So I get all these ideas for problem solving lessons, but a lot of times I struggle with pulling the trigger to try it out. I get excited about it, but then I hem and I haw, I get worried that it wont work, I debate about the implementation and fret about the logistics (planning class is tough when you have an intense type A personality but also a creative streak).
And then wonder if the thing I’m teaching could be better taught in a more straightforward manner.
So this is a time when I chickened out. To connect Riemann Sums with the physics of velocity and displacement so that the introduction of the integral is a meaningful and motivated as possible, I wanted them to use a video of my speedometer as I drove around a well known circle at our school to calculate the distance I drove. This is a modified idea from somewhere on Real Teaching Means Real Learning (though I can’t find the original post that inspired it, bah!). The video is not all that interesting but here it is:
I wanted to give them a video and let them just struggle through the task. They know that distance is velocity times time, so I wanted them to sense why this simple equation is far more difficult when the velocity is constantly changing as a motivation of why an integral is so important. I wanted them to get the idea that in order to calculate it we would have to split the trip up into much smaller segments, like a Riemann Sum, and that we could make it more accurate by doing the time at smaller and smaller intervals, but it would always still be an estimate until we did some sort of limiting process.
But then, I got worried. What if I asked them to bring computers and they didn’t? What if they got so hung up on the km/hr to m/s conversion that they couldn’t focus on the other stuff? What if they did something crazy instead of a Riemann Sum type thing? What if they couldn’t figure it out at all and we wasted a class? These are the questions that I get hung up on all the time with trying to implement #WCYDWT and #anyqs type instruction (though this is certainly different because I was asking a specific question).
So, instead we watched the video and talked as a class about the difficulty of the task with the constantly changing velocity (which meant I have no idea for how many students this really sunk in). Then I gave them this to help them solve the question:
We practiced drawing and calculating an applied Riemann sum with this, and used units to discuss why the area under the velocity vs. time graph. I think most students came away with an understanding of at least the idea that area under a velocity vs. time graph gives you displacement, but I don’t think they had the deep understanding I was hoping for, and especially not the deep motivation for integrals (which I could really tell when I tried to explain what the dx signified). It was more efficient, sure, but perhaps less effective…
…but importantly for my lesson planning was that I knew that with the scaffolding that it would work, but I wasn’t convinced it would work otherwise. How do I escape those thoughts, especially with 30 some odd teenagers staring at me for guidance every day and a tight yearly plan?