Braille Mathematical Literacy: A Comparative Analysis of Available Math Codes
CTEBVI 56th Annual Conference "Share your vision in our changing times"
Workshop 701 Saturday March 21, 12:45pm – 2:15pm Burlingame California
Presented by Robert Stepp, Ph.D.
Synopsis Via comparative examples, we will explore braille math notations in the Taylor math code, the Nemeth code, UEB, and NUBS. The study is that of the quality of the re-expression of print and the utility of the code to the blind scholar. Each code has strengths and weaknesses. It is now very timely that we all know just where UEB sits in such a comparison.
About the Author Dr. Robert Stepp has an A.B. degree in math and physics, an M.S. degree in physics and computer science, and a Ph.D. degree in computer science and artificial intelligence. He has taught both undergraduate and graduate courses in computer programming, computer engineering, analog circuit design, digital microprocessor design, and artificial intelligence. He created the world's first braille editing system for a personal computer (1978) and received an award for that system in 1981 from Johns Hopkins University. He is not certified for any braille code, but is no stranger to the details of braille. He was personally acquainted with Dr. Abraham Nemeth. He is the developer of Braille2000 (software) and the owner of Computer Application Specialties Company.
Overview You may not know anything about the Taylor code or the NUBS code, and you may know more or less about Nemeth and UEB. Of course the four codes have some elements in common and many differences. What is interesting is how they differ on four critical aspects, Comprehensiveness, Transparency, Efficiency, and Automatability. These factors are defined and related to the four math codes in the narrative below. It is a finding of the comparative analysis that UEB is distinctly less efficient than the others, at times sufficiently so as to be an encumbrance to the user.
We note that an often overlooked aspect of braille is its utility in the hands of the blind student or professional author, for doing homework and writing reports and taking notes. Examples in this paper demonstrate the apparent clumsiness of UEB for those purposes, for example, writing down the derivation of the general quadratic equation (example 2 on page 12). Math is often repetitive, and thus the inefficiencies build up and up. It may be that some students and professionals will be capable of inventing their own notational solution, inventing some kind of grade 3 notations or using eight-dot braille on their refreshable note-taking device in some creative way they invent. But neither of those shorthand solutions can offer braille-to-print translation by which good-looking print documents can be rendered from braille. A standard 1 braille code, such as UEB or NUBS can. There just isn't any good way to escape the need for comprehensiveness, transparency, efficiency, and automatability: they are fundamental.
The switch to UEB was dictated by the Braille Authority of North America in a politically charged atmosphere with little trial to see what effect it would have on foreign language and math. Most of the braille world is aware of the limitations of English Braille American Edition (EBAE), but it is a valid point to say that consideration of alternatives (say using Nemeth for all literature or switching to NUBS) was tilted against those alternatives by ad hoc motivations, one element of which was the last-minute retention of Nemeth code for math, even when the full meaning of that statement was unknown.
At the moment, the details of foreign language braille and math braille remain up in the air. A statement of "Provisional Guidance" for math has been adopted by BANA (see Appendix Two) with certain details that remain murky to say the least, and most interesting of all, a strategy that totally undermines the main reason for UEB, that of moving to a "unified" code. According to those guidelines, math materials would be transcribed in UEB narrative with all math expressions (essentially anything other than plain text) transcribed in uncontracted Nemeth set off from the UEB by enclosure within two-cell "begin Nemeth" and "end Nemeth" indicators. The guidelines do not provide for extended sections in Nemeth (other than displayed formulas), the idea being to promote UEB as much as possible for prose and switch in and out of Nemeth via embedding for individual math expressions. Note that this is likely to be many times per page of braille, mixing together on the same page upper- and lower-cell digits, UEB and Nemeth punctuation, UEB and Nemeth emphasis indicators, and spawning a debate over whether you switch or not for a single special symbol, such as the fancy "ℝ" that denotes the set of real numbers (a symbol that is likely to occur in the narrative portion as well as in a math expression).
It appears that at this time, Nemeth is being used only in the United States. Although Canada has been following BANA guidelines in the past, it appears that Canada will not be using Nemeth in the future, thus implying that BANA guidelines apply now only to the U.S.. And on top of that, UEB itself is an international standard maintained by ICEB (International Council on English Braille), and thus it seems that BANA has no authority over it. The scope of BANA's influence seems to be at an all-time low.
It is less and less clear who will be following the Provisional Guidance, or whether what is now "provisional" will ever become standard. It is equally unclear whether there will ever be anyone transcribing according to those guidelines or even that anyone will be certified in them, in part because all the details are still significantly undefined and incompletely documented, to say nothing about having training manuals for them.
As of now, UEB math is defined for use outside the U.S. while our math transcribers continue to generate Nemeth transcriptions for students in the U.S., all trained in Nemeth. So which is worse: not being able to read math literature from any other country (not being able to read UEB math), or continuing to read only Nemeth/EBAE, or reading from a stew of both UEB and Nemeth cluttered with four cells of embedding at every turn. Was it worth disrupting the old ways to get to this? Perhaps the comparative analysis below will help you decide for yourself.
There is much to learn, and hence the role of this paper to show by comparative example what the use of UEB means for math, what the use of the Provisional Guidance means to math, and 2 what alternatives exist, especially now, when all blind school children already know Nemeth and none know UEB. If you are a VI specialist, you can think about whether you will teach Nemeth to first graders or not, and at what grade level you will expect to see embedded Nemeth in the braille textbooks. At this moment there is no standard, so what are you going to do to prepare your blind students? This paper may help you decide.
