I really enjoyed doing final projects with the kids this year (which may be patently obvious considering that this is my 6th post on the topic). It’s such a fun way to end the year, seeing them get excited about doing something interesting with Calculus and coming up with ideas about math that I never would have even dreamed of.
But projects can also be very frustrating, and hard to implement. Here are the things I struggled with this year. I’d love any feedback or tips.
- Since these projects were very open-ended, some students felt a bit lost, and I struggled a lot with how much guidance to give and similarly, how much to let them struggle. I just find it so hard when we have such a short amount of time to see them getting thrown off in a crazy direction, especially if it’s going to lead them to a lot of useless work. I tried so hard to “be less helpful” but I just couldn’t resist sometimes! Part of me feels like I am stealing a bit of a learning opportunity from them and part of me feels like I am just advising them to help guide their crazy teenage thought process. Also, some students just started working on their projects without really knowing why they were doing what they were doing (they just wanted to do “something about optimization”). I wanted to help them do something for their idea without turning it into my idea, but I’m not sure how well I did at that.
THOUGHTS: I think that I am going to try to have them submit proposals next year where they present some sort of thesis, or a guiding question they are going to answer in their project. This might get them to plan out their project a bit better before starting, give me a chance to give good feedback and also give them an overall question which will really guide their whole project.
- One thing that I was continually frustrated throughout the week in class that I gave them to work on the projects was that students did not work very efficiently, leaving much of it for the end. Part of it was that they just had so much time in class, but part of it was that I have no idea how to help them structure their own project to use class time well. I had tons of students show up without materials to work on their projects, and even some who would sit there and do nothing telling me that they were just going to finish at home.
THOUGHTS: I wanted to do a midpoint deadline of some sort, but because all the projects were so different, it seemed really weird to me to organize something like that. I might try having them make a schedule in the beginning of the project, but I’m not sure how to help them stick to that, or if that is even worth all the work that it would be.
- Similar to supporting them in organizing their time, I struggled helping them work well together with each other. I think group work like this is crucial in high school to learn how to structure time with someone else and communicate about a project, but the students were terrible at this. They would do things like not show up to class without telling their partner, even though they had all the materials. I even had to mediate an email war between two girls who were flipping out at each other about who was doing less for the project.
THOUGHTS: Maybe this isn’t something that I need to do something for, and maybe this is something they just have to learn by doing the project, but perhaps I could find ways to help them structure their roles in the project beforehand, or maybe just do more long-term projects like this over the course of the year.
- Last, I really want them to show off their work to each other, but I’m not sure how to make class presentations anything but the boring yawn fest that they tend to be. Students did some really cool things, but were really bad at explaining those things in a way that the class could understand. Also, it’s really hard to listen to two full class days of presentations, even for me, and it’s really hard for students to get anything out of the presentation when they are not really expected to engage in a meaningful way (not one of the presentations was interactive in any way).
THOUGHTS: I’m looking for some sort of other structure to make it more interesting. Maybe some sort of gallery walk type structure? And I also want some formal way to get those listening involved so that they really pay attention and learn – some sort of commenting system, or interactive component. It’s very hazy in my head, but this is something I am going to try to flesh out over the summer.
Any ideas would be greatly appreciated!
(Also, below is my rubric for grading these projects)
A student’s mother is completing the Hajj this year, the pilgrimage that Muslims take to Mecca. This is one of the five pillars of Islam (along with prayer, fasting, charity and testifying that there is only one God). All physically and financially capable Muslims must carry out this pilgrimage at least once in their lifetime. This student based her whole project on the Hajj and calculated many different things about it. Specifically, she calculated:
- How long it would take to complete each part of the Hajj (once you get there, there are certain rituals during which the pilgrims walk to various places). She used aerial photographs and official information to measure the distance (around 40 km!) and then used an average person’s walking speed to estimate that each pilgrims walks for around 10 hours during the Hajj.
- How many people can be expected to attend the Hajj in the future given data from the past 10 years and assuming exponential growth. She used previous data and the basic exponential growth model to make predictions for the next 30 years.
- How large the current area around the Kaaba is (the holiest site of Islam around with the Hajj is based). She used GeoGebra and Google Earth software to measure the area.
