Questions tagged [mathematics-education]

For questions in Mathematics Education as a scientific discipline. For more hands-on questions on teaching Mathematics, please use the tag teaching. There is also a Stack Exchange community http://matheducators.stackexchange.com/

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What is the best way to introduce tensor products to undergrads?

What is the best way to introduce vector space tensor products to undergrads? We have a number of options, given in no particular order. The universal property proving uniqueness and not existence. ...
Zoltan Fleishman's user avatar
35 votes
4 answers
3k views

Psychological test for Euclidean geometry [closed]

There is the so-called FCI test. It contains a list of questions such that anyone who can speak will have an opinion. Based on the answers one can determine if the answerer knows elementary mechanics. ...
Anton Petrunin's user avatar
1 vote
1 answer
340 views

Book on analysis and algebra at the undergraduate level [closed]

I am writing this post because I would like to know what are your references concerning math book showing the interplay between analysis and algebra at an undergraduate-advanced undergraduate level. ...
3 votes
3 answers
476 views

Solving interval problems without outer measure

Is it possible to solve the following two problems on intervals using elementary methods, without using the outer measure ? Problem 1 If $(I_n)$ is a disjoint sequence of subintervals of interval $I$ ...
Ross Ure Anderson's user avatar
-4 votes
1 answer
435 views

Amount of mathematical knowledge required for starting Ph.D. in pure mathematics [closed]

How much mathematics should one know before starting a Ph.D. program in pure mathematics? For example what topics one must understand well to pursue a Ph.D. in US University in Number Theory (...
SARTHAK GUPTA's user avatar
1 vote
0 answers
95 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 "
zdo0x0's user avatar
  • 11
2 votes
0 answers
171 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} \...
Lili Si's user avatar
  • 105
1 vote
0 answers
145 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 ...
Yaroslav Bulatov's user avatar
9 votes
1 answer
624 views

Popular mistakes in probability

$\DeclareMathOperator\Var{Var}\DeclareMathOperator\Bern{Bern}\DeclareMathOperator\Pois{Pois}$Question: What not-trivial mistakes do students often make when solving problems in probability theory, ...
18 votes
9 answers
5k views

How does a Masters student of math learn physics by self?

I am a Masters student of math interested in physics. When I was an undergraduate, I took the introductory course of physics, but it is just slightly harder than high school physics course. To be ...
LZB's user avatar
  • 183
0 votes
1 answer
1k views

Alternative proofs of Euclid-Euler theorem

What are some alternative methods of proof for the necessity direction of the above theorem, ie $n$ an even perfect number $\Rightarrow n$ is of form $2^{a-1} (2^a - 1)$ where $2^a - 1$ is a Mersenne ...
Ross Ure Anderson's user avatar
10 votes
4 answers
647 views

Reference for shortest educational path to (Riemannian) hyperbolic plane

I am teaching an undergraduate class for math majors on axiomatic geometry, culminating in the proof that hyperbolic geometry is equiconsistent* with Euclidean geometry. I would like to make an end-of-...
David Steinberg's user avatar
15 votes
4 answers
5k views

How do you generate math figures for academic papers?

Good day! I am looking for any tool that would allow me to generate a figure similar to the figures embedded in the paper by King et al. (2020) titled "Trigonometry: a brief conversation." ...
Aidre Cabrera's user avatar
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 (...
3 votes
0 answers
124 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 ...
Anton Petrunin's user avatar
29 votes
15 answers
5k views

Lunch seminars for PhD students

The problem that I would like to ask about is metamathematical, but I hope the question is appropriate. I would like to know if there exist mathematical departments that run a regular seminar for all ...
47 votes
6 answers
7k views

Are hypergeometric series not taught often at universities nowadays, and if so, why?

Recently, I've become more and more interested in hypergeometric series. One of the things that struck me was how it provides a unified framework for many simpler functions. For instance, we have $$ \...
33 votes
5 answers
6k views

How should you explain parallel transport to undergraduates?

The title is a bit deceiving, because what I really mean is the parallel transport that corresponds to the Levi–Civita connection. This is in the vein of many other questions on mathoverflow: What is ...
Andrew NC's user avatar
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3 votes
0 answers
775 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 ...
Michelle's user avatar
  • 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?...
-7 votes
1 answer
354 views

Why is it impossible to find work of John Tate online? [closed]

The work of John Tate belongs to mankind. Why is not online in pdf´s? Who is dirty enough to earn money on HIS work?
Ola sande's user avatar
194 votes
18 answers
16k views

Great graduate courses that went online recently

In 09.2020 by pure chance I discovered the YouTube channel of Richard Borcherds where he gives graduate courses in Group Theory, Algebraic Geometry, Schemes, Commutative Algebra, Galois Theory, Lie ...
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 ...
James Fife's user avatar
2 votes
0 answers
160 views

A taxonomy of proof methods [closed]

I am looking for a taxonomy of proof methods in mathematics. For basic proof methods I would think of proof by contradiction, mathematical induction, structural induction (yes I am a computer ...
Gergely's user avatar
  • 291
1 vote
0 answers
303 views

Online courses for mathematics [closed]

I'm sorry if I'm posting this in the wrong forum. My background is in biology and medicine. I am looking to re-learn undergraduate-level mathematics, in particular discrete mathematics, calculus, and ...
Ansel Lim's user avatar
3 votes
2 answers
221 views

Which W W Sawyer titles exist in non-English language editions?

