some minor rewording and an extra higher math example

Author: dginev

gap-analysis/index.md

@@ -34,7 +34,7 @@ In the following sections, this document discusses the current state of Web math
 
 ## Current State
 
-Mathematical expressions involve text and graphical symbols. Many of the techniques used to make text accessible can be applied to math, but many don't work well due to multiple ways to read the same mathematical expression and also because the words used to speak the math are distinct from braille codes for mathematics that are based on the notations, not the words.
+Mathematical expressions leverage text, graphical symbols, grids and symbol decorations. Sources of extra difficulty come from having multiple ways to read the same mathematical expression and also because the words used in speech are distinct from braille codes for mathematics that are based on the graphical notations, not the words.
 
 ### Math Accessibility details on the Web
 
@@ -49,22 +49,23 @@ The platform APIs in general have not provided much support for mathematical not
 
 MathML and SVG live in somewhat parallel worlds in their relationship to HTML. [SVG Accessibility API Mappings](https://www.w3.org/TR/svg-aam-1.0/) (working draft, May 2018) gives details on SVG accessibility. In general, the document recommends adding ARIA to enhance the accessibility of SVG. Specifically, it states that shape elements (circle, etc) among many others do not go into the accessibility tree unless given semantics via ARIA (e.g, by aria-label). Also, more germane to MathML, elements that do not render visually should never be in the accessibility tree. For MathML, these invisible elements include the non-presentational part of `semantics`, `maction`, etc. Unlike math, there is no specialized braille language for graphics, nor is there an expected way SVG objects should be spoken in the absence of ARIA enhancements.
 
-Ideally, platform APIs should allow those tags and attributes of MathML that have semantic value to be exposed in a straightforward manner. Most MathML tags have semantic value; some attributes do also. Examples of attributes that have semantic value are token elements’ “mathvariant” attribute and mfrac’s “linethickness” and “bevelled” attributes (binomial coefficient if equal to 0 and Nemeth code difference, respectively).
-
+Ideally, platform APIs should allow those tags and attributes of MathML that are informative in narration to be exposed in a straightforward manner. We also refer to such pieces as ones of "semantic value". Most MathML tags indeed participate; some attributes do also. Examples of attributes that have semantic value are the “mathvariant” attribute of token elements, as well as the “linethickness” attribute of mfrac. To illustrate, a common binomial coefficient notation uses a “linethickness” of zero, while the same MathML expression with a positive “linethickness” can be a fraction or the Kronecker symbol.
 
 ### Accessibility of Mathematical Content
-For years, the accessibility of mathematics in print, and later on the web and in other formats has been a large pain point to both those who produce accessible content and to those who consume it. Typically, inaccessible images were used. Even when alternative text was provided, the text could not be converted to braille or navigated in a useful manner; only word-by-word navigation was possible.
+For years, the accessibility of mathematics in print, and later on the web and in other formats, has been a large pain point to both those who produce accessible content and to those who consume it. Typically, inaccessible images were used. Even when alternative text was provided, the text could not be converted to braille or navigated in a useful manner; only word-by-word navigation was possible.
 
 The use of MathML has dramatically reduced these problems. In addition to speech, using MathML allows the specialized braille formats used for mathematics to be generated. Unlike text, these formats are *not* based on the words used to speak the expressions, but on (mostly) the notations used to represent them. Critically, MathML allows readers to explore the mathematical structure of an expression when it is too complicated to be understood when read from start to end.
 
-A remaining problem with the accessibility of MathML occurs because mathematical notations can be ambiguous. For example, $(x,y)$ could be the coordinate of a point or it could be the open interval from x to y. Braille math codes such as Nemeth and UEB encode them the same. Speech could do so also with the literal reading "open paren x comma y close paren". However, this is not how someone would typically read it. Instead they would say something like "the point x comma y" or "the open interval from x to y". There is a supposition that semantic readings are "better", but this has not been confirmed by research for people that are blind; studies do show that semantic reading styles are better for individuals with dyslexia and other non-visual print disabilities. Nonetheless, it is widely assumed that semantic speech is better because people/teachers use semantic readings and listeners are used to hearing them. In many cases, the semantic reading is shorter and therefore uses less working memory. Some examples are:
+A remaining problem with the accessibility of MathML occurs because mathematical notations can be ambiguous. For example, $(x,y)$ could be the coordinate of a point or it could be the open interval from x to y. Braille math codes such as Nemeth and UEB encode them the same. Speech could do so also with the literal reading "open paren x comma y close paren". However, this is not how someone would typically read it. Instead, they would say something like "the point x comma y" or "the open interval from x to y". There is a supposition that semantic readings are "better", but this has not been confirmed by research for people that are blind; studies do show that semantic reading styles are better for individuals with dyslexia and other non-visual print disabilities. Nonetheless, it is widely assumed that semantic speech is important because people/teachers use such readings often and listeners are used to hearing them. In many cases, the semantic reading is shorter and therefore uses less working memory. Some examples are:
 * $x^2$ -- "x squared" versus "x superscript 2 end superscript"
 * $\hat{x}$ -- "x hat" versus "x modified above with circumflex"
 * $\big(\begin{smallmatrix} 1 & 0\\\\ 0 & 1\end{smallmatrix}\big)$ -- "the 2 by 2 identity matrix" vs "open paren start 2 by 2 table;  row 1, column 1, 1, column 2, 0, row 2, column 1 0, column 2 1, end table, close paren"
 
 While all AT speaks $x^2$ semantically, some do not speak $\hat{x}$ semantically, and none currently recognize an identity matrix.
 
-Some notations such as fractions and square roots sometimes require bracketing words or tones to indicate the start and end of some notations for someone who can't see the notation. For example, $\sqrt{x+1}$ is unambiguously spoken as "the square root of x plus 1 end root". Without these bracketing words, "the square root of x plus 1" could also be interpreted as $\sqrt{x}+1$. However, for someone who is dyslexic and uses AT, the extra words are a distraction and shouldn't be used. This difference between the needs of users means that any solution that is adopted should allow flexibility based on the reader -- simple text strings generated by authors may be in appropriate for some readers. 
+Higher mathematics adds an additional tier of complexty. In it, even $x^2$ can be ambiguous, and may warrant a different reading than ordinary. One such case is $L^2$ read simply "L2" in the domain of Lebesgue spaces. In such texts, the author can often find themselves to be one of a only a handful of practitioners who have full grasp of their technical terminology, making manual annotation a frequent need for good AT.
+
+Some notations, such as fractions and square roots, may require bracketing words or tones to indicate the start and end of an expression, for readers who can't see the presentation. For example, $\sqrt{x+1}$ is unambiguously spoken as "the square root of x plus 1 end root". Without these bracketing words, "the square root of x plus 1" could also be interpreted as $\sqrt{x}+1$. However, for someone who is dyslexic and uses AT, the extra words are a distraction and shouldn't be used. This difference between the needs of users means that a real solution should allow flexibility based on the reader -- literal text strings annotated by authors may be inappropriate for some readers.
 
 #### Large Example
 To fully appreciate the difference between presentational and semantic speech, below is a question from a [MathCounts middle school math competition](https://www.mathcounts.org/sites/default/files/2020%20Chapter%20Competition%20Solutions.pdf).