Mathematics Education TED Talk

I'll be teaching Calculus I at Emory University this fall, so I found this TED talk to be particularly interesting and useful. TED is a small nonprofit devoted to ideas worth spreading. In this TED talk, Dan Meyer, a Math educator, proposes a Mathematics curriculum makeover in America. He breaks down Mathematics into two categories: computation and math reasoning.

He claims while computational methods may be easily forgotten, as educators, we should focus on math reasoning. He outlines successful procedures to promote mathematical intuition and discussion in the classroom to foster math reasoning. This, in turn, will help students get involved in the formulation of the problem, a useful skill for students who continue to study math, and those who don't.

Can't wait to incorporate his suggestions in my classroom! Have a look and get inspired:

Chinese Abacus as Home Decor?

We've lived in our condo for a year now, and somehow our walls remain white and bare. As I was surfing online to find some wall art inspiration, I was intrigued by the abacus wall art available at Pottery Barn and CB2. I thought this would be THE perfect souvenir for two math fans to bring home from China. While other travelers may be searching for jade, porcelain, and magic carpets, thus started my endless search for a giant abacus on our holiday to the Orient.

The Chinese invented the abacus in 2nd century B.C. as a counting instrument. To my surprise, the abacus is very much in use by store clerks and accountants even today. In fact, I learned that the suanpan, the Chinese name for abacus, can be used for multiplication, division, addition, subtraction, square root, and cube root calculations as well. According to Feng Shui, the abacus placed on desk or cashier will multiply profits and enhance business dealings. For those seeking for knowledge in fields of such as mathematics, accountancy, science and engineering, placing an abacus on the study table can enhance learning.

Although my dreams of finding an ginormous abacus in China did not come true (yet!), I brought home functional abacuses (or is it abaci?) to place on our study table. Looking forward to many GREAT academic years ahead!

Wendy's Biggie Fries Container

Here is one to try on your next fast food date.  My husband and I were in the airport Wendy's enjoying our fries with ranch dressing and noticed the statement on the container read, "We figured out that there are 256 ways to personalize a Wendy's hamburger.  Luckily someone was paying attention in math class."  So naturally, we asked each other how many condiments does Wendy's offer to result in 256 combinations of burgers? 

I immediately answered 8 condiments, using the notion of power sets.  2 to the what equals 256?  Meanwhile, my husband attempted to solve the equation nC0 + nC1 + ... + nCn = 256 and also got 8 (after what seemed like hours after I gave my answer).  Alternatively, I suppose, you can just take a walk over to the fixings bar and count them.  Thanks Wendy's for making our layover so entertaining!

Chuck Close Self-Portrait at the High Museum

I can't think of a better way to relax after sitting for the Algebra PhD Qualifying Exam, than to spend an evening at Friday Jazz at The High Museum in Atlanta. Galleries and special exhibitions are open for extended hours, while live jazz is performed in the Atrium lobby every 3rd Friday of the month. The past Friday was our first time attending, but certainly not our last.

High Museum's permanent collection ranges from African to European, modern to contemporary, and folk to photographaphic art. Special exhibitions have included works of artists Leonardo DiVinci, Salvadore Dali, and others. In particular, I was attracted to a self-portrait by Chuck Close. Turns out, almost all of his work is based on a grid structure for the respresentation of an image, and in this case, himself.  The result is a geometric illusion such that viewed from afar the image appears real, but as the viewer steps closer, the geometric dots and dashes and intentional tones are more apparent and the image is lost.   Amazing!

FIFA World Cup Predictions and Outcome

Congrats to the 2010 champions, ¡Viva España! What were the chances?

Well I do not know much about soccer, but I do know that JP-Morgan, UBS and Goldman Sachs issue in depth reports every fourth year calculating the next champion based on predictive modeling. Turns out, England, the team JP-Morgan's 70 page report referred to as champs lost in the Round of 16, while UBS and Goldman reports predicted Brazil would end up on the top. C'mon... did you really need a complicated algorithm to predict that?

After reviewing the reports, the questions are: Where did Wall Street go wrong in their models (and why isn't Congress investigating it)? What was the data set, assumptions, independent and confounding variables? Is it necessary to include GDP in the calculation, and how about including team performance outside of the World Cup to the data set? Can you think of a better methodology to predict a winner? Clearly, there is no absolute truth in predictions, but we can try to improve their ability to foresee the future.

Or perhaps we should save ourselves the trouble and just depend on Paul the Octopus to call the shots?! Now I'll toot my vuvuzela to that!

