As I have previously mentioned, I am really excited to be creating Udacity style videos as resources for students in my classes. With my VideoPress upgrade in effect, I have included some of the two minute videos I put together for Calculus class tomorrow introducing limits. We have already spent some time exploring the concept of limits by graphing and evaluating numerically, but these videos are the start of a more formal treatment of evaluating limits algebraically.
I am very interested in feedback, so let me know what you think in the comments.
Excellent quality! I think I would enjoy being one of your students. I really like that you stress that you are making a definition at the end and that your students should refer to it when answering a question about the new terminology. Of course it is always possible to nit-pick, and you did express interest in feedback.
My nit-picks are: you did mention in the second video that we don't actually want to let x reach 3 when considering the limit as x approaches 3, but the comment really just slips in there. This is soooooooo essentially, and causes soooooo many problems for calculus students. I can ruefully recall dozens of less successful students trying to plug in c in their final exam "limit as x approaches c" problem. I don't think you can overdo how much this should be emphasized, and I think you could emphasize it more in this video, especially since the example they are to do is a function that is not defined at the point. (Of course, if your goal was indeed just play down this issue so that students would discover it and own it for themselves when doing the problem, then that's different --- even then, I think the point needs to be made again and again...and again.)
In the first video when you have the sliding rectangle, you say "suppose I want to be a certain distance away from 4...well that means I have to be a different distance away from 2. The closer I want to be to 4, the closer my x-value needs to be to 2." Because the cursor isn't moving back and forth between the x and y axis as you say this, I think it could be confusing to students when you say "you" want to be a certain distance from 4. Perhaps it would be clearer if you said you want the function (or the y-coordinate of a point) to be a certain distance from 4, move the cursor up by the y-axis, and pause so students would recognize geometrically what's happening with 4.
Again, these are minor --- on the whole I am really blown-away. What software did you use in producing these videos?
Hi Barry,
I really appreciate the comments – the best reason to share is to get feedback like yours! I completely agree about the point of limits being that we don't actually want to evaluate the function at a value. The students don't see why you can't just evaluate the function, a problem compounded by the fact that most textbooks introduce limits by evaluating functions that can be evaluated.
I've tried creating contrived situations where the students can't actually directly evaluate the function (Don't hit the 3 key on your calculator!) but the simpler solution is just to get students to use functions that have points where they are undefined. That gets it across the easiest. Since I greatly pared down my precalculus review this year, we spent a day exploring limits and getting comfortable with the concept of zooming in or approaching a value. They really got into it, and based on the quiz today, they understood it for the most part.
As for your point about my poor pronouns (I want to get close to 4 vs. the function) you are spot on – not something I could get just with editing. Improving the clarity is something I will continue to work on – thanks for calling me out on it, since you saw it it means it is noticeable.
I made the video using Camtasia for Mac, Sketchbook Express, and Geogebra. At the moment I'm working on reducing the total ratio of recording/editing time to video length, but that will come with practice. It is great to get such positive feedback – thank you!
In the first video, I appreciated the way you use the Epsilon-delta definition of a limit without ever mentioning the words "epsilon" or "delta", which tend to make kids soil themselves.
In the second video, I could imagine students wondering why you chose those particular numbers - the starting guess of 2.5 seemed particularly arbitrary. Granted, being familiar with the material, I could see where you were going with this, but I couldn't help but feel a student might be a touch confused. I wonder if it would have been more intuitive to start with 3 and work your way down first?
Finally, I realize these are meant to be direct instruction, and perhaps you set the stage in your class prior to assigning these videos, but could it hurt to include some sort of context for this introduction? When would a student be interested in limits?
On a different note, what are you using to record your screencast? Was getting set up difficult?
Thanks for sharing!
Hi Dave,
I was scared off when I did epsilon/delta notation myself learning Calculus - I didn't have a good sense of what I was actually doing at the time, and it seemed like a very arbitrary exercise. I'm glad you saw how I avoided it here, hopefully well enough to get the idea across without doing too much.
I have already exposed students to the trick of getting close to a value of x to evaluate a function, but the video focuses on the concept of approaching in a more methodical way than I have previously shown them. I sat with my finger ready to record for a long time wondering whether it would make sense to go the other way, but my rationalization became this: then I wouldn't be approaching a value, I would be running away from it. That made less sense.
The context for the limits has been touched on in a couple places - we've already looked at local linearity and zooming in to a graph to find slope near a point, and at area under a curve using rectangles. In both cases we've talked about getting a better and better approximation, and how the value for slope or area does seem to approach a value as we get closer. The idea for limits actually came after both of these activities, which is different from how I normally do it. I'm really happy with how it went, but agree that even more context within the video (and reminding the viewer about those concepts) would make it nan even better piece.
I made them using Camtasia for Mac, Sketchbook Express (for the background/writing) and Geogebra for the graph. THanks so much for the good ideas!
Best,
Evan