MAA Distinguished Lecture Series

If you live in the DC area and you like math, you have no excuse! Come to the MAA Distinguished Lecture Series.

These are one-hour talks, complete with refreshments, all for free due to the generous sponsorship of the NSA. The talks are at the Carriage House, at the MAA headquarters near Dupont Circle.

Here are some of the great talks that are on the schedule in the next few months (I’m especially excited to hear Francis Su on May 14th).

I’ve been to many of these lectures and always enjoyed them. Robert Ghrist‘s lecture was out of this world (here’s the recap, but no video, audio, or slides yet) and was so very accessible and entertaining, despite the abstract nature of his expertise–algebraic topology.

And that’s the wonderful thing about all these talks: Even though these are very bright mathematicians, they go out of their way to give lectures that engage a broad audience.

Here’s another great one from William Dunham, who spoke about Newton (Dunham is probably the world’s leading expert on Newton’s letters). Recap here, and a short youtube clip here:

(full  talk also available)

So, if you’re a DC mathophile, stop by sometime. I’ll see you there!

USA Science and Engineering Festival

If you’re local, you should go check out the USA Science and Engineering Festival this weekend. It’s on the mall in DC and everything is free.

USA-Science-and Engineering-Festival LogoThey will have tons of booths, free stuff, demonstrations, presentations, and performances. Go check it out!

For my report on the fest from two years ago, see this post. The USA Science and Engineering Festival is also responsible for bringing to our school, free of charge, the amazing James Tanton!

I ♥ Icosahedra

Do you love icosahedra?

I do. On Sunday, I talked with a friend about an icosahedron for over an hour. Icosahedra, along with other polyhedra, are a wonderfully accessible entry point into math–and not just simple math, but deep math that gets you pretty far into geometry and topology, too! Just see my previous post about Matthew Wright’s guest lecture.)

A regular icosahedron is one of the five regular surfaces (“Platonic Solids”). It has twenty sides, all congruent, equilateral triangles. Here are three icosahedra:

icosahedron coloringsHere’s a question which is easy to ask but hard to answer:

How many ways can you color an icosahedron with one of n colors per face?

If you think the answer is n^{20}, that’s a good start–there are n choices of color for 20 faces, so you just multiply, right?–but that’s not correct. Here we’re talking about an unoriented icosahedron that is free to rotate in space. For example, do the three icosahedra above have the same coloring? It’s hard to tell, right?

Solving this problem requires taking the symmetry of the icosahedron into account. In particular, it requires a result known as Burnside’s Lemma.

For the full solution to this problem, I’ll refer you to my article, authored together with friends Matthew Wright and Brian Bargh, which appears in this month’s issue of MAA’s Math Horizons Magazine here (JSTOR access required).

I’m very excited that I’m a published author!

Welcome James Tanton!

image stolen directly from mathcircles.org

Today we have the special privilege of hosting the one and only, Dr. James Tanton. He will be our guest speaker today and he’ll be talking with our students about his love for math, and hopefully spark in them an appreciation for mathematical play.

We’ll have 800 students at the assembly. And James will be armed with nothing but paper and pen (and a document camera). Bold man! :-)

If you’ve never checked out James’ materials, go visit his website, take a look at his prolific youtube channel, or follow him on twitter @jamestanton.

James is the author of 10 books on mathematics and math education. He is currently a Mathematician in Residence at the Mathematical Association of America, right here in Washington DC. He comes to us by way of the USA Science & Engineering Festival and its sponsors (Lockheed Martin, Northrop Grumman, Scientific American, Popular Science, and others). Thank you, USA Science & Engineering Festival!

We’re very excited to have James with us!

Math on the web

Here are two items that have been shared with me in the last 24 hours:

Item 1: Want To Be Better At Math? Use Hand Gestures! Jeremy Shere of Indiana Public Media. Check out this very short audio news that suggests that math instruction has been shown more effective with gestures. I flail around in front of my classroom all the time, so I guess that makes me a good teacher, right? I’d sure like to think so! :-)  (HT: Tim Chase)

Item 2: How to Fall in Love With Math. Manil Suri, professor at a small school down the road from me (University of Maryland…maybe you’ve heard of it?), has a very nice piece on why math is a worthy object for our affection. It’s been heavily shared in the circles I travel–and for good reason. He reminds us that people fall susceptible to two very common errors when casually speaking about math: (1) We reduce it to arithmetic, as in “come on guys, do the math” or (2) we elevate it to something so ethereal that it’s impossible to grasp, as in “that mathematician talks and I don’t understand a word he says. I never was good at math.” Math, Suri says, is much more than arithmetic and much more accessible than people give it credit for. People can appreciate it without understanding every difficult nuance, just as they do art. (HT: Beth Budesheim)

 

 

 

Progress Toward Twin-Prime Conjecture

This nice article came through on wired today:

Unknown Mathematician Proves Surprising Property of Prime Numbers

By Erica Klarreich, Simons Science News

Image: bwright923/Flickr

On April 17, a paper arrived in the inbox of Annals of Mathematics, one of the discipline’s preeminent journals. Written by a mathematician virtually unknown to the experts in his field — a 50-something lecturer at the University of New Hampshire named Yitang Zhang — the paper claimed to have taken a huge step forward in understanding one of mathematics’ oldest problems, the twin primes conjecture.

Editors of prominent mathematics journals are used to fielding grandiose claims from obscure authors, but this paper was different. Written with crystalline clarity and a total command of the topic’s current state of the art, it was evidently a serious piece of work, and the Annals editors decided to put it on the fast track.

Just three weeks later — a blink of an eye compared to the usual pace of mathematics journals — Zhang received the referee report on his paper.

“The main results are of the first rank,” one of the referees wrote. The author had proved “a landmark theorem in the distribution of prime numbers.”

