April 2010 Archives

U Substitution

Just last week I gave the lecture for "u substitution" to our class of first year calculus I students. In contrast, the the lecture that I gave on u substitution was a bit easier to present than the lecture for the Mean Value Theorem. By nature the two lectures are dissimilar in the sense that one is a technique for evaluating integrals and the other is something of a theoretical tool which does not get used much in a first year calculus class. Not only was u substitution easier to present but it was also much easier to prepare. I do feel, however, as someone who is new to teaching, that there are aspects of my style that need some improvement. While Barry did reassure me that my lecture on u substitution was indeed "very well done," I want to highlight some characteristics of a spring semester course in calculus I at UMM through my experiences teaching.

I'm not sure if I have made mention of the differences in mathematical maturity between students who take calculus I in the fall as opposed to those who take it in the spring, but in order to give some direction to this blog I will say that students who take calc in the fall are typically stronger than those who take it in the spring (at least this is true here at UMM). Since our calculus class is taking place in the spring, I have had the chance to witness and experience some of the difficulties that some of the students in a spring semester of calculus face. While a large part of the students can succeed, success is somewhat contingent on the presentation of the material. Barry was well aware that he need to treat our spring semester class just slightly different (with respect to how he presented the material) and he made sure I understood this. While I was aware of the students abilities, I did not quite catch on to how Barry has managed to polish his teaching style so that these students could improve in such a way that would prepare most of them to succeed in a section of calculus II.

I believe the way that this happens begins with how Barry lectures. For example, when I presented the material on u substitution I hurried through some of the tedious algebraic steps, believing that the students would follow what I was doing. And for some of the students did follow, and I feel that most who didn't follow right away could have followed if I had quickly jotted down the of the rule of exponents that we made use of. This may have taken just 10 seconds and would have ensured that all the students knew what I was doing. While Barry felt strongly about my presentation on u substitution, he made note of the fact that I should have made quick mention of the rule of exponents that I was using. While this is just a small detail that would have made my lecture more self contained, such details would benefit some of the students in our calculus I course.

For the most part much of the feedback that I received from Barry pertained to small details that, as Barry said, "Come with experience." It is my impression that Improving on these small yet important parts of my recent lectures would greatly improve the clarity and allow students to better grasp the material presented.


Preparing a Lecture

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I was given the opportunity to give our calculus class a lecture on the mean value theorem. I'd like to share the process of preparing this lecture.

As part of class assignments I've given in class presentations, most of which usually included a power point presentation. I knew teaching a small class of first semester calculus students the mean value theorem wouldn't exactly merit the same type of preparation as I had done previously for class presentations. And so with some experienced guidance from Barry I began to prepare. I started to prepare almost a week before the day of the lecture. For a while I thought I might be going overboard by giving myself that much time to prepare, however, I found that plenty of time to think about the mean value theorem beneficial. I found myself explaining to a fictional audience related theorems, such as Fermat's theorem, and thought about how clear my thoughts would have been had they been vocalized to a real audience. Eventually I started to plan what exactly I wanted to say in class with the hope that the evolution of the lecture would give way to a clear understanding.

After deciding on what I wanted to say and what examples I wanted to mention, I only needed to internalize the order in which I would present the ideas and the formal statement of the mean value theorem. I wrote down in a spare notebook the things I write on the board. All the details of what I would say to clarify the intuition behind the theorem had already been established from my pseudo-lectures in front of the fictional audience I had been practicing in front of for the past week.

When it came time to lecture I hardly needed my notebook, except occasionally to verify what I was writing on the board agreed with what I had in my notebook. I definitely had things I could improve on, such as my use of the whiteboard. While I enjoyed explaining the ideas behind the mean value theorem, Barry informed me that a bit more enthusiasm would improve the overall presentation.

Hopefully when I give the next lecture on the substitution rule for integration I can come through with a bit more charm and organization on the whiteboard. We'll see how it goes the second time around...    

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