American Action Fund for Blind Children and Adults*
Future Reflections* Special Issue: The Individualized Education Plan (IEP) STEM

by Al Maneki

**From the Editor: "I hate math!" a blind college student told me recently. "Last semester I dropped out of pre-calculus, but I have to get through it or I won't be able to graduate. I just don't understand what the professor is talking about, and nobody knows how to explain it to me." Like this young woman, many blind students struggle with math courses—not because they are inherently unable to master the subject matter, but because they do not have proper access to visually presented material. In this article, adapted from http://www.maa.org/node/790342 and updated since 2015, Dr. Al Maneki, a blind mathematician, draws upon a lifetime of experience and suggests ways that college-level math classes can be made fully accessible for blind students. Dr. Maneki's suggestions can easily apply to secondary and even elementary math instruction. Al Maneki serves as senior STEAM advisor (science, technology, engineering, art, and mathematics) at the National Federation of the Blind Jernigan Institute. He can be reached at mapmaneki@earthlink.net.**

by Dr. Martha Siegel

Professor Emerita, Towson University

A Project NExT online discussion about the availability of aids to blind and visually impaired students in college-level mathematics courses generated discussion with a current Towson University graduate student, Natalie Shaheen. (The Mathematical Association of America's Project NExT, New Experiences in Teaching, is a professional development program for new or recent PhDs in the mathematical sciences.) Ms. Shaheen is a doctoral candidate, and one of her areas of expertise is making STEM subjects accessible to blind students. She has been teaching STEM to blind students for nine years. Though she generally teaches K-12 students, she has advised many college professors on this subject. I found her recommendations insightful and spoke with her about publicizing them. Ms. Shaheen suggested we involve Dr. Al Maneki, who holds a PhD from Illinois Institute of Technology and is a 2007 retiree from the National Security Agency. With more than fifty years as a blind mathematician, Dr. Maneki has a unique perspective on how we might do a better job of educating our blind and visually impaired students in mathematics. The Committee on Undergraduate Programs in Mathematics (CUPM) of the Mathematical Association of America (MAA) has added his sage advice to its website.

The news that a sighted mathematics faculty member in a two- or four-year college program will have a blind student is all too often greeted with disbelief, panic, or resentment. The common assumption is that it is impossible for a blind student to learn or comprehend mathematical material, given the supposedly highly visual nature of this subject. This initial reaction immediately creates a barrier to the effective teaching of mathematics to a blind student by a sighted faculty member. It is important for us to overcome the difficulties that these negative stereotypes place in the effective teaching and learning of mathematical subjects.

In this article our aim is to provide a set of guidelines to assist two- and four-year college faculty members to provide meaningful mathematics instruction to blind and visually impaired students. It is generally assumed that the student will encounter less difficulty in learning mathematics if she or he possesses some usable vision. However, this is not necessarily the case. We should not assume that the student can depend upon limited vision for undertaking the study of mathematics or any other discipline. Most successful low-vision students have acquired a variety of skills of blindness in order to work efficiently. They use large print if and when it works for them; they use Braille when needed; they use synthetic speech on a computer or other notetaking device; and they use human readers. These guidelines are directed to faculty members who occasionally find themselves teaching mathematics at all levels to blind students, regardless of visual acuity. We will use the terms blind students, blind and visually impaired students, and students interchangeably.

Students as well as instructors may wish to read these guidelines. Students and instructors must work closely together to achieve the level of communication that will be most beneficial to the student and to make the best use of the instructor's time.

In these guidelines we assume that blind students are adequately trained in the effective use of the tools of blindness, as described above. It is not the responsibility of university faculty members to teach Braille or any other learning or study techniques that blind students need. Instead, it is important for faculty members to think about how mathematics can be communicated without the use of print media. This can indeed be done. In my own experience I have encountered mathematical discussions taking place in situations where pencil and paper pad, or even paper napkins, are not handy. In spare moments, how many of us find ourselves formulating mathematical arguments without writing them down?