About Math This paper is about braille codes for math. When you ask a child "what did you learn in math" the answer would never be "a,b,c" (the alphabet), it would always be something like "1+1=2". Numbers are central to math because they quantify so many vital real-world concepts such as length, angle, mass, distance, velocity, time, volume, volts, and on and on. But math is not just about numbers. It pertains to the conceptual relationships between things (such as distance and time) expressed as variables, expressions, equations, and systems. The formalism known as calculus provides for truth-preserving rewriting rules (symbol substitution) for manipulating expressions/equations/systems in order to derive new relationships to explain the real world and/or prove or disprove a conjectured theorem. In the typical math utterances there is a tight mixture of symbols, especially letters and digits. One should not need to argue the absolute necessity in today's world for math and math education.
On Being "First Class" This paper uses the term "first class" to talk about symbols that can nestle together in braille without intervening context switches. In braille, in general, the letters a-z are first class but the digits 0-9 are not (at least when mixed with letters such as a-j). To express a symbol that is not first class, some indicator of context switch (e.g., the braille number sign) must be imposed. This not only lengthens the braille statement but also imposes a mental burden on the reader (context begun must end at some point, and thus needs to be mentally tracked). This is different from symbols whose braille representation is multiple cells: using multiple cells does lengthen the statement but does not impose any context switch. Because math expressions are often complex and yet need to be absorbed efficiently (because of the limited short-term memory in humans, re. Miller's Magic Number), the length of statements has to be of some concern, although with a small braille alphabet of just 63 cells, optimum minimization of length just isn't possible. This is the attribute we call efficiency.
Math Braille Codes used in North America (ordered by date of first publication) Taylor (Braille Mathematical Notation) Originally British from 1894, revised for American English in 1920 and 1942. Taylor code is virtually unknown today and certainly not going to be revived. It is included here because it is British (UEB also has British influences), uses upper digits, and has the same issues with complex superscripts and subscripts that UEB has. It is thus similar in certain ways to UEB. (It is unknown to the author whether it was in any way an explicit role model for UEB.) Mr. Taylor was going blind at the time he created his code, and he himself used it for teaching math. Because Mr. Taylor was a math teacher, his code transcribes his understanding of math (the semantics), and not necessarily what it looks like in print. Nemeth (The Nemeth Braille Code for Mathematics and Science Notation) Created in the 1940s, published in 1952 and revised in 1956, 1965, 1972, and 1983. Nemeth has been taught to all braille-reading students in North America for the past 50-plus years. It has also been used for a time in New Zealand but is otherwise unknown to the rest of the world. Dr. Nemeth was a math student at the time he created his code and he used it when teaching math.
3 UEB (Unified English Braille) Created in 2004, officially adopted for use in North America in 2012 As the "fix" for the well-known limitations in EBAE, UEB grew out of a BANA study to find a "unified" code, not in the international sense of unified but in the literary plus math sense. The task evolved into an international committee project that drew it away from being strictly for the U.S. and Canada (referring to the use of British contraction rules that change the spelling of about 9% of all contracted words, a handicap to North American readers). UEB was created by a committee and although debated intensively, the committee did not depend on it to study or teach math themselves. With the adoption of the Provisional Guidance (see Appendix Two), the goal of a "unified" code via UEB has collapsed back into a two-system, non-unified approach. NUBS (Nemeth Uniform Braille System) A revision to Nemeth, completed in 2011 (maintained at www.all4braille.org) Dr. Nemeth created NUBS when he perceived that ICEB was not going to give further consideration to using lower digits, the cornerstone of Nemeth braille. He thought NUBS might have been what ICEB could have created had they adopted lower digits, and it has essentially the same attributes as UEB with respect to being non-ambiguous (i.e., being computer automatable) and unified (handling both literary and math transcriptions). Although largely ignored, NUBS is a Nemeth-like alternative to UEB and small-scale studies have shown that it is sufficiently like EBAE and Nemeth to be read by both students and adults without re-training. Noting the present track in the U.S. towards a UEB and Nemeth two-code world, NUBS is the only "unified" code of the four codes in this paper. Perhaps an interest in NUBS may yet emerge.
The codes are compared in order of their creation: Taylor, Nemeth, UEB, NUBS. A few real- world extended examples are given using Nemeth, UEB, and NUBS (Taylor is shown here to reveal past notations, but not as a contender for modern use). The codes are vast, as needed to cover the myriad special symbols found in printed math. Many aspects, such as shapes, arrows, adornment (typeface, overbars, underbars, text color) are not mentioned for lack of space.
Criteria of merit The following aspects would seem to be salient for effective math transcription. We note that a math code defines a universe of braille math literature to parallel print math literature, to be used for both reading and writing. Note that although it is possible to read or even write math in a challenging way, such as by audio narration or computer markup code (both mentioned in the table below), the Miller's Magic Number effect (that states the limited capacity of the human mind to retain short-term background information is "7±2" items) puts a premium on the shortness (efficiency) of expressions. Humans can memorize more than seven items, but it is much more difficult. The more information you can pack into a line of braille, the better.
