Sunday, October 21, 2012

Congratulations, Zubin!

Zubin Mukerjee, at right, with other members of the Upstate New York All-Star Math Team at the national high school math tournament, ARML, in June

Zubin Mukerjee, a veteran leader of Albany Area Math Circle who has also organized and led a satellite math circle of his own for younger students, has blazed yet another a new trail for others to follow.

Zubin, a Guilderland High School senior, who is taking advanced classes in math and economics at SUNY Albany, and his co-author, Uthsav Chitra from Delaware, have won semifinalist honors in a prestigious national research contest for high school students.  Zubin and Uthsav worked on an original research project in number theory, "Random Involutions and the Number of Prime Factors of an Integer,"under the guidance of a mentor at PROMYS in Boston last summer.

Research presents new challenges as well as new rewards compared to the contest problems with which many math circle students are familiar.  When you work on a contest problem, it may be very hard, but you KNOW that you are working on a problem that somebody else has already solved and that there must be a clever elegant solution to the problem.   It is indeed exciting to have the Aha! moment when you find the solution to a contest problem, but such moments pale compared to those you can experience in math research, the thrill of discovering an answer to a problem nobody else has ever found before. Sometimes the results are negatives ones, not exactly the ones you were hoping for originally, but even those disappointing results can provide important clues to promising new lines for exploration.

It takes passion, perseverance, and luck to find original new research results, far moreso than in contests.  When doing original research, there are no guarantees at the outset that the problem will even have a solution at all, let alone that it will yield interesting results worth sharing with others.   Even once the problem is solved, it takes excellent writing skills to write up your research results in a way that will allow others to appreciate the importance and validity of what you have discovered.  Zubin's years of helping to write power round solutions for our math circle teams as well as his prize-winning entries in history day competitions have certainly polished his expository writing skills.

Here are the abstract and executive summary for Zubin and Uthsav's research project.

Abstract:  For hundreds of years, mathematicians have tried to find good approximations for the function d(n), which counts the number of prime factors of an integer n. In this paper, we examine using random involutions to approximate d(n) by comparing the number of fixed points of a random involution on F22g(n) to the number of fixed points of a specific involution, τ(n). We find and prove that the expected number of fixed points of a random involution converges, so that d(n) cannot be approximated using this method; moreover, we use this to show that the involution τ(n) is not random, as it has more fixed points than a random involution.

Executive summary:  The natural numbers are perhaps the most familiar to humans. They are the counting numbers: 1, 2, 3, etc. A divisor of a number is something that divides evenly into that number. For example, 3 and 14 are divisors of 42, but 42 is not a divisor of 3 or 14. A prime number is a natural number whose only positive divisors are 1 and itself. The first few primes are 2, 3, 5, 7 ... there are also infinitely many of these. There is a well-known function that returns the number of prime divisors of a number n, given that number. We denote this function d(n). Our goal in this project is to further research on modeling d(n).

Our mentor proposed a possible method of modeling d(n) by looking at special functions called involutions that act on the surface of modular curves. In particular, we studied the involution τ(n), which is related to d(n), by comparing it to random involutions. We were able to conclude,through a series of proofs and derivations as well as some graphical analysis using Mathematica, that d(n) cannot be modeled by τ(n) and that, as a result,  τ(n) is not random. In other words, there is something special about τ(n) that makes it so we can’t model d(n).

The consequences of this result are not yet fully clear. Nevertheless, this result can lead the way to studying other types of involutions, some of which may be able to model d(n). An accurate model for d(n) would be incredible, as it would make finding the prime factorization of large numbers much easier; this would have many applications in cryptography and computer science. Much research remains to be done on involutions though; perhaps one day, a closed-form expression for d(n) will be found through random involutions.

Zubin and Uthsav's research mentor was Dr. Kirsten Wickelgren, an American Institute of Mathematics fellow at Harvard University.  Here is a link to a copy of the background document including the problems she suggested they investigate as well as definitions of some key concepts and a helpful list of the supplementary references with which she initially launched them on their way.  If you are interested in understanding more about their work, you may want to take a look at those references yourself.  Students who have not yet studied much number theory will also find the Art of Problem Solving's textbook on introductory number theory very helpful.  [Added later:  Zubin also passed along another recommendation of a classic number theory book, Hardy & Wright's Introduction to the Theory of Numbers, endorsed by PROMYS Director Glenn Stevens as "clear and concise."  Zubin also notes that Hardy & Wright cover many topics in number theory in their book, some relevant to their project and some not.]

You will also note that Zubin and Uthsav used Wolfram Mathematica computer software to help create graphs to give them insights into their problem analysis.  Thanks to Wolfram's sponsorship of contests such as American Regions Math League (ARML), Harvard-MIT Math Tournament (HMMT), and Princeton University Math Contest (PUMaC), Zubin and all our veteran math circle students who have participated in one or more of those contests have received free student licenses to use this very powerful software.  Those licenses will remain valid as long as they are students, including college and graduate school years ahead.

Saturday, September 29, 2012

Math Circle travel contest dates this year

Math contests are a fun (but entirely OPTIONAL) part of our Albany Area Math Circle experience.

To help our members plan ahead, I will describe the entire calendar of contests in which members can participate below.  I have broken them down into two categories: travel contests and locally administered contests.  This post describes the travel contests we plan for this coming year.  My next post will describe the locally administered contests we plan.

Harvard-MIT November Tournament (HMNT):  
Saturday November 10 at Harvard University in Cambridge, MA   
This is an ideal travel contest for our younger veterans who have done at least NYSML in the past, but who have not yet done much proof-writing.   Note that participating in this contest rules out participating in PUMaC the following weekend.  It also rules out participating in the (much harder) February 2013 HMMT.  If you are seriously considering this contest but not yet sure of your schedule, please email Mr. Babbitt ASAP to let him know of your tentative (or definite) interest.
HMNT November Cost per student:  $10 registration fee plus modest costs for lunch.  AAMC can put parents in touch with one another to arrange carpools so that parents can share gas and tolls.  Math contests at Harvard and MIT have been done as a daytrip by AAMC mathletes who were comfortable getting up early and sleeping in the car.   If your parents agree, it may also be possible for students to arrange to stay overnight in a Harvard or MIT dorm with a college student host or hostess.   (You would need to bring a sleeping bag, and you will probably be sleeping on the floor.  Note that dorms can be noisy on Friday nights!)  Parent drivers may want to work out arrangements to share hotel rooms (unless they are driving as a day trip.)   Although we will put parents in touch with each other to facilitate whatever arrangements seem most suitable to them, it is entirely up to the parents involved to work out mutually satisfactory travel arrangements.