I do not mean to imply that mathematics can be done without the aid of written media. I believe that ultimately we cannot understand a mathematical idea until we can write it down. For the blind student or blind mathematician, this usually means writing it in Braille. While we have heard much about auditory modes of learning, I do not believe that mathematics can be done in purely auditory form. However, there is much to be said for the ability to speak mathematical equations aloud while writing them down in the classroom. This combination of writing and speaking will enhance the experience for both blind and sighted students.

The guidelines I present here are primarily the result of my own experiences, both as a blind student and a professional mathematician working in higher education and in the federal government. They are not the result of extensive surveys or field testing by professional educators or psychologists. As they gain a wider circulation, others may want to make additions to, or include different perspectives on, these guidelines. Please submit your comments and ideas to me. Some sighted instructors may believe that these guidelines do not cover their particular situation. If this is the case, please contact me directly so that I may be of further assistance.

I have written these guidelines to correspond with the two primary functions of mathematics education: the delivery of information and inspiration in the classroom and the demonstration of acquired knowledge by solving problems and taking tests. I have included some background information primarily to give sighted faculty members a better understanding of the learning environment in which today's blind or visually impaired student must operate.

Blind students should not be exempted from satisfactorily completing required courses in mathematics and the hard sciences. Exemption from a required course is not a reasonable accommodation. Such exemptions simply deprive blind students and other students with disabilities of a learning experience that has been determined to be beneficial for all other students.

Much of the material included here may be well known to faculty and students. This section may be omitted by such readers.

LaTeX was initially invented as a typesetting language for mathematical notation. It is text-based and non-graphical in nature. LaTeX-based graphics packages even can be used for drawing pictures and creating diagrams. By typing standard text on a keyboard, one can represent all of the mathematical symbols, from the most elementary to the most advanced.

In recent years many books on mathematics have been published using LaTeX as the underlying typesetting or mark-up language. This development has been most beneficial to blind readers, as I will explain in the next section.

The Nemeth Braille Code was developed around 1950 as a method to represent mathematical symbols in Braille. It has been revised several times to improve earlier versions. Because the Braille cell is made up of only six dots, at most sixty-four dot combinations, including the blank or space character, can be produced from a single Braille cell. To allow for the many mathematical characters, the Nemeth Code employs well-defined sequences of Braille characters to represent one mathematical symbol. The Nemeth Code includes special symbols to represent complex structures such as subscripted subscripts, subscripted superscripts, complex fractions, and other nonlinear mathematical notations.

Despite the complexity of the Nemeth Braille Code, it is extremely useful. Books on mathematical subjects, when they exist in Braille at all, are produced using the Nemeth Code. Given the existence of LaTeX-based textbooks along with LaTeX-Nemeth Braille translation software, it is now much simpler to produce a Braille textbook when it is needed. Liberal copyright laws permit the production of books in alternative formats, such as Braille, for students with disabilities. However, publishers still are reluctant to release their LaTeX source code for fear that it will end up in the wrong hands and be abused. All of the math and science articles in Wikipedia are LaTeX-based. As an introductory step we may easily produce an article from Wikipedia in Nemeth Braille.

Currently, LaTeX-Nemeth Braille translation software does not permit the conversion of graphical images into tactile ones. Automated conversions present problems that have yet to be solved. The degree of resolution that the human eye can distinguish is much finer than that which human fingers can interpret. For a complex tactile drawing to be meaningful, it must be much larger than a drawing intended for the human eye. Furthermore, the sense of touch has no direct equivalent to the distinction of colors. The translation of visual graphics into tactile images means the production of line drawings, and this conversion requires human intervention.

University-level recorded textbooks have been available from Learning Ally (http://learningally.org) for decades. If a textbook is not available in Learning Ally's catalog, blind students may request Learning Ally to produce an audio edition of that book. Learning Ally recruits volunteers to record books in its studios, which are located throughout the country.

