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/.

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6 answers
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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 ...
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 ...
Aaron Meyerowitz's user avatar
16 votes
6 answers
3k 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 ...
16 votes
5 answers
1k views

Permission to use Online Notes

I am a new professor in Mathematics and I am running an independent study on Diophantine equations with a student of mine. Online I have found a wealth of very helpful expository notes written by ...
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 ...
Amir Asghari's user avatar
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16 votes
2 answers
1k 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. ...
rrrrrrr's user avatar
  • 161
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 ...
Joseph O'Rourke's user avatar
15 votes
4 answers
3k 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 ...
Yuhao Huang's user avatar
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15 votes
1 answer
691 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 ...
14 votes
9 answers
4k 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 ...
Hailong Dao's user avatar
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14 votes
5 answers
2k 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 ...
14 votes
7 answers
5k views

Freshman's definition of sin(x)?

I would like to know how you would rigorously introduce the trigonometric functions ($\sin(x)$ and relatives) to first year calculus students. Suppose they have a reasonable definition of $\mathbb{R}$ ...
Qfwfq's user avatar
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14 votes
3 answers
3k 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 ...
14 votes
2 answers
7k 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 ...
14 votes
9 answers
2k 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 votes
1 answer
959 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 ...
Allen Knutson's user avatar
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....
13 votes
7 answers
34k 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 ...
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+\...
Denis Serre's user avatar
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13 votes
3 answers
2k 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 ...
Giuseppe's user avatar
  • 193
13 votes
1 answer
3k 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 ...
Charles Staats's user avatar
13 votes
4 answers
2k views

"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'...
aglearner's user avatar
  • 14k
13 votes
1 answer
599 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 $\mathbb{P}^3$ ...
David E Speyer's user avatar
13 votes
1 answer
2k 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 ...
James Propp's user avatar
  • 19.4k
13 votes
2 answers
2k 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 ...
pro's user avatar
  • 534
12 votes
11 answers
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 ...
12 votes
17 answers
3k 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 ...
12 votes
9 answers
6k views

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 ...
12 votes
1 answer
518 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 ...
Francois Ziegler's user avatar
12 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 on,...
B. Bischof's user avatar
  • 4,782
12 votes
1 answer
755 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 ...
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 ...
11 votes
3 answers
713 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 $\...
Lutz Mattner's user avatar
11 votes
4 answers
848 views

Interesting examples of systems of linear differential equations with constant coefficients

In this paper, Gian-Carlo Rota wrote: A lot of interesting systems with constant coefficients have been discovered in the last thirty years: in control, in economics, in signal processing, even in ...
Michael Hardy's user avatar
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 ...
Uwe Franz's user avatar
  • 2,191
11 votes
1 answer
1k views

Teaching Experience for Graduate Students. [closed]

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 ...
11 votes
2 answers
1k 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 ...
11 votes
3 answers
434 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 ...
David E Speyer's user avatar
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 ...
Eugene Lerman's user avatar
11 votes
1 answer
2k 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 ...
11 votes
0 answers
1k 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 ...
David E Speyer's user avatar
10 votes
7 answers
2k 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 ...
10 votes
5 answers
3k views

Assessing effectiveness of (epsilon, delta) definitions [closed]

There is much discussion both in the education community and the mathematics community concerning the challenge of (epsilon, delta) type definitions in calculus and the student reception of them. The ...
Mikhail Katz's user avatar
  • 15.1k
10 votes
3 answers
3k 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 ...
user1447's user avatar
  • 299
10 votes
8 answers
2k 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 ...
10 votes
3 answers
1k 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....
Konstantinos Kanakoglou's user avatar
10 votes
4 answers
1k 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 ...
David E Speyer's user avatar
10 votes
3 answers
717 views

Calculus Teaching: Is it possible or desirable to give a severely abbreviated treatment of series convergence tests?

I will be teaching Calculus 2 this fall at a large U.S. state university. Our incoming students tend to have a limited or inconsistent background, which limits the amount of material we can cover. ...
9 votes
3 answers
1k views

Books on the relationship between the Socratic method and mathematics?

Apart from books on heuristics by George Polya. When trying to engage with and understand mathematical concepts and when applying abstract mathematical concepts to model "continuum" or real ...
James Fife's user avatar
9 votes
4 answers
2k 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 ...
candl's user avatar
  • 93