**6**

votes

**7**answers

11k views

### Real analysis has no applications?

I'm teaching an undergrad course in real analysis this Fall and we are using the text "Real Mathematical Analysis" by Charles Pugh. On the back it states that real analysis involves no "applications ...

**12**

votes

**13**answers

2k views

### Applications of connectedness

In an «advanced calculus» course, I am talking tomorrow about connectedness (in the context of metric spaces, including notably the real line).
What are nice examples of applications of the idea ...

**120**

votes

**6**answers

42k views

### Where to buy premium white chalk in the U.S., like they have at RIMS?

While not a research-level math question, I'm sure this is a question of interest to many research-level mathematicians, whose expertise I seek.
At RIMS (in Kyoto) in 2005, they had the best white ...

**8**

votes

**3**answers

2k views

### The harmonic (series) beetle: live illustrations of mathematical theorems

In my analysis class I use the following problem to illustrate the divergence
of the harmonic series (consider this as a hint for solving it).
Exercise.
A beetle creeps along a 1-meter infinitely ...

**11**

votes

**5**answers

2k views

### Introducing Cryptology to Undergraduates

This summer I am going to give some lectures to some REU students. I am still tossing around ideas for what I am going to talk about, but one thing I would at least like to give one or two lectures ...

**13**

votes

**1**answer

1k views

### An elementary proof that the degree of a map of spheres determines its homotopy type

I'm helping to teach an undergraduate algebraic geometry course (out of Hatcher's textbook). We have recently defined the degree of a map of spheres using homology, and the professor and I thought it ...

**6**

votes

**3**answers

1k views

### The etale fundamental group of a field

Background and motivation:
I am teaching the "covering space" section in an introductory algebraic topology course. I thought that, in the last five minutes of my last lecture, I might briefly sketch ...

**26**

votes

**13**answers

9k views

### How to draw knots with Latex?

I am writing an exam for my students, and the topic is intro knots theory. I have no idea how to put knots into the file, but I know many MO users who can draw amazing diagrams in their papers.
Can ...

**12**

votes

**7**answers

2k views

### What should be taught in a 1st course on Riemann Surfaces?

I am teaching a topics course on Riemann Surfaces/Algebraic Curves next term. The course is aimed at 1st and 2nd year US graduate students who have have taken basic coursework in algebra and manifold ...

**10**

votes

**7**answers

2k views

### How to motivate the skein relations?

I am teaching an advanced undergraduate class on topology. We are doing introductory knot theory at the moment. One of my students asked how do we know to use this skein relation to compute all these ...

**21**

votes

**14**answers

6k views

### “Homotopy-first” courses in algebraic topology

A first course in algebraic topology, at least the ones I'm familiar with, generally gets students to a point where they can calculate homology right away. Building the theory behind it is generally ...

**2**

votes

**1**answer

730 views

### Text/structure for an analysis course for students with pre-existing understanding of some applied aspects of analysis

Greetings,
I'm teaching a one-off course (perhaps never to be repeated) in a curriculum that's in transition, and I'm looking for advice on a textbook, or stories from people who have taught similar ...

**8**

votes

**4**answers

2k views

### How to teach introductory statistic course to students with little math background?

Next semester I will teach an elementary statistic course for the first time (which I am actually quite excited about). A brief description can be found here. I am told to expect very little math ...

**12**

votes

**1**answer

580 views

### Teaching Methods and Evaluating them

Hey,
As a lowly graduate student, I'm on a committee (I'm not sure how important my role really is) trying to evaluate how effective different approaches teaching undergraduates. We are looking at ...

**2**

votes

**4**answers

3k views

### Undergraduate Derivation of Fundamental Solution to Heat Equation

It is well known that the 1-dimensional heat equation $$\frac{\partial}{\partial t} u(x,t)=a\cdot\frac{\partial^2}{\partial x^2} {u(x,t)}$$ has the fundamental solution $$K(x,t)=\frac{1}{\sqrt{4\pi a ...

**51**

votes

**6**answers

3k views

### What does it take to run a good learning seminar?

I'm thinking about running a graduate student seminar in the summer. Having both organized and participated in such seminars in the past, I have witnessed first-hand that, contrary to what one might ...

**15**

votes

**13**answers

5k views

### Looking for an introductory textbook on algebraic geometry for an undergraduate lecture course

I am now supposed to organize a tiny lecture course on algebraic geometry for undergraduate students who have an interest in this subject.
I wonder whether there are some basic algebraic geometry ...

