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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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. ...
Hannay's user avatar
  • 1
20 votes
6 answers
6k 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 ...
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 ...
9 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 ...
32 votes
20 answers
5k 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 ...
3 votes
2 answers
941 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 ...
Federico Poloni's user avatar
35 votes
19 answers
9k 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 ...
24 votes
7 answers
7k 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?
51 votes
9 answers
25k 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) ...
51 votes
22 answers
18k 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 ...
24 votes
7 answers
4k 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 ...
Ilya Grigoriev's user avatar
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 ...
5 votes
3 answers
763 views

Euclidean function of Euclidean domain defined at 0

In a few places where I have looked the Euclidean Function of a Euclidean Domain is only being defined for non-zero elements. I am teaching an undergraduate course and I am trying to make things as ...
24 votes
11 answers
8k 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 ...
86 votes
44 answers
20k views

Demystifying complex numbers

At the end of this month I start teaching complex analysis to 2nd year undergraduates, mostly from engineering but some from science and maths. The main applications for them in future studies are ...
33 votes
20 answers
5k views

Do names given to math concepts have a role in common mistakes by students?

Perhaps this question overlaps with similar ones, ... but I want to focus on a particular possible cause of confusion. I notice that students are often confused by the concepts of "infinite" and "...
27 votes
5 answers
7k views

References for "modern" proof of Newlander-Nirenberg Theorem

Hi, I'm starting to prepare a graduate topics course on Complex and Kahler manifolds for January 2011. I want to use this course as an excuse to teach the students some geometric analysis. In ...
20 votes
9 answers
5k views

Mathematics and autodidactism

Mathematics is not typically considered (by mathematicians) to be a solo sport; on the contrary, some amount of mathematical interaction with others is often deemed crucial. Courses are the student's ...
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 ...
23 votes
15 answers
5k 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 of ...
154 votes
7 answers
83k views

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

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 ...
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
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
7 votes
3 answers
2k 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 ...
Charles Staats's user avatar
40 votes
13 answers
19k 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 ...
Hailong Dao's user avatar
  • 30.3k
20 votes
7 answers
3k 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 ...
jlk's user avatar
  • 3,234
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
  • 30.3k
40 votes
16 answers
11k 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
889 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 ...
9 votes
4 answers
4k 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 ...
Hailong Dao's user avatar
  • 30.3k
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 ...
2 votes
4 answers
6k 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 ...
vonjd's user avatar
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51 votes
6 answers
5k 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 ...
18 votes
12 answers
10k 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 ...
7 votes
2 answers
1k 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 ...
Ben Weiss's user avatar
  • 1,588
68 votes
20 answers
18k 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 ...
17 votes
12 answers
10k views

How seriously should a graduate student take teaching evaluations? [closed]

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
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
46 votes
10 answers
4k 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, ...
23 votes
13 answers
7k 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 ...
24 votes
4 answers
4k 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 ...
Tommi's user avatar
  • 648
26 votes
18 answers
32k 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 ...
148 votes
31 answers
69k 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 ...
32 votes
11 answers
13k 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 "...
42 votes
11 answers
16k 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. ...
Tyler Lawson's user avatar
  • 51.1k
2 votes
2 answers
6k 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 ...
John D. Cook's user avatar
  • 5,147
9 votes
3 answers
1k 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 ...
18 votes
10 answers
109k 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 ...