Matematika | Középiskola » Observing Your Mathematics Teachers, Some Personal Reflections

Source: http://www.doksinet 12. Observing Your Mathematics Teachers: Some Personal Reflections Professor Steven Krantz, in his highly regarded book, How to Teach Mathematics, asks the question “Why do we need mathematics teachers?” He then proceeds to answer that question and then to elaborate upon his reply. He provides three principal reasons. A mathematics teacher: 1) Sets a pace for the students; 2) Teaches the students how to read mathematics; 3) Helps engage the students in the learning process. I urge you to read this material, for it does an excellent job of describing and discussing his question and their answers. In addition, the article listed below that appeared in a 2006 issue of the Notices of the American Mathematical Society discusses the process of forming a philosophy about your teaching. I encourage you to read it Many of you will need to develop a statement of your teaching philosophy as a part of your applications for teaching positions. 1 Source:

http://www.doksinet Grundman, Helen G. Writing a Teaching Philosophy Statement Notices of the American Mathematical Society, December 2006, Volume 53 , Number 11, pp. 1329-1333 http://wwwamsorg/notices/200611/200611-tochtml On a more personal level, all of us have had direct experience with mathematics teachers who have had great influences on us, as well as with others who were less effective in influencing us – although they might have had more influence on other students. It can be valuable to try to extract the qualities and techniques that have made some teachers effective for your personal learning, and see if you can incorporate these into your own teaching. You can also learn from observing the most successful of your colleagues. In addition, you may be able to discern why other teachers appear to be less effective in their callings. In this chapter, I will provide some examples from my own learning experiences over a period of 60 years, both as a student and as a colleague

of other mathematics teachers. I encourage you to do this yourself, and to discuss these with your own colleagues. My Personal Experiences as a Student Rural Primary School, 1945-48 (Grades 1-3) Rural U. S Elementary Schoolroom 1940’s I attended Grades 1-3 in a small rural elementary (i.e primary) school in the U S state of Ohio. The photograph above is not of my school, its classroom and students we looked just like you see in the photo. I started to school when I was five years old, usually wearing bib overalls, and riding a school bus from a small farmhouse to a school in the countryside. There were only four teachers in that 4-room school, which included Grades 1-8. Each teacher taught two grades simultaneously in their classrooms. The school had outside pit toilets, no gymnasium, and very few amenities. Years later, the school district built larger and more centralized schools The new owners of the school converted it into a grain storage facility for surplus corn, and years

later tore it down. However, it provided a very good start to my life’s education, and generated a strong interest in mathematics. 2 Source: http://www.doksinet Because the school year started in September, and my birthday was in December, I was permitted to start Grade 1 when I was only 5 years old, even though the normal starting age was 6. Consequently, throughout my early schooling, I was younger than the others in my class. During Grades 1-2 I learned to read, and no learning experience has ever surpassed that, even mathematics. In this task I received much encouragement at home. Although neither of my parents had gone beyond high school, both were very helpful. My father frequently read or quoted traditional American poetry to me, and I still remember such classics as “Paul Revere’s Ride”, “Snowbound”, and others. He also encouraged me to read the newspapers, especially sports and articles relating to politics. Regarding the latter, he had me read the front page

and editorials, and explained why the Democrats were much better than the Republicans. My mother had taken some business classes, and it is from her that I probably first generated an interest in mathematics, as she was always giving me practical computations to do. I recall two special gifts from my parents. For Christmas when I was 10 years old I got my own desk, two sets of literary books, and a book of poetry. To this day there is nothing, not even mathematics, that I enjoy more than reading traditional printed material. Although we were not wealthy, my family always subscribed to three newspapers. Reading was important, as was keeping up on civic and sporting events The other memorable gift was on my 12th birthday, when they gave me my own Remington 20-gauge shotgun. It may seem strange, but I grew up in a traditional culture that still emphasized hunting. It also provided a way for children to bond with their parents’ generation, as we often would hunt with relatives and family

