Blind Mathematicians? Certainly!
by Alfred P. Maneki
From the Editor: As we encourage participation in science, technology, engineering, and mathematics, many blind students tell us stories about being discouraged from these fields. They want to know how to do what they’ve often been told can’t be done. For this reason the Braille Monitor is devoting some space to the how-tos of successfully getting the education and becoming employed in some of the most exciting fields the current job market has to offer. Here is what a Federation leader and successful blind mathematician has to say about his training and employment:
Introduction
If I date the beginning of my career in mathematics to September 1964 when I began my graduate studies at Illinois Institute of Technology in Chicago, I can say that I have been in the math biz for forty-eight years. During those years I have studied mathematics, taught it to college students, and worked as a mathematical scientist in the U.S. government. Since my retirement from the federal government in 2007, I have devoted my time to tutoring and advising blind students, as well as engaging in the development of tools to help blind people perform tactile drawings better. After almost a half century of work in mathematics, I not only have a deeper understanding of myself as a blind person, I have also seen the changes that have taken place in this profession. I believe that I am in a unique position to help others enter and become successful in this magnificent field of study and research.
Two years ago I began my comprehensive math survey for the National Federation of the Blind. The results were quite revealing as I have reported elsewhere in NFB literature. In my initial report I asked for additional responses to my survey, especially from people who have encountered difficulties in their mathematical studies. To get to my survey and to read my initial report, go to <http://www.nfb.org/Images/nfb/Publications/bm/bm11/bm1109/bm110909.htm>.
When the editor of the Braille Monitor asked me to write this article, the thought occurred to me that I was guilty of not complying with my own request of soliciting survey responses. Therefore I am writing this article partly as a response to my own survey. This may raise the eyebrows of professional survey takers, objecting to contaminating my sample data with my own biased opinions. Be that as it may, I trust that what I have to say in this article will be of value to some readers.
My Educational Experience
I have said elsewhere and should say again that my mathematical background prior to enrolling at the University of Hawaii (UH) for my undergraduate studies was minimal. My secondary school teachers did everything to discourage me from studying mathematics because they thought that blind people could never make it in this field. Needless to say, when I entered UH, I had some serious catching up to do. But, with the help of encouraging faculty and fellow students, I got by, making up for lost time with intensive summer semester courses. Textbooks were read to me by fellow students who saw this as an opportunity for additional study time to improve their understanding of the subject matter. From their reading I took Braille notes for my own review. I patiently dictated my homework assignments to these same students, who took my dictation, including all of my errors, which I asked them not to correct. These arrangements worked fairly well, especially when I received help from students who were not in my classes. My instructors administered course examinations orally, or I dictated my answers to other readers. In my day Braille textbooks in mathematics were nonexistent.
In graduate school I was fortunate to encounter the same support that I had had as an undergraduate, except that now fellow students recorded some of my textbooks. I also started to borrow audio tapes from Recordings for the Blind, today known as Learning Ally. I continued to take notes in Braille. Braille textbooks remained nonexistent. As I proceeded to my doctoral studies, I depended more on student readers to help me with library research and reading journal articles. Fortunately, the vocational rehab agency in Hawaii supported me with funds for paid readers. Rehab funds also paid for having my thesis manuscript professionally typed and prepared.
My experiences in studying math at the undergraduate and graduate levels are very much in keeping with the comments others have made in my survey and elsewhere. I quickly learned about the ambiguities inherent in spoken mathematics. Of course, to the person doing the reading, there are never any ambiguities because the reader is only reciting from written words and symbols. To the reader the rendering of that material into spoken language is always clear. If I was using a live reader, I could always stop the reading to ask for a clarification. I could insist that materials be read in a specific way. I also learned that different readers had different ways of reading similar material. As long as they remained consistent, I could pick up these particularities, and the reading proceeded smoothly.
In the case of recorded materials, the ambiguities became a guessing game. For fractions, where exactly was the break between numerator and denominator? For exponents, what specific quantity did the exponent cover, or, if the exponent was a long expression, where did it end? And, what did one do with exponents of exponents? Functions of more than one variable also presented their own difficulties, especially when the variables were themselves expressions of other variables. One instructive task in listening to recorded material is to be found in developing the ability to consider the possible interpretations of that reading and to eliminate those that make no sense in the context of the parts that make sense. This type of forced listening was excellent training for me, although it was time-consuming and often frustrating.