Princeton University Math Contest (PUMaC):  
Saturday November 17 at Princeton University in Princeton, NJ 
This is a very challenging contest which includes a collaborative team round.   Only experienced veteran students should consider this contest.  It is considerably harder than the AIME, so students who have not yet taken the AIME should not consider it.  But even the AIME is not sufficient preparation, because the team round requires mathematical expository skills.  The team round is proof-based and essay style.  Students need to be prepared to work collaboratively (and remotely) on a google document from Nov 10 through November 16.  In addition, there will be mandatory on-line practice in October for all team members on our AoPS forum.  Non-team members are also welcome to join the on-line practice forum.  Our Math Circle veteran leaders, Matt Babbitt and Zubin Mukerjee, will provide further information about what is expected of team members on the AoPS forum.  If you are seriously considering this contest but not yet sure of your schedule, please email Mr. Babbitt ASAP to let him know of your tentative (or definite) interest.
Cost per student:  $12.50 registration fee (which includes lunch).  AAMC can put parents in touch with one another to arrange carpools so that parents can share gas and tolls.  Parent drivers will probably want to work out arrangements to share hotel rooms.  Due to the distance involved as well as the need to finalize the power round with a final team meeting in Princeton on Friday evening, this event is NOT doable as a day trip.  Although we will put parents in touch with each other to facilitate whatever arrangements seem most suitable to them, it is entirely up to the parents involved to work out mutually satisfactory travel arrangements.

Harvard-MIT Math Tournament (HMMT)
Saturday February 16, 2013 at MIT in Cambridge, MA
This contest has always been extremely hard, and we expect that trend to continue to unprecedented epic levels of difficulty this year.  Only those with extreme tolerance for mathematical endurance should consider this event.  However, mathematical strength is not enough to be selected for this team, since collaboration with other team members will be very important.  It is very important that all members of our HMMT team know one another's strengths and weaknesses so that the "whole can be more than the sum of the parts."  It is also important that all team members encourage one another when the going gets tough, because--believe me!--it will get tough!

Our advanced veteran students who wholeheartedly participate in fall math circle activities including our PUMaC on-line practices on the AoPS forum and regularly attend (most of) our Friday meetings with an extremely positive and supportive spirit that encourages others not to give up when the going seems hopeless will be given first preference in team selection.   All other factors being equal, seniors will be given preference over younger students.   For younger students, participation in previous years' HMMT on-line events or travel events will be a plus factor.  All other factors being equal, younger students who have done well in prior year on-line HMMT events will be given preference over those who attended the travel event and did equally well.

Cost per student for Feb HMMT:  $10 registration fee plus modest costs for lunch.  AAMC can put parents in touch with one another to arrange carpools so that parents can share gas and tolls.  The event can be done as a daytrip if your mathletes are comfortable getting up early and sleeping in the car.   If your parents agree, it may also be possible for students to arrange to stay overnight in a Harvard or MIT dorm with a college student host or hostess.   (You would need to bring a sleeping bag, and you will probably be sleeping on the floor.  Note that dorms can be noisy on Friday nights!)  Parent drivers may want to work out arrangements to share hotel rooms (unless they are driving as a day trip.)   Although we will put parents in touch with each other to facilitate whatever arrangements seem most suitable to them, it is entirely up to the parents involved to work out mutually satisfactory travel arrangements.

New York State Math League (NYSML)
Saturday April 13, 2013 at Biram Hills  High School in Westchester County
NYSML is an ideal "first" travel competition for our students.  It is the statewide math championship for high school students.  It is held in a different place each year.  We (Albany Area Math Circle) were the host league in 2010, Suffolk County in eastern Long Island hosted in 2011, Southern Tier was the host league in 2012.  The 2013 host will be the Westchester-Putnam Math League.    All high school students who attend meetings regularly and demonstrate mathematical and behavioral maturity as well as enthusiasm and persistence are welcome to join our NYSML teams.  Unlike the other travel tournaments, we expect to be able to take as many 15-person teams as we would like to NYSML.  (In 2010, we had three teams with almost 45 students.  If a team is a bit short of 15 students, we can use alternates from other teams and vice versa at NYSML.)

Cost per student for NYSML:  Unknown at this time, but probably around $20 per student (including lunch).  It is a potential day trip (for mathletes who don't mind getting up early and sleeping in the car.)   Some parents and mathletes may prefer to stay in a hotel.  As with all travel contests, we will facilitate parents getting in touch with each other to make their own carpool/travel arrangements.   Although we will put parents in touch with each other to facilitate whatever arrangements seem most suitable to them, it is entirely up to the parents involved to work out mutually satisfactory travel arrangements.

American Regions Math League (ARML)
Saturday June 1, 2013 at Penn State (depart Albany morning of 5/30 and practice with team on 5/31 at Penn State, so students need to miss several days of school for this.)
ARML is the national high school championship event.  It has a format similar to NYSML but the problems are harder.  Albany Area Math Circle students can apply to be on the Upstate NY ARML team.   You will want to wait until you have your AMC, and (if applicable) AIME and NYSML scores before you apply to the team.  NYSML experience and performance will also be a big plus for ARML team selection.

Important info about conflict with SAT testing date:  if you are thinking of doing ARML, please plan ahead for the SAT conflict.  June 1 is an SAT administration date.  It is expensive and time-consuming and may not be possible to arrange a makeup date.  It is MUCH better if you plan to take your SAT subject tests in May rather than June, and/or your SAT reasoning tests in January or March rather than June.  The SAT schedule for this coming year is here.