Criteria: How well does it represent print (syntax versus semantics) Comprehensiveness (does it handle all details) Transparency (does it convey versus distort the print medium) Efficiency (is it tidy and terse) How well does it support creativity (as a vehicle for writing and symbolic manipulation, as in doing homework and writing papers) Comprehensiveness (does it handle all details of authorship) Efficiency (quick to write)
4 How well does it automate (support computer translation) non-Ambiguity (avoiding expressions having multiple print interpretations) non-Subjectivity (avoiding multiple ways to represent the print copy)
The main strategies compared generally: Comprehensive Transparent Efficient Automation Commentary Narration if detailed no no possible, but reading via talking (audio and by enough difficult book and writing via prose) dictation Raw LaTex, yes no no yes intended to be read by Tex, algorithms, not humans MathML Taylor limited partly* fairly* ambiguous invented in 1894 and and used in the U.S. until subjective 1956 Nemeth yes yes yes ambiguous invented in 1942 UEB yes mostly* fairly* yes same flavor as Taylor but much more comprehensive NUBS yes yes yes yes same flavor as Nemeth but non-ambiguous *by "fairly" we note the many extra cells (indicators) needed to mix digits with letters *by "partly" and "mostly" we refer to the situations in which braille constructs must be used that have no analog to the print, such as the braces, parentheses, and times signs inserted into Taylor code and the grouping symbols inserted into UEB. None of those devices distorts the semantics of the math, but it might be better not to transcribe semantics but instead offer a descriptive transcription of what the print looks like (in this regard, the Nemeth concept of up/down/baseline seems more transparent than the grouping symbol approach)
The obvious elephant in the room is UEB and its ongoing adoption in the U.S.. There are many lovely and intuitive elements in UEB. It is the judgment of the author that UEB has everything you would want once you propose that no blind child should study advanced math (a view said to exist in some other English-speaking countries that shaped the destiny of UEB) or that a blind math scholar can and will create his/her own personal code to overcome UEB to succeed(!). You may or may not agree with this conjecture after studying the comparisons below. But note that the author's view could not exist absent the decades of use of the Nemeth system. It is the fact that we have and currently do use Nemeth with its first-class math mode that makes UEB relatively inferior. UEB is a complete and unified system that, in the U.S. may never be used to its full potential because via the Provisional Guidance, see Appendix Two, its math portion will be sidestepped in favor of isolated Nemeth utterances. As you read this paper, please consider the relative merits of the codes. When compared side-by-side, which one would you, as a blind student/scholar/professional, like to be reading and which one would you like to be writing as you do your homework, take notes, and write drafts of your research?
A note about mode With six-dot braille there are just 63 cells plus the space and obviously the print world has far more than 63 possible glyphs (Unicode comprises millions of glyphs, although most are irrelevant to math). To cover even ordinary things, much more than 63 concepts are needed. Using the notion of mode, as signaled by indicator cells, the 63 cells can mean different things in different modes. This is required for capitalization, emphasis, special symbols, and for special
5 prose such as music (embedded) or formulas. In each mode, multi-cell constructs represent each character (glyph), uniquely so if the code is non-ambiguous.
What is ideal would be to have a mode in which the glyphs you need are all available at the same time. For example a mode to embed music, in which everything is music notation without needing constant mode switching.
This situation is precisely the allure of the so-called French Numbers. The digits are formed by adding dot 6 to the letters a-i and using (3,4,6) for zero. The important property is that in uncontracted English braille, the digits stand as themselves without needing any mode shift, i.e., needing no indicator cells at all. The simplicity is elegant, but the comprehensiveness is limited (you just can't do very much without modes).
We would say that in uncontracted braille using French numbers, the digits are "first class" symbols, i.e., symbols that are always themselves. A second class symbol would be one that requires a special mode. In all the braille codes listed above, digits are second class symbols along with Greek letters. But note that in UEB, Nemeth, and NUBS, there are modes for passages in which digits are temporarily first class symbols and those modes offer enhanced efficiency. Unfortunately, in UEB the extended number mode also removes the letters a-j that are often needed in the math expression, whereas the extended number mode in Nemeth (unnamed) and the one in NUBS (called "notational") simultaneously supports letters and digits as first-class symbols and that is a powerful advantage.
The digits conundrum Having been derived from French braille (in which there just weren't enough distinct cells to have digits as first-class symbols, given the need for accented vowels), digits have grown up as second-class objects, and as such they overlay something else. In EBAE and Taylor and UEB they overlay the letters a-j. In Nemeth and NUBS they overlay punctuation. In the French number scheme they would overlay common contractions (and hence the role of French numbers mostly in uncontracted braille). There is no reasonable scheme without some tension and fighting and clumsiness. In normal (non-math) prose, digits are fairly incidental and having them be second-class symbols is probably a good trade-off.
There is nothing sacred about how digits are done. Effective systems have existed for decades to handle the overlay of a-j (reference to EBAE) or equally well handle the overlay of punctuation (reference to Nemeth). It is far more than academic exercise to evaluate one versus the other, and this difference sets Taylor and UEB apart from Nemeth and NUBS. For many decades young children (first, second, third graders) have easily learned to read both "upper" (Taylor and UEB) numbers as well as "lower" (Nemeth and NUBS) ones. They are the same dot patterns, just higher or lower in the cell. Nobody should assert that one versus the other is "hard" or "easy" to read; little children have been doing both. Having first-class digits in a math expression has direct impact on notational efficiency and for advanced math, this is extremely important.
