Foundations of Mathematics
W. S. (Bill) Mahavier, Spring 2008, Emory University
Having spent countless hours discussing mathematics and the teaching of mathematics with Mahavier (my father), having taught Analysis using notes modified from materials he penned, and having watched this entire course, I provide the following observations about the course.
1. Mahavier’s students’ problems and presentations parallel the problems and presentations of research mathematicians.
Harder problems sometimes remain outstanding for multiple class periods or even weeks and 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. In addition to stating problems, which are basically theorems or lemmas, Mahavier poses four
questions during the semester to further introduce the students to research, since answering a question where one does
not know the answer a priori is significantly more difficult.
2. Mahavier has the ability to hone a concept down to its bare essence and to present it with a minimum of notation. The result is that only the language and logic dictate a student’s ability to solve it. These notes, and particularly his introduction to induction, provide examples of this.
3. Early in the course, many proofs are joint efforts with multiple students working on the same problem at the board, one after the other, and with suggestions from the audience.
4. Later in the course, Mahavier often does not encourage the students at their seats to give hints.
5. When a student does not complete a problem Mahavier sometimes lets another student attempt it that day, sometimes lets the first student hold the problem for the next class, and sometimes gives the first presenter a choice. I believe he factors in the number of successes of the interested parties, the likelihood of success the next day and the likelihood that allowing a student to “claim” the problem will result in a confidence boost for that student upon success.
6. Even though Mahavier has a note in his syllabus allowing a certain level of collaboration and reference material, there is no evidence in the videos or in his diary that students collaborated or accessed external material.
7. 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. Mahavier occasionally adds a problem (lemma) to help clarify an issue students are struggling with.
8. Mahavier appears equally content to see failures as successes at the board. 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. 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.
9. Mahavier does not require that problems be completely written out at the board. A personal surprise to me was how little was written on the board in terms of quantifiers and complete sentences. At the board, he is getting them to understand, but the written training is in the weekly written work.
10. During presentations, Mahavier focuses almost exclusively on the student at the board. Sometimes providing support and guidance and sometimes questioning the logic and language, but always focusing on making sure that the student at the board understands what s/he has right and what s/he has wrong to assure success on the problem in a future class. He will ask the class for questions or input, but is primarily focused on the presenter. Personally, I wondered if this is not caused by his severe hearing loss as I don’t recall this when I sat in on his classes 20 years ago.
11. 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. While the last problems are closer in level to a junior real analysis class, these students make good progress through them.
12. There are times when students at their desks appear not to be following the work at the board. Sometimes they are working at their desks, perhaps on this problem or something that was said in class, perhaps on another problem. Mahavier appears unconcerned with this. Even if they are working on another problem that they want to resolve, they are working on the class materials and if a student is not following what is occurring at the board, then it is fine with Mahavier as long as they are working enough to present regularly. Because regular presentations and weekly written work are the primary measures of student performance, Mahavier gives the students significant freedom to use their time as they see fit.
13. Mahavier makes sure that he has problems at a wide range of difficulties throughout the semester to assure that even a student who has not presented early in the semester will have the opportunity to do so throughout the semester.