friends. I suppose it engrained some applied physics, too, as one had to learn to lead a peasant in flight or an erratically running rabbit in getting off a successful shot. The first teacher that I especially recall was Ms. Montgomery, who taught the combined Grade 3-4 class when I was in Grade 3. There were only 16 students in the combined class, and the students in the two grades sat on opposite sides of the room. Usually those in one of the grades studied at their desks while the teacher conducted a lesson for the other class. I recall Ms Montgomery as being great in encouraging us to be creative (she knew that I liked poetry, so she encouraged me keep a book of the poems that I wrote), to do well, and to develop at our own paces. In many ways, her techniques worked well because of the small size and the nature of the classroom. However, her attitude of encouraging each of us individually undoubtedly was what proved so beneficial to me. It is something that I always try to keep in

mind in my own teaching. In any case, three of the class members (2 girls and me), who I guess she had recognized as being somewhat gifted, were encouraged to do our lessons not only with the other third graders, but also with the fourth graders when it was there turn. That was a special gift for me. Besides doing the third grade arithmetic, which I truly liked, I also got to do the fourth grade arithmetic, along with geography and history (also favorite subjects of mine – especially examining the maps), and reading. We were encouraged to memorize notable pieces of literature, the Bible, and history, as well as the multiplication tables. Many years later I found out that the teacher had scheduled us to skip the fourth grade. However, late in the school year my family moved. 3 Source: http://www.doksinet What was special in my initial schooling? I think it was the individual attention that the teachers gave the students, and their ability show an interest in every subject.

Learning was fun! We were encouraged to go ahead. I still recall the excitement of discovering on my own that in looking at the integral multiples of 9, each leading digit went up by 1 while the second digit down by 1 at each step. That may not seem like much now, but it was exciting to me then. Then, when I showed this to Ms Montgomery, she had me show it to the class on the board. What great positive reinforcement that was. We also did lots of board work. This was in a way a competitive exercise, because we always wanted to be the first to finish correctly. However, I do not recall that the slower ones felt uneasy. That was because the teacher always put those of who were at same level at board at the same time. Again, she gave us individual attention, incorporated fun competition, encouragement us to advance as fast as we could, and showed care for all of her students. Suburban Primary School, 1948-52 (Grades 4-6) Grandview Heights (Ohio) Edison Elementary School Near the end of

the 1947-48 academic year, my family move to a small, middleclass suburb of the city of Columbus, Ohio. The city featured outstanding schools (about 80% of high school graduates of era attended universities, and many went on to earn PhD, MD, and DDS degrees), able teachers, but was organized so that students progressed at a more traditional, standard pace. The fourth grade simply repeated mathematics that I already knew. I do not remember much mathematics from these years, but I do recall that the process of dividing fractions always bothered me. I could do that as well as anyone else, but inside of me I always thought of the division process as making numbers smaller (true if dividing by integers), but darn it, dividing by fractions such as ½, ¼, and others between 0 and 1 resulted in larger numbers. This always seemed counterproductive to me. A principal teaching insights that stem from this period is to always try to find ways to provide students with the reason behind

computations, algorithms, and the like – at a level that they can internalize and understand. Certainly, if I were to be a primary teacher, clarifying this mystery about the division by fractions less than 1 would be high on my list of priorities. I highly recommend 4 Source: http://www.doksinet that you read an excellent book that discusses how a young Chinese teacher did this, as well as describing her insightful work at Stanford University on the teaching of primary school mathematics: Ma, Liping. Knowing and Teaching Elementary Mathematics Lawrence Erlbaum Associates,1999. These final three years of my primary school curriculum consisted primarily of more and more arithmetic and drill. There was no algebra or pre-algebra I assume that there must have been some informal geometry, but I cannot recall it. The subjects that I recall most fondly were reading, history, and geography. This was also the period of the first years of television, and I remember watching the nightly

