Alumni Newsletter: Spring '00
"A teaching philosophy?"
Professor Naomi Jochnowitz received the Goergen Prize for Artistry in teaching in 1999. As part of the nomination procedure, she was asked to submit a summary of her teaching philosophy. She submitted the following statement.
I had been staring at a blank piece of paper for over an hour, trying to come up with a teaching philosophy statement sufficiently brilliant to be worthy of a Goergen nomination, and slowly reconciling myself to the fact that I didn't really know what my teaching philosophy was, when the phone rang and I found a distraught student on the other end. He was not even my current student, although he was someone I had taught in the past, and he was confused and panicky over some mathematical subtleties that had arisen in connection with a course he was taking. Without a moment's hesitation, I answered all his questions, and we proceeded to talk about other math related topics. When, exhausted after an hour and a half of animated conversation, I finally hung up the phone, I remarked to myself that although I didn't seem to know what my teaching philosophy was, there was absolutely no doubt that I knew how to teach.
It is not just that I have trouble articulating a teaching philosophy, but rather that on matter of strong principle, I believe in not having one. Having a philosophy indicates cognitive thought, whereas teaching to me is largely a matter of feeling. When I teach, I get up in front of the class and instinctively react, reading the faces of my students and intuitively responding. Teaching is something that is second nature to me, and I have been described as a "born teacher" ever since I was a young child.
That doesn't mean that I put no thought into the preparation of my lectures, that I just stand in front of my class and spontaneously emote. On the contrary, I spend hours upon hours preparing my classes, and I fill entire notebooks writing and rewriting prospective lectures. But when I do this, I am trying to listen to what these lectures would sound like from the student's point of view, to hear them in my "mind's ear" as my students will hear them, to imagine how they would sound to me if I were on the receiving end, and to fine tune them accordingly until I reach a perfect pitch. Of course, I don't read them from my notes when I am standing up in front of my students, but having written them up in advance, and anticipated potential difficulties makes it easier for me to relax, ad lib, and be receptive to subtle nuances when I am actually teaching and interacting with my class.
The award which you give is for artistry in teaching. But not all artists are able to articulate their philosophy of art. Some of them can, of course, and quite brilliantly at that. Some artists are very aware of everything that they do, and can meticulously describe every component of their art. However, others though just as artistically gifted, are less willing or able to dissect their art. To them, art is an activity as natural as breathing. It is something to do, not something to analyze.
Having said all this, let me nevertheless try to illustrate, at least a little, some of the things I think I do right when I teach. Most of the guiding principles which I could state would be obvious to anyone, and certainly none of them are very profound. But I believe I find ways to implement them that are uniquely my own, and I accomplish certain things that others do not.
Perhaps the thing I do best is to "deintimidize" the subject. Too many people are afraid of mathematics, and even people who are very mathematically talented are often held back to some degree by a subtle element of fear. One has to find a way of getting past this fear before real learning is able to occur.
The way I accomplish this in my teaching is to first of all put everyone at ease by being myself as much as possible when I am in front of my class. I don't try to hide behind some professional mask, but instead let my personality, my humor, and to some degree my vulnerabilities shine through. This definitely enhances the atmosphere for learning, and is noticed and appreciated by the students whom I teach. In fact, my all time favorite teaching evaluation began with the line, "If Woody Allen could be a math professor..."
I don't try to teach from a "professorial position", and if I need help while working out a problem, I'll turn to my students and I'll ask for their help. I tell them that it is all right to panic when first reading a math problem, that sometimes my first reaction is also to panic, but that then I go on and I work out the problem. I don't teach from above, but rather I approach my students as equals.
Someone once asked me if I thought that the secret to my success was that I could "get down to the student's level" while teaching my course. I remember recoiling at the words which they used for their very formulation would imply that I thought I was coming from a superior position to begin with. I believe if that was my attitude, I would not be able to teach.
Even very well meaning people can subtly convey feelings of superiority when they explain mathematical concepts to others. I saw this when my nephew, a very sensitive young teen, was trying to help his sister, my niece, with her ninth grade math homework. He was doing a fine job of patiently explaining everything, when my niece suddenly burst out crying saying she was too stupid to understand. Despite my nephew's good intentions, he had somehow conveyed to my niece that he was smarter than she was, and reinforced her own insecurities, and her lack of confidence in her own abilities.
When my nephew in exasperation beckoned me to take over, I approached my niece and told her that I didn't understand her math problems any better than she did (which, since her math book was in Hebrew, was probably the truth), but that I was sure that the two of us were smart enough to be able to figure them out together. At one point, I suggested that we work out the problems separately, and then compare answers to check if we had done them right, and I think it helped a lot that I was the one to make the first mistake. What was interesting to me was that my niece knew all along how to work out the problems. She was just using an approach slightly different than her brother's, and in fact her method was more original and showed greater creative thought than did his.