6 Comparative Notation The expression in the first column is shown in the four braille codes. Some entries have reference notes and those notes are listed immediately following the table. Note: some braille expressions below wrap merely because of column width; read them as if on the same line Taylor Nemeth UEB NUBS 123 #ABC #123 #ABC #123 number 1,000 #A1JJJ #1,000 #A1JJJ #1*000 number 1 - 2 #A-B #1-2 #A-#B #1-2 range note 1 1.2 #A.B #1.2 #A4B #1]2 decimal 1/2 #A/B #1_/2 #A_/#B #1_/2 fraction note 4 ½ #A/B or #A2 ?1_/2# #A/B ;?1/2# fraction note 2 2½ #B-A/B #2_?1_/2_# #B#A/B #2?1/2# two and a note 3 note 4 half 7a #G*A #7A #G;A #7A note 5 note 6 1 c [#A/BOC ?1/2#C #A/B;C ;?1/2#C 2 note 7 note 8 half c a c [A/#BOC ?A/2#C (A./#B)C ;?A/2#C 2 note 9 half a times c 2 X< X^2 X;9#B ;X^2 x note 10 x squared y 3 Y% Y^3 Y;9#C ;Y^3 y cubed note 10 x2yz34 X 7 Taylor Nemeth UEB NUBS x6 X@F X^6 X;9#F ;X^6 x to the 6th x−2 X@9B X^-2 X;9"-#B ;X^-2 x to the -2 xm X^M X^M X;9M ;X^M x to the m xm+1 X@8M5A0 X^M+1 X;9 e x E;X E;X E;5X ;E;X e sub x e x+1 E,8X5A0 E;X+1 E;5 x 1 ,x2 X1'1X2 X1,X2 ;;X5#A1 ;X1*X2 x sub 1 X5#B comma x sub 2 x m X;M X;M X;5M ;X;M x sub m $1 4#A @S1 @S#A ;@S1 one dollar π Π 'P 'P .P .,P .P ,.P .P _P pi note 17 note 17 ∞ 9F ,= #= @8 infinity 8 Taylor Nemeth UEB NUBS = ≠ 33 3W .K /.K "7 "7@: = .= equals not- equals < > 3- 3C "K .1 @< @> @< @> less than greater than ≤ ≥ 37 3G "K: .1: _@< _@> _< _> less than or equal greater than or equal + - 5 9 + - "6 "- + - plus minus × ÷ * // @* ./ "8 "/ .[ ./ times divide 3 >#C >3] %#C+ ;>3[ square root of 3 y >Y >Y] %Y+ ;>Y[ square root of y − y >9Y >-Y] %"-Y+ ;>-Y[ square root of -y xy >[XYO >XY] %XY+ ;>XY[ square root note 18 of xy xy22+ >[X<5Y 9 Taylor Nemeth UEB NUBS [ ] ( ) @( @) .< .> @( @) brackets { } 8 0 .( .) _< _> .( .) braces Notes 1. Only UEB requires a second number sign 2. Clever (?) use of both upper and lower digits 3. Hyphen (not in print) separates whole number from fraction 4. Additional number sign required 5. Because a is not first-class after a number, multiplication sign (not in print) is added 6. Because a is not first-class, G1 indicator (anti-number-sign) is added 7. The ½ is enclosed in braces (not in print) 8. Because a simple fraction cannot contain letters, the letter c is not in the denominator. 9. Grouping symbols (not in print) are needed to keep c out of the denominator 10. Uses "French" number without number sign 11. Uses "baseline" indicator 12. Uses "index" indicator (dot 4) and multiplication sign (not in print) 13. Uses G1 indicator to separate letter from upper digit 14. Uses braces (not in print) 15. Uses grouping symbols (not in print) 16. Uses "up up" to explicitly give elevation of second superscript 17. Nemeth and UEB differ on the position of the capital sign for Greek letters 18. Uses parentheses (not in print) 19. Needs notational indicator unless already in that mode It is interesting to note the same ways Taylor uses braces (note 14) and UEB uses grouping symbols (note 15). Also note that UEB usually requires the most cells. Efficiency (a main criterion) As shown above and then below in some everyday examples, UEB almost always requires more cells. This is, in general, because of two aspects: (a) no mode in which letters and digits are both first-class and thus the need for number signs and letter signs, sometimes shifting back and forth again and again, and (b) the predominance of two- and three-cell symbols rather than one- and two-cell symbols. The table below shows the math print symbols that are one-cell and two-cell, with the two-cell math print symbols grouped by the first (prefix) cell, either (4), (5), (46), or (456) . Math symbols not shown in the table are more than two cells. Notice that UEB has no one-cell math symbols while the other codes have several commonly occurring symbols expressed by just one cell. This is one reason UEB is less efficient. 10 Common math symbols expressed as one or two cells Taylor Nemeth UEB NUBS one-cell symbols digits in some digits, comma, digits, comma, other than letters circumstances, decimal pt,(), +, - decimal pt, (), +, - (), [], {}, +, -, :, , = ×, ∞ two-cell beginning currency symbol, currency currency symbol, @ <, >, and, or, not symbol, and, or <, >, ∞ two-cell beginning <, > ditto, +, -, =, ×, " ÷, *, () two-cell beginning Greek, =, ÷ Greek, [], _ Greek, dash, {}, /, . ×, ≠ two-cell beginning / {}, / /, *, ≤, ≥ _ Some everyday examples (the numbers in parentheses are the cell counts) 1. Algebra (factorization) "solve 9a2-4b2 = 0" (the solution is two lines in the a/b plane) the solution involves factoring the expression by deriving the equality 9a2-4b2 = (3a-2b)(3a+2b) and noting that for this to be zero we have either that (3a-2b) = 0 or that (3a+2b) = 0. And thus the solution lies along the line a = 2/3 b or along the line a = -2/3 b. The equality 9a2-4b2 = (3a-2b)(3a+2b) in braille is Nemeth (29) #9A^2"-4B^2 .K (3A-2B)(3A+2B) UEB (53) ;;;#I;A9#B"-#D;B9#B "7 "<#C;A"-#B;B">"<#C;A"6#B;B">;' NUBS (28) #9A^2"-4B^2 = (3A-2B)(3A+2B) To work out this solution, the student would need to rewrite these expressions in several intermediate forms, including at some point the terms resulting from the multiplication of (3a-2b)(3a+2b) yielding (3a)2+(-2b)(3a)+(3a)(2b)-(2b)2. Then the terms would be combined to show that this is indeed 9a2-4b2 (the original expression). The statement (3a)2+(-2b)(3a)+(3a)(2b)-(2b)2 in braille is Nemeth (33) (3A)^2"+(-2B)(3A)+(3A)(2B)-(2B)^2 UEB (65) "<#C;A">;9#B"6"<"-#B;B'>"<#C;A">"6 "<#C;A">"<#B;B">"-"<#B;B">"9#B NUBS (34) ;(3A)^2"+(-2B)(3A)+(2B)(3A)-(2B)^2 11 2. Algebra (quadratic equation) "solve ax2+bx+c = 0" (a, b, and c are arbitrary constant numbers) If ax2+bx+c = y is drawn on the x/y plane, it is a parabola that crosses the x axis (where y=0) at two places. The student will learn the general form of the solution as: −b±−b2 4ac x = which in braille is 2a Nemeth (24) X .K ?-B+->B^2"-4AC]/2A# UEB (36) ;;;X "7 ("-B_6%B9#B"-#D;AC+./#B;A);' NUBS (24) ;X = ?