All Questions
495 questions
16
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13
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Do you find your students are less competent in basic algebra and arithmetic, and, if so, do you believe that this is due to overuse of calculators at an early level? [closed]
So first I gave my class the quiz problem: Compute $$\lim_{h\rightarrow 0} \frac{\frac{1}{3+h} - \frac{1}{3}}{h}.$$ Upon finding that they could not do that (no real surprize) I asked them to compute $...
16
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 ...
- 30.5k
16
votes
5
answers
3k
views
Integrating powers without much calculus
I'll jump into the question and then back off into qualifications and context
Using the definition of a definite integral as the limit of Riemann sums, what is the best way (or the very good ways) to ...
- 30.1k
16
votes
6
answers
3k
views
How to mentor an exceptional high school student?
I have a unique and, quite truthfully, humbling opportunity. The parents of an exceptionally talented high school freshman have reached out to me and asked if I might be able to help.
This kid is ...
16
votes
10
answers
6k
views
Undergraduate Topology
I am developing an introductory topology course for undergraduates, and I am wondering what topics to cover. At my institution, real analysis is not a prerequisite for the course, so it is more than ...
16
votes
7
answers
2k
views
Unexpected applications of the fact that nth degree polynomials are determined by n+1 points
I had a funny idea for proving an identity in Euclidean geometry. While it didn't end up being a very nice proof strategy in my case, I would still like to collect nice examples of where the proof ...
16
votes
5
answers
1k
views
Permission to use Online Notes
I am a new professor in Mathematics and I am running an independent study on Diophantine equations with a student of mine. Online I have found a wealth of very helpful expository notes written by ...
16
votes
7
answers
2k
views
Uppercase Point Labels in High-School Diagrams: from Euclid?
I wonder if the convention of labeling points in geometric
diagrams with uppercase symbols ultimately derives from
Greek mathematics, which was originally written in
"majuscule" (uppercase) Greek ...
- 151k
16
votes
1
answer
2k
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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 ...
- 2,437
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
16
votes
5
answers
2k
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"Classical" consequences of Bezout's theorem in dimensions $>2$
By Classical I mean something that could have been found before 1900 (say).
A well known consequence of Bezout's theorem for plane curves is Pascal's theorem http://en.wikipedia.org/wiki/Pascal'...
- 14.3k
16
votes
2
answers
2k
views
There are two points on the Earth's surface that ... ?
At every moment in time, there are two points on the Earth's surface that have the same $\lbrace x, y, z, ... \rbrace$...?
What is the strongest, most impressive statement one can make here? The ...
- 151k
16
votes
1
answer
978
views
Pedagogically intuitive reformulation of Zorn's Lemma for functional analysis
While teaching an applied functional analysis class, I’ve noticed that students often struggle to develop an intuitive understanding of Zorn’s lemma. It’s relatively straightforward to explain why ...
- 5,824
16
votes
2
answers
952
views
Where and when did "transition to abstraction" courses start?
I often find myself debating the content and structure of such courses and I would find it useful to know the basic history.
I don't remember any such offerings during my own undergraduate days in ...
- 17.6k
16
votes
3
answers
2k
views
Solving a modified birthday problem at a glance
Modified Birthday Problem: a bunch of people line up, and the winner is the first person who shares their birthday with someone lined up ahead of them. What position in the line is optimal?
Three (...
- 7,831
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
15
votes
7
answers
6k
views
Freshman's definition of sin(x)?
I would like to know how you would rigorously introduce the trigonometric functions ($\sin(x)$ and relatives) to first year calculus students. Suppose they have a reasonable definition of $\mathbb{R}$ ...
- 23.4k
15
votes
5
answers
3k
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Why is a topology made up of 'open' sets? Part II [closed]
Because the display was getting quite cluttered, I thought I'd post a second part to this question separately. I hope the Gods of Math Overflow don't take too much offense. I'll go now into some ...
15
votes
2
answers
5k
views
What areas of algebra could be interesting to probability theorists?
I would like to find some topic of algebra (beyond linear algebra; algebraic number theory is fine) that would be interesting both to a student that wants to specialize in probability theory and to me ...
- 16.9k
15
votes
4
answers
3k
views
How does one motivates the method of separation of variables when teaching PDE's?
I'm not sure if this question is appropriate for MO. Add comments if it is not. Thanks.
How to explain/motivate the method of separation of variables for PDEs to undergraduates? What's the real math ...
- 5,052
15
votes
1
answer
757
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 ...
14
votes
11
answers
35k
views
Why does undergraduate discrete math require calculus?
Often undergraduate discrete math classes in the US have a calculus prerequisite.
Here is the description of the discrete math course from my undergrad:
A general introduction to basic
...
14
votes
2
answers
5k
views
A certain mathematical competition in the UK
There is a foreword, written by professor Snow, to the book A mathematician's apology.
In the foreword, it is written some thing like the following:
"Hardy was opposed to a certain mathematical ...
- 366
14
votes
7
answers
6k
views
Usage of set theory in undergraduate studies
I would like to ask my colleagues their thought on good practices concerning
set theorical framework in undergraduate studies. For example, have there been any attempt to use another mathematical ...
14
votes
4
answers
5k
views
Which edition of Philosophiae Naturalis Principia Mathematica of Isaac Newton would you recommend to me?
