27
votes
Rings before groups in abstract algebra?
My favorite textbook for an undergraduate course in Abstract Algebra, Ted Shifrin's Abstract Algebra: A Geometric Approach, uses a rings-first approach. The primary pro is that students are much more ...
- 2,581
17
votes
What makes cosets hard to understand?
I strongly suspect the difficulty is not with cosets specifically, but with working with equivalence relations generally, especially when combined with objects that they have only recently become ...
- 2,581
17
votes
Rings before groups in abstract algebra?
I have taught both groups first and a rings first course.
When I was a post-doc at Rutgers University, I taught their standard introduction to modern algebra course using Hungerford's undergraduate ...
- 626
16
votes
Simple examples that violate group axioms
Combining colored paint is an interesting example of a non-associative operation.
Define $Paint_1 * Paint_2$ to be the paint obtained by mixing the two paints in a $1:1$ ratio. It is easy to see that ...
- 1,506
16
votes
Is MacLane and Birkoff's "Algebra" suitable today as either an undergraduate or graduate text in abstract algebra?
Preface. Birkhoff & Mac Lane's Algebra is a brilliant book. I should probably spend some time with it again, actually. Also, I apologize for such a long response. I think too much about algebra ...
15
votes
Accepted
Impossibility of trisecting the angle, doubling the cube and alike, what are reasons for or against discussing them in a course on algebra?
Let me first say how I have taught this, and then why it was worth doing.
Here is the stripped down version I speak of. The first block of bullet points is one day.
Forget about straight edge and ...
- 5,108
15
votes
Examples of basic non-commutative rings
The quaternion ring is a pretty simple example of a non-commutative ring (a skew-field, even).
- 3,740
14
votes
What are the "best" groups to use as examples while learning new concepts in algebra?
Symmetry groups of basic regular polygons and polyhedra. For one thing, they are noncommutative; for another, sometimes they coincide with other "known" groups; for a third, there can be physical ...
- 5,952
14
votes
Rings before groups in abstract algebra?
I've done it both ways, although I do rings-first now and for the foreseeable future. I think the pros and cons have a lot to do with the audience, especially if there are a lot of pre-service ...
- 1,962
14
votes
The term "unique" for functions and operations
I don't think there's a lot of educational value to fixating on trying to word definitions in exactly the perfect way.
Students have trouble with the notion of a function because it's hard. The way ...
- 11.5k
12
votes
Accepted
When to cover other algebraic structures in an abstract algebra course?
I think you are correct to worry about an overload of algebraic structures, especially if they are not well-motivated. I would strongly encourage you to keep the naming of various algebraic ...
- 2,581
11
votes
Simple examples that violate group axioms
I really like the example of the game Rock-Paper-Scissors, thinking of it as a binary operation $\star$ on the set {Rock, Paper, Scissors}.
The rules are:
Rock$\star$Paper $\mapsto$ Paper
Rock$\...
- 8,856
11
votes
Accepted
How to teach abstract algebra for the first time?
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 ...
- 2,594
11
votes
Where do students learn to solve polynomial equations these days?
Students learn linear and quadratic equations in high school algebra. And then, if they have forgotten it, re-learn it in college, in courses called "pre-calculus" or something.
Unless they ...
- 7,273
10
votes
Impossibility of trisecting the angle, doubling the cube and alike, what are reasons for or against discussing them in a course on algebra?
Perhaps rather than spend time establishing that trisecting an angle is impossible via Euclidean (ruler-compass) constructions,
you could instead (a) Make that claim without proof,
and (b) ...
- 29k
10
votes
Examples of basic non-commutative rings
Why not turn $M_2(\mathbb{R})$ into a multiplication on $\mathbb{R}^4$? Here's a fun, somewhat weird, ring:
$$ \left[ \begin{array}{cc} a & b \\ c & d \end{array} \right] \left[ \begin{array}{...
- 10.7k
9
votes
At what point in the curriculum should the tensor product be introduced?
(This pertains to U.S. universities.) Times may have changed, but when I was in graduate school (several places, and yes I know this is unusual, but I mention it because I'm talking about more than ...
- 5,698
9
votes
Accepted
Where do students learn to solve polynomial equations these days?
The rational root theorem, synthetic division, the remainder theorem, Descartes rule of signs, and similar lower level topics were fairly widely taught in U.S. high school algebra-2 courses before and ...
- 5,698
9
votes
Introductory real analysis before or after introductory abstract algebra?
Despite the names of these fields, as a student I found real analysis more
abstract than abstract algebra: real analysis was less real and more abstract to me than abstract algebra. I don't think I ...
- 29k
8
votes
What are the differences between popular undergraduate abstract algebra books?
Judson's Abstract Algebra: Theory and Applications is different in that it is an open source textbook that is available at no cost. I haven't used it (yet), but I think it's worth pointing out for the ...
- 4,596
8
votes
What makes cosets hard to understand?
One of the most famous mathematics education theoretical constructs, namely APOS, offers an answer for this question, in fact, the exact same question. A quick search leads you to some papers and even ...
- 4,368
8
votes
Examples of basic non-commutative rings
How about $\mathbb{Z}[i]$ together with conjugation, thought of as a ring element? Conjugation and multiplication by $i$ do not commute.
- 4,915
8
votes
How to teach abstract algebra for the first time?
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 ...
- 4,915
8
votes
Introductory real analysis before or after introductory abstract algebra?
Is 'at the same time' an option? I mean, by junior year, math majors should be taking at least two math classes per semester, right?
When I was an undergraduate at Penn State, these two courses were ...
- 333
8
votes
Accepted
Undergraduate-level abstract algebra books or courses that don't start with groups or rings
Not sure if this counts for the OP's purpose, but I started with Hungerford's Abstract Algebra: An Introduction. The first 3 chapters go like this:
Arithmetic in $\mathbb Z$ revisited
Congruence in $...
- 23.6k
7
votes
Simple examples that violate group axioms
For something that's not commutative, try rotations about axes through the origin in 3D. A simple rectangular book can illustrate this very well. Try rotating the book in 90 degrees around vertical ...
- 8,953
7
votes
How should normal subgroups be introduced?
I like Gowers' fake history of normal subgroups.
This is also good. Especially if you can relate it to change-of-basis, and Weyl's famous quotation "The introduction of a coördinate system is ...
- 167
7
votes
Rings before groups in abstract algebra?
I think groups first is the right approach. My reasoning is that, groups are very "strange" structures in that they don't feel natural to work with at first. Rings are much more intuitive because as ...
- 249
7
votes
What are the "best" groups to use as examples while learning new concepts in algebra?
You mentioned the linear groups over the real numbers, which I think is a useful example, with the potential drawback that it is infinite.
To "remedy" this one can consider the linear groups over ...
quid♦
- 7,632
7
votes
Simple examples that violate group axioms
I can't believe no one has mentioned (positive, say) integers under exponentiation as a non-associative operation. It's by far my favorite, though of course this example is also not commutative and ...
- 5,952
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