Philosophy of Teaching Statement Karoline Pershell
Math is important as an independent subject because it teaches the mind to think, process and prove in a way that is not possible with words. An undergraduate education is essential today to introduce students to the complex ideas and technologies that will make each an informed global citizen, but I must warn you that I am biased: I love math. I enjoy the beauty of its simultaneous simplicity of notation and complexity of ideas and work to instill an appreciation of the subject in my students: a love of math for math's sake. I understand that many students will not become math professors, and so also strive to pull math out of a vacuum, recognizing it as a tool in other disciplines – from chaos or geometry in art, to differential equations in physics to logic in philosophy or law – and thus preparing my students to become mathematically cognizant of their respective discipline. Teaching is my opportunity to share my enthusiasm with students who have yet to find that beauty in math, and so will be my most important duty as a college professor. My experiences in teaching as an undergraduate, my research into teaching, and my graduate teaching experience have each molded my pedagogical philosophy and given me practical ideas for applying my philosophy in the classroom. I was lucky enough as an undergraduate mathematics major to lecture for College Algebra and Business Calculus for four semesters at the University of Tennessee at Martin. The students were varied in both their mathematical backgrounds and goals, forcing me to sift through the mathematical jargon - which was by then entrenched in my vocabulary - to find the building blocks that each student needed to create his or her own mathematical foundation for future classes. The students spanned the spectrum in age, in time since they had last had a math course, and in their course and career goals. To address everyone's needs in the course, I surveyed the class regarding their goals at the beginning of each semester, tailored examples to fit their disciplines, and created an open, questioning atmosphere in which students were comfortable saying "I don't understand." Teaching these 'remedial' math classes was very important to my education as a teacher, because mathematics had always been easy for me. Teaching in Tennessee helped me to understand the thinking process of students who do not immediately grasp the concepts. This has provided me with a better intuition of what material will be difficult for students, helped me to think like a non-math person and taught me how to anticipate questions. Graduate school is a place for students to learn math, to learn how to teach themselves math, and to learn how to create new math through research. Rice University is unique in that there are opportunities for graduate students to learn how to become effective professors as well as researchers, and I have sought out three in particular. The Mathematics Graduate Teaching Seminar provides graduate students the opportunity to present mock lessons to other graduate students and receive immediate feedback regarding one's quality of board work, speed of the lesson and examples the speaker chose for the lesson. The seminar also brings in faculty members to give lectures which have covered topics like how to write a good calculus test, how decide on homework problems, and how to choose a textbook for a math course. My six semesters in this has been a helpful introduction to the nuts and bolts of teaching, but I have sought out other pedagogical
opportunities to enhance my abilities as an educator. I have been certified through the Rice Teaching Workshop, which is a series of lectures from professors campus-wide, that addresses issues of being a university professor that are irrespective of the discipline. Topics have included: how to deal with cheating, how to spot it, and how to set up your class to minimize occurrences; how to include undergraduates in research; what is a professor's role as a mentor to students; and how to be active in the university through committees and clubs. These lectures have brought to light administrative details of which many grad students are unaware. Finally, I enrolled myself in a course through the Department of Education, Theory and Methods of Teaching Mathematics. This course is intended for students who plan to teach high school math in Texas, but was beneficial to me as well. First, it was a reminder as to what students are learning in high school, and the ways in which they have been learning math. I noticed the differences in high school math textbooks and college textbooks, and was reminded what subjects students should know (and not know!) when they enter college calculus. Second, I was reminded of the intimate level in which high school teachers interact with students by engaging students throughout the lesson, and pulling questions out of the struggling students who are too timid to ask them. Third, I was reminded that many high school lectures are done inductively as opposed to deductively to keep the students involved and thinking, rather than just robotically taking notes. Fourth, high school teachers are constantly assessing students' understanding of the material, by walking around as students work on a problem in class, to calling people to the board so that students can learn from their peers, to giving quizzes and tests more frequently than twice a semester. Finally and most importantly, I was reminded of the role of the teacher as model mathematician, and the impact of singling out students to say, "You are good at this. Have you thought about majoring in math?" Teaching as an undergraduate, Graduate Teaching Seminar, the extracurricular pedagogical guidance I have sought out, and being a student of mathematics for 22 years have each contributed to shaping my teaching philosophy. Whether the class is remedial or advanced, high school or college, the same principles apply: you learn math by doing math. To be able to do math means first thinking like a mathematician. Thus, I always start with definitions