Teaching is not so much telling someone something, but rather helping bring them to the point where they can figure it out for themselves. I try to do this in my lectures, even when I am in front of a very large class. Realizing that students within the same class have a wide range of interests and abilities, I have learned to speak on several different levels at the same time. I try to include something for everyone in my lectures, so that the quicker students will catch on early and anticipate what I am leading up to well in advance, whereas other students will be gradually brought along until they reach the point where they too can see the conclusion for themselves. I would like everyone to be able to sense what I am building up to before I get to the point where I explicitly spell it out. The trick is to be able to do this in such a way so as to stimulate and maintain the interest of the quicker students, while simultaneously making the lecture understandable and intriguing to those who need a bit more time.
Moreover, good teaching must always be a dialogue between the instructor and the class. The communication must be two sided, and not flow in only one direction. The means of dialogue are different in a large lecture, than in a small class, but even in a large lecture, one must elicit constant feedback from the students. Teaching is dynamic. It is not static over time. It is for this reason that even though I sometimes teach courses I have taught many times in the past, each time is a unique experience completely different from every other.
Math is a very beautiful subject. It is sweepingly powerful and charmingly graceful at the same time. It is more an art than it is a science, although unfortunately it is an art that is feared by too many and appreciated by all too few. It is also the only subject where no one can tell you that something is true, unless they are able to furnish an iron clad logical proof. I think this last property is what made me fall in love with mathematics.
I want my students to see my love of mathematics, to feel my enthusiasm for the subject, and to sense my excitement for the field. More importantly, I want them to develop their own passion for mathematics, to appreciate its beauty, and to delight in the challenges it can pose. One of my greatest pleasures is seeing my students' faces light up when they feel the power inherent in a mathematical argument, the magic that occurs when the pieces of a proof coalesce, or the thrill of coming up with creative ideas of their own. Moreover, I try to sprinkle my lectures with thought provoking questions, assign homework problems that are lots of fun to do, and even compose exams that are pleasant learning experiences for my class.
But perhaps the most important thing that I do right when I teach is to make my students understand that I am always on their side. I am not here to judge them, but rather to encourage them to do their best. My students don't have to prove to me how good they are in math. If anything, I see it as my responsibility to prove to them how good they can be.
When I told one of my students in casual conversation that I was having trouble deciding what to include in my teaching philosophy statement, he said, "Just make sure that you tell them that you care. It is important that they know how much you really care." My students all know that I care and that I want them to succeed. Moreover, I believe very strongly in positive reinforcement.
If they make a wrong statement, I will show them what part of it is right, and I'll reinforce their creativity and ingenuity of thought. If one of their hypotheses is wrong, I'll of course point this out, but at the same time, I'll try to salvage as much of their argument as one possibly can. I'll even show them how their reasoning might have been correct if a certain fact were different than it is, and perhaps I'll try to find an alternate context where an argument like theirs could be successfully applied. If they make a mistake, I'll show them why it was a natural mistake to make, and why I might have fallen into the same trap myself if I hadn't seen similar problems many times before. I sometimes remind them that it is better to make a mistake than never to have tried, and the students in my class always eagerly volunteer answers even when they know there is a good chance they might be wrong.
I also always encourage my students to work together with one another, and tell them that they can learn more from each other than they can from just me. Whereas this may make it slightly harder for me to get an accurate picture of their individual level of skills, I tell them that what they gain is much more valuable than what I lose, and that even research mathematicians make the most progress when they are able to meet and exchange ideas. I'll help my students set up study groups which meet in their dorms, announcing the times and places for these in my class, and I'll also match up study partners who I think would work well together. This is in addition to the workshops (for honors calculus), and my office hours and evening help sessions, which are also all set up to foster group interaction.
Most of my math department colleagues and I are approximately the same age, so in that sense there are similar factors that have exerted an influence on our lives. Except that girls who were good at math in the fifties were brought up differently than boys who were good at math back then, and I believe this gives me a unique perspective, somewhat different than that of my colleagues.
I recently found a junior high school math book from the late fifties that proudly boasts as one of its main selling points that in order to make math more socially relevant, it contains separate problems for boys and for girls. Moreover, the problems for boys are the kind that give you a stomach ache just looking at them, for they are made to look much harder than they actually are, whereas the problems for girls are simplistically stated and deal with questions like, "Jane weighs 130. Her ideal weight is 125. How much weight does Jane have to lose?" I sometimes entertain my classes with "dramatic readings" from this book, for it really does sound quite funny when read in the light of modern day sensibilities. Yet I sober up a little when I am reminded of the fact that I was a product of some of the same environmental factors that led to the production and successful marketing of a book such as that.
