# Notes on Blind Mathematicians

Subjects:

- The World of Blind Mathematicians
- Article on Successfully Studying
- NFB Math Survey: A Report of Preliminary Results

Related Topics:

No attempt was made to verify the correctness or accuracy of email addresses, websites, claims or statements of fact made by any contributor to Blindmath Gems.

## 1.THE WORLD OF BLIND MATHEMATICIANS

By: Allyn Jackson

Reprinted from: Notices of the AMS (American Mathematical Society)

November 2002; Volume 49, Number 10, Pages: 1246-1251

Editor’s Introduction: The credit for bringing this article to our attention goes to Judy Fisher, who posted it in the Blindmath listserv on June 1, 2005. She included this article in a thread on blind people working in the mathematical sciences. The original article was downloaded from www.ams.org/notices/200210/comm-morin.pdf. Since this article is only available in PDF format, some effort was required on Ms. Fisher’s part to include it in the Blindmath listserv. We took her post and did further work to refine her file into a more presentable text version for these gems. This is an outstanding article that will be of considerable interest to anyone who wants to know how blind mathematicians do their work. The discussion of technology in this article is dated, but still valuable.

A visitor to the Paris apartment of the blind geometer Bernard Morin finds much to see. On the wall in the hallway is a poster showing a computer-generated picture, created by Morin’s student François Apéry, of Boy’s surface, an immersion of the projective plane in three dimensions. The surface plays a role in Morin’s most famous work, his visualization of how to turn a sphere inside out. Although he cannot see the poster, Morin is happy to point out details in the picture that the visitor must not miss. Back in the living room, Morin grabs a chair, stands on it, and feels for a box on top of a set of shelves. He takes hold of the box and climbs off the chair safely—much to the relief of the visitor. Inside the box are clay models that Morin made in the 1960s and 1970s to depict shapes that occur in intermediate stages of his sphere eversion. The models were used to help a sighted colleague draw pictures on the blackboard. One, which fits in the palm of Morin’s hand, is a model of Boy’s surface. This model is not merely precise; its sturdy, elegant proportions make it a work of art. It is startling to consider that such a precise, symmetrical model was made by touch alone. The purpose is to communicate to the sighted what Bernard Morin sees so clearly in his mind’s eye.

A sighted mathematician generally works by sitting around scribbling on paper: According to one legend, the maid of a famous mathematician, when asked what her employer did all day, reported that he wrote on pieces of paper, crumpled them up, and threw them into the wastebasket. So how do blind mathematicians work? They cannot rely on back- of-the-envelope calculations, half-baked thoughts scribbled on restaurant napkins, or hand-waving arguments in which “this” attaches “there” and “that” intersects “here”. Still, in many ways, blind mathematicians work in much the same way as sighted mathematicians do. When asked how he juggles complicated formulas in his head without being able to resort to paper and pencil, Lawrence W. Baggett, a blind mathematician at the University of Colorado, remarked modestly, “Well, it’s hard to do for anybody.” On the other hand, there seem to be differences in how blind mathematicians perceive their subject. Morin recalled that, when a sighted colleague proofread Morin’s thesis, the colleague had to do a long calculation involving determinants to check on a sign. The colleague asked Morin how he had computed the sign. Morin said he replied: “I don’t know—by feeling the weight of the thing, by pondering it.”

### BLIND MATHEMATICIANS IN HISTORY

The history of mathematics includes a number of blind mathematicians. One of the greatest mathematicians ever, Leonhard Euler (1707–1783), was blind for the last seventeen years of his life. His eye-sight problems began because of severe eyestrain that developed while he did cartographic work as director of the geography section of the St. Petersburg Academy of Science. He had trouble with his right eye starting when he was thirty-one years old, and he was almost entirely blind by age fifty-nine. Euler was one of the most prolific mathematicians of all time, having produced around 850 works. Amazingly, half of his output came after his blindness. He was aided by his prodigious memory and by the assistance he received from two of his sons and from other members of the St. Petersburg Academy.

The English mathematician Nicholas Saunderson (1682–1739) went blind in his first year, due to smallpox. He nevertheless was fluent in French, Greek, and Latin, and he studied mathematics. He was denied admission to Cambridge University and never earned an academic degree, but in 1728 King George II bestowed on Saunderson the Doctor of Laws degree. An adherent of Newtonian philosophy, Saunderson became the Lucasian Professor of Mathematics at Cambridge University, a position that Newton himself had held and that is now held by the physicist Stephen Hawking. Saunderson developed a method for performing arithmetic and algebraic calculations, which he called “palpable arithmetic”. This method relied on a device that bears similarity to an abacus and also to a device called a “geoboard”, which is in use nowadays in mathematics teaching. His method of palpable arithmetic is described in his textbook Elements of Algebra (1740). It is possible that Saunderson also worked in the area of probability theory: The historian of statistics Stephen Stigler has argued that the ideas of Bayesian statistics may actually have originated with Saunderson, rather than with Thomas Bayes [St].

Several blind mathematicians have been Russian. The most famous of these is Lev Semenovich Pontryagin (1908–1988), who went blind at the age of fourteen as the result of an accident. His mother took responsibility for his education, and, despite her lack of mathematical training or knowledge, she could read scientific works aloud to her son. Together they fashioned ways of referring to the mathematical symbols she encountered. For example, the symbol for set intersection was “tails down,” the symbol for subset was “tails right,” and so forth. From the time he entered Moscow University in 1925 at age seventeen, Pontryagin’s mathematical genius was apparent, and people were particularly struck by his ability to memorize complicated expressions without relying on notes. He became one of the outstanding members of the Moscow school of topology, which maintained ties to the West during the Soviet period. His most influential works are in topology and homotopy theory, but he also made important contributions to applied mathematics, including control theory. There is at least one blind Russian mathematician alive today, A. G. Vitushkin of the Steklov Institute in Moscow, who works in complex analysis.

