All Questions
495 questions
19
votes
14
answers
4k
views
Excellent uses of induction and recursion
Can you make an example of a great proof by induction or construction by recursion?
Given that you already have your own idea of what "great" means, here it can also be taken to mean that the chosen ...
5
votes
1
answer
1k
views
Is Diagonalization worth to be taught? [closed]
When students come to the College (first two years of the University system in most of the developped countries) to train in mathematics, they get a linear algebra / matrix analysis course. After a ...
11
votes
1
answer
1k
views
Teaching Experience for Graduate Students. [closed]
I am currently a graduate student, who will (hopefully!) graduate in the next year (or two..). I have slowly come to realize that I enjoy teaching, and consequently want to do more of it! My main ...
24
votes
2
answers
2k
views
Direct proof that the centralizer of $GL(V)$ acting on $V^{\otimes n}$ is spanned by $S_n$
Let $V$ be a finite dimensional vector space over a field of characteristic zero. Let $A$ be the space of maps in $\mathrm{End}(V^{\otimes n})$ which commute with the natural $GL(V)$ action. Clearly, ...
- 156k
16
votes
6
answers
3k
views
How to mentor an exceptional high school student?
I have a unique and, quite truthfully, humbling opportunity. The parents of an exceptionally talented high school freshman have reached out to me and asked if I might be able to help.
This kid is ...
17
votes
17
answers
3k
views
Readings for an honors liberal art math course
Our university has an Honors section of our "liberal arts mathematics" course. Typically 10-20 students enroll each Fall, with most of them extremely bright, but lacking the interest and/or ...
24
votes
9
answers
9k
views
How to motivate and present epsilon-delta proofs to undergraduates?
This would seem to be a common question, but I am surprised not to see it already asked and answered on MO!
I am teaching an undergraduate course, and I want to teach them to construct basic epsilon-...
2
votes
2
answers
1k
views
Decomposition of $K_{10}$ in copies of the Petersen graph
It is a well-known and cute exercise in algebraic graph theory to show that $K_{10}$ cannot be written as the edge-disjoint union of three copies of the Petersen graph $P$. Indeed, the graph $G$ whose ...
- 10.9k
17
votes
6
answers
7k
views
Explaining the concept of projective space: notes for students
This is a question on teaching.
I am teaching at this moment a course in algebraic geometry for master students on a very basic level. Today (this was the fourth lecture) I discovered that only four ...
7
votes
4
answers
2k
views
Help me find good math questions for my students [closed]
I am a teacher at 西铁一中。 I teach mathematics in English for students going abroad.
Now this is my problem, there are few mathematics books written in English that are at the level of high school, ...
8
votes
4
answers
2k
views
Differential Equation Examples for Calculus Students
I've been teaching calculus courses for a while now, and something always bothers me each time I teach it. Students always seem to have trouble connecting with the differential equation material for ...
16
votes
7
answers
2k
views
Unexpected applications of the fact that nth degree polynomials are determined by n+1 points
I had a funny idea for proving an identity in Euclidean geometry. While it didn't end up being a very nice proof strategy in my case, I would still like to collect nice examples of where the proof ...
16
votes
5
answers
1k
views
Permission to use Online Notes
I am a new professor in Mathematics and I am running an independent study on Diophantine equations with a student of mine. Online I have found a wealth of very helpful expository notes written by ...
11
votes
4
answers
3k
views
Topological examples of profinite groups
I am preparing a course on profinite groups, to be delievered to early graduate students. The first part of the course will discuss the equivalent characterizations of profinite groups. I will first ...
- 113
2
votes
3
answers
274
views
learning sources about Ihara Coefficient
Do we have any good sources(lecture notes or books) for learning about $Ihara$ Coefficient?
Is there any relation between $Ihara$ Coefficient and the eigenvalues of graphs?
Thanks for any help.
- 4,784
1
vote
0
answers
430
views
Professional skills advising for math jobs [closed]
Hi,
I am a postdoc at the University of Nottingham (UK) and I am beginning to apply for Assistant Professor positions in US.
I would like to receive a feedback on the material that I am sending (...
