**52**

votes

**12**answers

10k 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 ...

**21**

votes

**5**answers

3k 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 ...

**9**

votes

**8**answers

3k 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 ...

**4**

votes

**3**answers

543 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

394 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 ...

**6**

votes

**9**answers

1k 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 ...

**16**

votes

**14**answers

2k 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 ...

**18**

votes

**21**answers

12k 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, ...

**7**

votes

**4**answers

864 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 ...

**13**

votes

**1**answer

1k 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 ...

**5**

votes

**5**answers

979 views

### Topics for a matrix analysis course

I recently taught a new (to my department) course titled "Matrix Analysis". For various reasons that I won't go into here, I was dissatisfied with the textbook I (loosely) followed, and with every ...

**8**

votes

**7**answers

1k views

### Mathematics seminar for “non-mathematicians”

Next term I am leading a seminar for students, who will become teachers for elementary school i.e. for kids of age 6-10. The students in the seminar will have no mathematical background beyond the ...

**11**

votes

**1**answer

1k views

### Good chalk in the UK

Sometime ago it was asked in Mathoverflow about good chalk in the US Where to buy premium white chalk in the U.S., like they have at RIMS?. I will be grateful for any recommendations on good chalk in ...

**34**

votes

**7**answers

11k views

### Collecting proofs that finite multiplicative subgroups of fields are cyclic.

I teach elementary number theory and discrete mathematics to students who come with no abstract algebra. I have found proving the key theorem that finite multiplicative subgroups of fields are cyclic ...

**9**

votes

**3**answers

675 views

### Reference for working with the implicit function theorem

I just had a student come to my office hours and ask me a ton of questions, the answer to all of which was "that's a slight variant to the implicit function theorem, which is proved by formal ...

**17**

votes

**4**answers

2k views

### What is the best way explain to undergraduates that all 1-dimensional manifolds are orientable?

Let's suppose that $M$ is a connected $1$-dimensional smooth manifold (Haussdorf and paracompact). We know that there are exactly two types, up to diffeomorphism (even up to homeomorphism), namely ...

**5**

votes

**1**answer

2k views

### Who is this guy : Z.A. Melzak (wrote Companion to Concrete Mathematics) ? [closed]

Author : Z.A. Melzak
Book Title : Companion to Concrete Mathematics.
Publication : Dover renewed 2004 2 volumes in one. Copyright 1972/1976.
I found this book extremely nice.
To whet your appetite ...

**44**

votes

**22**answers

13k views

### Interesting Calculus Questions/Exercises

I am in the process of redesigning the calculus course that I have taught five or six times. What I would like to know is if anyone has some really good examples or exercises that I could either do ...

**3**

votes

**3**answers

854 views

### Pedagogical question concerning $\Gamma(z)$

Pedagogically speaking, I see two problems with defining
$\Gamma(z)$ (at least for real $z$) by the limit
$$\Gamma(z)=\lim_{m\to\infty}\frac{m! m^z}{\prod_{i=0}^m (z+i)}$$
as compared with the formula
...

**1**

vote

**0**answers

3k views

### What is the geometric meaning of the third derivative of a function at a point? [closed]

What is the geometric meaning of the third derivative of a function at a point?
This question is now asked on the sister site: ...

**4**

votes

**2**answers

747 views

### Terminology question on covering spaces

I'm teaching an elementary class about fundamental groups and covering spaces. It was very useful to use "fool's covering spaces" of a space $X$, defined as
functors $\Pi_1(X)\to Sets$, where ...

**1**

vote

**1**answer

472 views

### What are some interesting grading/curving systems you have seen for a course? [closed]

It seems like every math course has something unique in how things are graded.
1) What are some interesting grading systems you have seen/used? (include curving types, etc.)
2) What are some pros ...

**11**

votes

**2**answers

990 views

### Social Reading Platform for Math or LaTeX texts

Social reading is considered to be one of the big trends that could be catalysing learning by reading. Features could include:
Highlighting or annotating paragraphs or single steps in a proof for ...

**24**

votes

**9**answers

11k views

### Applications of knot theory

An answer of André Henriques' inspired the following closely related CW question. Parts of the following is extracted from his answer and my comments.
I regularly teach a knot theory class. ...

**50**

votes

**11**answers

4k views

### How to introduce notions of flat, projective and free modules?

In the coming spring semester I will be teaching for the first time an introductory (graduate) course in Commutative Algebra. As many people know, I have been plugging away for a while at this ...

**10**

votes

**4**answers

2k views

### How does one motivates the method of separation of variables when teaching PDE's?

I'm not sure if this question is appropriate for MO. Add comments if it is not. Thanks.
How to explain/motivate the method of separation of variables for PDEs to undergraduates? What's the real math ...

**4**

votes

**0**answers

674 views

### Almost linear ODE: how node becomes a spiral

Most introductory ODE books contain a discussion of almost linear systems, and there are two cases when the behavior of an almost linear system near an equilbrium point can differ from the behaviour ...

**18**

votes

**1**answer

1k views

### Resources for teaching arithmetic to calculus students

Every time we teach calculus we discover that a significant portion of our students never understood arithmetic. I don't mean that they can't multiply numbers, but rather that they don't know ...

**18**

votes

**5**answers

3k views

### Varieties as an introduction to algebraic geometry / How do professional algebraic geometers think about varieties

This really is two questions, but they are kind of related so I would like to ask them at the same time.
Question 1:
In a question asked by Amitesh Datta, BCnrd commented that it is important to ...

**2**

votes

**4**answers

2k views

### Best way to introduce the Chinese Remainder Theorem (to a high school student)

What do you think to be the most effective way to teach the Chinese remainder theorem to a smart high school student, which is supposed to only have a soft idea about how modular arithmetic works, and ...

**7**

votes

**2**answers

3k views

### What is the dual concept to “annihilator” called, and do any linear algebra textbooks discuss this concept first?

When introducing dual spaces for the first time, most linear algebra textbooks proceed in what seems to me a rather backwards fashion: the annihilator $\{f\in V^*: f(u)=0\quad \forall u\in U\}$ of a ...

**58**

votes

**10**answers

7k 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

1k 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 ...

**7**

votes

**7**answers

3k views

### Freshman's definition of sin(x) ?

I would like to know how you would rigorously introduce the trigonometric funcions ($sin(x)$ and relatives) to first year calculus students. Suppose they have a reasonable definition of $\mathbb{R}$ ...

**67**

votes

**32**answers

44k 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 ...

**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

1k 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.
...

**20**

votes

**6**answers

4k 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 ...

**7**

votes

**8**answers

1k 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 ...

**7**

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 ...

**22**

votes

**17**answers

3k 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 ...

**2**

votes

**2**answers

635 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 ...

**31**

votes

**19**answers

7k 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 ...

**18**

votes

**7**answers

5k 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?

**37**

votes

**8**answers

11k 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) ...

**30**

votes

**21**answers

10k 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 ...

**22**

votes

**7**answers

3k 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 ...

**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 ...

**20**

votes

**11**answers

5k 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 ...