# Questions tagged [teaching]

For questions related to teaching mathematics. For questions in Mathematics Education as a scientific discipline there is also the tag mathematics-education. Note you may also ask your question on http://matheducators.stackexchange.com/.

226
questions

**0**

votes

**0**answers

108 views

### What benefits of math can be conveyed to mid/high schoolers? [closed]

I'm teaching mathematical proof writing to a few of math teachers (in the US) this summer. In the beginning of class, I send a survey asking them why they are here. Most of them are here for getting ...

**30**

votes

**3**answers

2k views

### What do we learn from the Wronskian in the theory of linear ODEs?

For a real interval $I$ and a continuous function $A: I \to \mathbb{R}^{d\times d}$, let $(x_1, \dots, x_d)$ denote a basis of the solution space of the non-autonomous ODE
$$
\dot x(t) = A(t) x(t) \...

**5**

votes

**0**answers

196 views

### Interesting things you learned while grading/marking? [closed]

What are some interesting mathematical things you have learned while grading (or marking, if you prefer) student work? For example, clever proofs that students came up with; nice counterexamples or ...

**3**

votes

**1**answer

146 views

### How to find eigenvalues following Axler?

Preparing my Linear Algebra lecture I like the determinant free approach of Axler because the proof that operators $T$ on an $n$-dimensional complex vector space have eigenvalues is so simple:
Fix ...

**3**

votes

**1**answer

196 views

### Elementary classification of division rings

Are there examples (other than the two mentioned below) of fields $K$ such that the classification of all finite dimensional division $K$-algebras is possible using only elementary theory (lets say a ...

**12**

votes

**1**answer

496 views

### Source of a quote by Ferdinand Rudio

I am looking for the source and context of this quote, found e.g. at St Andrews:
Only with the greatest difficulty is one able to follow the writings of any author preceding Euler, because it was ...

**9**

votes

**0**answers

410 views

### How many ways are there to teach class field theory?

I will soon have to teach class field theory (I do not know whether it will be local or global yet:)) to postgraduate students. I wonder, which approaches to this subject(s) exist now.
I definitely ...

**14**

votes

**1**answer

427 views

### Teaching cohomology via everyday examples

This question is a "sequel" to my similar questions about the fundamental group and homology. All of these questions were inspired by seeing a talk, by Tadashi Tokieda, about the interesting physics ...

**2**

votes

**1**answer

185 views

### Defining integrals by residue theorem

I have always been interested in alternative definitions of mathematical objects. I wonder if one can craft an useful definition of definite integral by using the Residue Theorem from complex analysis....

**7**

votes

**4**answers

733 views

### Different derivations of the value of $\prod_{0\leq j<k<n}(\eta^k-\eta^j)$

Let $\eta=e^{\frac{2\pi i}n}$, an $n$-th root of unity. For pedagogical reasons and inspiration, I ask to see different proofs (be it elementary, sophisticated, theoretical, etc) for the following ...

**7**

votes

**1**answer

283 views

### Theory of surfaces in $\mathbb{R}^3$ as level sets

Is there a book that treats the classical theory of surfaces in $\mathbb{R}^3$ from the point of view of level sets of a function? I seem to remember someone telling me that such a book exists, but I ...

**6**

votes

**2**answers

645 views

### Pages from a known textbook on Euclidean geometry?

Do you recall having seen the attached pages in a textbook once? If so, would you be so kind as to share its bibliographic record (or the main items in it) with me below?
A teacher provided us xerox ...

**0**

votes

**1**answer

84 views

### Name of a matrix with one column and row removed [closed]

I am looking for the exact name of a matrix where the i-th column and rows have been removed.
I cannot remember how it is called in linear algebra, does anyone got an idea?
Thanks!

**13**

votes

**3**answers

2k views

### Teaching Prime Number Theorem in a Complex Analysis Class for Physicists

This is a question about pedagogy.
I want to sketch the proof of the prime number theorem or any other application of complex analysis to number theory in a single lecture, in a complex analysis ...

**8**

votes

**2**answers

329 views

### Big ideas and big ways of thinking in statistics?

I'm moving to a new university for the fall semester, and I'll be teaching a statistics class for the first time. I'm familiar enough with doing statistics (my dissertation in math ed was a mixed-...

**75**

votes

**16**answers

7k views

### Short papers for undergraduate course on reading scholarly math

(I know this is perhaps only tangentially related to mathematics research, but I'm hoping it is worthy of consideration as a community wiki question.)
Today, I was reminded of the existence of this ...

