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Description:
This was a 15-minute lecture I made for mock teaching. The goal of the lecture was to introduce derivatives in calculus.
At the beginning of lecture, I asked students what a slope is and how they would calculate it. (This is a safe question to ask, because slopes as a concept get pounded in students' heads in grade school.) After that I told them to imagine that Madison is organizing the next winter olympics, and alpine skiing will be organized on Bascom Hill. To make sure that Bascom Hill is suitable for alpine skiing, we need to know its slope. So students were instructed to imagine that they are the ones responsible for measuring the slope (using any tool or method they preferred) and to form small groups to discuss their ideas. After about a minute we discussed some ideas as a class. In almost all cases, the method was based on the definition of the slope (rise/run), where two sample points were taken: one at the top of the hill and one at the bottom.
The story then continued. Much to everyone's sadness, a meteor hits Bascom Hill. Fortunately nobody is hurt, but a large crater is created, making the hill unsuitable for alpine skiing. However, the organizers have the idea that maybe it will be a good place for ski jumping. The idea is that skiers reach the bottom of the crater with high speed and use that momentum to jump on the top of the Historical Society building at the bottom of the hill. The important question is: is it possible for them to do this? Obviously, nobody would want skiers to fly into the side of the building. One thing one needs to calculate for this is the slope at the point where the skier detaches from the ground. The next challenge for students in small groups was to figure out how they would do this.
As everyone guessed it right, the key is to use the slope formula, but with sample points that are much closer to each other than before. And what if it is still curvy between the points that are already close? Well, we just take the points even closer! And this got us to the limit definition of the derivative: limh->0 (f(x+h)-f(x-h))/2h.
A wrong definition, since this formula would mistakenly define the derivative to be zero at a sharp peak. So next, I pointed out what a better definition was, and we ended there.
Reflection:
According to my fake students (none of whom was an expert in mathematics), the lecture was very engaging and entertaining up until I got to the formulas. They liked that the lecture was interspersed with small-group discussions and that students had the opportunity to share their ideas. Their complaint with introducing the formulas was that they appeared suddenly from nowhere, so it was hard to relate them to the story. The notes seen above are actually for a revised version of this lecture, where I have already added notations like x, x+h, x-h to the pictures of the stories, hoping to make the connection from the story to the formulas smoother.
I am not certain how effectively this lecture would teach students the concept of the derivative in a real classroom. Some entertainment in a lecture certainly helps grab their attention, and hopefully students will remember a story they can relate to derivatives. For a more thorough understanding, however, a lot more additional practice is needed.
Delta pillars:
teaching as research: I gave this lecture twice during the College Classroom class with Nick Balster and Christen Smith. I got feedback the first time, and I made changes to the lecture based on this feedback to make improvements for the second time.
learning communities: small group discussions during the lecture to allow students exchange ideas.
learning-through-diversity: several groups got to share their thoughts with the whole class, thus enriching the discussion with diverse ideas.
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