The selection of audio books on mathematics in Learning Ally's catalog is quite extensive. Historically, producing an audio book has been the quickest and cheapest way to get that book into the hands of blind students. The primary drawback I have found with audio math books is the inconsistency in which equations and formulas are read and the manner in which diagrams are described. Since several volunteers may be used to read a single title, there is not necessarily consistency in how fractions, subscripted variables, and exponents are read. This inconsistency can leave a listener puzzled and bemused about just what a mathematical expression means. For this reason, I do not recommend audio math books. However, when a given textbook is not available in any other formats, the desperate student may have no choice but to borrow the audio version from Learning Ally.

For these reasons I have found it most satisfactory to have textbooks read by a live reader. My reader and I can agree on the manner in which material is to be read. If I still encounter ambiguities, I can stop the reader and get immediate clarification. I can also take Braille notes as the text is being read.

Some courses permit or require the use of graphics-based software or graphing calculators as teaching aids. These tools may pose vexing problems for the blind student, as they are not always nonvisually accessible. If such tools are required in a particular course, faculty and students must cooperate to find acceptable alternatives. It is unwise to list such alternatives in these written guidelines, as the field of nonvisual access is changing rapidly. Various websites, listservs, discussion groups, and blogs are dedicated to nonvisual access. Some of these are listed at the end of these guidelines.

The American Printing House for the Blind (www.aph.org) sells the Orion TI-84 Plus Graphing Calculator for six hundred dollars. It is identical to the standard TI-84 Plus Graphing Calculator, except that synthetic speech and audio graphing features have been added. I have no experience working with this calculator, and a quick search on Google yielded no results for reviews of this product.

The situation may be even bleaker for the symbolic mathematical tools such as Mathematica, Maple, MyMathLab, and SageMath. Some users may have devised roundabout solutions to achieve a degree of nonvisual access to these packages. Though these languages are text-based, their development environments are not always friendly for the blind user. One common problem area lies in the debugging messages one gets when attempting to compile code written in these languages. For the nonspecialist, the student who needs to complete a one- or two-semester math requirement, the effort to achieve nonvisual access to these tools may be too much to undertake.

For statistical software, a number of blind users have had success with the R Project (https://www.r-project.org/). However, I don't believe that the use of R is for the faint-hearted. It has a steep learning curve, and only people who will be doing serious statistical work should undertake the study of R.

More satisfying at the introductory levels may be the use of Microsoft Excel and the mathematical functions it contains. The developers of the JAWS screen reader have put considerable effort into access to Excel. While graphs generated by Excel may not be reproducible in nonvisual form, the mere ability to read table values may be sufficient to give the blind user a good idea about what the data looks like.

If all else fails, there is always access to the output of software tools through the help of a live reader. Your reader simply executes your instructions and describes the results to you.

Graphs and other pictorial images have always been troublesome for blind students. Without access to tactile diagrams, blind students have been forced to rely on verbal descriptions. Such descriptions may be problematic. Sometimes there are simply no words or phrases that can accurately describe a diagram.

A usable system for tactile graphics should satisfy two functions: a read-only function in which a tactile graph or diagram is drawn for a blind student to examine and a read-write function that allows a blind student to draw a new diagram. A read-write function for tactile graphics would permit both functions simultaneously. This means that the tactile diagram should permit erasures so that an existing drawing may be corrected. A major problem for read-write function tactile graphics arises with the use of scribing tools that press down into the drawing medium, usually a sheet of paper. To feel a raised-line diagram, one must turn the drawing sheet over and examine the drawing on the reverse side.

For quite some time the Sewell Raised-Line Drawing Kit has been marketed as a limited read-write function, right-side-up drawing system. A plastic sheet is used on a board covered with a thin sheet of rubber. When a pointed scribing tool is moved over the mounted plastic sheet, a raised line or curve is created that can be felt right-side up. This drawing kit has primarily been a read-only function system because there was never an easy way to erase and correct tactile diagrams. More recently the SenseSational Blackboard has permitted same-side drawing and reading of graphics using standard paper.