France has produced outstanding blind mathematicians. One of the best known is Louis Antoine (1888–1971), who lost his sight at the age of twenty-nine in the first World War. According to [Ju], it was Lebesgue who suggested Antoine study two- and three-dimensional topology, partly because there were at that time not many papers in the area and partly because “dans une telle étude, les yeux de l’esprit et l’ habitude de la concentration remplaceront la vision perdue” (“in such a study the eyes of the spirit and the habit of concentration will replace the lost vision”). Morin met Antoine in the mid-1960s, and Antoine explained to his younger fellow blind mathematician how he had come up with his best-known result. Antoine was trying to prove a three-dimensional analogue of the Jordan-Schönflies theorem, which says that, given a simple closed curve in the plane, there exists a homeomorphism of the plane that takes the curve into the standard circle. What Antoine tried to prove is that, given an embedding of the two-sphere into three-space, there is a homeomorphism of three-space that takes the embedded sphere into the standard sphere. Antoine eventually realized that this theorem is false. He came up with the first “wild embedding” of a set in three-space, now known as Antoine’s necklace, which is a Cantor set whose complement is not simply connected. Using Antoine’s ideas, J. W. Alexander came up with his famous horned sphere, which is a wild embedding of the two-sphere in three-space. The horned sphere provides a counterexample to the theorem Antoine was trying to prove. Antoine had proved that one could get the sphere embedding from the necklace, but when Morin asked him what the sphere embedding looked like, Antoine said he could not visualize it.

### EVERTING THE SPHERE

Morin’s own life story is quite fascinating. He was born in 1931 in Shanghai, where his father worked for a bank. Morin developed glaucoma at an early age and was taken to France for medical treatment. He returned to Shanghai, but then tore his retinas and was completely blind by the age of six. Still, he has a stock of images from his sighted years and recalls that as a child he had an intense interest in optical phenomena. He remembers being captivated by a kaleidoscope. He had a book about colors that showed how, for example, red and yellow mix together to produce orange. Another memory is that of a landscape painting; he remembers looking at the painting and wondering why he saw three dimensions even though the painting was flat. His early visual memories are especially vivid because they were not replaced by more images as he grew up.

After he went blind, Morin left Shanghai and returned to France permanently. There he was educated in schools for the blind until age fifteen, when he entered a regular lycée. He was interested in mathematics and philosophy, and his father, thinking his son would not do well in mathematics, steered Morin toward philosophy. After studying at the École Normale Supérieure for a few years, Morin became disillusioned with philosophy and switched to mathematics. He studied under Henri Cartan and joined the Centre National de la Recherche Scientifique as a researcher in 1957. Morin was already well known for his sphere eversion and had spent two years at the Institute for Advanced Study by the time he finished his Ph.D. thesis in singularity theory in 1972, under the direction of René Thom. Morin spent most of his career teaching at the Université de Strasbourg and retired in 1999.

It was in 1959 that Stephen Smale proved the surprising theorem that all immersions of the n-sphere into Euclidean space are regularly homotopic. His result implies that the standard embedding of the two-sphere into three-space is regularly homotopic to the antipodal embedding. This is equivalent to saying that the sphere can be everted, or turned inside out. However, constructing a sphere eversion following the arguments in Smale’s paper seemed to be too complicated. In the early 1960s, Arnold Shapiro came up with a way to evert the sphere, but he never published it. He explained his method to Morin, who was already developing similar ideas of his own. Physicist Marcel Froissart was also interested in the problem and suggested a key simplification to Morin; it was for the collaboration with Froissart that Morin created his clay models. Morin first exhibited a homotopy that carries out an eversion of the sphere in 1967.

Charles Pugh of the University of California at Berkeley used photographs of Morin’s clay models to construct chicken wire models of the different stages of the eversion. Measurements from Pugh’s models were used to make the famous 1976 film Turning a Sphere Inside Out. Created by Nelson Max, now a mathematician at Lawrence Livermore National Laboratory, the film was a tour de force of computer graphics available at that time. Morin actually had two different renditions of his sphere eversion, and at first he was not sure which one appeared in the film. He asked some of his colleagues who had seen the film which rendition was depicted. “Nobody could answer,” he recalled.

Since the making of Max’s film, other sphere eversions have been developed, and new movies depicting them have been made. One eversion was created by William Thurston, who found a way to make Smale’s original proof constructive. This eversion is depicted in the film Outside In, made at the Geometry Center [OI]. Another eversion originated with Rob Kusner of the University of Massachusetts at Amherst, who suggested that energy- minimization methods could be used to generate Morin’s eversion. Kusner’s idea is depicted in a movie called The Optiverse, created in 1998 by the University of Illinois mathematicians John M. Sullivan, George Francis, and Stuart Levy [O]. Sculptor and graphics animator Stewart Dickson used the Optiverse numerical data to make models of different stages of the optiverse eversion, for a project called “Tactile Mathematics” (one aim of the project is to create models of geometric objects for use by blind people). Some of the optiverse models were given to Morin during the International Coloquium on Art and Mathematics in Maubeuge, France, in September 2000. Morin keeps the models in his living room.

