How I teach topology: an inquiry-based learning approach

Recently I’ve had a number of people ask for more information about how I teach topology. I’ve taught it five times using a “modified Moore method” or “inquiry-based learning” approach. I’ve modified it each time, trying to work out the bugs. I think it is pretty successful now.

Context. At our college all math majors are required to take a 400-level class. Typically it is Complex Analysis or Topology (they are taught in alternating years). Thus, while topology is not a required class, many math majors have to take it in order to graduate. Some of them find it extremely challenging. On the other hand, I often have very talented students in the class. In fact, often the juniors (or even sophomores) who take the class are among the best students. Typically I have 8-15 students in the class (depending on whether we offer one or two sections). The other important piece of information is that the college wants the 400-level course to be writing intensive. Many departments have a “senior seminar” that fills this role. We don’t, so that’s an extra reason to teach topology in the way that I do.

Brief description. In this course the students do not have a textbook; in fact, they are forbidden from using outside sources at any time. Instead, they are given the skeleton of a textbook. It has definitions, statements of theorems, some explanatory text, and some problems. They must prove the theorems, solve the problems, and type their work into the empty textbook. By the end of the semester they have created their own textbook. I have it printed and bound at the college’s print center and they take it home with them. (I like to have 5-10 people contribute to each textbook, so if the class is on the large side I have them create two textbooks in parallel.)

Textbook. The textbook I use came from the topology course that I took as an undergraduate—from Dick Bedient at Hamilton College. I’ve modified the book quite a bit over the years, but I couldn’t have done this without the excellent book I began with. It began as a Word file, but I moved everything over to LaTeX years ago. I maintain two copies of the book—one has space for the students to write in (like a workbook) and the other has no spaces (this is the version that they edit). I have the college’s print center print the workbook one sided (in case they need to write on the backs of the pages), three-hole punch it, put it in a binder, and sell it in the bookstore (for the cost of printing only). Sample pages of both versions are shown below.

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Homework. Each class period I tell the students what theorems/problems they should work on. It is a guessing game on my part—I have to estimate how far we’ll get in the next class period. They do the work in the workbook at home. They are free to work together, they are free to talk with me, but they are forbidden from looking at outside sources.

Presentations. When I first taught the class I had students volunteer to present the proofs/problems in class. The problem was that only the best students volunteered. So I moved to a system in which I assigned (randomly) presentations to the students. Now the students know ahead of time what “their problem” will be. If a student comes to class unprepared or presents an incorrect proof, then I take volunteers to complete the proof (or the student can stay at the board and repair the proof with the help of the rest of the class). This is also a time to talk about how the proof is written. They know that what goes on to the board will ultimately go into the textbook, so the class often gives wording, notation, or organization suggestions. (At the beginning of the semester I talk about how to give appropriate feedback. We don’t want hurt feelings.) Often the weaker students are nervous about presenting a solution that is wrong, so I tell them that they are free to come to my office in advance to “check out” their proof.

Daily sheets. The students are required to participate in class. Each day I bring the sheet below and keep track of who presents and who makes comments. I also leave myself notes, like “great proof!” or “volunteered to present the theorem when Billy was sick” or “came to class 15 minutes late.” (I number each column by the theorem/problem number.) I use these sheets to help assign a class participation grade at the end of the semester. (Excel file)

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Green pen/homework. At the end of class, the students turn in the pages that we just discussed. I grade each problem on a simple V (very good work), G (good work), L (little work), N (no work) scale. Because we talked about each problem at length, I do not write any comments on their work. One great idea I had a few years ago was to require the students to bring a green pen to class. During class they can take as many notes and make as many corrections as they please, BUT they must use the green pen. This makes my grading especially easy. I can see exactly what they wrote at home and what they wrote in class.

Secretary. Each day one student is the “secretary”—this position rotates through the class. (If the class is broken into two groups, there will be two secretaries—one for each textbook.) The secretary is responsible for writing down everything clearly, taking note of all editorial suggestions. The secretary must type all of the new content into the textbook before the start of the next class.

