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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.

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As a fellow liberal arts college person, I have to ask: What exactly do you mean by average undergraduate math major? I am not sure it will really change the nature of the answers that you get here, but it should certainly change the nature of the answers you give to your students. – Thierry Zell Nov 1 '11 at 22:59
@Thierry: Many of our math majors will become secondary math teachers, some will become actuaries and will go into industry or computer science. At my institution, I think the mean' would lie in the secondary education region. I didn't emphasize this in asking the question, since I didn't want a list of answers aimed at applications potentially useful for future teachers. I'm more interested in giving our students experience with living mathematics' than I am with their finding it `useful' in their future endeavors. – Jon Bannon Nov 2 '11 at 0:33
Is there a "nice" proof to the Herstein problem? (i.e., one where you don't first show by computation that always ab^2 = b^2 a or something similar) – Frank Thorne Nov 2 '11 at 1:46
When I was a student, the book we used for our first algebra course was Stillwell's Elements of Algebra, which I thought was very nice and suitable for a liberal arts college. In one semester one introduces fields, groups and enough Galois theory to get to the insolubility of a quintic, construction of n-gons and trisection of angles. – Kimball Nov 2 '11 at 4:30
@Frank: A few friends and I were thinking about just this the other day. So far, though, there is always a tad of brute force tinkering involved. I'd love to find a nice solution. If someone does, please don't post it...but perhaps send it via e-mail. – Jon Bannon Nov 2 '11 at 12:05

12 Answers 12

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.

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@Steven Gubkin: I think you mean "high school", not "middle school", at least in the U.S. – Dave L Renfro Nov 2 '11 at 16:43

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.

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This works well with certain populations but would go down in flames with others. – Alexander Woo Nov 1 '11 at 18:58
Which populations are you referring to? Not saying I don't believe you, of course - just curious as to what you mean specifically. – Daniel Fleisher Nov 1 '11 at 19:03
Thanks for your answer, Daniel. Here's something I'd like to know: What is a nice problem that would force a student to discover the fundamental theorem of finite abelian groups? I remember this being a bit of a jump from playing around with cyclic groups, when I was an undergrad. – Jon Bannon Nov 1 '11 at 19:05
@Daniel: I am thinking of students who are conservative in the sense of Burke, meaning having significant respect for established traditions and rules. This is probably more prevalent among students from certain religious traditions as well as prospective high school teachers; it also probably varies geographically. ("God made the integers; all else is the work of man." -- Kronecker) I'm an anarchist like you :), but not everyone, not even every college student, is. – Alexander Woo Nov 2 '11 at 0:37
Understood. I was thinking more of the stereotypical "student at a small liberal arts college". – Daniel Fleisher Nov 2 '11 at 16:50

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.

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Funny, I was looking at Artin's text as an alternative this morning! – Jon Bannon Nov 1 '11 at 19:32
@Jon: It's the opinion of several people far more experienced than I that Artin is too difficult a text for the average student to read. – Alexander Woo Nov 1 '11 at 21:37
You may be right about Artin's book. But, at the very least, I vouch for his approach using matrices. – Christopher A. Wong Nov 1 '11 at 23:33
@Alexander: Unfortunately, this is perhaps true. Are there any undergraduate-level analogues out there? – Jon Bannon Nov 2 '11 at 0:36
Hm, funny, this seems like "the hard way" to me. But I guess that means it's good to say something about both the discrete and the geometric points of view... – Frank Thorne Nov 2 '11 at 17:43

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.

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The viewpoint you are describing may not be "irrational", but it is an uninformed viewpoint. There are loads of instances where knowledge of powerful mathematical machinery and a more abstract approach help one solve completely concrete real life problems. – Alex B. Nov 2 '11 at 2:14
1) It is not clear these solutions to concrete problems would not have been found without abstract mathematics. It would undoubtedly have taken longer, but if you want to make a utilitarian argument, then you have to weigh the cost of not having these solutions, for a longer time or at all, against the cost of producing all the abstract mathematics that is not used. 2) Knowing about Shakespeare is not useless for communication either. – Alexander Woo Nov 2 '11 at 6:37
That would be true if we lived in the 19th century, not the 21st. Today a typical engineering question looks like "Consider the polynomials with coefficients $\pm 1$ on the unit circle. How to build a large (of size $2^{n-O(\log n)}$, say,) and easy to recognize family such that the $L^\infty$ norm is never greater than $A\sqrt n$ with as small $A$ as possible?" (it is a real question asked by an electric engineer and I still can't answer it despite I've got a fairly decent training in Fourier analysis. So, my training is insufficient, not excessive, to face the today's applied problems. – fedja Nov 2 '11 at 14:04
The implicit advice (to build the exposition around the applications the students will face later or, at least, around some problems they can comprehend but not solve without any special theory) is good though. I wish we had a few textbooks like that. That's something all those "Stewarts" and "Thomases" should really think of instead of making new color pictures for the same (rather terrible) text every year. – fedja Nov 2 '11 at 14:14
Well, what in your opinion would be the "right thing" for problems like that? The guy who asked looked at Kahane's book on random series and squeezed all he could from there, so they currently use some randomized procedure to encode. "Wrong" does not exist without "right" and in this case there is no "right" like there is no "right" training for the Riemann hypothesis. As to the definition of limit, it is useful to learn to write the alphabetic characters without sweating before trying either business letters, or poetry. – fedja Nov 2 '11 at 18:44