newscasts that usually had films (several days old) of the war in Korea. I still recall clearly my first introduction to where Korea was through the initial retreat in Pusan, General McArthur’s attack on Inchon, the drive to the Yalu River, the subsequent intervention of the Chinese and the resulting retreat and negotiations at Panmujam (all of these spelled here in the 1950’s English version). Junior High School, 1952-54 (Grades 7-8) Grandview Heights (Ohio): Edison Elementary and Middle Schools http://grandviewschools.org/eims/indexhtml 5 Source: http://www.doksinet This period called “Junior High School” in those days, but called “Middle School” now, was a complete academic and social disaster for me. While I had been a leader in primary school, I found it hard to adapt to junior high socially, where I was not as mature as my classmates, and academically, where we had separate classes for each subject for the first time. I liked history, but just tolerated most of

the others (English – all grammar, health – boring to the nth degree, science – memorization). In mathematics, I went continually downhill. Our classes consisted of continuing amounts of boring arithmetic, “practical” applications (such as changing bushels to pecks and acres to square yards), and some elementary geometry. I was so alienated by the subject and by my social distress that I ended up earning grades of D in 8th grade mathematics. The teachers were carrying out the designated curriculum, but they were not reaching me. There was neither anything new nor interesting Of course, there were no computers or calculators in those days, either. These were indeed two grim years for me. I do not know how much of my experience at this period has influenced my university teaching, but I know that it is always important to introduce both strong and weak students to new and interesting mathematical ideas, even if they are not part of the regular curriculum. Senior High School,

1954-58 (Grades 9-12) Grandview Heights (Ohio) High School http://grandviewschools.org/ If junior high school was a disaster for me, then high school provided my mathematical salvation, not that everything was great. We had a choice of two languages to study: Latin or French. I took Latin for four semesters, and my grades for the successive were A, B, C, D. But, other subjects provided me with wonderful things to pursue, especially history and English, especially at the latter now provided a wealth of literature as well as grammar and writing. In my view sciences such as physics and chemistry were “OK”, although I hated biology slide identification (although I did not fare quite as badly as James Thurber in the short story “University Days” in his book My Life and Hard times, a story I suggest that you read some day). However, in the sciences I learned the use of my first computational device, the slide rule. Its construction and use always seemed contrived and inexplicable

until one of 6 Source: http://www.doksinet my mathematics teachers explained the underlying ideas (logarithms), and provided me with new and useful insights. Both, by far, the greatest experience in high school was algebra. It was a life changing experience to have an excellent Grade 9 Algebra I teacher, Mrs. McCaughey. She and husband (a civil engineer with the state highway department) were friends of parents. I liked almost everything in the class: factoring, solving equations, graphing, etc. Actually, I never liked word problems, but did learn to set them up and solve them. Only later in life did I really find a good way to visualize word problems, first in my teaching, and then especially in creating spreadsheet models and implementations. Again, the class had a lot of work at the blackboard, and I received encouragement from my teacher to enter the state mathematics contest in Algebra I. I did very well in it, and for the first time I ever received recognition for academics

skills, having had only moderate success in sports, the true measure of success among one’s peers at that time. My 10th grade mathematics course, plane geometry, was a setback. I was not excited by proving theorems, and even less so be being required to memorize each theorem by its number as well as its words. Along with the second year of Latin, Grade 10 was certainly not a positive experience. As a teacher, you need to recognize that not everyone finds formal proofs exciting or can even see where the ideas for a proof come from. Being able to lead students through proofs so that they can see how one might conceive of the technique is a valuable skill to possess. In addition, it can help some students to visualize the setting before describing a proof, to illustrate the conclusion of the theorem in a variety of settings, or to explore the ideas of the theorem interactively using a computer. In Grade 11 I was excited to be back doing algebra, studying Algebra II with Mrs. Peterson,

another wonderful teacher. Again, I had a teacher who used new and creative ways to approach mathematics, was full of encouragement, and adopted subtle humor, such as her statement, “If I ever hear of one of my students enrolled in bonehead mathematics (i.e remedial mathematics) in a university, I will disown them”. She continued a caring, personal involvement later in life, by coming to see me at my parents’ funerals. She was also my teacher in 12th grade for Solid Geometry and Trigonometry. She always did things a little differently, things that I remember 50 years later. For example, in Solid Geometry, when it was clear we were not keeping up with the work, she wrote theorems on index cards, shuffled them in front of the class, and had a student draw one from the deck. That was our quiz question for the day. It worked; we were more faithful in keeping up to date 2+6 =3 2+2 Only worms cancel terms 7 Source: http://www.doksinet She also used both clever and silly signs