A few years ago a Vermont-based company, E.A.S.Y., LLC, developed an improved drawing board, the inTACT Sketchpad, along with an eraser for corrections. The eraser gives the inTACT Sketchpad true read-write functionality. While there are other tactile graphics boards on the market, the inTACT Sketchpad is the only one having true erasing capabilities. [Disclosure: Since the inception of E.A.S.Y., LLC., I have advised company officials on the design of the company's inTACT product line. The company currently keeps me on retainer.]

Disabled Student Services offices exist on nearly every university campus. Their role is to facilitate the learning experience for all students with disabilities. When it comes to mathematics, DSS personnel may attempt to secure Braille textbooks, but they are not always successful, given the prohibitive cost of producing Braille. They may attempt to provide "note-takers," people who take notes during lectures and somehow reproduce these notes in a usable form for blind students. DSS officers also may negotiate with instructors about test-taking arrangements. While DSS officers can be helpful, they should not interfere with instructor-student communications. In any subject, it is best for students and their instructors to work directly together.

DSS personnel often complain that instructors are much too slow in selecting their textbooks. Instructors with blind students enrolled in their courses should be mindful of the need to announce their textbook selections as early as possible. The blind student needs sufficient time to secure the textbook in an accessible format.

The most frequent criticism from blind students is that sighted math instructors do not verbalize enough of the material that is presented visually during a class lecture. It is all too easy for a teacher to write something down and point at it when referring back to that particular statement or expression. Often the lecturer may believe that a formula or other expression is too complicated to be read aloud or that speaking it will take too much time. There needs to be a balance here. We all recognize that there is only so much time in a lecture period. Yet for the purposes of classroom delivery, I believe the notation can be simplified without loss of preciseness. If the notation is too cumbersome to be spoken in its entirety, enough of it can be spoken to give the listener the gist of what is going on. With this basic explanation, the student can make reasonable sense of the arguments that are being presented. An ideal comfort level exists between not enough speaking and speaking too much. When a blind student understands a style of delivery, he or she can pretty much guess at the items that are not being spoken. This understanding may be acquired by frequent "offline" chats between student and instructor.

As an illustration, many years ago, I took a course in Lisp programming. In Lisp all ordered pairs are set off in parentheses, such as (—, —). Needless to say, Lisp allows for endless sequences of nested parenthetic expressions within these structures of ordered pairs. Once I understood that all ordered pairs are surrounded by parentheses, it was not necessary for the instructor to speak aloud all of the sequences of left and right parentheses.

As the costs of reproducing printed materials have decreased, the quantity of printed classroom handouts has escalated. Gone are the days of mimeographs and ditto sheets. In the past the only recourse for blind students was to have these handouts read by a live reader. Now, if the handouts are produced by LaTeX, students may have them in Braille or read them with synthetic speech on a computer. Mathematical materials translated into synthetic speech may be filled with the same ambiguities that I discussed earlier. These handouts may be enlarged for the student with low vision. If handouts cannot be produced in Braille, then they are best read to the student by a live reader.

When producing Braille notes for personal use, students tend to simplify the Nemeth Code for convenience and speed. The shorthand Nemeth that I write for myself is definitely context-sensitive and filled with inconsistencies. Since I have devised my own shorthand Nemeth, I am the only one who can read and make sense of my notes.

A personalized shorthand Nemeth-Braille system also works for solving problems and completing assignments. A student may also use a Sketchpad to produce diagrams that may be needed in the solution of a particular problem. I did not have this luxury in my student days. A Sketchpad is simple enough for anyone to use. In an office-hour discussion, an instructor may even use the student's Sketchpad to illustrate a point.

No Nemeth editing software exists to help a blind student produce completely accurate Nemeth code. There is no translating software to take a Nemeth-Braille file input and produce LaTeX output. For these reasons the blind student must spend considerable time preparing assignments and examinations for sighted instructors. I found it most efficient to dictate mathematical content to a sighted reader.