I'm searching for a good edition of Philosophiae Naturalis Principia Mathematica of Isaac Newton in English. Which edition of the Principia can you suggest me? If it's possible, cheap and similar to ...
- 141
14
votes
4
answers
2k
views
The ten most fundamental topics in geometric group theory
What are the ten most fundamental topics in geometric group theory?
This is a pedagogical question prompted by the fact that I am teaching geometric group theory to undergraduates. They are expected ...
14
votes
3
answers
1k
views
Where can I read reviews of mathematical theories? [closed]
I'm really enjoying the AMS column "What is ..." (http://arminstraub.com/math/what-is-column) and The Princeton Companion to Mathematics.
I am looking for something similar. I'd like to acquire some ...
14
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 topology 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 ...
- 7,338
14
votes
5
answers
17k
views
Reference letters for graduate school after a couple years in the industry
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 ...
14
votes
3
answers
3k
views
Open source LaTeX lecture notes/slides/books [closed]
In the mathematics community it's quite common for professors to write their own notes for the classes they are teaching. The notes are then usually published in both PDF and PS form on the course ...
14
votes
2
answers
7k
views
What is the dual concept to "annihilator" called, and do any linear algebra textbooks discuss this concept first?
When introducing dual spaces for the first time, most linear algebra textbooks proceed in what seems to me a rather backwards fashion: the annihilator $\{f\in V^*: f(u)=0\quad \forall u\in U\}$ of a ...
14
votes
9
answers
2k
views
math circles video lectures for school children?
Hello,
I am from India. I find the mathoverflow amazing.
I have a question: Are there any good quality video lectures on school math topics?
There are a lot of high quality lectures available on ...
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
13
votes
11
answers
5k
views
Math History books
I'm teaching a course over the summer (it's a sort of make-your-own course for non-majors) and I'm planning on organizing it as a math history course, hitting on major threads through about 1900, and ...
- 16k
13
votes
7
answers
35k
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 ...
13
votes
3
answers
1k
views
Teaching polarisation formula
When teaching about Hilbert spaces, one begins with a polarisation formula, which allows us to reconstruct the scalar product from the norm:
$$\langle u,v\rangle=\frac14(\|u+v\|^2-\|u-v\|^2+\imath\|u+\...
- 52.4k
13
votes
17
answers
3k
views
Short Course Suggestions For High School Students
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 ...
13
votes
5
answers
2k
views
How to make a lecture series useful
I have been to a number of advanced lecture courses (of between 3 and 10 lectures) over the years, given (in principle) by experts to graduate students and experts in neighbouring fields. Examples of ...
13
votes
2
answers
6k
views
What is the history of $\sqrt{}$
Why we use the symbol $\sqrt{}$ when we take square roots ? Anybody knows the history ?
user30695
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
13
votes
3
answers
2k
views
History surrounding Gauss Theorema Egregium and differential geometry
I am teaching a class on elementary differential geometry and I would like to know, for myself and for my students, something more about the history of Gauss Theorema Egregium, that is the Gaussian ...
- 193
13
votes
4
answers
1k
views
Simple groups with the same cardinality as PSL_2(Z/p)
In an undergrad honors algebra course it's sometimes shown that when $p$ is prime and $>3$ then
$PSL_2(Z/p)$ is simple of of order $p(p-1)(p+1)/2$. But that this is the "only" simple group
having ...
13
votes
2
answers
3k
views
The probabilistic method - reference to less challenging questions
I am teaching a course in combinatorics and large part of it is dedicated to the probabilistic method especially in the case of graphs. The course is an undergraduate level (almost none of the ...
13
votes
1
answer
605
views
A funny factorization of the Jacobian coming from the lines on the Fermat cubic
Here is something which came up in my algebraic geometry class, and I'm wondering if it has a deeper explanation. Let $F(w,x,y,z) = w^3+x^3+y^3+z^3$ and let $X$ be the cubic surface in $\mathbb{P}^3$ ...
- 156k
13
votes
1
answer
1k
views
Classroom platonism
I'd like to know whether any form a certain hypothesis about the
learning of higher mathematics has entered the mathematical or
educational literature. I'll frame the hypothesis here but not defend
...
- 17.6k
13
votes
1
answer
2k
views
conditional equality symbol
Is there a standard notation (perhaps $A \stackrel{\leftarrow}{=} B$) meaning "in all situations where $B$ is defined, $A$ is defined and equals $B$"?
The kind of situation in which such a notation ...
- 19.7k
13
votes
2
answers
2k
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teaching higher algebra
Has anyone ever (successfully or unsuccessfully) taught a course in higher algebra (in the $\infty$-categorical sense)?
I'm asking out of curiosity (and also hoping for more resources).
The kind of ...
- 534
12
votes
12
answers
2k
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What are fun elementary subjects in probability?
I have to read several lectures on probability or applications of probability for high school students (of high level). There is no necessary part I must lecture, that is, my aim is just advertisement....
12
votes
11
answers
2k
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Giving a math talk with no blackboard or projector
I need to give a math talk to a group of undergraduates. I am asking for advice because this talk will take place at a department picnic and there will be no blackboard or projector. I would like to ...
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.....