As I listened to different mathematical materials, it became obvious that I would have to take Braille notes for myself, since most of the calculations and proofs were much too intricate and complex to commit to memory. Initially I faced the overwhelming temptation to write everything verbatim. It soon became obvious that this was too tedious. The trick, as I soon discovered, was to distill the primary elements of a calculation or proof and to copy only those items. This forced me to fill in the gaps as I studied my notes later, again providing additional mental discipline in mastering the subject matter. While studying at UH, I picked up a copy of a one-volume guide to the Nemeth Braille Code. I realized that the Nemeth Code is extremely cumbersome because it must resolve all possible ambiguities. To save time, I invented my own shorthand Nemeth-like notation. I relied on the context of the subject matter to understand the meaning of my Nemeth shorthand. I tossed out all of my notes from those early years in my many moves to different locations. I’m sure that, if I had these notes today, I could not read them for myself; my shortcuts are long forgotten.
As for writing mathematical proofs and performing calculations, there is no substitute for hard copy Braille done on a manual Braillewriter. One then has the ability to perform operations in a non-linear order, i.e., inserting a line of Braille that is based on calculations made in the lower lines. This is exactly what happens in long division, which is still taught today. While no one has to do long division for a livelihood, it is still a skill worth having.
What if a computation or proof is too large for a standard sheet of Braille paper? In my graduate studies, when I was required to do such computations and proofs, I relied on my readers to serve as my pencil. I would have my readers write down the expressions that I needed to use. Then I would ask them to read portions of these expressions back to me in the right order so that I could do the calculations mentally and dictate these to them. Although these large computations are now done by digital devices, it is still useful to do some of this manually, if only to verify that you have given the proper instructions to your computer.
My Work Experience
Midway through my graduate studies I found employment as an instructor in the math department at North Dakota State University (NDSU) in Fargo. I was most fortunate in coming to the job market when our nation’s colleges and universities were desperately looking for people to staff their math departments. NDSU made me an offer I simply could not refuse. I was given all the teaching help I needed, from reading course textbooks, to preparing lectures on transparencies, to proctoring tests, to grading papers, and a little time to read research materials. The early ‘70s were glorious days for academic mathematicians because state legislatures were much more generous to state-run institutions of higher learning.
As good as life was in North Dakota, I knew that other challenges and greater opportunities awaited me. In August 1974 I packed up my belongings and moved to the environs of Fort Meade, Maryland, where I had accepted a position as a research mathematician with the Department of Defense. In some ways the decision to move was an easy one. The salary was higher, and the work would be very challenging. As it turned out, I spent the rest of my career here, working on mathematical problems in cryptology; data analysis; and, later, network security. The research I did was not of the academic variety that I had previously done. It was primarily concerned with devising efficient methods for testing assumptions about digital data that were given to us. We knew very little about the underlying structures that caused statistical peculiarities in what we were observing. Looking back on my experiences here, my mathematical training was extremely valuable for this work, even though I did not directly apply the mathematical subject matter that I had learned.
A few paragraphs earlier I mentioned the need to resolve ambiguities inherent in spoken recorded mathematics and how I was able to resolve these ambiguities by judicious guessing. I’m now convinced that this mental discipline went a long way to help me in the work I did at Fort Meade. Once I worked out the problem-solving techniques I needed, the rest of my work involved programming my ideas on a mainframe computer. It was much later in my career when we began to use networked personal computers. We had synthetic speech and Braille printers. These facilitated my programming efforts immensely. When I needed to write papers for publication, the offices I worked in always provided the necessary support. Without that support it was well understood that these papers would not be written.
In the mid-90s research in cryptology and network security was coming into vogue in both industry and academia. Because of the possible implications for the federal government, we were encouraged to have greater interactions with these external specialists. Limited research collaboration also became possible. As a result of these activities, I participated in and presented papers at several conferences among government, industry, and academia.