-B_+8>B^2"-4AC[/2A# but it is one thing to simply read the general solution or recite it, it is another to derive the solution for yourself. "by completing the square, derive the general solution for x in ax2+bx+c = 0" That is done by showing that equivalent problems are x2+bx/a = -c/a and also x2+bx/a+b2/4a2 = -c/a +b2/4a2 (by the truth-preserving manipulation of adding the same expression to both sides of an equation) and then showing that x2+bx/a+b2/4a2 can be rewritten b−cb2 as (x+b/2a)2. You then take the square root of both sides to get x +=±+and then 24aaa2 you push things around to get the general form shown above. This kind of problem is the essence of algebra and symbolic manipulation of expressions. This is plain-Jane math but very few students can do the manipulations just in their head… it needs to be written down. So lets see what this looks like to the blind student who is doing this for homework and thus authoring his homework solution in braille. The derivation written by a sighted student might well be this: ax2+bxc+=0 bc xx2 +=− aa bb22cb xx2 ++=−+ a44a22aa bcb2 ()x +2 =−+ 24aaa2 bcb2 x +=±−+ 24aaa2 12 bcb2 x =−±−+ 24aaa2 b 1 x=−±(−+4)acb2 2aa(2)2 b 1 x=−±−b24ac 22aa −b±−b2 4ac x = 2a Here is that same homework authored by the blind math student. Nemeth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lgebra (systems of equations) a matrix with omissions | a11 a12 . . . a1n | | a21 a22 . . . a2n | | ...... | | am1 am2 . . . amn | 14 Nemeth (92) ,\A11 A12 ''' A;1N",\ ,\A21 A22 ''' A;2N",\ ,\'''''''''''''''''',,\ ,\A;M1 A;M2 ''' A;MN",\ UEB (143) ""=;;; ,_\A5#AA A5#AB 444 A5<#AN>,_\ ,_\A5#BA A5#BB 444 A5<#BN>,_\ ,_\ 4 4 444 4 ,_\ ,_\A5 NUBS (100 ) @,\A11 A12 ,,' A;1N"@,\ @,\A21 A22 ,,' A;2N"@,\ @,\,,' ,,' ,,' ,,' @,\ @,\A;M1 A;M2 ,,' A;MN"@,\ 4. Calculus 3 "evaluate the expression (2x+1) dx " ∫2 The steps for the student amount to successive rewritings, as follows 3 xx2 + 2 (32 + 3) – (22 + 2) 12-6 6 Nemeth (52) !;2^3"(2X+1)DX @(X^2"+X@);2^3 (3^2"+3)-(2^2"+2) #12-6 #6 UEB (85) ""=;;; !5#B9#C"<#BX16#A">DX 15 . NUBS (55) ;!;2^3"(2X+1)DX ;@(X^2"+X@);2^3 ;(3^2"+3)-(2^2"+2) #12-6 #6 5. Chemistry (reaction balance equation) CH4 + 2Cl = CH3 Cl + HCl Nemeth (28) ,C,H4+2,CL .K ,C,H3,CL+,H,CL UEB (40) ;;;,C,H5#D"6#B,CL "7 ,C,H5#C,CL"6,H,CL;' NUBS (28) ;,C,H4+2,CL = ,C,H3,CL+,H,CL You are encouraged to read through some of the longer lines of braille to get an appreciation for the multitude of number signs and letter signs in UEB, and for the number of dot-5's in UEB. They all add up and lead to UEB requiring many more cells to express the same thing, compared to the other codes. It is not clear whether the Proposed Guidance will have simple superscript and subscript expressions transcribed in embedded Nemeth, but if not, those UEB indicators will be easily mistaken for 5 and 9 and the UEB letter signs will be easily mistaken for "down". The point is that when mixing two dissimilar codes on the same page, much more care is required to read them. Try reading a line of UEB above and then a line of Nemeth—it's a considerable challenge, and that kind of thing is what our children are going to see on every page. What about using a modern markup language in lieu of a braille code? This paper has presented a wide variety of notational situations showing how each of the four braille math codes represent print. In today's world there are several print markup codes for math. Why not just give the blind reader direct access to that markup and call it "done"? Consider the quadratic equation in the form of MathML markup: (260) 16 Most math students would not wish to read this kind of markup (although all the information is there in a well-documented understandable manner). The issue is efficiency… it surpasses Miller's Magic Number in look-ahead/look-back in order to parse the statement. Summary Analysis Today's blind students (grades K-12 plus college) and blind professionals, those that use braille, they have all learned Nemeth. Almost nobody in North America knows UEB (especially UEB math notations), and given the clutter and inefficiency shown above, one would wonder if they would find it worth the trouble to learn, if they were given the choice, especially in contrast to the Nemeth they already know, a code that is demonstrably more nimble and efficient. This finding is precisely why the Provisional Guidance to infiltrate Nemeth into math transcriptions is being considered. Nemeth, the old way, made efficient reading but it was ambiguous and had trouble when one wanted to generate a printed document from braille. There are proposed extensions to Nemeth for report-writing, but with the advent of NUBS, a non-ambiguous flavor of Nemeth that does back-translation, there is no interest in them. It is worth noting that the Provisional Guidance that describes embedding uncontracted Nemeth into a UEB transcription does result in a non-ambiguous whole document, i.e., back-translation for report printing is supported in theory, as also supported in UEB math. You would think that braille standards should be based on the very best system possible, as in the slogan and book title by President Jimmy Carter, Why Not The Best? (The phrase is said to have originated with Admiral Rickhover, who asked of all naval cadets, "in your life was there ever a time in which you did less than the best?" and he would follow up that question with "Why not the best?"). The math codes have multiple attributes, and in some cases there has to be a trade-off amongst them. When searching through those trade-offs, we can identify several important traits: Comprehensiveness: does it handle all math discourse, Transparency: does it express the print without distortion or new artifacts, Efficiency: does it get to the point without a lot of finger-tip clutter, and Automation Potential: is the code non-ambiguous with a grammar that lends itself to computer translation, both print-to-braille and braille-to-print (for math, when we say "print" we mean to and from a widely-used print markup language (such as MathML or Tex) that other widely-used tools (such as browsers or word-processors) will render into good-looking math on the screen or on paper). What we should not consider is the degree to which we in the United States braille community are in conformity with international standards, especially if that compromises "Why not the best". The United States with its Americans With Disabilities Act is the world leader in special education: we set the standards, we do not need to follow outside notions of compromise, no matter how politically correct. It seems vital that a math code be nimble enough to be written easily and efficiently. Do you really think a hard-working blind student or professional would put up with a clumsy code? No way. He or she would create their own short-hand, just as a young Abraham Nemeth did in 1946. Look back at the comparative examples in this paper, particularly the homework-type problem of example 2… there is a huge difference in notational efficiency between UEB and Nemeth/NUBS, potentially enough of a difference to cause the student to choose a code other than UEB, even if he or she has to invent it. (The author has been told that around 1942, it was the insufficiency of the Taylor code that caused Abraham Nemeth to invent the Nemeth code.) In previous