Great short math videos

Many of our math circle students are mentoring or coaching younger students. Here are some excellent model examples of presentations to inspire you.

Ina Petkova (Columbia math PhD student) gave a very elegant, simple and clear exposition of a proof of the Pythagorean theorem that middle school students should readily be able to understand and later to recreate for themselves. The Girls' Angle program in Cambridge Massachusetts created the video of her presentation as the first video in their series. I look forward to seeing more on their website in the future.

Richard Rusczyk has many excellent video presentations which you can watch on the Alcumus section of the Art of Problem Solving website, but you will need to solve some problems first in order to earn the right to see them. What I like about Richard's presentations is that he doesn't just demonstrate the "nice" way to do a particular problem, he also shows some of the not-so-nice ways we might initially think about trying. It's important to acknowledge to students that we don't always know at the outset exactly what approach is going to work best for a given problem. There's a certain amount of groping about that is often necessary to get insights into a good method to solve a problem. We need to demonstrate that problem solving is a process, not just a magic bag of tricks.

A puzzle celebrating Martin Gardner's 95th birthday

John Tierney of the New York Times begins his feature article on the prolific and always delightful recreational mathematician Martin Gardner with a puzzle:

For today’s mathematical puzzle, assume that in the year 1956 there was a children’s magazine in New York named after a giant egg, Humpty Dumpty, who purportedly served as its chief editor.

Mr. Dumpty was assisted by a human editor named Martin Gardner, who prepared “activity features” and wrote a monthly short story about the adventures of the child egg, Humpty Dumpty Jr. Another duty of Mr. Gardner’s was to write a monthly poem of moral advice from Humpty Sr. to Humpty Jr.

At that point, Mr. Gardner was 42 and had never taken a math course beyond high school. He had struggled with calculus and considered himself poor at solving basic mathematical puzzles, let alone creating them. But when the publisher of Scientific American asked him if there might be enough material for a monthly column on “recreational mathematics,” a term that sounded even more oxymoronic in 1956 than it does today, Mr. Gardner took a gamble.

He quit his job with Humpty Dumpty.

On Wednesday, Mr. Gardner will celebrate his 95th birthday with the publication of another book — his second book of essays and mathematical puzzles to be published just this year. With more than 70 books to his name, he is the world’s best-known recreational mathematician, and has probably introduced more people to the joys of math than anyone in history.

How is this possible?

For the answer to the puzzle and the fascinating story of a mathematical legend, see For Decades, Puzzling People With Mathematics in today's New York Times.

You can also find more great Gardner puzzles on John Tierney's NYT blog here.

As the article notes:

“Many have tried to emulate him; no one has succeeded,” says Ronald Graham, a mathematician at the University of California, San Diego. “Martin has turned thousands of children into mathematicians, and thousands of mathematicians into children.”

Happy Birthday indeed! And many happy returns of the day!

Congratulations on AMC12 Team Standings!

The photo above shows Prof. Moorthy celebrating with the Albany Area Math Circle team of Dave Bieber, Yipu Wang, and Andrew Ardito. Their outstanding team score of 387 on the AMC12B was second place in New York State, and nipped at the heels of first-place powerhouse Stuyvesant (391.5) closer than ever before!



The photo above shows Albany Area Math Circle team of Jay White, Matthew Babbitt, and Andrew Ardito celebrating their excellent third place in New York performance on the AMC12A date.

Here are the top team score standings for New York State on both dates:


AAMC students who took the A-date contest at their schools also contributed to Merit Roll honors for team scores of 300 or above at Bethlehem High School, Emma Willard School, Guilderland High School, and Niskayuna High School.

Congratulations to all the schools listed above, and to all the hundreds of excellent high schools all over the state and thousands of schools across the country that offered the AMC contests to their students! If your school doesn't offer the AMC contests, check out this link and then talk to your math teacher!

The complete list of AMC honors for the 2009 AMC10 and AMC12 contests are available in the PDFs linked below. Students working on college applications who wish to look up their honor roll scores for the past six years can search for their names and/or school names in these PDFs.

