Of Lines, Curves, and Points....

 Patrick Honner looking at lines and curves, and making a few rational (and irrational) points along the way:

https://www.quantamagazine.org/how-simple-math-reveals-rational-points-on-curves-20210722/

Weekend Potpourri

So many mathy reads to choose from this weekend, if you've missed them (really enough for a couple of weekends!)....

1) Brand new from Sol Lederman, a wonderful podcast with mathematician/writer Erica Klarreich:

http://www.buzzsprout.com/5316/78729-erica-klarreich-inspired-by-math-22

I've previously mentioned my belief that interviewing lesser-known math figures is almost more enticing than interviewing the titans in the field because readers/listeners already know so much about the 'big names' out there that much of what they say may seem repetitious (even if important) of things they've voiced before. Lesser-known folks have a fresher appeal as there is so much new to learn about them, and I think this podcast demonstrates my point. I was fascinated hearing Erica's views and experiences on a wide range of topics. See if you don't agree.
And below, some of Erica's prior writings for the Simons Foundation:

https://simonsfoundation.org/?s=erica+klarreich&submit

2) MIT physicist Max Tegmark is famous for his belief that the Universe is "built" of mathematics. He expresses his viewpoint in this straightforward interview from the ScienceNow site… and attracts a lot of comments in the process (including cynical ones):

http://news.sciencemag.org/sciencenow/2013/02/do-we-live-inside-a-mathematical.html

3) Julie Rehmeyer, who I just posted about a few days ago, has a new piece on Fermat's Last Theorem and its axiomatic basis, in ScienceNews here:

http://tinyurl.com/a4ujnx5

4) Patrick Honner again laments a question from a NY State Regents Math exam that entails unstated assumptions and in so doing short-circuits deep mathematical thinking:

http://mrhonner.com/2013/02/21/regents-recap-january-2013-unstated-assumptions/

5) Some bloke named Keith Devlin has a fantastic longread on math games in American Scientist, leading up to release of his own company's new animated game for math learning. He sets forth the criteria or qualities a video needs to possess to be successful as a math instruction tool (and explains why MOST games FAIL):

http://www.americanscientist.org/issues/pub/the-music-of-math-games

From the description, it sounds to me as if the prospective player (young person) of these new games will learn math in a manner reminiscent of the original Karate Kid (the movie) learning karate without ever knowing it from his master Mr. Miyagi. Keith's discussion of the "symbol barrier" and symbol manipulation in math is especially enlightening, but the entire piece is GREAT. He employs a music (piano) metaphor to explain what a successful math game should be like.

6) The brilliant Barry Mazur, recent recipient of the National Medal of Science, gives us a rich (and philosophical) piece called "Shadows of Evidence" on what constitutes "evidence" in the realm of mathematics. The essay ends with a quote from Chris Anderson essentially arguing that "modeling," as traditionally used in science, is becoming obsolete (because ALL models are, technically, flawed), and that with the advent of computer number-crunching ability, only "correlation" derived from huge data sets will be needed. To which Mazur responds that, "correlation alone will never replace the explanatory power of mathematics.":

http://tinyurl.com/bhzrjgf

This essay in turn leads to an even longer Mazur read entitled "What Is Plausible" here (pdf):

http://www.math.harvard.edu/~mazur/papers/Plausibility.Notes.3.pdf

7) Finally, perhaps appropriately after all of the above, I'll end with 50 varied quotations just put up by Guillermo Bautista on what mathematics is:

http://mathandmultimedia.com/2013/02/23/50-mathematics-quotes/

a few of my favorites:

"Mathematics is no more computation than typing is literature." – John Allen Paulos

"Mathematics, in the common lay view, is a static discipline based on formulas…But outside the public view, mathematics continues to grow at a rapid rate…the guide to this growth is not calculation and formulas, but an open ended search for pattern."  -- Lynn A. Steen

"Mathematics compares the most diverse phenomena and discovers the secret analogies that unite them." — Joseph Fourier 

.... ADDENDUM:  Experimenting with shorter time segments, Sol Lederman has already put up another podcast interview (22 mins.), this time with Jason Ermer, creator of the "Collaborative Mathematics" project (who I referenced a bit ago):

http://www.buzzsprout.com/5316/78788-jason-ermer-inspired-by-math-23

Mathematicians have been at the forefront of bringing productive Web collaboration to academic subjects, and now Jason is attempting it at lower levels. Check it out!