- And how much the area will have to increase in future years to accommodate the extra pilgrims. Based on her predictions of the increase in the number of pilgrims, she mapped out how big the area around the Kaaba will have to be for the pilgrims to all have the same amount of area. She thought it was cool they they would have to restrict the number of pilgrims, or knock down highways in order to keep the area per person the same.
The math wasn’t perfect and there were some crazy assumptions made, but I absolutely loved this project. It was from someone who had told me in the beginning of the year that math wasn’t her thing, and it was really cool to see her get excited about the project because it applied to something really interesting. All the math was very well motivated and taken from a wide range of things that we did this year. Great stuff!
NEXT YEAR: I could see doing some sort of city planning project involving Google Earth that somehow involves population growth. It would be really cool to look at current rates on population increases in areas and see what that would mean for the physical space. I am so happy that a lot of these final projects have translated into great teaching ideas!
For their final project, one group decided to make a twitter account and track how many followers they gained over time. The account was called “UknowURatKings” (King’s is our school… so YOU KNOW YOU’RE AT KING’S for those who hate txtspeak). They tweeted inside jokes about the school that you would only really get if you were pert of our community. I was following them, which was good because they ventured into inappropriate territory once (it was a nice mini experiment in social networking with students!). Here was my favorite tweet of theirs:
They had predicted that the followers function would follow a logistic model. Using a few data points, they created a logistic model of their own: they thought they would max out at around 100 followers (the size of the senior class population on twitter plus some extras), they originally told 13 people, and after one day they had something like 40 people (unfortunately, I can’t find where they uploaded their project ahh!). Based on that they created their logistic model. Then, they tweeted furiously for about a week and recorded how many followers they had each day. At the end, they compared their results with their model…
They were way off. Though they had chosen the right model, the number of followers increased slower than they thought and maxed out around 60, not 100. My favorite part of their project was that they didn’t try to fudge their numbers or make the data fit their model – instead, they talked about their assumptions that may have been flawed, their tweeting behavior skewing the results, and inconsistencies in data collection. I ❤ data.
NEXT YEAR: I thought that this was a really fun and simple project, and it might be something that I try to do with my whole class when we study exponential models next year (I swear I could teach a whole term on just the logistic function). I think we could have an awesome discussion about modeling with all the different inconsistencies that will arise, and we could even add a competition component, to see who can get the most followers for their account under certain constraints… Too many ideas, too little time.
**The next few posts are going to be spotlights of final projects that students did that I thought were cool or interesting and then a few reflections on doing final projects in general. I could picture doing a lot of my student’s projects as a whole class!**
If I had one more week in my non-AP Calculus class, we would study volumes of revolution. That’s probably the biggest weakness of my course right now, and I am trying to figure out a way to include that next year. A junior who is in my regular class and is taking AP next year was a bit lost when coming up with an idea, so he asked me for a topic that we do in AP but did not do in our class so he could be a bit prepared. I suggested volumes of revolution and after a lot more nudging and guidance and idea planting than I did for other students, we decided that a good project for him would be to recreate an interactive 3D model of a solid of revolution using GeoGebra and Winplot. (actually it works with solids of known cross section too).
Here’s how it works…
1. Upload a picture into GeoGebra (he chose a huge vase from the art room). Fit functions to the edges of the object on the part that will be revolved.
2. Recreate the same exact functions in Winplot (which has much better 3D capabilities than GeoGebra does).
3. Use Winplot’s revolving capabilities to revolve the surface around an axis (any axis!). And then, voila, you have a 3D model of your object that you can use the arrows on the keyboard to rotate in any direction. It actually ends up being really impressive – my student told me that he left the model up on his computer and every time he would turn it on he would rotate his vase a bit.
After I saw the success of this project, I suggested the same one to a few students in my AP class (who were required to do a much more low key, shorter version of a final project because of time restraints). They decided to recreate a bunch of sports equipment using the program, which I thought was a really cool idea! Their rotate-able objects:
NEXT YEAR: I made an instruction sheet for those AP kids because they had less time, but I’m glad I did because this was a really cool project and is something that I can see myself doing with a whole class next year. Here it is below. If you haven’t tried making any 3D models (not necessarily real objects) with Winplot, definitely try it out – it’s super cool!