TEACHING MATHEMATICS AS A BLIND PERSON

Good afternoon! Well, clearly this is an undergraduate class. Had everyone written that greeting down in his notebook, I would have concluded that this is a graduate class. To understand how a blind person can successfully teach mathematics, I must take you back to my early childhood. I was born congenitally blind and thus have no personal memory of the world of sight. I grew up, surrounded by love, in an extended family with two parents, four grandparents, and lots of uncles, aunts, and cousins. My father would take me by the hand as a little boy and would keep up a constant conversation to keep me in touch with my environment. "Right now we are walking west," he would say, "and as soon as we make a left turn, we will be walking south." He would ask me to take note of the direction of traffic flow on the street along which we were walking, and would explain to me that when we got to the next street, the traffic would be flowing in the opposite direction. He would apprise me of the names of all the cross streets along our route of travel. He would lead me from home to our usual destinations always by a different route so that I would develop a mind's-eye map of the neighborhood in which we lived. He would let me feel the metal lettering on mailboxes, police and fire call boxes; he would let me play with building blocks with large embossed letters; he would let me assemble large rubber stamps with raised letters; he would let me feel the neon tubing on the front of store windows so that I could learn the shapes of fancy script letters.

My mother would send me on errands. She would give me a list of four or five grocery items to bring home. But she was safe -- the grocery store was just around the corner with no streets to cross, and the grocer was my grandfather. I had good memory training by remembering grocery lists, and good training in mental arithmetic by adding up a few small numbers and figuring out the right change.

I attended the regular New York City Public School System during my grade school and high school years. In these schools, there was one room called a resource room staffed by a teacher trained in blindness skills. I attended regular classes with my sighted friends for such subjects as geography, history, arithmetic, and spelling, but when the subject was art or penmanship, I returned to the resource room where the teacher taught me braille, typing, and other skills of blindness. I was a competent typist by the age of nine.

I was always interested in mathematics, even in elementary school. But when I entered Brooklyn College, I heeded the advice of my counselors who told me that a field like psychology was much more realistic for a blind person than mathematics. Accordingly, I followed their advice and in due course received an M.A. degree in psychology from Columbia University. But I couldn't get math out of my system. As an undergraduate, I took all the math courses that my schedule would allow. Even after I was married, I took math courses at Brooklyn College in the evening until there were no more math courses in the catalog to take. Meanwhile, having no success in finding employment in my field of psychology, I supported myself and my wife by taking one unskilled job after another during the day. One day, my wife asked me if I wouldn't rather be an unemployed mathematician than an unemployed psychologist. So I gave up my daytime job and enrolled at Columbia University as a candidate for the Ph.D. degree in mathematics; my wife went to work.

In 1946, men were coming home from the War in large numbers. At Brooklyn College, hundreds of men were registering for the second semester of calculus, having taken and passed the first semester before being distracted by the War. I leave it to you to guess how much of that first semester they remembered a War later. They needed help. A special workshop was set up for that purpose. A large classroom was set aside with blackboards all around except for where the windows were. Each student, with a book in his hand, was assigned a blackboard panel on which to set down as much of his calculus problem as he could do. Many blackboard panels were blank. A corps of volunteers circulated among these men helping them to complete the problem. I was one of the volunteers. I would ask the student to read me his current problem from the book. Then I would ask him to read as much of the solution as he had managed to write on the board. After analyzing the problem and discusssing the next step, I picked up a piece of chalk and wrote the next (sometimes it was the first) mathematical expression. My father's training in acquiring the skill of writing and printing came in handy. Unknown to me, the department chairman who was supervising this workshop was watching me with interest. One Friday night, I received a telegram from the department chairman telling me in telegraphic style that one of his staffers had become ill, that he would be out for the rest of the semester, and would I take his classes until he was able to return? That weekend, I scurried around, found the relevant textbooks, and boned up on all the material I would have to teach the following Monday evening, and that's how my career as a math teacher was launched.