My thirty-three years of federal government service were most gratifying. Not only was I involved in interesting research, but I worked with some of the most respected individuals in this field. Toward the end of my career I had the opportunity to mentor some of the younger mathematicians entering this field of research.
My Advice
Looking back on my mathematical experiences, I would say that the primary driving force for my continued endeavors was, and still is, a consuming passion for the subject. While I was growing up, with all the discouraging remarks thrown my way about entering the scientific professions, I was forced to consider other alternative courses of study. I thought about pursuing the law or foreign languages or economics or political science. As mightily as I tried, these subjects could never ignite the same spark of enthusiasm as mathematics did for me. I believe now that my greatest doubt resulted from the failure to be seriously challenged by my teachers until I began my university studies. My first advice to anyone contemplating a mathematical career is to be sure that you are consumed by an overwhelming passion for it.
Concerning the formative years of my mathematical training, it was the worst of times and the best of times. It was the worst of times because there were virtually no blind mathematicians anywhere. Braille books in mathematics did not exist; the technology to assist with mathematical computations and the writing of proofs was still far in the future; and employment prospects for blind mathematicians were bleak. But it was the best of times because the overall job picture for the mathematical sciences was excellent; there was less need for early specialization than there is today; and vocational rehab agencies were better able to finance graduate studies for their blind clients. What is so astonishing to me now is that the rehab agency in Hawaii was quite willing to fund my graduate studies in Chicago once I established my academic record at UH. The question of what I would do professionally with my advanced degree in mathematics once I earned it did not appear to be uppermost in the mind of my rehab counselor. I guess that, once I had earned my degree, my case would have been successfully closed, job or no job. Looking back, I have no complaints. I would do it all over again if I had to.
I was fortunate to have entered government service at the time when applicants were not expected to have any background in cryptology. All of us who entered government service at Fort Meade in the mid-70s would receive the necessary training for the tasks that were assigned to us. Today I strongly recommend that those contemplating work in cryptology and network security have strong backgrounds in these subjects during their undergraduate or graduate studies.
Similar advice applies to mathematicians who wish to work in other fields. In the mathematics of finance, for example, one should have a background in subjects such as risk management, contingency analysis, stock market models, or general financial planning.
Many other fields (biology, chemistry, physics, meteorology, computer science, and electrical engineering) now require special mathematical knowledge in their undergraduate curricula. It is no longer enough merely to have a strong background in mathematics to enter these fields. As for teaching and doing research at a university, bear in mind that these positions are extremely competitive because of limited budgets.
Today, in an extremely tight job market, it is critical for blind students to have strong academic backgrounds, excellent grade averages, and related work experience as they enter the professional job market. These must-haves are not as intimidating as they were in my student days. Textbooks in properly formatted Braille, including Nemeth Code and tactile graphics, are somewhat more available today, even though automated print-to-Braille translation is still not in our grasp. Although not quite perfected yet, there are a few somewhat screen-reader-friendly computer programs to assist with complex mathematical calculations. We have tools such as LaTeX to assist us with the preparation of documents. Most exciting of all, we are beginning to see job opportunities for blind undergraduate and graduate students in science and engineering.
Blind students of today enjoy the luxury of standing on the shoulders of the blind scientists who preceded them. They will already find blind people gainfully employed in mathematics, biology, chemistry, physics, and engineering. If you are thinking about entering one of these professions, please call upon those who are already there. The best avenue to reach these professionals is through the NFB’s Science and Engineering Division. You can go to the NFB’s website <www.nfb.org> to obtain all of the necessary contact information under Fast Facts.
As I look back on my long career in mathematics, I am pleased by the thought that those who start today will go much further in their forty-eight years than I went in mine. This is as it should be. Of course technology is responsible for some of this. But the most essential ingredient for this future progress is to be found in the vibrant way in which we as blind people have come together as an organized community in the National Federation of the Blind. By working together, we will achieve a level of success for all of us that none of us could achieve by ourselves.
I did not write this article with the intention of misleading you or frightening you. For those of you who take it, the road ahead is difficult. But I hope that those of you with the proper passion and ability will be challenged to take it. You will be well rewarded for your successes. If I can help you in any way, please contact me by phone at (443) 745-9274 or send email to <[email protected]>.