# Foundations of Mathematics

#### W. S. (Bill) Mahavier, Spring 2008, Emory University

**FAQ**'s

**Q.**
Why use the Moore Method?

**A.**
Watch DeMitchell’s attempt on Day 3 and Winterhalter’s attempt on
Day 5, followed by both of their successes on Day 10.
These show how quickly students can develop mathematically under the method.

**Q.**
What do you do on a day when students have nothing to present?

**A.**
See Day 30 to see how Mahavier gently queries students about what they have to present and,
when they don’t, encourages them to show what they are working on. See the next FAQ to see his mini-lectures which
typically occur when students don’t have material to present.

**Q.**
Do Moore Method instructors lecture?

**A.**
Yes, although Moore Method lectures are not the focus of the course. They are brief and likely given to introduce a
single concept or as a reaction to student successes. You can see the types of mini-lectures that Mahavier gives on
Day 10, Day 13, Day 15,
Day 18, Day 19, Day 21,
Day 29, Day 32, and Day 33.

**Q.**
Does the Moore Method work better (only) for “brighter” students?

**A.**
On the second day of class, DeMitchell visits Mahavier’s office and he notes in his diary that she needs help.
On Day 3 she attempts a problem, but manages only a special case.
On Day 6 and Day 10 she presents nice arguments.
On Day 31 she presents a proof of the intermediate value theorem.
On Day 13 Zahn presents a problem, but is clearly not comfortable at the board.
He goes on to produce nice work as illustrated by his presentation of Rolle’s Theorem on Day 30.
“Were these ‘weak’ students? Mahavier did not believe that we, as mathematicians, are capable of determining which are
‘weak’ and which are ‘strong.’ He believed that given a fair opportunity to develop, all students would make progress and
some would become mathematicians.

**Q.**
How does one decide who goes to the board?

**A.**
On Day 31 we see how Mahavier carefully gauges what each student is working on and what they
have and how gently he encourages them to talk about what they have, even late in the semester.

**Q.**
What should one do when a student is completely stuck on a problem?

**A.**
On Day 34 we see how Mahavier doesn’t provide a hint or give the first few lines or leads, but rather
listens to the students attempt and then states a problem that he believes will clarify the point of
mis-understanding.

**Q.**
Can a Moore Method class cover enough material?

**A.**
This is a transitions course, so the goal is to teach students to prove theorems rather than to cover a body of material,
as Mahavier states in his syllabus. Yet, the course ends up covering much of what an analysis
course might cover including limit points, sequences, subsequences, continuity, derivatives and integrals.
We can see dramatic change in the level of the material by viewing any early day, perhaps Day 5,
where the problems are so very elementary. Later clips include the proofs of Rolle’s Theorem
Day 30, the Intermediate Value Theorem Day 31, and a precursor to the Extreme
Value Theorem Day 36. One can imagine how these students will perform in Abstract Algebra,
Real Analysis, Topology and subsequent theorem proving classes.

**Q.**
Does the Moore Method encourage collaboration?

**A.**
Sometimes there is collaboration in the classroom, depending on who implements the method.
See Day 9, Day 20 and Day 29 for examples
of the class collaborating on a problem. See Day 21 for collaboration on

**Q.**
Should I “sell” the method to the students?

**A.**
Because Moore Method classes are so different from classes students are accustomed to, offering encouragement and some
explanation can be helpful. Mahavier always compliments arguments with a “nice job” at the end of a presentation.
On Day 6 he passes out student comments from a previous course evaluation and even
reads one of the student comments that he particularly likes in order to further encourage students to go to the board.

**Q.**
What should I do when I must miss a class?

**A.**
One of the nice things about the Moore Method is that students learn to work as a class as well as individually.
On Day 10 Mahavier leaves the room to get some papers and the class solves a nice problem
without him. Many Moore Method practitioners simply allow the students to conduct class independently
when they are out of town at conferences.

**Q.**
How do you handle students who fail at the board?

**A.**
Because Mahavier creates an environment where any student is welcome to go to the board and present an idea on a problem,
simply to see what it is they understand and what it is they do not, “failure” at the board does not really exist.
They are simply at the board to communicate their ideas and see where it leads them. Thus, “failure”
is viewed as a necessary precursor to success. Following Phelan’s progress provides a good example of this.
On Day 7 he goes to the board and fails, frustrated. He never resolves this problem.
On Day 9 he returns to the board well-prepared and makes a nice presentation.
On Day 20 he goes to the board and fails, but gives a beautiful argument on the next day,
Day 21. One can see from Day 31 that there is no embarrassment or stigma associated
with making a mistake and sitting down.

**Q.**
Is the competition an integral part of the Moore Method?

**A.**
I have always felt that competition in Moore Method classes arose naturally and in a friendly way.
I believe that Day 27 illustrates this nicely.