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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-8 votes
0 answers
159 views

Is there a more descriptive term for Characteristic Classes? [closed]

Recently I went along to a conference on Communication in Mathematics where a number of people, me amongst them, complained about talks where one only understood 5% of what was going on. A part of ...
16 votes
8 answers
3k 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 ...
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1 vote
0 answers
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References to learn modern functions applied to integration and numerical series problems and how to teach them to Calculus students [closed]

I think most of us have met integration problems concerning the trigonometric, polynomial, exponential, hyperbolic and power functions in the calculus courses. But many of the problems in this website ...
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10 votes
4 answers
591 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-...
13 votes
4 answers
2k 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." ...
48 votes
8 answers
4k 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
98 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 ...
27 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 ...
44 votes
6 answers
6k 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 $$ \...
31 votes
5 answers
5k 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 ...
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3 votes
0 answers
334 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 ...
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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
286 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?
186 votes
18 answers
13k 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 ...
2 votes
0 answers
146 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 ...
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1 vote
0 answers
230 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 ...
3 votes
2 answers
205 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
154 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 ...
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24 votes
3 answers
2k 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 ...
2 votes
1 answer
579 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 ...
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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 (...
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1 vote
1 answer
176 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{\...
24 votes
3 answers
2k 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}_{...
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6 votes
2 answers
680 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 ...
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9 votes
2 answers
574 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 ...
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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 ...
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54 votes
6 answers
11k 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 ...
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7 votes
0 answers
203 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 ...
1 vote
1 answer
365 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 $...
113 votes
1 answer
9k 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 ...
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42 votes
12 answers
8k 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 ...
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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 $(\...
15 votes
2 answers
747 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 ...
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 ...
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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
523 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 \...
67 votes
23 answers
16k 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
139 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 ...
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14 votes
4 answers
4k 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 ...
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91 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 ...
15 votes
3 answers
1k views

Axioms for constructive Euclidean geometry

In the summer I will be teaching a course in (plane) Euclidean geometry to future high school teachers and I am looking for a suitable axiom system (unlike College (Euclidean) geometry textbook ...
20 votes
4 answers
1k views

Problems for developing mathematical visualization expertise

Einstein stated that he often explored and reasoned visually and spatially, and only after achieving understanding cast his insights into algebraic form. He could just "see" the answer. There are ...
252 votes
29 answers
87k 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 ...
13 votes
4 answers
914 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 ...
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1 vote
3 answers
1k views

How to be a Great mathematician in prison/without a master? [closed]

Is it possible to be a great mathematician in our home with a laptop+poor internet+electronic books+some books+a little food +a little money or not? without having a constant job without studying P.H....
-1 votes
1 answer
477 views

Are manifolds typically taught to undergraduates outside mathematics (and possibly theoretical physics) tracks? [closed]

I'm writing my dissertation on symplectic structure-preserving algorithms for Hamiltonian systems simulation, and I'm trying to figure out how much exposition is necessary for it to be readable by ...
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2 votes
3 answers
7k views

How can I combine my interests for pure mathematics and computer science in college? [closed]

I’m a high school senior who's gone through quite the self-introspection the past few months while applying for college, and I have a bit of a dilemma. All my life, I've loved & excelled at ...
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7 votes
0 answers
530 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 ...
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