Cost per student for ARML:  Unknown at this time, but in recent years the cost has been around $300 to $325.  That included a chartered bus from Binghamton to Penn State and back as well as two night's housing in Penn State dorms and some, but not all, meals.  Students also needed about $25 to buy additional meals on their own.  Thanks to Cecilia and Gili's moms efforts in fundraising the NYSSPE donated funds that helped to offset about $100 of the costs per student.

In the next post, I will describe the contests that we expect to be offered locally, without the need to travel.

Tuesday, June 12, 2012

Upstate NY All Stars!

Albany Area Math Circle members of the New York State All Star Math Team thank the New York State Society of Professional Engineers for their support:  
Yujun (Etheal) Chen, Emma Willard School
Sherry He, Emma Willard School
Gili Rusak, Shaker High School
Jien Ogawa, South Colonie High School
Cecilia Holodak, Niskayuna High School
Matt Babbitt, heeg
Zubin Mukerjee, Guilderland High School
Ziqing (Bill) Dong. Farnsworth Middle School

Complete 2012 Upstate NY AllStar Math team roster for 2012 ARML, including students from all over Upstate New York (defined as New York State minus {New York City, Nassau County, Suffolk Count})

Matthew Babbitt heeg
Jemmin Chang Somers
Yujun Chen Emma Willard
Yiqun Cui Brighton
Matt DeCross Pittsford-Sutherland
Pranav Devarakonda Brighton
Matt Dobbins Corning West
Ziqing Dong Farnsworth
Ryan Gao Brighton
Huajun Gu Corning West
Jeff Guo Penfield
Sherry He Emma Willard
Cecilia Holodak Niskayuna
Doug Knowles Churchville-Chili
Jeremy Koob Corning West
Benjamin Lei Arlington
Devin Li Corning West
Allen Liu Penfield
Ben Lowenstein Pittsford-Sutherland
Dongze Lu Harley-Allendale-Columbia
Evan Lustick Canandaigua
Zubin Mukerjee Guilderland
Jien Ogawa South Colonie
Jordan Roeder Canandaigua
Gili Rusak Shaker
Perry Wang Corning West
Felix Weilacher Penfield
Yunmei Zhang Harley-Allendale-Columbia
Simon Zheng Somers

Guerrilla Math venues?

Schenectady--and more broadly, the Capital District, have many wonderful free outdoor activities.  They are fun family places to go and enjoy our beautiful summers.  And they are great places for you and your families  to do Guerrilla Math!  Just bring some sidewalk chalk or a portable whiteboard and markers, and your ideas for fun math explorations to share with families waiting for the event to begin or during intermissions, afterwards. 

Here are few links I got from a colleague at Union College.  If you can suggest other links, please post in the comments below.

Sunday, June 10, 2012

Guerrilla Math Circle at the Youth Peace Rally

It is a beautiful day for more sidewalk chalk Guerrilla Math Circles.  I will be doing it at today's Youth Peace Rally at Jerry Burrell Park in Schenectady.

As I wrote in my last post:

Jerry Burrell Park is a place that is near and dear to my heart, because one of my former students at Union College, Jeremy Taglieri, spent a year working along with two friends to raise $84,000 in funds and recruiting 140 volunteers to renovate the park.  They called their project Project SKIP, where SKIP = "Schenectady Kids Imagine and Play."  Jerry Burrell Park is also the site of a SICM free lunch site, and in fact, volunteers from my own church will take a turn serving lunch for a week there later this summer. 

All math circle families, siblings, friends, and neighbors are cordially welcome to join us.  But--as I also wrote in my last post--I am mindful that not all parents will be comfortable with their families coming to this neighborhood.  It is a decision that should be made with full information--this is one of the highest crime neighborhoods in our area, and in 2010, just a few weeks after the playground's renovation, a 17-year-old shot and seriously wounded an 11-year-old boy and his 18-year-old older brother.  Each family needs to make a choice they are comfortable with.

But today, I choose to join those who hope to bring and keep peace in this community.  And I will bring what I have to share--the beauty and excitement of creating and discovering mathematical patterns.

Directions:  Jerry Burrell Park is located at the corner of Hamilton and Schenectady Streets in the Hamilton Hill neighborhood of Schenectady.

Friday, June 8, 2012

Pascal's triangle in sidewalk chalk

First 15 rows of Pascal's triangle (mod 2) in sidewalk chalk

Imagine that you throw a dart at Pascal's triangle.  What are the odds you will hit an even number?

Paul Zeitz raised this fascinating question at the MSRI Great Circle's workshop three years ago.

It's a fun one to explore in sidewalk chalk--perfect for Guerrilla Math Circles.

A good way to start thinking about this problem is simply to begin coloring in the triangle--and what could be more fun that the idea of creating a really large one with sidewalk chalk?  And engaging any curious passersby in helping you.  How big can we make it if we get lots of folks helping us?  (And of course, we could also start some people on exploring Pascal's triangles with multiples of other prime numbers colored in.)

I started one the other day in Schenectady's Central Park and I will be back there this evening (Friday June 8) to do more.  There is a nice large stretch of blacktop that is blocked off from automobile access.  It is adjacent to Tiny Totland and not far from the Music Haven (site of many wonderful free concerts) and also near the picnic pavillion area where Schenectady Inner City Mission serves free lunches in the summer.  Central Park is the "crown jewel" of Schenectady's park system, designed by Frederick Olmsted and with a world class rose garden.  To get to where I will be tonight, enter Central Park via the main entrance off Central Parkway and then turn to the left when you get to the lake.  Park in the parking lot near Tiny Totland and look for me in the blacktop area between Tiny Totland and the lake.

On Sunday afternoon, from 1 to 5 p.m. I will be doing more Guerrilla Math Circles at a different park in Schenectady, Jerry Burrell Park.  Jerry Burrell Park is a place that is near and dear to my heart, because one of my former students at Union College, Jeremy Taglieri, spent a year working along with two friends to raise $84,000 in funds and recruiting 140 volunteers to renovate the park.  They called their project Project SKIP, where SKIP = "Schenectady Kids Imagine and Play."  Jerry Burrell Park is also the site of a SICM free lunch site, and in fact, volunteers from my own church will take a turn serving lunch for a week there later this summer.

