All Questions
495 questions
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What does a mathematician expect from mathematics education? [closed]
Consider that my question is not a personal and/or subjective question. Perhaps, you have hired a mathematics educator in your department and you are interested in finding a way to communicate with ...
5
votes
4
answers
1k
views
Lecture on Fractals for Middle School Students
I'm going to have a one-hour lecture for middle school students next Monday. It will be about fractals. The students know virtually nothing about this subject.
I'll show some fractal images and a few ...
- 87
11
votes
4
answers
2k
views
Why do mathematicians prefer one definition over the other when they both define the same concept?
Here is a basic, though very important, example:
Hilbert takes as primary the notion of “congruence” (or “equal”) between segments. His first axiom of congruence “requires the possibility of ...
- 2,437
12
votes
2
answers
2k
views
Reference for a nice proof of "undetermined coefficients"
I'm teaching an honors differential equations class and have been using linear algebra heavily. I thought it would be interesting to include a proof of the method of undetermined coefficients along ...
- 7,273
9
votes
1
answer
1k
views
Topology, the board game
Edit: I am reposting this question fom math.stackexchange.com; there may be some professors here who have more experience teaching topology.
This is a math education question that I've been thinking ...
5
votes
2
answers
800
views
Faculty Handbook: Mentoring Undergraduates in Research and Scholarship
A few days ago I was asked by the director of the Center for Undergraduate Research and Scholarship at Georgia Regents University (formerly known as MCG and Augusta State) to contribute an article for ...
7
votes
3
answers
1k
views
Higher dimensional Bezout via Hilbert polynomials: a reference
For the purposes of teaching my elementary course in algebraic geometry I am looking for a reference (or notes) that contains a complete proof of a higher-dimensional weak Bezout theorem. I only want ...
- 14.3k
19
votes
3
answers
2k
views
Research level applications of "row rank = column rank"?
No less an authority than Gilbert Strang frames "row rank equals column rank" (and a couple of other facts) as "The Fundamental Theorem of Linear Algebra."
I'd simply like to assemble (for teaching ...
2
votes
1
answer
806
views
Math major at 36 [closed]
I decided to go for math at 36. Is this idea possible? I studied literature, political science and international relations and still I am not really sure what I am doing.
Since I was kid, I was not ...
16
votes
5
answers
2k
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"Classical" consequences of Bezout's theorem in dimensions $>2$
By Classical I mean something that could have been found before 1900 (say).
A well known consequence of Bezout's theorem for plane curves is Pascal's theorem http://en.wikipedia.org/wiki/Pascal'...
- 14.3k
11
votes
2
answers
3k
views
Good examples of random variables whose image is not a measurable set?
Are their simple/natural examples of real-valued Borel-measurable random variables whose image is not a Borel set? Something that occurs "naturally"?
I am teaching Doob's lemma (for two real-valued ...
- 2,201
25
votes
11
answers
5k
views
Learning through guided discovery
I have been working through Kenneth P. Bogart's "Combinatorics Through Guided Discovery". You can download it from this page: http://www.math.dartmouth.edu/news-resources/electronic/kpbogart/
I've ...
13
votes
2
answers
6k
views
What is the history of $\sqrt{}$
Why we use the symbol $\sqrt{}$ when we take square roots ? Anybody knows the history ?
user30695
34
votes
13
answers
6k
views
Elementary applications of linear algebra over finite fields
I'm teaching axiomatic linear algebra again this semester. Although the textbooks I'm using do everything over the real or complex numbers, for various reasons I prefer to work over an arbitrary ...
11
votes
1
answer
1k
views
Teaching stacks to differential geometry students
Does anyone have any experience teaching stacks over the category of manifolds to students whose background is, say, a semester-long course on manifolds? Does anyone know of any publicly available ...
- 1,710
14
votes
7
answers
6k
views
Usage of set theory in undergraduate studies
I would like to ask my colleagues their thought on good practices concerning
set theorical framework in undergraduate studies. For example, have there been any attempt to use another mathematical ...
0
votes
2
answers
562
views
Lines on degree 2n-3 Fermat hypersufaces
It is well known that a generic hypersurface of degree $2n-3$ in $\mathbb CP^n$ has finite number of lines. I would like to ask a couple of questions about lines on Fermat hypersurfaces and their ...
