All Questions
495 questions
61
votes
10
answers
10k
views
Teaching proofs in the era of Google
Dear members,
Way back in the stone age when I was an undergraduate (the mid 90's), the internet was a germinal thing and that consisted of not much more than e-mail, ftp and the unix "talk" command ...
10
votes
7
answers
2k
views
Proof that bases etc. exist in early linear algebra course?
I'm currently struggling to teach a 2nd course on linear algebra (in the UK, not at an Oxbridge quality university: the students have done a 1st course which concentrated upon algorithms you can apply ...
36
votes
11
answers
10k
views
Categories First Or Categories Last In Basic Algebra?
Recently, I was reminded in Melvyn Nathason's first year graduate algebra course of a debate I've been having both within myself and externally for some time. For better or worse, the course most ...
15
votes
7
answers
6k
views
Freshman's definition of sin(x)?
I would like to know how you would rigorously introduce the trigonometric functions ($\sin(x)$ and relatives) to first year calculus students. Suppose they have a reasonable definition of $\mathbb{R}$ ...
- 23.4k
114
votes
34
answers
86k
views
Why do we teach calculus students the derivative as a limit?
I'm not teaching calculus right now, but I talk to someone who does, and the question that came up is why emphasize the $h \to 0$ definition of a derivative to calculus students?
Something a teacher ...
6
votes
3
answers
1k
views
Publishing with Undergraduates
Is doing research with a student considered to be good for a dossier? Is it okay to have few research publications but a lot of student projects? I am finishing up a grad program and am looking at ...
- 61
3
votes
6
answers
2k
views
Teach a course in 1 month
I need to teach an intro course on number theory in 1 month. I was just notified. Since I have never studied it, what are good books to learn it quickly?
0
votes
1
answer
2k
views
Dual of Zorn's Lemma? [closed]
It seems to me that the dual of Zorn's Lemma should be true: if $S$ is a non-empty partially ordered set and every chain of $S$ has a lower bound in $S$, then $S$ has at least one minimal element.
...
- 1
25
votes
3
answers
7k
views
Analysis from a categorical perspective
I have not studied category theory in extreme depth, so perhaps this question is a little naive, but I have always wondered if analysis could be taught naturally using categories. I ask this because ...
- 5,759
19
votes
6
answers
6k
views
an engineering Ph.D. teaching math in college
I have a friend who has been teaching college-level math (e.g., all levels of calculus)
for about 4 years, although all of his education, including his Ph.D., was in engineering.
Now he is ...
8
votes
2
answers
679
views
To what extent can algorithms in undergraduate linear algebra be made continuous/polynomial/etc.?
I feel like many of the algorithms that I learned — indeed, that I have taught — in undergraduate linear algebra classes depend sensitively on whether certain numbers are $0$. For example,...
- 54.6k
18
votes
17
answers
6k
views
What is your favorite isomorphism? [closed]
The other day I was trying to figure out how to explain why isomorphisms are important. I pulled Boyer's A History of Mathematics off the bookshelf and was surprised to find that isomorphism isn't ...
10
votes
8
answers
2k
views
Undergraduate Probability Topics
I am teaching undergraduate probability this semester, and I am looking for some suggestions about inspiring applications that could be reasonably covered over the course of two one-hour lectures or ...
40
votes
21
answers
16k
views
Journals for undergraduates
Are there math journals that are aimed for undergraduates? I don't mean here journals where students can publish their papers, but journals that publish introductory articles that an undergraduate can ...
9
votes
4
answers
2k
views
Applications of Math: Theory vs. Practice
I have a problem: I learned about a lot of the applications of mathematics from academics. Neither they nor I have had much contact with the "real world" to go and see for ourselves how mathematics ...
37
votes
1
answer
3k
views
Community experiences writing Lamport's structured proofs
About two years ago, I came across this paper by Lamport
http://research.microsoft.com/en-us/um/people/lamport/pubs/lamport-how-to-write.pdf
on writing proofs hierarchically. It changed how I wrote ...
5
votes
3
answers
2k
views
Graphical representation of mathematical structures (in the spirit of unified modeling language)
In software engineering the unified modeling language ("UML") is a well established technique for providing overview of complex systems and an efficient means of communicating about them. There are ...