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.

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You wouldn't believe how many students get a degree without knowing what Jordan normal form is. – Alex B. Nov 2 '11 at 2:15
I don't think much of the importance of Jordan normalization. What is certainly important is diagonalization of the generic matrix and of symmetric matrices. Most of the proofs I have seen in algebra that use the Jordan form can be done without, and they don't get more difficult this way. Then again, I tend to avoid case-distinction type results, so my results are skewed (after all, it's the case-distinction type results that usually are proven with Jordan normalization). What I think is really criminal about Linear Algebra education is the omission of tensor products! – darij grinberg Nov 2 '11 at 2:50
@darij: I generally agree with your perspective. I think the actual form of Jordan normalization is less important than the question that it answers. Ideally, a student who takes courses in linear algebra and abstract algebra would not come away with the idea that the subjects have nothing in common (other than the word 'algebra' in their name) -- but my suspicion is that most students come away with just that conclusion. – Michael Joyce Nov 2 '11 at 4:00

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.

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This sounds solid. The only thing that bothers me a bit is that I would have hoped equivalence classes would be already clear coming into such a class, but I guess it depends how your curriculum works. – Thierry Zell Nov 3 '11 at 0:31
It's interesting that you think equivalence classes would be easy, because I assure you that this is the most difficult part of the course. Not simply the definition of an equivalence relation, but how to work with a quotient set in the proper way. For example why does multiplication work out for Z/nZ in the naive way, but not exponentiation? – Steven Spallone Nov 3 '11 at 14:02
By the way, Thierry, I notice you got your degree from Purdue. I'm referring to the M453 class which I taught there many times. – Steven Spallone Nov 3 '11 at 14:59

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.

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I don't really see what's going on in Polya enumeration geometrically. Certainly one could draw pictures of necklaces, but that's it... But the keyword Polya enumeration is a good one - it can motivate a lot of modern combinatorics (Witt vectors, symmetric functions, necklace rings). – darij grinberg Nov 2 '11 at 2:45
@darij: I meant it the other way around: use geometric examples for Polya enumeration, like coloring the faces of a cube. I'm not sure that there is much geometric intuition to be used in that case, but I would hope at the very least that the geometric setting would be familiar and appealing. – Thierry Zell Nov 2 '11 at 2:50
Ah. Also, confusing, but this is probably a good thing - it shows that algebraizing the problem is a good idea. – darij grinberg Nov 2 '11 at 2:59

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.

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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.

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This may be a productive approach. – Jon Bannon Nov 2 '11 at 12:03
This reminds me of Lindsay Childs' "A Concrete Introduction to Higher Algebra." He certainly starts with elementary number theory. – J W Nov 2 '11 at 17:07

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".

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I love this book...but would the average undergraduate? – Jon Bannon Nov 2 '11 at 14:08
The advantage is that you see a lot more "honest theorems" than your standard undergrad analysis course. At least I felt like my undergrad analysis was basically repeating calculus, onyl with all the proofs this time (And my calculus course had all the proofs to start with!). So Calculus on Manifolds would at least show that there is more calculus to learn, and the algebra of differential forms would show how "algebra" can be very useful in a geometric/analytic context. – Steven Gubkin Nov 2 '11 at 14:15
I envy you your average undergraduate, Steven! – Mariano Suárez-Alvarez Nov 2 '11 at 14:19

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.

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Thanks for the answer, Peter. My choice of nomenclature came from the idea that we should neither aim at the "bottom" nor the "top" of our class when teaching, and this seems really tough to do with the courses in question. Any suggestion for improvement would be welcome! – Jon Bannon Dec 4 '11 at 12:35

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.

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I believe that Jon does have grad-school bound students in his classes, and so presumably the idea is not to design a radically different curriculum, but simply to help motivate the existing one (warts and all). – Thierry Zell Nov 3 '11 at 0:18

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