**31**

votes

**2**answers

976 views

### Is it consistent with ZF that $V \to V^{\ast \ast}$ is always an isomorphism?

Let $k$ be a field and $V$ a $k$-vector space. Then there is a map $V \to V^{\ast \ast}$, where $V^{\ast}$ is the dual vector space. If we are in ZFC and $\dim V$ is infinite, then this map is not ...

**5**

votes

**2**answers

1k views

### Examples of analytic functions to motivate a first course in complex variables

[Changed title as a plea to re-open the question.]
If one is to motivate a course in complex variables, what specific analytic (holomorphic/meromorphic) function of one variable would you cite as an ...

**16**

votes

**1**answer

2k views

### A conjecture in which both “if” and “only if” are near misses

[Migrated from Math Stack Exchange]
More than a year ago, I posted the following on the Math Stack Exchange.
Consider $2^n-1$. Based on checking a few small numbers for $n$ (in
fact, the first ...

**16**

votes

**2**answers

742 views

### Teaching Steenrod Operations

I am teaching a class on topology and want to introduce Steenrod Operations. I have talked about simplicial sets and classifying spaces of groups but have not talked about Eilenberg–MacLane spaces. ...

**10**

votes

**3**answers

322 views

### Easy proof that reflections generate $N(T)/T$ for connected compact group?

I'm teaching a course on Coxeter groups and I'd like to provide an overview of the connection to compact Lie groups. Let $G$ be a compact connected Lie group, $T$ a maximal torus and $N(T)$ the ...

**6**

votes

**3**answers

920 views

### Problems reducing to a graph-theory algorithm

This is essentially a question in pedagogy -- the answers could be useful to teach (or rather, motivate) graph theory, and especially the algorithmic side of it.
I have been very impressed with this ...

**34**

votes

**4**answers

1k views

### Important open exposition problems?

Timothy Chow, in his article A beginner's guide to forcing, defines an open exposition problem as a certain concept or topic in mathematics that has yet to be explained "in a way that renders it ...

**236**

votes

**29**answers

83k views

### Mathematical games interesting to both you and a 5+-year-old child

Background: My daughter is 6 years old now, once I wanted to think on some math (about some Young diagrams), but she wanted to play with me...
How to make both of us to do what they want ? I guess ...

**7**

votes

**0**answers

449 views

### How necessary is the knowledge of Lebesgue integral for non-analysts? [closed]

Recently I have learned that at some math department the introductory course to Lebesgue integration not obligatory. Thus in another course on introduction to Hilbert spaces the $L^2(0,1)$ space is ...

**9**

votes

**3**answers

603 views

### About the classification of commutative and of cocommutative, fin. dim. Hopf algebras

I want to prove that the cocommutative finite dimensional Hopf algebras over an algebraically closed field of characteristic zero are group algebras (for some finite group) and that the commutative f....

**6**

votes

**2**answers

347 views

### Applications of isotropic quadratic forms

I will soon be teaching an introductory course on bilinear algebra and quadratic forms. I will likely spend most of the time and effort on positive definite quadratic forms and euclidean spaces. These ...

**13**

votes

**3**answers

1k views

### Teaching polarisation formula

When teaching about Hilbert spaces, one begins with a polarisation formula, which allows us to reconstruct the scalar product from the norm:
$$\langle u,v\rangle=\frac14(\|u+v\|^2-\|u-v\|^2+\imath\|u+\...

**2**

votes

**1**answer

249 views

### Defining negation

I'm currently coauthoring a book intended to teach first-year students basic proof techniques. One of the chapters, written by my coauthor, is about basic logic. In that chapter the negation of a ...

**6**

votes

**1**answer

443 views

### How to talk about certain “free” categories?

Given two categories $\mathcal{C}$ and $\mathcal{D}$, we can describe the following category $\mathcal{E}$. It is the initial category whose object set contains $\mathrm{Obj}(\mathcal{C}) \times \...

**5**

votes

**0**answers

1k views

### A course on modern algebraic geometry from “The Stacks Project”

I hope this question is viable for this site. I'm sincerely sorry, if you think it isn't.
For a lot of time, "EGA" by Alexander Grothendieck and Jean Dieudonne was "the" reference on the basics of ...

**45**

votes

**16**answers

10k views

### Why do we need random variables?

In this MathStackExchange post the question in the title was asked without much outcome, I feel.
Edit: As Douglas Zare kindly observes, there is one more answer in MathStackExchange now.
I am not ...

**1**

vote

**1**answer

256 views

### proof without words for logarithms [closed]

Does anyone know of any PROOF WITHOUT WORDS for logarithmic functions?
The only one I've seen in calculus based and I need one for high school math kids in MATH 1,2,3.
Any suggestions would be ...

