A Brief Historical Overview Of Tactile And Auditory Aids For Visually Impaired Mathematics Educators and Students

"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 I had difficulty understanding concepts such as how four walls meet the ceiling, and I actually stood on a chair to study this."

- Bev Wieland, Programmer/Analyst, University of Delaware

"Teachers, as well as institutions, need to understand that if the tools are provided for people with disabilities, they become abled. I believe that the question should be answered in this way. If the tools are provided for people with disabilities, standards will have to be raised, not lowered. We are a pretty hungry bunch."

- Dick Banks, Adaptive Technology Consultant, Library Learning Center (University of Wisconsin, Stout)

For many blind individuals, learning math can be a frustrating endeavor. Few educators are prepared to teach mathematics to a child who is blind, and as mainstreaming efforts expand, increasing numbers of teachers will encounter blind students. Hopefully, this article will help to better prepare teachers for the challenge presented by mathematics education for blind students.

The National Council of Teachers of Mathematics Curriculum and Evaluation Standards advocates the use of graphing calculators. As a visual learning tool, the graphical calculator has stirred a good deal of excitement. Unfortunately, graphing calculators are not accessible to the student who is blind or visually-impaired. This article chronicles a portion of the history of mathematical and graphing aids for the blind and presents a glimpse of the future.

Computational Aids for the Blind

The Taylor slate was one of the earliest manipulative/tactile aids for visually impaired students of mathematics. It measures about 11 by 17 inches, has a tray to hold pegs and an array of holes to contain these pegs. There are 22 holes across and about 20 holes in a column. These holes are shaped similar to a plus sign overlying a "X" and the pegs have the same shape. Additionally, at each end of the one inch pegs, to one side, is a bar and at the other end two conical projections. The angle of insertion determines the numerical value, with the bar side up the positions run one through 8 and flipping the pegs over the values 9 and 0, with the remainder of the positions determining the operation. The Taylor slate was used through the 1930s and early 1940s (Garvin 1994).

The Brannan slate came into use in the late 1940s or early 1950s and is still available. This slate consists of a 16 by 16 array of square holes. These hold small dice-like cubes which have raised Braille numbers. The position of the cube determines what Braille number is being used. One side of the cube indicates the operation. Thus, blind children can set up their own problems. If the problem entails carrying or borrowing, the top row is left empty (Garvin 1994).

Circa 1960, Tim Cranmer adapted the abacus to serve as a calculating tool for the blind and visually impaired. This abacus is snugly mounted on a red felt board which prevents the beads from unintentionally slipping. The use of white beads as counters provides high contrast for the visually impaired (Garvin 1994).

Currently, Nemeth math code is used in Braille texts for the blind. If the individual is a proficient Braille reader, this code can be used for computations. The developer, Dr. Abe Nemeth, who is blind, taught mathematics at the University of Detroit. Dr. Nemeth developed the code which was adopted by the Braille Authority of North America (BANA) in 1961 (Garvin 1994).

Graphing Aids for the Blind

Large scale tactile drawings can be accessible. These drawings are usually 2 to 4 times larger than "normal". This type of graphic is usually prepared by a sighted person. Blind and visually-impaired students do not have easy access to the means of creating their own graphs. Some large-scale, inaccurate graphs can be created by a blind individual, but, first they must know that these systems are available. Graphing aids for the blind or visually-impaired student take on many dimensions; they range from embossed Braille paper to graphing utilities that can access a tactile printer which provides raised-line drawings.

Embossed Braille paper forms with grid sizes ranging from 1/2 to 1-inch grids are mounted by the student onto cork boards; push pins locate the coordinates and are connected with rubber bands (Garvin 1994). Additionally, a teaching aid called Wicky Sticks, made of wax in long strips, can be used with the embossed Braille paper. These strips are very flexible and can be broken into small dots to represent points on a graph. Another method of inaccurate graphing involves the use of the Sewell raised-line drawing kit. This simple tool has contributed to the understanding of graphs by blind/visually-impaired individuals pursuing mathematics. This kit consists of a clipboard with a rubber pad, stylus, and mylar sheets. A mylar sheet is inserted under two thumbscrews, then the stylus is used to create a picture. A raised line is formed because of the stretching capability of the thin mylar.

Graphing aids for the blind student also include a complete drawing kit comprised of a stylus, a compass (produces up to a 5.5 inch diameter circle), a protractor (with 15 degree markings), a 12 inch ruler, a square and 100 sheets of mylar. The protractor and ruler have pegs mounted on their underside. These pegs fit into holes which are located around the perimeter of the 11.5-inch square board with a surface of neoprene rubber. The student mounts the mylar sheets and draws with the stylus or compass, thereby producing a raised-line drawing (Garvin 1994; Trace Center 1994).