Far from detracting from his extraordinary visualization ability, Morin’s blindness may have enhanced it. Disabilities like blindness, he noted, reinforce one’s gifts and one’s deficits, so “there are more dramatic contrasts in disabled people,” he said. Morin believes there are two kinds of mathematical imagination. One kind, which he calls “time-like”, deals with information by proceeding through a series of steps. This is the kind of imagination that allows one to carry out long computations. “I was never good at computing,” Morin remarked, and his blindness deepened this deficit. What he excels at is the other kind of imagination, which he calls “space-like” and which allows one to comprehend information all at once. One thing that is difficult about visualizing geometric objects is that one tends to see only the outside of the objects, not the inside, which might be very complicated. By thinking carefully about two things at once, Morin has developed the ability to pass from outside to inside, or from one “room” to another. This kind of spatial imagination seems to be less dependent on visual experiences than on tactile ones. “Our spatial imagination is framed by manipulating objects,” Morin said. “You act on objects with your hands, not with your eyes. So being outside or inside is something that is really connected with your actions on objects.” Because he is so accustomed to tactile information, Morin can, after manipulating a hand-held model for a couple of hours, retain the memory of its shape for years afterward.

### GEOMETRY: PURE THINKING

At a meeting at the Mathematisches Forschungsinstitut Oberwolfach in July 2001, Emmanuel Giroux presented a lecture on his latest work entitled “Contact structures and open book decompositions”. Despite Giroux’s blindness—or maybe because of it—he gave what was probably the clearest and best organized lecture of the week-long meeting. He sat next to an overhead projector, and as he put up one transparency after another, it was apparent that he knew exactly what was on every transparency. He used his hands to schematically illustrate his precise description of how to attach one geometric object to the boundary of another. Afterwards some in the audience recalled other lectures by Giroux, in which he described, with great clarity, certain mathematical phenomena as evolving like the frames in a film. “In part it’s my way of doing things, my style” to try to be as clear as possible, Giroux said. “But also I’m often extremely frustrated because other mathematicians don’t explain what they are doing at the board and what they write.” Thus the clarity of his lectures is in part a reaction against hard-to-understand lectures by sighted colleagues, who can get away with being less organized.

Giroux has been blind since the age of eleven. He notes that most blind mathematicians are or were working in geometry. But why geometry, the most visual of all areas of mathematics? “It’s pure thinking,” Giroux replied. He explained that, for example, in analysis, one has to do calculations in which one keeps track line-by-line of what one is doing. This is difficult in Braille: To write, one must punch holes in the paper, and to read one must turn the paper over and touch the holes. Thus long strings of calculations are hard to keep track of (this burden may ease in the future, with the development of “paperless writing” tools such as refreshable Braille displays). By contrast, “in geometry, the information is very concentrated, it’s something you can keep in mind,” Giroux said. What he keeps in mind is rather mysterious; it is not necessarily pictures, which he said provide a way of representing mathematical objects but not a way of thinking about them.

In [So], Alexei Sossinski points out that it is not so surprising that many blind mathematicians work in geometry. The spatial ability of a sighted person is based on the brain analyzing a two-dimensional image, projected onto the retina, of the three-dimensional world, while the spatial ability of a blind person is based on the brain analyzing information obtained through the senses of touch and hearing. In both cases, the brain creates flexible methods of spatial representation based on information from the senses. Sossinski points out that studies of blind people who have regained their sight show that the ability to perceive certain fundamental topological structures, like how many holes something has, are probably inborn. “So a blind person who has regained his eyesight can at first not distinguish between a square and a circle,” Sossinski writes. “He just sees their topological equivalence.

On the other hand he sees immediately that a torus is not a ball.” In a private communication, Sossinski also noted that sighted people sometimes have misconceptions about three-dimensional space because of the inadequate and misleading two-dimensional projection of space onto the retina. “The blind person (via his other senses) has an un-deformed, directly 3-dimensional intuition of space,” he said.

As noted in [Ja], attempts to understand spatial ability have a long history going back at least to the time of Plato, who believed that all people, blind or sighted, have the same ability to understand spatial relations. Based on the ability of the visually impaired to learn shapes through touch, Descartes, in Discours de la méthode (1637), argued that the ability to create mental representational frame- works is innate. In the late eighteenth century, Diderot, who involved blind people in his research, concluded that people can gain a good sense of three-dimensional objects through touch alone. He also found that changes in scale presented few problems for the blind, who “can enlarge or shrink shapes mentally. This spatial imagination often consisted of recalling and recombining tactile sensations [Ja].” In recent decades, much research has been devoted to investigating the spatial abilities of blind people. The prevailing view was that the blind have weaker or less efficient spatial abilities than the sighted. However, research such as that presented in [Ja] challenges this view and appears to indicate that, for many ordinary tasks such as remembering a walking route, the spatial abilities of blind and sighted people are the same.

