Motivating Algebra and Analysis for Average Undergraduates I work at a small liberal arts college, where many of our mathematics majors will not attend graduate school in mathematics. My hope in asking the following question is to gather innovative ideas for motivating an average student for the development of theory usually found in first courses in algebra and analysis.

Question: What are the most productive ideas for motivating algebra and analysis for average undergraduate math majors not destined for graduate school?

I'm interested in hearing about motivation of the subjects in general, and bits of theory in particular. 
On the first front, around half way through a typical undergraduate linear algebra course (around when abstract vector spaces appear) there is an abrupt change in the focus of the material from highly computational exercises meant to produce what the students think of as an "answer", to proofs of basic theorems. (Of course good theory facilitates computation, but without experience students probably can't even appreciate the questions driving the theory!) What are some ideas for smoothing this transition?
On the second front, how do we motivate individual concepts? Ideally it would be nice if there were collections of questions that made the need for theory absolutely clear for students, e.g. the very nice problem found in Herstein's Topics in Algebra: 

Let $G$ be a finite group whose order is not divisible by 3. Suppose that $(ab)^{3}=a^{3}b^{3}$ for all $a,b \in G$ prove that $G$ is abelian.

Students can compute until they are blue in the face with this one, but are basically forced to consider the properties of homomorphisms to solve the problem. I'd like to be pointed to questions like these.
Addendum: MO may be an ideal place to quickly synthesize "problem hikes" through theory, as many of us have encountered favorite problems that illuminate important ideas. Please feel free to submit your favorite problems as answers to this question. It would be great if this site could generate undergraduate problem books in (or at least problem hikes through) algebra and analysis.
 A: For the mathematics educators, it is worthwhile to note that most "graphing" done at a middle school level is done by transforming the plane.  If you want to graph $y = x^2 + 6x + 10$, you try to find a way to recognize it as a transformed version of $ y = x^2$, but you only have certain allowable transformations (usually you are looking at the group generated by translations and vertical/horizontal stretching).  Completing the square is a technique for putting an equation into a form where you can read off the transformations needed. This explicit understanding of the transformation lets you understand everything about the graph (where is the vertex, does it open up or down, where are the roots?).  So having a very explicit understanding of a group of symmetries of space is of utmost practical importance for middle school students.
A: 1) For everyone except mathematicians (and, prior to 1820 or so, everyone except George Berkeley), the ultimate reason for believing calculus is that it helps engineers build bridges that don't collapse.  Analysis does not justify calculus; applications do.  Do some naive manipulations with series get one into trouble?  Yes, but it is not necessary to develop analysis to deal with the problem; one just has to learn by experience not to do those manipulations.  (This is not a new point of view.  I am just trying to explain what Wittgenstein tried to explain to Turing in 1946.)
2) In the same sense that literature is an unnecessary, parasitic phenomenon upon ordinary language, "higher mathematics" is an unnecessary, parasitic phenomenon upon ordinary calculation.  English majors study Shakespeare because it is a great historical achievement of our civilization and its study teaches us various useful skills.  Math majors study analysis and algebra for the same reason.  (EDIT: I reread this, and realized that it's possible for people to misread it.  I have the greatest respect both for the study of literature and the study of higher mathematics and think both are worthwhile pursuits.  I think the viewpoint that denigrates these pursuits is a bad viewpoint, but at the same time I don't think it is an irrational viewpoint.)
3) In my experience, although students may ask for motivation, what they are really looking for is something with which they are familiar to which they can compare the new stuff they are learning, so that they can build a context for the new concepts.  I hope you will get some answers answering this implied question, but since I believe in brutal honesty with students, I think my above points needed mentioning.
A: I find that motivating algebra is best done with cyclic groups for the following reason. 
Students under a certain age - whether or not they care much for math - seem to like just about anything that involves breaking down their old ideas about math. They like seeing that what they took for granted or was simply taught as the truth is really just one [arbitrary] choice. 
Introducing finite cyclic groups as simply a new way of adding the numbers they've been dealing with for years seems to be pretty stimulating. For one, they can add in small cyclic groups simply by drawing arrows around a circle of n points. If you're decent at drawing, there's also the intuitive bonus of describing this new way of adding numbers as moving around a helix that was once the regular integers - here I mean thinking of the integers as an infinite bit of string and wrapping it around itself. 
I've found this sort of thing is very interesting, even for the most disaffected students. From there it's pretty easy to motivate just about any other reasonably tangible algebraic structure as "just another way of adding." 
Anyway, my point is that the idea of algebra (and analysis I suppose, but I try not to think about it) as something anarchic is a powerful tool when getting through to college students. Confirming their middle/high school suspicions about "all of this crap being arbitrary" is psychologically very comforting, and from there you can go on to tell them that you get a whole host of interesting ideas simply by changing your point of view.
A: This isn't a complete answer, but taking the counterpoint to Daniel's answer, I think that group theory often is best first introduced in the context of matrix groups. I'm fully aware that not all introductory abstract algebra texts have this emphasis, though, if I recall, Michael Artin's algebra textbook has this emphasis.