I invite Albany Area Math Circle families (students, parents, siblings) to join me in sharing the fun, beauty, and awesomeness of mathematics with a wider world, a world of folks who have yet to discover how much delight can be had in exploring the wilds of mathematics.

All of you can do these activities wherever your parents feel comfortable with your doing them.  I grew up in a city myself, and have always been comfortable having my children play in Schenectady parks when they were growing up, with--of course--sensible precautions taken about when and where to go.  Personally, I am comfortable being in city parks when SICM volunteers from churches all over the area are serving free lunches, but not when they are mostly deserted.  And this Sunday's event at Jerry Burrell Park is part of a larger event organized by city officials.

Nothing is ever 100% safe--and there are risks to just sitting in your own living room (a car might come crashing through the wall), but I recognize that these are decisions that math circle parents needs to make for their own children.  I feel compelled to point out that--sadly--just a few weeks after the renovation of the Jerry Burrell Park in 2010, a 17-year-old boy shot and seriously wounded an 11-year-old and an 18-year-old boy who were walking to the park from just a block away.

But--remember that you and your sons and daughters can also do these same activities WHEREVER you happen to be--on the sidewalk in front of your own home or the park just down the street from you in your own neighborhood.

No matter where you are comfortable with participating in our outreach activities, you can work with us to brainstorm ideas and post links to photos in the comments below.  Albany Area Math Circle members can also post about their experiences in our private discussion forum on the Art of Problem Solving.

Thursday, June 7, 2012

Home of the math champions: Upstate New York!

Upstate NY All-Star Math Team celebrates the announcement that Allen Liu is the national high school math champion!

Hundreds of mathletes and coaches applaud Allen as he walks down to center court at Penn State's Bryce Jordan Center.  Allen's victory is all the more impressive because he is only an eighth grader.  He attends Bay Trails Middle School in Penfield, New York, but he has already been excelling in math classes at the University of Rochester.  He is currently attending the Math Olympiad Summer Program, along with high school students on this year's International Math Olympiad team and others considered strong prospects for future year teams.  As a USA Math Olympiad  qualifier every year since the fifth grade and a top-10 finisher at ARML since sixth grade, Allen has already compiled an extraordinary track record, yet he is an outstanding example of modesty and a generous spirit always willing to share what he knows with others.

Upstate NY All-Star Coaches Jen Vibber, Brenda Munch, and Tom Zink beam with pride as Allen becomes the newest member of their Upstate NY dynasty of national champions. No other ARML team anywhere in the country (or world!) can boast about so many national champions.  ARML went national in 1984. Since that time, 26 distinct individuals have won ARML.   Of those 26 champions, four are from the Upstate NY team:

Robert Kleimberg (1992)
Jeremy Bem (1993)
Aaron Pixton (2004)
Allen Liu (2012)
Coach Jen Vibber has been Allen's local coach throughout his middle school years, as well as a member of the Upstate NY Math Team coaching staff.  She is also the incoming head coach for the Upstate NY Math Team.  That's outgoing Upstate NY head coach George Reuter looking over her shoulder in the photo!  With the foundation built by George, Jen, and the other dedicated veteran coaches on the Upstate NY staff, we all know the Upstate NY team is in great hands going forward.  Thanks to all the tremendously dedicated coaches for the many hours of tireless work they put into creating an awesome community experience for mathematically passionate kindred spirit students all over Upstate NY.   It is obviously a labor of love for each and every one of them, as they are all volunteers.   They are outstanding teachers who inspire their mathletes with their character as well as their knowledge of problem-solving methods.

2012 Upstate NY Math Team Coaching Staff:

George Reuter (MCML/Canandaigua Academy) 
Doug Becker (MCML/Gates-Chili HS), Treasurer
Brenda Munch (MCML/Brighton High School)
Patty Pragel (MCML/Greece Arcadia High School)
Jen Vibber (MCML/Penfield High School)
Glen Stephenson (Corning West High School)
Tom Zink (AAMC/MicroStrategy)
Bill Babbitt (AAMC/RPI)
Rita Biswas (AAMC/UAlbany)

(If you would like to learn more about the "Parent Function" cited on the Upstate NY team t-shirts, check out Pat Ballew's blog post describing the fascinating history of the term's usage.)

Monday, May 21, 2012

NY team scales new heights at MATHCOUNTS Nationals

The New York State MATHCOUNTS team raised the bar to the highest heights in recent memory with a fourth place finish at the National MATHCOUNTS competition held in Orlando, Florida last week.  The photo above shows the team celebrating on Expedition Everest at Disney World after the contest.

Three-year team veteran Allen Liu from the Rochester area reached the National Countdown for the third consecutive year, the first New Yorker ever to accomplish this feat, and reached the semifinals for a top four individual finish.

Albany Area Math Circle's own Alex Wei had one of the best finishes ever for an Albany Area mathlete, coming in a very impressive 18th place among the 224 national finalists.  Alex's first MATHCOUNTS coach, who launched him on his way back when he was in sixth grade, was another former national mathlete from the Albany Area, Dave Bieber.  (Dave, in turn, had been coached by yet another former national mathlete from the Albany Area, founding AAMC member Alison Miller, when he was in sixth grade!)

For the first time anyone can recall, all four members of the team, which also included two-time veteran Peter Mizes from the Rochester area and Calvin Lee, the Manhattan MATHCOUNTS chapter champion, all earned top 56 recognition.  Most impressively of all, they were able to come in very strong on the collaborative team round, allowing their team to rank higher than the traditional powerhouse states of California and Texas.

Because of the very large distances separating the mathletes and the fact that there is no travel budget to enable the team to practice together as a team prior to nationals, the students seized the initiative, took ownership of their learning, and organized their own on-line collaborative practices prior to nationals, without requiring adult assistance.  Essentially, they coached themselves.  The team clearly made New York State proud in many ways--their friendly collaborative work together carried on a fine tradition to new heights and surely will inspire future mathletes from our state to new heights, as they stand on the shoulders of these mathematically collaborative and resourceful giants.

