Another year, another D4 table. Or D8 table, if you prefer.
You may (or may not) have noticed that I have been totally absent from PhD Plus Epsilon for the last 6 weeks. If saw my office right now, you’d definitely notice that it looks like someone ransacked the place and threw an alarming amount of paper, books, and chalk dust on every surface. What happened? (You may or may not ask.) The answer is that for the last month*, I have been teaching Abstract Algebra out of a new book and that’s pretty much the whole story. I am the one who ransacked my office, slowly, one hurried search for keys and dash to class at a time. I am the one who assigned the homework—fair, at only one submitted problem per chapter—but also mind boggling: 27 TeXed proofs in 3 weeks from 25 students. I am the one who effectively disappeared the last month of my life. However, from this end, it was definitely worth it. And if you read no further, know that I am convinced that rings first really works for Abstract Algebra. I have been converted, at least for now.
I have taught Abstract Algebra three times before, so this probably wouldn’t have been such a rollercoaster if I had just used my materials and plans from the last several iterations. But my colleague Marlow Anderson is (with Todd Feil) the author of “A First Course in Abstract Algebra: Rings, Groups, and Fields” (AFCIAA). Marlow is teaching Abstract Algebra II out of the book in the coming month, and it seemed like a good idea to use the same book for the first course. AFCIAA begins with the integers: how do they work, and what is so great about them? The first 5 chapters cover well ordering, integer division and the Euclidean algorithm, and introduce polynomials over the integers, rationals, and complex numbers. This provides a natural setting to see how the ideas of factorization, irreducibility, and primeness generalize from the integers to new settings. In chapter 6, we are introduced to the definition of a ring, and the next 10 chapters cover rings, ideals, factor rings, and homomorphisms, including a discussion of maximal and prime ideals and ring isomorphism theorems (I was impressed–I definitely didn’t see these until graduate school). Groups begin on Chapter 17, with a presentation of symmetry groups and permutations. By the time abstract groups are introduced, we had a lot of examples: the additive group and multiplicative unit group of every ring already provide plenty to work with. We got through the Fundamental Isomorphism Theorem for groups (Chapter 26) by the end of the block.
This course was co-taught with Hanson Smith, a CC alum who is currently a graduate student at the University of Colorado Boulder, studying Number Theory with Kate Stange. Hanson did a senior thesis with me as an undergraduate, so we knew that we spoke the same mathematical language. We asked Hanson to come teach at CC for the block because the course was initially supposed to have 32 students, above our enrollment cap. Hanson was/is a great student and deeply interested in teaching, and he already had lots of experience as a student on the block plan. I took the lead on some aspects of the course, but Hanson did a great job. He put in a ton of work, made strong contributions, and it would not have been possible to do what we did this block without him. He is already a good teacher, and I have great faith that he will become an amazing teacher as he gains more experience. THANKS SO MUCH, HANSON!! We also had help from Marlow Anderson, who acted as a “course buddy” and contributed some extra office hours to the course, and two student learning assistants, Hanbo Shao and Bob Kuo. Hanbo and Bob led extra help sessions in the evening, which was a great help and stress reducer for the students. They all made enormous contributions to the students’ success, and I am very grateful to all of them.
A Breakdown of the Course
Notable strengths of the book: I really liked this book.
- Motivates rings very thoroughly through Number Theory. This is particularly useful and fitting at Colorado College, since our students are required to take Number Theory before declaring the major. It serves as an introduction to proof-based mathematics at CC, so theoretically all students come into Abstract Algebra well versed in factorization and the Chinese remainder theorem. Continuing this line of thought in Abstract Algebra seemed natural and beautiful.
- Is very readable, so suitable for pre-reading and in-class discussion. My teaching method for upper level classes is generally to assign pre-reading and reading questions, then discuss and work problems in the classroom. This text reads really well—most students had the big picture by the time they walked into the classroom, and after discussion they could go back and review the details at any time.
- Has a good range of exercises, from warm-up through fairly difficult results. I assigned the warm-up exercises for reading questions, had students to some problems in class, and assigned one more challenging problem from each chapter for homework.
- Includes historical notes at end of each chapter. I enjoyed these and actually learned quite a few new things reading these pieces.
The best parts about rings first: In my judgement, these outweigh the hard parts listed below.
- Natural connections with number theory to rings. I loved this part. As math majors and minors, the students know and understand a lot about the integers. Really using this knowledge in Abstract Algebra really grounded the course for me, and rescued it from the arbitrariness that Algebra can carry.
- Deeper coverage of rings than in usual Abstract Algebra I class. As I mentioned before, I didn’t see many of these results until graduate school. I feel that these students will be well prepared to face a wide variety of algebraic objects.
Hard parts about rings first:
- Didn’t have time to go deeply into permutation groups. The students who go on to take Abstract Algebra II will see parity of permutations and alternating groups, but those who do not go on just won’t see this stuff.
- Ring proofs are more complicated than group proofs. There are about twice as many things to prove, which makes some of the “greatest hits” proofs take longer in the rings case. It sort of front-loads the class, so the group versions just seem easier at the end. But that extra difficulty at the beginning is a little more daunting for students who are shaky with proofs.
- The students had more problems keeping notation straight than I have seen before. In particular, cyclic subgroups and principal ideals were constantly confused. This is probably just because I have not spent nearly as much time in rings in the course before, so I didn’t have the opportuntity to get stuck in this.
The components of our course:
- Prereading and reading questions: Most nights, two chapters of reading were assigned. The students were asked to take notes, especially on the definitions and theorems, and they had a few reading questions from each chapter that were to be turned in at the beginning of class. The questions were graded for completeness, not correctness. Some students did the bare minimum (of course), but others made really thoughtful connections here.
- Sticky note questions: Each student was responsible for writing one question per day on a sticky note (with their name on the back), which they stuck to the board at the beginning of class. We asked the students to arrange the notes by topic, and to read other people’s questions. These sticky notes guided me as I led discussion, and I could take a particularly interesting question up with the individual student during break.
- Discussion and class work: Class time was a mix of lecture (clarifying points and doing examples), discussion, and group work on problems. This was fun, and having two teachers really made the group work feasible as we could help all the groups.
- Homework: One problem from was assigned as homework from each chapter. We required that these problems were typeset in LaTeX and formatted as formal proofs. This was a tall order but the students rose to the occasion and I was very impressed in the end.
- Honor problems and proof portfolio: We created a list of “honor problems” (mostly one from each chapter, again), and the students were assigned to choose five of these during the block to complete. They were not allowed to work with others on these problems, or to seek help from any source except the instructors and Marlow. After getting feedback on these problems, the students were asked to revise the proofs to near perfection and submit them as a proof portfolio at the end of the course.
- Two tests, closed book and no notes. The first test was at the end of two weeks, the second a week and a half later.
What I would do differently:
- Less submitted homework. This took us far too long to grade, and most of the students were really pushed to their limit by the number of assignments. Fewer submitted problems would have given them a chance to look back more and make connections. I really believe in the power of homework, but I need to cut down on the block plan. It is not the same as a condensed semester long course, and this aspect is especially hard to adjust to in teaching this way.
So there went 3.5 week active learning Algebra. Wow! I’m tired, and still grading, but recovering after watching about 20 hours of Olympic figure skating. Thoughts on how to make this better? What do you do in your Abstract Algebra courses? Olympic figure skating fan? Let me know in the comments!
* Why three and a half weeks? I teach at Colorado College, where all courses are taught in three and a half week blocks. Students take one course at a time. There are eight blocks per year, and this year I teach four of them.