Lecture-less Modern Algebra and Foundations, Part II

This semester I decided to move away from the lecture format in my Foundations and Modern Algebra courses. I wrote about this back in March so I will refer readers to my earlier blog post for the reasons and outline of the course. That entry was Part I—the idea and how it looked at mid-semester. It seems like it’s now time for Part II—what it looks like a few days after grades have been submitted, and what I would change and keep on second pass.

THE GOOD. I would keep (with slight modification):

Required before-class reading, reading assignments, and notes. The students were truly prepared for class and ready to engage with material when they walked in the door. I sensed some irritation with the amount of work, but students more often seemed actually happy to be ready for class. It was important that I collected the assignments and checked notes. This got to be difficult as the semester went on, but I would actually be more diligent in checking notes if I did this again.

Blogs. I really worked to sell the blogs to my students by talking about the importance of being able to communicate mathematics, both to their peers and to non-mathematical audiences. Not every blog post was interesting, or even wholly mathematically accurate, but some students really hit it out of the park, and just the effort of considering how to communicate seemed to have an effect on everyone. Many students didn’t know that math blogs existed, and hadn’t thought much about how to communicate about math outside of the classroom, so when I assigned them to read and respond to some pieces by Steven Strogatz, it felt like some students saw math and its possible place in the world differently. I think I would make the posts due every two weeks, instead of every week as I first planned, and I would more reliably set aside a short time for discussing their work. I did this several times but not often enough.

LaTeX and Proof Portfolios. The students did not love learning LaTeX. Some students never became wholly proficient with it, and saw it as a burden to have to write up their work this way. However, most students came to appreciate its capabilities and enjoyed creating really professional looking work. The vast majority of the students made substantial revisions to their old work and submitted very strong portfolios. I was so proud of what they had done! Several students told me that they were also impressed with their own work and seemed surprised by what they were capable of. All I would change is actually putting a little more time into teaching/demonstrating LaTeX in class at the beginning. I gave a brief demo and gave the students a template, but some people did not get the idea on the first pass. I should have made a short assignment due by the end of class one day, just to get everyone working in one place. Also, I used Share LaTeX for the course and it worked pretty well, but I might use the LaTeX sharing capabilities of Sage Math Cloud in the future.

Tests. I decided to keep traditional exams as a component of the course and I am glad that I did. I felt that the tests that were a good combination of definitions, computation, and proof, were not too difficult to grade, and gave students some solid feedback on what they should revisit. The final was cumulative and many students had mastered the material they had missed on the first round. They also were a traditional aspect in a generally non-traditional course, so gave students something familiar to work on.

THE BAD. I would drastically modify:

The homework structure. In both classes, I collected and graded weekly homework. This was reasonable for Foundations, which had 13 students. Grading the homework for my 31-person Algebra class was a Sisyphean task. I could not keep up, no matter how many hours I graded. I returned one enormous stack of assignments only to pick up another. I felt a responsibility to give a lot of feedback, perhaps because I really wanted to help people learn to write proofs, and I was nervous about the new structure of the class. But grading became a black hole that sucked in all my time and energy. It kept me from spending time outside of class doing other preparation that might have been more rewarding for the students. I could have spent more time preparing for discussion about the blogs, or devising mini-lessons on important topics, or meeting one-on-one with students. Or, dare I say, doing research or playing the accordion or just not grading? In the next iteration I will absolutely get away from this homework structure. I think that giving a list of suggested problems with posted solutions is probably the key. I will only collect one or two problems a week from the students, and I will focus on the proofs.

The daily in-class routine. The plan was that the students would spend the class-time everyday working on problems with their classmates. I thought there would be a daily reading assignment, and I would check their notes at the beginning of class. I would then answer any questions that they had about the reading assignment and let them loose on problems. The reading would have prepared them to get started, and I would be there to guide and help them as they worked. This sort of happened, but it sure wasn’t what I imagined. Checking notes every day took forever, so I switched to doing it during group work. However, this interrupted their group work and also kept me busy checking notes when I could have been answering questions and really getting to know my students.