Textbook. When I was a student the textbook was a MS Word file that lived on a 3.5″ floppy disk. The students would pass it around as needed (this was in 1992—before the web, and at a time when I had to go to the basement of the library to check my email). When I first taught the class the textbook was a Word file that students emailed back-and-forth. Later I turned it into LaTeX (which meant there was a whole directory of files). At first I had the secretaries edit the files on a dedicated computer in the math major’s research room. The last two times I taught the class I placed the files in a shared DropBox folder—the students could edit the book at home or on one of the school machines.  That worked very well, but not flawlessly—a few times we had problems with files being forked. But we’d come a long way. Next time I teach the class I plan to use one of the new online LaTeX websites. I think this will be fantastic—collaboration will be a breeze and the students won’t have to install LaTeX to do their work. (Note: I’ve had my students use LaTeX in many of my classes—by and large they seem to really like it. I made this LaTeX cheat sheet which helps flatten the learning curve a little.)

Editing. The textbook is broken down into chapters. At the start of the semester I create “editorial teams” for each chapter. After we complete a chapter (and the secretaries have typed in the proofs), the editorial team is responsible for “cleaning it up.” They do that, then I take a pass at it, making my corrections and suggestions (hand written on the pages), and then the editorial team implements my changes. If there is a weakness in my system it is at this stage. The editorial team does not make many changes on their own. I think they are uncomfortable editing the work of a classmates. The draft I get generally needs a lot of work. I need to strategize about how to improve this. (Suggestions? Leave them in the comments!) There is also an editorial team that is responsible for the entire book. They coordinate the editing of the full book at the end of the semester.

Images. The book requires many images—open and closed sets in \mathbb{R}^n, the torus, the Klein bottle, the Möbius strip, etc. I have yet to find a good drawing program that is easy to use, powerful enough to produce good images, and free (or cheap). I’ve tried many different pieces of software and have had mixed results with all of them. I’d be happy to hear suggestions for what to use.

Exams. I either give two midterms and a final exam or three midterms and no final exam.

Peer evaluations. I have the students evaluate each other (and themselves) at the end of the semester. I use this information to help me assign their class participation grades. I’ve found that sometimes students who are quiet in class will get rave reviews from their classmates (“she was always extremely helpful with editing,” “he was very good at explaining the concepts to me,” etc.). On the flip side, some of the more “active” members of class could be bad collaborators (“he never showed up for our editing sessions,” “her secretarial work was never done on time,” etc.). Here is an example of the evaluation sheet. (Excel file)

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Final textbook. At the end of the semester, after the textbook has been edited multiple times, I give it a grade. The entire group gets the same grade on the book. Usually the grade is high, but occasionally (rarely) I’ll have a group that does not put in the required effort. When it is complete I quickly send it to our print center to have it printed and bound. The turn-around time is very short, so most students are able to pick it up before they leave for the semester. Typically the students are THRILLED to hold this book. They love to share it with their friends and parents. It is neat to see their pride. However, I used to have a few books weren’t picked up. So (and this also coincided with the recession and subsequent budget cuts) I now have them pay for the book (in advance)—which is usually $5, or so. This way we only print the book for students who want it (usually 75% of the class).

Final thoughts. I love teaching this class and the students (for the most part) enjoy taking it. The material is very challenging and some of the weaker students struggle with it. But I think that a faster-paced traditional class would be even worse. I think that the students really learn the material using this approach. I like that the students spend time presenting mathematics at the board. This is missing from most other mathematics classes. And the emphasis on writing and revising is very valuable. Finally, I love seeing their faces light up when they pick up the finished book at the end of the semester.

Source files. Professors at other institutions have asked me for copies of the blank textbook. Unfortunately, because I was not the original creator of the book I don’t feel comfortable distributing it. However, there is a lot of good IBL content online. For example, look at the Journal of Inquiry-Based Learning in Mathematics and the links on their website.

What’s next? I’ve enjoyed this method of teaching so much that I’ll be teaching my Discrete Math/Intro to Proof class using this technique next semester. I’m very excited about it (and somewhat nervous about it too). I’m using these notes (by Dana Ernst) as a starting point. I’m planning to do things similarly, but slightly differently, in this class than in my topology class. But that’s a topic for another blog post.


