# 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/

248
questions

**21**

votes

**6**answers

4k views

### Euclid with Birkhoff

I'm looking for an short and elementary book which does Euclidean geomety with Birkhoff's axioms.
It would be best if it would also include some topics in projective (and/or) hyperbolic geometry.
...

**12**

votes

**2**answers

940 views

+150

### 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 (...

**86**

votes

**9**answers

24k views

### How do you not forget old math?

I am trying to not forget my old math. I finished my PhD in real algebraic geometry a few years ago and then switched to the industry for financial reasons. Now I get the feeling that I want to do a ...

**111**

votes

**7**answers

21k views

### Is the boundary $\partial S$ analogous to a derivative?

Without prethought, I mentioned in class once that the reason the symbol $\partial$
is used to represent the boundary operator in topology is
that its behavior is akin to a derivative.
But after ...

**63**

votes

**21**answers

19k views

### How should one present curl and divergence in an undergraduate multivariable calculus class?

I am a TA for a multivariable calculus class this semester. I have also TA'd this course a few times in the past. Every time I teach this course, I am never quite sure how I should present curl and ...

**106**

votes

**25**answers

15k views

### “Mathematics talk” for five year olds

I am trying to prepare a "mathematics talk" for five year olds from my daughter's elementary school. I have given many mathematics talks in my life but this one feels
very tough to prepare. Could the ...

**789**

votes

**250**answers

212k views

### Examples of common false beliefs in mathematics

The first thing to say is that this is not the same as the question about interesting mathematical mistakes. I am interested about the type of false beliefs that many intelligent people have while ...

**45**

votes

**14**answers

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 ...

**23**

votes

**16**answers

11k views

### Useless math that became useful

I'm writing an article on Lychrel numbers and some people pointed out that this is completely useless.
My idea is to amend my article with some theories that seemed useless when they are created but ...

**1**

vote

**1**answer

93 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{\...

**29**

votes

**7**answers

8k views

### On starting graduate school and common pitfalls…

Hi,
I'll be starting graduate school soon, and when I look back at my college career, there are certain things I wish I could have done differently. In hindsight, I wished I wasn't in such a rush to ...

**6**

votes

**2**answers

632 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 ...

**5**

votes

**0**answers

634 views

### Probability in Math Education [closed]

Why is probability an under-emphasized subject in most math programs? Why does it seem that the hot topics these days are category theory and algebra? What do you think about the following: A student ...

**69**

votes

**16**answers

6k 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 ...

**5**

votes

**4**answers

859 views

### Lecture on Fractals for Middle School Students

I'm going to have a one-hour lecture for middle school students next Monday. It will be about fractals. The students know virtually nothing about this subject.
I'll show some fractal images and a few ...

**19**

votes

**1**answer

1k 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}_{...

**147**

votes

**28**answers

44k views

### Cool problems to impress students with group theory [closed]

Since this forum is densely populated with algebraists, I think I'll ask it here.
I'm teaching intermediate level algebra this semester and I'd like to entertain my students with some clever ...

**47**

votes

**6**answers

8k 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**

votes

**1**answer

1k views

### What topics should be included in a calculus-for-the-liberal arts course?

I have in mind a course taken by liberal-arts students who will probably never take another math course. I would like such a course to convey some of the way mathematical thinking is done (i.e. not a ...

**63**

votes

**20**answers

12k views

### What should we teach to liberal arts students who will take only one math course?

Even professors in academic departments other than mathematics---never mind other educated people---do not know that such a field as mathematics exists. Once a professor of medicine asked me whether ...

**28**

votes

**9**answers

5k 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 ...

**39**

votes

**11**answers

3k views

### effective teaching

Eric Mazur has a wonderful video describing how physics is taught at many universities and his description applies word for word to the way I learned mathematics and the way it is still being taught, ...

**8**

votes

**2**answers

450 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 ...

**342**

votes

**78**answers

120k views

### Video lectures of mathematics courses available online for free

It can be difficult to learn mathematics on your own from textbooks, and I often wish universities videotaped their mathematics courses and distributed them for free online. Fortunately, some ...

**76**

votes

**11**answers

17k views

### Is Euclid dead?

Apparently Euclid died about 2,300 years ago (actually 2,288 to be more precise), but the title of the question refers to the rallying cry of Dieudonné, "A bas Euclide! Mort aux triangles!" (...

