*Jack Brown wrote a Ph.D. thesis in analysis under the direction of H.S. Wall. He is Professor of Mathematics at Auburn University.*

April 30, 1996

To Whom It May Concern:

I am just writing down some comments concerning my own experiences as a student in the "3rd Floor Group" of the Department of Mathematics at The University of Texas during the period 1956-1967 and the effect those experiences have had on the way I have been teaching mathematics at the university level during the period 1967-1996.

#### Undergraduate Experiences:

I went to UT in 1956 as a Physics major. I took algebra, trig, and analytic geometry my freshman year. I signed up for Wall's calculus fall semester my sophomore year but dropped that course when I found out that there was no text and changed to the calculus course taught by possibly the worst Professor on the 2nd floor. I only went to class on exam days. In fact, I was called into the Dean's office near the end of the semester because of excessive absences. The Dean asked me how I was doing in there. I told the Dean I had a 100 average. He said "Why don't you just go to class the rest of the semester," which I did.

On the advise of two Graduate Teaching Assistants I had as instructors (Howard Cook and Tom Hayden), I changed over to Dr. Moore's calculus the second semester. Needless to say, this was drastically different from my experience fall semester. I didn't skip classes and had to work very hard. That course was not a "Texas Style" Theorem-Proof course but was a standard differential and integral calculus course. It was probably more rigorous than most courses at that level but the emphasis was on using calculus to solve "word problems." Moore gave exams. I think we had two hour tests plus the final exam. I made an A.

My junior year I took Moore's Introduction to Analysis and Ralph Lane's Probability Theory course. These were both "Texas Style" courses. I didn't do so well in Moore's course but became really attracted to Lane's approach. I got sick and missed Moore's final exam (I'm sure you've heard this before) and then had to study by myself (rather than with a friend with whom I usually studied) to prepare for a make-up final on which I made a C. I made an A in Lane's course, of course. I made A's in courses the second semester.

My senior year, I took Lane's Mathematical Statistics course and differential equations from Ettlinger. Lane's course was taught "Texas Style" and Ettlinger's course was a standard D.E. course with a standard textbook. I liked Ettlinger a lot. He was also my boss in his role as coordinator of the graders for the department. When I made Phi Beta Kappa, Ettlinger called me up (at about 5 AM) and told me he wanted me to wear his key until I got mine.

#### Master's Work:

I worked under Lane for my Master's degree. I took my first course (322) from Wall at this time and really liked him, too. I also took Real Variables from Wall at this time. That course was probably what determined the direction my career would take.

Wall's teaching style was quite different from Moore's. Of course, he stated theorems and problems and gave the students the chance to work out their own proofs. However, unlike Moore, Wall did not give examinations. He asked everyone to write up a "term paper" at the end of the semester and turn it in. I think grades were based on whether you turned in a term paper and maybe what you had presented during the semester. As a result, half the graduate students (including myself) were usually in the position of having an "Incomplete" in some course of Wall's for which they still had not turned in a term paper. I must say that I didn't like this aspect of Wall's teaching method.

#### Ph.D. Work:

Right when I finished my Master's thesis, Lane resigned his position at Texas and took a position at Iowa State. It was my intention to follow him to Iowa State and pursue my Ph.D. under his direction. He told me I should stay at UT for another year to take Moore's 688 and also to give him a chance to get settled. I did this. Lane died that first year at Iowa State.

I did well in Moore's graduate topology and in Wall's graduate course that year (which was probably Normed Linear Spaces). I took Moore's course in Lebesgue measure the following summer and was probably the best student in there. I liked this subject matter the best of any I had seen up to that time. That was the summer he filmed the "Moore Movie."

I took Moore's second year topology course and another course from Wall the fall semester of the following year. It was at that time that I decided to work with Wall and asked him if that was OK. I continued working pretty much exclusively with Wall after that until I finished a dissertation.

#### My Own Teaching Method:

I believe the "Texas Style" of teaching contributed greatly to my own development as a mathematician. Maybe I would have done just as well in some other kind of graduate study environment but I was sold on the environment on which I was trained. Therefore, I adopted many of the teaching methods of Moore, Wall, and Lane into my own way of teaching. One of the main reasons I came to Auburn University in 1967 was that it was made clear to me that I could use this teaching approach if I so desired.

I have never used the "Texas Style" in any courses except the senior-level analysis and graduate-level analysis. All other courses (probability, statistics, linear algebra, differential equations, etc.) which I have taught have been taught with a standard textbook and more or less standard examination and grading schemes.

In those upper level analysis classes in which I use a modified "Texas Style" approach, I provide the students with typed notes which include all of the definitions, axioms, and statements of theorems to be proved, counterexamples to be found, problems to be solved, and questions to be answered. I always use standard names for concepts studied and for Theorems which have names associated with them. I give the students a chance to prove theorems and solve problems. If they get stuck, I give hints. If they stay stuck, I show them the solution and we go on.

My method of grading is more like Moore's than Wall's or Lane's. I tell the students I will be computing two grades. A "class participation grade" is based upon the number of (and difficulty of) problems they present in class. A "written material grade" is based upon a standard averaging process using (1) weekly written homework assignments which are turned in and graded, (2) one or two hour exams given during the quarter, and (3) the final exam grade. The student gets the higher of these two types of grades that he or she earns.

During my 28 years at Auburn I have supervised 5 Ph.D. students and 14 Master's students. Every one of them has gone on to professional occupations based upon their mathematical training and has been pretty successful. About half of them have taken applied jobs and about half are in teaching. Indeed, I have spent about a quarter of my own time in applied jobs through temporary appointments (mostly summers) at such locations as Bell Labs, Sandia Labs, National Security Agency, IDA Center or Communications Research, TRW, and elsewhere. I think the students I have trained probably chose to work with me because of my teaching. I would guess that many of them would feel more strongly than I do that the "Texas Style" of teaching is the way to go.

Whatever anyone else may or may not think, I feel that the method worked well for me and my students. In any case, if I were unable to teach my upper division analysis classes this way, I wouldn't be in the teaching profession.

J.B.B.

UT classes of '60, '62, '67