(more)

This is very exciting news, and the whole story has a fantastic David & Goliath feel–“little known mathematician delivers a crushing blow to a centuries old problem” (not a fatal blow, but a crushing one). It’s such a feel-good story, almost like Andrew Wiles and Fermat’s Last Theorem. Here’s my favorite part of the article:

…during a half-hour lull in his friend’s backyard before leaving for a concert, the solution suddenly came to him. “I immediately realized that it would work,” he said.

Just chillin’ in his friend’s backyard…and it came to him! Anyone who has worked on math problems or puzzles has had this experience, right? It seems like an experience common to all people. This has definitely happened to me lots of times–an insight hits me out of nowhere and unlocks a problem I’ve been working on for weeks. It’s one of the reasons we do mathematics!

TI Calculator Emulators

Online Emulator

Check out this online TI-83 Plus emulator! This just came across my radar from Hackaday.

It requires that you upload a (legally acquired) rom, but once you do, this seems like it would be a very good ‘on-the-go’ resource for presentations, teaching, or just any other time and place when you might need at graphing calculator.

I don’t have a TI-83 plus rom lying around, and I tried a regular TI-83 rom (which I did happen to have) but it didn’t seem to work for me. Hmm.

Mobile Devices

In other emulator news, I have successfully gotten a TI-83 Plus and a TI-89 emulator on my Android phone. Maybe it’s not news to you, but I was delighted when a student told me about this. Here’s how to do it:

These instructions come (verbatim) from our student, Jim Best. Thanks, Jim!

Android Products

This is completely free and legal. (I mention legal because sometimes different ROMS can be illegally pirated. But this one isn’t.)

  1. Download the app “Andie Graph” on your market for the phone
  2. Go to this link on the phone to download the ROM.
  3. There will be ads promoting other products so click on the bottom link that says DOWNLOAD.
  4. Once this has finished downloading, Run “Andie Graph”
  5. Go to settings by pushing the little icon on the phone itself that looks like a garage door or a tool box.
  6. Go down to ROM and select the ROM you downloaded. If the app doesn’t find the ROM, then you can search for it from the app in the phone.

Apple Products

There is an app that is a type of TI-83. It is called RK-83 on the app store for apple products such as the iPhone and iPod touch. This is a $0.99 app that has the same functionality as a TI-83. It does not have the best of reviews but for $0.99, its worth a shot. There is also an app by the same creator that has better reviews but it is an 89.

Plain-old Software

And as far as plain-old installation software goes, here are some great emulators if you haven’t already found them:

I actually prefer the Rusty Wagner emulators to the TI-SmartView emulator, even though our school has purchased copies for all of the math teachers.

Product Failure

I’ve been taking a grad course in statistics this semester and so I’ve been thinking about all sorts of real world examples of math, including the classic product-failure example that’s a mainstay of most stat classes.

One of the simplest continuous distributions is the exponential distribution which is a pretty decent way to model product failures. The probability of failure f(t) after time t is given by

f(t)=\frac{1}{\lambda}e^{-t/\lambda}.

I read this great article about product failure and testing in Wired this week. I encourage you to check it out. Read the last page of the article especially, where it talks about how cutting-edge companies are modeling minute variations in materials using an electron microscope and some statistics. Instead of actually testing the product over and over again using a fatigue machine, they can create surprisingly accurate models of the materials using computers. Prior to this, the behavior of materials was somewhat unpredictable.

Of course I was excited to see this figure in the article, which shows the Weibull distribution modeling failures of steel bars in a fatigue machine.

The Weibull distribution, unlike the exponential distribution, takes the age of a product into account. If the parameter k is greater than zero, than the rate of product failure increases with time. The probability of failure f(t) after time t is

f(t)=\frac{k}{\lambda}\left(\frac{t}{\lambda}\right)^{k-1}e^{-(t/\lambda)^k}.

The first obvious thing to note is that the exponential distribution is just a special case of the Weibull distribution, with k=1. The next thing to say is that this distribution is single-peaked. So how is the above a Weibull distribution? The article says it is, but I think it might be a linear combination of two Weibull distributions, don’t you? Whatever–normalize, and you’ve got yourself a probability distribution.

[pun warning!]

The real question is, if this is TWO Weibulls, would you settle for the lesser of two Weibulls?

Sorry. I had to.

Don’t flip out!

Well, not yet at least. Everyone’s flipping the classroom, but is it really worth it? Yes and no, as NCTM president Linda Gojak explains in her column this week. I don’t always highlight her column, but I especially appreciated the nuanced way in which she approached this trendy subject. There’s something more fundamental that we need to aim for: engaging our students in mathematics and problem solving. Whether we flip or not may be immaterial, as Linda points out.

Here are a few excerpts from her article, which you should check out in full here.

To Flip or Not to Flip: That Is NOT the Question!

By NCTM President Linda M. Gojak NCTM Summing Up, October 3, 2012

A recent strategy receiving much attention is the “flipped classroom.” Innovative use of technology to enhance student learning makes flipping possible and motivating for students and teachers.

I believe that we need to go further. As we consider effective instruction that leads to student learning, we must remind ourselves of the characteristics of mathematically proficient students.

Rich mathematical tasks provide students with opportunities to engage deeply in mathematics as opposed to a lesson in which the teacher demonstrates and explains a procedure and the student attempts make sense of the teacher’s thinking. Communication includes good questions from both teacher and students and discussions that develop in students a deep understanding by wrestling with the mathematical ideas.

Although the flipped classroom may be promising, the question is not whether to flip, but rather how to apply the elements of effective instruction to teach students both deep conceptual understanding and procedural fluency.

(more)

All that being said, I still DO want to try flipping my classroom on a small scale, one-lesson at a time basis. I promise I’ll try it someday.