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/.
250
questions
2
votes
1
answer
219
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?
46
votes
7
answers
7k
views
Zorn's lemma: old friend or historical relic?
It is often said that instead of proving a great theorem a mathematician's fondest dream is to prove a great lemma. Something like Kőnig's tree lemma, or Yoneda's lemma, or really anything from this ...
- 17.3k
7
votes
1
answer
251
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{ ...
- 455
1
vote
1
answer
106
views
Resources on blended teaching and flipped classroom in undergraduate mathematics education [closed]
I'd like to learn about the implementation of "blended teaching" in general and "flipped classroom" in particular for the teaching of undergraduate mathematics. Can anyone ...
- 141
26
votes
2
answers
2k
views
What is the origin/history of the following very short definition of the Lebesgue integral?
Typical courses on real integration spend a lot of time defining the Lebesgue measure and then spend another lot of time defining the integral with respect to a measure. This is sometimes criticized ...
- 26.5k
1
vote
0
answers
86
views
what belongs in a first university-level geometry course? [closed]
I know this is not really a research question, but I would like to ask it of research mathematicians, to see if there is a consensus. In a recent discussion on this topic, someone suggested that if ...
- 89
7
votes
2
answers
1k
views
Mathematics of sustainable development and energy sobriety in the classroom
Faculty members are encouraged to highlight the connection between the courses we teach and climate change, and raise awareness of the issue in our lectures, across subjects in my university. I am ...
0
votes
0
answers
182
views
Is Baire's theorem stronger than needed for functional analysis?
Many classic theorems in functional analysis involve using Baire's theorem to prove facts about topology that relate to maps between Banach spaces (or, more generally, F-spaces). The application ...
- 109
22
votes
2
answers
2k
views
Can one deduce the fundamental theorem of algebra from real calculus and linear algebra?
Motivation: let $A\in\mathbf{R}^{n\times n}$ be symmetric. Then by the method of Lagrange multipliers, a maximum of $x\mapsto x^tAx$ on the compact unit sphere $\mathbf{S}^{n-1}$ must be an ...
- 337
0
votes
0
answers
137
views
About the theorem of Weierstrass?
Is $E=Vect\{1,x,x^2,...,x^{2^n},...\}$ dense in $C([0,1])$ for the uniform norm?
While looking for a short proof for Weierstrass' theorem, I came across this justification(*) (which shows this result)...
- 3,365
5
votes
1
answer
592
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 ...
48
votes
8
answers
5k
views
Ideas for introducing Galois theory to advanced high school students
Briefly, I was wondering if someone can suggest an angle for introducing the gist of Galois groups of polynomials to (advanced) high school students who are already familiar with polynomials (...
22
votes
1
answer
3k
views
What is so special about Chern's way of teaching?
First of all sorry for this non-research post.
I was watching Jeffrey Blitz Lucky documentary movie and it was interesting to me that a winner of Lottery was a math Ph.D. from Berkeley.
In the movie ...
- 4,125
5
votes
0
answers
125
views
Examples of partial adjoints
Recall that a functor $$R: D \to C$$ is said to have a partial left adjoint $L$ defined at an object $X \in C$ if the functor
$$D \to Sets, Y \mapsto Hom_C(X, R(Y))$$
is corepresentable by some object ...
- 1,824
22
votes
6
answers
2k
views
What is the standard 2-generating set of the symmetric group good for?
I apologize for this question which is obviously not research-level. I've been teaching to master students the standard generating sets of the symmetric and alternating groups and I wasn't able to ...
- 4,134
5
votes
1
answer
191
views
Seven Bridges of Königsberg for hypergraphs
I am teaching a course involving hypergraphs. I would like to have a physical analogy/motivating problem for hypergraphs similarly to how the Seven Bridges of Königsberg motivate graphs. Can you help ...
- 51
3
votes
0
answers
456
views
Hard problems solving tricks
This question is motivated by this one that I posted on math.stackexchange.
When I fail to solve a hard math problem (like the ones I presented in the linked post), I read a solution and I noticed ...
- 161
46
votes
10
answers
11k
views
What kid-friendly math riddles are too often spoiled for mathematicians?
Some math riddles tend to be spoiled for mathematicians before they get a chance to solve them. Three examples:
What is $1+2+\cdots+100$?
Is it possible to tile a mutilated chess board with dominoes?...
-4
votes
2
answers
221
views
An elementary-looking integral inequality
This might seem a bit easy but I still like to ask it for pedagogical reasons.
QUESTION. Is this inequality true for non-negative integers $n$?
$$\frac{\pi}2\int_0^1x^n\sin\left(\frac{\pi}2x\right)dx\...
- 40.2k
23
votes
14
answers
4k
views
Math talk for all ages
I've been asked to give a talk to the winners of a recent math competition. The talk can be entirely congratulatory, or it can contain a bit of actual mathematics. I'd prefer the latter. I'd also ...
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
0
votes
1
answer
97
views
Are there search algorithms that are competitive against (gradient based) optimization routines for continuous problems?
Suppose that $f: \mathbb{R}^n \to \mathbb{R}$ is a continuous function for which we want to minimize. We may arbitrarily impose good conditions for $f$, such as Lipschitzness, smoothness, convexity, ...
- 459
8
votes
2
answers
657
views
Seeking a combinatorial proof for a binomial identity
Let $n\geq m\geq0$ be two integers. The below binomial identity is provable by other means:
$$\sum_{j=0}^m(-1)^j\binom{n+1}j2^{m-j}
=\sum_{j=0}^m(-1)^j\binom{n-m+j}j.$$
QUESTION. Can you provide a ...