If you would like to see the problems they tackled, you can find them posted on the Art of Problem Solving discussion forum here.   One of my favorite problems from this year's national contest is here--it is a beautiful illustration of how probability problems can often be solved by viewing them as geometry problems.

Sunday, May 6, 2012

Guerrilla Math Circles brainstorming

Dan Zaharapol, the founding director of the extremely awesome Summer Program in Mathematical Problem Solving (SPMPS), has been blogging about low-cost high impact ideas to spread math opportunity.

Now, if you don't already know about Dan's program, you should definitely check it out, especially the link to the nice New York Times article covering the program, A Sleepaway Camp where Math is the Main Sport.

But, as Dan acknowledges in his post, although the program is terrific and serves a historically very underserved population (inner city middle school students), its reach is very small due to the expense and logistics involved in running a residential summer program.  The program started with 20 students last summer and--after a major fundraising effort led by its sponsors at Art of Problem Solving, the program will double in size to 40 students this summer.

And even if there were funds to serve every middle schooler, the problem is that we need outreach to children long before they reach middle school age, long before they are ready to head off to a residential summer program.

And, as Dan writes:

Too many students are ready to do more mathematics but 
  • do not know where to do it, and
  • do not even know that such opportunities exist or that they should be doing it. 
The students don’t know; their parents don’t know; their teachers don’t know. They have no way to discover that their peers, successful math students from other communities, do more than just what they see in school. 
It’s not just that they need to be told about it; it needs to be part of their culture. It’s not just that they must know that such programs exist and that successful people do them; they should feel it is expected of them, that lots of people they’ve known and admired do math beyond school. 
How can we possibly create this culture and community where it does not already exist?
This is exactly the same problem that I have been thinking about for years, but three weeks ago at the Julia Robinson Math Festival, a brainstorm hit me, which I have been calling Guerrilla Math Circles.

What do I mean by Guerrilla Math Circles?  The idea for the name comes from guerrilla marketing, which in turn originated with guerrilla warfare.  It is all about low-cost, informal, unconventional, low overhead, and non-bureaucratic approaches.  It is all done on a shoestring, small scale, but very easy to replicate, emulate, and improve by trial and error and sharing on blogs, videos, and social media.

How do Guerrilla Math Circles happen?

Guerrilla Math Circles do not involve signing kids up for anything.  They do not involve any kind of custodial responsibility for the children--because they happen in places and times where and when children are already with their parents or caretakers.   They do not require parents to take their children to any particular special places where they would not already be going.

Basically, the guiding principal of Guerrilla Math Circles is we bring the same kind of fun engaging low-cost activities done at Julia Robinson Math Festivals to places where the children who need them most already are.  And we do it on a shoestring, including elements of math as performance art on the street and math as theatrical improv or maybe even flashmobs.  They are low-tech and high touch, person-to-person and interactive.

Remember Tom Sawyer whitewashing the fence and how he convinced all his friends that this would be a fun thing to do?  That's kind of the spirit of Guerrilla Math Circles.  Start doing some interesting math in  a highly visible public place in a way that invites others to join in.

Another inspiration for Guerrilla Math Circles comes from the long-time example of chess outreach programs in public parks and other public spaces.  In our area, the local chess club sponsors a "Chess under the Marquee" program under the marquee of Proctor's Theater in downtown Schenectady.  The marquee provides shelter from sun and rain and the broad sidewalk provides space to set up tables for drop-in folks to engage in pickup chess games.  Why not ask Proctor's if we can have "Math under the Marquee" at another time or day?

My head is virtually exploding with ideas--they come to me faster than I have had time to organize them all, so I will just use this post to add them as I have time to do so.

Where can Guerrilla Math Circles happen?

urban playgrounds
urban parks, especially those that serve free lunches in the summer
urban swimming pools
urban street festivals
urban farmer's markets

urban daycare programs
urban libraries

waiting rooms at social services agencies (where parents often may have bored children in tow, and would be happy to have them entertained while they wait)

urban bus stops (where parents may also be stuck waiting with bored children in tow)

Places where children are with their parents are especially good, because we have the opportunity to get parents as well as children engaged!

What can happen at a Guerrilla Math Circle?

Use simple materials (paper, yarn, beads) to make beautiful polyhedra using the ideas from Geometric Delights.   Or use balloons and these ideas from Vi Hart's blog.

Introduce kids to the unsolved "Million Dollar Math Problems" at, these typically require only simple materials like sidewalk chalk or dice.  (Two of the million dollar math problems have special local resonance for our neck of the woods.  A mathpickle problem accessible to 7 or 8-year olds introduces the ideas behind the Graceful Tree Conjecture, which is related to our own Professor Krishnamoorthy's research.  I'll be blogging about this soon.  Another mathpickle game accessible to slightly older children introduces them to the ideas behind the RSA algorithm.   Ron Rivest, the "R" in RSA, attended a local high school, located in the affluent and highly educated suburban town where our math circle holds its weekly meetings.  A number of our current math circle students come from that town, and other students in our math circle come from affluent and highly educated towns up to 45 minutes or an hour away at rush hour. But just ten minutes away from our regular meeting place at Ron Rivest's old high school are desperately poor neighborhoods, filled with children whose parents have limited and largely negative experiences with formal education, neighborhoods which have never sent a single student to our math circle--and those are the students who most need our outreach efforts!)

Share/read aloud mathematical picture books like Anno's Mysterious Multiplying Jar or Melisande or The King's Chessboard or Powers of Ten.  Use simple props to draw kids into acting out the roles.

Tell stories about famous mathematicians (like Archimedes and the bathtub or Gauss and the evil Prussian schoolmaster--admit they might be aprocryphal but there is math involved!) or read mathematical poetry (Theoni Pappas' Math Talk: mathematical ideas in poems for two voices is great fun!) or sing mathematical songs (like these from Tom Lehrer) or get children acting out little skits involving logic problems like this.

