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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