**5**

votes

**2**answers

761 views

### Maximal Ellipsoid

John's Theorem can be stated as "To every compact, convex body, there is a unique inscribed ellipsoid, whose volume is maximal among all inscribed ellipsoids." It goes on to classify this maximal ...

**34**

votes

**20**answers

6k views

### Fun applications of representations of finite groups

Are there some fun applications of the theory of representations of finite groups? I would like to have some examples that could be explained to a student who knows what is a finite group but does not ...

**18**

votes

**12**answers

7k views

### How seriously should a graduate student take teaching evaluations?

Pretty much the question in the title. If a grad student gets bad reviews as a TA, how much does that hurt them later? How much do good reviews help? What if the situation is more complex? (For ...

**14**

votes

**1**answer

767 views

### Founding of homological without quite involving derived categories

I am looking at the foundations of homological algebra, e.g. the introduction
of Ext and Tor, and am unsatisfied. The references I look at start with
"this is called a projective module, this is ...

**31**

votes

**10**answers

2k views

### effective teaching

Eric Mazur has a wonderful video describing how physics is taught at many universities and his description applies word for word to the way I learned mathematics and the way it is still being taught, ...

**18**

votes

**13**answers

4k views

### Pedagogical question about linear algebra

Last semester I taught a linear algebra class that is intended to introduce young students (at a sophmore-junior level) to "abstract mathematics". It seems that a major conceptual hurdle for many of ...

**21**

votes

**4**answers

3k views

### Curriculum reform success stories at an “average” research university

Greetings all,
There's a never-ending story that many of us have sunk our teeth into. How do we go about teaching subjects like calculus and analysis "well?" Most universities that I'm familiar ...

**0**

votes

**5**answers

2k views

### How to teach addition of negative numbers? [closed]

I have a friend with dyscalculia and was teaching her some some mathematics (namely, solving a linear equation, simplifying certain expressions, and what (affine linear) functions are).
She ...

**15**

votes

**17**answers

12k views

### Undergraduate Differential Geometry Texts

Can anyone suggest any basic undergraduate differential geometry texts on the same level as Manfredo do Carmo's Differential Geometry of Curves and Surfaces other than that particular one?
(I know a ...

**71**

votes

**25**answers

26k views

### What are the most misleading alternate definitions in taught mathematics?

I suppose this question can be interpreted in two ways. It is often the case that two or more equivalent (but not necessarily semantically equivalent) definitions of the same idea/object are used in ...

**24**

votes

**11**answers

7k views

### Lecture notes on representations of finite groups

Next term I am supposed to teach a course on representation of finite groups. This is a third year course for undegrads. I was thinking to use the book of Grodon James and Martin Liebeck ...

**22**

votes

**11**answers

7k views

### Blackboard rendering of math fonts

I learned most of my math font rendering from watching others (for example, I draw ζ terribly). In most cases it is passable, but I'm often uncomfortable using fonts like Fraktur on the board. ...

**1**

vote

**2**answers

5k views

### Examples of random variables

I'm looking for a list of examples of random variables to use in teaching a measure-theoretic probability course. For example, the Rademacher functions are an explicit construction of independent ...

**8**

votes

**3**answers

897 views

### Where can I find questions motivating important ideas in math?

I would like questions that demonstrate why a mathematical tool or technique is useful, and which can be used to introduce that idea. Ideally, this would be a compilation of problems organized by the ...

**17**

votes

**10**answers

64k views

### What are the qualities of a good (math) teacher? [closed]

In forming your answer you may treat the qualifier math or maths as optional, since part of the question is whether there is anything peculiar to the subject of mathematics that demands anything ...

**8**

votes

**2**answers

2k views

### Math History Question about the exponential function

While tutoring a student recently, I have come across the situation of explain logarithms by first introducing functions of the form $$f(x)= a^x$$ where $a \ge 0,x\in \mathbb{R}$. My student then ...

**1**

vote

**0**answers

465 views

### How much of math could be taught without using mathematical notation? [closed]

Given that mathematics is not about number, and that it is not even about the cryptic notation used to describe mathematical problems, how much of mathematics could be taught without reference to ...

**15**

votes

**13**answers

4k views

### Category theory sans (much) motivation?

So I have a friend (no, really) who's taking algebra and is struggling to gain intuition for it. My story is as follows: I used to hate abstract algebra, with pretty much a burning passion, until I ...

**36**

votes

**6**answers

7k views

### Teaching statements for math jobs?

What is the purpose of the "teaching statement" or "statement of teaching philosophy" when applying for jobs, specifically math postdocs? I am applying for jobs, and I need to write one of these ...