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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 ...
Gerry Myerson's user avatar
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 ...
Joseph O'Rourke's user avatar
25 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 ...
Joseph O'Rourke's user avatar
24 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-...
mweiss's user avatar
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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'...
Daniel R. Collins's user avatar
13 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 ...
Fareed Abi Farraj's user avatar
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 ...
Gerry Myerson's user avatar
12 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 \...
Gaussler's user avatar
  • 221
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 ...
user1815's user avatar
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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 ...
Joseph O'Rourke's user avatar
10 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 ...
Robin's user avatar
  • 209
10 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. ...
Peter - Reinstate Monica's user avatar
10 votes

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

Mistakes Allow Thinking to Happen
Burt's user avatar
  • 694
10 votes

Big list of "interesting" abstract vector spaces

The set of solutions to a system of linear homogeneous ODEs is a vector space, and the dimension of this vector space is equal to the total order of the system. The idea that every solution is the ...
Michael Seifert's user avatar
10 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 \mathbin{\Delta} ...
Gerald Edgar's user avatar
  • 7,607
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 \...
Adam Latosiński's user avatar
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 ...
Jasper's user avatar
  • 3,178
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 ...
NiloCK's user avatar
  • 5,020
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 ...
Dave L Renfro's user avatar
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,...
Joseph O'Rourke's user avatar
8 votes

Big list of "interesting" abstract vector spaces

Below are two elementary/expository papers that I know for generating strange examples, although how interesting they are will vary from person to person (however, a connection with the special theory ...
Dave L Renfro's user avatar
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" ...
Chris Cunningham's user avatar
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 ...
Sebastian Schoennenbeck's user avatar
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 $\theta/\pi$ ...
Kostya_I's user avatar
  • 1,411
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 ...
Joseph O'Rourke's user avatar
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 ...
Joseph O'Rourke's user avatar
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 ...
svavil's user avatar
  • 324

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