Stuart Ambler gave a lecture that would take place in a Linear Algebra class for humanities students. He said that he thought these students learn concepts well, but may have more trouble with calculations. He began by asking students what they thought the term "linearity" meant. After getting several responses, he then described his notion of the word and compared his answer with students' answers through examples of linear real-world situations.
After Stuart's talk, there was a discussion about what kind of background knowledge he expected his students to have. If students had already taken calculus, they might have been frustrated with this lesson. It was also suggested that all examples should be worked out ahead of time to make sure they work because students get frustrated when they have to change their notes. Alex Himonas liked that Stuart got his class to really think about what the concept of linearity meant. He also pointed out the importance of coming to class early to make sure that all desired equipment is there, and he would have liked if today's session was conducted to better simulate a real class.
John Harper then gave a lecture on change of variables that would take place in Calculus III. He first gave an idea about what the term meant through a series of pictures, and then he more precisely gave the conditions required to perform the change of variable process as he had depicted. He gave an example using polar coordinates to show that this new idea of change of variables agreed with what they had already done with polar coordinates. Then he gave another example that used a change of variable to simplify a problem about integrating over an ellipse to integrating over a circle. He pointed out at the end that over the real numbers (instead of R^2), change of variables agrees with substitution.
During the discussion after John's talk, it was suggested that he write bigger, especially in pictures, so that details can be seen by all students, even those in the back. It was also suggested that he should have talked slower, and maybe he shouldn't have tried to cover so much material. The main point of conversation then turned to different philosophies on interaction with students. Alex commented that a teacher needs to make a connection with students so they don't get bored. Others thought that John's lecture was very good, but perhaps an easy way that John could have interacted with his students would have been to work in questions about material that they already should know (like what they expect to happen in the polar coordinates example). It was agreed upon that time management in a classroom is hard; it takes practice to learn how to work in questions and still get through the material.
(reported by Stacy Hoehn)