2009 AMC10/12 Honors and Award Summary Book

2008 AMC10/12 Honors and Award Summary Book

2007 AMC10/12 Honors and Award Summary Book

2006 AMC10/12 Honors and Award Summary Book

2005 AMC10/12 Honors and Award Summary Book

2004 AMC10/12 Honors and Award Summary Book

Hard Problems: The Road to the World's Toughest Math Competition

Public television stations across the country will be showing Hard Problems, the documentary about the six American students who represented the United States at the 2006 International Math Olympiad. In the Albany area, WMHT (channel 17) will be showing movie on Sunday October 18 at 6 p.m. More information about the movie, including air times in other cities, is available here. Here's the movie's trailer:

Good luck and best wishes to the new math circle starting up in Princeton NJ

A new math circle for middle school and high schools students is starting up today in Princeton, New Jersey! The legendary Professor John Conway is going to be giving a talk and Albany Area Math Circle founding member Alison Miller, now a math grad student at Princeton, will lead a problem session afterwards.

AAMC high school students who are leading middle school math circles may not be able to get John Conway to appear in person, but that doesn't mean you can't introduce the mathematical excitement of his work to your students.

Here's a video of John Conway talking about his work on cellular automata and The Game of Life:



If you don't know about his work on cellular automata and The Game of Life, you and your students are in for a treat. Visit this link to learn more about the concept and play an interactive version of the game.

Understanding the real world policy applications:

Conway's Game of Life relates very nicely to some extremely influential work done by economist Tom Schelling, who won a Nobel Prize for his work applying game theory to real life. Schelling's books are extremely well-written, lucid and thought-provoking. They don't require a lot of background. Micromotives and Macrobehavior is a good place to start. Schelling started working on models of racial segregation patterns in the 1950s and 1960s by creating simple cellular automata rules with pennies on a chessboard. Another great Schelling book has a title that I love and can (deeply!) relate to: Choices and Consequences: Perspectives of an Errant Economist. Here's a video from economist Tim Harford briefly presenting and discussing Schelling's model, using eggs to demonstrate cellular automata:



To explore the math more deeply:

Check out the links on this page at Wolfram demonstrations. The first page allows you to change parameters and visualize different ways that his Game of Life will play out, but make sure to check out the links on the right side of the page as well.

John Conway is co-author, along with Richard Guy and John Berlekamp, of a great series of books on mathematical game theory, called Winning Ways for Your Mathematical Plays. It's in the fun but deep and sometimes hard category--highly recommended for those who want to delve more deeply into this subject. Google Books also offers a limited preview of volume 2 of the series which I've linked below. If you follow the google books link, you can find what local libraries near you own the book.

Geometry brain warmup for high school students

The standard American high school math curriculum seems to regard geometry as something to be gulped down as quickly as possible, rather than savored and thought about in order to develop geometric intuition. Many bright high school students take geometry in 8th or 9th grade and their teachers then believe they are "done" with geometry.

Award-winning math teacher Pat Ballew, who taught in schools in Japan and England for 16 years, has a different perspective.

Mr. Ballew posted the following:

In the January, 1929, issue of the American Mathematical Monthly, there appears a problem submitted by J. Rosenbaum of Milford Connecticut. The problem begins, "It is well known that the radius of the inscribed circle of a right triangle is equal to half the difference between the sums of the legs and the hypotenuse." I ... suggest that the theorem suggested may be less well known now than it might have been in the past.

If it's been a while since you've taken geometry, and you're feeling uncomfortably rusty, see if you can prove this so-called "well-known theorem" by playing around with it, then look at Mr. Ballew's very nice presentation and discussion of several approaches to proving this theorem. Even if you came up with your own proof, you may find you get new insights by looking at his approaches.

If you liked working on this problem, there are more where that one came from.

Advice for new student coaches

Several new student high school student coaches have written to me for coaching advice, so I've decided to post a few of my thoughts on this subject. I'll be posting more thoughts this month, as I'm preparing to give a talk to Bard College students who are planning to start up math circles near Kingston.

I welcome suggestions from experienced student coaches as well--feel free to post yours in the comments.

Dear new student coaches and math circle mentors:

I'm delighted to hear that you have decided to work with some younger students in a MATHCOUNTS group or small middle school math circle. And I can completely understand that it can be tough to keep the energy going at first.

