42

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 if the base were constant the answer would be $x^x\log x$, so she added the two together to get $x^x+x^x\log x$. I was just about to mark the answer as wrong, ...


26

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 Einstein is weak. See quoteinvestigator.com.) Jo Boaler: "When I have tutored people in math, I've always started by saying, 'By the way, I just want you to know ...


20

Perhaps: The discovery a year ago 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 discovery): Here is a nice article on the discovery in The Guardian, by Alex Bellos. As Alex says, the problem has been studied for $100$ years now, since Reinhardt in ...


19

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-semester linear algebra course: a. It has a natural inner product on it, given by $\langle f, g \rangle = \int_a^b f(t)g(t) \, dt$ b. It contains the (infinite-...


14

"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 failures are people who did not realize how close they were to success when they gave up." Thomas A. Edison


11

“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 gladly, if he could, expunge it from his memory. And yet he ought not entirely to regret it, because he cannot be certain that he has indeed become a wise ...


11

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. As I said over there, I liked that a lot and it stuck with me. Making your own mistakes is a sign of growing up.


9

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 past the low point). 1D: The curve described by a hanging chain/flexible rope (in equilibrium). 2D: The surface of a soap film (in equilibrium). Generally, ...


9

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 lignin should be removed from wood, if one wishes to squeeze the wood into a cheap structure similar to carbon fiber with an optimum strength-to-weight ratio? When ...


9

Mistakes Allow Thinking to Happen


8

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 like, perhaps in stringing together several mini-sessions. Can every triangle tile the plane? (Yes.) Form parallelograms, then argue that a parallelogram tiles ...


8

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, but a vector has no fixed position. Points are positions in space. Vectors are displacements. It makes no sense to add two points, but it does make sense to ...


8

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 \colon V \to V $, taking the class of a function to the class of an antiderivative. So suddenly, the indefinite integral operation is well-defined. Note that it ...


7

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" but doesn't actually have any prerequisites. It's not as quick of a payoff as your ideas in the question, but the outline goes like this: Show them how to ...


7

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 random, but this interesting bias is suddenly discovered. The amazing thing about it is that the observation could have been made by pretty well any beginner ...


6

If you know what you are doing, then you are wasting your time. Anonymous


6

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 Quality SEER Cancer Incidence Bureau of Labor Statistics Bureau of Economic Analysis IMF Economic Data Dow Jones Weekly Returns Data.gov.uk Enron Emails Google ...


6

"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, changing your world. You're doing things you've never done before, and more importantly, you're doing something." Neil Gaiman "Failure is simply the opportunity ...


6

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


6

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 Teller, in Dr. Edward Teller's Magnificent Obsession by Robert Coughlan, in LIFE magazine (6 September 1954), p. 62.


6

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 words pushing in the direction you want versus a raging river pushing the other way. Even Michael Jordan's quote about being cut from his high school basketball ...


6

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 $\theta/\pi$ irrational. The proof I knew was generalizing equidistribution from indicators of intervals to arbitrary (say, continuous) functions and then using Fourier transform. ...


6

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 consequence of the linearity of the Schrodinger equation. The computer science slant on this situation uses the word "qubits."


6

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 $$ E \Delta F = (E \cup F)\setminus (E \cap F) $$ as addition, and $$ 0E = \varnothing,\qquad 1E=E $$ as scalar multiple. Any finite-dimensional vector space over $\mathbb F_2$ has ...


6

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 classes. There's not a consolidated repository though: past exams are hosted under individual course webpages. For example here are the past exams for Math215 ...


5

I customarily use: Gillison, Maura L., et al. "Prevalence of oral HPV infection in the United States, 2009-2010." Jama 307.7 (2012): 693-703. This has a nice mix of CI's, P-values, and statements about distribution shapes (bimodal) in the abstract. At one point I used this article: Peck, Peggy, "Long Work Hours Increase Hypertension Risk", MedPage Today. ...


5

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 maximal volume. The rectangle might be specialized to a square, as below. See also The Math Forum.                 ...


5

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 much exactly what you ask for. Research mathematicians present some "recent developments" in their field in a way that is understandable for high school ...


5

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 the truth, but you don't know which one is which. What one question can you ask to find out which path you should take? Three boxes are labeled "Apples", "...


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