When I arrived at Edinboro University the students in the upper division mathematics courses regarded abstract mathematics as an enterprise of dubious value and a sure impediment to the maintenance of the grade point average required of the candidates for certification as secondary school teachers. The courses were generally approached with a mixture of fear and trepidation. In my third year in the tenure track and my twenty-ninth as a teacher, I was invited to teach Introduction to Analysis. I decided to structure the course to reflect my love of mathematics together with my conviction that mathematics students, especially those headed toward teaching or business careers, should develop into independent learners and presenters of mathematics, a pursuit which requires excellent reading skills and good public self-expression.

Before I studied mathematics I spent fifteen years as a performing musician and music teacher. So, drawing on my experience as a chamber music performance coach I developed a student-centered approach to the course in which I directed the students in the preparation and delivery of all of the lectures for the course. We used the expository text, A Radical Approach to Real Analysis by David Bressoud, which is very different from their customary theorem-proof-exercise style texts. Each presentation covered a clearly delineated subsection of a chapter and included a worked out example relevant to the theory. Meanwhile the class as a whole was assigned homework problems for each section of the text and these problems were expected to be prepared thoroughly and presented to the class on designated days. To start the semester?s presentations I randomized the student list and assigned a solo presentation to each student in the new ordering. After I had assessed their capabilities I allowed presenters to work in self-selected teams of up to four members and assigned their presentations according to my perceptions of their mathematical ability and their confidence with public speaking. Course grades were assigned based on these presentations, on participation in the discussions which grew out of them, on presentations of correct homework solutions, on end-of-chapter in-class exams and on a comprehensive final exam.

Dynamic interaction between me and the students developed as future presenters would meet with me for mandatory coaching. During these seminar-type meetings, the presenters and I would comb through the text to discern the important points and discuss how they were developed by the author. Once comfortable with the material, students would work up their outlines and prepare any visual aids. I placed great emphasis on the students? fluent and fluid incorporation of the correct vocabulary of real analysis into their presentations. In particular we developed what came to be known as "Sprague?s No 'It' Rule."

The idea sprang from the experience of preparing, together with two of my friends, for our dissertation defenses. My friends each had a tendency to answer any question by waving a hand at an overhead while saying something like, "Because this one "or, "It?s a space" while leaving the details to the imagination of the interrogator. Our PhD adviser often sat impassively waiting for a presenter to construct adequate verbal phrases involving mathematically sound terminology couched in complete sentences. We came to understand that we were to resist all temptation to use any of the words: "it," "this" and "that one." My friend and colleague Dr. Michael McConnell, Clarion University of PA, provides a familiar-sounding sample of the sort of communication which we frequently encounter in discussions of mathematics by young mathematicians:

"... Instead of saying ?The function is increasing and concave down because the first derivative is positive and the second derivative is negative? they say things like "It's increasing and it's concave down because it's positive and it's negative." At the time, they may know what each of those "its" refers to, but very soon they get muddled."

As I worked with the Analysis students I adopted the same standard of fluent accurate mathematical expression that my adviser had imposed. I knew I was on the right track when one day a presenter responded to a classmate's question by saying, "I'm not sure what you mean. Please say that again without the 'it'!" Most of the students by then had heard this very phrase from me and the class burst into laughter. The No ?It? Rule became the class standard from that moment forth.

Three of my former Intro to Analysis students responded to my request for reactions to the Rule. They wrote:

- (from a Math major) "I like the "No it" rule. Sometimes we need a little prodding to learn to actually say what we're thinking."
- (from a Math/Secondary Ed major) "I remember the "No it" rule. That rule drove me crazy, but I know that explaining exactly what you mean requires not using certain words. (I had to rewrite this because the message had it twice)!"
- (from a Computer Science Major) "The "No 'It'" Rule actually comes in handy in programming, in logic, and in anything in which I'm working on a function. There is a limit, however; in Object-Oriented Programming, there's a point at which you MUST abstract your function to an "it". The very fact that you must consider this transition has held me in good stead."

In the Fall of 2008 one of the homework problem sessions was observed by a member of the departmental Development and Evaluation Committee as part of my annual tenure review. The observer wrote:

"?I arrived early and the students were already writing their work on the board. ? Dr. Sprague had each student stand up and present his work. ... Dr. Sprague insisted the students use proper and concise mathematical terms when explaining their work. I was impressed with the students? preparation and the discussions that followed."

I?d like to end this exposition with a few success stories:

- In Fall 2008 there were twenty-seven students of whom twenty-four earned C or higher.
- One of Fall 2008 students was awarded a fully-funded research assistantship in a physics graduate program because his graduate adviser, who called me for a reference, was pleased to obtain a mathematically literate student.
- In the Spring of 2010 all of the fifteen students who began the course completed it with a grade of C or better.
- Several Spring 2010 students voluntarily elected to submit posters based on material provided by Bressoud about the analysis of the computation of pi to our departmental "Pi Day" competition.
- Two of those Spring 2010 students amplified their posters into an expository talk "Calculating Pi" which they delivered at the April 2010 meeting of the Allegheny Mountain Section of the MAA.
- The fourteen students in Spring 2011 who completed the course finished with a grade of C or better.
- A team of three students from the Spring 2011 class expanded their homework presentations into an expository presentation: "Exploring Limits: Some Interesting Examples" which they brought to the Allegheny Mountain Section Meeting of the MAA in April, 2011.

The success and confidence of the Spring 2011 presentation team, which was built upon the successes in previous semesters, has inspired no fewer than half a dozen of our returning mathematics and math education students in a wide variety of upper division math classes, to return to us in Fall 2011 clamoring for research topics to prepare for presentation at the Spring 2012 MAA section meeting. My colleagues and I look forward to a rich and rewarding year interacting with engaged and confident students.

Emily Sprague (esprague@edinboro.edu) received a B.Mus from the New England Conservatory, and an M.Mus from the University of Texas at Austin. Following a fifteen year career as a symphony cellist and music teacher she earned a B.A. in Mathematics from Castleton State College in Vermont and went on to earn an M.A. and a PhD (2003) from Kent State University. She is currently a tenured Assistant Professor of Mathematics at Edinboro University of Pennsylvania. Her mathematical interests lie in Measure Theory while her pedagogical interests tend toward collaborative learning. She designed a First Year Experience course, "The Mathematics of Musical Consonance" which was approved for offering in Fall 2011 and Fall 2012. The idea to write this piece grew from a conversation with Dr. Bonnie Gold at MAA Math Fest, 2011, in Lexington, KY.

*The Innovative Teaching Exchange is edited by Bonnie Gold.*