49
votes
Interesting things you learned while grading?
I once asked students to find the derivative of $x^x$ (with respect to $x$). One student figured that if the exponent were a constant then the answer would be $xx^{x-1}$ which is to say $x^x$, while ...
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26
votes
What are some recent, interesting, accessible pieces of mathematics
Perhaps: The discovery a year ago
in 2015 of a new tiling of the plane by
a convex polygonal tile, found by
Mann, McLoud, and Von Derau (the latter of whom was an undergraduate at
the time of the ...
- 29.5k
26
votes
Accepted
Quote to show students don't have to fear making mistakes
Johann Wolfgang von Goethe: "By seeking and blundering we learn."
Original German, 1825.
Albert Einstein: "Anyone who has never made a mistake has never tried anything new." (However, attribution to ...
- 29.5k
21
votes
Accepted
Big list of "interesting" abstract vector spaces
Here are some more examples:
$C[a,b]$, the set of continuous real-valued functions on an interval $[a,b]$. This abstract vector space has some very nice properties that make it very good for a first-...
- 17.3k
17
votes
Interesting things you learned while grading?
Possibly not what you're looking for, but: the things I've learned while grading are mostly pedagogical, not new mathematical facts (in fact, teaching at a community college as I do, I'm not sure that'...
- 24.4k
14
votes
Quote to show students don't have to fear making mistakes
"I have not failed. I've just found 10,000 ways that won't work."
"Our greatest weakness lies in giving up. The most certain way to succeed is always to try just one more time."
"Many of life's ...
13
votes
Interesting things you learned while grading?
Reading the answer posted by Daniel R Collins reminded me of something else I learned while marking student work. Not exactly something mathematical, more something about constructing math exams.
I ...
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12
votes
Where can I find public repositories of past math exams?
Quite a few universities publicly post the math exams their faculty write:
UC Berkeley hosts an archive of their past exams, sorted by course.
University of Michigan hosts past exams for some ...
12
votes
Educators, what resources have you built to better serve your students?
Riemann Sum visualization
Fundamental Theorem of Calculus Explanation
Error bound on integral for increasing function
"Secant Circles" and "Tangent Circles"
Some students of mine ...
11
votes
List of realistic extremum problems
Some example types:
Minimizing potential energy of any realistic physical system. Examples:
0D: The point where a rolling ball/flowing water might* come to rest (*might not, if momentum carries it ...
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11
votes
Quote to show students don't have to fear making mistakes
“There is no man,” he began, “however wise, who has not at some period
of his youth said things, or lived in a way the consciousness of which
is so unpleasant to him in later life that he would ...
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11
votes
Quote to show students don't have to fear making mistakes
There was a post over at academia where somebody essentially said that after starting by investigating other people's mistakes:
Eventually I got better — I started making my own mistakes.
...
10
votes
Good Examples of Questions to have Students Ponder Over Without Paper
A possibility, requiring one definition: What is a tiling of the plane with an infinite supply of congruent copies of a single tile (technically,
a monohedral tiling). This can go as deep as you'd ...
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10
votes
10
votes
Pi Day is approaching: What are some interesting math questions whose answer is exactly $\pi$?
Regarding your second example. Not only $\int_{-\infty}^\infty \frac{\sin x}{x} dx = \pi$, but also
$$\int_{-\infty}^\infty \frac{\sin x}{x} \frac{\sin (x/3)}{x/3} dx = \pi $$
$$\int_{-\infty}^\infty \...
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9
votes
List of realistic extremum problems
Here are some optimization problems that were harder than a simple homework problem:
Given the pavillion angle of a diamond, what crown angle produces optimal light return?
What percentage of the ...
- 3,148
9
votes
What are some recent, interesting, accessible pieces of mathematics
The 2016 result about Unexpected biases in the distribution of consecutive primes.
This is really pretty simple to understand - the distribution of primes had been supposed to be unconditionally ...
- 4,980
9
votes
Big list of "interesting" abstract vector spaces
The vector space $V = C^{\infty}(\mathbb{R},\mathbb{R})/\mathbb{R}[x]$ of smooth functions modulo polynomials. Note that $ d/dx \colon V\to V $ is an isomorphism, so that we have a nice inverse $\int \...
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9
votes
Accepted
What are examples of math-themed sci-fi appropriate for students?
On the off-chance you don't know about it, I recommend looking through entries at Alex Kasman's website Mathematical Fiction. This has been around quite a while. I don't know when I first learned ...
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8
votes
Examples of informal explanations that cause misconceptions
A point (say, in $\mathbb{R}^n$) is a vector. Vectors and points are really no different. They are both $n$-tuples in $\mathbb{R}^n$.
The difference between two points (in $\mathbb{R}^n$) is a vector,...
- 29.5k
8
votes
Big list of "interesting" abstract vector spaces
Let $\Omega$ be a set, and let $\mathcal A$ be an algebra of subsets of $\Omega$. Then $\mathcal A$ is a vector space over the field $\mathbb F_2 = \{0,1\}$, with the operation
$$
E \Delta F = (E \...
- 7,369
7
votes
Accepted
what are some surprising result of math for a kid?
I had a lot of luck when running a primary school math club (ages 9-10) with the game of Nim. It's not much harder than tic-tac-toe to play, but there is a lot more math underneath that sounds "hard" ...
- 21.2k
7
votes
What are some recent, interesting, accessible pieces of mathematics
Not a single answer but rather a resource:
The snapshots of modern mathematics from the mathematical research institute Oberwolfach (http://www.mfo.de/math-in-public/snapshots/) aim to provide pretty ...
7
votes
Interesting things you learned while grading?
I gave an advanced course on Probability that contained some ergodic theory. In exercises, I outlined the usual proof of the equidistribution of $e^{in\theta}$ on the circle, for $\theta/\pi$ ...
- 1,164
6
votes
List of realistic extremum problems
There are many volume-of-a-box questions. I like this one, simpler than
what the OP cites:
Given a rectangle,
cut out squares from the corners so you can fold it up to a box, without a top,
of ...
- 29.5k
6
votes
Accepted
Favorite datasets to use when teaching statistics
19 public data sets, from Springborg blog, curated by T.J. DeGroat.
Summaries and links for each in DeGroat's page.
United States Census Data
FBI Crime Data
CDC Cause of Death
Medicare Hospital ...
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6
votes
Quote to show students don't have to fear making mistakes
If you know what you are doing, then you are wasting your time.
Anonymous
- 10.3k
6
votes
Quote to show students don't have to fear making mistakes
"I hope that in this year to come, you make mistakes. Because if you are making mistakes, then you are making new things, trying new things, learning, living, pushing yourself, changing yourself, ...
6
votes
Quote to show students don't have to fear making mistakes
Is a quote about failing in topic? I think so. Here we go.
“Ever tried. Ever failed. No matter. Try again. Fail again. Fail better.”
Samuel Beckett
More about this quote on booksonthewall
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