around the classroom, such as: “only worms cancel terms”. It generally looked like the one below This seems silly I suppose, but I still recall it today. As a result, here is another suggestion – to keep students engaged, continually try something new, do not just get stuck in a rut. At the university level, you need to be careful about doing silly things, but sometimes they work there, too. One of the top mathematics professors at Iowa State used to like to demonstrate the concept of center of mass by balancing a chair on his chin. It may not work for you, but it did for him All in all, my high school mathematics teachers did an excellent job of preparing me for my upcoming university studies. This is another point for you to remember You need to ensure that you equip you students not only to pass the course that you are teaching, but also prepare them to be able to succeed in courses for which your prepares a foundation. Undergraduate Study, 1958-62 (4 years) Mathematical

Sciences Building: Bowling Green State University http://www.bgsuedu/departments/math/ My 4 years of undergraduate study represented a highlight of my mathematical life. My method of deciding which university to attend was anything except scientific The usual thing to do would be to search to find the best universities, visit them in person, compare amounts of financial support, etc. I did none of those I lived in Columbus, the location of Ohio State University, where most graduates of my high school attended. However, my parents encouraged me to “go away” to a university out of town, and frankly, I did not want to attend a huge university. However, because of our limited family finances, it was probably necessary for me to attend a state university. After Ohio State, the best known of these was Miami University in Oxford, where many of my relatives had gone and thrived. However, I wanted to be myself, so without knowing anything about it, I choose Bowling Green State University.

At that time BGSU, in a small city northwestern Ohio, had an enrollment of 5,000 students. It did not have a well-known reputation, except in basketball, but I could not have chosen better. It offered a traditional that was just staring to change into a more modern one. Its faculty were primarily teachers, rather than researchers, and related well to their students. One of my favorites was Dr Krabil, who formerly taught at the 8 Source: http://www.doksinet United States Military Academy. He was a serious man who taught rigorous courses, who quickly became my favorite. I had him for classes in numerical analysis, differential equations, and statistics. While there were no computers at BGSU in those days, he still emphasized a computational approach that excited me. In my junior year, our numerical analysis class took a field trip to see a computer, but otherwise we used pencil-and-paper or simple electrical calculating machines. For many years after I started teaching on my own, I

modeled my teaching after him. Since then I have continually adjusted to come up with my own style, but I have always used his businesslike approach, with a computational emphasis. Like my favorite teachers of earlier years, he constantly exposed me to new ideas and techniques, and led me to expand my horizons. He also did something else that was new Occasionally, he invited his students to his house. He was the only one who did that It is a practice that I have usually tried to follow in my own teaching. Incidentally, my first year English teacher had another interesting practice: he visited his students in their dormitory rooms! Although I have tried to do that on occasions (I once spent several hours with a student who’s roommate had committed suicide in their room), doing this would take more time that I felt that I could afford. Another special professor was Dr. Long who I had for analysis courses I was not a great student of analysis, but he encouraged me to take a graduate

level course with him that proved to be a great benefit for my future studies in graduate school. One of the most memorable times of my career when invited to give an invited hour address at MAA in Denver 1985, and he attended my talk. Not only that, but he also gave a presentation that I was able to attend. The experience of being on the same program is unforgettable. As a faculty member, I have experience same thing either presenting with former students, or observing them. At KAIST, I have enjoyed co-presenting with two graduate students. Another professor was Dr. Graue, from whom I had classes in calculus and abstract algebra. It was the first time for the university to offer a abstract algebra course, and at that time I was a rank novice. However, this became my research area in graduate school. His leadership and encouragement led me in a good direction Another wonderful professor was Dr. Tinnapple, who taught foundations of theoretical mathematics and was advisor of the

