configuration for the braille letter “U” is the first cell of the symbol, and the reversed image of the braille letter “U” is the second cell. This might be confusing, however, for a student with reversal problems.
• To teach the concept of intersection to a student who is blind, use an example similar to the following. List the students who are in the band. Make a roster of the students who are in the chess club. Then determine which students are in the band and also in the chess club. Those students then comprise the intersection of the set of band members and the set of chess club members. Other sets of students can be developed to meet the requirements of the particular situation. This concept can be presented later with the use of raised line diagrams.
• To teach the concept of intersection at a more symbolic level, the teacher can use two rings about 8 inches across, with cellophane stretched across one frame and a very thin fabric across the other. Label one ring A and the other B. Have the student examine each frame separately, then slide one over the other so that only a portion overlaps. Clip the frames together so that the student can now examine the area which overlaps. This area is the intersection. The student can express this with the statement A ? B.

  The teacher can later place braille labels on each of the materials covering the frames; two of the labels should be duplicated on both frames. The duplicated labels should be positioned toward the edge of the frame so that, when the frames are partially overlapped, these duplicated labels will be included in the overlapped portion. A raised line drawing of circle A, circle B, and the partial overlapping of A and B can later be presented to the student for examination.

• To demonstrate the concept of union, the student can place several items in each of two boxes (e.g., Box A contains a stylus, pen and paper clip, while Box B contains a safety pin, key ring and slate). The student then empties the contents of both boxes into a third box or into a tray. The result is the union of the two boxes. Later, duplicate items can be included, but when the two boxes are combined, all duplicate items must be removed before identifying the elements forming the union.
• At a more symbolic level, the rings A and B described above can be examined, first separately, then again in the partially overlapped position. This time, however, the union is represented by all the area covered by either or both the frames. If the rings are used with the labels, then the union includes all the labels on either or both of the overlapped frames, minus any duplicate labels that may be present. The student can express this with the statement A U B. Later, a raised line drawing can be used to represent this concept.
• A mnemonic device for remembering factor polynomials is FOIL:

  F = first
  O = outside term
  l = inside term
  L = last term
  

• A “number line” made of stairs can be used to teach the concept of signed numbers. Take the students to a landing between floors, with stairs going up and down. The landing is zero. The stairs going up from the landing are positive numbers, while the stairs going down are negative numbers. Have the student go to “positive seven” (+7), or seven steps up. Then ask the student to add a “negative nine.” To do this, ask the student which way is negative? When the student responds “down”, ask the student to move down nine steps, and to tell where he is in relation to the landing [he is at “negative two” (-2)]. The student can relate what he has experienced with the number sentence “+7+-9=-2.” This process can be continued with additional addends of both positive and negative value.
• In algebra, to apply the concept of equality as being two amounts which are the same, use a balance scale which can hold two dishes. Have the left dish represent the left side of an equation while the right dish represents the right side of the equation. If the scale has a needle, it must be perpendicular to show equality (serving as the equal sign). If the scale does not have a needle, the fulcrum can serve as the equal sign.

  Have the student place a weight in one dish and try to balance it by placing an equivalent weight in the other. Students could start with a 2 g ram weight in each dish; later they could place a 10 gram weight in the left dish, and use smaller weights in the right dish to balance. Students can later add and subtract equal values to the two balanced dishes.

• To learn about combining like terms, students can use what they already know about place value. The teacher can make a place value chart applicable to algebra by using library pockets arranged in columns, and labeled as below in braille. A set of cards can then be brailled with values for each column: 0, +1, +2, +3, +4 +5 +6 +7 +8 +9, +0x, +1x, +2x, +3x, +4x, +5x, +6x, +7x, +8x, +9x, +0x2, +1x2, +2x2, etc. On the reverse side of each card, the same value should be brailled with a negative sign.

  First, students can review place values and their application to algebra by going over the chart. The teacher should stress that values in a column may only be combined with values in the same column, and that their “value” is formed by multiplying the place’s value by the amount in that place. Just as place values can accommodate decimals (fractional values with powers of ten as the denominators), this chart can be expanded to include variables raised to negative exponents, using pattern analysis to continue the series with ascending negative exponents (e.g., +x2, x, etc.).