**17**

votes

**4**answers

2k views

### Languages beyond enumerable

A language is a set of finite-length strings from some finite alphabet $\Sigma$.
It is no loss of generality (for my purposes) to take $\Sigma=\{0,1\}$; so a language is a set of bit-strings.
...

**42**

votes

**5**answers

4k views

### How do you mentor undergraduate research?

Lets say you had an undergraduate who wanted to do some advanced work and some research, possibly for a thesis, or things like that.
There are two slightly more specific groups of questions I have ...

**45**

votes

**14**answers

10k views

### Applications of the Cayley-Hamilton theorem

The Cayley-Hamilton theorem is usually presented in standard undergraduate courses in linear algebra as an important result. Recall that it says that any square matrix is a "root" of its own ...

**13**

votes

**12**answers

2k views

### What are fun elementary subjects in probability?

I have to read several lectures on probability or applications of probability for high school students (of high level). There is no necessary part I must lecture, that is, my aim is just advertisement....

**27**

votes

**10**answers

3k views

### What (fun) results in graph theory should undergraduates learn?

I have the task of creating a 3rd year undergraduate course in graph theory (in the UK). Essentially the students will have seen minimal discrete math/combinatorics before this course. Since graph ...

**22**

votes

**4**answers

1k views

### Technical issue in the approach to Lie groups taken in a book

I'm teaching Lie groups and Lie Algebras out of Brian C. Hall's book (Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Springer), which I've enjoyed using. I'm confused about ...

**12**

votes

**2**answers

1k views

### teaching higher algebra

Has anyone ever (successfully or unsuccessfully) taught a course in higher algebra (in the $\infty$-categorical sense)?
I'm asking out of curiosity (and also hoping for more resources).
The kind of ...

**8**

votes

**0**answers

352 views

### Pedagogical question on Lie groups vs. matrix Lie groups

There are two common approaches taken in introductory texts on Lie groups: studying all Lie groups, or focusing only on matrix Lie groups. The main advantage of the latter approach is that one can ...

**53**

votes

**4**answers

4k views

### Advice for PhD Supervisors

My first PhD student is having his viva tomorrow. Hence, I began contemplating a bit about the whole process of supervising. One thing I realized is that while there seems to be plenty of advice for ...

**5**

votes

**1**answer

326 views

### How to teach generalizing the induction hypothesis? [closed]

I just finished teaching a class on using proof assistants (in this case, Agda) to write provably correct programs. Reflecting on how it went, the biggest difficulty I noticed the students having was ...

**33**

votes

**15**answers

3k views

### Historical (personal) examples of teaching-based research

The phrase "teaching-based research" brings to mind research about teaching, though important, it is not what I mean. Unfortunately, I couldn't come up with a better phrase, thus please bear with me ...

**85**

votes

**2**answers

3k views

### History of $\frac d{dt}\tan^{-1}(t)=\frac 1{1+t^2}$

Let $\theta = \tan^{-1}(t)$. Nowadays it is taught:
1º that
$$
\frac{d\theta}{dt} = \frac 1{dt\,/\,d\theta} = \frac 1{1+t^2},
\tag1
$$
2º that, via the fundamental theorem of calculus, this is ...

**11**

votes

**3**answers

601 views

### Why does inconstructibility of $\sqrt[3]{2}$ imply impossibility of cube doubling? [closed]

In this question "constructing" and "doubling" is meant in the compass-and-straightedge sense.
On my desk I have five Basic Algebra texts treating constructability in the plane $\mathbb{C}$ or $\...

**9**

votes

**2**answers

2k views

### Which universities teach true infinitesimal calculus? [closed]

My colleague and I are currently teaching "true infinitesimal calculus" (TIC), in the sense of calculus with infinitesimals, to a class of about 120 freshmen at our university, based on the book by ...

**19**

votes

**2**answers

2k views

### Teaching stochastic calculus to students who know no measure theory (or PDE, or…)

I've got quite a challenge as my teaching assignment for the next Fall (not that I want to get rid of it, quite the contrary, but I still feel like asking for advice won't hurt :-)).
I'm to teach the ...

**13**

votes

**1**answer

556 views

### A funny factorization of the Jacobian coming from the lines on the Fermat cubic

Here is something which came up in my algebraic geometry class, and I'm wondering if it has a deeper explanation. Let $F(w,x,y,z) = w^3+x^3+y^3+z^3$ and let $X$ be the cubic surface in $\mathbb{P}^3$ ...

**9**

votes

**0**answers

983 views

### Total spaces of tangent/cotangent bundles in a course where all varieties are quasi-projective

$\def\PP{\mathbb{P}}$In a course where all varieties are quasi-projective (as in Shafarevich Volume I), I am trying to figure out whether I can justify talking about the total spaces of the tangent ...