All Questions
495 questions
19
votes
10
answers
6k
views
Research Experience for Undergraduates: Summer Programs
Some time ago, I found this list of REU programs held in 2009.
The main aspects that characterize such programs are: (a) a great deal of lectures on specific topics; and, admittedly more importantly,...
165
votes
28
answers
56k
views
Cool problems to impress students with group theory [closed]
Since this forum is densely populated with algebraists, I think I'll ask it here.
I'm teaching intermediate level algebra this semester and I'd like to entertain my students with some clever ...
57
votes
34
answers
13k
views
Are there any books that take a 'theorems as problems' approach?
Are there any books that present theorems as problems? To be more specific, a book on elementary group theory might have written: "Theorem: Each group has exactly one identity" and then show a proof ...
9
votes
4
answers
10k
views
Applications of Euler-Cauchy ODEs
The Euler-Cauchy ODE (2nd order, homogeneous version) is:
$$
x^2 y'' + a x y' + b y = 0
$$
Looking in various books on ODEs and a random walk on a web search (i.e. I didn't click on every link, but ...
- 26.8k
11
votes
2
answers
2k
views
Teaching and students
Sometimes I get stumped by students' questions in my classes I teach. I am an algebraist by training and have just started teaching. Sometimes I have to teach analysis courses. My question is: Is it ...
- 127
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 ...
- 1,588
50
votes
4
answers
7k
views
Motivation for concepts in Algebraic Geometry
I know there was a question about good algebraic geometry books on here before, but it doesn't seem to address my specific concerns.
**
Question
**
Are there any well-motivated introductions to ...
- 12.1k
12
votes
5
answers
9k
views
How seriously do professors take teaching evaluations? [closed]
Do they ever know who writes them? How seriously do departments take teaching evaluations? If a professor knows which student wrote a particular evaluation....would they be biased (e.g. be nicer, etc.....
25
votes
6
answers
25k
views
What are the advantages and disadvantages of the Moore method?
Describe your experiences with the Moore method. What are its advantages and disadvantages?
46
votes
15
answers
11k
views
Strong induction without a base case
Strong induction proves a sequence of statements $P(0)$, $P(1)$, $\ldots$ by proving the implication
"If $P(m)$ is true for all nonnegative integers $m$ less than $n$, then $P(n)$ is true."
for ...
30
votes
6
answers
11k
views
Mathematics for machine learning
I would like to know what mathematics topics are the most important to learn before actually studying the theory on neural networks.
I ask that because I will start to learn about neural networks and ...
- 419
69
votes
20
answers
19k
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 ...
6
votes
8
answers
1k
views
Reference for elementary and "cool" statistics or financial math
I signed up for a Math Mentorship Program (for high school students) this term, but one of the students assigned to me is more interested in Statistics and Finance - something that would help him to ...
16
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
961
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 ...
- 27.9k
81
votes
18
answers
24k
views
Depressed graduate student. [closed]
How does a depressed graduate student go about recovering his enthusiasm for the subject and the question at hand?
Edit: I am not that grad student; it is a very talented friend of mine.
Moderator's ...
45
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 ...
23
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 ...
97
votes
17
answers
17k
views
What's a nice argument that shows the volume of the unit ball in $\mathbb R^n$ approaches 0?
Before you close for "homework problem", please note the tags.
Last week, I gave my calculus 1 class the assignment to calculate the $n$-volume of the $n$-ball. They had finished up talking about ...
124
votes
37
answers
12k
views
One-step problems in geometry
I'm collecting advanced exercises in geometry. Ideally, each exercise should be solved by one trick and this trick should be useful elsewhere (say it gives an essential idea in some theory).
If you ...
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 ...
- 648
26
votes
18
answers
34k
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 ...
150
votes
31
answers
70k
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 ...
8
votes
4
answers
2k
views
Choice of adviser
Not sure how to tag this one so feel free to edit and add tags.
When I initially started graduate school my choice for an area of study was quite nebulous. I had only figured out enough to know that ...
15
votes
13
answers
23k
views
Math journal for high school students?
I recently discovered The College Mathematics Journal and enjoyed reading through some of the articles on fun applications of mathematics. I'd like to send some of the articles to my younger sister, a ...
- 169
168
votes
37
answers
207k
views
Too old for advanced mathematics? [closed]
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, ...
33
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
17k
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. ...
- 52.7k
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 ...
- 5,227
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 ...
17
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 ...
16
votes
7
answers
6k
views
How have mathematicians been raised? [closed]
Many of us have -- or at some point want to have -- children, and wonder how we can do our best to fulfill the "nurture" component of helping them develop mathematical talent... not because we want ...
35
votes
11
answers
5k
views
Are there elementary-school curricula that capture the joy of mathematics?
UPDATE: Wow, thank you everyone for the great insights!
A couple of months ago I stumbled across Paul Lockhart's essay A Mathematician's Lament and it made perfect sense to me. I'm not meaning to ...
9
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 ...
- 297
12
votes
3
answers
1k
views
Is formal proof (formalized mathematics) interesting to practicing mathematicians? To educators? [closed]
Formalizing mathematical proofs so that they can be checked for correctness and manipulated by computer is a recurrent proposal, most notably stated in the QED manifesto (1994). The December 2008 ...
17
votes
5
answers
5k
views
Pacing for learning new material [closed]
I'm beginning to run into work where I have to do a significant amount of learning of math by myself, with a book rather than with a teacher. Now, I do know that doing problems tends to be the best ...
71
votes
10
answers
20k
views
Relating category theory to programming language theory
I'm wondering what the relation of category theory to programming language theory is.
I've been reading some books on category theory and topos theory, but if someone happens to know what the ...
- 1,801
44
votes
42
answers
15k
views
What should be offered in undergraduate mathematics that's currently not (or isn't usually)? [closed]
What's one class that mathematics that should be offered to undergraduates that isn't usually? One answer per post.
Ex: Just to throw some ideas out there
Mathematical Physics (for math students, not ...
2
votes
0
answers
526
views
How much of math could be taught without using mathematical notation? [closed]
Given that mathematics is not about number, and that it is not even about the cryptic notation used to describe mathematical problems, how much of mathematics could be taught without reference to ...
1
vote
2
answers
1k
views
An "Elementary" Math Question Generalized (Ring Theory Perhaps)
The following question is posed in the book "The USSR Olympiad Problem Book: Selected Problems and Theorems of Elementary Mathematics"
"Prove that if integers a_1, ..., a_n are all distinct, then the ...
- 1,801
195
votes
30
answers
78k
views
Real-world applications of mathematics, by arxiv subject area?
What are the most important applications outside of mathematics of each of the major fields of mathematics? For concreteness, let's divide up mathematics according to arxiv mathematics categories, e.g....
2
votes
2
answers
349
views
Characterizing triangles unembeddedly
The mathedu mailing list has a recent longish thread at
http://www.nabble.com/Why-do-we-do-proofs--to25809591.html
which discussed among other things whether we should teach triangles as labeled or ...
- 1,160
22
votes
13
answers
8k
views
Category theory sans (much) motivation?
So I have a friend (no, really) who's taking algebra and is struggling to gain intuition for it. My story is as follows: I used to hate abstract algebra, with pretty much a burning passion, until I ...
80
votes
7
answers
20k
views
Teaching statements for math jobs?
What is the purpose of the "teaching statement" or "statement of teaching philosophy" when applying for jobs, specifically math postdocs? I am applying for jobs, and I need to write one of these ...