I only have a bit of time right now, but here are a few ideas off the top of my head:

1) Remember that encouragement and cheerleading is really important. MATHCOUNTS or AMC8 problems can be so much tougher than what the students are used to in their regular math class. It's easy for them to get discouraged.

Empathize with that--tell them that you also found MATHCOUNTS problems when you began yourself. When they make silly mistakes, tell them that even the best of students do that sometimes. If it's true for you, tell them you make silly mistakes sometimes too, and share your strategies for avoiding them.

It's certainly true for me. I make lots of silly mistakes, and I'm not shy about sharing that! Mistakes--and half-baked ideas--and learning from them--is part of the Aha! problem-solving process.

2) Ask the parents of your students if they'll take turns sending in refreshments! That helps a lot, especially if you try to hold practices right after school.



3) Get your students to savor and enjoy and celebrate their Aha! experiences. The picture above comes from the legendary Harvard physics professor, Howard Georgi, who has a wonderful approach of going around to different groups of students during physics problem-solving sessions, watching students develop cool insights, and he'll periodically exclaim "Eureka!" That might sound corny to you, and maybe it's not exactly your particular personal style, but you can figure out something else that works for you.

4) If some of the students are veterans and some are rookies, get the veterans to help the rookies.

5) Tell the rookies not to be shy about asking for help. Everyone was a rookie once. Tell the rookies that they are actually helping the vets when they ask questions, because the vets deepen their own understanding by explaining things.

6) Build in games as breaks in your practices. I posted some ideas on the Math Circle blog and I'll post more when I have time. Hangmath is a fun and easy game to run. The directions are here.

7) Another good game is the Factors game. The version of the Factors game at this link is a computerized version, but once you yourself have played it against the computer a few times, you can see that it's easy to figure how to run the game with pairs of students playing it on a chalk board (with two different colors of chalk) or just on a piece of paper (with two different colors of writing instruments.)

There are lots of good games (that just require pencil and paper) in Marilyn Burns' classic book, About Teaching Mathematics K-8. It's a relatively inexpensive paperback and it's chock full of great problem solving games and activities.

8) Tell stories. One of my favorite stories (possibly apocryphal, but still fun!) is to talk about the famous mathematician Karl Friedrich Gauss. Here's a great opening to the story from Jim Loy:

There is a story about Carl Friedrich Gauss. Supposedly, when he was a little boy, his teacher asked the class to add up the numbers one through a hundred (1+2+3 etc., all the way up to 100). The teacher wanted to get some work done, or get some sleep, or whatever. Anyway, to the teacher's annoyance, little Gauss [Here the lecturer holds his hand out to show that little Gauss was about 2 feet tall, to the amusement of the audience]... To the teacher's annoyance, little Gauss came up to the teacher with the answer, right away. The teacher probably had to spend the rest of the class time verifying little Gauss's [2 feet tall] result.

Some people find that story hard to believe, even impossible. I think that the story has the ring of truth to it. I believe that the story is true, or close to it. There are versions of the story, in which the numbers are one to a thousand [murmur in the audience].

I think that you people can duplicate little Gauss's [2 feet tall] trick [doubt in the audience]. I'm going to give you two very small hints. But, that's all you will need, to be just like little Gauss [2 feet tall].

Then you can show them the trick. I really like the way Jim Loy does it, because he first tries a few really inefficient ways first before getting to the snappy efficient way. That's a great way to motivate problem-solving.

9) Another story I like to tell students who are getting discouraged because they keep trying things that don't work is about Albert Einstein. Supposedly when he was first coming to work at Princeton, they asked him what furniture he needed for his office. His answer: all he needed was a desk and a "very large trash can," for all his false starts and mistakes!

10) You can get more great stories from reading Richard Feynman's books--he's a great storyteller, and I think part of being a great teacher is being able to talk about the process of problem-solving, the stories of the mistakes you made and the insights you got from making them. Popular math biographies can be great sources of inspiration and stories too--see what you can find at your local library.

11) You can find lots of little gems to share in Martin Gardner's recreational math books. The Aha! Insight and Aha! Gotcha! books are great places to start.

I've got a bazillion more ideas to share, and no time to write more at the moment. But I welcome other ideas in the comments. Experienced coaches can contribute ideas that have worked for them. Even if you're not an experienced coach, think back to when you were a new student, what ideas worked for you?