### CHALLENGES OF ANALYSIS

Not all blind mathematicians are geometers. Despite the formidable challenges analysis presents to the blind, there are a number of blind analysts, such as Lawrence Baggett, who has been on the faculty of the University of Colorado at Boulder for thirty-five years. Blind since the age of five, Baggett liked mathematics as a youngster and found he could do a lot in his head. He never learned the standard algorithm for long division because it was too clumsy to carry out in Braille. Instead, he figured out his own ways of doing division. There were not many textbooks in Braille, so he depended on his mother and his classmates reading to him. Initially he planned to become a lawyer “because that’s what blind people did in those days.” But once he was in college, he decided to study mathematics. Baggett says he has never been very good in geometry and cannot easily visualize complicated topological objects. But this is not because he is blind; in visualizing, say, a four-dimensional sphere, he said, “I don’t know why being able to see makes it any easier.” When he does mathematics, he sometimes visualizes formulas and schematic, suggestive pictures. When he is tossing around ideas in his head, he sometimes makes Braille notes, but not very often. “I try to say it aloud,” he explained. “I pace and talk to myself a lot.” Working with a sighted colleague helps because the colleague can more easily look up references or figure out what a bit of notation means; otherwise, Baggett said, collaboration is the same as between two sighted mathematicians. But what about, say, going to the blackboard to draw a picture or to do a little calculation? “They do that to me too!” Baggett said with a laugh. The collaborators simply describe in words what is on the board.

Baggett does not find his ability to calculate in his head to be extraordinary. “My feeling is that sighted mathematicians could do a lot in their heads too,” he remarked, “but it’s handy to write on a piece of paper.” A story illustrated his point. At a meeting Baggett attended in Poland in the dead of winter, the lights in the lecture hall suddenly died. It was completely dark. Nevertheless, the lecturer said he would continue. “And he did integrals and Fourier transforms, and people were following it,” Baggett recalled. “It proved a point: You don’t need the blackboard, but it’s just a handy device.”

Blind mathematics professors have to come up with innovative methods for teaching. Some write on the blackboard by writing the first line at eye level, the next at mouth level, the next at neck level, and so on. Baggett uses the blackboard, but more for pacing the lecture than for systematically communicating information the students are expected to write down. In fact, he tells them not to copy what he writes but rather to write down what he says. “My boardwork is just an attempt to make the class as much like a normal lecture as possible,” he remarked. “Many of [the students] decide they have to learn a different way in my class, and they do.” He makes up exams in TEX and has a Web page for homework problems and other information. For grading, he can use graders “but I lose personal feedback,” so he uses a variety of schemes, such as having students present oral reports on their work. It is clear that Baggett’s devotion to teaching and concern for students overcome any limitations imposed by his disability.

### MEANS OF COMMUNICATING

When he was growing up in Argentina in the 1950s, Norberto Salinas, who has been blind since age ten, found, just as Baggett did, that the standard profession for blind people was assumed to be law. As a result, there was no Braille material in mathematics and physics. But his parents would read aloud and record material for him. His father, a civil engineer, asked friends in mathematics and physics at the University of Buenos Aires whether his son could take the examination to enter the university. After Salinas got the maximum grade, the university agreed to accept him. In a contribution to a Historia-Mathematica online discussion group about blind mathematicians, Eduardo Ortiz of Imperial College, London, recalled examining Salinas in an analysis course at University of Buenos Aires. Salinas communicated graphical information by drawing pictures on the palm of Ortiz’s hand, a technique that Ortiz himself later used when teaching blind students at Imperial. Salinas taught mathematics in Peru for a while and then went to the United States to get his Ph.D. at the University of Michigan. Today he is on the faculty of the University of Kansas.

Salinas said that he would often translate taped material into Braille, a step that helped him to absorb the material. He developed his own version of a Braille code for mathematical symbols and in the 1960s helped to design the standard code for representing such symbols in Spanish Braille. In the United States, the standard code for mathematical symbols in Braille is the Nemeth code, developed in the 1940s by Abraham Nemeth, a blind mathematics and computer science professor now retired from the University of Detroit. The Nemeth code employs the ordinary six-dot Braille codes to express numbers and mathematical symbols, using special indicators to set mathematical material off from literary material. Standard Braille was clearly not intended for technical material, for it does not provide representations for even the most common technical symbols; even integers must be represented by the codes for letters (a = 1, b = 2, c = 3, etc.). The Nemeth code can be difficult to learn because the same characters that mean one thing in literary Braille have different meanings in Nemeth. Nevertheless it has been extremely important in helping blind people, especially students, gain access to scientific and technical materials. Salinas and John Gardner, a blind physicist at Oregon State University, have developed a new code called GS8, which uses eight dots instead of the usual six. The two additional dots, which are reserved for mathematical notation, provide the possibility of representing 255 characters rather than the sixty-three that are possible in standard Braille. In addition, the syntax of GS8 is based on LATEX, making it feasible to convert GS8 documents into LATEX, and vice versa.

Computers have opened up a whole world of communication possibilities for blind people. Screen reader programs, such as Jaws or SpeakUp, translate text on-screen into spoken words using speech synthesizers. Unfortunately, these programs generally do not work well with text containing mathematical symbols, and some blind mathematicians tend to use the programs only for reading email or surfing the Web (which is becoming more complicated for the blind due to the heavy use of graphics). A blind computer scientist at Cornell University, T. V. Raman, has developed a program called AsTeR, which accepts a TEX file as input and as output produces an audio file that contains a synthesized vocalization of the document, mathematics and all. Gardner has developed a program called TRIANGLE that has a speech synthesizer that is more basic than AsTeR and also includes a program for converting between LATEX and the GS8 code.

Some blind mathematicians actually read TEX source files directly; Giroux does so using a refreshable Braille touch-screen. He said it is more comfortable to have an audio recording of a paper, but before having a recording made, he wants to know whether the paper is really interesting to him. Reading TEX files provides quick and direct access to the documents. Of course, TEX files are meant to be read by computers rather than humans and are therefore cumbersome and verbose. Nevertheless, Giroux said that their easy availability through electronic preprint servers and journals represents “huge progress” in his ability to stay in touch with current research. Books are a bigger problem than papers; although TEX is the standard way of publishing mathematics books, obtaining the TEX files from publishers is not a straightforward process.

