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Hi gang:

Abe Nemeth has asked me to post the following paper.

Incidentally, Abe, you can post to nfb-se@nfbcal.org

directly and you have been added as a member of the list.

Read carefully what Abe has to say.

I know it has been a problem for me to train a reader to talk "mathspeak".

You need a system that is quick to teach, compact, and unambiguous.

Abe has captured it all here.

Take this as a launching pad,

how do you guys teach a reader to read tech?

What do you like, and what drives you crazy?

This is a popular FAQ.

I want to give an answer with some "group think" put into it.

Thanks Abe,

John

From: Abraham Nemeth

To: Members of the R&D and the S&E committees of NFB

Subect: Mathspeak

Date: August 5, 1995

When I was studying math at the college and post graduate levels,

I used sighted readers for accessing text materials, since

braille materials in math were almost entirely unavailable. Even

as a college professor, particularly during my early years, I

continued to use sighted readers, volunteers, paid, and teaching

assistants, largely for the same reason.

No standard protocol exists for articulating mathematical

expressions as it does for articulating the words of an English

sentence. Therefore, it was necessary for my reader and me to

come to some agreement as to the most efficient method for

conveying mathematical text to me in an unambiguous manner. Since

I needed to turn in assignments as a student, and since I often

prepared handouts for my students and for faculty seminars as a

professor, it soon became clear that the same protocol should be

used when I was dictating and my reader was doing the writing.

Little by little, a surprisingly simple protocol evolved. I can

teach it to a new reader in about 15 minutes. The most important

feature of this protocol is the principle that anything that was

read earlier or anything that will be read later should never

affect what I write now in response to my reader's current

utterance. I appropriated this principle from the Nemeth Code

which I had already developed and which was, of course, the code

that I was using for the writing of mathematical text.

At our recent convention in Chicago, this matter of "mathspeak"

came up at our Science ane Engineering meeting. At the conclusion

of this item, ohn Miller, our Chairman, gently but firmly

instructed me to prepare a writeup of this method and post it on

the Internet. What follows is my attempt to comply. I am posting

it on the R&D "listserve and am asking John to forward it to the

S&E listserve since I do not know the E-mail adress of the

latter. And, John, if you will give me that address, I will

subscribe.

It had never occurred to me before now that the method was

sufficiently substantial to warrant a formal description in

writing. Without further ado, here's how "mathspeak" works.

Letters:

Lowercase letters are pronounced at face value without

modification. They are never combined to form words. In

particular, the trigonometric and other function abbreviations

are spelled out rather than pronounced as words. We say "s i n"

rather than "sine," "t a n" rather than "tan" or "tangent," "l o

g" rather than "log," etc.

A single uppercase letter is spoken as "upper" followed by the

name of the letter. If a word is in uppercase, it is spoken as

"upword" followed by the sequence of letters in the word,

pronounced one letter at a time.

For a Greek letter, my reader is taught to say "Greek" followed

by the English name of the letter. If he knows its Greek name, he

simply speaks that name. Thus, my reader might say "Greek e" or

"epsilon." Uppercase Greek letters are pronounced as "Greek

upper" followed by the English name of the letter, or "upper"

followed by the name of the Greek letter. For new readers, I had

a card with the lowercase and the uppercase Greek letters and

their names. Pretty soon my readers did not have to refer to the

card, or they did so only for infrequently occurring Greek

letters.

I never devised a formal method for pronouncing letters that were

printed in italic type, boldface type, script type, sanserif

type, or that were underlined. However, I do not anticipate any

difficulties in this area, should the need for a formal

"mathspeak" protocol become desirable.

Digits and Punctuation:

Digits are pronounced individually; never as words. Thus, 15 is

pronounced "1 5" and not "fifteen." Similarly, 100 is pronounced

"1 0 0" and not "one hundred." An embedded comma is pronounced

"comma," and a decimal point, whether leading, trailing, or

embedded, is pronounced "point."

The period, comma, and colon are pronounced at face value as

"period," "comma," and "colon." Other punctuation marks have

longer names and are pronounced in abbreviated form. Thus, the

semicolon is pronounced as "semi," and the exclamation point is

pronounced as "shriek." I borrowed the latter from the APL

programming language in which "shriek" is the standard way of

referring to the exclamation mark.

The grouping symbols are particularly verbose and require

abbreviated forms of speech. Thus, we say "L-pare" for the left

parenthesis, "R-pare" for the right parenthesis, "L-brack" for the

left bracket, "R-brack" for the right bracket, "L-brace" for the

left brace, "R-brace" for the right brace, "L-angle" for the left

angle bracket, and "R-angle" for the right angle bracket.

Operators and Other Math Symbols:

We say "plus" for plus and "minus" for minus. We say "dot" for

the multiplication dot and "cross" for the multiplication cross.

