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I am looking for resources for designing undergraduate mathematics classes that are not lecture-based. (Bonus points if the design is for an introduction to proof course).

For example, Robert Talbert blogs about flipped classes (calculus, and intro-to-proof). Do you know of others?

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    $\begingroup$ How often do your courses meet? It goes without saying that most k12 courses are not lecture-based, and besides for culture it seems like one of the big barriers facing college instructors moving away from lecture is amount of instructional time. $\endgroup$ – Michael Pershan Mar 14 '14 at 17:22
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    $\begingroup$ MTWT and MTWF for 50 min each day. $\endgroup$ – David Steinberg Mar 14 '14 at 17:50
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    $\begingroup$ The AIBL has a list of course resources for Inquiry-Based Learning (IBL/Moore Method). $\endgroup$ – François G. Dorais Mar 14 '14 at 19:20
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    $\begingroup$ One well-studied alternative to lecture classes is to use the Moore method (though this might only be suitable for advanced proofs-based classes). I have no personal experience with the Moore method, but there is quite a lot of literature on it. $\endgroup$ – Jim Belk Mar 14 '14 at 21:56
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    $\begingroup$ there is some math that can be taught via hands-on computers/computer experiments etc, so called "empirical mathematics"... its very good for statistics & monte carlo simulations etc ... have not seen a lesson plan but think there must be some out there.. $\endgroup$ – vzn Mar 27 '14 at 17:10
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I think you will definitely want to check out the The Journal of Inquiry-Based Learning in Mathematics. It is dedicated to being a "resource for designing undergraduate mathematics classes that are not lecture-based."

Another place to start that is overwhelmingly huge and mostly aimed at the K-12 level, but definitely has a lot of good ideas that are applicable to this question, is the huge community of math teacher bloggers. Here is one place to start that exploration.

I also second Michael Pershan's recommendation to examine the PCMI problem sets. They provide a magnificent model for what it looks like to develop content through problems, and it's a little different from much of the stuff you'll find at the Journal of Inquiry-Based Learning, which is coming out of the Moore Method tradition.

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There are a few ways to do this. I recently did it in Carnegie Mellon's introductory proofs course. I'll try to outline here a few different solutions that various places have tried.

Carnegie Mellon (15-151): We alternated introducing new topics (usually via lecture) with what I called workshops. A workshop is during normal lecture time. Students are given 3-5 questions that range from conceptual questions to difficult proofs. We created the groups randomly, and we changed them every few workshops. We gave them whiteboards (you can buy big sheets of whiteboards at home depot for like $5 and have them cut them down for you) and markers, and asked them to work together on the proofs in order. As the lesson progressed, the TAs and I would walk around the room answering questions and reviewing proofs. Depending on how many resources you have this could be very difficult, but we decided to split the large lectures down into smaller ones so this was feasible.

MIT: Albert Meyer in 6.042 takes a different approach. The classes are still smaller, and I believe there are still "lab assistants" running around the room helping out, but he does the flipped classroom every lecture. He's done it in several different ways over the years. I believe originally, he gave mini lectures for the first 20ish minutes, and then the students worked for the rest. He might have moved to asking the students to watch the lectures at home--in a true flipped classroom format--now. One other thing to note is that the book Albert wrote is absolutely fantastic and completely open. It's a great resource for good problems, good explanations, and expository material for students to read.

Stanford: I can't remember the course number or name right now. I will do my best to follow up later by adding a link. Stanford takes a different approach than either of the other two. They have a standard lecture series, and sometimes, they offer workshops in the evening (which I believe are optional). This approach is particularly nice in the college setting, because it requires fewer resources and less contention for classroom space. Both of the CMU and MIT courses used a "special room" which allows for collaboration. At CMU, 15-151 is held in the CTC, and at MIT, 6.042 is held in the TEAL rooms.

There are other various strategies to do this, but these are the ones I've seen most commonly (and successfully) used for introductory proofs courses. Albert's resources are very useful, and I would be happy to share some of mine as well.

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  • $\begingroup$ That sounds great. As a rule, the more mathematics you can help your kids do, the more they'll learn! $\endgroup$ – Michael Pershan Mar 14 '14 at 17:46
  • $\begingroup$ I found the whiteboard group activity to be pretty helpful when it can be feasibly resourced. But I would hesitate to use it all the time in a course. (Hi Adam) $\endgroup$ – Brendan W. Sullivan Mar 15 '14 at 1:54
  • $\begingroup$ @brendansullivan07 Yeah, the resources issue can definitely be a problem. I think, in the future, I'll do the workshops in the evening possibly. (Hi Brendan!) $\endgroup$ – adamblan Mar 15 '14 at 1:59
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I really think that the Park City Math Institute's courses are fantastic, both as resources and as models for what it could mean to have students spend their time in class solving interesting problems.

They run 3-week courses in the summer, often on a topic relating to number theory. For example, check this set of problems out:

http://mathforum.org/pcmi/hstp/resources/course2009.html

Or, more generally, here: http://mathforum.org/pcmi/hstp/resources.html

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Currently, Jim Fowler and I are running a hybrid linear algebra/multivariable calculus/multilinear algebra course at

http://ximera.osu.edu/course/kisonecat/m2o2c2/course/

It is totally based around completing activities, and discovering mathematics by doing mathematics. Check it out!

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