### MATHEMATICS ACCESSIBLE TO THE BLIND

It is easy to understand how well-meaning people who know little about mathematics might assume that the subject’s technical notation would create an insurmountable barrier for blind people. But in fact, mathematics is in some ways more accessible for the blind than other professions. One reason is that mathematics requires less reading because mathematical writing is compact compared to other kinds of writing. “In mathematics,” Salinas noted, “you read a couple of pages and get a lot of food for thought.” In addition, blind people often have an affinity for the imaginative, Platonic realm of mathematics. For example, Morin remarked that sighted students are usually taught in such a way that, when they think about two intersecting planes, they see the planes as two-dimensional pictures drawn on a sheet of paper. “For them, the geometry is these pictures,” he said. “They have no idea of the planes existing in their natural space.” Because blind students do not use drawings, it is natural for them to think about the planes in an abstract way.

The most famous blind American mathematician right now may be Zachary J. Battles, whose extraordinary story was even covered in People magazine. Blind almost from birth and adopted from a South Korean orphanage when he was three years old, Battles went on to earn a bachelor’s degree in mathematics and a bachelor’s and master’s in computer science from Pennsylvania State University. He also traveled to Ukraine twice to teach English as a second language and worked as a mentor for other disabled students. He is now studying mathematics at the University of Oxford on a Rhodes Scholarship. Like so many other blind mathematicians, Battles is an inspiration to the sighted and the blind alike.

Caption: Bernard Morin with one of Stewart Dickson’s models, at the International Colloquium on Art and Mathematics in Maubeuge, France, in September 2000. Photograph courtesy of John M. Sullivan, University of Illinois.

- References
- [A] P. S. ALEKSANDROV, V. G. BOLTYANSKII, R. V. GAMKRELIDZE, and E. F. MISHCHENKO, Lev Semenovich Pontryagin (on his sixtieth birthday), Russian Math. Surveys 23 (6) (1968), 143–52.
- [FM] GEORGE K. FRANCIS and BERNARD MORIN, Arnold Shapiro’s eversion of the sphere, Mathematical Intelligencer 2 (1979/80), no. 4, 200–3.
- [H] BRIAN HAYES, Speaking of mathematics, American Scientist, March–April 1996, describes T. V. Raman’s program AsTeR.
- [I] HORST IBISCH, L’œuvre mathématique de Louis Antoineet son influence, Exposition. Math. 9 (1991), no. 3, 251–74.
- [Ja] R. DANIEL JACOBSON, ROBERT M. KITCHIN, REGINALD G.GOLLEDGE, and MARK BLADES, Rethinking theories of blind people’s spatial abilities, available at http://garnet.acns.fsu.edu/djacobso/haptic/nsf-und/3theory.PDF.
- [Ju] GASTON JULIA, Notice nécrologiquesur Louis Antoine, Comptes Rendus de l’Académie des Sciences de Paris, tome 272 (March 8 1971) Vie Académique, pp. 71–4.
- [MP] BERNARD MORIN and JEAN-PIERRE PETIT, Le retournement de la sphère, Pour la Science 15 (1979), pp. 34–49.
- [OI] Outside In, http://www.geom.uiuc.edu/locate/oi.
- [O] The Optiverse, http://new.math.uiuc.edu/optiverse/.
- [P] ANTHONY PHILLIPS, Turning a surface inside out, Scientific American, May 1966.
- [So] ALEXEI SOSSINSKI, Noeuds: Genèse d’une Théorie Math-ématique, Seuil, 1999.
- [St] STEPHEN M. STIGLER, Who discovered Bayes’s theorem? , Am. Stat. 37 (1983), 290–6.
- [Str] DIRKJ. STRUIK, A Concise History of Mathematics, Dover, 1987.

## 2.Article on Successfully Studying

Karen Keninger karen.keninger@blind.state.ia.us

Wed Dec 5, 2007

I'm working on an article for my statewide newsletter on methods for a blind student to succeed in a math and/or science curriculum at the university level. I'd be very interested in responses to these questions:

- How can a blind person access math textbooks? Are they available digitally? Is there software that can scan and convert mathematical symbols and scientific notation effectively?
- How do you use Braille?
- Do you know the Braille math code--Nemeth? If so, how did you learn it? How do you use it now?
- How do you gain access to figures, graphs and similar visual representations in your math and science books?
- How do you output your math assignments? What software do you use? Is it compatible with screen readers?
- What computer software do you use? What function(s) does it serve? (e.g.. graphing calculator, etc.)
- How do you do your math assignments?
- What level of math have you studied?

We have received Karen Keninger’s permission to reproduce the article she wrote in response to this survey. This article was never posted on the Blindmath listserv.

### Blind Students Study Higher Mathematics

By Karen Keninger

Published in the January, 2008 White Cane Update, the newsletter of the Iowa Department for the Blind

According to the Fall 2006 Salary Survey, National Association of Colleges and Employers, (www.careervoyages.gov/pdf/occquarterly-stemoccupations.pdf - ) average starting salaries for graduates with a Bachelor’s degree in science, technology, engineering and mathematics (STEM) fields range from 44,672 to $51,313. Average salaries in these fields overall in 2005 were $64,560 while average salaries for the total workforce were $37,870. Obviously, careers in STEM occupations pay well compared to other fields, and all indications are that opportunities in these fields will continue to grow while the available pool of qualified STEM professionals may lag behind demand.

