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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429
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186answers
109k 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 ...
202
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72answers
81k 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 ...
131
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35answers
29k views

Why is a topology made up of 'open' sets? [closed]

I'm ashamed to admit it, but I don't think I've ever been able to genuinely motivate the definition of a topological space in an undergraduate course. Clearly, the definition distills the essence of ...
116
votes
29answers
36k 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, ...
98
votes
28answers
24k 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 ...
96
votes
37answers
72k views

Too old for advanced mathematics? [closed]

Kind of an odd question, perhaps, so I apologize in advance if it is inappropriate for this forum. I've never taken a mathematics course since high school, and didn't complete college. However, ...
93
votes
7answers
9k views

How to memorise (understand) Nakayama's lemma and its corollaries?

Nakayama's lemma is mentioned in the majority of books on algebraic geometry that treat varieties. So I think Ihave read the formulation of this lemma at least 20 times (and read the proof maybe ...
89
votes
53answers
20k 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 ...
80
votes
26answers
11k 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 ...
67
votes
15answers
6k views

Why do we care about L^p spaces besides p = 1, p = 2, and p = infinity?

I was helping a student study for a functional analysis exam and the question came up as to when, in practice, one needs to consider the Banach space $L^p$ for some value of $p$ other than the obvious ...
65
votes
20answers
8k 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 ...
62
votes
18answers
14k views

Depressed graduate student. [closed]

How does a depressed graduate student go about recovering his enthusiasm for the subject and the question at hand? Edit: I am not that grad student; it is a very talented friend of mine. Moderator's ...
58
votes
16answers
5k 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 ...
57
votes
73answers
11k views

Elementary+Short+Useful

Imagine your-self in front of a class with very good undergraduates who plan to do mathematics (professionally) in the future. You have 30 minutes after that you do not see these students again. You ...
56
votes
6answers
2k views

Good ways to engage in mathematics outreach?

Greetings all, I have often heard that it would be good if we as a community did more in the way of mathematics outreach: more to explain what it is we do to the community at large, more to expose ...
55
votes
10answers
10k views

Is Euclid dead? [closed]

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!" ...
53
votes
10answers
6k views

Teaching proofs in the era of Google

Dear members, Way back in the stone age when I was an undergraduate (the mid 90's), the internet was a germinal thing and that consisted of not much more than e-mail, ftp and the unix "talk" command ...
52
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20answers
7k 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 ...
50
votes
15answers
7k views

What's a nice argument that shows the volume of the unit ball in $\mathbb R^n$ approaches 0?

Before you close for "homework problem", please note the tags. Last week, I gave my calculus 1 class the assignment to calculate the $n$-volume of the $n$-ball. They had finished up talking about ...
49
votes
9answers
3k views

Taking “Zooming in on a point of a graph” seriously

In calculus classes it is sometimes said that the tangent line to a curve at a point is the line that we get by "zooming in" on that point with an infinitely powerful microscope. This explanation ...
47
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14answers
5k views

How to write popular mathematics well?

Recently, some classmates and I were lamenting the fact that our classmates in other disciplines had almost no conception of what we did, despite the large mathematics population at Waterloo. Instead ...
45
votes
13answers
5k views

How do you approach your child's math education? [closed]

My son is one year old, so it is perhaps a bit too early to worry about his mathematical education, but I do. I would like to hear from mathematicians that have older children: What do you wish you'd ...
45
votes
9answers
10k views

Relating Category Theory to Programming Language Theory

I'm wondering what the relation of category theory to programming language theory is. I've been reading some books on category theory and topos theory, but if someone happens to know what the ...
44
votes
4answers
2k views

What algorithm in algebraic geometry should I work on implementing?

This summer my wife and one of my friends (who are both programmers and undergraduate math majors, but have not learned any algebraic geometry) want to learn some algebraic geometry from me, and I ...
43
votes
5answers
7k 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 ...
43
votes
7answers
6k views

What is Lagrange Inversion good for?

I am planning an introductory combinatorics course (mixed grad-undergrad) and am trying to decide whether it is worth budgeting a day for Lagrange inversion. The reason I hesitate is that I know of ...
42
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18answers
13k 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 ...
41
votes
42answers
11k views

What should be offered in undergraduate mathematics that's currently not (or isn't usually)? [closed]

What's one class that mathematics that should be offered to undergraduates that isn't usually? One answer per post. Ex: Just to throw some ideas out there Mathematical Physics (for math students, not ...
41
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46answers
14k views

An example of a beautiful proof that would be accessible at the high school level?