In this community question asking about books that teach the practice of mathematics, the author mentions the works of W W Sawyer. Which of Sawyer's books were translated into languages other than ...
user avatar
1 vote
0 answers
185 views

Studying the vast world of Number Theory [closed]

I'm a high school student, interested in mathematics, especially in number theory. While preparing for the IMO test, and thinking about generalizations or the root of many olympiad problems led me to ...
Junsukim's user avatar
  • 141
25 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 ...
3 votes
1 answer
716 views

What are some problems for research in functional analysis that can possibly be solved by someone with basic knowledge of the subject? [closed]

I wanted to know are there any problems in Functional Analysis (FA) that can possibly be successfully tackled by someone like me who does not have any expertise in this area but is only familiar with ...
J. Doe's user avatar
  • 167
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 (...
kotlinski's user avatar
  • 181
1 vote
1 answer
242 views

Generalized Fourier integral and steepest descent path, saddle point near the endpoints

I am looking forward to solving the integration in the following equation with the assumption that $ka$ is very large \begin{align} H = 2jka\int_{-\pi/2}^{\pi/2}\cos{(\varphi-\phi)}e^{jka[\cos{\...
gemmy9492's user avatar
26 votes
3 answers
3k views

Why is the standard definition of a $(p, q)$-tensor so bizarre?

At time of writing the first definition of a $ (p, q) $-tensor on the Wikipedia page is as follows. Definition. A $ (p, q) $-tensor is an assignment of a multidimensional array $$ T^{i_1\dots i_p}_{...
Arthur's user avatar
  • 1,379
6 votes
2 answers
702 views

Books on the History of math research at European universities

Are there good books that cover the history of math and mathematical science (ex. physics, chemistry, computer science) PhD programs in the Occident? My primary motivation is to figure out how the PhD ...
Aidan Rocke's user avatar
  • 3,827
9 votes
2 answers
619 views

Constructivist defininition of linear subspaces of $\mathbb{Q}^n$?

Let me preface this by saying I'm not someone who has every studied mathematical logic or philosophy of math, so I may be mangling terminology here (and the title is a little tongue in cheek). I (and ...
RBega2's user avatar
  • 2,458
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 ...
Kim's user avatar
  • 4,124
63 votes
6 answers
12k views

Why isn't integral defined as the area under the graph of function?

In order to define Lebesgue integral, we have to develop some measure theory. This takes some effort in the classroom, after which we need additional effort of defining Lebesgue integral (which also ...
user57888's user avatar
  • 1,229
7 votes
0 answers
212 views

Do cocycles “break” symmetry?

In an article by A. Borovik, “Is mathematics special?”, he talks about the fact that carrying is a cocycle. He then says [A student] discovered that carry is doing what cocycles frequently do: they ...
mosuem's user avatar
  • 71
1 vote
1 answer
372 views

Why is $n_{n^2-1}$ the smallest graph that clearly shows the structure of multiplication by $n$?

Initially, I wanted to ask this question as a puzzle. Consider a regular $m$-gon. Let $0$ be the lower corner and count the corners clockwise. Let $n_m$ be the multiplication-by-$n$-graph of $...
Hans-Peter Stricker's user avatar
113 votes
1 answer
10k views

What happened to Suren Arakelov? [closed]

I heard that Professor Suren Arakelov got mental disorder and ceased research. However, a brief search on the Russian wikipedia page showed he was placed in a psychiatric hospital because of political ...
Bombyx mori's user avatar
  • 6,141
45 votes
12 answers
9k views

How to explain to an engineer what algebraic geometry is?

This question is similar to this one in that I'm asking about how to introduce a mathematical research topic or activity to a non-mathematician: in this case algebraic geometry, intended as the most ...
29 votes
2 answers
2k views

Why did Dedekind claim that $\sqrt{2}\cdot\sqrt{3}=\sqrt{6}$ hadn't been proved before?

In a letter to Lipschitz (1876) Dedekind doubts that $\sqrt{2}\cdot\sqrt{3}=\sqrt{6}$ had been proved before: quoted from Leo Corry, Modern algebra, German original: Why did Dedekind doubt that $(\...
Hans-Peter Stricker's user avatar
15 votes
2 answers
1k views

What kind of computer tools topologists/geometrists 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 ...
stressed out's user avatar
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 ...
guest17's user avatar
  • 253
15 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 ...
9 votes
1 answer
609 views

De-Nesting Absolute Value Function into Linear Combination of Absolute Value Functions

Context: In formulating problems for secondary school mathematics teachers (and students) about absolute value functions, which we define as functions $\mathbb{R} \rightarrow \mathbb{R}$ that send $x \...
Benjamin Dickman's user avatar
69 votes
24 answers
18k views

PhD dissertations that solve an established open problem

I search for a big list of open problems which have been solved in a PhD thesis by the Author of the thesis (or with collaboration of her/his supervisor). In my question I search for every possible ...
6 votes
0 answers
151 views

Is there Cauchy-Goursat for $1$-cycles without invoking winding numbers?

Depending on one's approach to Complex Analysis in One Variable, Cauchy's Integral Theorem is one of the first interesting results about holomorphic functions in any course. There are several related ...
M.G.'s user avatar
  • 6,730
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 ...
Davide's user avatar
  • 141
93 votes
20 answers
10k 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 ...

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