We say "star" for the asterisk and "slash" for the slash. We say

"superset" in a set-theoretic context or "implies" in a logical

context for a left-opening horseshoe. We say "subset" for a

right-opening horseshoe. We say "cup" (meaning union) for an

up-opening horseshoe and "cap" (meaning intersection) for a

down-opening horseshoe. We say "less" for a right-opening wedge

and "greater" for a left-opening wedge. We say "join" for an

up-opening wedge and "meet" for a down-opening wedge. The words

"cup," "cap," "join," and "meet" are standard mathematical

vocabulary. We say "less-equal" and "not-less" when the

right-opening wedge is modified to have these meanings. We say

"greater-equal" and "not-greater" under similar conditions for

the left-opening wedge. We say "equals" for the equals sign and

"not-equal" for a cancelled-out equals sign. We say "element" for

the set notation graphic with this meaning, and we say "contains"

for the reverse of this graphic. We say "partial" for the round

d, ane we say "del" for the inverted uppercase delta. We say

"dollar" for a slashed s, "cent" for a slashed c, and "pound" for

a slashed l. We say "integral" for the integral sign. We say

"infinity" for the infinity sign, and "empty-set" for the slashed

0 with that meaning. We say "degree" for a small elevat$ circle,

ane we say "percent" for the percent sign. We say "ampersand" for

the ampersand sign, and "underbar" for the underbar sign. We say

"crosshatch" for the sign that is sometimes called number sign or

pound sign. We say "space" for a clear space in print.

Although the above is a long list, it is not intended to be

complete. It does, however, cover most of the day-to-day material

with which we usually deal.

Fractions and adicals

We say "B-frac" as an abbreviation for "begin-fraction," and

"E-frac" as an abbreviation for "end-fraction". We say "over" for

the fraction line. Even the simplest fractions require "B-frac

and E-frac. Thus, to pronounce the fraction "one-half" according

to this protocol, we say "B-frac 1 over 2 E-frac." By this

convention, a fraction is completely unambiguous. If we say

"B-frac a plus b over c + d E-frac," the extent of the numerator

and of the denominator are completely unambiguous.

A simple fraction (which has no subsidiary fractions) is said to

be of order 0. By induction, a fraction of order n has at least

one subsidiary fraction of order n-1. 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 fraction.

To communicate the presence of a complex fraction, we say

"B-B-frac," "O-over," and "E-E-frac" for the components of a

complex fraction, in the manner of stuttering. For a hypercomplex

fraction, the components are spoken as "B-B-B-frac," "O-O-over,"

and "E-E-E-frac," respectively. You can see that the speech

patterns are designed to facilitate transcription in the Nemeth

Code, according to the rules of that Code.

Radicals are treated much like fractions. We say "B-rad" and

"E-rad" for the beginning and the end of a radical, respectively.

Thus, we say "B-rad 2 E-rad for the square root of 2.

Nested radicals are treated just like nested fractions, except

that there is no corresponding component for "over." Thus, if we

say "B-B-rad a plus B-rad a plus b E-rad plus b E-E-rad," the

braille reader is immediately alerted to the structure of the

notation just as the sighted reader is by mere inspection, and

the expression is unambiguous.

Subscripts and Superscripts

We introduce a subscript by saying "sub," and a superscript by

saying "sup" (pronounced like "soup.") We do not say "x square;"

instead we say "x sup 2." We say "base" to return to the base

level. The formula for the Pythagorean Theorem would be spoken as

"z sup 2 base equals x sup 2 base plus y sup 2 base period."

Whenever there is a change in level, the path, beginning at the

base level and ending at the new level, is spoken. Thus, if e has

a superscript of x, and x has a subscript of i+j, we say "e sup x

sup-sub i plus j." And if e has a superscript of x, and x has a

superscript of 2, we say "e sup x sup-sup 2." If the superscript

on e is x square plus y square, we say "e sup x sup-sup 2 sup

plus y sup-sup 2." If an element carries both a subscript and a

superscript, we speak all of the subscript first and then all of

the superscript. Thus, if e has a superscript of x, and x has a

subscript of i+j and a superscript of p sub k, we say "e sup x

sup-sub i plus j sup-sup p sup-sup-sub k."

If a radical is other than the square root, we speak the radical

index as a superscript to the radical. Thus, the cube root of x+y

is spoken as "b-rad sup 3 base x plus y E-rad."

We say "underscript" for a first-level underscript, and we say

"overscript" for a first level overscript. We say "endscript"

when all underscripts and overscripts terminate. Thus we say

"upper sigma underscript i equals 1 overscript n endscript a sub

i." We say "un-underscript" and "O-overscript" for a second-level

underscript and a second-level overscript, respectively. We speak

all the underscripts in the order of descending level before

speaking any of the overscripts. Each level is preceeded by

"underscript" with the proper number of "un" prefixes attached.

Similarly, we speak the overscripts in the order of ascending

level. Each level is preceeded by "overscript" with the proper

number of "O" prefixes attached.

Conclusion:

The speech generated by this protocol is not exactly what a

professor in class would use, but it is absolutely unambiguous

and results in a perfect Nemeth Code transcription. It avoids

largely unsuccessful attempts by a reader to describe the

notation he sees, accompanied by the shouting and gesturing that

such attempts at description engender. When each of a number of

readers abides by this protocol, it is a snap to record the

information, and makes much better use of the time spent with a

reader.

As Raised Dot Cumputing has amply demonstrated, Nemeth Code can

be converted into correctly formatted print notation. And this

print notation can be converted into speech by using the protocol

described in this paper. Thus, speech to Nemeth Code to print to

speech demonstrates that these three systems are notationally

equivalent. The ability to convert from one form to another is a

distinct benefit to a blind person engaged in the field of

mathematics, whether as a student, a teacher, or as a worker in a

professional field, and gives him a competitive edge.

These considerations strongly suggest that serious thought be

given to refining "mathspeak" and making it a standard by which

complex mathematical notation can be communicated to a blind

person in verbal form. It could also serve as the basis for

transmitting mathematical notation electronically when ASCII is

not capable of conveying the notation (many math symbols have no

ASCII representation) or when ASCII codes which represent

notation are likely to be unfamiliar to the recipient.

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