Mathematics Club, of which I was president in my senior year. He helped me develop leadership skills, as well as mathematical insights. My advisor was Dr Vogeli, who was fluent in Russian He encouraged me to study Russian and to do some translation for an article that I submitted to a journal. Although the journal chose not to publish the article, the experience was valuable nonetheless, and it generated a lifelong interest in publishing my ideas. Thus, through their individual attention, skills in teaching, and encouragement each of these people helped in my professional development. I believe that you will be wise to adopt these attitudes in your teaching, too. Finally, when I entered my universities, I started out to become a high school mathematics teacher. Consequently, I spent my final semester in taking Education methods courses and in student teaching. Unfortunately, the methods courses were quite forgettable and of little value. The practical classroom teaching in an area high

school was invaluable. I taught under highly experienced teachers, with classes in Algebra II, Geometry, and Physics. This allowed me to see two excellent teachers in 9 Source: http://www.doksinet action, and to get first hand advice and criticism from people that wanted to help me develop into a skilled teacher. Here I learned the value of being fully prepared for each class, how to anticipate difficulties, how to design good tests, and techniques in classroom management. Practice teaching under a highly qualified mentor is valuable; in graduate school as a teaching assistant, it would be beneficial to receive similar supervision from top professors. One additional life-changing thing happened during my final semester at BGSU. I met my future wife in April, 1962. We became engaged in June, and were married and moved to Seattle in September where I began my graduate studies in mathematics and she completed her undergraduate degree in Home Economics Education. Although one

ordinarily does not recommend such a short time between meeting and marriage, this September in Korea we celebrated our 44th wedding anniversary. In between, our marriage has even survived my tutoring her in mathematics classes, and having her as a student in a computer science course. Moreover, we have twice given a joint presentation on Mathematics, Computing, and Art at national meetings of the Mathematical Association of America. While working as a teaching assistant, of course you must neglect neither your studies nor your teaching, but it is possible to have a viable social and family life, too. Graduate Study, 1962-67 (5 years) University of Washington http://www.washingtonedu/ I attended graduate school at the University of Washington in Seattle, starting in 1962. Seattle is a beautiful city, surrounded by water, with ferries, many distinct cultures, and many things to do. I chose my graduate school pretty like my undergraduate university. This time I did first get information

from many large universities, finally settling on three: University of Southern California (expensive, with no teaching assistantship or financial aid), University of Arizona (affordable, with a $1800 9-month teaching assistantship), and University of Washington (most affordable, with a $2400 9-month teaching assistantship, tuition reduced from $600/year to $100/year). Accordingly, I went with the big money. Again, it turned out to be a very good decision In my case, I needed a teaching assistantship to support my studies financially. However, in the United States it is valuable to have been a teaching assistant, even if 10 Source: http://www.doksinet you have adequate financing. This is because most universities that hire new PhDs in mathematics prefer them to present evidence of successful teaching. I had mostly teachers of a good, standard quality during my graduate studies. However, the quality of undergraduate teaching that I observed was very mixed. Some were outstanding,

while others were quite bad. In those days, the university had no program of teaching evaluation, so the student organization did their own. Some of the mathematics teachers earned the “grade” of F, and some damning, often unprintable, individual comments. Usually these focused on classes conducted without preparation, failure to show up at classes, and a lack of caring. Nonetheless, others earned A’s. However, I did not acquire pointers on exceptional undergraduate teaching techniques during my five years. As a graduate student, I did receive some excellent insights into the profession. However, unlike my undergraduate teachers, most faculty were impersonal. One exception was the chair, Dr. Pierce, who provided individual encouragement, had departmental picnics, drank beer with graduate students, and was a good teacher who had a sense of humor. His first comment to new teaching assistants was about their attire. He said that he preferred males to wear a tie, so that the students

could tell them from the regular faculty. (Many of the regular faculty members were young, and only few ever wore ties or suits. This was at the start of the “hippy” era of informal dress) Of the 100 graduate students at that time, probably only 4 or 5 were women. That ratio has changed dramatically since then. Dr. Charles Hobby was my dissertation advisor in the area of finite p-groups He always found time for me, helped me to become knowledgeable in the area, and aided me in settling on a dissertation project. He also provided invaluable help in finding my first academic position. I have always remembered this, and his willingness to help has played a major part in my willingness to help students with their need for evaluations, recommendations, and the like. One of my teachers had received his PhD under Professor R. L Moore at the University of Texas. Professor Moore was famous for producing successful researchers and for his use of the “Moore Method” of teaching (see the