Additionally, blind or visually impaired mathematics students can use a special paper which, when heated, provides a tactile image. One system involves printing or copying an image onto the paper, then heating the paper with a Tactile Image Enhancer. The second system uses a special "hot pen" which is sold in Germany. With this pen, the student can draw a raised picture on the tactile paper. Presently, these systems are costly; the tactile paper sells for about $1.00/sheet; the image enhancer retails around $800.00; and the hot pen is available for around $200.00.

Drawings produced by others for the blind can be made using a compass and stylus with tracing wheels. These produce reverse images. Other mathematical tools available for blind/visually impaired students include raised number lines, geometric shapes which introduce a child to different polygons, etc.

Technological Auditory Aids

Talking Calculators were one of the earliest auditory aids. The first breakthrough in talking calculators was made by Telesensory, whose "Speech Plus" was released around 1974. This model had five functions and one memory. Sharp Electronics released a talking clock and a talking calculator in 1979 or 1980; a later model combined these two features (Garvin 1994). Dr. Thomas Blenham, a blind physicist, started Science Products for the Blind which mounted scientific calculators onto voice boxes. The first - an HP calculator model - was followed by Texas Instrument's TI-66. This type is difficult to use proficiently as the keys do not speak - the user must either memorize the key pad or use the Braille key guide. Science Products also adapted Canon's BUSINESS model which was the first talking version with business functions.

A scientific calculator and a financial calculator with speaking keys were released in 1994. The scientific calculator has trigonometric, algebraic, clock, calendar, and other features. The Louis Herbet Center for the Blind has provided a scientific calculator since 1992. This one utilizes a female voice in a French or English version (Trace Center 1994).

Software packages, used in conjunction with an adapted computer, have enhanced accessibility for the visually impaired. Most have scientific functions including trigonometric, logarithms, exponents, and roots. All entries, functions and answers are spoken. These are available through the Internet at various FTP sites.

Another software package converts the computer-like system of Braille-n-Speak 640S and it's companion Type-n-Speak. This shareware alters these notepads into scientific calculators -- no inverse functions. The software is free and available via anonymous ftp from:

handicap.afd.olivetti.com in the /pub/braille directory as CALCBNS.ZIP.

Computers with voice output, referred to as text-to-speech, have been available since the early 1980s. Mathematics is not interpretable by these programs since much of mathematics involves the use of symbols and/or the equations take up more than one line. The symbols used in mathematics are not deciphered in a recognizable form yet. These DOS programs are referred to as text-to-speech and can only work with ASCII characters.

The Nomad Pad, an interactive audio-tactile graphics system developed in Australia, was operational in 1987 and marketed in 1989 for PCs and in 1992 for Apple and MacIntosh computers. This touch pad enables a student to study a series of prepared graphs independently. This is a learning medium which, used in conjunction with stored graphics, allows independent study. It does not allow blind/visually-impaired students to create their own graphics. Someday blind students will be able to create their own graphics, allowing them to share the joy of discovery with their sighted classmates.

Telesensory, Inc. has marketed a product named Oscar which contains a scanner and software. The scanner "reads" in a ready- made graph which is then output to a TSI Braille printer (Graham 1994). The resolution on these graphs is limited, and can "square" round curves.

The Future

Currently under investigation are various shareware graphing utilities which can provide output to a laser or tactile printer. The two requirements of this type of software include its ability to work easily on a 386 or 486 PC with voice output, and that it be "user-friendly" enough for a middle-school student. Much current research in the area of accessible mathematics is geared toward graphing and mathematic or science texts in electronic form. One notable project is a voice synthesizer-based system called AsTeR by its designer, T.V. Raman, who developed the system as a graduate student at Cornell University. Currently this system is implemented on a PC under a Linux operating system, using a Multivoice speech synthesizer. This software will read in the math text, analyze the information, create an abstract model, then provide output using pauses, tones and pitch to indicate the type of mathematical information is on screen. The sound board provides tonal/musical clues, which can indicate section or paragraph beginnings. "Stereo effects are used to make sounds appear in different parts of space. This is useful for reading aloud tables where columns can sound next to each other in space." (Barry, et al)

Additionally, Dr. Gardner has developed DotsPlus which is a "tactile method of printing technical literature for blind readers that incorporates both Braille and graphic symbols in a manner that retains the same structure as a document printed for a sighted person. Some of the more easily recognized symbols such as plus, minus, the division line in fractions, etc. are enlarged and printed as raised images, while Braille is used for alphabetic characters, numbers, punctuation marks and other symbols that are hard to recognize as raised symbols (Barry, Gardner, & Raman)." These systems require more work: AsTeR needs to be more accessible. Plans are underway to move AsTeR from Linux to a common operating system. A better tactile printer needs to be developed for DotsPlus, and last, but not least, books need to be made available in an electronic format which can be easily interpreted by these programs, e.g. SGML or TeX/LaTeX.

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