23

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 ...


18

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 ...


11

"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


10

“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 ...


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

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.


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 ...


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 ...


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

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

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.


5

Some physics examples: -- Given that the range of a projectile is $R=(v^2/2g)\sin\theta\cos\theta$, prove that the maximum range is achieved for $\theta=45$ degrees. An electrical transmission line of resistance $x$ is in series with a load of resistance $y$. For a fixed voltage $V$, the useful power dissipated in the load is $P= V^2y(x+y)^{-2}$. Show that ...


5

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


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

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", "...


5

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


5

Mistakes Allow Thinking to Happen


4

Since Benjamin Dickman mentioned The Tokyo Puzzles in the comments, I'll include a couple of the questions from that book here that I thought fit the prompt nicely; neither requires a pen and paper to think about, and a student can just ponder them mentally. Two brothers decided to run a 100-meter race. The older brother won by 3 meters. In other words,...


4

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 ...


4

I did something very similar as a project when I taught AP statistics a few years ago. It was relatively effective and my students left that class with that project as one of the ones they felt was the most impactful, since it showed them how statistics was actually used in real life. I believe that the most recent topics for this project were linear ...


4

I always enjoy bringing in Bortkiewicz’s data on the annual deaths by horse kicks in the Prussian Army from 1875-1894.This is what inspired the discovery of the Poisson distribution. Sometimes, students enjoy thinking about how to collect data about rare events: How often does the anaesthetist has fallen asleep during surgery? And other medical “Never ...


4

"Human is the only animal that trips twice over the same stone" - Anonymous "Next time you trip over a stone, instead of stepping over it, place a big flashing sign that will remind you where you fell to the ground. The next time you travel the same path you will remember your past mistake and be able to avoid the hazard" - Me This is trying to be funny to ...


4

The road to wisdom? — Well, it's plain and simple to express:         Err         and err         and err again         but less         and less         and less.” — Piet Hein


4

For a much lower-level topic, consider explaining to beginning algebra students why "like terms" can be combined. On a few occasions, I have resorted to reasoning with students that adding algebraic expressions is like adding quantities with units. [Our curriculum begins with units and geometry before algebra, so this is usually safe ground in my class.] If ...


4

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 ...


3

Edit (June 2019): I used this final project with minor tweaks again this year. You can find links to some of the output for Spring 2019 - both students' graphs and write-ups - here and here. I used a Desmos Make-A-Graph prompt for an Algebra 2 class' final project last year. The results were quite good; so, I expect to incorporate at least one similar ...


3

Another math circle activity that is fine for kids (warning: I have not tried this specifically on primary school, so the patience required might be too high): Modular arithmetic modulo small numbers is required, but kids are more than happy to accept it. Write on a piece of paper that (only on this piece of paper) 3 = 0. Then talk about it; if 3 = 0, then ...


3

There are elementary results in mathematics being produced regularly either in the sense that the ideas don't need a lot of background to understand them or sometimes that even the "details" of the proofs are comprehensible. An example, is that of the de Bruijn graph, which has ties to the theory of Eulerian circuits in directed graphs and has found many ...


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