- 14.3k
30
votes
15
answers
17k
views
Useless math that became useful
I'm writing an article on Lychrel numbers and some people pointed out that this is completely useless.
My idea is to amend my article with some theories that seemed useless when they are created but ...
16
votes
5
answers
3k
views
Integrating powers without much calculus
I'll jump into the question and then back off into qualifications and context
Using the definition of a definite integral as the limit of Riemann sums, what is the best way (or the very good ways) to ...
- 30.1k
16
votes
2
answers
2k
views
There are two points on the Earth's surface that ... ?
At every moment in time, there are two points on the Earth's surface that have the same $\lbrace x, y, z, ... \rbrace$...?
What is the strongest, most impressive statement one can make here? The ...
- 151k
5
votes
3
answers
2k
views
Continuous change of basis (and on the definition of determinant) [closed]
Let $(u_1, \ldots, u_n)$ and $(v_1, \ldots, v_n)$ be two ordered bases of $\mathbb R^n$. The orientation of the first basis is defined as the sign of the determinant of $[u_1 \cdots u_n]$, and ...
- 1,174
23
votes
12
answers
15k
views
Textbook for undergraduate course in geometry
I've been assigned to teach our undergraduate course in geometry next semester. This course originally was intended for future high-school teachers and focused on axiomatic, Euclid-style geometry (...
6
votes
1
answer
4k
views
Examples of separable ordinary differential equations in economics
I'm currently teaching an integral calculus course for business students, and we're just about to discuss differential equations. They've worked hard, and I'd like to reward them with some economic ...
- 1,665
6
votes
2
answers
935
views
Surface Laplace-Beltrami without coordinates, exterior calculus?
Let $f: M \rightarrow \mathbb{R}^3$ be an immersion of a surface $M$. For pedagogical purposes (i.e., I'm teaching a class!) I am looking for an expression for the scalar Laplace-Beltrami operator $\...
- 3,059
12
votes
4
answers
2k
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Seeking a Geometric Proof of a Generalized Alternating Series' Convergence
Let $z \in \mathbb{C} \backslash \lbrace 1 \rbrace$ with $|z| = 1$. We consider the following infinite series, which necessarily converges:
$$S(z) := \sum_{n = 1}^{\infty}\frac{z^n}{n}$$
Note that $S(...
- 7,831
28
votes
4
answers
3k
views
The function $\sum_{0}^{\infty} x^n/n^n$
The function $F(x) = \sum_{0}^{\infty} x^n/n^n$ may be familiar to many readers as an example sometimes used when teaching tests for absolute convergence of entire functions defined by power series. I ...
- 679
23
votes
4
answers
5k
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Is $\ x\! \cdot\!\tan(x)\ $ integrable in elementary functions?
I'm teaching Calculus and my students asked me to calculate the integral of $\ x\! \cdot\!\tan(x)$.
I spent quite a lot of effort to do this, but I'm now even not sure if the integral could be ...
- 1,437
123
votes
25
answers
18k
views
"Mathematics talk" for five year olds
I am trying to prepare a "mathematics talk" for five year olds from my daughter's elementary school. I have given many mathematics talks in my life but this one feels
very tough to prepare. Could the ...
12
votes
9
answers
6k
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Topics for an Undergraduate Expository Paper in Number Theory
I am teaching an undergraduate course in number theory and am looking for topics that students could take on to write an expository paper (~10 pages). No new results are expected of them. Many of the ...
3
votes
2
answers
1k
views
Function with all but mixed second partial derivatives twice differentiable?
Let $f(x,y)$ be a a real valued function on an open subset of $\mathbf{R}^2$ with continuous partial derivatives $\frac{\partial^2 f}{\partial x^2}$ and $\frac{\partial^2}{\partial y^2}$. Is $f$ twice ...
- 107
8
votes
4
answers
1k
views
Multivariable Calculus Lecture Ideas
I am teaching a course in multivariable calculus this semester. We are covering the basics about $\mathbb{R}^n$, including dot products and cross products, curves, and quadric surfaces. After that ...
- 822
20
votes
2
answers
2k
views
Bitcoin Research
I have recently been assigned to advise a student on a senior thesis. She has taken linear algebra, introductory real analysis, and abstract algebra. Her interest is in cryptography. And she has a ...
