48 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 ...
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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 ...
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20 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-...
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15 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'...
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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 ...
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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 ...
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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  • 219
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. ...
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9 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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  • 4,569
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 ...
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  • 2,871
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 ...
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  • 4,850
9 votes

Quote to show students don't have to fear making mistakes

Mistakes Allow Thinking to Happen
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  • 654
8 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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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,...
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8 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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8 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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  • 181
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" ...
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  • 19.1k
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 ...
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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 $\theta/\pi$ ...
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  • 689
7 votes

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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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
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  • 7,575
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

"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, ...
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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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  • 169
6 votes

Quote to show students don't have to fear making mistakes

The one I'm using is: An expert is a person who has found out by his own painful experience all the mistakes that one can make in a very narrow field. Attributed to Niels Bohr, quoted by Edward ...
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  • 324
6 votes

Quote to show students don't have to fear making mistakes

There are plenty of quotes here... But if you're trying to increase willingness to make and learn mistakes, I doubt that quotes will change a lot of minds, because a quote will be a tiny trickle of ...
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6 votes

Big list of "interesting" abstract vector spaces

The spin states of an electron form a two-dimensional vector space over the complex numbers. Designate "spin up" and "spin down" for a basis. The vector space structure is a ...
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  • 7,575
6 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 \...
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  • 6,171
5 votes

Good Examples of Questions to have Students Ponder Over Without Paper

Perhaps logic puzzles would work in this case. Some classic examples are: You're traveling along a road and arrive at a fork. Two guides are posted, but one always lies and the other always tells ...
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