  1. Bob says:

    This is great! I wish more classes were taught like this. It would have great to learn linear algebra and calculus like this!

  2. mathematrucker says:

    Thirty years ago (1982-83) I had the good fortune to attend the undergraduate analysis sequence taught by ergodic theorist D. A. Lind at the Univ. of Washington.

    Lind’s method (modified Moore? he called it that, but I’m not sure…) went like this:

    First of all we used a textbook: “Mathematical Analysis” by Tom Apostol.

    For the homework problems (all from the textbook), we could use any sources at all, including the library and collaboration with anyone. (The information superhighway was still a decade away.)

    During class, homework solutions were presented at the board by students selected individually at random (the whole class cycled, then a new cycle would begin).

    We also handed our solution sets in and they were graded carefully by Professor Lind himself, not some grad assistant. It clearly behooved students to do all the work themselves.

    Each quarter (to this day the UW still uses the quarter system) we were given a final exam made up of questions by Lind, not the textbook – though one of Lind’s final exam questions was to find and correct an error in one of Apostol’s proofs.

    For the final exams we were not allowed to consult anything – no library, no collaboration, no homework solutions – nothing except our thinking cap. However I do remember bumping into Lind one morning down at the rec center and getting a tad bit of “Whadya Know” info out of him regarding an approach to a final exam question: he said “sounds promising”, which of course helped a lot! We had a week to solve all the problems.

    To this day I still remember the first two students called to the board to present homework problems. In order, they were: (1) UCLA math physicist Barry Merriman, and (2) myself.

    Barry was a hard act to follow. I didn’t have a full solution to my problem, but did thankfully at least have a partial one to present.

    This was the single best math course I’ve ever taken.

    1. Sounds great. Thanks for sharing! (I’ve never met Lind, but I love his book (w/Marcus) on symbolic dynamics.)

  3. Thank you for sharing this! Could I interest you in posting a link to this at the Mathematics Teaching Community h ? There is a tag there on inquiry based learning and a couple of brief posts on it.

    The Mathematics Teaching Community  is a new online community for those of us who are passionate about teaching mathematics. It’s a place where we can learn with and from each other and build a repository of knowledge to help us improve math education. Everyone who teaches (or taught) mathematics at any level from PreK through college is invited. Use the tags to search for topics of interest. Post submissions, which can be anything for or about mathematics teaching, such as activities, questions, or links to useful resources. Vote for postings that you find helpful or interesting. Further information about the site can be found in the FAQ and in postings with the “meta” tag.  

  4. Dana Ernst says:

    Thanks for sharing the details of how you run your topology class. I hope that my notes work out well for you next semester.

  5. Thank you for this. I have many powerful memories of learning complex variables & this approach would have helped me, especially as the only young woman in the class during the late 70s, early 80s. Your students are fortunate to have you as a teacher! Inquiry-based learning rocks. . . we use it in my Learn 2 Teach, Teach 2 Learn program that works to get Boston youth into STEM!

  6. Toshi says:

    Very good aproach, but unfortunately it’s not applicable to classes with many students (100+). There is only one list with those who are going to take up this class, usually consisting of a big number of students that sometimes can even reach 500. But as I said only a portion of them come in the end of the semester to take the exam, and an even smaller portion (still many though) attend the lectures. Wish it was applicable to bigger classes too.

  7. Alison says:

    I also took Topology with Bedient at Hamilton College – small world. Some other Hamilton professors have adopted this method too. I loved it, and still have the textbook we created. I made a print of torus retraction in my printmaking class for the cover, but the images are definitely the weakest part of the book. Some students figured out how to make images in Mathematica, and those were definitely a cut above.

    1. Small world indeed. After taking Dick Bedient’s topology class I took a second topology class as an independent study from Larry Knop (also Moore method). I also took an algebra seminar from Sally Cockburn that was in a similar style. Good memories!

  8. Sol Lederman says:

    Your blog readers might be interested in the podcast you and I made where you discuss inquiry-based learning and much more!

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