The background of my question comes from an observation that what we teach in schools does not always reflect what we practice. Beauty is part of what drives mathematicians, but we rarely talk about ...
40
votes
18answers
5k views

How can an extremely mathematically talented young person be helped to fulfill his/her potential?

Obviously, this question is not a research level mathematics question at all. But, I've just met an extremely mathematically talented 11 years old student and I don't know how I can help him. For ...
39
votes
11answers
5k views

Interesting results in algebraic geometry accessible to 3rd year undergraduates

On another thread I asked how I could encourage my final year undergraduate colleagues to take an algebraic geometry or complex analysis courses during their graduate studies. Willie Wong proposed me ...
39
votes
10answers
6k 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 ...
39
votes
7answers
5k 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 ...
38
votes
19answers
5k 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 ...
37
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32answers
5k views

Are there any books that take a 'theorems as problems' approach?

Are there any books that present theorems as problems? To be more specific, a book on elementary group theory might have written: "Theorem: Each group has exactly one identity" and then show a proof ...
35
votes
8answers
4k views

Possibility of an Elementary Differential Geometry Course

I have to admit I'm not sure if this is an appropriate question. It's related to research in math education, but not directly to math. I've found that in talking to professional physicists and ...
34
votes
12answers
7k views

Examples of undergraduate mathematics separation from what mathematicians should know

I'm looking for examples of four kinds of things: Material that is usually covered in standard undergraduate mathematics courses and/or in first-year graduate work (or tested in qualifying ...
34
votes
15answers
6k views

Strong induction without a base case

Strong induction proves a sequence of statements $P(0)$, $P(1)$, $\ldots$ by proving the implication "If $P(m)$ is true for all nonnegative integers $m$ less than $n$, then $P(n)$ is true." for ...
34
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1answer
2k views

Hilbert's Hotel

Hilbert's Hotel is a famous story about infinity attributed to David Hilbert (1862-1943). Is it documented that Hilbert's Hotel is in fact due to Hilbert, and if yes, where?
34
votes
7answers
2k views

Informal online seminars or reading groups via videoconferencing?

Does the following exist, and if not, does anyone besides me wish it did? A web site where a mathematician (say) could find other mathematicians who want to study the same book or paper, and arrange ...
34
votes
1answer
3k views

Probability that a stick randomly broken in five places can form a tetrahedron

The following problem was brought to my attention by a doctoral dissertation on Mathematics Education, but - as far as I know - the solution remains unknown. I have already asked this question on ...
33
votes
6answers
2k views

What is the simplest, most elementary proof that a particular number is transcendental?

I teach, among many other things, a class of wonderful and inquisitive 7th graders. We've recently been studying and discussing various number systems (N, Z, Q, R, C, algebraic numbers, and even ...
32
votes
7answers
3k 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 ...
31
votes
4answers
3k views

Motivation for concepts in Algebraic Geometry

I know there was a question about good algebraic geometry books on here before, but it doesn't seem to address my specific concerns. ** Question ** Are there any well-motivated introductions to ...
30
votes
12answers
9k views

Teaching undergraduate students to write proofs

In my experience, there are roughly two approaches to teaching (North American) undergraduates to write proofs: Students see proofs in lecture and in the textbooks, and proofs are explained when ...
29
votes
21answers
8k 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 ...
28
votes
6answers
2k views

Taylor's theorem and the symmetric group

Anytime I see an $n!$ in some formula, my instinct is to look for the symmetric group on $n$ letters coming in somewhere. I have never done this seriously with the $n!$ in Taylor's theorem. Question: ...
28
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10answers
2k 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, ...
28
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2answers
957 views

Schubert calculus, as lowbrow as possible

Starting in a week I'm going to be an instructor at a summer program for exceptionally mathematically talented high school students, and I'm going to be teaching a class on Schubert calculus. The ...
27
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11answers
3k views

Are there elementary-school curricula that capture the joy of mathematics?

UPDATE: Wow, thank you everyone for the great insights! A couple of months ago I stumbled across Paul Lockhart's essay A Mathematician's Lament and it made perfect sense to me. I'm not meaning to ...