workshops I have been asked if I thought that some students might simply not study 17 math and/or science because of the reading/writing difficulties. The Provisional Guidance tries to address this, at the expense of maintaining an un-unified two-code system and confusing the reader and transcriber with two mutually incompatible intertwined systems of notation. The efficiency of the embedded approach is compromised by all the begin/end indicators, but it is much better than UEB math itself. If only we had a unified approach with all the good characteristics. What the comparative analysis shows us about Nemeth (in a math context) The table of simple expressions as well as the selection of typical textbook uses show that Nemeth is much more tidy, transparent, and efficient than UEB. The race isn't even close. Nemeth (with its underpinning of EBAE) handles all English literature (prose plus math) except computer notations (things that would be in embedded CBC). The need for CBC and the fact that Nemeth is ambiguous are its weaknesses along with its dependence on EBAE narrative (the problem is its outdated symbol set). But given that everyone knows Nemeth, it is truly powerful and quite elegant. When transcribing superscript and subscripts, Nemeth uses a highly transparent position notation informally called "up", "down", "up up", "down down",… and "baseline". These concepts have explicit scope and thus Nemeth (and likewise NUBS) does not have the scope-less problems of both Taylor and UEB, for which they use grouping symbols not in print. Given that superscripts are raised up and subscripts are lowered down, this notation is more transparent than grouping notions. It is often more efficient as well (saves cells). Nemeth scores high on efficiency and transparency; not high on automatability. What the comparative analysis shows us about UEB (in a math context) UEB is a comprehensive and non-ambiguous code. It has all the features for prose and math that one would want and it seems to be very solid in that it hangs together as a unified body without any obvious inconsistencies. But for math (and sometimes even in prose), the clutter and multi- cell symbols for commonly occurring things make it much less efficient. In algebra (situations with superscripts, subscripts, special operators, etc.) the clutter rises to the level of stifling the enjoyment of the literature. In equations (such as those in this paper) with 2 and b in close proximity, you can be sure that the reader will go back and forth over the braille making sure to account for each number sign and each G1 (letter) sign, trying to be sure about dots (12) being a letter versus a digit. The braille is crystal clear, but it is definitely cluttered and that is a challenge to anything complicated (such as algebra, calculus, and the mathematics of physics, things like Laplace Transforms and the like). The modern notion about transcribing is to "capture print", not the transcribers interpretation of its meaning (i.e., notation but not semantics). The table of simple expressions shows what happens to codes whose constructs lack scope, such as superscript, subscript, and radical in Taylor code. In 1894 the notion was to transcribe the meaning (semantics) of the math and not worry about deviating from print. That is not what we do today, particularly when we want automated transcription. When comparing UEB with Taylor, we see the same kind of effect: compound superscripts and subscripts are bundled by grouping symbols not in print. In UEB this is a markup device (in Taylor it was a semantic device). The markup works, but it connotes the wrong kind of thing, better for xab+ (in which a+b is likely to be evaluated as a group, as in x()ab+ ) than for xab, (a notational device denoting, perhaps some vector behavior). UEB scores high on automatability, pretty high on transparency (slight penalty for invisible grouping symbols), and low on efficiency. 18 What the comparative analysis shows us about NUBS (in a math context) The introductory remarks talked about the necessity for mode in braille codes. Also discussed above is the issue of digits and how there just aren't enough distinct cells in six-dot braille to give them their own cells in all modes. The overlap between digits and punctuation was never a problem, because in Nemeth unannounced digits are preceded by the number sign, and then they aren't punctuation. The use of signs for math constructs (begin/end fraction, radical, integral, equals sign, other operators and symbols) that are otherwise contractions made Nemeth code highly ambiguous. In NUBS that difficulty was repaired by using a mode (called notational) that is analogous to the G1 mode of UEB (and given the need for mode in all codes, it is no better and no worse than using G1 mode in UEB). The table of simple expressions demonstrates many of the cases where, to achieve mode-based non-ambiguous notation, some cells do differ between NUBS and Nemeth. You can see that the general flavor of the two remains the same. You can also see that NUBS has just about the same efficiency of expression as Nemeth. And we note that by way of the notational mode (as with G1 mode in UEB), there is no need for CBC. NUBS is every bit as comprehensive as UEB, but unlike UEB it is built on EBAE spellings. NUBS scores high on transparency, efficiency, and automatability. Although not a legitimate criteria, NUBS scores low on political correctness with few people understanding that it is a viable candidate for a truly unified braille code. Conclusion An efficient, transparent, comprehensive, computer-automatable braille notation for mathematics is a most worthy goal. None of the four codes studied is perfect in these regards, but some are better than others. If you valued computer-automation above all else, then you would want to choose UEB or NUBS, and at the moment UEB is commercially available and NUBS is not. Print-to-braille works for Nemeth, but not braille-to-print and the latter is instrumental to allow a blind student to generate reports in print notation on paper from braille. UEB can do that, unless the student gets lost in the myriad indicators needed and enters the math wrong with correspondingly bad print results. UEB with embedded Nemeth can to that too, and is undoubtedly easier for today's Nemeth-trained users to write. The verbosity of UEB is daunting at times. It is especially tedious when in the hands of the blind student doing homework or research: why should that person need all those indicators and dot 5's just to jot down some letters and numbers and operators? And what if the "proper" way is one way in narrative and a