article http://en.wikipediaorg/wiki/Moore method), whereby students learned everything on their own by discovery. The professor used a version of this approach with us, but to me it seemed quite unsuccessful. In your teaching, it is natural to adopt many of the techniques of your favorite teachers, but you should be aware that what works well for someone else may not work for you. In such a case, you need to be prepared adjust your teaching techniques if you see that they are not working as they should. I also wrote my first computer program in graduate school. The University of Washington did have a computer science department. Besides teaching classes, it provided consulting for Seattle area industries, which in tern provided job opportunities for graduate students. After I finished all of the requirements for my PhD, I attended a free short course for university staff. It consisted of four 60-minute sessions on programming in Fortran, complete with punched cards and batch

processing. The mathematics faculty’s opinion of this at that time seemed to be, “Why should a mathematician bother with computing?” However, it turned out to be a valuable experience for me. Later in life, I was able to teach programming and other topics in computing, often spending at least half of my time in this direction. My brief 11 Source: http://www.doksinet experience in graduate school told me that I could do it, and gave an indication of how I could use computers in mathematics. Moreover, it was fun! While in graduate school, I encourage you to find time to learn things in such areas as computer science, technology, different subfields of mathematics, and at least one other discipline that uses mathematics. If you keep count of the years above, you will see that by the time I earned my PhD in 1967, I had been in school for 21 years (even without kindergarten, which became common in Ohio only after I had started school). I remember this because I attended graduate

school during the Vietnam War. At this time, young males were subject to the draft, and many males of my age were drafted. Those who went to graduate school in disciplines such as science, mathematics, or engineering could usually get exemptions from the draft. An increased number of men went to graduate school for that reason. That was not what I did, but still I was happy enough to receive an exemption. However, in my final year I did have to undergo a physical exam and other testing to see if I was qualified for the army. On one form, I needed to check off how many years of education I had. The options only went through 20 I already had more than any others there, and I have always felt conscious about this. As a final topic, how did I get my first position? Again, I did so much in the same way that I had selected my universities – sight unseen. Early in 1967, I asked my adviser when I could expect to finish. He went away to give that some thought, after which he came back and

said, “You probably could learn more and become a better researcher by staying here for another year. However, with the declining job market, after that it may be much harder to get a job, so let’s aim for finishing by the end of summer.” He then asked where I would like to obtain a job I said within a one-day drive of my home state of Ohio, but not in that state. This would allow us to be near our families, but not so near as to be a distraction. So he called ten universities in the Midwest, and – sight unseen (no interview, no application, nothing but his phone calls) – I received three job offers as an Assistant Professor. They were, together with their annual salaries: Iowa State ($10,000), Nebraska ($9,500), and Georgia ($9,500). So again, without knowing much about any of them, I went for the “big” money and chose the Iowa State University of Science and Technology. And again, I have never regretted my choice. Teaching in a Large Public University, 1967-74, 1976-7

(8 years) 12 Source: http://www.doksinet Carver Hall: Iowa State University http://www.mathiastateedu/ As you work in a university, you well may be able to observe other teachers, and perhaps even attend some of their classes. I highly recommend doing this After a few years of teaching in my first position in a research-oriented department at Iowa State University, I wanted to expand my teaching capabilities to include applied areas. A number of faculty members in both the Mathematics Department and other departments granted my requests. I selected the teachers to ask based upon the recommendations of graduate students and other faculty members. Within the Mathematics Department, I received significant insights into the teaching of ordinary and partial differential equations from Professor James Dyer, an excellent teacher with insights from his experience in industry, and operations research from Professor A. M Fink, another gifted teacher possessing broad academic and nonacademic