Meanwhile, as the mathematical concepts, and therefore the notation, became increasingly intricate, I found that the braille techniques for expressing mathematical notation were either inadequate or non-existent -- just as my counselors warned me. Little by little, however, I began to improvise new braille techniques to make it possible for me to write down all the notation that was buzzing around in my head. My mother's early training in sending me to the grocery store was excellent, but there was a limit to one's mental capacity. Finally, I settled on a braille system that was both consistent and served my needs. One day, a colleague who was a nuclear physicist and who was blind asked me if I had a table of integrals. I told him that I had one, but that it was written in a private braille code that he would not be able to read. Would I brief him on the code, he implored; he needed the table of integrals desperately. Within a half hour, he was having no difficulty reading the table of integrals with all its fractions, radicals, superscripts, Greek letters, and all the other arcane notation involved. Impressed, he asked me to write up a short expository paper of how the code worked, highlighting its underlying principles. I complied, but the result was not as short as he had wished. Nevertheless, he invited me to present my code before the braille standard-setting body of the time, and after about a week of testing, the code was adopted as the stedard mathematics code in the United States and was thereafter known as the Nemeth Braille Code for Mathematics and the Sciences. It subsequently went through several revisions, each one extending the capabilities of the previous one. Meanwhile, the Code was adopted in Canada and in New Zealand. Several years ago it was translated into French for use in the French-Canadian provinces of Canada. Today there are tens of thousands of braille volumes in the Nemeth Code. Braille transcribers who study the Nemeth Code, who pass a qualifying test, and who submit a manuscript to demonstrate their proficiency are certified in it by the Library of Congress. The two major volunteer organizations with a combined membership approaching five thousand routinely conduct Nemeth Code workshops at their national and regional meetings, and they routinely publish Nemeth Code information in the skills columns of their respective publications.

Let me now consider how a blind person receives his mathematical training. Many people who are legally blind still have varying degrees of residual vision. Many of them can use ordinary print effectively. These are not the people I have in mind, unless the prognosis of their eye condition is poor, in which case they should learn braille early. Still others, while they can use print to some extent, require special lighting, need to assume a special posture for reading, are limited to the length of time they can read, etc. People in this category are also better off learning braille. It is the braille users that I have in mind. By far, the best results are obtained when the blind student has in braille the same math textbook that his class is using. This is not always possible due to frequent changes in textbooks. Also, different school districts and different colleges use different textbooks for the same subject. Failing this ideal, the second best solution is to have in braille an earlier edition of the textbook if available. A third useful alternative is a parallel textbook which covers the same subject, if the previous two alternatives are not available. Some educators believe that modern technology is the solution. They believe that an audiocassette or a computer can be just as effective as braille. This is a mistaken view. The educators who promote it either do not know braille or are too lazy to learn it. Just suppose you are listening to a math text on an audio tape, and the reader suggests that you consult formula (17b) on page 46. First of all, you can't be certain that page 46 is on the cassette you are currently using. Even if it is, imagine fast-forwarding or rewinding the tape with no easy clue as to how close to page 46 you are. There are some techniques that make such a search easier, like tone-indexing, whereby you can hear a page boundary when in fast-forward or in rewind mode, but otherwise such a signal is inaudible in play mode. Nevertheless, the search is clumsy at best. If you are lucky enough to find the formula in question, you must then return to the place at which it was referenced. I hope you have the mathematical content well in mind while you are doing all this fiddling with your electronic device.

I can't imagine taking notes in a math class other than in braille. It is a mistake to record the whole lecture and then listen to the recording later. First of all, it takes just as long to replay the lecture as to listen to the original. Second, the available time for study does not permit such a leisurely approach.

How does a blind student take a test? There are many formats, but I will describe the one I used as a student. At the beginning of the semester, earlier if possible, I would make an appointment with my professor and discuss with him the accommodations I would need as a student in his course. One of the issues I discussed with him was that of taking tests. I suggested that, at the time of the test, we find an empty classroom or office space. Either he or his TA would dictate the questions to me as I wrote them in braille on my braille woiter. This would be done about 15 or 20 minutes before the scheduled test time. I would work out the test in braille on my braille woiter, using blank braille paper I brought with me. When finished, I would put my worked-out test into an envelope that I also brought with me and, at the end of the test, hand it either to the professor or to his TA. The only extra time I needed was the time to copy the questions into braille. Some professors wanted me to read what I had written to them personally; some asked me to read my answers to his TA for grading; some wanted me to read my answers to another professor. Some professors proposed giving me the test orally. I always discouraged this approach; it may convey my understanding of the basic concepts, or lack thereof, but an oral test does not reveal any skill I have in manipulating mathematical expressions. This manipulative skill is carefully nurtured and cultivated from the earliest grades, and it should pay off when performing a substantial differentiation or integration or in solving a differential equation. This manipulation cannot be done efficiently during an oral test.