**172**

votes

**69**answers

37k views

### What are your favorite instructional counterexamples?

Related: question #879, Most interesting mathematics mistake. But the intent of this question is more pedagogical.
In many branches of mathematics, it seems to me that a good counterexample can be ...

**5**

votes

**9**answers

3k 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 ...

**32**

votes

**11**answers

8k views

### Categories First Or Categories Last In Basic Algebra?

Recently, I was reminded in Melvyn Nathason's first year graduate algebra course of a debate I've been having both within myself and externally for some time. For better or worse, the course most ...

**42**

votes

**4**answers

2k views

### When exactly and why did matrix multiplication become a part of the undergraduate curriculum?

The story about Heisenberg inventing matrices and matrix multiplication in 1925 is very well known and well documented. A few weeks later, Born and Jordan read this work and recognized matrix ...

**74**

votes

**11**answers

20k views

### How misleading is it to regard $\frac{dy}{dx}$ as a fraction?

I am teaching Calc I, for the first time, and I haven't seriously revisited the subject in quite some time. An interesting pedagogy question came up: How misleading is it to regard $\frac{dy}{dx}$ as ...

**177**

votes

**30**answers

69k views

### Real-world applications of mathematics, by arxiv subject area?

What are the most important applications outside of mathematics of each of the major fields of mathematics? For concreteness, let's divide up mathematics according to arxiv mathematics categories, e.g....

**6**

votes

**0**answers

176 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 ...

**46**

votes

**22**answers

15k views

### Why linear algebra is fun!(or ?)

Edit: the original poster is Menny, but the question is CW; the first-person pronoun refers to Menny, not to the most recent editor.
I'm doing an introductory talk on linear algebra with the ...

**50**

votes

**27**answers

9k views

### Nontrivial question about Fibonacci numbers?

I'm looking for a nontrivial, but not super difficult question concerning Fibonacci numbers. It should be at a level suitable for an undergraduate course.
Here is a (not so good) example of the sort ...

**2**

votes

**1**answer

333 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**

votes

**1**answer

7k 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 ...

**138**

votes

**28**answers

22k views

### How To Present Mathematics To Non-Mathematicians?

(Added an epilogue)
I started a job as a TA, and it requires me to take a five sessions workshop about better teaching in which we have to present a 10 minutes lecture (micro-teaching).
In the last ...

**76**

votes

**20**answers

7k views

### One-step problems in geometry

I'm collecting advanced exercises in geometry. Ideally, each exercise should be solved by one trick and this trick should be useful elsewhere (say it gives an essential idea in some theory).
If you ...

**33**

votes

**15**answers

3k views

### Historical (personal) examples of teaching-based research

The phrase "teaching-based research" brings to mind research about teaching, though important, it is not what I mean. Unfortunately, I couldn't come up with a better phrase, thus please bear with me ...

**49**

votes

**8**answers

8k views

### Why the Killing form?

I'm teaching a short summer course on algebraic groups and it's time to talk about the Killing form on the Lie algebra. The students are all undergrads of varying levels of inexperience, and I try to ...

**29**

votes

**2**answers

1k 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 $(\...

**45**

votes

**13**answers

9k 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 ...

**17**

votes

**7**answers

1k 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 ...

**68**

votes

**22**answers

10k views

### Are there proofs that you feel you did not “understand” for a long time?

Perhaps the "proofs" of ABC conjecture or newly released weak version of twin prime conjecture or alike readily come to your mind. These are not the proofs I am looking for. Indeed my question was ...

**7**

votes

**2**answers

1k views

### How should you respond to a student who asks whether a very nice physical example constitutes a proof? [closed]

"Is this really a proof?" is the exact question e-mailed to me today from an undergraduate mathematics student whom I know as a highly competent student. The one sentence question was accompanied with ...

**8**

votes

**7**answers

4k views

### Review papers in mathematics

Are there review papers, literature reviews in mathematics that describe the recent developments in various fields for a newcomer? Or is the prerequisite knowledge always provided in research ...

**12**

votes

**2**answers

475 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 ...

**74**

votes

**24**answers

17k views

### Parodies of abstruse mathematical writing

Perhaps under the influence of a recent question
on perverse sheaves,
in conjunction with the impending $\pi$-day (3/14/15 at 9:26:53),
I recalled a long-ago parody of abstruse mathematical language
...

**13**

votes

**3**answers

2k 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 ...

**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 ...