totally different way in "equations" (reference the "Provisional Guidance", see Appendix Two). This system, so absolutely burdened and finicky, will drive the serious math students nuts, particularly those that already use Nemeth. If the Proposed Guidance become a standard, then the inefficiency of UEB math may not be felt in the U.S. but we won't really have a "unified" braille code. UEB is a comprehensive and solid braille notation, quite suitable for general literature. Absent awareness of Nemeth code, it would even seem "OK" for math, but when contrasted to the efficiency and transparency of Nemeth, UEB is dull, and awkward, and tedious. Given that no code is perfect in every dimension, we could look to other, say, political advantages for UEB. According to the UEB marketing strategy, the attractiveness revolves around these aspects: a. Internationalization of English braille (but the amount of braille, especially educational literature, flowing across international borders is miniscule, and with many states having local textbook adoption procedures, it is not likely to rise), b. It has a robust structure, particularly when compared with EBAE (this is true), 19 c. It is the first "unified code" bringing together literary and math notations (no, Nemeth code did that fifty years ago; and with embedded Nemeth, there is no unified code in the U.S.), d. It will improve education (no, an arbitrary change to 9% of standard spellings cannot be considered an improvement, most of the rest of the improvements over EBAE can be handled via the Special Symbols Page to describe those symbols EBAE lacks, when such are found to be needed in the transcription; an improved EBAE is needed, but it shouldn't be so costly to achieve as switching to UEB), e. It will improve automation of braille production (true), f. It is required to mesh with other English-speaking countries (why should the tail wag the dog; the U.S. with its Americans with Disabilities Act is the leader in special education, and it is we who should wag the rest of the world; in particular, why should our blind kids have to relearn 9% of word spellings to switch to the British way?) You can make a good case for UEB, but it was oversold to folks in a blatantly misleading and deceitful way with myriad loose ends including foreign language and math, still up in the air. For math, the comparative analysis indicates that NUBS is the code that fits the times: it doesn't change spellings, it is non-ambiguous, and it is efficient. The next effective approach would seem to be the UEB with embedded Nemeth strategy of the Proposed Guidance. Mixing two dissimilar codes intimately together is absolutely crazy, but it still comes out in second place. So who cares anyway (this is author opinion, please judge for yourself) Complacency about braille math is not new—it has been my observation (for what it's worth) that a "literary transcriber" seems to feel uninvited to do math, even simple math, as if there were a special "Nemeth club" for math transcribers. Perhaps my observation is simply wrong. But during all the UEB discussion and debate, the voices objecting to the inferior math notation of UEB were few and without general support, again as if the majority was not in the "club" and need not therefore be the least bit concerned. That attitude (if real or partly real) might have been permissible fifty years ago, but in today's world, the world in which STEM (science, technology, engineering, mathematics) studies and concepts permeate everyday life, complacency over braille math, by anyone in the braille community, is a horrendous disservice to the core constituency of braille literature: children in grades K-12. Perhaps you see something I do not, but I do not see how the adoption of UEB and the Provisional Guidance aids education. I refer specifically to the sudden changes to spellings and notations for just about everything else and the mixing of Nemeth piece-meal into UEB. If you stop and think through all the issues and changes, you must simply shake your head and wonder how in the world we wound up with such poor choices for the most important braille-reading community. Kudos to the folks proposing the Provisional Guidance: they are trying to salvage a bad deal. But why are we in such a compromised (non-optimal) position in the first place? Are we actually asking our braille community to accept "in your life was there ever a time in which you did less than the best?" This paper contains notational examples from the various braille math codes. Every attempt has been made to give accurate transcriptions of the formulas shown. If you find any errors, corrections will be greatly appreciated and warmly received. The purpose of the paper is to stimulate thought. I encourage you to form your own assessment and opinions, but please do not accept the status quo just because you are afraid to rock the boat. I hope this paper provides useful starting information for that endeavor. 20 Appendix One – Brief History of the Codes Taylor Henry Martyn Taylor (1842-1927) taught math at Trinity College, Cambridge England. He developed a braille code for math in 1894 when he himself became blind. That code was revised in the U.S. in 1920 and again in 1942 by a sub-committee of the Commission for Uniform Type for the Blind (a predecessor of BANA). The Braille Mathematical Notation code uses upper-digits (although not exclusively) and like UEB has scope-less superscripts and subscripts. Unlike the others, it also has a scope-less radical. For compound superscripts and subscripts, braces (indistinguishable from print braces) provide scope. For compound radicals, parentheses (indistinguishable from print parentheses) provide scope. Digits are never first-class symbols. There are notations for math elements up through trigonometry and then an incomplete set of symbols to support calculus. EBAE English Braille American Edition (not a math code) was imported from British braille and adopted in 1959 and revised in 1962, 1968, 1994, and 2002. EBAE is ambiguous and differs from British braille in the avoidance of contractions in contexts where the contraction would blur (span) the edge of a prefix, root, suffix, or main syllable. Nemeth Abraham Nemeth (1918-2013) taught math at the University of Detroit until 1985. He published his code for math in 1952 and it was revised in 1956, 1965, 1972, and 1983. The Nemeth Braille Code for Mathematics and Science Notation uses lower digits and positional (up/down/baseline) indicators for notating superscripts and subscripts, thereby avoiding any issues of scope. Although the term "Nemeth" is sometimes used to denote only the math