- 4,014
32
votes
20
answers
6k
views
What are your favorite puzzles/toys for introducing new mathematical concepts to students?
We all know that the Rubik's Cube provides a nice concrete introduction to group theory. I'm wondering what other similar gadgets are out there that you've found useful for introducing new math to ...
5
votes
2
answers
2k
views
How to study a math text [closed]
Hello,
recently I've been trying various attempts regarding how to approach a math book to learn in the best way. Should one memorize the theorems and proofs so that one can recite them? I tend to ...
3
votes
2
answers
957
views
Simple definition of the Hausdorff measure using squared paper
I am giving a "non-technical" seminar in which I would like to give an elementary introduction to the Hausdorff dimension and measure.
For simplicity, I was hoping to give a more intuitive ...
- 20.2k
35
votes
19
answers
9k
views
Interesting applications (in pure mathematics) of first-year calculus
What interesting applications are there for theorems or other results studied in first-year calculus courses?
A good example for such an application would be using a calculus theorem to prove a ...
24
votes
7
answers
8k
views
How do professional mathematicians learn new things? [closed]
How do professional mathematicians learn new things? How do they expand their comfort zone? By talking to colleagues?
52
votes
9
answers
26k
views
Is Galois theory necessary (in a basic graduate algebra course)?
By definition, a basic graduate algebra course in a U.S. (or similar) university with
a Ph.D. program in mathematics lasts part or all of an academic year and is taken
by first (sometimes second) ...
51
votes
22
answers
19k
views
Why linear algebra is fun!(or ?)
Edit: the original poster is Menny, but the question is CW; the first-person pronoun refers to Menny, not to the most recent editor.
I'm doing an introductory talk on linear algebra with the ...
24
votes
7
answers
4k
views
Why are two notions of Gaussian curvature are the same - what is the simplest & most didactic proof?
This question is still wide open - all of the answers so far rely on magical calculations. I've only accepted an answer because, by bounty rules, otherwise one would be accepted automatically. I can't ...
- 3,203
60
votes
8
answers
13k
views
Why the Killing form?
I'm teaching a short summer course on algebraic groups and it's time to talk about the Killing form on the Lie algebra. The students are all undergrads of varying levels of inexperience, and I try to ...
- 7,273
74
votes
9
answers
16k
views
What is Lagrange Inversion good for?
I am planning an introductory combinatorics course (mixed grad-undergrad) and am trying to decide whether it is worth budgeting a day for Lagrange inversion. The reason I hesitate is that I know of ...
13
votes
2
answers
3k
views
The probabilistic method - reference to less challenging questions
I am teaching a course in combinatorics and large part of it is dedicated to the probabilistic method especially in the case of graphs. The course is an undergraduate level (almost none of the ...
7
votes
2
answers
2k
views
Vinogradov's Elements of Number Theory
I can't be the only person here who has fond memories of the problems in Vinogradov's Elements of Number Theory. (For people who have not read it - the text itself is just a concise basic number ...
5
votes
3
answers
799
views
Euclidean function of Euclidean domain defined at 0
In a few places where I have looked the Euclidean Function of a Euclidean Domain is only being defined for non-zero elements. I am teaching an undergraduate course and I am trying to make things as ...
24
votes
11
answers
8k
views
The role of the mean value theorem (MVT) in first-year calculus
Should the mean value theorem be taught in first-year calculus?
Most calculus textbooks present the MVT just before the section that says that if $f'>0$ on an interval then $f$ increases on that ...
2
votes
0
answers
1k
views
Good sources for linear algebra for convex optimization and graph analysis?
What are some good sources for linear algebra for convex optimization and graph analysis?
In Particular, is Gilbert Strang's MIT course suitable, or some other online course? I prefer online courses (...
- 2,383
18
votes
12
answers
10k
views
Theorems in Euclidean geometry with attractive proofs using more advanced methods
The butterfly theorem is notoriously tricky to prove using only "high-school geometry" but it can be proved elegantly once you think in terms of projective geometry, as explained in Ruelle's book The ...
9
votes
4
answers
2k
views
Problem suggestions for polymath for undergraduates research
I'm inspired by the polymath project. It might be great for few undergraduates to work together on a research topic.