A final note on using Braille: Since refreshable Braille notetakers such as the BrailleNote, PAC Mate, and Braille Sense became available, many blind students have opted to take classroom and study notes using these devices. Braille notetakers are very powerful tools with helpful features such as spell checkers, online dictionaries, and word processing editors. The primary drawback is that the Braille display consists of a single line of Braille containing a maximum of forty Braille cells.

Compare this to a standard page of Braille produced by a Braillewriter, a page consisting of forty cells per line and twenty-five lines. As one writes on a Braillewriter, the Braille characters are raised so that it is easy to read what one has written. The complexity of Nemeth Code, with multiple Braille cells representing one math character, means that very little of a mathematical expression is exposed at any given time on a Braille notetaker. This limitation definitely can impede comprehension of the underlying mathematics. Furthermore, when we look at math in Braille, we want to view an entire expression by using the fingers of both hands, where each hand may rest on a separate Braille line. For example, consider the problem of checking nested parenthetical expressions.

Some refreshable Braille notetakers are loaded with full scientific calculators that may be quite useful. In the best of worlds a blind student should possess a refreshable Braille notetaker as well as a standard Braillewriter.

If a student has a firm grasp of LaTeX, it is possible for him to complete problem-solving assignments and submit answers to examinations by writing answers in LaTeX. Using a LaTeX editor, the student compiles LaTeX source code and submits readable mathematics. While this sounds simple enough, as we all know, debugging source code in any language can be time-consuming. Some instructors may accept the LaTeX source code and may be willing to work through it to evaluate a student's work. The difficulty here for the instructor and the student is to determine whether an error in the student's work is due to an incorrect solution or to an error in LaTeX coding.

Some students have had remarkable success by using an editor to type mathematical notation in entirely linear form. With judicious use of parenthesis, square brackets, and curly brackets, it is possible to type complex expressions in linear form. Greek alphabet letters may be spelled out instead of being represented as symbols. LaTeX achieves this by spelling out the Greek letter and preceding it with a backslash symbol. For example, the integral (x)dx may be expressed as integral (a,b)f(x)dx. Since there are no agreed-upon conventions for such linearizing, instructor and student must agree on the conventions that will be used here. The linear text equivalent for ∞ may be represented as \infty, adopting the LaTeX convention. To save space, one may adopt a shorter notation such as (00).

In my pre-technology student days, when Braille math books were nonexistent, classmates or math majors ahead of me read my textbooks and took dictation for my problem assignments and tests. Although there was no money to pay them, my classmates saw this reading time as an opportunity for us to study together and to benefit from the ensuing discussions. As they read, I made Braille notes to help me with problem-solving assignments and test preparation. I worked out problem solutions in Braille, then dictated them to my readers. I completed tests in the same fashion. My instructors were very cooperative in finding unoccupied spaces where I could take tests with a reader. Sometimes we just used the instructor's office. It was understood that my reader would scrupulously write down what I dictated.

Unfortunately, the use of live readers in any subject is no longer in vogue. This is a pity, especially in mathematical subjects. The DSS offices are quite explicit about paying for notetakers, people who will take notes for students in class, but they generally will not pay for readers.

Chelsea Cook, a blind student who graduated with a degree in physics from Virginia Tech in 2015, uses a purely dictation method that she has developed for homework and exams in her higher-level classes. Since there is no widely accepted standard for spoken mathematics, individual students are encouraged to develop their own dictation methods. Obviously, dictation methods of this type are highly context sensitive. They are intended to work only between individual blind students and their live readers.

With the advent of tactile graphics drawing boards, diagrams from textbooks may be drawn to present graphical ideas and constructions to the blind student. Since it is not always easy to render a printed image into a tactile one, images from textbooks must be drawn by hand. A blind student does not need to have every graphic image reproduced. My experience has been that, for the most part, I get a good idea of what is represented pictorially from the accompanying textual description.

