All Questions
495 questions
36
votes
11
answers
10k
views
Categories First Or Categories Last In Basic Algebra?
Recently, I was reminded in Melvyn Nathason's first year graduate algebra course of a debate I've been having both within myself and externally for some time. For better or worse, the course most ...
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 "...
37
votes
6
answers
4k
views
Taylor's theorem and the symmetric group
Anytime I see an $n!$ in some formula, my instinct is to look for the symmetric group on $n$ letters coming in somewhere. I have never done this seriously with the $n!$ in Taylor's theorem.
Question: ...
- 12.1k
3
votes
1
answer
234
views
Geometric construction exercises
Many of you know dynamic geometry exercises in Euclidea; if not, here is one example.
It lets you do a geometric construction and sends a message once you achieve the result.
I am looking for a way to ...
- 45k
96
votes
5
answers
8k
views
Is there a database for tracking the dependencies of mathematical theorems?
Given a proof for a result, one could denote the proof as a node on a graph, and then draw arrows to the node from axioms and previous results that the proof uses, and then draw arrows from the node ...
- 2,680
58
votes
4
answers
5k
views
Advice for PhD Supervisors
My first PhD student is having his viva tomorrow. Hence, I began contemplating a bit about the whole process of supervising. One thing I realized is that while there seems to be plenty of advice for ...
24
votes
9
answers
9k
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 epsilon-...
19
votes
3
answers
1k
views
What kind of computer tools topologists/geometers use to visualize the objects they deal with?
I have recently started to read a bit about geometry and topology. Hopf fibration, Lense spaces, CW complexes, stuff that are discussed in Hatcher's Algebraic Topology and other things that require ...
- 351
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 ...
- 107
45
votes
14
answers
13k
views
Examples of undergraduate mathematics separation from what mathematicians should know
I'm looking for examples of four kinds of things:
Material that is usually covered in standard undergraduate mathematics courses and/or in first-year graduate work (or tested in qualifying ...
37
votes
18
answers
5k
views
Insightful books about elementary mathematics
What are some books that discuss elementary mathematical topics ('school mathematics'), like arithmetic, basic non-abstract algebra, plane & solid geometry, trigonometry, etc, in an insightful way?...
36
votes
3
answers
3k
views
What do we learn from the Wronskian in the theory of linear ODEs?
For a real interval $I$ and a continuous function $A: I \to \mathbb{R}^{d\times d}$, let $(x_1, \dots, x_d)$ denote a basis of the solution space of the non-autonomous ODE
$$
\dot x(t) = A(t) x(t) \...
- 12.6k
34
votes
18
answers
20k
views
Interesting and accessible topics in graph theory
This summer, I will be teaching an introductory course in graph theory to talented high school seniors. The intent of the course is not to establish proficiency in graph theory, per se. Rather, I hope ...
59
votes
5
answers
25k
views
Are there any "related rates" calculus problems that don't feel contrived?
I just finished teaching a freshman calculus course (at an American state university), and one standard topic in the curriculum is related rates. I taught my students to answer questions such as the ...
7
votes
1
answer
723
views
Alternate algorithms for Chinese remainder theorem
I was teaching Discrete this semester and set the students loose on a system of linear congruences. One of them came up with this solution. Say $$ x \equiv 1 \textrm{ mod } 3 $$ $$ x \equiv 3 \textrm{ ...
- 525
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 ...
1
vote
0
answers
109
views
Problems Correction of "Algebra, Topology, Differential Calculus, and Optimization Theory For Computer Science and Machine Learning "' [closed]
Where I can find the problems correction of this book " Algebra, Topology, Differential Calculus, and Optimization Theory For Computer Science and Machine Learning "
- 11
27
votes
8
answers
5k
views
Conceptual algebraic proof that Grassmannian is closed in Plücker embedding
I'm planning lectures for my intro algebraic geometry course, and I noted something awkward that is coming up. We're starting projective varieties soon. Of course, we'll prove that projective maps are ...
- 156k
67
votes
9
answers
7k
views
Taking "Zooming in on a point of a graph" seriously
In calculus classes it is sometimes said that the tangent line to a curve at a point is the line that we get by "zooming in" on that point with an infinitely powerful microscope. This explanation ...
- 12.1k
2
votes
1
answer
295
views
Examples of new results found via exams [closed]
I suspect that there have been many instances throughout history where a new proof of an existing result has been discovered by a student while taking an exam. Does anyone have an example of this?
61
votes
13
answers
9k
views
How do you approach your child's math education? [closed]
My son is one year old, so it is perhaps a bit too early to worry about his mathematical education, but I do. I would like to hear from mathematicians that have older children: What do you wish you'd ...
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 ...
34
votes
13
answers
6k
views
Elementary applications of linear algebra over finite fields
I'm teaching axiomatic linear algebra again this semester. Although the textbooks I'm using do everything over the real or complex numbers, for various reasons I prefer to work over an arbitrary ...
