The majority of students who have taken my classes have done so to fulfill a requirement for their programs of study. With these students, I believe my teaching should aim to portray mathematics as an enjoyable, worthwhile subject rather than as a prerequisite or a burdensome hurdle. I consider that the best way to improve my students' attitudes about mathematics is to educate them about what people in the mathematical sciences really do. In interactions in the classroom and office, I try to convey the delight felt by every mathematician at having reached an elegant result. While I realize that my enthusiasm at completing a mathematical problem often produces curious stares among my students, it is this curiosity I most want to develop in them; I want them to reflect, "if math is so awful, then why is my teacher so happy about it?" When opportunities arise, I give historical background to specific topics, so that my students can better understand my area of research (the history of mathematics) and can sense the exciting historical developments in mathematics that tend to be muted in the organized pages of their textbooks. Mathematics seems to be complete and immutable to many new to the subject; I want my classes to discover its dynamic, innovative facets. I factored my research directly into one introductory statistics course, where I used results from my dissertation to explain how to present and read statistical information in tabular form. Former students from this class still ask how my research is going. Though most may never become mathematics majors, they have learned to appreciate better what goes on in their university's mathematics department.

  Besides getting my students acquainted with what I and others in mathematics do outside of class, I encourage them to become active participants in the learning process. For one applied calculus course, after sensing a lack of preparedness by my students for each class, I began to call on them to present homework problems at the board at the beginning of class. I would give a "play-by-play" account of the board work and would get other students involved. After enjoying a good response to this alteration in my lectures, I extended this technique to the review sessions held before each examination. The tactic encouraged the class to work steadily rather than to cram before tests and gave these participants confidence to present mathematics before a group in a clear manner. I also refined my students' presentation skills by requiring them to write review sheets for upcoming examinations. I feel that my understanding of a concept increases tenfold when I teach it. By involving students in the teaching process, I believe their understanding grows as well.

  I employed group work constantly in my direction of the inaugural year of the Virginia Mathematical Scholars Program, an intensive, year-long calculus workshop that a select group of students take in tandem with their first-year calculus sequence. Using resources from the summer training session I attended at the University of Texas, Austin, reform calculus textbooks, and problems of my own, I fashioned worksheets for my students to tackle for each two-hour class. These challenging worksheets required cooperation among group members, and they often gave tantalizing previews of what these students might find in advanced mathematics courses. For one worksheet, each group made conjectures about Euler's formula by triangulating models of manifolds; another challenged students to consider the Borsuk-Ulam theorem from topology. The aspect of these worksheets most enjoyable for me, but that initially represented the most upsetting aspect for my students, was the lack of an answer key. For the first few weeks, my students repeatedly implored me for prepackaged answers to our worksheet problems. Accustomed to answer keys in the backs of their books, this group balked but eventually warmed to the idea that doing mathematics is not a prefabricated process with one pretty answer. In getting them beyond the final answer and orienting them towards the mathematical processes, I felt like I was really getting them closer to the stuff of mathematics. Beyond facilitating group work, I presented facets of mathematics that my students were unlikely to encounter otherwise by arranging for a statistics professor to speak about careers in actuarial science and showing documentaries about Fermat's Last Theorem and the life of Paul Erdös. I also learned about diverse mathematical subjects, such as the mathematical traditions from the Native Americans of South and Central America, through my students' in-class presentations.

  Every mathematics class I have taught, including the Virginia Mathematical Scholars Program, contains students with varying levels of preparation and different learning styles. In class, I strive to engage advanced students while not leaving students with less mathematical training behind. However, I believe that my availability outside of class is necessary to bridge the diverse mathematical backgrounds of my students. I have also used my office hours as a comfortable environment in which to address my students' questions and concerns about effective study strategies (such as how to read a mathematics textbook), future mathematics courses in which they are interested, and career paths. In the office, the student and I can roll up our sleeves, get covered in chalk, and really work through problems together. I love thinking on my feet in these situations and celebrating when we solve a particularly troublesome problem.

  When all else fails, I have found that food is a particularly effective motivational tool. Truly remarkable amounts of mathematical thinking get done when a candy bar is on the line. Even a 3:00 PM Friday class can be enlivened by a plate of chocolate chip cookies. I find that conversations about concerns I may have about a student's performance or concerns my student may have about mathematical obstacles flow more easily over a cup of coffee. Making the effort to nourish my students' stomachs while challenging their minds is one way to let them know that I want to teach them and not just talk to a blackboard. Outside of class time or office hours, I have periodically hosted a class dinner or picnic at my home to strengthen the ties between my students and help them feel comfortable to call on each other when studying.

  I believe that students leave my class with answers to the ubiquitous question, "what good is mathematics anyway?" Through encouraging them to take an active role in class and addressing special concerns during office hours or individually, I motivate them to take pencil to paper and begin the fundamental task of learning mathematics.