If drawing a graph or other line diagram is required in a problem assignment or on a test, the blind student may use a raised-line drawing board rather than giving verbal instructions to a reader/scribe. Tactile graphs can easily be read by a sighted person.

Since I was never exposed to drawing at an early age, my drawings were very crude. However, they were good enough to convince the harshest examiner that I had mastered the concepts they represented.

Mathematical subject matter is not inherently beyond the capabilities of blind or visually impaired students. What is important is the communication channel through which knowledge is transmitted. The most efficient communication channel is one that is worked out directly between instructor and student. Instructors can be most helpful by providing the blind student with out-of-classroom time. When thinking about learning arrangements and the communication channel, remember these watchwords: "keep it simple!" For example, the same methods should be used for both completing homework assignments and taking tests. As I stated in the introduction, if there are matters in these guidelines that I have not covered to fit a particular situation, please get in touch with me.

These guidelines have been written primarily from my own perspective and experience. Others, both faculty and students, may wish to share different ideas and experiences. I envision a website where these guidelines and other articles can be posted and new ideas and resources can be added over time. As we gather the collective mathematical experiences of blind students and their instructors, the reasons for exempting blind students from math requirements will disappear.

I wish to acknowledge the valuable help and encouragement of many people who led me to write these guidelines. Natalie Shaheen from the National Federation of the Blind Jernigan Institute introduced me to Dr. Martha Siegel of Towson University. Dr. Siegel engaged me in lengthy discussions about teaching math to blind and visually impaired students. These guidelines are the result of her generous encouragement. Dr. Michael Boardman of Pacific University in Oregon succeeds Dr. Siegel as chairman of CUPM at the end of January. His help will be valuable in presenting these guidelines to a wider mathematical audience. Dr. Doug Ensley, deputy executive director of the Mathematical Association of America, along with Dr. Siegel and Dr. Boardman, is responsible for including this paper in the MAA's literature collection. Chelsea Cook, a blind graduate in physics from Virginia Tech, reviewed the original draft and pointed me to the paper she has written on this subject.

Access2Science, http://access2science.com

Blindmath listserv, http://nfbnet.org/mailman/listinfo/blindmath_nfbnet.org

Independence Science, http://www.independencescience.com

National Center for Blind Youth in Science (NCBYS), http://www.blindscience.org

A Selection of Postings from the Blind Math Listserv, http://www.blindscience.org/blindmath-gems-home

Cook, C. (unpublished) "Math 2974: Mathematical Visualization."

To request copies, contact mcook2010@vt.edu.

Jackson, A. "The World of Blind Mathematicians." http://www.ams.org/notices/200210/index.html

Maneki, A. (2012) "Blind Mathematicians? Certainly!" *Braille Monitor,
*July. https://nfb.org/images/nfb/publications/bm/bm12/bm1207/bm120702.htm

— "Can We Erase Our Mistakes? The Need for Enhanced Tactile Graphics." (2012) *Braille Monitor,* June.

https://nfb.org/images/nfb/publications/bm/bm12/bm1206/bm120602.html

— "NFB Math Survey: A Report of Preliminary Results." (2011) *Braille Monitor,* October.

https://nfb.org/images/nfb/publications/bm/bm11/bm1109/bm110909.htm

— (2013) "The Dawn of the Age of Tactile Fluency: Let the Revolution Begin!" *Braille Monitor,* November.

https://nfb.org/images/nfb/publications/bm/bm13/bm1310/bm131003.htm

— "Handling Math in Braille: A Survey." (2011) *Braille Monitor,* February.

https://nfb.org/Images/nfb/Publications/bm/bm11/bm1102/bm110208.htm

— (2014) "The Tactile Fluency Revolution: Year Two." *Braille Monitor,* December.

https://nfb.org/images/nfb/publications/bm/bm14/bm1411/bm141113.htm

Maneki, A. and A. Jeans. (unpublished) "A Simple LaTeX Tutorial." To request copies, contact

mapmaneki@earthlink.net.