139
votes
Mathematical games interesting to both you and a 5+-year-old child
One evening at the dinner table, when my oldest daughter was 3 or 4, I was in a teasing mood, and I called her a goose. She didn't want to be a goose, so she refuted the claim, "I am not a goose!" ...
99
votes
Mathematical games interesting to both you and a 5+-year-old child
The game "Set" seems to fit the bill. It's a card came where there are cards that show images which have four different features, each of which comes in three possibilities:
number (1, 2, or 3 ...
73
votes
Mathematical games interesting to both you and a 5+-year-old child
Another topological game: Sprouts.
Rules:
The game is played by two players, starting with a few spots drawn on a sheet of paper. Players take turns, where each turn consists of drawing a line ...
70
votes
Accepted
Ideas for introducing Galois theory to advanced high school students
I have now twice taught Galois theory to advanced high school students at PROMYS. This is a six week course, meeting four times a week, for students who already are comfortable with proofs and, in ...
62
votes
Accepted
Zorn's lemma: old friend or historical relic?
I agree with almost everything in your post. But still, I believe I know why people use Zorn's lemma.
My answer. Zorn's lemma encapsulates succinctly many of the consequences of AC via transfinite ...
- 236k
49
votes
Mathematical games interesting to both you and a 5+-year-old child
Knot or not?
The topological game involves a projection of a knot, drawn onto paper, such as:
Player 1 picks an intersection and assigns a crossing (which segment of the curve is "above" and which "...
46
votes
Accepted
Is it consistent with ZF that $V \to V^{\ast \ast}$ is always an isomorphism?
No, it’s not consistent.
Let $V=k^{(\omega)}$ be the vector space of finite sequences of elements of $k$. Then $V^*$ can be identified with the vector space $k^\omega$ of all sequences, and elements ...
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42
votes
Zorn's lemma: old friend or historical relic?
I agree with the existing answers, but I personally like Zorn's lemma both pedagogically and mathematically for an additional reason: the "poset of partial solutions" that it introduces is a ...
- 21.1k
40
votes
Short papers for undergraduate course on reading scholarly math
I think this is a delightful paper:
Hull, Thomas C. "Solving cubics with creases: The work of Beloch and Lill." The American Mathematical Monthly 118, no. 4 (2011): 307-315.
(PDF download.)
It ...
40
votes
Accepted
What is the standard 2-generating set of the symmetric group good for?
Let $f\in\mathbb{Q}[x]$ be an irreducible polynomial of prime degree $p$, with exactly $2$ non-real roots.
You can view the Galois group of $f$ (i.e., the Galois group of the splitting $f$) as a ...
- 2,809
38
votes
Accepted
What is so special about Chern's way of teaching?
Louis Auslander has described his experiences on S.S. Chern as a teacher:
Somehow Chern conveyed the philosophy that making mistakes was normal
and that passing from mistake to mistake to truth was ...
- 188k
32
votes
What kid-friendly math riddles are too often spoiled for mathematicians?
Here's a few, two I got to solve myself as a kid and one (a trickier one, in my opinion) that was spoiled for me.
There are $1000$ lights all in a line and turned on. At time $n$, person $n$ comes by ...
30
votes
Mathematical games interesting to both you and a 5+-year-old child
Dots and Boxes
Is a pencil-and-paper game for two players. It's quite simple to explain but quite hard to play. Five year olds should be able to learn it and with some training maybe also being good ...
30
votes
Can one deduce the fundamental theorem of algebra from real calculus and linear algebra?
I don't think so. Indeed the result that every symmetric matrix is diagonalizable is true for some orderable non-real-closed field $K$ (see this answer by Will Sawin to Over which fields are symmetric ...
- 63.9k
28
votes
Mathematical games interesting to both you and a 5+-year-old child
There is an infinite indexed family of family-friendly, $\geq2$-player, perfect-information, draw-free-if-finite, cheap-to-construct, two-player combinatorial1, solved, sequential games. They are ...