5
votes
2
answers
2k
views
Any suggestions for a course in Mathematical Logic?
I am teaching a topics course for Mathematics majors (at Temple), and am considering Logic as the topic. I was wondering if people (a) have suggestions for an appropriate text and (b) how much might ...
13
votes
17
answers
3k
views
Short Course Suggestions For High School Students
I am planning to teach a course for talented high school students at a summer camp and I need suggestions for possible topics. The students usually have different backgrounds but most of them are ...
74
votes
14
answers
9k
views
How to write popular mathematics well? [closed]
Recently, some classmates and I were lamenting the fact that our classmates in other disciplines had almost no conception of what we did, despite the large mathematics population at Waterloo. Instead ...
32
votes
9
answers
21k
views
Interesting applications of the classical Stokes theorem?
When students learn multivariable calculus they're typically barraged with a collection of examples of the type "given surface X with boundary curve Y, evaluate the line integral of a vector field Y ...
59
votes
5
answers
25k
views
Are there any "related rates" calculus problems that don't feel contrived?
I just finished teaching a freshman calculus course (at an American state university), and one standard topic in the curriculum is related rates. I taught my students to answer questions such as the ...
13
votes
5
answers
2k
views
How to make a lecture series useful
I have been to a number of advanced lecture courses (of between 3 and 10 lectures) over the years, given (in principle) by experts to graduate students and experts in neighbouring fields. Examples of ...
37
votes
6
answers
4k
views
Taylor's theorem and the symmetric group
Anytime I see an $n!$ in some formula, my instinct is to look for the symmetric group on $n$ letters coming in somewhere. I have never done this seriously with the $n!$ in Taylor's theorem.
Question: ...
- 12.1k
27
votes
17
answers
9k
views
Using slides in math classroom
I am toying with the idea of using slides (Beamer package) in a third year math course I will teach next semester. As this would be my first attempt at this, I would like to gather ideas about the ...
17
votes
12
answers
5k
views
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 ...
7
votes
2
answers
830
views
Virtual algebraic calculation within proofs
It seems to me that the undergraduates I teach have particular difficulty with proofs that involve reasoning about algebraic calculations that arise only theoretically. Since I have in mind doing ...
28
votes
6
answers
2k
views
Means of Promoting Mathematics in Young Countries!
We all know mathematics is life, this question is for Mankind. It's mathoverflow here when some parts of the world we have mathunderflow! I think we can do something through ideas. A similar ...
3
votes
3
answers
2k
views
What to teach in a second graduate course in algebra? What textbook to use?
There is a standard syllabus for a first graduate course in algebra. One teaches groups,
rings, fields, perhaps a little bit of Galois theory, perhaps a little bit of
category theory, perhaps a ...
67
votes
9
answers
7k
views
Taking "Zooming in on a point of a graph" seriously
In calculus classes it is sometimes said that the tangent line to a curve at a point is the line that we get by "zooming in" on that point with an infinitely powerful microscope. This explanation ...
- 12.1k
5
votes
2
answers
1k
views
Is beauty at the high school level even possible? [closed]
This question is a follow up to 74841, and follows from a suggestion by Gian-Carlo Rota that beauty as judged by the educated public differs from that experienced by mathematicians (he gives Euclidean ...
12
votes
11
answers
2k
views
Giving a math talk with no blackboard or projector
I need to give a math talk to a group of undergraduates. I am asking for advice because this talk will take place at a department picnic and there will be no blackboard or projector. I would like to ...
7
votes
5
answers
2k
views
Commutative algebra final project
I'm looking for a topic for a final project in commutative/homological algebra, for first year master's students (in a decent European university). During the course, they will cover the following ...
74
votes
51
answers
28k
views
An example of a beautiful proof that would be accessible at the high school level?
The background of my question comes from an observation that what we teach in schools does not always reflect what we practice. Beauty is part of what drives mathematicians, but we rarely talk about ...
103
votes
13
answers
37k
views
How misleading is it to regard $\frac{dy}{dx}$ as a fraction?
