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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Solve trigonometric equation for 0° to 360° and give the answer in 4 significant figures [closed]

Could somebody help me to solve this? $ 2cos x- \sqrt{3} = 0 $ $ 2cos x = \sqrt{3} $ $ cos x = \frac{\sqrt{3}}{2} $ By reason $cos x > 0 $ , it lies on the quarters 1 and 4. reference angle $= cos^...
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How to solve product of Rayleigh quotient problem to get a single solution?

\begin{align} \text{(P1):} &\ \ \ \ \ \ \ \max\limits_{\mathbf f} \log_2\left(\prod^K_{i=1} \ \ \frac{ \mathbf f^H \mathbf {\mathbf E_i} ( \mathbf W_i, \Theta, \tau_i) \mathbf f} { \mathbf f^H \...
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2answers
153 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 ...
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Classical papers in linear algebra suitable for an undergraduate reading group?

I'm interested in collecting some of the classical papers in linear algebra over the years. To be more specific, I'm looking for interesting and useful results that extend beyond what is typically ...
2
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0answers
127 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 ...
25
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3answers
1k 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
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1answer
372 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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4answers
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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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1answer
115 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{\...
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1answer
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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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2answers
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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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2answers
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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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9answers
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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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6answers
9k 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 ...
6
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0answers
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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 ...
2
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1answer
340 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 $...
110
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1answer
8k 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 ...
28
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9answers
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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
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2answers
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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 $(\...
14
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2answers
540 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 ...
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3answers
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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 ...
15
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3answers
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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 ...
8
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1answer
364 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 \...
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13answers
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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 ...
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0answers
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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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4answers
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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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16answers
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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
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3answers
731 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 ...
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4answers
868 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 ...
238
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29answers
83k 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 ...
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4answers
870 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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3answers
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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....
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1answer
341 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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3answers
5k 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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0answers
452 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 ...
13
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1answer
2k 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 ...
7
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1answer
211 views

Five cubes, Hadamard and Shklyarskiy

Here is my(=bad) translation of from the paper about Shklyarskiy by Golovina: ... in 1937/38 Dodik presented to school students a complete proof of Abel's theorem about equations of degree 5. He ...
1
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1answer
108 views

Expectation of changing the gift choice [closed]

Suppose we are given two boxes, with one of gift valued $n$ dollars and the other one valued twice as much. We can pick a box, and after open it we have the choice of switching to another box. Shall ...
10
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1answer
463 views

A proof without derivatives that a real polynomial of degree $n$ has at most $n-1$ local extrema

This question is about math education and is not research level, so do not hesitate to delete it if it feels inappropriate. I already asked it here a year ago: https://math.stackexchange.com/...
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14answers
4k views

Interactive model of the hyperbolic plane for a general public lecture

The following is not quite a research level question, but I still find this site appropriate for asking it. I hope I get it right here. I am preparing a talk for a general public and I want to ...
88
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5answers
6k 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 ...
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1answer
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Is there a way to embed Clifford algebras into the corresponding tensor algebra?

$\newcommand{\talg}{\mathcal{T}(V)}$$\newcommand{\clalg}{\mathcal{Cl}_q(V)}$$\newcommand{\qalg}{\mathcal{I}_q(V)}$Is there a way to embed Clifford algebras into the corresponding tensor algebra? There ...
3
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1answer
589 views

Cambridge Mathematical Tripos papers from late 19th century

Are the scanned images of Cambridge Mathematical Tripos papers from late 19th century available anywhere on Internet?
15
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2answers
3k 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 ...
2
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2answers
580 views

Math and social commitment [closed]

I am a master's student and am looking for ways that link a certain social commitment with serious math. Since I have not found such an overview yet and in order to raise public awareness of such ...
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0answers
84 views

Discrete Mathematics Uses [closed]

I am trying to explain how and why discrete maths is used in areas such as programming, correctness, data types, state transistion and conditionals. I'm having a really hard time articulating it ...
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5answers
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 ...
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1answer
1k views

Everyday, real-life applications of mathematical concepts, and human intuition vs mathematical analysis [closed]

I'm working on an educational project about the applications of reasonably 'lofty', high-ish-level mathematical concepts in the real world. I've already scoured these links (1) (2) (3) after ...
10
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1answer
588 views

Problems which use S₄ → S₃

I need examples of problems which use, directly or indirectly, the homomorphism $S_4\to S_3$ in the solution (its kernel is $\mathbb{Z}_2\oplus\mathbb{Z}_2$). Obvious candidates: Lagrange resolvent (...
2
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2answers
407 views

A logarithmic cotangent inequality

I must be a terrible googling searcher but I cannot find a reference to the following inequality: $$ \forall_{\phi\in(0;\frac \pi 4)}\ \ln(\cot(\phi)))\, <\, \cot(2\!\cdot\!\phi) $$ I have just ...

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