Also, students often needed to just sit and read the problems for a while. It wasn’t really that useful to put them immediately into groups. I also found that the students craved more lecture, more examples, than I had planned to share. Responding to this, I would often end up talking for a long time (I am not always great at shutting up), then breaking them into groups, only to find that the end of class was near. So they didn’t always get to the place where I have found collaboration is most useful—when I have understood the problem, tried it on my own, and found a way to get stuck.

I wish that I had somehow gotten the students to engage with the actual problems before they broke into groups. Next time, instead of having a daily reading assignment, I would maybe try  longer, weekly assignments that the students would work on over the weekend. Imagining a 3-day class week, we would then have different structures on each day of the week. Monday would be a discussion of the reading assignment, questions, perhaps examples. At the end of the first day I would give the homework questions and some kind of assignment that they needed to start on by themselves before the next class. Wednesday the students would work in groups most or all of the time. On Friday, I would really like to have more group work, presentations, and a group discussion to sort of tie things together. I think a weekly routine would have given the class some rhythm.

The way I calculated grades. My syllabus included

  • 30% exams
  • 30% homework
  • 10% blogs
  • 10% notes/reading assignments
  • 10% proof portfolio
  • 10% presentations

There were just too many things to grade! I didn’t have time to be critical on the blogs, so these all got an A for effort. Same with the notes/reading assignments. The proof portfolio was easy to grade and most students did very well, but it seemed to be worth too much in comparison to the homework they had stressed over. The presentations didn’t even happen—there was no time for real in-class presentations, so I changed tactics and had the students create chapter review materials for the final. I would make the exams worth more and merge the blogs and proof portfolio into the homework portion. This would give me more flexibility on how to weigh these. I would keep the notes and reading, and the presentations, but only if I managed to incorporate… real presentations.


Overall, I have mixed feelings about the semester. I liked my students a lot. I believe they learned an enormous amount, more than they would have learned in a lecture class with me. Most of them are now impressively competent at writing proofs. However, I worked an unsustainable amount every week and still didn’t feel that I was doing enough to do justice to the teaching ideas. Some of the students didn’t like the course, and probably didn’t agree with the choices I made. I worry that my evaluations will be lower than I would like.

I will definitely revisit this style of somewhat-flipped classroom in future courses, because it worked. And all of the things I included in the course were worthwhile activities/good ideas. However, I think it was too much for one course. Whatever I do in future semesters, I will certainly do less of it, and make sure there is enough time for everything. I now pledge to carefully consider that every good idea that I include in the syllabus is something I have to design, help with, and grade. I will remember that I can do everything, but that means I will do nothing really well. Why is that so hard to remember? Maybe this time I will learn. Putting it on my calendar now, for future, full-of-good-ideas-and-energy me: reread this in August.



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2 Responses to Lecture-less Modern Algebra and Foundations, Part II

  1. Anne Ho says:

    Love the ideas of the blogs! I snuck in some similar ‘nonstandard’ material in my Coding Theory class this semester, but it’d be neat to have more of this in my other classes too and on a regular basis.

  2. Ben Braun says:

    For the amount of new things you were trying, it sounds like the class went remarkably well! One thing I have found useful to link the reading assignments with in-class work is to have the students start class by discussing the reading with several explicit prompts, eg analyze this proof, or discuss this particular computation, etc. This lets students immediately start talking based explicitly on the prep work they did for class. I wrote an article about one of the prompts I use “hallway proofs” in the most recent MAA Focus. With homework, when I first started teaching at this level I swamped myself with grading. I’m now convinced that less is more in regards to homework and grading, as long as the homework assignments are carefully designed to complement the in-class activities. Thanks for sharing your thoughts on your class! It’s interesting to read and reflect on.

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