experience. Stemming from both of these, I acquired new mathematical skills, teaching approaches, and insights into applications that I adopted into my own teaching later in life. Both of these individuals had the ability to explain concepts clearly, and to provide real world applications, as well as some history of the development of various topics. Similarly, I benefited from attending classes taught by instructors in such disciplines as statistics, industrial engineering, and computer science. All of these individuals shared similar traits: their classes were interesting, they were always wellprepared, they had acquired outside experience, and were familiar with many realworld applications. It is important for you to know how other disciplines use mathematics, and over the years to build up your insights and expertise in this direction. Teaching in a Small Private College, 1977-84, 1986-88, 1990-95 (18 years) Whitworth College http://whitworth.edu After ten years of teaching, I a

moved to Whitworth College, a respected, small, church-related, liberal arts college in Spokane, Washington. Here I also learned from experience teachers, and incorporated additional insights and techniques into my own teaching. Professors John VanderBeek and Howard Gage were both outstanding teachers. Moreover, in addition to requiring its faculty to be excellent teachers, the 13 Source: http://www.doksinet college expected them to get to know their students, to mentor and counsel them, and to share their Christian faith. Both of these individuals provided me, and other teachers, with excellent role models. In such an environment, you can learn better how to focus your teaching and to better develop interpersonal skills. In addition to teaching mathematics, the department at Whitworth expected me to teach in the area of computer science. To do this I had to do extensive amounts of study, mostly on my own. However, as well as learning new material individually, I had the opportunity

for many discussions with Dr. Gage, who, although also selftaught in computing, was an excellent teacher who had required a reputation throughout the region for his computing skills by working summers in some of the leading companies of the area. I attended some of his computer science lectures, as well as those of the Director of Computer Services, Bob McCroskey. As with most people in my generation, these people were self-taught, but were highly motivated to share their hard-gained knowledge with others who were learning. Because I benefited so much from such individuals, it has always been my policy throughout my career to do the same. I frequently volunteer to run classes for other faculty and staff members in the use of computers. Initially this involved programming, but currently focuses on the use of computer applications software, primarily spreadsheets, where my greatest talents lie. As I mention, the late Professor Gage was self-taught in computing, but each summer he worked

at various jobs in computer-related jobs. Doing so provided him with (a) skills in computer science, (b) applications and insights that were invaluable in teaching, (c) unexcelled contacts with which to provide internships and future jobs for Whitworth graduates. Teaching in Papua New Guinea, 1974-76, 1989, 1995-2000 (9 years) University of Papua New Guinea Graduation (1989) http://www.upngacpg In my nine years of teaching at the University of Papua New Guinea, the principal thing that I observed was that the most successful teachers, as was the case elsewhere, were those who took an interest in the particular students that they were teaching. My nine years there were the most rewarding of my career. It was been gratifying to see a number earn advanced degrees and achieve senior academic positions themselves, as 14 Source: http://www.doksinet well as other who now occupy significant positions in business and government. As you progress your careers, teaching overseas can add a

special component to your career. The language of education in PNG is English, so your significant investment of time in learning English would be most important in working there. While at UPNG, I frequently invited my classes to our house for informal social visits, just as Dr. Krabil had done for his classes while I was an undergraduate student at BGSU. Doing this provided excellent opportunities to get to know my students better, and allowed them to visit with their professor in a different setting. In my second and third appointments at UPNG, I also found that my use of spreadsheets in teaching mathematics classes made a positive impact with the students, and that my voluntary classes in the use of spreadsheets for both academic and non-academic staff members were received most positively. Teaching in Other Overseas Nations, 1985, 2000, 2004-06 (4 years) In addition to the nine year I taught in Papua New Guinea, I also taught for one year in Australia, for one semester in Austria,

and now, for two and one half years in Korea. These were rewarding experiences, both professionally and personally My wife and I felt highly welcomed by not only the university communities, and by the general populace. Again, I highly recommend that you consider doing this during your teaching careers. In all of these countries, I taught in English. In the latter two countries doing this creates some stress on both professor and students, as English was not the first language most of the students. This has special implications for a teacher A discussion of these aspects appears elsewhere in this material. You Can Teach Yourself a Lot, But It is generally true that someone with a graduate degree should have acquired the skills needed to be able to continue to learn on their own for the remainder of their careers. However, after you have completed your formal graduate studies, you will probably still be most effective in your continued educational growth if you can continue to learn