60
votes
8
answers
13k
views
Why the Killing form?
I'm teaching a short summer course on algebraic groups and it's time to talk about the Killing form on the Lie algebra. The students are all undergrads of varying levels of inexperience, and I try to ...
- 7,273
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 ...
30
votes
15
answers
5k
views
Making sure that you have comprehended a concept
I have a question that I've been thinking about for a long time.
How can you assure yourself that you've fully comprehended a concept or the true meaning of a theorem in mathematics?
I mean how can ...
2
votes
0
answers
237
views
Solve the recurrence relation with 2 variables
We have the following recurrence relation:
\begin{equation}
f(n,m) = f(n-1,m) g_{\alpha, \gamma}(n,m) + f(n,m-1) g_{\beta, \gamma}(n,m) \\
g_{\alpha, \gamma}(n,m)= \sum^{n}_{i = 0} \sum^{m}_{j = 0} \...
- 105
17
votes
5
answers
3k
views
Teaching prime number theorem in a complex analysis class for physicists
This is a question about pedagogy.
I want to sketch the proof of the prime number theorem or any other application of complex analysis to number theory in a single lecture, in a complex analysis ...
- 253
24
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 ...
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 ...
27
votes
19
answers
26k
views
Good combinatorics textbooks for teaching undergraduates?
Hello, can anyone recommend good combinatorics textbooks for undergraduates? I will be teaching a 10-week course on the subject at Stanford, and I assume that the students will be strong and motivated ...
27
votes
5
answers
6k
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 ...
- 22.1k
24
votes
3
answers
4k
views
What aspects of math olympiads do you find still useful in your math research?
I was rereading the book Littlewood's Miscellany and this passage struck me:
It used to be said that the discipline in 'manipulative skill' bore
later fruit in original work. I should deny this ...
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 ...
41
votes
3
answers
3k
views
Can the unsolvability of quintics be seen in the geometry of the icosahedron?
Q1. Is it possible to somehow "see" the unsolvability of quintic polynomials
in the $A_5$ symmetries of the icosahedron (or dodecahedron)?
Perhaps this is too vague a question.
Q2. Are there ...
- 151k
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 ...
43
votes
9
answers
29k
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. ...
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 ...
4
votes
1
answer
1k
views
Chalkboard eraser [closed]
I just started my first year of university and because I'm visually impared I have trouble seeing what's written on the chalkboard.
I've partially solved this problem by purchasing chalk from hagoromo ...
43
votes
7
answers
12k
views
On starting graduate school and common pitfalls...
Hi,
I'll be starting graduate school soon, and when I look back at my college career, there are certain things I wish I could have done differently. In hindsight, I wished I wasn't in such a rush to ...
19
votes
14
answers
4k
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 ...
87
votes
2
answers
4k
views
History of $\frac d{dt}\tan^{-1}(t)=\frac 1{1+t^2}$
Let $\theta = \tan^{-1}(t)$. Nowadays it is taught:
1º that
$$
\frac{d\theta}{dt} = \frac 1{dt\,/\,d\theta} = \frac 1{1+t^2},
\tag1
$$
2º that, via the fundamental theorem of calculus, this is ...
- 31.5k
17
votes
4
answers
2k
views
Some interesting and elementary topics with connections to the representation theory?
I'm going to give a talk to talented high school seniors (for nearly 1.25-1.75 hours, maybe a little bit longer). They know some abstract algebra (groups, rings, modules...), linear algebra (...
- 181
5
votes
9
answers
7k
views
Applications of basic linear algebra concepts to computer science? [closed]
I'm trying to explain linear algebra to some programmers with computer science backgrounds. They took a course on it long ago, but don't seem to remember much. They can follow basic formalism, but ...
- 4,164
1
vote
0
answers
155
views
Introducing generating functions to engineer audience?
What is a good way of summarizing when "generating function" approach is useful to an audience of practitioners?
I'm giving a talk on training neural networks (see Velikanov, Kuznedelev, and ...
- 2,252
49
votes
5
answers
4k
views
How do you mentor undergraduate research?
Lets say you had an undergraduate who wanted to do some advanced work and some research, possibly for a thesis, or things like that.
There are two slightly more specific groups of questions I have ...
13
votes
4
answers
998
views
Source for analysis of identification of structures in learner's mind and mathematical structures?
Concerning the structure of the learner's mind, psychologist Piaget claimed that
There exists, as a function of the development of intelligence as a whole, a spontaneous and gradual construction of ...
- 16.6k
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 ...
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, ...
61
votes
10
answers
10k
views
Teaching proofs in the era of Google
Dear members,
Way back in the stone age when I was an undergraduate (the mid 90's), the internet was a germinal thing and that consisted of not much more than e-mail, ftp and the unix "talk" command ...