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This is very similar to a question I asked at MSE a while back ago but I think it is far more appropriate here. https://math.stackexchange.com/questions/632139/examples-of-open-ended-calculus-class-project-ideas. However, I'd like to throw in a twist. I want to emphasize writing about mathematics (or applied mathematics), so I want them to write a paper. In order for this to work, I'd like the open ended project ideas to be less like harder/more involved HW questions than what I got on that site, and more like open ended projects where the students really have some freedom in where their research takes them.

Has anyone tried requiring a final report/paper for a calculus class? If so what did you ask them? and how did it go over in general? Would you do it again?

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    $\begingroup$ I'd like to hear ideas for this, too, especially for a calc course geared towards business and social sciences. $\endgroup$ – Alexander Gruber May 7 '14 at 20:05
  • $\begingroup$ I don't have a specific example, but what I have in mind is an open-ended question that can be solved using various calculus tools, but which isn't fully specified. So the students would have to make and justify assumptions or decisions about how they were going to approach the situation, and write about those justifications, as well as the calculus they did once they had turned it into a solvable problem, and then talk about whether the results they computed actually seemed suitable for the original purpose, etc. $\endgroup$ – PurpleVermont May 12 '14 at 23:22
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I have offered papers for a variety of classes. For most students, the idea of writing a math paper is completely foreign to them. As a mathematics educators, we should encourage the learning of all of the mathematics out there, and I have taken this past opportunity to offer historical papers. I think it would otherwise be difficult for mathematics students to write a mathematically rigorous paper on the mathematics that they are just learning now.

Depending on the level of students, I have required different lengths of papers. Below is a full copy of the assignment I have offered to have them write on, though I leave open the option for them to find other topics.

Please write a (minimum) five-page (double spaced, 1 in margins all around, 
Times New Roman 12-point font, APA formatting) paper emailed to me by 11:59PM,
May 16. List all your sources. Each of these topics below has more than 
enough material to fill five pages, but there are lots of other topics that 
can also be written about. You must confirm your topic with me before you 
begin.

One possible outline:

•   Introduction: Give a general overview of the subject
•   Body: Explain the topic. Explain why it is important. How does it related
 to anything in the world today?  What other fields is it connected to?
 Expand on one (or more) particular idea, and go over it in detail.
•   Conclusion: Explain what you learned doing the research. Was there 
anything that was familiar you came across during your research?  List all
sources used. What was the most interesting thing in the research to you?


Examples of Extra Credit Topics:

The number e, its history, and uses
The number π, its history, and uses
The Fibonacci numbers
Magic Squares
Prime numbers
Numeral systems
Complex Numbers
Perfect Numbers
The Mandelbrot Set
Inductive Proof
Pigeon Hole Principle
Modular Arithmetic
Computus
Adrien-Marie Legendre
Carl Friedrich Gauss
Joseph Louis Lagrange
Augustin Louis Cauchy
Jean Baptiste Joseph Fourier
Bernhard Riemann
Peter Gustav Lejeune Dirichlet
Archimedes
Euclid
Rene Descartes
Leonhard Euler
Georg Cantor

Outcomes: It has been a few years since I incorporated this in a class, but as I recall, most students chose not to write a paper. Many expressed that they thought it was impossible to write 3-4 pages on a mathematics topic. There were some who chose to write the papers though. Those who did, saw how much was out there, they were amazed at how much they had to cut out.

Did Students Learn Things? This is difficult to quantify from my assignment. The main goal I was trying to accomplish was to have my students learn that there is a lot of other math out there; that it's not just pushing numbers around. I would say that those students who took the assignment seriously did see that there was at least some other aspect of mathematics.

What would I do differently? After first giving this assignment, I found that many students were drawn to write about the well-known mathematicians. After reading 3 papers on Newton that provided nothing new for me to learn about him, I eliminated him, along with a few others perhaps (I think Einstein may have been on my original list).

Would I do this again? I just gave this to a remedial class the other day. When I receive the papers, I will post on more results. I will continue to offer this as an extra credit assignment for appropriate classes until I find something more valuable.

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    $\begingroup$ I wonder how one avoids the student simply looking up the topic in Wikipedia, and summarizing that article? $\endgroup$ – Joseph O'Rourke May 11 '14 at 17:09
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    $\begingroup$ You may want to include some women on that list. May I recommend Ada Lovelace (mathemetician and creator of the first computer program more than 100 years prior to the first computer ever built). Other options are Emmy Noether, Sofia Kovalevskaya, Sophie Germain, Maria Gaetana Agnesi. $\endgroup$ – WetlabStudent May 12 '14 at 7:47
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    $\begingroup$ @MHH, thanks for the names! I'll add them to my list for next time. $\endgroup$ – Andrew Sanfratello May 12 '14 at 18:34
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    $\begingroup$ Also searching for famous Asian, African, or South American mathematicians might be good. Just having a name on the list is good for promoting diversity even if no-one chooses them. $\endgroup$ – WetlabStudent May 12 '14 at 18:40
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    $\begingroup$ @MHH See also matheducators.stackexchange.com/q/1817/262 for women mathematicians. (The list includes all those who you mentioned.) $\endgroup$ – Benjamin Dickman May 12 '14 at 20:40
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Not my own experience, so I can't comment on how it went (and therefore this doesn't really answer the question), but Gavin LaRose at the University of Michigan has a page of Calculus writing prompts on his website.

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Perhaps this type of question could work, supplemented with some structuring guidelines...?

Where and how is calculus used in epidemiology today? Please try to find explicit examples, and explain the use of calculus in these examples.

And then replace "epidemiology" with: {genetics, exoplanet search, drug delivery, atmospheric pollution monitoring, oceanography, stockmarket predictions, global warming, ...}.

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    $\begingroup$ -1; if you actually did this, then please edit to talk about how it went, and whether you would do it again. $\endgroup$ – Chris Cunningham May 12 '14 at 0:13
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    $\begingroup$ @ChrisCunningham: I have not actually attempted this. I have never taught calculus---I must be rare in that regard! Which is why my first word was "Perhaps"! $\endgroup$ – Joseph O'Rourke May 12 '14 at 0:29
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    $\begingroup$ @ChrisCunningham Does this answer differ from the top-voted one in some meaningful way? The other answer has a longer list and an assurance to provide more info on the outcomes later, but otherwise these are both lists of final prompts for a Calculus class. (What I think both are missing are the supplemental "structuring guidelines" JO'R alludes to, and more details than just a topic name). $\endgroup$ – Benjamin Dickman May 12 '14 at 0:36
  • $\begingroup$ I agree that giving a specific application topic helps focus, and prevents just copying text from elsewhere. Just be careful that material is readily available, of the right level. $\endgroup$ – vonbrand May 14 '14 at 2:41

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