This semester I decided to take a (maybe temporary) break from lecturing. Don’t get me wrong, I love lecturing, and I love going to good lectures. However, there have always been a few aspects of my own lecture classes that are not satisfying. If I take enough time to explain some topics very carefully, we don’t have enough time for other topics I want to cover. Many students in any lecture are focused entirely on copying down my blackboard work and are not actively engaged with the material. If I do group work in the classroom or try to incorporate interesting activities, we don’t get through enough material and I end up rushed later. If I assign reading to make up the difference, students will only rarely complete the reading and then often complain a lot about it. Also, articles like this one and this one also got me thinking that there might be good philosophical/ethical reasons to look beyond lecturing. So this semester I decided to go all the way and turn it around.

What instead? The best choice for me, and these classes, seemed to be something very structured, that made use of existing resources as much as possible. The plan:

- Assign reading before each class.
- Give students a reading assignment and check their notes for each section.
- In class, discuss the reading assignment and take questions on the material.
- Work in groups on homework assignments from the text.

I am currently teaching two classes this way. The first is Foundations, Villanova’s bridge to higher mathematics course. Second is Modern Algebra, required for all math majors, but not a prerequisite for many other courses in the catalog. In both cases, the students already have some mathematical maturity, so reading a book was not as distasteful to them as it might have been to a Calculus I class.

Another reason that this was a good semester to try something different is that my good friend and colleague Katie Haymaker is also teaching Foundations, and we both wanted to take a very hands-on approach to the course. As Katie says, “for me, one really important aspect of this format is that students are practicing proving things during class. I feel like intro to proof classes in particular suffer from the osmosis issue – students think that because they saw you prove something and understand every step, that they understand how to do it. But then when they are asked to prove something themselves, they freeze up and have no idea how to start. This is why I much prefer this format in the proof-heavy classes, but why I don’t feel compelled to do things this way in Calc 2 (in addition to the fact that no one wants to curl up and read Stewart after a long day of classes).” We have been planning our Foundations classes together and comparing notes on what worked and didn’t. This has been incredibly helpful. I can also get Katie to talk to me about my Algebra class, since she is teaching that class next year and is now considering how to organize that.

For Foundations, Katie and I selected *Book of Proof* by Richard Hammack, which we really like. All the students have reported that it is very readable, it has a lot of exercises in each section, and the odd problems have solutions, which provide many extra examples for the students. Also, it is available as a low-cost paperback or as a free pdf from the author’s website. Most of my students bought the paperback, but they always have access to the book online. For Modern Algebra, I chose *A Book of Abstract Algebra**,* by Charles Pinter. (I apparently like these direct titles; titles that let you know “I’m a book! About this!”) Pinter’s book is also extremely readable and has many exercises. A few more solutions and examples would be great, but there are a few full solutions in the back of the book. My students have reported that they like the book. Both of these books are divided up into reasonably short sections, something I think is important for this model. The students can read an entire section or chapter, then answer questions relevant to just this material before we move on.

The reading assignments are generally short, and I aim for them to be relatively easy, given the reading. Sometimes I overshoot and ask more challenging questions, but it seems to work out since we go over the reading assignments in class. Each class I also check for 1-2 pages of notes on each section. The reading questions and notes are worth 10% of the students’ final grades. Weekly homework is worth 30%, tests are worth 30%, and the rest of the grades come from: presentations (10%), a proof portfolio (10%), and weekly-ish blog entries (10%). I am not sure now if I would distribute the points the same way (more on that another week).

The blog posts were inspired by a blog post (meta!). One of the main goals I had for the course was to help my students become great at both communicating mathematics and communicating *about* mathematics. Writing proofs is important, and we put a lot of energy into that, but I think it’s almost as important that the students can have a satisfying conversation about mathematics with either a mathematical peer or a non-mathematician. The blog posts ask them to practice explaining mathematical ideas in everyday terms, or reflect on especially difficult concepts in the course.