I am teaching Calc I, for the first time, and I haven't seriously revisited the subject in quite some time. An interesting pedagogy question came up: How misleading is it to regard $\frac{dy}{dx}$ as ...
27
votes
5
answers
6k
views
The Matrix-Tree Theorem without the matrix
I'm teaching an introductory graph theory course in the Fall, which I'm excited about because it gives me the chance to improve my understanding of graphs (my work is in topology). A highlight for me ...
- 22.1k
9
votes
3
answers
12k
views
An image of the hierarchy of algebraic structures
Hello! Does anybody know an image of a graph featuring the hierarchy of algebraic structures? Something rather complete.
So far I've found similar images describing the hierarchies of classes/...
- 441
7
votes
8
answers
4k
views
Mathematical Advice for Interested Highschool Students
This may not be a research level math question, but I believe it is still relevant to Math Overflow.
What general resources exist for students in highschool who are very interested in Mathematics?...
5
votes
3
answers
647
views
Looking for ideas concerning the teaching of lower-division differential equation courses...
I'm looking for problems/lessons plans that could be used in a lower-division differential equations course that involve discerning properties of solutions of an equation, IVP, or BVP, without looking ...
3
votes
0
answers
431
views
Concrete questions that turn into math problems [closed]
I'm writing an article about the way we teach math, trying to find out why so many people are discouraged from learning, and have no interest for math and logic.
At some point, I want to show that ...
- 131
5
votes
9
answers
2k
views
Suggestions for teaching advanced high school students
Hi all,
I'm a grad student and just joined a mentoring program in which I will visit a group of advanced year ten high school students (around 16 years old) from a group of schools in the area. I don'...
18
votes
14
answers
3k
views
Teaching a pedagogy course
At my institution incoming graduate students must take a semester long course on pedagogy taught by current grad students. I may soon be in the position of having to teach this course and I'm looking ...
34
votes
18
answers
20k
views
Interesting and accessible topics in graph theory
This summer, I will be teaching an introductory course in graph theory to talented high school seniors. The intent of the course is not to establish proficiency in graph theory, per se. Rather, I hope ...
18
votes
1
answer
2k
views
Looking for an appealing counterexample in probability
There is a commonly-encountered-but-wrong rule of thumb that says something like
If a probability distribution is positively skewed, its mean is greater than its median.
(You sometimes also see it ...
- 1,180
8
votes
6
answers
1k
views
Seemingly emergent structures in mathematics
I rather suspect that this must have come up here on MO already, but my handful of searches didn't turn up the thread, so...
I'm curious about examples of mathematical structure that seems to arise "...
158
votes
8
answers
7k
views
Resources for mathematics advising.
This question is possibly ill-advised. (If it is not right for this site I will delete it.)
I, suddenly, have students.
It is very clear to me that there is nothing in my education that has ...
34
votes
23
answers
29k
views
Textbook recommendations for undergraduate proof-writing class
I am teaching the proof-writing class (for the 3rd time) in the Fall and plan to buck the party line and use a different text than the default Bond and Keane. My parameters are as follows:
Logic, ...
9
votes
4
answers
1k
views
Characterization of the Poisson law
This semester, I teach an introduction to probability course tailored for students with no science background and so with very very little prerequisites. We started with the basics of analytic ...
- 10.9k
222
votes
8
answers
35k
views
How to memorise (understand) Nakayama's lemma and its corollaries?
Nakayama's lemma is mentioned in the majority of books on algebraic geometry that treat varieties. So I think Ihave read the formulation of this lemma at least 20 times (and read the proof maybe ...
- 14.3k
13
votes
4
answers
1k
views
Simple groups with the same cardinality as PSL_2(Z/p)
In an undergrad honors algebra course it's sometimes shown that when $p$ is prime and $>3$ then
$PSL_2(Z/p)$ is simple of of order $p(p-1)(p+1)/2$. But that this is the "only" simple group
having ...
13
votes
1
answer
2k
views
conditional equality symbol
Is there a standard notation (perhaps $A \stackrel{\leftarrow}{=} B$) meaning "in all situations where $B$ is defined, $A$ is defined and equals $B$"?
The kind of situation in which such a notation ...
- 19.7k