  The student can arrange the brailled cards to work specified problems. As they work problems, they combine like terms by collecting only the cards in one column at a time, adding them together, and recording the results in descending order of the power of the variable (e.g., 17x2-6x+2y-9).

• To develop the concept that variables represent unknown amounts, students can use a balance scale which holds two dishes. The teacher brailles a variable (“y”) on a light weight bag, places a gram weight in the bag, and seals it shut (the student does not know the weight placed in the bag). The bag, along with one or more other weights (e.g., 7 grams), is placed into one dish, while enough weights to balance (e.g., 37 grams) are placed in the other dish.

  To observe the process of isolating variables, the student must remove the same number of gram weights from both sides of the scale until only the bag is isolated in one dish. By deduction, the student can determine the amount of grams that are in the bag by counting the weights in the other dish which balances it. The student can then open the bag and weigh the weight to confirm its value. The student can then braille the process he or she has carried out in an algebraic equation.

• To teach the concept of the distributive property for X and demonstrate how a term outside parentheses is distributed to each term inside parentheses, teachers can use a painting activity. First, the teacher labels a box with braille mathematics parentheses (dots 1-2-3-5-6 and 2-3-4-5-6). The teacher then places a selection of items (e.g., an eraser, paper cup, ruler) in the box, with each item labeled as a variable (x2 for the eraser, +7x for the cup, -12 for the ruler). Then the student can work the problem

  2x(x2+7x-12)

  The student is directed to brush all the items in the box with water (represented by 2x), and then leave the items out of the box “to dry”. When asked how the items outside the box are different from when they were in the box, the student can identify that now all the items are wet and that each item was altered by the water. In the same manner, all terms (represented by the items) are multiplied by the term (represented by the water) outside the parentheses (represented by the box).

  Therefore, as the student applies water to each item,

  2x(x²) = +2x³
  2x(+7x) = +14x²
  2x(-12) = -24x
  Arranged horizontally: 2x³+14x²-24x
  

• To observe that angles are not affected by the length of their rays, students can place items such as the long cane perpendicular to the floor and use their braille protractor to measure the right angle formed, noting that one ray (the cane) is much shorter than the other (floor). Students can also use differing lengths of yarn and align them as rays to form specified angles.
• To demonstrate the concept of correct movement of the decimal point in metric, the student can use a paper plate as a “dancing decimal point.” A group of students can stand in a row, each with an assigned number. The paper plate decimal point is moved between each student, either to the left or to the right, depending on whether there is a change to a larger metric unit or to a smaller metric unit. For example, students could be named with each of the following digits: 2, 5, 9. To have the number represent 259 meters, the decimal point plate can be placed to the right of the 9. Then, to change to kilometers, the decimal point plate is moved three places to the left, before the 2, representing 259 kilometers, and so on. If there are not enough students, lined up chairs could also be used.
• To teach the concept of hierarchal mathematics operations, the teacher can use a set of steps made with boxes, arranged as below, with one side open so that the “rules” for each step or level can be inserted. For example, using these steps, students can perform operations in a problem in the correct sequence. If a problem has more than one of the same level of operations (e.g., a division and a multiplication operation), the operations should be performed starting from the left side of the problem.
• To remind students that rules which apply to one level of an operation do not necessarily apply to another, short “help” cards can be brailled and placed in the hierarchal boxes described above, or in a notebook for quick reference. For example,

  Rules applying to fractions include:

  • For addition/subtraction, the denominators must be the same; add the numerators
  • For multiplication, multiply numerators across; multiply denominators across

  Rules applying to directed (signed) numbers (when working with only 2 numbers at a time) include:

  • For addition, if the signs are the same, add the values; the sum gets the sign of the original numbers. If the signs are different, subtract the values; the difference gets the sign of the larger (absolute value) original number.
  • For multiplication and division, like signs (+x+, -x-) result in a positive number, while unlike signs (+x-, -x+) result in a negative number.

The examples described above and other like activities can be used to good advantage in clarifying fundamental concepts in upper level mathematics.

References

Dick, T., & Kubiak, E. (1997). Issues and aids for teaching mathematics to the blind. Mathematics Teacher, 90, 344-349.