The Time I Chickened Out… [dealing with fears of lessons not working]
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?
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Posted on April 6, 2012, in Calculus and tagged fundamental theorem of calculus, integration, lesson design, lesson planning, odometer, Riemann Sums, speedometer. Bookmark the permalink. 11 Comments.
Bowman in Arabia 
What if you eased them into it? Do a video of constant speed for a certain amount of time to get them thinking about how it works (and deal with conversions and the like). Then do one where you go a certain speed for a certain amount of time, then quickly change to another speed and hold that for a bit, then maybe change a third time and hold. Then finally use this one.
Anyways, I say you just go for it at some point. If they go in the wrong direction, you can not-so-gently nudge them back in the right one. Or the “worst” that can happen, you jump in with your original explanation. Together, you WILL get there, the only real thing that worries me about these lessons is the amount of time it takes to get them there on their own.
Wow, perfect suggestions, thank you! You’re right, the whole time consuming part is one of my biggest worried, but I think this one could be totally worth it (I mean the foundational underpinning of the integral deserves some time!). Definitely going to use your suggestions to scaffold the lesson w videos and make it completely student focused. Thanks!
>when I tried to explain what the dx signified
I’d like to hear more about that.
I’m less courageous than you, so I don’t have any answers to your question. I’ll be looking forward to the comment thread.
Well trying to explain the whole idea of adding up the vel x time at every single instant… The way i try to do it is by connecting the set up of a Riemann sum to the notation of the integral and using that to motivate why we need it (if the v is constantly changing we need to use smaller and smaller intervals but it’s always going to be an estimate until we use an infinitely small interval… dx, the limit of a Deltax)
I also like the idea of using a graphing story for motivating concepts in calculus, and will likely use one for integrals just as I did for derivatives. I’ll use this video: http://www.youtube.com/watch?v=bVDwAOgbc6s&feature=results_video&playnext=1&list=PLD593525B85261D94
because it starts of nice and easy with constant speed and then becomes more complicated. That way I’ll ease them into it, but without using several videos.
All in all I think there are three components to understanding integrals, one has to do with why we’d want to find areas under curves. Another is that you can find them by adding infinitely small somethings (rectangles, trapezoids, etc), and the third is that you can find them by anti-differentiation. I think the graphing stories approach can target at least the first two of these components, and perhaps the third as well since students should connect that if speed is derivative of distance then distance is anti-derivative = area under curve of speed. I’d like best to find/make a graphing story that has constant acceleration so the speed fits a linear function. Then the students could find the distance traveled in x seconds and see that it is the anti-derivative of the speed. Awesomeness would surely follow.
Regarding those thoughts, however. I don’t see why you’d want to get rid of the doubts. Anticipating problems is part of good lesson planning, and the response should be to plan your reactions to those possible problems. Sometimes we don’t have the time or resources (computer lab would be nice) to deal with likely problems, in which case the best move is as you did, to change the plans into something perhaps less stunning, but at least more manageable.
thanks for your thoughtful comments again, I really appreciate them. I noticed you have a blog but haven’t updated in a while – do you still blog somewhere else? Dave brought up the idea of scaffolding with an “easier” video and I think you guys are both spot on with this. That’s very helpful, and I really like your idea of doing it all in the same video. I’ll just hop in my car again!
Oops, now that I’ve updated the link in my name (decided to go with blogger rather than wordpress) you’ll see it’s a bit more updated. 🙂 I even have a post on graphing stories in calculus recently. Thanks for pointing it out!
Thanks for your reflection on the effectiveness of the teaching (not knowing how much they truly understand) and how you are planning your teaching. I felt the same way when I was lesson planning – trying to be innovative and creative but also wanting things to be successful.
One thing that came to mind for me was that you could have let them grapple with it individually, then maybe in small groups, and monitor how things were going, keeping your full class discussion of the problem only if they needed it. That way you would get to see their preconceived notions about velocity and cars and speedometers, which they need to understand before even getting to Riemann. Besides, if we always coach them through the messier problems, they will end up thinking that all problems have a right approach and a neat answer that someone else already figured out.
So my opinion is to go for the messy albeit more nerve-wracking route, with a rescue plan in your back pocket!
The comments on this post have been so encouraging… you have made me realize that the plan I used should have been the “rescue plan,” as you described it, in the first place. I’m definitely going to take the leap with this one next year.
I have my students do a project leading up to this lesson where they go as a group of four and record their speeds every set amount of time(ten or twenty seconds). And they record their odometer as well. Then I have them make a graph and do a left and right Riemann sum. They mostly realize they need to convert units when they see they’re not getting the same distance as their odometer said. I give them over Christmas break and a couple weeks after so that they have plenty of time to figure out a time they can all get together. I have a worksheet that has all the requirements and they present their findings in class. Having this project in their past experience really helps when talking about the integral as used to accumulate a rate. I talk about it all the time when doing rate or velocity problems.
Your project caught my eye because I was thinking of doing a follow up lesson of taking a video of the school parking lot and then my speedometer while I drive to a familiar nearby place. The obvious question when I stop the video is where am I? This would involve them doing similar work and probably using google maps.
Thanks for sharing your video.
great idea on the project – I have a bit of a tough time getting my students to do independent work as it is, so not sure that would be the best route for me, but I would love to see the worksheet you give them if you are willing to share! bowman.g.dickson@gmail.com