Math Methods Versus Math Tricks

This week's Sunday reflection comes from Jim Propp in a recent Web piece:

"Mathematicians are people who like solving problems, and have the persistence to work on problems that take time to solve, and have collected a mental tool-kit consisting of methods that have helped them solve problems in the past. Some mathematicians distinguish between methods and tricks. A method is a tool that solves more than one problem, while a trick is a tool that applies to only one. Under this definition, I’d say that there are no tricks in math, and part of the discipline of getting good at math is to study every trick you encounter until you see the method hiding inside it."

Ben and Jim Deliver (good Wednesday stuff!)

These are both toooooooo good to hold them until the Friday potpourri, so passing them along now:

1) Ben Orlin, a bit more serious than some Wednesdays, on “the three phases of the mathematical life” (competition, mentorship, and collaboration):

https://mathwithbaddrawings.com/2017/05/24/the-three-phases-of-the-mathematical-life/ …

2)  Jim Propp on “Math, magic, and mystery” in this 36-min. video describing math being “liberated from (physical) reality”:

A couple of pieces I think might also make good adjunct readings to Jim’s talk are this Evelyn Lamb piece from a couple years back on epsilons and deltas:

https://blogs.scientificamerican.com/roots-of-unity/the-subterfuge-of-epsilon-and-delta/

…and this old Terry Tao piece on rigor and intuition in math:

https://terrytao.wordpress.com/career-advice/there’s-more-to-mathematics-than-rigour-and-proofs/

Either Jim Propp is Santa Claus or Just a Great Blogger

I wrote shortly ago that I expected Jim Propp to have an interesting take on self-referential sentences sometime this month, and that exceedingly-rich post is now up, full of fascinating ideas (...I’ll admit ‘self-reference’ isn’t everyone’s cup-of-tea of a topic, but for those it is, Jim’s thoughts and many great links and endnotes are must-reading, including Jim’s own self-referential ‘test,’ if you’re in the mood for a real mind-twister):

https://mathenchant.wordpress.com/2016/11/17/breaking-logic-with-self-referential-sentences/

[One, of many, small interesting sidelights for me was discovering that Janna Levin, better known for her physics writing, had written a 2007 novel intertwining the lives of Kurt Gödel and Alan Turing. 

-- p.s. here's a link to Dr. Levin's appearance at "The Moth," which Brainpickings' Maria Popova once called "the greatest story ever told on The Moth").]

Props to Propp...

Everyone probably knows Jordan Ellenberg's fabulous bestseller from a few years back, "How Not To Be Wrong." A few days ago at "Mathematical Enchantments" Jim Propp put up one of the odder math posts I've seen in quite awhile (and a bit longish, but nonetheless entertaining) entitled (taking off from Jordan), "How To Be Wrong." It has to do with the benefit of teachers, for the sake of teaching, making mistakes and facing them square-on... and, not being too dogmatic or pompous along the way; what he calls "the art of being wrong":
 
https://mathenchant.wordpress.com/2016/01/16/how-to-be-wrong/

At one cogent point Propp writes:

"I stress to my students that the earlier you make your mistakes, the better.  Every mistake you make in the classroom, or on your homework, is a mistake you probably won’t make on the exam, where mistakes can really hurt you.  And, carrying this idea further: a mistake you make in college or graduate school is a mistake you’re less likely to make after you graduate, when you’re building bridges or designing cancer treatment protocols. So my answer to the question 'How to be wrong?' is: 'Early and often!'”

By name or initials he brings a number of interesting folks into the post, as well as many interesting ideas/examples. I'm actually surprised there haven't been more responses in the comments section to the post, though maybe it is so odd that folks aren't sure just how to respond to it.
Anyway, check it out, especially if you're a math teacher.

Also, all of this emphasis on "mistakes" reminds me of an older Keith Devlin post on the importance of "failure":
http://mooctalk.org/2013/12/15/maththink-mooc-v4-part-2/

It too, should NOT be missed, nor should the two commencement addresses Dr. Devlin links to at the end (Steve Jobs and J.K. Rowling). Great stuff!