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I always read about the Moore method with great enthusiasm. Somehow I always felt that it should be how we do it in an ideal world, but it is impossible to use because of time and other constrains.

The most important point is that users of the method claim that the students may have less factual knowledge of the material, but their understanding and the deepness of their knowledge pays for everything: they can easily master any missing part afterwords when needed. This seems to me impossible to do with a fixed syllabus.

The wikipedia page, however, has an impressive list of places where this method is still in use. Do you use the Moore method? Are there (preferably online) resources helping to learn how to do it as an instructor? How do you overcome the problem that your students have to master all the material cited in the syllabus at the end of the semester?

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    $\begingroup$ I fear that to pull this off successfully requires somebody intimately familiar with the material (much more than your average teacher), so to set up themas to be handed out and direct the ensuing discussions. $\endgroup$ – vonbrand Mar 23 '14 at 22:19
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    $\begingroup$ Related mathoverflow.net/questions/12070/… $\endgroup$ – Willie Wong Mar 24 '14 at 12:16
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    $\begingroup$ Moore Method, or IBL, may be more suited for some courses than others. I teach an Intro to Proofs course that uses IBL with, I think, great success. (In particular, I have confidence that the future teachers in the room really understand what a proof is. That seems to me to be important.) $\endgroup$ – Jim Hefferon Apr 19 '14 at 22:08
  • $\begingroup$ Just testing, just testing...... $\endgroup$ – András Bátkai May 6 '14 at 20:30
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The Moore Method is alive and well, and so are a great many variants. These days the community is more likely to use the term Inquiry-Based Learning (IBL), because the Moore Method can be seen as a restrictive set of practices, and people use the same underlying ideas in a lot of different class structures.

If you want to learn more and meet people who are using IBL in college classrooms, I suggest the following resources:

1) The Academy of Inquiry Based Learning has information about workshops and obtaining mentors

2) The Journal of Inquiry Based Learning in Mathematics has sets of course notes that have been class-tested and peer-reviewed.

3) Each summer there is a conference called the Legacy of RL Moore Meeting. This summer we meet in Denver on June 19-21, 2014. (Full disclosure: I am program co-chair for the upcoming meeting.) This meeting is sponsored by the Educational Advancement Foundation and the Mathematical Association of America.

4) Many of the larger meetings of the MAA (MathFest & JMM) have had sessions devoted to IBL teaching in the last few years. This is the case for at least the next year.

As to the meat of your question: I use IBL in all of my courses in some form. This includes proof-based courses for majors (where what I do looks a lot like a Moore method classroom), down to classes for liberal arts students who are required to take one quantitative reasoning course (where my approach is similar in spirit, but very different in practice).

Right now, I am learning to do IBL in a Linear Algebra course. It is definitely more challenging to figure out what to do in a course where the list of content goals is long and the focus is not on writing complete proofs. The key insight I have so far I got from a mentor (Ed Parker): find the appropriate level of rigor. Basically, you have to answer the question: “What counts as a convincing argument for students at a pre-rigorous stage of development?” For this term, I am experimenting with the idea of building examples or non-examples as a replacement for formal proof writing.

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    $\begingroup$ Is there anyone using this at the community college level? in developmental math courses? $\endgroup$ – Kara Nov 20 '14 at 5:48
  • $\begingroup$ I think the answers are "yes and yes." Drop me an email and i can try to connect you with people doing these kinds of things with those audiences. $\endgroup$ – TJ Hitchman Nov 20 '14 at 6:51
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    $\begingroup$ I'd be really interested to know how this Linear Algebra course turned out, a year later haha. Any willingness to share your notes? $\endgroup$ – Zach Haney Apr 10 '15 at 1:40
  • $\begingroup$ I am happy to share, but my work is far from done. I am easy to find on the internet. A search for me name and linear algebra should get you close to what you want to see. $\endgroup$ – TJ Hitchman Apr 10 '15 at 1:45
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The Moore method is used at the University of Chicago in some sections of "Honors Calculus", which is really an introductory real analysis course for top incoming freshmen. I assisted with it a couple of times and taught it on my own once. It absolutely depends on having a well-constructed sequence of notes to use; we started with a truly excellent set of questions that develop the topology of the continuum from three or four simple axioms and I was amazed at how successful it was. Designing such a set of notes does require intimate familiarity with the material; teaching from them only a little less so.

As for covering material, the class in question is a whole year long (a three-quarter sequence), and although we spent a lot of time in the fall quarter on topology and learning to write proofs while the other sections of Honors Calculus were zipping on through limits and derivatives, we pretty well caught up to them in winter and spring. Perhaps we didn't go quite as deeply into the winter and spring topics as we might have, but the students were so solid in the foundations that I didn't worry about their ability to pick up anything we missed.

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I took a number theory class at University of Cincinnati that was taught using a modified Moore method. The class size was pretty small, and credit-wise it was an "upper level mathematics elective" (i.e. not a core class), so our professor had some room to be experimental.

Like with the Moore method, the students presented all course material. The instructor sat in back, asked questions every so often if the student was not clear enough, helped us if we got impossibly stuck, and participated in the discussion when he had something insightful to say. We rotated teaching between three groups of two or three people. Our presentations were allowed to last multiple class sessions.

The class differed from the classic Moore method by loosening the requirement that students prove all theorems themselves, to address the time/constraints issue. We did have a book, but it was a computer science oriented text that left large holes in the proofs (or omitted them completely). We had to complete proofs at least well enough to field questions from each other and the professor, which often forced us to create our own lemmas. If anyone couldn't prove something at the board, the whole class had to finish it for homework. We were also encouraged to come up with our own examples, and add in whatever cool applications, algorithms, or observations we could find. Among other things, I ended up proving Chebyshev's theorem (the one leading up to the prime number theorem) after a whole bunch of lemmas, at a depth which I never would have otherwise encountered.

It turned out that most students learned the material from their own sections very deeply, but did not understand others' presentations as well as their own. I guess that's inevitable. Another complication was that some students were better at communicating their ideas than others. I bet that this method would work particularly well with a group of motivated students that had some previous practice working together. In any case, I would say I learned more in this class than other traditional classes at around the same level, so I support this adaptation of the method. It could use more development, but at the very least, I don't think it turned out worse than standard instruction.

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