For questions related to teaching mathematics. For questions in Mathematics Education as a scientific discipline there is also the tag mathematics-education.

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2
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3answers
333 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 ...
11
votes
3answers
2k views

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 ...
9
votes
3answers
946 views

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 ...
11
votes
9answers
1k views

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 ...
14
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12answers
2k views

Excellent uses of induction and recursion

Can you make an example of a great proof by induction or construction by recursion? Given that you already have your own idea of what "great" means, here it can also be taken to mean that the chosen ...
4
votes
1answer
656 views

Is Diagonalization worth to be taught? [closed]

When students come to the College (first two years of the University system in most of the developped countries) to train in mathematics, they get a linear algebra / matrix analysis course. After a ...
10
votes
1answer
815 views

Teaching Experience for Graduate Students.

I am currently a graduate student, who will (hopefully!) graduate in the next year (or two..). I have slowly come to realize that I enjoy teaching, and consequently want to do more of it! My main ...
19
votes
2answers
783 views

Direct proof that the centralizer of $GL(V)$ acting on $V^{\otimes n}$ is spanned by $S_n$

Let $V$ be a finite dimensional vector space over a field of characteristic zero. Let $A$ be the space of maps in $\mathrm{End}(V^{\otimes n})$ which commute with the natural $GL(V)$ action. Clearly, ...
14
votes
6answers
2k views

How to mentor an exceptional high school student?

I have a unique and, quite truthfully, humbling opportunity. The parents of an exceptionally talented high school freshman have reached out to me and asked if I might be able to help. This kid is ...
15
votes
17answers
2k views

Readings for an honors liberal art math course

Our university has an Honors section of our "liberal arts mathematics" course. Typically 10-20 students enroll each Fall, with most of them extremely bright, but lacking the interest and/or ...
18
votes
9answers
4k views

How to motivate and present epsilon-delta proofs to undergraduates?

This would seem to be a common question, but I am surprised not to see it already asked and answered on MO! I am teaching an undergraduate course, and I want to teach them to construct basic ...
2
votes
2answers
363 views

Decomposition of $K_{10}$ in copies of the Petersen graph

It is a well-known and cute exercise in algebraic graph theory to show that $K_{10}$ cannot be written as the edge-disjoint union of three copies of the Petersen graph $P$. Indeed, the graph $G$ whose ...
7
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6answers
2k views

Explaining the concept of projective space: notes for students

This is a question on teaching. I am teaching at this moment a course in algebraic geometry for master students on a very basic level. Today (this was the fourth lecture) I discovered that only four ...
6
votes
4answers
2k views

Help me find good math questions for my students

I am a teacher at 西铁一中。 I teach mathematics in English for students going abroad. Now this is my problem, there are few mathematics books written in English that are at the level of high school, ...
8
votes
4answers
1k views

Topological examples of profinite groups

I am preparing a course on profinite groups, to be delievered to early graduate students. The first part of the course will discuss the equivalent characterizations of profinite groups. I will first ...
1
vote
0answers
379 views

Professional skills advising for math jobs [closed]

Hi, I am a postdoc at the University of Nottingham (UK) and I am beginning to apply for Assistant Professor positions in US. I would like to receive a feedback on the material that I am sending ...
4
votes
2answers
947 views

Any suggestions for a course in Mathematical Logic?

I am teaching a topics course for Mathematics majors (at Temple), and am considering Logic as the topic. I was wondering if people (a) have suggestions for an appropriate text and (b) how much might ...
12
votes
17answers
2k views

Short Course Suggestions For High School Students

I am planning to teach a course for talented high school students at a summer camp and I need suggestions for possible topics. The students usually have different backgrounds but most of them are ...
15
votes
8answers
5k views

Interesting Applications of the Classical Stokes Theorem?

When students learn multivariable calculus they're typically barraged with a collection of examples of the type "given surface X with boundary curve Y, evaluate the line integral of a vector field Y ...
33
votes
4answers
5k views

Are there any “related rates” calculus problems that don't feel contrived?

I just finished teaching a freshman calculus course (at an American state university), and one standard topic in the curriculum is related rates. I taught my students to answer questions such as the ...
12
votes
5answers
1k views

How to make a lecture series useful

I have been to a number of advanced lecture courses (of between 3 and 10 lectures) over the years, given (in principle) by experts to graduate students and experts in neighbouring fields. Examples of ...
21
votes
17answers
3k views

Using slides in math classroom

I am toying with the idea of using slides (Beamer package) in a third year math course I will teach next semester. As this would be my first attempt at this, I would like to gather ideas about the ...
3
votes
3answers
1k views

What to teach in a second graduate course in algebra? What textbook to use?

There is a standard syllabus for a first graduate course in algebra. One teaches groups, rings, fields, perhaps a little bit of Galois theory, perhaps a little bit of category theory, perhaps a ...
12
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11answers
2k views

Giving a math talk with no blackboard or projector

I need to give a math talk to a group of undergraduates. I am asking for advice because this talk will take place at a department picnic and there will be no blackboard or projector. I would like to ...
6
votes
5answers
1k views

Commutative algebra final project

I'm looking for a topic for a final project in commutative/homological algebra, for first year master's students (in a decent European university). During the course, they will cover the following ...
40
votes
44answers
13k views

An example of a beautiful proof that would be accessible at the high school level?

The background of my question comes from an observation that what we teach in schools does not always reflect what we practice. Beauty is part of what drives mathematicians, but we rarely talk about ...
37
votes
10answers
5k 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 ...
15
votes
4answers
2k 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 ...
8
votes
9answers
2k 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
3answers
499 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
0answers
370 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 ...
5
votes
9answers
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
14answers
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 ...
12
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19answers
7k 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
4answers
793 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
1answer
985 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
5answers
895 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
7answers
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 ...
10
votes
1answer
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 ...
17
votes
7answers
5k 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 ...
8
votes
3answers
613 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 ...
15
votes
4answers
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 ...
4
votes
1answer
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 ...
31
votes
18answers
8k 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
3answers
822 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 ...
4
votes
2answers
712 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
1answer
450 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 ...
10
votes
2answers
968 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 ...
16
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9answers
7k 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. ...
40
votes
11answers
3k 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 ...