In the fall, I am teaching one undergraduate and one graduate course, and in planning these courses I have been thinking about alternatives to the "standard math class". I have found it much easier to find different models for undergraduate classes than for the graduate classes. There are many blogs and other online resources focusing in large part on math pedagogy, often taking "nonstandard" approaches to the math course (Moore method, inquiry based learning, flipped or inverted classroom, Eric Mazur's "Peer instruction",...). For instance,...Read more

I'm going to be a teaching assistant for an undergraduate class in abstract algebra next semester, for students who have not taken abstract algebra before. It will deal with group theory and linear algebra, but the students have already had a semester of linear algebra so I'm thinking about how to deal with group theory.I'd like to present some examples of applications of group theory that motivate the theoretical questions the course will deal with. For instance, "symmetries of three-dimensional objects form groups. Crystals have symmetrical s...Read more

Kind of an odd question, perhaps, so I apologize in advance if it is inappropriate for this forum. I've never taken a mathematics course since high school, and didn't complete college. However, several years ago I was affected by a serious illness and ended up temporarily disabled. I worked in the music business, and to help pass the time during my convalescence I picked up a book on musical acoustics.That book reintroduced me to calculus with which I'd had a fleeting encounter with during high school, so to understand what I was reading I fig...Read more

In my experience, there are roughly two approaches to teaching (North American) undergraduates to write proofs:Students see proofs in lecture and in the textbooks, and proofs are explained when necessary, for example, the first time the instructor shows a proof by induction to a group of freshman, some additional explanation of this proof method might be given. Also, students are given regular problem sets consisting of genuine mathematical questions - of course not research-level questions, but good honest questions nonetheless - and they get...Read more

Without prethought, I mentioned in class once that the reason the symbol $\partial$is used to represent the boundary operator in topology isthat its behavior is akin to a derivative.But after reflection and some research,I find little support for my unpremeditated claim.Just sticking to the topological boundary (as opposed to the boundaryof a manifold or of a simplicial chain),$\partial^3 S = \partial^2 S$for any set $S$.So there seems to be no possible analogy toTaylor series. Nor can I see an analogy with thefundamental theorem of calculus.T...Read more

Why we use the symbol $\sqrt{}$ when we take square roots ? Anybody knows the history ?...Read more

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 familiar with single variable calculus and very basic linear algebra over the reals. The teaching format will be two hours per day, six days per week and two weeks in total. Suggestions for one week courses are also welcome.There are two things I want about this course. First, it should have a direction and a final goal. So it shouldn't be based on is...Read more

I'd like to know whether any form a certain hypothesis about thelearning of higher mathematics has entered the mathematical oreducational literature. I'll frame the hypothesis here but not defendit since this is not a blog-in-disguise; likewise I'm not solicitingdebate.This hypothesis opposes to some degree the common shibboleth whichholds that "mastering abstraction" constitutes the single majorplateau which undergraduate mathematics students must, but often donot, scale.For the sake of making the distinction, I'll first flesh out what Imean...Read more

(I know this is perhaps only tangentially related to mathematics research, but I'm hoping it is worthy of consideration as a community wiki question.)Today, I was reminded of the existence of this paper: Terminating Decimals in the Cantor Set. It is a concise paper (3.5 pages) that employs nothing too sophisticated, just some modular arithmetic and careful casing. I thought it would be a wonderful to spend a few class meetings with undergraduate math majors reading this paper for understanding. We could practice reading the dense writing and fi...Read more

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 beauty in the teaching of school mathematics. I'm trying to collect examples of good, accessible proofs that could be used in middle school or high school. Here are two that I have come across thus far:(1) Pick's Theorem: The area, A, of a lattice polygon, with boundary points B and interior points I is A = I + B/2 - 1.I'm actually not so interest...Read more

Does the following exist, and if not, does anyone besides me wish it did? A web site where a mathematician (say) could find other mathematicians who want to study the same book or paper, and arrange to meet via videoconference, and run their own informal seminar around that topic, and then disband when they're done.The reason I ask is that I have plenty of material I'd like to learn this way. I have an interesting job in industry, with occasional mathematical challenges, but there are more topics I would like to learn outside of work, and that ...Read more

There are many different styles of lecturing, and many different aspects that are blended together to give a whole "lecturing style". That said, I'm particularly interested in hearing people's experiences with so-called "handouts". At one extreme lie the lecturers who "dictate" a set of notes (usually not actual dictation, but by writing on a board) whilst at the other are lecturers who distribute complete lecture notes in advance.As this is math overflow, I realise that it is extremely unlikely that it will be possible to answer the question...Read more

Is it possible to be a great mathematician in our home with a laptop+poor internet+electronic books+some books+a little food +a little money or not? without having a constant jobwithout studying P.H.D or going to university or having a good master or traveling to other countries , in real life? i mean the life with all of it's problems?it's not the whole problem: you don't have to be necessarily a person like "Sir Andrew Wiles" in a short time like 5 years studying. i mean consider a person who is not beginner in math. consider a normal person ...Read more

A few days ago I was asked by the director of the Center for Undergraduate Research and Scholarship at Georgia Regents University (formerly known as MCG and Augusta State) to contribute an article for the new Faculty Handbook: Mentoring Undergraduates in Research and Scholarship which is to be written along the lines of the University of Alaska Anchorage Handook. In my short time at Georgia Regents University I was indeed one of the first mathematicians who engaged in "Undergraduate Research" in part due to the nature of my subject Dynamical Sy...Read more

How does one return to graduate school after spending a couple years in the industry? In particular, what are ways of getting good recommendations? I'm not concerned about the "adjustment" to the grad student lifestyle, but rather about the application process if the goal is a top school.I was a CS/math major at MIT for undergrad, but wasn't really sure at the time if I wanted to go into academia, so I ended up doing more software and machine learning. For a while now, though, I've been realizing that I miss the academic life, so I've been thin...Read more