- 40.2k
1
vote
0
answers
128
views
What benefits of math can be conveyed to mid/high schoolers? [closed]
I'm teaching mathematical proof writing to a few of math teachers (in the US) this summer. In the beginning of class, I send a survey asking them why they are here. Most of them are here for getting ...
- 4,570
34
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) \...
- 10.4k
6
votes
0
answers
274
views
Interesting things you learned while grading/marking? [closed]
What are some interesting mathematical things you have learned while grading (or marking, if you prefer) student work? For example, clever proofs that students came up with; nice counterexamples or ...
3
votes
1
answer
379
views
How to find eigenvalues following Axler?
Preparing my Linear Algebra lecture I like the determinant free approach of Axler because the proof that operators $T$ on an $n$-dimensional complex vector space have eigenvalues is so simple:
Fix ...
- 14.4k
3
votes
1
answer
242
views
Elementary classification of division rings
Are there examples (other than the two mentioned below) of fields $K$ such that the classification of all finite dimensional division $K$-algebras is possible using only elementary theory (lets say a ...
- 24.4k
12
votes
1
answer
514
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 ...
- 29k
9
votes
0
answers
676
views
How many ways are there to teach class field theory?
I will soon have to teach class field theory (I do not know whether it will be local or global yet:)) to postgraduate students. I wonder, which approaches to this subject(s) exist now.
I definitely ...
- 16.1k
15
votes
1
answer
612
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 ...
2
votes
1
answer
312
views
Defining integrals by residue theorem
I have always been interested in alternative definitions of mathematical objects. I wonder if one can craft an useful definition of definite integral by using the Residue Theorem from complex analysis....
- 331
8
votes
4
answers
772
views
Different derivations of the value of $\prod_{0\leq j<k<n}(\eta^k-\eta^j)$
Let $\eta=e^{\frac{2\pi i}n}$, an $n$-th root of unity. For pedagogical reasons and inspiration, I ask to see different proofs (be it elementary, sophisticated, theoretical, etc) for the following ...
- 40.2k
7
votes
1
answer
339
views
Theory of surfaces in $\mathbb{R}^3$ as level sets
Is there a book that treats the classical theory of surfaces in $\mathbb{R}^3$ from the point of view of level sets of a function? I seem to remember someone telling me that such a book exists, but I ...
- 6,617
6
votes
2
answers
1k
views
Pages from a known textbook on Euclidean geometry?
Do you recall having seen the attached pages in a textbook once? If so, would you be so kind as to share its bibliographic record (or the main items in it) with me below?
A teacher provided us xerox ...
- 8,492
0
votes
1
answer
93
views
Name of a matrix with one column and row removed [closed]
I am looking for the exact name of a matrix where the i-th column and rows have been removed.
I cannot remember how it is called in linear algebra, does anyone got an idea?
Thanks!
- 111
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
8
votes
2
answers
412
views
Big ideas and big ways of thinking in statistics?
I'm moving to a new university for the fall semester, and I'll be teaching a statistics class for the first time. I'm familiar enough with doing statistics (my dissertation in math ed was a mixed-...
92
votes
19
answers
9k
views
Short papers for undergraduate course on reading scholarly math
(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 ...
35
votes
2
answers
1k
views
Is it consistent with ZF that $V \to V^{\ast \ast}$ is always an isomorphism?
Let $k$ be a field and $V$ a $k$-vector space. Then there is a map $V \to V^{\ast \ast}$, where $V^{\ast}$ is the dual vector space. If we are in ZFC and $\dim V$ is infinite, then this map is not ...
- 146k
8
votes
2
answers
1k
views
Examples of analytic functions to motivate a first course in complex variables
[Changed title as a plea to re-open the question.]
If one is to motivate a course in complex variables, what specific analytic (holomorphic/meromorphic) function of one variable would you cite as an ...
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 ...
- 2,259
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. ...
- 161
11
votes
3
answers
417
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 ...
- 146k
6
votes
3
answers
2k
views
Problems reducing to a graph-theory algorithm
This is essentially a question in pedagogy -- the answers could be useful to teach (or rather, motivate) graph theory, and especially the algorithmic side of it.
I have been very impressed with this ...
- 2,125
39
votes
4
answers
2k
views
Important open exposition problems?
Timothy Chow, in his article A beginner's guide to forcing, defines an open exposition problem as a certain concept or topic in mathematics that has yet to be explained "in a way that renders it ...
256
votes
29
answers
88k
views
Mathematical games interesting to both you and a 5+-year-old child
Background: My daughter is 6 years old now, once I wanted to think on some math (about some Young diagrams), but she wanted to play with me...
How to make both of us to do what they want ? I guess ...
7
votes
0
answers
556
views
How necessary is the knowledge of Lebesgue integral for non-analysts? [closed]
Recently I have learned that at some math department the introductory course to Lebesgue integration not obligatory. Thus in another course on introduction to Hilbert spaces the $L^2(0,1)$ space is ...
- 20.3k
10
votes
3
answers
912
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....
- 7,468
6
votes
2
answers
496
views
Applications of isotropic quadratic forms
I will soon be teaching an introductory course on bilinear algebra and quadratic forms. I will likely spend most of the time and effort on positive definite quadratic forms and euclidean spaces. These ...