Now let me jump ahead to the time a blind person, trained as a mathematician, is seeking employment. His resume should contain a short section about his blindness and the alternative techniques he uses to achieve the same result as his sighted colleagues. But this description should not overwhelm the remainder of the resume. In all other respects, his resume should conform to what is expected of any other job asplicant.

Our blind job applicant is now being intervieed. He should answer all questions forthrightly and pointedly. However, he should also be alert to the type of question which exppicitly asks one thing and implicitly asks another. When I was interviewed for my entry level position at the University of Detroit, I was asked if I had been to downtown Detroit and what impression did I have of that experience. I realized that my interviewer was not interested in a tourist's perspective of downarown Detroit. What he really wanted was information about my ability to be mobile an travel about independently. I began by telling him that my first impression of downtown Detroit was that it was not unlike the King's Highway shopping center in Brooklyn, by which I intended to convey the idea that the problems in getting around in downtown Detroit were not overwhelming and that they were well within my ability to cope with them. I then elaborated, but not at great length, on my ability to travel comfortably around New York City, somewhat larger than Detroit, by using its bus and subway systems effectively.

Our blind job seeker has now been offered a job and has accepted it. He must now consider the issue of classroom conduct and devise strategies for each of its several aspects.

Let me first consider the issue of classroom decorum. This is almost always the very first question with which an interviewer is concerned. The answer is simple. Gain the respect of your students early. A teacher who has the respect of his students experiences very few problems in classroom decorum. The one or two would-be trouble-makers are quickly subdued by the law-abiding majority. If a teacher does not have the respect of his students, it doesn't matter if he is blind or has a thousand eyes with 20/20 vission in each one; he is going to encounter disciplinary problems.

The next issue is handling the blackboard. As I indicated earlier, my father's patience and foresight made me comfortable in the smill of writing, and this skill has stood me in good stead during my working years and beyond. What remained was to acquire a disciplined approach to writing coherent straight lines. I used my anatomy as a guide. On a blank panel, the first line would be written at the level of my forehead, the next line at the level of my nose, the next at the level of my mouth, the next at the level of my chin, the next at the level of my neck, the next at the level of my chest, etc. When one panel was filled, I would move to the next panel and proceed in the same manner. I erased the board just as systematically. Frequently, a student in the first row would volunteer to erase the board and it became his full-time job for the duration of the course. Sometimes two front-row students took turns.

I encouraged my students to ask questions and otherwise participate in classroom discussions. Since hand-raising was ineffective in my case, I made it clear that it was not uncooth to interrupt me at the end of a sentence. I suggested that the interruption be in the form of calling out the student's own name. Calling out my name gave me no new information; I already knew who I was. But calling out the student's own name allowed me to know the identity of the speaker and possibly to recognize his voice on subsequent occasions.

Let me tell you how I handled my lecture notes. I will do so by means of describing a typical situation. I was teaching a class in Fourier analysis. I had prepared my lecture by writing the pertinent formulas in braille on 5 by 8 cards, one formula to a card, in the order in which those formulas would arise in the course of the lecture. I assembled a small file of about 12 or 15 cards which I put into my left-hand jacket pocket. As the lecture got under way and I approached the first formula, I nonchallatly reached my left hand into my jacket pocket and with the right hand wrote the formula on the board that my left hand was reading. As I proceeded to deliver the expository connective information leading up to the next formula, I would casually remove the top card from the file, place it at the back of the file, thereby exposing the next formula. I put the updated file back into my jacket pocket where the next formula was ready for me to read with my left hand and write with my right hand at the appropriate time. I proceeded in this manner to deliver a coherent lecture. My students were impressed. They agreed that only a genius could put all those formulas on the board, with all the integrals, limits, and summations in proper order. Since the statute of limitations has run out, I hereby admit that I never did anything to disillusion them.

Finally, I would like to stress the importance of full participation in departmental activities and in the university community itself.

The premise of my presentation is that a blind person who is adequately trained in his field and who has mastered the skills of blindness can function as competently and as productively as anyone else as a mathematics teacher, and that his blindness need not be an obstacle in choosing that career, as it was in my case. My career was delayed at least five or six years because of the advice I received from my counselors, and I hope that my experience will be of benecit to other blind people who seek mathematics teaching as an honorable and achievable goal. Thank you.