Traditionally, blind students have been steered away from these fields because of the obstacles they present in terms of coursework. However, some blind people are succeeding in these fields of study and work and are reaping the benefits of higher salaries and rewarding and interesting careers.

Mathematics comprises a significant component of study and work in these fields. Every science, technology, engineering and (obviously) math student is required to take a lot of math courses. Mathematics, with its whole universe of symbols, its graphs, its figures and formulas, presents unique challenges to blind students. However, according to several I corresponded with, these challenges can be met with the right combination of skills and technology.

Reading the textbooks is one significant consideration. Braille, specifically Nemeth Braille in the United States, is the best way to read a mathematics textbook. This presupposes knowledge of and practice in reading Nemeth code. One student told me that he learned very basic Nemeth code (numbers and simple arithmetic like the plus sign) in from his Teacher of the Visually Impaired (TVI), but he had to teach himself more advanced applications. The Nemeth code was developed by a mathematician specifically for mathematics and scientific notation, and has been extended over the years to include all higher mathematics. But most college math books are not available in Braille, and the likelihood of getting a transcriber certified in Nemeth and able to produce materials on time is not 100%.

Some new and highly capable electronic alternatives are emerging to join the traditional human reader and recorded textbook. Two electronic file formats hold promise for math students—LaTex and MathML. LaTex is based on a standard format used by printers to publish books, and adapted to accommodate mathematical and scientific notation. Professors format handouts, tests, etc. in LaTex because it can be readily converted to a printable form. In its basic form, LaTex can be read as a text file and interpreted in the context of its tags. Using a LaTex file gives a blind student direct access to the materials.

According to Aaron Cannon, an Iowa student currently studying mathematics and actuarial science, “The best thing a blind student of math can do for themselves after learning Braille is to learn LaTex. Fortunately, it’s quite simple to learn (Took me only about an hour to master the basics) and requires no special software; I read and write it with a standard text editor." Others say LaTex is not really simple, but it is very doable. Aaron goes on to say: "I have had reasonably good luck contacting the authors and publishers of my text books. Many authors write their books in LaTex, and they are sometimes willing to pass those files along. In addition, LaTex is often used by math professors in preparing their lecture notes, quizzes, and exams.” Duxbury can convert LaTex to Nemeth code. Aaron says "I use Duxbury to translate LaTex to Braille. It’s not always perfect, so sometimes I have to refer to the LaTex file to see what's going on, but it’s nice because I'm not tied to the computer.”

MathML is an alternative format based on extended mark-up language (XML) similar to the HTML that displays web pages, but adapted to display math and science notation. MathML has been incorporated into the Daisy standard, which is a very positive step toward fuller accessibility. Files can be read in a variety of ways, including with a Daisy player programmed to accept the MathML code. GH LLC MathSpeak (www.mathspeak.org) is one effort to make MathML accessible audibly at the level of precision required for serious study in math. According to Neal Kuniansky of Duxbury Systems, the next major release of Duxbury, due out sometime this year, will be able to convert MathML into Nemeth Braille.

Scanning math is also becoming a reality with InftyReader developed by the Infty Project in Japan (www.inftyproject.org). This software converts a scanned image containing math symbols to several optional formats, including MathML and LaTex.

Audio graphing calculator software is also available for purchase from ViewPlus Technologies (www.viewplus.com) as well as one developed by NASA called MathTrax which can be downloaded free (http://learn.arc.nasa.gov/mathtrax/download.html. This software adds audio signals and verbal descriptions to its graphs, and the ViewPlus software can export its graphs to tactile image output.

Blind students and professionals must also prepare their work for use by sighted professors or colleagues. Several students told me that their preferred method for presenting their work is to write it in LaTex, either using a standard text editor or a more dedicated LaTex editor, and then converting the results into PDF or HTML for the completed product. Because LaTex was designed by the printing industry to handle layout, fonts, and other printing requirements readily, it lends itself very well to this kind of output.

Software is also being developed to back translate Nemeth Code, created on a note taker or computer, into digital text. BackNem, WInsight, and Nemetex are emerging solutions for students doing their homework in Braille. Information is available at http://braille2print.org/backnem2.htm

With the combined experience of blind scientists and mathematicians, the advances in access technologies, and the renewed emphasis on teaching Braille to blind students, mathematics has become an approachable and very doable discipline for blind students and professionals, pushing wider the doors that open on the STEM disciplines and the interesting and rewarding careers available there. For additional information, and a wealth of resources on this topic, visit http://www.nfbnet.org/mailman/listinfo/blindmath

## 3.NFB Math Survey: A Report of Preliminary Results

By Al Maneki

From the Editor: In the February 2011 issue we published an article by Al Maneki containing a survey, the results of which Al reported to the parents division in July. Because we know how many jobs, both current and future, require proficiency in math and in order to provide some context for Al’s report, we are again including his survey. We strongly urge everyone who can do so to complete the survey now so that we can understand more clearly what separates those who succeed at mastering mathematics and those who have concluded that it is simply beyond them. Here is the survey, followed by Al’s report based on initial responses to his original request:

Who Should Complete This Survey?

We would like to hear from any blind or visually impaired person who has taken or is taking at least one math or math-based science course at the secondary or post-secondary school level. We would also like to hear from any parent or teacher who has advised or assisted a blind or visually impaired child with at least one math or math-based science course. Furthermore, we are interested in students' experiences learning geometry or elementary school arithmetic.

There is no restriction on when or how long ago you or your child took a math course. We want to learn about the methods of handling math that worked best for you. We are equally interested in methods that were not particularly successful or useful.

