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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10
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8answers
3k views

Leibnizian calculus textbook

Where can I find a calculus textbook that emphasizes differentials? Is there such a book that I could realistically require my calculus students to use? I want a textbook that supports me when I tell ...
42
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34answers
6k 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 ...
17
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6answers
3k 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. ...
40
votes
1answer
3k views

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

Edit (June 2015): Addressing this problem is a brief project report from the Illinois Geometry Lab (University of Illinois at Urbana-Champaign), dated May 2015, that appears here along with a ...
46
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46answers
15k 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 ...
47
votes
7answers
7k 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 ...
49
votes
8answers
7k 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 ...
4
votes
1answer
87 views

Numerical equality testing

I am working on developing an online homework system. One thing I would like to have is something which compares a student's answer (like $2\sin(x)\cos(x)$) with the intended answer (maybe ...
3
votes
1answer
383 views

What questions should -ologists of mathematics ask, in order to improve maths researcher training? [closed]

This question is part of a project funded by the International Council for Science, supported by the IMU (among other bodies). Answers gathered here on mathoverflow may be included in the final ...
471
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193answers
121k 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 ...
59
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7answers
9k 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 ...
32
votes
21answers
7k 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 ...
51
votes
9answers
11k 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 ...
4
votes
1answer
153 views

How to teach generalizing the induction hypothesis? [closed]

I just finished teaching a class on using proof assistants (in this case, Agda) to write provably correct programs. Reflecting on how it went, the biggest difficulty I noticed the students having was ...
18
votes
12answers
6k views

How seriously should a graduate student take teaching evaluations?

Pretty much the question in the title. If a grad student gets bad reviews as a TA, how much does that hurt them later? How much do good reviews help? What if the situation is more complex? (For ...
27
votes
12answers
2k 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 ...
25
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12answers
2k views

Where have you used computer programming in your career as an (applied/pure) mathematician?

For background: I'm working on a book to help mathematicians learn how to program. However, I need to see some examples from people in the field that have done different kinds of things than I have. ...
25
votes
3answers
2k views

Is “problem solving” a subject to be taught?

I am witnessing a new curriculum change in my country (Iran). It includes the change of all the mathematics textbooks at all grades. The peoples involved has sent me the textbook for seven graders (13 ...
46
votes
17answers
10k 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 ...
126
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30answers
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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, ...
5
votes
1answer
1k views

In studying maths, do you tend to go through books one at a time? [closed]

Or do you read a lot of books on the same subject (say Topology)? Consider these two cases: Case 1: Suppose you want to study point set topology. You pick up a good book and work thoroughly on it. ...
62
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 ...
34
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 ...
6
votes
13answers
6k 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 ...
2
votes
0answers
100 views

Applications of Freiman's theorem?

What are some interesting applications of Freiman's theorem or, better-yet, its recent generalizations (eg Green-Ruzsa) that could be included in a graduate course in additive combinatorics? I'm ...
18
votes
14answers
2k views

Insightful books about elementary mathematics

What are some books that discuss elementary mathematical topics ('school mathematics'), like arithmetic, basic non-abstract algebra, plane & solid geometry, trigonometry, etc, in an insightful ...
18
votes
10answers
3k views

Research Experience for Undergraduates: Summer Programs

Some time ago, I found this list of REU programs held in 2009. The main aspects that characterize such programs are: (a) a great deal of lectures on specific topics; and, admittedly more ...
30
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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, ...
99
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55answers
22k 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 ...
20
votes
6answers
4k views

an engineering Ph.D. teaching math in college

I have a friend who has been teaching college-level math (e.g., all levels of calculus) for about 4 years, although all of his education, including his Ph.D., was in engineering. Now he is ...
48
votes
12answers
8k 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 ...
41
votes
10answers
5k 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 ...
1
vote
1answer
177 views

finding permutation matrix I which minimizes TRACE( I* C*( I^T)* D) matrix

I have a problem that is really important for my thesis and i am not studding math so i will be very glad if you help me in this case... thanks for your help in advance I want to find permutation ...
47
votes
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 ...
3
votes
11answers
4k views

Your experience of Computer Science/Programming in Mathematics Education? [closed]

This is a survey question, which seeks to produce a list of answers from the audience of mathematicians. Motivation: I'm doing research in mathematics education. I'm particularly interested in ...
30
votes
21answers
9k 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 ...
26
votes
2answers
1k views

When exactly and why matrix multiplication became a part of undergraduate curriculum?

The story about Heisenberg inventing matrices and matrix multiplication in 1925 is very well known and well documented. Few weeks later Born and Jordan picked this and recognized the matrix ...
105
votes
7answers
10k 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 ...
2
votes
2answers
291 views

Characterizing triangles unembeddedly

The mathedu mailing list has a recent longish thread at http://www.nabble.com/Why-do-we-do-proofs--to25809591.html which discussed among other things whether we should teach triangles as labeled or ...
4
votes
2answers
601 views

Power series with funny behavior at the boundary

Consider a power series $$ \sum_{n=0}^{\infty}a_nz^n $$ where $a_n$ and $z$ are complex numbers. There is radius $R$ of convergence. Let us assume that is a positive real number. It is well known that ...
33
votes
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?
1
vote
1answer
77 views

Distance between two distribution of image

I am looking for a common distance method to compare two distribution (ex: histogram of image). Please suggest to me some common method to do it. I found some method ex: Bhattacharyya distance , K-L ...
32
votes
7answers
4k 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 ...
5
votes
1answer
813 views

Can one live without actual infinity? [closed]

The title of this question is the exact title of one of the sections of a book written by Alexandre Borovik: Mathematics under the Microscope. Under the title, we read: How should we approach the ...
68
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 ...
1
vote
0answers
360 views

Is Independent University of Moscow recognized? [closed]

What graduate schools recognize the degree from Independent University of Moscow? It is not a university strictly speaking and their degree doesn't have any official status in Russia, but they claim ...
41
votes
18answers
6k 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 ...
27
votes
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 ...
106
votes
37answers
87k 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, ...
16
votes
4answers
3k views

Is $x \, \tan(x)$ integrable in elementary functions?

I'm teaching Calculus and my students asked me to calculate the integral of $x \, \tan(x)$. I spent quite a lot of effort to do this, but I'm now even not sure if the integral could be presented in ...