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William (Bill) Duke,
CSUN Graduate Student
Summary: Introducing a new language for easily and effectively communicating mathematics information between blind students and their teachers and fellow students not familiar with Braille.
The author's objective is to introduce the audience to a new and simple language for communicating mathematics information between blind students familiar with Braille and their teachers, fellow students, and parents who may not be familiar with and not able to use Braille.
Those of you who are blind or are teachers or parents of blind students are strongly encouraged to experiment with this new language, improve it over time, and in the process increase the mathematical knowledge of blind students. Mathematics is the language of science and technology. With this necessary knowledge of mathematics, blind students will develop into even more capable, self-sufficient, contributing citizens of the twenty- first century.
There are just two major requirements for this new mathematics language. First the language must be the same when spoken as when written. And second the language must be the same when written in Braille as when printed.
While these two requirements sound simple they require fundamental changes in the way mathematics expressions are written. Fortunately these changes impact only the form of written mathematics and not its underlying meaning. By analogy, mathematics is fundamentally the same whether written in the German, French, Arabic, Chinese, Spanish, Greek, or English language.
For the past several hundred years mathematics has been typically presented on the printed page in a two dimensional format with symbols printed not only horizontally left to right across the page but also vertically above and below in superscripts, subscripts, and sub-subscripts. Admittedly this allows the traditional printed form of mathematical expressions to be quite compact. However, because speech is restricted by time to a single linear dimension, it is necessary to similarly restrict this new mathematics language when written to a single horizontal linear dimension across the page so that the language is the same when spoken as when printed.
The second requirement, that mathematical expressions are the same when written in Braille as when printed, is accomplished by limiting the number of single character symbols in the new language to only sixty-four, the number of six-dot Braille characters. Traditional mathematics uses nearly a thousand special symbols developed over several hundred years, many of which are retained to this day in their original form in honor of the famous mathematicians that first published those particular mathematical ideas.
This new mathematical language for blind students uses for simplicity the sixty-four characters of ASCII Braille. ASCII is the abbreviation for the American Standard Code for Information Interchange. Those interested in but unfamiliar with ASCII Braille are referred to the Internet where searching on the words "ASCII Braille" will provide multiple hits illustrating the specific relationship between the sixty-four six-dot ASCII Braille symbols and their printed equivalents. During the presentation of this paper handouts describing ASCII Braille will be available to the audience and a transparency will be projected using the overhead projector.
Briefly ASCII Braille includes the same twenty-six lower case English alphabetic characters as all other American and English Braille. However, the fifth character line of ASCII Braille is allocated to the ten digits one through nine and zero rather than adding a prefix character to the alphabetic characters a through j of the first Braille character line. Having unique Braille symbols for the ten digits is very useful and makes numerical expressions written in ASCII Braille much simpler. This new math language writes whole and decimal fraction numbers in the usual way where four hundred eighty three and a quarter is written as 483.25 and spoken as four, eight, three, point, two, five with no spaces between these six characters.
Besides the twenty-six alphabetic characters, the ten digits, and the all important blank space character, ASCII Braille includes twenty- seven common printed symbols usually found on most American typewriter and computer keyboards. These printed symbols and their spoken names are more than sufficient to write and speak all mathematical expressions provided one follows the same writing style as a novelist. The novelist is required to name and introduce the characters in the story before using their names or nicknames to subsequently and efficiently refer to these people. While several of the twenty-seven symbols such as the plus sign and the equal sign will have well understood global definition, many of the other symbols must be defined locally by the writer before their use in mathematical expressions for the mutual understanding of the writer and reader.
Just as taught in high school algebra, one can use an alphabetic character to represent a variable, an element of a set, or a set of elements. When properly predefined, the twenty-six lower-case alphabetic characters can be used to represent any mathematical expression. And just as in narrative writing, one can use predefined pronounceable single or multi-syllable combinations of ASCII Braille characters (often referred to as words or nicknames) to provide all the mathematical symbols one may ever need.
To avoid confusion one must separate every mathematical symbol or word from its neighbors with the all important blank space character. Therefore the product of the two variables x and y cannot be written in the usual way by placing the x next to the y without a space between them, but must be written as x space star space y where the asterisk, nicknamed star, represents the multiplication symbol.
In the following examples the symbols for the four operations of addition, subtraction, multiplication and division will be given the nicknames of plus, minus, star (for the asterisk), and slash respectively. Also the left and right parentheses, nicknamed open and close respectively, are used to specify the order in which one does the operations starting first with the inner most parenthetical expression. In order to handle exponents while remaining on a single printed line one can use the caret or up-arrow and the underscore symbols, nicknamed up and down respectively.
At the presentation of this paper the author will write out the following two expressions for the audience on the easel paper pad. To express four plus one-half, one uses the following sequence of symbols: 4 + ( 1 / 2 ) which is spoken with pauses replacing the spaces as "four plus open one slash two close". To express that the square root of sixty four equals eight one may write the following sequence of symbols: ( 64 ^ ( 1 / 2 ) _ ) = 8 which is spoken with pauses replacing the spaces as "open six four up open one slash two close down close equals eight".
Yes, these linear representations are unable to utilize the compactness of traditional two-dimensional mathematical expressions; however this minor disadvantage is greatly over overcome by the feature of this new mathematics language that the spoken, printed, and Braille forms are identical, which allows blind students and their teachers to be able to communicate any and all types of mathematical expressions as long as they have locally predefined the elements of their expressions.
Several types of moderately priced equipment on display here at the CSUN Technology and Persons with Disabilities Conference are available to assist blind students and teachers in converting narrative text and mathematical expressions between their spoken, printed, and Braille forms. Where graphs or drawings are helpful in understanding mathematical expressions and concepts, these must be drawn as very simple tactile drawings and complete narrative descriptions of these drawings must precede each tactile drawing. Equipment on display here at the conference is also available for converting between tactile and printed drawings.
It is also strongly recommended that authors of written mathematics learning material for blind students limit their vocabulary to the one to two thousand words used both by Voice of America to communicate with listeners with limited knowledge of English and used by sign language to communicate with those with hearing and speaking disabilities. While it is important to familiarize students with the traditional and historical terms for mathematical concepts, after introducing these complex terms authors should substitute common nicknames for them from the above-mentioned limited vocabularies.
Because change is always difficult, it is anticipated that some traditional mathematicians may resist using this new mathematics language and may even be offended by it.
The author of this paper is currently working on developing a textbook for blind students covering arithmetic and algebra using this new mathematics language and has coined a nickname for the language. The language is called "mith" spelled m, i, t, h, which stands for Mathematics In The Head.
Stop and realize that you, the audience, have learned this new language, mith, in less than a half hour. Please use it and improve upon it to help blind students learn mathematics.
Permission is granted by the author to reproduce this paper in any form for any purpose that will contribute in any way toward helping blind students learn mathematics.
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