Statement of Teaching Philosophy (2006)

sarah-marie belcastro

My teaching is careful, considered, and energetic; I think through every aspect of my teaching, continually try to improve, and approach my interactions with students with gusto. Here I describe both my teaching philosophy and current teaching practice. (This unfortunately makes the document rather long!) The content falls into three sections: pedagogy (active learning), assessment (both of students and of faculty), and education of the whole student. More details on how I developed my teaching philosophy and what exactly happens in my classroom are availablee in the text of a presentation I gave to the 2004 Project NExT cohort.

Pedagogy

My guiding pedagogical principle is that students learn most deeply and retain concepts most completely when they participate actively in the learning process and engage with the subject material. I encourage my students to construct as much knowledge as they can on their own, then help them progress when they have reached their limits. I also help them learn how they can construct knowledge that is new to them. In having high expectations of my students, I keep pushing them to learn, and they work hard in response. As a result, I am rarely disappointed. For example, while I do not expect my gifted high-school students to know definitions, they can prove almost any theorem given enough definitions. The process requires a great deal of interaction with students, which takes place in various settings including collaborative learning, class discussions/presentations, and email.

Among my expectations is that my students will read the text that I assign. They quickly learn that this is a real expectation: I do not lecture over the assigned material, so in order to keep up with class, they must read the text. To make this a realistic expectation, I am very careful to choose only the most readable of texts. (Three such choices are Marsden’s Elementary Classical Analysis, Gallian’s Contemporary Abstract Algebra and Crauder/Evans/Noell’s Functions and Change.) Furthermore, because the coverage of material in my courses is then usually quite text-dependent, I make sure that the curricular philosophy of the author(s) matches my own.

I change the role of lecture depending on the level, topic, and size of my class. In small Real Analysis classes, I do not lecture at all and instead direct the students in discussing the text and associated problems. In a Linear Algebra course for math majors, I prefer to deliver short lectures-on-demand during class when students express confusion on particular concepts. In large service courses, I begin each section with an interactive lecture on an illustrative problem.

I do not spend the majority of class time at the board. Instead, I like to have the students work in small groups, and to have the groups report frequently to me and to the whole class. In many classes, I ask the students to solve interesting problems posed in the text. While the students work, I circulate among the groups. I check to see that each group is on-task, coax them to explain their current reasoning to me (and thus to each other), and ask them leading questions which are likely to help them make progress.

One of my favorite approaches to mathematics is to ask students to examine examples and make conjectures, or to investigate conjectures made by their peers. This is a natural part of my upper-level classes. In order to extend this approach to all levels of student, in 2004 I designed and began teaching six inquiry-based courses at the general education level. The topics include graph theory, topology, higher-dimensional geometry, and the mathematics in The Number Devil.

I also encourage collaborative learning outside the classroom. In lower-level classes such as calculus, I require that all homework be done in a structured group setting. Not only can the students help each other (allowing me to assign more conceptually difficult problems), but they are assigned roles to help them reflect on their learning processes. As they take on different duties, the students concentrate on different aspects of the collaboration. In beginning proof classes such as linear algebra, I often require both individual and group homework, and in upper-level classes such as real analysis, all homework must be written up individually. However, I always advocate that students work together on individual homework.

A benefit of collaboration which may not be apparent to the students is that their mathematical and technical writing skills improve. When students are required to work together, they must discuss the write-up format so that the person producing the final draft will know how to accurately reflect the group’s conclusions. I grade the writing quality in order to nurture this process; while at first it may be nearly unreadable, group homework invariably becomes well-written. When students are working on proofs together, the topic of how to properly explain their reasoning often arises even if they need not collaborate on the writing.

I think it is very important for students to learn to effectively communicate mathematics, because no matter how well a student has learned a concept, this knowledge is meaningless unless s/he can communicate that concept to others. To this end, I not only give careful feedback on the writing of homework, but also encourage students to share their thoughts in class. This encouragement takes many forms: in some classes, I require formal presentations either of course material or of independent projects; in other classes, students report on in-class work or on simple homework problems done in preparation for class; in still other classes, a discussion-style format allows the students to regularly practice both informal and formal communication of mathematics.