Give a child a large sheet of newspaper and challenge the child to see how many times they can fold it in half--then talk about powers of two and why it is so hard to fold with regular paper.  (Until ten years ago, it was thought to be impossible to fold more than seven times, but then a high school student in California managed to surprise everyone by getting to 12 folds, and just last year a group of Massachusetts high school students and their advisor set a new record at 13 folds, but they had to use over half a mile of very thin toilet paper.)   Then, get them thinking about an allowance that starts at a penny a week--and doubles every week.   Supposing their dad is Bill Gates or Warren Buffett, how long before their allowance bankrupts him?  (Give them a long strip of adding machine tape and a pencil and get them to start doubling.  This is also fun to do with sidewalk chalk.)

Speaking of pennies, you could start a kid flipping pennies and charting strings of heads and tails and finding patterns.  Or casting dice and charting totals and discovering that not all totals come up equally often (and figuring out why that's true--and some of the implications for which properties are the best investments in Monopoly, along the lines here.)   You could also teach kids how to play a fun game called Prime Number Monopoly or HangMath.  (Hmm, and we could actually talk about the mathematics of lottery probabilities right in the shadow of the New York State lottery headquarters, which are located in downtown Schenectady!)

Who can make guerrilla math circles happen?

anyone with a love of recreational mathematics and a willingness to share it!   That would be all kinds of folks--for starters, the students in formal math circle programs like ours, college and high school math clubs, their friends, supporters, and advisors.

Okay--enough.  It is a beautiful spring day and the Schenectady Green Market has finally moved outdoors!  It is calling my name so I can scope it out for possibilities for our Guerrilla Math Circle initiatives.  Here is a video created by a Union College student that captures a little of the vibrant spirit of the Schenectady Green Market atmosphere and suggests why it might be a good place for guerrilla math circle activities:

More later about the details!

Thursday, May 3, 2012

Hats off to the NYSSPE!

Albany Area Math Circle is grateful to the New York State Society of Professional Engineers for their donation to supporting our students selected for the AllStar Upstate New York Math Team, which will represent Upstate New York at the national high school championships early next month.

NYSSPE is a particularly awesome sponsor for our Upstate NY AllStar Math Team high school students because the Professional Engineers are also founding sponsors of MATHCOUNTS and run the chapter and state programs which gave many of our local students their initial taste of the excitement of math team experiences.  We are also very happy to report that many of our high school students have continued their involvement with MATHCOUNTS by volunteering to coach middle school math teams. 

Sunday, April 29, 2012

Keeping math contests in perspective

It is sometimes easy to lose sight of the fact that math contests are meant to be fun and engaging, but not ultimate ends in and of themselves.

 The main value of math contests is not to be found in the honors and awards (which are very nice, of course, but somewhat beside the point). The main value of math contests is to be found in wonderful mathematics it can draw students into discovering. Not only do they discover mathematics, but they can also discover the joys of working hard towards shared goals with a community of kindred spirits.

 Math contests are a little bit like mountain climbing. The views from the top are nice, but even spectacular views do not justify the extreme efforts it can take to get there. What can justify those mountain-climbing efforts are the joys of the journey, especially if shared along with others, and the beautiful things you can learn about the natural world and the power within yourselves as you help one another to discover hidden potentials and problem solving abilities within you that you did not know existed.

Just as you should "stop and smell the flowers" as you climb the mountain, so too should you remember to "savor the problems" you encounter in math competitions. By construction, those contest problems have already been solved, but reflecting on them after the contests are over may inspire you to create new and fresh ones of your own devising--or you may discover exciting new ways to solve old problems.

 A recently revived discussion on the Art of Problem Solving discussion forums reminded me of a remarkable essay written three years ago by a student on his contest experiences, which has resonated with many readers. The essay is very moving, thoughtful, and beautifully written. Both the essay (linked in the first post on the thread) and the discussion which followed it are readings that I would highly recommend to students, parents, and math teachers and coaches.

I would also recommend reading an inspiring speech given by our alumna speaker at Math Prize for Girls last fall, which also addresses these issues, and gives some excellent advice well worth bearing in mind for making the most of your math contest adventures.

You may also want to explore the links to the authors' blogs, which discuss some of the wonderful mathematics they have been inspired to learn and share with others. There is more recommended reading on these topics here. Some of our older alumni may also find useful advice and insights in a similar vein here.

Monday, April 23, 2012

Drawing new circles

The folks at the First Unitarian Society of Schenectady celebrate this poem:
He drew a circle that shut me out 
Heretic, rebel, a thing to flout.
But Love and I had the wit to win: 
We drew a circle that took him in! 
— "Outwitted" — Edwin Markham
This poem resonates with me in many ways, but today I want to focus on the way it connects to my vision for guerrilla math circles.

Many people feel "shut out" of the mathematical community.  They see math as a superpower that others have but that they hopelessly lack.  They can't imagine math as a joyful and empowering activity, as hard yet rewarding work.  They see the world as divided into non-intersecting circles of "People who can do math," and "People who can't do math."  They place themselves squarely in the latter and can't imagine that they could ever find joy and empowerment in visiting the other circle.  They may even be inclined to disparage or make fun of others who claim to enjoy math.

Can we find a way to draw math circles in a way that draws those folks in? I think we can.  That is where my concept of "Guerrilla Math Circles" come in.

To be continued ...

Guerrilla Math Circles, Math Super Powers, Math as Performance Art, and Math for the 99%?

I love the dual messages encoded in the T-shirt logo design above:

  • Math is a superpower!
  • Share with it with everybody!
    (∀ is a mathematical symbol that means "for all.")
If you also love this logo (designed by mathematician Cindy Traub of the always awesome St. Mary's College math department), please go here to vote for it.  If the design wins the contest, then mathematicians from all over the country attending this summer's Math Fest will get t-shirts with this logo, and I think it would be outstanding to have this message spread far and wide.

Mathbabe has been blogging about how math is a superpower, and that wonderfully evocative and inspirational phrase has been reverberating around in my head ever since I encountered it for the first time on her blog.