and examples. Definitions provide a common vocabulary, allowing one to speak intelligently about the subject. Examples, the backbone of any math course, give each person a mental file cabinet of pictures or diagrams which illustrate the definitions more concisely and precisely than words often will allow. Already, I want the students to be doing math, so I may ask of them: "What does it mean for a function to be continuous?" Guiding the students through to the correct definition of continuous may be more time consuming than just stating it on the board, but the loss of classroom time is made up for in two ways. First, I have found that the students have a better understanding of continuous and are less likely to forget it when they have been asked to put some thought into creating the definition. Second, students are empowered when they realize that they are doing higher math. I structure my classroom time to maximize understanding and retention of the material by carefully considering the progression of the class, and by having the students use each other to learn. The progression of math throughout a class period is important, as it should
walk the students through increasing difficulty and increasing levels of abstraction of the material. To this end, I believe it is imperative to develop an interesting and informative pyramid of examples. Before each class, I want to see that my examples each do what I intend, that the numbers work out like I intend, that my progression of examples is in the right order, that I do not have redundancies, and that I prepare the students for the homework. I also write each example completely and correctly, so as to illustrate how good math is written, and so that note-taking is beneficial for the students. Additionally, daily homework problems should be thought out in the same way: a logical progression, increasing in difficulty and abstraction, allowing the students to utilize and reinforce what they learned that day in class. Students also learn form each other, and probably more so than they learn from me. It is important that they see other students work examples and problems, so I encourage group work on homework, and I want students to present problems at the board to other students. The way I set up board problems is unique to each class, depending on the size and willing participation of the students. As a graduate student, I was able to incorporate these philosophies into my classroom when I taught Calculus 101 in May 2006 for 3 weeks, meeting 3 hours every day. At the beginning of each class, I asked each student to write-up a specific problem from the homework at the board, and then each student presented her solution to the class. If a student had not been able to do the problem, we worked it out together at the board, and then the student presented it to the class. Having the students present forced each one to write math and talk math. The class asked questions, as did I, to flesh out the technicalities of the problem, and I also used this opportunity to have them edit their notation at the board. Other students saw common mistakes that they themselves might have made. I did not collect homework problems, and did not assign a large number of problems, but rather a smaller problem set that hit home the main points, and progressed sufficiently in difficulty. I then presented three sections of the text in about 2 hours, laden with examples that explained the techniques of each section and teased out the difficulty, all the while getting questions and answers from the audience. I ended each class with a 'quiz,' an ungraded set of problems that I gave the students 10 minutes to work on which served as a means for me to assess how much they had understood from the day's lecture, and address any issues on which I had not been clear. This was an immediate assessment and feedback for the lecture, and in a manner similar to the homework, I had each student explain a problem at the board. It allowed me to walk around the room and answer questions, and the quizzes made the students think, with questions like "In your own words, explain what a limit is." I know my methods of teaching were useful, homework and quizzes complemented each other and a lot was learned in only three weeks, based on my student's performance, and their feedback. One students commented on her class evaluation, "I think the reasons I have learned is because Karoline stresses that we teach others, which is what we do with quizzes and homework, I like that a lot, I hope we do more of it. I felt like she always tries to make sure we understand, which was extremely helpful considering the pace of this course. And I like that she considers our input helpful." Another student bullet-pointed some things that she liked about my teaching:
"Explaining concepts clearly and thoroughly, especially with the use of graphs/other pictorial representation that put the new math into context with what we already knew "Explaining the background behind types of problems, why we were doing them, practical applications, etc. "Being very open to questions after explanations." The students and courses I have taught have varied greatly from undergraduate lecturing to teaching as a graduate student, but the ideals I teach by work across all: start with examples and definitions and let them do it themselves by setting them up for success with the necessary tools. Even though mathematics is my world, I understand that it may not be my student's world. For those that are not math majors, it is important that mathematics is not a separate discipline on a pedestal, but rather that they understand the workings of math as a tool in the context of their discipline. For all math students, majors and nonmajors, students should come away with an understanding of how rich and diverse the field is. Students learn by example, so each class will be an example that the problems are exciting and challenging; that we do math as a means to an end in everyday problems, but also that the process of solving, of flexing that mental muscle, is something in and of itself to enjoy.