Analysis

W. Ted Mahavier, Fall 2011 - Spring 2012, Lamar University

Course Observations

Having taught this course approximately eleven times over the past seventeen years, I offer a few comments on the course, the notes and this particular class.

The Course


1. Early in the course, many proofs are joint efforts with one student working at the board and others assisting from their seats. As the semester wears on, I wean them off help from the audience and push for complete proofs by the individual at the board. When a student is not successful at the board s/he can fix it at the board, pass the problem to another student, or hold the problem till the next class period.

2. The problems and presentations parallel the problems and presentations of research mathematicians, with harder problems remaining outstanding for multiple class periods or even weeks. Students, like mathematicians, can judge the difficulty and worthiness of a problem by how long it has been outstanding. When presenting, students parallel mathematicians as well, not writing everything on the board, but presenting the key ideas and addressing questions as they arise. I pose some problems as questions, rather than as stated facts to prove. This helps to further introduce the students to research, since resolving a question by proof or counter-example is decidedly harder when one does not know the validity of the claim a priori. A favorite sequence of questions I ask address the existence of a set with these properties: P1 = the set contains all its limit points, P2 = the set has no limit points not in the set, and P3 = the set contains no intervals. Then I ask, can you think of a set with property P1? With property P2? With properties P1 and P3? This can be done even before they know the formal definitions of closed and perfect. Of course, one must assure that they have certain tools such as the nested interval theorem.

3. I am equally content to see failures as successes at the board and encourage students to go to the board with ideas, if they have thought about the problem and have a start. Failures indicate that the students are working at problems at the periphery of their ability (educators might call this working in the zone of proximal development). This attitude trains the students that failure is often an integral precursor of success, and we see many days where failure on one day precedes success on the next. I strive to make my students feel very comfortable expressing their ideas, even incomplete ideas, at the board as part of the learning process, so it does not appear that being seated is equated with having failed, but rather with having learned.

4. During presentations, I encourage discussions and questions, but my primary focus is to assure that the student at the board understands what s/he has right and what s/he has wrong to increase the odds of success on the problem in a future class period.

5. There are three major transitions that occur during the course. First the students transition from an inability to speak or write correct mathematical statements to the ability to do so. Second, the students transition from writing correct mathematics, but not having theorems fully proved, to knowing when they have a proof and stating before they go to the board if they believe they have it or if they are stuck. Even when stuck, they are willing to present what they have. From this point forward, the majority of times a student goes to the board, s/he either has a proof or recognizes the mistake and presents the problem correctly at the next class. Finally, there is the transition to more difficult problems that test the limits of the students and often requires multiple trips to the board to have both a correct proof and understanding by the majority of the class. The problems progress linearly from simple to hard, but with some elementary problems sprinkled throughout.

6. What goes on the board is not always a well written proof. The weekly writeups will assure that the students can write mathematics, the board work is to encourage them to experiment with the mathematics.

7. There is a wide range of difficulty in the problems to assure that even a student who has not presented early in the semester will have the opportunity to do so throughout the semester.

The Notes


1. The notes are a never-ending adaptation of a set of notes my father passed on to me in 1995 which he developed at Emory University in the 1970s and which no doubt had roots at the University of Texas under R.L. Moore.

2. The notes are simple, containing minimal extraneous definitions or axioms. Only what is needed is presented to assure that students are not confused by superfluous material.

3. When I hit continuity and differentiablility, I give multiple definitions for each. In hindsight, I think that it might have been wiser to have given the first definition. A later problem could have been to show the equivalence between this and another definition. I work around this by simply insisting at the beginning that we use the geometric definitions in our proofs to the class and later I discuss the equivalence of the multiple definitions when students struggle with them.

4. When I decided to publish these notes in The Journal of Inquiry-Based Learning in Mathematics I broke the material into chapters rather than have one long problem sequence. Still you will find that the problems are mixed between the chapters. For example, standard problems on continuity show up in later chapters. I discuss at length the idea of having multiple threads intertwined in The Moore Method: A Pathway to Learner-Centered Instruction. In brief, keeping multiple topics intertwined (perhaps sequences and limit points) increases the chances that multiple students will have success. If one student picks up on one definition and the next ten problems all depend on that, then this student will likely not be caught by the rest. If there are two or three definitions and threads intertwined, then this doubles or triples the odds of success for each student.

5. My goal in the first semester is to get to some integration. This is why I move subsequences and Cauchy sequences later than most books and why I don't treat the real numbers or set theory in detail.

6. In a typical two semester course, we cover all of Chapters 1, 2, 3, 4, 5, 7, 8 and portions of 5, 6, 9.

This Class


It is ironic that the year I filmed was one of the most challenging sequence I ever taught. This course is always a challenge simply because Lamar is a regional university and most of our students are first-generation and are either on financial aid or work at least part time. This year, there were an unusual number factors that effected the class. Two students did not have a "C or better" in the pre-requisite course and one who did had taken that course fifteen years earlier. On top of these ill-prepared students, an usually large number of students had valid student-life crises. One student who worked full time was switched to nights and unable to attend further. One was working as an assistant coach to the woman's basketball team while signed up for eighteen hours of mathematics. Another, who helped his grandmother care for his grandfather, became the full-time caregiver when his grandfather fell and shattered an elbow. A casualty of the recession, one talented student worked full-time the first semester and dropped out during the second semester because she became the primary breadwinner for her household and her parent's household. And these are simply the problems that I somehow became aware of. Additionally, two of my best students from the first semester were unable to attend the second semester due to course scheduling conflicts with their second majors which, had they taken my course, would have set their graduation back by a year. The top student in my first semester course was accepted to the mathematics Ph.D. program at N.C. State so I pulled him from the second semester class and met with him independently. This allowed me to accelerate his progress and to work with him at a higher level and make connections to graduate-level mathematics. Lastly, for the first time in my history of teaching this course, I had a student look to outside materials, violating the class rules. Never in my experience of teaching this course have I faced so many hurdles.