- 822
3
votes
2
answers
395
views
Integration in several variables and elementary applications
This fall I'm teaching the "second half" of the standard entry-level undergraduate multivariable calculus course: the focus is on double and triple integrals, path integrals, Green's theorem, Stokes' ...
7
votes
5
answers
6k
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Advantages of the sequence definition of limits
I will be teaching an introductory analysis course in the coming semester. In it the students will learn about limits of real sequences, and then will learn about limits of functions in terms of ...
4
votes
0
answers
286
views
MathJax (or something like it) as a classroom “blackboard”
(I tried this first at https://math.stackexchange.com/questions/187265/mathjax-or-something-like-it-as-a-classroom-blackboard , but didn't get satisfactory responses.)
What is the best desktop ...
- 17.6k
7
votes
6
answers
1k
views
Another chicken or egg: sequence or series
This is a side question which is more motivated by teaching than research.
First, I am trying to convince myself that sequences appear before series (as numerical approximations to "interesting" ...
16
votes
10
answers
6k
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Undergraduate Topology
I am developing an introductory topology course for undergraduates, and I am wondering what topics to cover. At my institution, real analysis is not a prerequisite for the course, so it is more than ...
42
votes
16
answers
5k
views
Justifying/Explaining math research in a public address
I have been chosen by my university to give a 1 hour public research lecture. Every year a researcher is chosen for this honour. Traditionally people explain their own research about designing ...
16
votes
3
answers
2k
views
Solving a modified birthday problem at a glance
Modified Birthday Problem: a bunch of people line up, and the winner is the first person who shares their birthday with someone lined up ahead of them. What position in the line is optimal?
Three (...
- 7,831
2
votes
3
answers
410
views
Pedagogical notes on line bundles on complex projective manifolds
I would like to find some notes (or book), that explains on a very basic level what is a line bundle on a complex projective manifold. Maybe even, what is a line bundle on $\mathbb CP^n$. It seems ...
3
votes
3
answers
3k
views
Battle of the brains; cultural mathematics [closed]
Firstly, I apologize if my question is long.
Three years ago, I watched a video with the name Battle of the Brains. It was a wonderful video about challenging some famous peoples to solve some ...
21
votes
10
answers
6k
views
Not especially famous, long-open problems which higher mathematics beginners can understand
This is a pair to
Not especially famous, long-open problems which anyone can understand
So this time I'm asking for open questions so easy to state for students of subjects such as undergraduate ...
11
votes
5
answers
4k
views
Applications of Liouville's theorem
I'm looking for "nice" applications of Liouville's theorem (every bounded entire map is constant) outside the area of complex analysis.
An example of what I'm not looking for : a non-constant entire ...
394
votes
115
answers
110k
views
Not especially famous, long-open problems which anyone can understand
Question: I'm asking for a big list of not especially famous, long open problems that anyone can understand. Community wiki, so one problem per answer, please.
Motivation: I plan to use this list in ...
13
votes
1
answer
1k
views
Classroom platonism
I'd like to know whether any form a certain hypothesis about the
learning of higher mathematics has entered the mathematical or
educational literature. I'll frame the hypothesis here but not defend
...
- 17.6k
1
vote
0
answers
631
views
Arguments against Reductio ad Absurdum [closed]
Could Reductio ad Absurdum not be consireded a valid proof method? Are there any compelling arguments against it, or at it's favor?
I feel like I am assuming some metamathematical hypothesis about my ...
- 11
14
votes
3
answers
3k
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Open source LaTeX lecture notes/slides/books [closed]
In the mathematics community it's quite common for professors to write their own notes for the classes they are teaching. The notes are then usually published in both PDF and PS form on the course ...
13
votes
3
answers
2k
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History surrounding Gauss Theorema Egregium and differential geometry
I am teaching a class on elementary differential geometry and I would like to know, for myself and for my students, something more about the history of Gauss Theorema Egregium, that is the Gaussian ...
- 193
14
votes
9
answers
2k
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math circles video lectures for school children?
Hello,
I am from India. I find the mathoverflow amazing.
I have a question: Are there any good quality video lectures on school math topics?
There are a lot of high quality lectures available on ...
5
votes
2
answers
2k
views
Advice on doing physics under the umbrella of mathematics and the converse
Note: This is a question directly copied from Theoretical Physics SE primarily to get the advice of people indulged in mathematics.
In the current scenario of research in QFT and string theory (and ...