If your child or you are considering taking math courses at any level, you should read these survey questions. They may help you get the information you need to complete your courses successfully.

Instructions

In your responses please provide contact information (name, address, email, phone) so that I can reach you for possible clarification and follow-up interviews. Please also include your age (closest to 5-year multiples, i.e., 20-25, 25-30, etc); the highest level of education you have completed; your primary reading medium; your current employment status and job title.

You need not answer all of the questions, since some of them may not be relevant to your experience. You do not have to answer questions separately. You may provide a narrative summary as your response to this survey.

If you require additional information about these questions please get in touch with me. You may contact me by email, phone, or snail mail. My contact information appears at the end of the survey.

You may submit your responses by email or snail mail (Braille or print please, no audio) to the addresses shown below. Please complete this survey by April 15, 2011. Those taking courses after this date may respond later, since I anticipate a continuation of this survey.

Your answers will not be used to judge your mathematical strengths or weaknesses. Any personal information you may reveal in your responses will remain confidential. Names, mailing addresses, email addresses, and phone numbers will not be distributed.

Survey Questions

Here are the questions to consider:

- What math or math-based science courses have you taken (elementary, secondary, community college/university, graduate school)? Specify the level of each course, and describe the subject matter that was included.
- Were classroom lectures useful to you? Since mathematics is generally communicated visually, tell us as specifically as you can what you actually learned from these lectures. If lectures were not helpful, tell us what you did to compensate for the missing information.
- Were you able to take classroom notes? If so, tell us what method you used: large print, hardcopy Braille, electronic or live notetakers, audio recordings, etc.
- How did you handle reading assignments? Tell us about your use of Braille textbooks, recorded textbooks, large-print textbooks, or live readers or tutors.
- How did you do homework assignments and take tests? Describe your use of large print, notetakers, hardcopy Braille, mental arithmetic, or dictation to a live reader. If you used Braille, describe your method of translating Braille into a medium accessible to instructors who do not know Braille. If you used Braille/print reverse-translation software of any kind, describe how this worked. In your answer to this question, tell us about any additional devices and technologies you have used, i.e., older devices such as the Taylor Slate, Cube-a-Rithm Slate, Circular Slide Rule, and Cranmer Abacus; and newer devices such as talking calculators or specialized learning software.
- Have you written papers containing mathematical content in an academic or professional setting? Describe how you did this, especially the use of human support.
- How did you work with line drawings, graphs, or charts? Explain how these were described to you or produced in accessible formats. If you had to construct these items, tell us how you accomplished this task.
- How familiar are you with the Nemeth Braille code? Describe the extent to which you use it for reading or writing.
- Are there any tools/devices/aids that you wish you had had that would have enhanced your mathematical experiences?
- How satisfied are you with your mathematical experiences? Would you like to make other comments about how blind and visually impaired people may read and do mathematics?

This is an informal survey. I am conducting it with the intention of using the results to help others who will be taking math and math-based science courses in the future. The results of this survey, after they have been compiled, may also prove useful to people who are accustomed to doing math in their own ways. These folks may find new ways of working more productively. It could further turn out that these responses will suggest altogether different ways of doing math, either by refining methods already in use or by suggesting the development of new techniques and technologies. I fervently hope that over time this survey will make it possible for blind and visually impaired people to learn and do mathematics more efficiently and with greater ease.

I plan to compile the first set of responses (received by April 15, 2011) into an article, ideally for publication in the newly established Journal of Blindness Innovation and Research. It is also my hope that this survey will be a continuing investigation. Additional articles pertaining to this survey will be published if they are warranted.

In preparing this article and survey, I received valuable help from Deborah Kent Stein, editor of Future Reflections, and from Mark Riccobono and Judith Chwalow of the NFB Jernigan Institute. Although they have left their marks on this article and survey, I assume responsibility for all shortcomings, errors, and omissions. I thank you in advance for helping me with this survey. I look forward to hearing from you.

Al Maneki

blmathgems@earthlink.net

(443) 745-9274, cell

9013 Nelson Way, Columbia, MD 21045

Now here is Al’s preliminary report to the parents:

As I said in my article which appeared in Future Reflections and the Braille Monitor, the seed for this survey sprouted from the workshop “I Survived Math Class,” which I moderated at our convention last year in Dallas. I’m pleased to come before you today to report that the responses I have received so far have been most gratifying. We received messages from several people saying that this survey was much needed and that, even though they were not part of the survey population, they were very interested in the results. We heard from a few teachers of visually impaired students, and we are keeping track of their responses. We even heard from a sighted professor of mathematics who expressed an interest in what we are doing.

Dr. Abraham Nemeth was kind enough to respond to my survey. He listed his age range as 90-95, is obviously retired, but went into some detail about his work experiences. Although he did not directly answer the question concerning familiarity with the Nemeth code, I thought that it would be safe to include him in this category. I understand that he is attending this convention.

So far we have received fifty-three responses. Of these

- 30 percent are students.
- 72 percent are Braille readers (a much higher average than for the general blind population); 10 percent are large-print readers; and 18 percent use speech.
- 64 percent considered their math experiences successful, while 36 percent did not (a biased result in my opinion).
- Among our responders who were not retired or did not list their employment status, 89 percent were employed (much higher than in the general blind population) and 11 percent were unemployed (also a biased result).
- Among our employed responders, 75 percent used Braille, 17 percent used speech, and 8 percent used large print.
- The vast majority of employed Braille readers also use a mix of electronic speech and live readers or large print.

