How to teach abstract algebra for the first time?

Question

I am a Ph.D student in computer science. I am TAing one course this semester, which requires the basics of abstract algebra like rings, fields, ideals, and basic theorems about them. I have done two courses in the mathematics department related to algebra (abstract algebra and commutative algebra).

Working as a TA, once a week I have to teach some basic algebra objects and their theorems to these students. This is the first time I am teaching things related to algebra. I feel like it is hard to explain these things to students. This is how I teach them: I first write the definition and then give them one or two examples. Many times I find out that these students say that they have understood the definition, but when I ask them the next day, they are blank. I want to simplify things, but I don't know if this will be a good way to teach algebra.

Question: How do I teach algebra for the first time?

Answer 1

I second Adam. Keep it with strong with examples if you can. There is a mathoverflow answer with a ton of group theory examples. You might want to force them to practice doing more "rote" exercises that often come in an intro abstract algebra book to help gather examples.

Also it should be mentioned that not all CS students have a strong background with proof based mathematics. If the course requires the use of proofs (as abstract algebra often does), you might want to highlight proof techniques as well. Any definition you give should come with a few examples, plus I like to include a non-example. This will help build their intuition. Also, theorems and the like should be presented as relating definitions or noticing trends in your now plethora of examples. Proofs are just there as a formal way to explain why things are true. I find that describing proofs this way removes the intimidation factors of having to do a proof from many students.

Answer 2

I think that you probably need to give them some hands-on experience before any of it will sink in. I assume that they are CS students? After you give them the definition and a few examples, you could have them work out -- for themselves -- the addition and multiplication tables for the field with $2^n$ elements, expressed as strings of bits. (Keep $n$ small, maybe $n\leq 4$.)

If you can, have them implement some ring in whatever programming language you or they prefer. The ring axioms are very much like a interface specification, after all.

Answer 3

Some years ago I worked as a "CA" (Course Assistant) for preservice/inservice teachers taking a first course on Abstract Algebra (mostly group theory, with a bit on rings and fields towards the end). After the weekly CA sessions concluded, I TeXed a summary of what was covered. I found that having this extra document was very helpful for students (especially those whose schedules precluded them from attending at all!) to refer back to, and I also made an effort to write out a sample midterm, some sample solutions, and a sample final. When discussing answers (and sometimes in the TeXed notes) I made an additional effort to give insight, when I could, into how I came up with certain ideas. Perhaps this becomes more difficult, or less relevant, when one's understanding of a subject has deepened, but I thought it was helpful (and still think it can be helpful!).

To clarify the sort of notes that I am talking about, and just in case it might be useful, I have uploaded all of these 2012 notes to dropbox: Folder Link. There are references to various problems in these notes for which I do not have the original materials; still, I hope making these available will be a net positive. (As always: Be on the lookout for errors!)

If you do not feel like digging through dropbox materials, then I can point you to at least one answer I put on MSE, about group theory, in which I tried to describe my thinking in a manner that could be helpful to someone learning the rudiments of the subject: MSE 1214719.