April 21, 2005: Prof Alex Hahn - Summary

Prof Alex Hahn, Director of the Kaneb Center for Teaching and Learning, spoke in today's seminar. According to Alex Himonas, Professor Hahn has worked very hard to improve the teaching of the math department. Professor Hahn reciprocated Professor Himonas' comments by indicating that Himonas has injected energy into the department with respect to teaching.

Professor Hahn began with the idea that if you think you are great in the classroom and that there is no room for improvement regarding your teaching ability, you are probably not so great at teaching. Teachers should always reflect on teaching - they should always strive for improvement. Professor Hahn's goal today was to get us thinking about how to teach when the time finally comes that we are called upon to do so.

Professor Hahn once overheard the statement "You haven't taught until the students have learned". This statement was made on a program about the legendary basketball coach John Wooden. Wooden's players all felt great esteem for their coach as he was somebody who cared about them from head to toe and taught them all he could - not just about basketball, but about life in general. In many ways, teaching is like coaching - there is more to it than just book knowledge; it is a matter of how the players play the game. In the case of teaching, it is not just a matter of teaching them the fundamental techniques - there is much more to any given subject than just the calculations.

According to Professor Hahn, it is NOT the case that, when teaching, you perform while the students behold (with admiration) your technical prowess. Rather, it is a matter of getting your students active in learning - get them to do the math and they will learn.

After his opening remarks, Professor Hahn handed out the first activity, which involved the audience individually (and shortly thereafter, collectively) brainstorming ideas of what we think science and engineering students should learn in first semester calculus. In other words, what should a student know in order to receive a stamp of approval that he or she "knows" first semester calculus. Here's a list of some of our ideas of what students should understand and/or be able to do:

• Limits - proficiency in computation as well as some exposure to epsilon/delta techniques.
• Derivatives - standard, partial, temporal, etc. (How to do them and when to use them.)
• l'Hopital's rule.
• Integrals - technical ability, application, and concepts.
• The fact that \infinity is not a number, and sort of how it behaves.
• Fundamental Theorem of Calculus.
• The idea that differentiation and integration are opposites (see the proof, apply the concept).
• Convert real-life situations into mathematics and then do the math.
• A little culture - how the broader world of math (outside of calculus) works (theorems, proofs, etc.).
• Given some idea or situation, some concept, how can you make math out of it? In general, the move from intuition to precision. After all, that is often how math research is done!
• Students should be able to understand you (the teacher). If you are concerned about this, contact Noreen Deane-Moran in the Graduate School. Professor Himonas pointed out that students will often overlook your accent if you convince them that you are trying to help them.
• It is the journey, not the destination (like the Odyssey). Who cares if you make it all the way to Ithaka - it is the journey that matters most. Some audience members thought that this idea might better suit courses for non-scientists since scientists often need the material they are being taught.
• Plant mathematical seeds that can be cultivated later, as necessary.
• Students should be able to articulate (at the board) exactly the ideas behind the math they are being taught.

Following this exercise, we worked on another. The goal of this one was to detemine how to teach to elicit the desired effects listed above. Here are a few of our ideas:

• Give candy every time a question is asked to get them to ask questions.
• Repetition, repetition, repetition. (not to be too redundant).
• Think about what they should be able to do - problems at the board? problems on their own? conceptual arguements?
• Have games (like the infamous derivative races). For this, a friendly, open atmosphere is needed.
• Assign homework problems whose origin the students don't know (so they don't know which techniques to use).
• Mix problem types so that the students need to keep thinking.
• Involving students is very important - remember the Atlas complex article! Student involvement is particularly well-suited for conceptual work.
• Discussion may not be allowed to become too wishy-washy or philosophical - it is important to stay on track.

By the time we finished these discussions, the period was about over. Professor Hahn handed out a sample set of learning goals from an MIT physics course that makes use of PRS's (personal response systems). These are zappers (like remote controls) that students can use to signal their responses to questions. Suppose you are giving a lecture on calculus material to a 400-person class and want to know how they are following. With this system, you could write a problem on the board with some multiple choice answers and have them dial in their answers - instant feedback! There is a small fee associated with these systems, but you can track all of this information semester-long (so they are good for attendence too), making it possible to pinpoint where individual students are having difficulties.

Professor Hahn closed by encouraging us to keep aiming towards the Ithaka of good teaching - keep trying to improve! Also, he mentioned that it is a good idea to break the ice early in the semester with some easy material that is good discussion fodder. He used this technique to loosen up one of his classes, and they have performed very well.

Finally, Professor Himonas reminded us that there is one more seminar this semester (click here for details). That will be next Thursday (4/28) at 11 a.m. and will cover teaching with technology. The department will begin a pilot program involving exactly this topic (via Professor Qing Han) next fall, so this is a good seminar to attend for everybody!

(reported by Dan Bates)