Infinity Set at Wimbledon?

Today was a historic day for tennis not to be forgotten. American John Isner was victorious over Frenchman Nicolas Mahut in a first round Wimbledon match lasting a world record 11 hours and 5 minutes spanned over 3 days. My hats off to these immortal athletes for their cool sportsmanship and their mindblowing endurance - they practically played the equivalent of 4 full tennis matches just among each other without a slam of the racquet or slip to the umpire! Sports broadcasters have dubbed the final set as the "Infinity Set", however how can it be an infinity set when the fifth set actually terminated at 70-68? Although a true infinity set is theoretically possible, I wouldn't doubt that Wimbledon officials reevalute the tiebreaking rules before that becomes a reality.

Corona Commercial

I can get used to watching NBA playoffs for a lifetime with my husband given the advertisments are Super Bowl caliber as they were during the Magic/Celtics game last night. In a commercial for Corona beer, a wiseguy orders his Coronas at the bar and he notices a sign that states, "Happy Hour: 5-8". He rotates the "8" in the sign by 90 degress so it is to appear as the infinity symbol and then rejoins his friends for the now never ending happy hour!

Monty Hall Problem

Yesterday I gave a mini-lecture in our teaching seminar on Bayes' theorem motivated by the "Monty Hall Problem":

Suppose you are on "Let's Make a Deal" gameshow, and you are given the choice of three doors: behind one door is a new car; behind the others are goats. You pick a door, say door #1, and Monty Hall, the host, who know knows what's behind the doors, opens another door, say door #3, and reveals a goat. He then says to you, "Do you want to stay with door #1, or switch to door #2?" Is it to your advantage to switch your choice?

If your initial guess is you have a 50/50 chance on winning the car regardless if you choose door #1 or door #2 knowing there is a goat behind door #3, think again. A simple application of Bayes' theorem will show that the probability of winning the new car by switching from door #1 to door #2 is 2/3. Therefore, it's to your advantage to switch your choice. Discuss.

Alice in Wonderland

Apparently "Alice in Wonderland" is not a recollection of an acid trip, but rather a parody of mid-19th century mathematics! This discussion has resurfaced given the recent release of "Alice in Wonderland" starring Johnny Depp as the Mad Hatter (BTW, highly recommend the movie in 3D IMAX). The New York Times Op-Ed follows:

March 7, 2010
http://www.nytimes.com/
Op-Ed Contributor
Algebra in Wonderland
By MELANIE BAYLEY
Oxford, England

SINCE “Alice’s Adventures in Wonderland” was published, in 1865, scholars have noted how its characters are based on real people in the life of its author, Charles Dodgson, who wrote under the name Lewis Carroll. Alice is Alice Pleasance Liddell, the daughter of an Oxford dean; the Lory and Eaglet are Alice’s sisters Lorina and Edith; Dodgson himself, a stutterer, is the Dodo (“Do-Do-Dodgson”).

But Alice’s adventures with the Caterpillar, the Mad Hatter, the Cheshire Cat and so on have often been assumed to be based purely on wild imagination. Just fantastical tales for children — and, as such, ideal material for the fanciful movie director Tim Burton, whose “Alice in Wonderland” opened on Friday.

Yet Dodgson most likely had real models for the strange happenings in Wonderland, too. He was a tutor in mathematics at Christ Church, Oxford, and Alice’s search for a beautiful garden can be neatly interpreted as a mishmash of satire directed at the advances taking place in Dodgson’s field.

In the mid-19th century, mathematics was rapidly blossoming into what it is today: a finely honed language for describing the conceptual relations between things. Dodgson found the radical new math illogical and lacking in intellectual rigor. In “Alice,” he attacked some of the new ideas as nonsense — using a technique familiar from Euclid’s proofs, reductio ad absurdum, where the validity of an idea is tested by taking its premises to their logical extreme.

Early in the story, for instance, Alice’s exchange with the Caterpillar parodies the first purely symbolic system of algebra, proposed in the mid-19th century by Augustus De Morgan, a London math professor. De Morgan had proposed a more modern approach to algebra, which held that any procedure was valid as long as it followed an internal logic. This allowed for results like the square root of a negative number, which even De Morgan himself called “unintelligible” and “absurd” (because all numbers when squared give positive results).

The word “algebra,” De Morgan said in one of his footnotes, comes from an Arabic phrase he transliterated as “al jebr e al mokabala,” meaning restoration and reduction. He explained that even though algebra had been reduced to a seemingly absurd but logical set of operations, eventually some sort of meaning would be restored.