What are some research problems with the following properties(Experimental ...
30
votes
6
answers
5k
views
Euclid with Birkhoff
I'm looking for a short and elementary book which does Euclidean geometry with Birkhoff's axioms.
It would be best if it would also include some topics in projective (and/or) hyperbolic geometry.
...
- 45k
86
votes
44
answers
21k
views
Demystifying complex numbers
At the end of this month I start teaching complex analysis to
2nd year undergraduates, mostly from engineering but some from
science and maths. The main applications for them in future
studies are ...
21
votes
6
answers
3k
views
Yet another 'roadmap' style request- a second bite of the cherry
Okay, so I know MO has had a recent proliferation of this kind of question, and I know MO is not really for this type of question (though I suspect perhaps this is a phenomenon that is likely to ...
5
votes
7
answers
12k
views
Undergraduate approach to learning math [closed]
I am going into my sophomore year as an undergraduate and I would like to ask the more experienced folks a couple questions about learning math and related things. What are your experiences and advice ...
27
votes
19
answers
26k
views
Good combinatorics textbooks for teaching undergraduates?
Hello, can anyone recommend good combinatorics textbooks for undergraduates? I will be teaching a 10-week course on the subject at Stanford, and I assume that the students will be strong and motivated ...
16
votes
7
answers
2k
views
Uppercase Point Labels in High-School Diagrams: from Euclid?
I wonder if the convention of labeling points in geometric
diagrams with uppercase symbols ultimately derives from
Greek mathematics, which was originally written in
"majuscule" (uppercase) Greek ...
- 151k
63
votes
20
answers
13k
views
What should we teach to liberal arts students who will take only one math course?
Even professors in academic departments other than mathematics---never mind other educated people---do not know that such a field as mathematics exists. Once a professor of medicine asked me whether ...
7
votes
4
answers
2k
views
What would be good to know before starting my undergraduate studies to become a good mathematician?
First of all, I'm sorry if this isn't the kind of question that should be made in MathOverflow. I read the FAQ and I didn't consider this (that) inappropriate. I couldn't resist! People here are ...
12
votes
10
answers
16k
views
Learning Algebra & Group Theory on my own [closed]
I'm learning Algebra & Group Theory, casually, on my own. Professionally, I'm a computer consultant, with a growing interest in the mathematical and theoretical aspects. I've been amazed with ...
33
votes
20
answers
5k
views
Do names given to math concepts have a role in common mistakes by students?
Perhaps this question overlaps with similar ones, ... but I want to focus on a particular possible cause of confusion. I notice that students are often confused by the concepts of "infinite" and "...
27
votes
5
answers
7k
views
References for "modern" proof of Newlander-Nirenberg Theorem
Hi,
I'm starting to prepare a graduate topics course on Complex and Kahler manifolds for January 2011. I want to use this course as an excuse to teach the students some geometric analysis. In ...
19
votes
9
answers
5k
views
Mathematics and autodidactism
Mathematics is not typically considered (by mathematicians) to be a solo sport; on the contrary, some amount of mathematical interaction with others is often deemed crucial. Courses are the student's ...
152
votes
18
answers
24k
views
Why do we care about $L^p$ spaces besides $p = 1$, $p = 2$, and $p = \infty$?
I was helping a student study for a functional analysis exam and the question came up as to when, in practice, one needs to consider the Banach space $L^p$ for some value of $p$ other than the obvious ...
22
votes
16
answers
6k
views
What are your experiences of handouts in mathematics lectures?
There are many different styles of lecturing, and many different aspects that are blended together to give a whole "lecturing style". That said, I'm particularly interested in hearing people's ...
43
votes
7
answers
12k
views
On starting graduate school and common pitfalls...
Hi,
I'll be starting graduate school soon, and when I look back at my college career, there are certain things I wish I could have done differently. In hindsight, I wished I wasn't in such a rush to ...
3
votes
4
answers
2k
views
Less-known conjectures of significant influence and the contrary
In mathematics, it is common that theorems/results and problems appearing dull in one generation get revitalized and become the center of research in another one.
Sometimes conjectures that are ...