Firstly, I think that linear transformations are still concrete enough that students are comfortable with them (since they should've seen linear algebra), and they are applicable enough that students should appreciate their study. Secondly, after building up all their intuitions about matrix groups, then you can discuss how, through representations, many abstract groups can be thought of as matrix groups.
I started learning group theory using finite groups (and thus, using the cyclic group approach of Daniel's), and found it difficult to comprehend. I think this had to do with the fact that the relationship between finite groups and combinatorics is very tight, and thinking about group elements as being permutations was not natural for me because I'm used to thinking geometrically (and continuously) rather than discretely.
A: As far as abstract algebra goes, I think you should make the analogy with your students that when they learned the four basic operations as a child that they did not at the time have full appreciation of what they were learning.  To explain to a seven year old the "application" of balancing a checkbook or calculating the tip on a check would have little meaning to them.  Similarly, when learning about groups or rings for the first time, they are entering a whole new world of calculations that they can perform.  At first, they will learn the rules without a full understanding of what they are doing.  In due time (i.e. later in the course), after much hard work on their part, their understanding will have increased and they will be ready to appreaciate the applications of their new skills.  I generally find students are willing to give you some slack, at least for a while, but you do need to make sure you fulfill your promise so that the students do get at least a glimpse of some interesting applications before the course is done.
As far as specific applications go, using Burnside's Theorem for enumeration problems is a fun application of finite groups that students enjoy once they "get" it.  It is also worthwhile, assuming the students have a linear algebra course in their background, to go back and reinterpret diaganolization and Jordan normal form in terms of group actions.  It is criminal how many math majors get a degree without having any concept of why diagonalization and Jordan normal form matter, despite how important eigenvalue-eigenvector analysis is in real world science and engineering applications.  It should be possible to introduce the basic idea of Klein's Erlangen program (in the familiar example of Euclidean geometry) for students to see another example of symmetry at work, though this might be a bit ambitious.  I would avoid problems such as classifying finite simple groups of order $< n$ (your choice of $n$).  I think that the few students who actually are stimulated by such problems are the ones who generally need the most encouragement to unleash their creative side to complement their appreciation for logical rigor.
A: I taught abstract algebra to a class dominated by education majors for a few years.  Much of the "applications" from the course were either algebraic facts which are familiar to calculus students, or modular arithmetic.  For example, we spent a couple weeks on Peano arithmetic.  The theory of polynomials illuminates the theory of partial fractions.  Many "individual concepts" appear along the way.  For example the concept of a ring is quite natural after you've been discussing numbers and polynomials in the same style.  While more difficult, the notion of equivalence classes appear when you define rational numbers or rational functions precisely.  I found that education students were sufficiently motivated by the clarity and authority that comes from proving such standard facts.  And modular arithmetic is just fun for everyone.  I have some notes on my OU webpage with a large "collection of questions" for this course.
A: I have yet to teach such a course, but I would motivate abstract algebra based on plane geometry, especially group theory. It certainly worked for me to some extent, though I suspect that's because I saw a lot more plane geometry in high school than our students ever do. Various flavors of plane transformations, dihedral groups, regular polyhedra and crystalline structures, you can even go into combinatorics with Polya enumeration$\dots $ I think a geometric picture would be valuable even after moving away from geometric problems.
As for analysis, I would go for the big theorems, which seems where the difference between analysis and calculus lies. For instance, after doing a bunch of limits of integrals depending on an integer parameter, wouldn't it be nice to know that limits and integrals can be interchanged when the convergence is uniform? The big theorems can be presented not for the sake of abstraction, but because they make boring computational tasks easier.
I don't know if this answers your question, though I'm putting it out there anyway; this is almost too obvious, you probably wanted more details or a different tack altogether.
A: If you color the Cayley table of a group $G$ according to what coset of some subgroup $H$ an element belongs to, then you get a nice pattern if and only if $H$ is normal.
A: I wonder if abstract algebra arose in response to certain questions in number theory.  E.g. Gauss's arguments for modular integers precede LaGrange's theorem about orders of subgroups.  I would try to introduce problems in number theory, then use modular arithmetic to treat them and then transition to group theory.
A: Representation theory has more practical applications than I would know how to list.  Maybe you could get students through the basics of group theory and try to get them on to representation theory as soon as possible?  Not sure how practical that would be.
A: Try to see the interconnectedness of algebra and analysis.  A great place to do this is when studying analysis on manifolds, in particular differential forms.  I would recommend Spivak's book "Calculus on Manifolds".
A: My experience is that people learn better if they have an intuitive understanding for what purpose they are learning all that stuff. This is why I would start with symmetry groups of crytsals. This ensures for instance also interdisciplinarity and something people can understand. M. Artin's book "Algebra" is really good there.
Normally, I would have advised you to tell them about the fascinating applications in differential geometry, such as the geometry of symplectic spaces. But, as you are dealing with "average undergraduates" as you named them I would use the above.
Remark: Although I would like to oppose to the nomenclature: Average undergrad at a samll college doesn't mean they shouldn't be motivated. Perhaps some of them will be fascinated by your course so intensely that they plan to go on to grad school.