Making Assumptions...

Patrick Honner, a NY award-winning math teacher who I previously interviewed here, gave this talk to Math For America about a month ago dealing with certain aspects and assumptions of standardized tests that he keeps an eye on (the audio isn't quite as crisp as one might like and may be better through earphones than over speakers):




In a slightly similar vein, Sol Lederman's latest podcast is with Glenn Van Brummelen who talks (and has a book) about "spherical trigonometry," which is a somewhat lost art that nonetheless remains valid and practical, even while planar trigonometry remains the brand we all learn routinely in school:

http://wildaboutmath.com/2013/01/27/glen-van-brummelen-inspired-by-math-18/

A Bit of Serendipity

Since the prior post was heavy with some Tim Gowers material I can't help but point out another cosmic synchrony ;-) that materialized on the Web… A short while back Patrick Honner (who I just interviewed recently) posted this note on his blog:

http://mrhonner.com/2012/10/24/this-is-not-a-trig-function/

...basically, he notes that a purported sine wave curve on a NY State math exam simply didn't look right to him at a glance ("too rounded"), and then goes on to show that his visual hunch was indeed correct upon closer inspection. The ensuing comments to his brief post are fascinating as well, but the capper is that about 10 days later, in a bit of serendipity, Tim Gowers writing on Google+ noted his immediate sense that a British air traffic control tower he'd passed by recently did not conform to a "mathematically natural shape," and wondered aloud "how acute this sense that mathematicians have is," at which point he stumbled upon Honner's post:

https://plus.google.com/u/0/103703080789076472131/posts/VhHE8TuJhcm

…it reminds me a wee bit of the old retort (applied when you succeed at something, and someone else says it was 'just luck'), that it seems 'the more I practice the luckier I get!' ...or, in-other-words, it's a skill developed from experience.
Having said that though I s'pose it's a bit difficult to know which is the cart and which is the horse here: i.e., do mathematicians have a special knack for recognizing certain patterns and forms because it's something their minds have experienced repeatedly (practiced)… or, is it that people who have a special or inborn aptitude for form and pattern recognition are more likely to end up in mathematics???


[p.s... if anyone hears the result of Tim Gowers' recent heart surgery and how he is doing, please report to us in the comments; I'm sure many would like to hear that all went well.]

ADDENDUM 11/10/12: Tim now has a detailed post up that he is back home recovering (with some discomfort) from the hopefully successful surgery:
http://gowers.wordpress.com/2012/11/09/what-actually-happened/

Patrick Honner of MrHonner

Math-frolic Interview #6

"What I enjoy the most about mathematics is the moment when a solution, a relationship, or a structure becomes evident.  It's a powerful feeling when you conquer a challenge, or see and understand the real essence of something for the first time." --Patrick Honner

Patrick Honner is an award-winning New York state secondary math teacher who is quite active on the Web.

He blogs at MrHonner.com and has given a TedxTalk as well, in addition to also being a contributor to the NY Times Learning Network (and is @MrHonner on Twitter).

He kindly answered my inquiries as follows:

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1) How did your interest in mathematics originally come about, and when did you realize you wanted to pursue math professionally?

I've always enjoyed math.  I recall participating in math contests in elementary school, and I consistently had good math teachers through junior high and high school.  When I started college I wasn't sure what I was going to do, but I knew I would keep taking math classes.  I never really did figure out what I was going to do, so I just kept taking math classes and moved on to graduate school.

2) What are your favorite aspects of mathematics that you most like studying/reading about?

What I enjoy the most about mathematics is the moment when a solution, a relationship, or a structure becomes evident.  It's a powerful feeling when you conquer a challenge, or see and understand the real essence of something for the first time.   I still regularly experience that feeling now, as I learn new mathematics, or learn to see old mathematics in new ways.  It's part of what makes teaching math so wonderful.

3) How do you go about selecting the topics you blog about? And what do you think is the strongest 'draw' for your blog, out of so many 'math education' blogs?

 I write about my mathematical experiences, which range from the mundane--like over-thinking prices in the supermarket--to the academic--like finding novel proofs and derivations of facts and theorems.  Mathematics plays a substantial role in how I understand and engage with the world, so it's always present in my mind.