26
votes
What kid-friendly math riddles are too often spoiled for mathematicians?
To make this suitable for MO rather than math.SE, perhaps we can define a "too often spoiled" puzzle to be one that can be recognized instantly by a mathematician even with what looks like ...
25
votes
Mathematical games interesting to both you and a 5+-year-old child
What about Spot It! (US), also known as Dobble (Europe)?
We are given a deck of 55 cards. Each card contains 8 different symbols, such that any pair of cards in the deck has precisely one symbol in ...
25
votes
Accepted
A conjecture in which both "if" and "only if" are near misses
False claim: A Hausdorff topological space is compact if and only if it is sequentially compact.
It's believable if your intuition of Hausdorff spaces comes entirely from metric spaces (where the ...
- 12.4k
24
votes
What is the standard 2-generating set of the symmetric group good for?
It might be interesting (to some) to see that every possible shuffle of a pack of $n$ cards can be achieved by a sequence of operations in which you either swap the first two cards or move the bottom ...
23
votes
Short papers for undergraduate course on reading scholarly math
I think Calkin and Wilf’s lovely short paper on what is now known as the Calkin-Wilf tree would be suitable.
Neil Calkin and Herbert S. Wilf, Recounting the Rationals. The American Mathematical ...
23
votes
Ideas for introducing Galois theory to advanced high school students
There is a nice book, specially written for high school students:
V. B. Alekseev, Abel's theorem in problems and solutions. Based on the lectures of V. I. Arnold (to high school students), and also ...
22
votes
Teaching polarisation formula
To me it seems most natural to show that the norm determines the scalar product via the two formulas
$$\Vert u + v \Vert^2 = \Vert u \Vert^2 + \Vert v \Vert^2 + 2\mathrm{Re}\langle u, v \rangle$$
and ...
- 40.2k
21
votes
Accepted
Source of a quote by Ferdinand Rudio
The quote is from a speech Rudio gave at the Town Hall in Zürich on the 6th December 1883; The German original is published in Felix Stähelin, Reden und Vorträge (1956, I have not found it online).
...
- 188k
21
votes
Books on the relationship between the Socratic method and mathematics?
An influential book on the teaching of mathematics via the Socratic method is Imre Lakatos, Proofs and Refutations. The full book can be browsed on Google, and individual chapters can be donwloaded ...
- 188k
21
votes
Zorn's lemma: old friend or historical relic?
Your choice (ha) to prove the existence of a basis by using a well-ordering in place of Zorn’s Lemma turns things around historically: the first proof of the existence of algebraic bases (using only ...
- 50.6k
20
votes
Accepted
Why do we need random variables?
One of your concerns is (let me quote from your question)
Often I read that there is the possibility of having a family X1,…,Xn of random variables on the same space.
I know no example—and would ...
- 2,583
20
votes
Teaching prime number theorem in a complex analysis class for physicists
"Newman's short proof of the prime number theorem" by Don Zagier might work, in particular since there is an extensive discussion of the steps in that proof in this MSE posting. "The proof has a ...
- 188k
20
votes
Accepted
What is the origin/history of the following very short definition of the Lebesgue integral?
This definition is due to Jan Mikusiński, see Mikusiński, Jan,
The Bochner integral. Basel, Stuttgart: Birkhauser, 1978.
Mikusiński has co-authored another book on integration with Hartman in 1961, ...
- 8,992
20
votes
Accepted
Pedagogically intuitive reformulation of Zorn's Lemma for functional analysis
As a logician myself, I have similar feelings to Joel in that I think transfinite induction and transfinite recursion are underappreciated as general techniques in mathematics.
That said it is ...
- 12.4k
19
votes
Accepted
What do we learn from the Wronskian in the theory of linear ODEs?
Here is a typical use in an undergraduate textbook: to prove that for distinct $\lambda_j$ the exponentials $e^{\lambda_jt}$ are linearly independent. It has some applications on the more advanced ...
- 91.8k
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