from another teacher. This is especially true if you are making a transition into another field of study. I can see this clearly in my own professional career, as I broadened my area of interest from mathematics and finite groups into areas of computer science. I grew up in a time when computers were just beginning to enter society, primarily in large industries and big universities. In my undergraduate studies at a regional university, my sole experience with computers occurred one day when my numerical analysis class took a field trip to the Ohio Oil Company in Findlay, Ohio to see a computer. The computer utilized thousands of vacuum tubes and filled a large room, with its various components connected with many wires. Large as it was, its computing capacity fell very far short of even the simplest of today’s laptops. My graduate university possessed a computer science department by the time I graduated, but at that time, mathematics students received no encouragement to take

classes in computing. Work in that area consisted primarily of programming using 15 Source: http://www.doksinet punched cards with an early version of FORTRAN. Nonetheless, one of my fellow graduate students worked part-time in the Computer Science Department, and he encouraged me to take a 4-session short programming course in FORTRAN. That was the entirety of my “formal” education in computing as a student. My next computing experience came seven years later while I was teaching at Iowa State University. Then the ISU Business College asked the Mathematics Department to incorporate the use of the BASIC programming language in the business mathematics course. As one of the teachers of that course, this meant that I needed to learn how to program myself. This was the beginning of a lifelong selfdirected study in the area of computing Later, at Whitworth College, the Mathematics Department was also responsible for teaching Computer Science. Because at that time no one in the

department had received any instruction in computing, we all learned from each other and on our own. Over the years there and elsewhere, I ended up spending thousands of hours in selfinstruction, first in learning how to program, then in learning concepts of structured programming, and finally in expanding into other areas of computer science. Initially, at Whitworth, I taught programming in BASIC and became reasonably talented in both programming and in teaching. However, by being largely self-taught, while I was quite good in some areas, I had large gaps of knowledge in many other areas, and often I would be unaware of the newest trends in the discipline. This is an inherent difficulty in learning on your own. Getting additional external input is vital Later, I spent further vast amounts of time in becoming able to program and teach programming in COBOL, FORTRAN, Pascal, and C++. It would have been both more efficient and more instructive to learn from a master teacher in these

areas, but at the time, that was not an option. I also spent hundreds of hours learning several word processing, spreadsheet, and database packages, the first of which tended to be inadequate and poorly documented. I did virtually all of this on my own time, on top of my other teaching, research, and service obligations. The one exception to this was attending a brief summer class in Apple Pascal taught by Professor Ray Hamel of Eastern Washington University. There I saw what a structured program really was In that one course taught by an expert teacher, I advanced much more quickly and more insightfully than I ever had done on my own. A good teacher is a godsend! In another direction, at the same time that I was learning how to use a spreadsheet (again on my own), I was also teaching a course in numerical analysis. I immediately realized that the spreadsheet – designed for business applications – could also become a natural tool for mathematics. I immediately devoted my research

efforts to this area, and became one of the pioneers and leading authorities on the subject. I gave numerous presentations, including two invited hour addresses to the Mathematical Association of America, and published books on my work. Thus, possessing good research training, you can learn on your own. 16 Source: http://www.doksinet So in conclusion, it is possible – in fact imperative – to be self-taught in your research since you are breaking new ground. However, many students will not be doing that, and even those that are will benefit from having a good teacher, as it greatly increases your efficiency in mastering a wide range of knowledge. You can do a lot on your own, but with a talented teacher • • • • • • • your learning is usually more efficient you will be directed to insights that you otherwise may be unlikely to find you can discover good ways for you to interact with others you can find new directions for further study you will better see how things

fit together you can learn a discipline’s standards from a mentor you may encounter more cutting-edge applications. Indeed, a good teacher can make a real impact in learning mathematics effectively. Exercises 1. Write short paper describing how an influential teacher has influenced you, and how you might adopt some of his or her teaching methods in your own teaching. 2. Create a list of some annoying habits that you have observed among teachers that you will avoid in your own teaching. 3. Write a short teaching philosophy of your own 17