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.
Best Answer
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.