Thus far we haven’t heard from very many people who had unsatisfactory math experiences. While it is often embarrassing to reveal one’s unsuccessful experiences, we need to hear about more of these to gain an accurate picture of the state of math education for blind people. Whether your math experiences have been successful or unsuccessful, please continue to respond to this survey. We need to hear from more of you.

While we must be wary of drawing conclusions from small samples, I want to share some of the impressions I have gathered up to now:

Some of our responders were fortunate enough to have Braille textbooks. Those lacking Braille, used recorded textbooks or live readers. With recorded texts, they had to cope with the inconsistencies in which mathematical material was read and the ambiguities of having diagrams and charts described orally. The most successful responders did not hesitate to seek clarification from their instructors and to get help from classmates, live readers, or tutors. The most successful responders were keenly aware of the way they used class time to ask their questions and the creative strategies they used to communicate their solutions in homework assignments and tests.

Responders commented about the significant amount of extra time required for their math classes. If they did their homework assignments and tests in Braille, time had to be spent on transcribing their work into print. If they received assignments and tests in print, these had to be read to them, and they had to read their solutions back to their readers.

A disconnect often develops between the functions of math instructors and TVIs or DSS staff. Some responders complained that too many instructors ignore the needs of their blind students, making the assumption that it is the job of the TVI or DSS office to teach the math, even though it is not. We want and need a true partnership here. The math instructor should creatively think of nonvisual ways to teach because these methods could also help sighted students. TVIs and DSS personnel should know enough math to ensure that materials are properly transcribed.

The vast majority of Braille readers claimed a degree of familiarity with the Nemeth code. I gained the distinct impression that Nemeth code has been taught as it was needed. This is as it should be. Some of the responders said that, given rudimentary knowledge of the code, they proceeded to invent Braille symbols for their own use. This practice should be encouraged because writing correct code can be cumbersome in note-taking. For personal notes we should use any shortcuts we like. For reading textbooks produced for general distribution, we must use correct Nemeth code.

More important than reading math texts is the need to work on exercises and problems to firm up one’s mastery of the concepts. This area proved to be most difficult for the responders. Electronic Braille notetakers don’t work because math solutions require the simultaneous examination of multiple lines of calculations and expressions. Writing with slate and stylus was also unsatisfactory. Most Braille readers have found the best success with mechanical Braillewriters, such as the Perkins, because embossing is right-side-up and paper can be shifted from line to line without disturbing the position of the embossing mechanism. Large-print readers, even the few who know Braille, used large print to write their solutions. Anyone who has developed sufficient mental abilities may simply dictate solutions to a live reader. None of the responders, except a few large-print readers, expressed confidence in their ability to construct diagrams and charts.

Technology, no matter how good, can only go so far in helping blind students learn math. By itself it will not make learning math easier. It will not turn any of us into brilliant mathematicians. But I think technology has a place in education. The people who responded to my survey share my belief that:

- We need a multi-line electronic Braille display;
- We need smoother mechanisms to move from print to spoken math or from print to Nemeth Braille and to go back from Nemeth Braille to print;
- Just as we learn Nemeth code incrementally with math subjects, we should be taught enough LaTeX, the math typesetting software, to handle our math assignments;
- Last, math software such as Mathematica, Maple, and Sage must be made accessible to us, because the use of these tools in the classroom is growing.

I hope to get more responses very soon, and I hope these responses will shed more light on our experiences with math courses. I’ve not yet decided how best to display a summary of responses while maintaining individual confidentiality. I’m thinking about editing the more substantive ones and posting them on my own web page with links from the NFB website. I also believe that with additional responses other articles will be forthcoming.

To conclude my report to you today, I want to quote the response from Sandi Ryan, a retired dietitian from Ankeny, Iowa, to survey question 10, giving her comments about how blind and visually impaired people can read and do mathematics.

I have her permission to quote her answer in its entirety. Here is what Sandi says:

I feel that, as I approach my sixties, I understand a lot more about math and science than I ever thought I would. I had always been given to understand that blind people couldn’t do math and science. I was fortunate to know a couple of blind electrical engineers, and they obviously had to do math. If I’d known, going in, how difficult my college career would be, I’d have backed out before I started. But I am glad I didn’t know. I learned a great deal from surviving math and science classes. I learned the concepts, of course, but I also learned that I can be pretty creative and innovative when it’s needed, and so can some sighted people who agreed to tutor me and used solid objects and hundreds of drawings to make sure that I understood what I needed to know. Incidentally, my tutors were always students—not in my class, but in the discipline I was studying—and they took their valuable time to do things, for which my payment was meager, to ensure my successful education. Several of them bought into my education as much or more than my teachers, and I am grateful that they believed in me when I wasn’t sure I did. I have had a wonderful career as a registered dietitian and look forward to the future because I have been successful.

A woman at my university who ran the Handicapped Student Office stated, in a talk during Woman’s Week, that not many disabled people are interested in math, science, technology, or engineering. I challenged her, and I still believe she is absolutely not right. I think many disabled people avoid these disciplines because they fear they cannot succeed. And I believe their well-meaning advisors and instructors encourage them to go another direction. I would love to be around when there are so many blind and otherwise disabled science, technology, engineering, and math professionals that we aren’t even pointed out as unusual. I doubt that will happen in my lifetime, but math and science are rewarding, and blind people shouldn’t miss out on the reward because they lack tools and education.

With thanks to all of you here today, and with thanks to our collective efforts in the NFB and the Jernigan Institute, we are turning Sandi’s vision and our vision of a brighter math and science future into reality.