Those messages especially reverberated in my head last weekend, when I was in Washington, DC helping out at the Julia Robinson Math Festival held at the Smithsonian during the Math Circles on the Road event.  

You will have to forgive me--my head is truly exploding with all the inspiration and ideas I brought away from that experience, so this post (and most likely my next few posts) will be rambling all over the place as I share them.   A giant brainstorm hit me at the end of the weekend, a new concept I will call "Guerrilla Math Circles," which I will explain in a later post.   I will get there...I promise.

The event at the Smithsonian was really wonderful, with 60 enthusiastic math circle leaders from all over the country (including Elizabeth Parizh and myself from Albany Area Math Circle) helping to run free and fun public math circle demonstration activities for hundreds of enthusiastic participants.

There was a strong theme of "math as performance art" running through many of the sessions, including the one that Elizabeth and I helped Anna Burago to run, along with Ashley Reiter Ahlin, Berhrooz Parmani, Yulin Qing, and Jack Reynolds.   We engaged a group of students (around 8 or 9 years old) in acting out mathematical logic problems set on an island inhabited by Knights (who always tell the truth), Liars (who always lie), and Tourists (who can go either way.)  Simple props (leis for the tourists and pennants bearing a K or an L for the knights and liars) added greatly to the engagement of the event.  (The kids loved waving the pennants--and it was a really nice way to get their heads inside the logic exercise.  We had originally intended to use paper hats bearing K or L, but after consideration of hygiene/sanitation issues that could arise from switching hats around, we decided to go with pennants instead.  Props to my ever-resourceful 80-year-old problem-solving mother for suggesting that chopsticks leftover from takeout orders work much better than drinking straws for constructing inexpensive pennants!  The kids loved waving the pennants around so much that drinking straws would have quickly drooped.)

Another great session I had the chance to observe also involved math as performance art.  Blake Thornton of the Washington University in St. Louis Math Circle adapted an idea from Terry Tao's blog into a great session on the Island of the Blue and Brown Eyes.  (Again, it was fascinating to see what a difference the use of a simple but concrete prop made with the young children, who were acting out the roles of islanders trying to reason through a problem of inferring their own "eye-colors" based on their observations of the "eye-colors" they observed on the other islanders along with a remark made by a clueless tourist who did not understand the island's taboo against discussing eye color.  In the first run of this activity, Blake and his assistants gave each child an index card that told the child how many of the other islanders in the room had blue eyes, and how many of the other islanders had brown eyes.  In the second run of the activity, Blake and his assistants asked all the children to close their eyes as they placed a colored sticker on each child's forehead to designate that child's "eye color".  When the children were told to open their eyes, they could then immediately observe the eye color of all the other islanders.  This very simple expedient worked *much* better for the students involved, and I was really impressed at the way the children were then able to reason through the problems presented to them.)

I also got the chance to observe a Math Wrangle organized by Tatiana Shubin from the San Jose Math Circle.   The wrangle involved six very impressive young members of the Fairfax Math Circle, who wore awesome t-shirts bearing a translation of a famous quote from Georg Cantor as they wrangled in front of an adult audience awed by their poise in presenting their mathematical reasoning.

Their shirts said:  "The essence of mathematics of mathematics is its freedom."

Freedom...yes, freedom and free were more words that reverberated in my head last weekend.  Math is free--you can do it with scratching in the sand or dirt (as Archimedes did) or even just in your head (as prisoners of war have done in order to maintain their sanity) or with the simplest of materials such as stones or paper and string or colored sidewalk chalk.

And I was troubled by that message.


Turnout at the free math festival and at the nearby free Math Alive! exhibit at the Smithsonian was excellent.  It was a beautiful spring day with a Cherry Blossom Parade that had brought huge crowds downtown.  

There were thousands of children eagerly passing through the Math Alive! exhibits, with hundreds of them checking into our math festival and staying to participate in an activity or game with us.

So why was I troubled?

Because, among the hundreds of students that I personally observed passing through the festival and the museum that day I did not see a single African-American child visit our festival--and this in a city where the overwhelming majority of public school students are African-American and where the black-white educational gap is the greatest in the country.   (I did hear a report from other attendees that they did see a few African-American students attending, but there was general agreement that they were very few in number.)

We were at the Smithsonian in a FREE math festival, held in a FREE museum, on a national mall surrounded by monuments and memorials in the capital of a country that cherishes FREEDOM.  Our society is far from perfect, and yet it represents a beacon of freedom and opportunity to the entire world.    The free museums of the Smithsonian and our free-to-the-public math festival were emblematic of that freedom.

The newest memorial celebrating freedom near the national mall just opened last fall, the Martin Luther King Memorial.  For me, it brought back many memories of my childhood growing up in Washington, DC in the 1960s.  I spent much of the summer of 1963 at my grandmother's apartment, where she was dying of cancer, and tenderly cared for by a much-beloved African-American woman.   I still remember us sitting together in the living room as we watched the black-and-white television in awe of the vast crowds assembled on the mall downtown and heard Martin Luther King's powerful words reverberate:   "Free at last!  Free at last!  Thank God Almighty, we are free at last!"  

I too have a dream.  And it began to take concrete shape last weekend as I contemplated all these memories that reverberated in my head last weekend in Washington, DC.

To be continued....

Monday, March 26, 2012

"Dumb" questions and STEM bullies

I hasten to point out that the folks in these pictures are most definitely NOT bullies!

They are the Stanford professors who have been teaching some wonderful on-line classes that I have been taking this year.  In the fall, I took Introduction to Artificial Intelligence with Professors Peter Norvig and Sebastian Thrun.  (They were amazing.  Among other things, Prof. Thrun headed the Stanford team that designed and built the driverless car that won the DARPA desert challenge.  He has an inspiring Ted Talk that I highly recommend.)  Now I am taking Probabilistic Graphical Models with Professor Daphne Koller.  (Her photo above is from wikipedia.  This NYT article tells a bit more about the cool work she does.)

There are tens of thousands of students all over the world taking these classes along with me, and students helping one another on the course discussion boards has been an essential and exciting part of the learning process.

