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Lectern lessons: a prof and a GSI share their classroom wisdom

The Professor: Steven Vogel, Political Science

Vogel is one of four faculty members honored in June with the 2005 Faculty Award for Outstanding Mentorship of GSIs. His statement of mentoring philosophy (adapted here) appears online at gsi.berkeley.edu/awards/mentor.html, as do statements from other recent winners.

The real secret to being a good GSI mentor is to have great GSIs. Beyond that, I rely on a few basic practical strategies.

I try to give my GSIs lots of information at the outset of the semester: lecture outlines, study questions, class announcements, exam guidelines, exams, and teaching evaluations from previous semesters. I also give them memos from previous GSIs for the same course, which address two questions: How could I (the professor) improve the course in future years? And what do future GSIs need to know to perform their best as instructors in this course?

By asking the GSIs to write these memos, I achieve several goals at once: 1) I improve my own teaching, 2) I allow my current GSIs to benefit from the experience of the previous GSIs, and 3) I signal to the GSIs that I take their input very seriously.

I try to convey to the GSIs that I view them as full partners in teaching the course. I consult with them in advance on issues that affect them directly, such as the scheduling of paper assignments and class debates. I tell the GSIs that they should feel free to challenge me or to raise questions in class or in online discussions.

I encourage but do not require the GSIs to give a guest lecture. I allow them to choose the topic and the date, and whether to use an entire class session or just part of one. I make sure to give them suggestions before the lecture and feedback afterward.

I visit the discussion sections early in the semester, and then meet with the GSIs to discuss teaching strategies. I make it clear that I need their input as much as they need mine.

With good communication and mutual respect, the teamwork tends to fall into place naturally. Good GSIs make teaching easier for the professor, and a good professor makes teaching easier for the GSI. I emphasize that I see the GSIs as partners in a joint enterprise, and I try to convey this attitude to the students as well as to the GSIs. I try not to ask the GSIs to do too many administrative tasks, because this could undermine the spirit of partnership.

I discuss the division of labor with the GSIs from the outset. I do not view the professor simply as a lecturer or the GSIs simply as discussion leaders. I incorporate discussion into the "lecture" sessions, and the GSIs integrate some teaching/ lecturing into the "discussion" sessions. To improve coordination between the two, I try to give the GSIs advance notice about what material I plan to cover, and which discussion questions I plan to raise in the lecture session. But I also tell my GSIs that some repetition from lecture to discussion can be useful. Normally, the students should encounter important concepts/ topics three times: first in the reading, then in lecture, and finally in the discussion section.

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The GSI: Aubrey Clayton, Mathematics

Clayton is one of 15 recipients of 2005's Teaching Effectiveness Awards. To read recent essays submitted by GSIs in departments ranging from rhetoric and music to physics and mathematics, visit gsi.berkeley.edu/awards/index_year.html.

Mathematics is ultimately nothing more than a language of human ideas, but these ideas are expressed in an extremely concise and oftentimes apparently indecipherable code. As a result, many Calculus 1B students are frustrated by the amount of rote memorization they think is expected of them. Indeed, many textbooks reinforce this misconception by presenting much of the material without any motivation. So, for example, it's natural for a student to feel overwhelmed by the definition of the limit of a convergent sequence of numbers: "A sequence s(n) is said to converge to a limit L if for all ε > 0 there exists N such that n > N implies | s(n) – L | < ε." My goal was to show the students that this definition is actually very natural and intuitive, and thus shouldn't be dismissed as another useless fact to memorize.

My strategy was to start with a problem and get the students to arrive at the definition without having seen it before. So, I challenged them to convince me that a concrete example of a sequence, the sequence 1/2, 3/4, 7/8, etc., was "tending" towards the number 1. Everyone agreed that it was, but no one could construct a persuasive argument. "What exactly does it mean to 'converge' to 1?" I asked. The conversation quickly focused on the differences between the sequence terms and the limiting value, and the class pointed out that these values were getting progressively smaller as the sequence continued. The difficulty, however, is that this value never reaches zero, and so the putative limit never actually appears in the sequence. The students struggled to find a convincing argument that it should still be the right answer, but without a precise definition, it became clear that there was no way to be sure of their claim, or even make sense of what their claim meant. This was an important moment in their understanding of the material.

To get them started, I asked a student with a calculator to compute some more terms in the sequence as I copied them onto the board. The critical event occurred when after a few terms the calculator could not distinguish between the sequence terms and the number 1. We soon agreed that this phenomenon would happen no matter how precise a calculator we bought, since they would all have some round-off error, however small. This was the convincing argument we had been searching for. At last, the test we came up with was that given a fixed round-off error, no matter how small, it was the case that eventually all the sequence terms were within that error of the limit. And this is exactly the definition of limit, just written out in words rather than symbols. Most students, once I showed them the book's definition, were amazed at the similarity.

In addition to learning about the need for a precise definition when making an argument, the class learned to make sense of the symbols in the book by rephrasing them in their own words. Also, they learned that mathematics actually uses a lot of common sense, even if it is sometimes apparently obscured by notation. As a result, I saw a dramatic improvement in their ability to read and understand proofs, and the students were confident and excited to learn, rather than frustrated at the prospect of a whole semester of memorization.

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