I received the same signals as did other women of my generation, causing us to feel that we were less good at math than we were, and although I was able to overcome these to a large extent, I cannot claim to have been completely immune. When I went to graduate school at Harvard, I was the only woman in my class, and in fact there was a five-year gap after I received my Ph.D. before the next woman received her Ph.D. in math from Harvard.
All these factors had an influence on my life, and are largely responsible for who I am today. In particular, they help explain many of the unique elements of my teaching which I have tried to describe up above.
They also help explain the special interest I have in encouraging women students as a group to continue on in mathematics. Although the overall picture is slowly improving, I believe that today's undergraduate women are still being subjected to powerful messages (albeit more subtle ones than in my day) that math past a certain point is primarily for men. Because of my background, I am especially sensitive to such messages, and I feel a personal responsibility to do everything I can to counteract them. Furthermore, this sense of responsibility is particularly strong since I am the only tenured (or tenure track) woman on the mathematics faculty. I believe that the percent of women in my honors calculus classes and the proportion of these continuing on to major in math is evidence of the success of my endeavors towards this end.
Moreover, partly as a result of my own experiences, I feel a special affinity for students who are creative and talented, yet not aware of their own strengths. I believe I have a kind of radar that enables me to detect such students and recognize their potential, often before others are able to do so. I am happy to provide all the support that I can, since I know that with the right encouragement, these students can develop the confidence they need to let their natural creativity shine through with quite spectacular results. Sometimes (although not always) these students are female, and I am able to see glimpses of myself at a younger age in them. By symmetry, these students are able to identify with me, and I can provide them with a role model and a much appreciated friend.
In any case, for one reason or another, there seem to be certain students whom I can inspire to do their best, that other teachers are not able to get through to. I remember one woman who got the highest grade one semester out of all three sections of Math 143 that were being offered that term. When I told her the good news, she ecstatically informed me that she had never gotten above a C+ in a math course before.
I remember another student who was earning a high A in the second semester abstract algebra course which I was teaching, although she had earned a C with another professor the first semester of the course. The other professor was aware of the fact that she was quite talented, but he said she just didn't care, and consequently didn't work very hard. This was echoed by a different professor who had also taught her in the past, and who assured me that she was someone who was very happy doing a minimal amount of work. One day when I was talking to her, she started to cry, and she admitted to me how scared she really felt. Apparently, her bravado was just a defense. If she acted like she didn't care, then she didn't have to try, and as a result she would never risk having to fail. She turned herself around, credited me with the change, and went on to enroll in a Ph.D. program in math at UC Berkeley two and a half years later.
Still another student was finishing her math major from another university, and needed one last course to finish her degree. She had tried to take this course before at her other university, but she had dropped it at the professor's urging when he told her she was just not capable of learning the material. When she took the course with me, she earned one of the highest grades in the class, and went on to earn her masters degree in math and to pursue a career teaching math at the community college level.
I have been at this University since 1982 and brought my own unique approach to teaching and to interacting with students. During this time, I have taught many courses together with my colleagues, and had numerous occasions to exchange ideas with them about specific students and/or specific situations that arose. This year, I am teaching Honors Calculus with Adrian Nachman, who will be teaching the course again next year together with someone else. I give Adrian copies of my notes in advance, and talk with him a great deal about what seems to work best in teaching this course. Thus, although perhaps I can't claim credit for any major institutional changes, in my own way I have had a significant impact on shaping the direction of the teaching of math at our school. In addition, I have had an important influence on the lives and careers of many of the students I have taught.
I feel it is appropriate to end this statement with the words of a student, words which mean a great deal to me. I was waiting on line at the campus post office one day when I was approached by a student I didn't even recognize. He said he was currently an officer in the military, having graduated ROTC the previous year, and he had taken my Math 142 class four years earlier when he was a freshman. The class was a large one consisting of over 110 students, and this was the only time I had ever taught a course of that size. Although I had never had any individual interaction with him, the student had very fond memories of my course, and he thanked me for teaching it the way that I did. Then he added something I will never forget. He said, "You know, I had a lot of professors at this University, but I think you are the only teacher I ever had."
Teaching to me means instinctively giving. In some sense, I have no choice but to teach. Put me next to someone who wants to learn, and that is what I will automatically do. I don't feel comfortable "tooting my own horn," but I have been teaching my heart out at this school for the past seventeen years, and I appreciate the opportunity to share my thoughts with you.
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