I think the depth and variety of my experiences, both in mathematics and in teaching, give me a unique perspective in this field.  And I think the way I look to celebrate and appreciate math--through compelling questions, interesting stories, and beautiful images--makes math accessible and enjoyable in a novel way for some readers.

4) A controversial editorial ran in the NY Times awhile back questioning whether algebra should be a required course for ALL high school graduates. Many responses appeared on the Web to that piece, and eventually YOU had a full response in the Times itself. Can you tell any backstory to that (or post-story for that matter)? Did you approach the Times about doing a reply, or did someone there approach you specifically for a response? And are you pleased with the overall discussion the episode generated?

My initial reaction to the editorial in question was to notice, as others did, that Andrew Hacker didn't really seem to understand what algebra is.  After all, he suggested that we replace algebra with, well, algebra.
I have been contributing math content (like lessons, activities, and quiz questions) to the New York Times Learning Network for several years, and I thought the controversy surrounding "Is Algebra Necessary?" created a perfect opportunity to demonstrate to students and teachers how the tools and techniques of algebra can be used to explore what anyone can find in the Times.

 The response to the piece was great.  Despite being up for less than a month, "N Ways to Apply Algebra with the New York Times" was one of the Learning Network's top-viewed posts of the past year.  And lots of teachers and students responded with comments.

5) You also took on the "establishment" with a series of blog posts about flaws in a NY state math examination… has anything substantive resulted from those critiques?
And you're actively involved in various efforts to improve secondary math education in the U.S. where there seems (to me, as an education outsider) to be a lot of disagreement/controversy over how best to proceed. How well do you think matters are proceeding, and are you optimistic that math education, nationwide, will be much improved for future generations, or will there always be unresolved controversy over methods?
 
Nothing substantive has resulted from my critiques of math problems on New York state exams, nor do I expect anything to happen.  I'm simply trying to raise the point that the quality and validity of these standardized exams are rarely, if ever, called into question.  If the tests aren't good, how can they possibly determine if teachers should keep their jobs or schools should remain open?
Public education will always be, in part, a political issue, and thus will always be subject to the controversies (both real and manufactured) that politics brings to everything it touches.  I think the best avenue for improving math education is a sustained focus on elevating the profession of teaching: more support for teachers at the ground level; more opportunities for growth in content-knowledge and pedagogy; and real opportunities for teachers to collaborate, share, and actively shape the profession itself.

I have been very fortunate to be a part of Math for America, an organization that does all of this in the most supportive, least restrictive way imaginable. It has made a huge difference in my career, and in the careers of many others.

6) What are some of your favorite math books to read for enjoyment, and how about math books you'd especially recommend to lay people with some math interest?

I enjoy doing math more than reading it.  I greatly enjoy solving problems, creating new problems to solve, or creating new ways to think about mathematical ideas.  In terms of books about math for lay people, I'm not sure anyone does it better than Steven Strogatz.  His "The Calculus of Friendship" and "The Joy of X" are both wonderful.  John Allen Paulos has also written several excellent, accessible books that I've enjoyed, like "A Mathematician Plays the Stock Market".

7) What online math resources do you find especially useful in the classroom? And do you use your blog or any social media in your classroom as well?

Alexander Bogolmony's Cut the Knot is one of the best math websites around for both teaching and learning.  He has a unique perspective on math and teaching, and he has produced many wonderful interactive mathematical explorations that I and my students enjoy.  I am a huge fan of Geogebra, and I find myself using Desmos more and more.

I try to use my website as a bridge between the classroom and the greater world of mathematics for my students, a place for us to continue our conversation and share new experiences.  I have students create blogs as part of projects, and have experimented with other social media technologies as teaching and learning tools as well.

8) Any parting words, not covered above, you'd want to pass along to a math-oriented audience?

Participation in the digital mathematics and math education communities has profoundly impacted me, both personally and professionally.  Thanks to everyone out there who reads, writes, tweets, and posts; this is truly a remarkable community to be a part of!

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-- Thanks Patrick for participating here, and even more-so for your active engagement in the wider field of math education. As they say, 'keep fighting the good fight!'

[...and if readers have anyone you would particularly like to see interviewed here let me know.]