Such loose mathematical reasoning would have riled a punctilious logician like Dodgson. And so, the Caterpillar is sitting on a mushroom and smoking a hookah — suggesting that something has mushroomed up from nowhere, and is dulling the thoughts of its followers — and Alice is subjected to a monstrous form of “al jebr e al mokabala.” She first tries to “restore” herself to her original (larger) size, but ends up “reducing” so rapidly that her chin hits her foot.

Alice has slid down from a world governed by the logic of universal arithmetic to one where her size can vary from nine feet to three inches. She thinks this is the root of her problem: “Being so many different sizes in a day is very confusing.” No, it isn’t, replies the Caterpillar, who comes from the mad world of symbolic algebra. He advises Alice to “Keep your temper.”

In Dodgson’s day, intellectuals still understood “temper” to mean the proportions in which qualities were mixed — as in “tempered steel” — so the Caterpillar is telling Alice not to avoid getting angry but to stay in proportion, even if she can’t “keep the same size for 10 minutes together!” Proportion, rather than absolute length, was what mattered in Alice’s above-ground world of Euclidean geometry.

In an algebraic world, of course, this isn’t easy. Alice eats a bit of mushroom and her neck elongates like a serpent, annoying a nesting pigeon. Eventually, though, she finds a way to nibble herself down to nine inches, and enters a little house where she finds the Duchess, her baby, the Cook and the Cheshire Cat.

Chapter 6, “Pig and Pepper,” parodies the principle of continuity, a bizarre concept from projective geometry, which was introduced in the mid-19th century from France. This principle (now an important aspect of modern topology) involves the idea that one shape can bend and stretch into another, provided it retains the same basic properties — a circle is the same as an ellipse or a parabola (the curve of the Cheshire cat’s grin).

Taking the notion to its extreme, what works for a circle should also work for a baby. So, when Alice takes the Duchess’s baby outside, it turns into a pig. The Cheshire Cat says, “I thought it would.”

The Cheshire Cat provides the voice of traditional geometric logic — say where you want to go if you want to find out how to get there, he tells Alice after she’s let the pig run off into the wood. He points Alice toward the Mad Hatter and the March Hare. “Visit either you like,” he says, “they’re both mad.”

The Mad Hatter and the March Hare champion the mathematics of William Rowan Hamilton, one of the great innovators in Victorian algebra. Hamilton decided that manipulations of numbers like adding and subtracting should be thought of as steps in what he called “pure time.” This was a Kantian notion that had more to do with sequence than with real time, and it seems to have captivated Dodgson. In the title of Chapter 7, “A Mad Tea-Party,” we should read tea-party as t-party, with t being the mathematical symbol for time.

Dodgson has the Hatter, the Hare and the Dormouse stuck going round and round the tea table to reflect the way in which Hamilton used what he called quaternions — a number system based on four terms. In the 1860s, quaternions were hailed as the last great step in calculating motion. Even Dodgson may have considered them an ingenious tool for advanced mathematicians, though he would have thought them maddeningly confusing for the likes of Alice (and perhaps for many of his math students).

At the mad tea party, time is the absent fourth presence at the table. The Hatter tells Alice that he quarreled with Time last March, and now “he won’t do a thing I ask.” So the Hatter, the Hare and the Dormouse (the third “term”) are forced to rotate forever in a plane around the tea table.
When Alice leaves the tea partiers, they are trying to stuff the Dormouse into the teapot so they can exist as an independent pair of numbers — complex, still mad, but at least free to leave the party.

Alice will go on to meet the Queen of Hearts, a “blind and aimless Fury,” who probably represents an irrational number. (Her keenness to execute everyone comes from a ghastly pun on axes — the plural of axis on a graph.)

How do we know for sure that “Alice” was making fun of the new math? The author never explained the symbolism in his story. But Dodgson rarely wrote amusing nonsense for children: his best humor was directed at adults. In addition to the “Alice” stories, he produced two hilarious pamphlets for colleagues, both in the style of mathematical papers, ridiculing life at Oxford.

Without math, “Alice” might have been more like Dodgson’s later book, “Sylvie and Bruno” — a dull and sentimental fairy tale. Math gave “Alice” a darker side, and made it the kind of puzzle that could entertain people of every age, for centuries.

Melanie Bayley is a doctoral candidate in English literature at Oxford.