Many of my courses are supported with technology, either in the form of graphing calculators (TI) or a computer algebra system (Mathematica). While the students may be using technology heavily, I do not emphasize it in class. I assume that the students will use technology when necessary to complete calculations, or to generate examples; it is a tool to expedite mathematical processes, not a substitute for knowing how calculations could be done by hand.

All of my courses are supported by two particular forms of technology: electronic mail and world wide web pages. I encourage students to contact me by email at any time; as I try to be logged in whenever I am in the same room as a computer, I respond quickly. I also make a web page for each course on which I archive handouts, post assignments, and list relevant resources. Furthermore, I use class email lists to send my students various types of information (career fair details, test anxiety workshop announcements, assignments of students to groups). Often students unconsciously gain practice in communicating about mathematics when they respond to mass emails with specific questions about homework problems.

Assessment

Because my classroom is not dominated by traditional lecture, the assessment techniques I use are appropriately modified. At the end of 2002, for example, I searched the educational literature to find ways to assess significant classroom participation in mathematics and found an excellent rubric which I have since successfully incorporated in many of my classes.

My courses focus on concepts, and so my homework sets and exams concentrate on applying these concepts and explaining how they are used. In upper-level courses, I use take-home exams in lieu of in-class exams. (In mid-level courses such as linear algebra, I generally use both in-class and take-home exams.) Take-home exams help the students learn more while being assessed, and help to lower the level of anxiety associated with assessment.

Often I require a paper or project in order to give the students the opportunity for a more extensive writing experience than a typical homework problem, as well as to allow me to assess their writing in a context where they have had a chance to write and revise (and thus improve). In some classes, these papers have substituted for a final exam.

Just as assessment of students can be both formative and summative (the students can use the feedback to make changes, while faculty can issue a report on achievement), so can assessment of faculty. In this vein, I have used two assessment techniques to improve my own teaching. One is customized evaluation forms, administered at mid- or end-of-term. The other is Small Group Instructional Diagnosis (SGID), in which a trained facilitator first observes a mid-term class period, then uses the end of class to solicit consensus feedback from the students, and finally has a feedback session with the instructor. I always reflect on the feedback and report back to the class. Evaluations of all forms are useful to me in planning future courses, but SGIDs are uniquely helpful in improving an ongoing course. I have learned to administer SGIDs, and am happy to do them for colleagues.

Educating the Whole Student

Finally, I would like to emphasize my strong commitment to the total education of undergraduates. Teaching a wide variety of mathematics courses appeals to me, and I have tried to teach as many different courses as possible. Interestingly, I find that working on my research improves my teaching as well. The experience of learning, struggling with and creating new mathematics echoes the challenges my students face, and thus I empathize with student stress. In 2004, I worked with an undergraduate student to produce original research in topological graph theory. I’m always on the lookout for subproblems and subprojects which would be appropriate for students—nontrivial, yet tractable and requiring minimal background. I would also like to teach occasional courses in other areas within my purview (examples: ballet, feminism and science).

I like to get to know my students and support them in other areas of their lives. For example, I try to attend their art exhibits and music performances. Other courses, campus life, and personal life affect not only overall student development, but also performance in my courses; because first-year students face academic and personal transitions, I try to attend to this in courses aimed at first-years. At the end of most terms, I have students over to my house for snacks, puzzles, and chatting. I have always lived close enough to campus that it is easy to be on campus in the evening. (This also means that students can easily find my house!) I have enjoyed my role as advisor, formal or otherwise, for individual students and for student groups. All these aspects come together in my summer work with talented high school students, where I live with the students, interact with them for more than ten hours each day, and advise some of them for years afterwards.

A friendly and fun atmosphere nurtures students academically and personally, and the resulting energy nurtures faculty in return. I am always excited to be with my students and work hard at making our interactions valuable for them. This often means providing a nontraditional and dynamic classroom experience, in which I ask the students to become highly involved in the construction of their knowledge. In turn, such pedagogy leads to alternative methods of assessment which are tailored to specific course structures. Both my pedagogy and assessment choices generate a great deal of contact with and feedback from my students. This improves my teaching, and the circle is complete.