notations themselves, in this paper "Nemeth" is the term for the entire transcription including the narrative parts (that are done in EBAE with a few changes). In a Nemeth transcription, there are no context switches between the narrative and math portions, and because of this the code is ambiguous. In math contexts, digits and letters are first-class symbols. UEB Having begun as a study project by the Braille Authority of North America (BANA) in 1991, the project was taken over by the International Council on English Braille (ICEB) in 1993. Dr. Nemeth was one of the braille scholars participating at that time however when ICEB decided to use upper digits and became explicitly disinterested in concepts from Nemeth code, Dr. Nemeth resigned from committee participation. Unified English Braille re-introduces into braille for North America the contraction usage rules from British braille. This changes about 9% of all English words (now using boundary-crossing contractions that EBAE avoided, e.g., using the st contraction in "mistake"). ICEB carefully analyzed the limitations of EBAE with respect to automation (principally the ambiguity) and 21 decided to improve things by eliminating seven contractions. For math notation, UEB is similarly structured as Taylor code, in that digits are never first-class symbols and superscripts and subscripts are scope-less, with scope denoted when necessary by "grouping" symbols that are essentially parentheses not in print but known to the braille reader as being invisible, i.e., an artifact only of the braille. This markup strategy might be confused with the semantics of the computation but such confusion is usually harmless. UEB is non-ambiguous. NUBS Having once hoped that UEB would integrate math notations of Nemeth code (which would at a minimum have required lower digits), Dr. Nemeth began work on NUBS when it became clear that UEB would use upper digits and thus borrow essentially nothing from Nemeth code. Although not thought of as a general code, Nemeth code already handled both narrative and math discourse except for computer notations. And thus Dr. Nemeth's goal for NUBS was to modernize Nemeth code to make it non-ambiguous and encompass computer notation. This was done principally by introducing a math-oriented uncontracted mode (similar to the G1 mode of UEB) which in NUBS is called notational mode (the opposite mode is called narrative mode). Nemeth Uniform Braille System has math expressions that are generally similar to Nemeth, and thus they have (in notational mode) first-class digits. Also like Nemeth, there is no scope issue for superscripts and subscripts. Word contractions are those of EBAE, with the elimination of "ble", "ation", and "ally" (for the same pragmatic reasons they are eliminated from UEB). NUBS is non-ambiguous. 22 Appendix Two – BANA's Proposal on Embedded Nemeth "Provisional Guidance" (a public statement from the BANA web site, February 2015) Approved November 2014 When Nemeth Code is to be used for mathematics, the actual math and technical notation should be presented in Nemeth code or the Nemeth-based chemistry code, as applicable, while the surrounding text should be presented in UEB. Basic Guidance on when to Switch 1. Any mathematical or chemical formula should be done in Nemeth Code. This includes fragmentary expressions, including isolated signs of operation or comparison when mentioned in reference to such formulas. 2. All other text, including punctuation that is logically associated with surrounding sentences, should be done in UEB. 3. Exceptions: Despite the above principles, it is nevertheless desirable not to overdo switching just for the sake of a simple item that is easily read in either code. On that basis, avoid switching in the following two cases: a. A mere mention of a number or a letter variable within an otherwise UEB context should be done in UEB. This “number” exception applies to both Arabic and Roman numbers and includes numbers that contain commas and decimal points, but not fractions, which should be done in Nemeth code. b. When a comma or semicolon occurs between items that are to be transcribed in Nemeth Code, even if they could logically be regarded as belonging to the sentence structure, the comma or semicolon should be transcribed in Nemeth Code. Likewise parentheses, brackets, or braces that enclose only material that is to be transcribed in Nemeth code may be transcribed as part of that material even if they could logically be considered as belonging to the larger sentence structure. 4. To avoid use of switch indicators when a single word standing alone occurs between two math expressions, a one-word switch indicator (6,3) may be used in Nemeth mode to indicate that the following word is in UEB. Contractions may be used in the subsequent word. The one-word switch indicator should precede the word whether or not it contains contractions. Otherwise, no contractions are used in Nemeth mode. Similar to the capital indicators, the one-word switch indicator is disregarded for purposes of the UEB lower-sign rule. (This symbol is experimental and not part of the Nemeth code specification). 5. A switch from Nemeth to UEB or from UEB to Nemeth terminates the effect of typeform and capitalization indicators without the need for explicit terminators. 23 Appendix Three – Accessibility to Math In the body of this paper there are numerous print examples of math expressions. The workshop proceedings are to be published to participants as accessible PDFs, but for math, accessibility is not even defined (a PDF file is just the recipe for drawing the pages on a printer, a device whose native actions include drawing text given a font and drawing pictures; there is no standard for drawing math except as a picture or as artfully arranged snippets of text, but if a fraction, nothing that says "fraction"). For this paper, the author has used a commercial typesetting tool for Word known as MathType. Internally the math is notated as Tex or MathML markup, neither of which makes it into the PDF (because that markup doesn't go to a physical printer either). So this paper, in PDF form, has impaired accessibility. Sorry about that. The point is that braille math code is of itself a marvelous markup language to express math in a way that is accessible, thereby heightening the attractiveness for the best braille math code possible. For talking books, there is the "Math Speak" project at Purdue University comprising verbal cues based on Nemeth concepts. If you have difficulty reading this paper because of inaccessibility via the PDF version, please contact the author at [email protected] for the RTF version. 24