What are some good ways to motivate and introduce reasoning abstractly about abstract algebra?

Question

I've found one of the hardest topics to introduce to students early on is abstract algebra. Even if they've already written proofs, it's hard for them to work directly from axioms. They seem to have problems with understanding why we view sets and operations abstractly.

Additionally, it can often be difficult to motivate why these topics are useful. One reasonable motivation that I've seen is RSA, but this generally compounds a tenuous grasp of number theory with a shaky understanding of group theory. This boils down to giving students who lack a mathematical background a reason to care about the topic.

What other strategies and motivations have people tried to (1) motivate how and why we think abstractly, and (2) why abstract algebra is useful in practice?

Answer 1

When I introduce groups I first go over a very (e.g., 15 examples) long list of particular groups. I go over each example and verify the group axioms without naming it such. Then I present the problem of studying all of these examples together, motivated by simply saving labour. Instead of proving things again and again for each case, we'd like a single uniform formalism in which to study all of them. From there the road to the definition of abstract group is quite short.

A bit further into the course, when we get to polynomials, I explain that polynomials are an excellent algebraic way to encode combinatorics and therefore it is important in lots of places (since counting is fundamental). I present the students with the question "Can you design two (biased) six-sided dice such that the probability for each sum between $2$ and $12$ is equal". The negative answer can be obtained by considering the generating polynomials.

Of course, if such theorems as the Burnside Lemma in group theory is covered, then lots of real-life applications can be given. I also like to very briefly mention some of the applications of group actions in chemistry.

I also find that many students try to visualize what an abstract group is. This is impossible of course, and for students who never took any abstract course, and are so used to just computing or drawing graphs, this is a big psychological hurdle. I strongly advise, several times, not to even try to visualize. Rather, they need to understand that they need to develop a feel for the axioms and the proofs, of course, through practice. But don't visualize!

Answer 2

In general, issues with abstraction and axioms can boil down to losing the ability to visualize what's going on. I'd suggest appealing to visual/tactile intuition as much as possible, especially in the beginning of the course. Present motivating examples of groups and other fundamental concepts that students can (quite literally) play with.

For instance, talk about the Rubik's Cube! Or symmetry groups of simple polygons, and fractals. Act out permutation groups in small (ahem) groups.

Also did some googling, and this looks interesting: PascGalois Project

Answer 3

I learned algebra from Hungerford's undergraduate text, with rings before groups. This is salient for your question in a few ways.

1. The examples are more concrete and more familiar. Students will have seen $\mathbb{Z, Q, R, C}$, and polynomials and matrices. That plus arithmetic modulo $n$ gives plenty of examples, all easier to explain than the symmetric or dihedral or alternating group on $n$ elements.
2. The variety of examples shows the advantages of a general theory that deals with all of them. This motivates "how and why we think abstractly".
3. The applications come sooner for number theory and cryptography. This shows "why abstract algebra is useful in practice."

Answer 4

I was introduced to abstract algebra through it's application to number theory, using Richman's "Number Theory: An Introduction to Algebra" (Brooks/Cole, 1971) as a textbook. Sadly, the book is long out of print (already was in 1984). Having "concrete" examples of rings, groups, ... was essential.