I got the idea for a proof portfolio from Chris Rasmussen at Wesleyan University. The assignment asks students to choose their ten best proofs from the whole semester, revise them to make them absolutely impeccable, and then submit the portfolio in the last week of class. I used proof portfolios in a previous number theory/intro to proofs course, and was really impressed with what the students submitted.

Teaching this way has been a roller coaster for me. I really like it, and think it is working great in many ways. However, it is stressful to ask students to invest in something they initially find foreign and annoying. If the students don’t buy in, it can’t work, and I often feel stressed that I am not really in control of this aspect of the class. It has required a huge investment from me, too, in developing new materials, lots of grading, and the straight effort of organization. That said, I think that teaching this way was a good choice. My students are taking responsibility for their own learning in a real way. They are submitting solid proofs and becoming clearer communicators. I just gave the students a mid-semester feedback survey, and their responses indicate that they find the general structure of the course really effective. Hooray!

As the semester winds down, I will report back with more in-depth discussion of the components of the course, what has worked and what hasn’t. In the mean time, please send me some good teaching vibes, and tell me what has and hasn’t worked for you in the comments!

For beguines mathematics appears horrifying and sometimes fear make them phobic for ever, there role of rare and unusual teachers/mentors stands crucial in turning those fears into intuitive fun—- here this method speaks great., thanks for innovating way to teaching…

I teach High School special education math, and take graduate math courses.

Your method assumes the student is a strong reader – reasonable at your level! Problematic for me to try. But – could work for, say, an honors course.

Modern Algebra, correct me if I’m wrong, isn’t a freshman course. So your students have some math maturity coming in. That also works in favor of your method.

As a student, I like your idea. I’m one of the ones that trying to write down everything you put on the board. I do it to try and keep myself engaged. If it’s material I’m on top of already, then I would have the luxury of trying to follow your logic as you offer it. For example, I’ve already had my physics, so when I listen to the Feynman lectures I can fully attend to and appreciate the nuances – but I’m not trying to learn it at the same time!

I seldom read the text, since the lecture is often the text written onto the blackboard. I use the text for reference. In your class I would read the text and then have a chance to clarify it. I like it.

I note your method assumes:

*good reading skills

*some math maturity

*intrinsic motivation, like you would see in an upper level math course

Keep on keeping on!

Sorry – I read your post in haste.

I see you acknowledge that it’s not a first course in math, and students come in with some mathematical maturity.

I feel your pain on the amount of grading. An idea to cut your work a little while providing a learning experience, make groups of 3 and have the group craft their best effort from their individual effort. If you give feedback on the result, 3 people benefit from your time. There’s a compromise between grading effort and confidence in the accountability of the group.

That is a good idea. I think I may try to incorporate more group grading next time around. The grading has been very time intensive, especially for Algebra. Luckily my students seem to be paying attention and responding to my comments, which is satisfying.

I love this idea. I’d like to know a little more: (1) what the difference in your first two billeted items (assign reading before class and give students reading assignments)? (2) when grading the notes they took from the reading, how close did you examine it? Just volume? Or a detailed examining of them? (3) Was the proof portfolio graded for completion only, or were the compiled proofs “re-graded”…perhaps even more stringently than the first time?

Thanks for sharing!

I assign a section to read. The students must take a couple pages of notes, to show they’ve actually done the reading. I/we (my colleague Katie Haymaker creates some of the assignments for Foundations) also assign 3-4 questions to get the students started on working with the material. These questions might ask students to explain a concept from the section, or do some computation, or follow an example in the text. The idea is that these questions should be doable even if the student doesn’t grasp all the nuances of the material yet.

For the notes, I usually just look them over quickly, but I do comment if there doesn’t seem to be much, and I praise students that take especially nice notes.

The proof portfolio is graded for quality. I give the students a sample of good/bad portfolio proofs early on–Chris Rasmussen shared his rubric with me, and I use something very similar. This lets me be a little easy on the proofs the first time around, and gives the students a chance to re-evaluate and improve on shaky spots.

Thanks for your comment! I will post more details about the different aspects of the class in the next several weeks.