I had this idea, and I wanted to throw it down before I lost it. It may be nothing, or it may be something awesome.
I have been mulling over if I should do a project in calculus in the fourth quarter. And I had a thought. I have been really trying to focus on the fundamental underlying ideas in calculus, and shooing away the algebraic gobblygunk. Why? Because my kids aren’t taking AP Calculus. Most won’t be taking math in college. So I want my kids to leave calculus saying: “Yes, I understood the ideas. Calculus is about ideas.”
I wonder if a good final project, which would force them to grapple with the Big Ideas, might be having students create a collective time capsule, which will be stored in some deep underground facility, and will be the only remnants of “Calculus” that may exist after some horribly apocalyptic disaster.
I’m not sure what would go in the capsule, but I like the idea that all students would be asked to contribute a few items. Maybe we’d break the course into chunks, and each student would be responsible for writing an accessible explanation of each chunk — and we bind these together into a book? And each student would create set of drawings/graphs/photograph/images that (for them) represent the Big Ideas of Calculus, and they have to explain each one of them… What’s the idea, and why is it so important?
In addition to these required items, students could have their choice of what else to contribute… Things like:
1) A video of the student explaining the weirdnesses/paradoxes/strange ideas of (or relating to) calculus
2) A short research paper on the history of calculus
3) A letter to the future explaining why calculus is an important swath of knowledge that shouldn’t be forgotten (including uses / applications of calculus)
4) A challenging calculus problem, and it’s solution
5) A “concept map” for calculus
6) Audio recordings of students reading quotations about calculus that resonated with them, and then students explaining why it resonted with them.
7) Designing a cover to the collective calculus book we bound together, and on the back cover, an explanation of how the cover exemplifies the course
Or other things?
I don’t know. It felt like a cool idea when it jumped in my head a few minutes ago, but now that I’m writing it, I can’t quite picture it … yet. Any ideas of how to take this idea and turn it into something good? Throw it in the comments!
Continuous Everywhere but Differentiable Nowhere 
I think this is really interesting and it’s a good take on the the classic sort of build your own review assignment. But how do you make this really serve as a time capusle? Could you have next year’s class open it on the first day of school and send responses to the seniors you created it?
When you mentioned the book idea, I immediately thought of book Math Curse http://www.amazon.com/Math-Curse-Jon-Scieszka/dp/0670861944. I’m at middle school, and my 6th graders loved my reading this book to them, then each student wrote and illustrated a page of their math curse and we bound it into a book. You could do a “calculus curse” book.
I like the idea of saving the “book” for the next class. I’m also thinking you could make this into a collaborative project, using complex instruction techniques with group roles, etc. For example, you could have one group do a page (or “chapter”) on rules on using derivatives to graph functions. One student in that group could write/draw about the first derivative, another about the second, a third about relating first and second to each other, and together they could write a summary about what derivatives teach us about functions (maybe include a procedural summary of how to solve problems based on the ideas in that chapter).
I like your time capsule idea, but when you mentioned putting it together into a ‘book’, I thought they might like to try their hand at explaining to other calculus students who are struggling now.
This seems especially cool in light of recent discussions of how badly publishers are doing math texts these days: http://open.salon.com/blog/annie_keeghan/2012/02/17/afraid_of_your_childs_math_textbook_you_should_be
What about creating a time-capsule for aliens, as if you were trying to explain calculus to someone who does not understand our language? You’d have to explain the concepts all through pictures, and you’d have to avoid using numerals as well. No easy feat. The Voyager space-craft has some examples of our mathematics that is intended for aliens to be able to explore.
Don’t forget to laminate the pages! We wouldn’t want the papers to decay post apocalypse. :) I like the idea!!!
For me, I would want every kid to do a skeletal book for every part, and then to choose ONE part to really flesh out in details (with a partner). This can ensure that they each learn ALL the big ideas but are really responsible for explaining / showing off one part!!
I love this idea of synthesizing the course by stripping away all of the technicalities and focusing on the big picture in creative ways. Your apocalyptic scenario seems fun, as does the idea of a children’s book (or struggling student reference).
I have some ideas about the project overall:
Since this is to be a cumulative project/assessment, I think you need to think a little more closely about what it is you want your students to understand and whether the project truly gives an accurate picture of that understanding. If you want them to understand the big ideas of calculus, then I’m not sure that focusing on one or two ideas per student will demonstrate an understanding of the big picture…. although certainly the end product has the potential to do that for someone else!
Coming to my mind immediately are two possible solutions:
1. Include some sort of analysis/synthesis aspect at the conclusion of the project – like, We’ve got 30 ideas, but we’ve just found out that the capsule can only hold 8. How can we combine/condense these into 8 overarching themes? or, Write a preface and table of contents that summarizes these ideas and organizes these pages with the best possible flow.
2. Give the task of coming up with a complete capsule/book/film/whathaveyou with ALL the big ideas. You can do this individually or in groups, but ultimately you will see a student’s (or group’s) understanding of the entire course and not just one or two ideas.
I’m sure whatever you come up with will be great, and I hope to see some pictures of the the final results!
Following up on dwees, I really like the idea of communicating without words. There’s a long history of this in math (http://www.cut-the-knot.org/ctk/pww.shtml and many more), but I’m also thinking of the directions for FLL (FIRST Lego League) building instructions. With their materials used around the world in areas for which a total of over 100 different languages are spoken, they provide directions using pictures only. Amazing. Can it be done here?
I was just checking out TED’s new TEDedu channel and immediately thought of you and your class project. How about the ultimate task of designing a TED lesson about the big ideas of calculus? You could start with individual chunks and incorporate them into a cohesive unit with some group evaluation. Here’s the link for idea submissions: http://education.ted.com/lesson/