There is absolutely no way that Professors Thrun, Norvig, and Koller or the few official teaching assistants who help them could answer all our questions.  There are so many unanticipated sources of confusion and technical difficulties (for example, some students live in countries where they use commas instead of periods to denote decimal points, people are using many different operating systems on their computers, for many students English is a second language, etc.)   I am once again struck by the spirit of generosity among my classmates.   While observing the rules of the Stanford Honor Code (which prohibit giving help on the substantive content of graded homework assignments), my classmates have generously provided assistance in dealing with various technical issues that have arisen with downloading and installing and running the required software.  This has been very helpful to many of us.

However, very occasionally there is an obnoxious comment posted on the discussion boards making a snide remark such as "Anyone who asks such a dumb question clearly does not belong in this class."

I cringe when I read remarks like these.  I think of the people who make such posts as STEM bullies.

My feeling is that the askers of the questions DO belong in the course.  The ones who do NOT belong are those who put others down for asking "dumb questions".

I feel the same way about our math circle as I do about the on-line classes I am taking.

Thus, I was heartened to read this powerful post on the subject of "dumb questions" by Professor Thrun--it captures my own beliefs so well that I wanted to share it--I will be reading this aloud at this Friday's math circle:

I really hope that this new digital medium makes it easier to ask "stupid" questions. Let me report on myself. I work with a 200+ people team at Google (reporting into me), I co-founded Udacity, I am an authority in my area of research. I ask many many "stupid" questions. I have learned that asking questions is power. The problem is if others respond to such questions with "you should have known." People rarely do this to me, but they do this to my students. I really dislike this, and I usually confront them. We should remember that there is NO learning without asking questions. In this class, there are people with many different levels of knowledge and skills. What brings us together at this point is that we are all 100% dedicated to make this class. be kind. Reach out to people asking questions whose answer appears trivial to you. Be a friend. And make a friend. remember the question that seems obvious to you once was non-obvious to you. You find that people respect you for being kind. Being kind is one of the highest levels of achievement. I will respect you for it, and so will the people around you. There will come the day when you are asking the stupid question - and you will appreciate the kindness of others.

Saturday, March 3, 2012

Young student honors

The American Mathematics Competition has a special national public honor list for students who score high on a contest designed for older students.

Congratulations to the following students who made those national honor lists this year:

(Students in tenth grade or below with scores over 90)
Cecilia Holodak (Niskayuna HS) 99

(Students in tenth grade or below with scores over 90)
Matt Gu (Guilderland HS) 93

(Students in eighth grade or below with scores over 90)

Alex Wei Van Antwerp MS 127.5
William Wang Farnsworth MS 123
Patrick  Chi Iroquois MS 120
Ziqing Dong Farnsworth MS 117
Junsu Park Albany Academies 113
Andrei Ahkmetov Van Antwerp 105
Liam McGrinder Van Antwerp 105
Gideon Schmidt Iroquois MS 102
Jason Tang Van Antwerp 100.5
Chenyang Wang Shaker JHS 99
Alex Cao Shaker JHS 93
Luke Lubel O'Rourke MS 90

(Students in eighth grade or below with scores over 90)

Alex Wei Van Antwerp MS 135
Ziqing Dong Farnsworth MS 126
William Wang Farnsworth MS 124.5
Jason Tang Van Antwerp MS 118.5
Andrei Akhmetov Van Antwerp MS 108
Alex Cao Shaker JHS 108
Patrick Chi Iroquois MS 106.5
Liam McGrinder Van Antwerp MS 103.5
Vladimir Malcevik Van Antwerp MS 99
Gwenda Law O'Rourke MS 97.5

Students with light blue backgrounds behind their names are members of Albany Area Math Circle and/or our affiliated middle school outreach programs.  (If you know any of the others--or any other local students who might enjoy our math circle activities, please invite them to subscribe to our email lists by sending an email to for high school students and their parents or for parents of middle school students.)   

Congratulations to our American Invitational Math Exam qualifiers

Congratulations to all the Albany area students who embraced the "extreme math" challenge of this year's AMC10 and AMC12 contests.

Here are the criteria for invitation to the AIME along with the honor lists:

American Invitational Math Exam (AIME) qualification:
115.5 or above on AMC10A
120 or above on AMC10B
94.5 or above on AMC12A
99 or above on AMC12B

Congratulations and best wishes to the following students from the Albany area who have qualified to take the AIME, the next step in a series of progressively more challenging mathematics exams leading to the International Mathematics Olympiad.

American Invitational Math Exam (AIME) 
AMC12B qualifiers:
Matthew Babbitt (heeg) 117

Wyatt Smith (heeg) 114

Elizabeth Parizh (Niskayuna HS) 99

American Invitational Math Exam (AIME)
AMC10B qualifiers:
Alex Wei (Van Antwerp MS) 135
Ziqing Dong (Farnsworth MS) 126
William Wang (Farnsworth MS) 124.5
Aniket Tolpadi (Niskayuna HS) 123

American Invitational Math Exam (AIME)
AMC12A qualifiers:
Matthew Babbitt (heeg) 130.5
Zubin Mukerjee (Guilderland HS) 102
Sherry He (Emma Willard School) 101
Cecilia Holodak (Niskayuna HS) 99
Wyatt Smith (heeg) 99

J Chung (Emma Willard School) 95
N Xie (Albany Academies) 95

American Invitational Math Exam (AIME)   
AMC10A qualifiers

Alex Wei (Van Antwerp MS) 127.5
Philip Sun (Shenendahoah HS East) 126
William Wang (Farnsworth MS) 123
Patrick Chi (Iroquois MS) 120

Vineet Velandula (Niskayuna HS) 120
Ziqing Dong (Farnsworth MS) 117

Gili Rusak (Shaker HS) 117

Students with light blue backgrounds behind their names are members of Albany Area Math Circle and/or our affiliated middle school outreach programs.  (If you know any of the others--or any other local students who might enjoy our math circle activities, please invite them to subscribe to our email lists by sending an email to for high school students and their parents or for parents of middle school students.)  

Please report any errors or omissions by sending email to mathcircle at gmail.