Category: Math Circles

  • How to catch a cheater

    I proctored the AIME II contest this week, and caught a cheater. Here are some details and thoughts about the occasion.

    At about 4pm the day before the contest, I started getting emails and phone calls from parents, from tutors, some students, and even my math colleagues at Bard who had been contacted as well, in desperate attempts to contact me. Their children had all planned to take the contest at Kean University in NJ, but for logistic reasons, the contest manager had to cancel, less than 24 hours in advance.

    Up until then, I only expected to proctor one student, since the other two AIME qualifiers from when we hosted the AMC 10B and AMC 12B contests had also qualified on the A schedule a few weeks earlier. But now I imagined a packed room with 20 or 30 AIME students! The AMC response (amcinfo@maa.org) was quick and responsive, and they granted permission to include as many students as I had AIME answer forms for (I counted, and had 20 forms), even though I had registered and paid for only 10 students.

    I passed on the good news to everyone, and quickly drafted out a liability and photo release form for the newcomers, along with detailed instructions on how to drive up to Bard College (more than 2 hours away from New Jersey) the next morning.

    But then, over the next few hours, all of them found closer places, and cancelled, thankful that the worst case scenario – a long drive up to the Hudson Valley – was there if needed, but the drive could be avoided.

    All but one, that is: the cheater. He is a high school freshman at one of New Jersey’s most prestigious public high schools. He was driven up by his math competition tutor, who described himself as a friend of the family. He coaches the cheater, and four other NJ students in competition math. All the rest of the students had also planned to come up to take the AIME II at Bard, but had found more convenient locations overnight.

    They arrived at Bard, and were welcomed in by the local math student (a middle school student) who had qualified, and his mom. When I arrived a little later, I set up the room, and welcomed the mom and the tutor to enjoy Bard College, a wonderfully secluded campus where they could stretch their legs, catch breathtaking views of the Catskill Mountains and the Hudson River, and explore eclectic architecture (including a Frank Gehry building in north campus). Or, they could set up their laptops with Bard’s wifi, and catch up on work. Bard is on spring break this week, so parking is plentiful, and so are study nooks.

    I helped the students bubble in the answer forms with their names and other information (kids these days don’t remember their street addresses), and made sure they understood the AIME contest rules. Then I started the exam by clicking on the countdown timer (Google search for “internet timer”) set to 3 hours.

    Ten minutes later, the cheater pulled a cell phone out of his pocket and put in on his lap. Did he not understand that electronic devices were strictly not allowed on the AIME? I walked over to him and confiscated his phone immediately. I let him continue the exam, because I saw that he hadn’t received any information from the phone.

    In the middle of the exam, he excused himself to go to the bathroom. He returned a few minutes later. Later, with half an hour to go, he excused himself again to go to the bathroom and left the room.

    A minute later, the other student’s mom opened the door to ask me a question. In the hallway, she told me that the NJ student had gone outside. He told her that he needed to get his photo ID, and she wondered if it was okay that her son didn’t have an ID with him. She was worried about that, but I reassured her that since her son had been to the AMC contests at Bard for several years in a row, I recognized him, and no ID was needed.

    Then the other student came back from outside, not from the bathroom. That was strange. Back in the testing room, the NJ student got back to work, and started to open up a folded piece of paper. I stood up and walked over to him, explaining that outside notes are not allowed on the exam, and asked him to hand it over. I unfolded the paper, and found a numbered list of 3 digit numbers. The answer to each AIME problem is a 3 digit number.

    I was shocked at how blatant this student was in his cheating, and how easy he was to catch.

    I decided to let him continue working, mostly to avoid distracting the other student, and also to maintain control over the situation.

    When the time was up, I asked the students to sign the statement on their answer form that all the work on the contest was their own. Both students signed. I collected the answer forms and excused them from the testing room.

    I told the NJ student that I wanted to speak with him and with his tutor, and he went outside to the parking lot. Then I took the opportunity to thank the local student and his mom, and to ask what problems he found interesting. They left soon after that.

    But the NJ student and his tutor didn’t come back. I waited and waited, and then gathered my things and started leaving. Funny – I still had the student’s cell phone and his cheat sheet. Right before I left the building – it seemed about half an hour after the contest ended – the tutor appeared.

    The tutor asked what had happened. We talked, and the tutor explained that he had given the student the answers on a sheet of paper, because he thought the exam was over (it was just after 12pm when I confiscated it).

    He told me that one of his other NJ students had finished the exam early, about 10am, at some NJ location, and texted an image of the exam to him, which he worked on in his car.

    The student showed up soon after that, and I talked with them both. The student admitted to cheating, and said that he wanted the opportunity to take the USAMO. He said that the tutor didn’t know about the cheating, but I don’t believe him. The tutor also claimed not to know about the cheating, even though he admitted to receiving an electronic copy of the contest around 10am, during the exam, working on it, and handing his student a neatly-written list of answers. He didn’t seem at all upset that his student had lied to him, and asked me to give his student a chance and to forgive the cheating.

    It was fascinating to interview a cheater and his accomplice, and to feel in control of the situation. I asked them question after question to collect information to share with the AMC. I was curious if they would show any remorse for what they had done, and so I asked them questions to open opportunities for that. But there was no remorse at all, so I ended the conversation and left.

    The cheater ran after me again to ask for his cell phone back, but I refused to, and told him to email me and I’d ship it back to him.

    The cheating incident is over, but it leaves me with some questions. If you have answers, please leave a comment!

    1. How important is it to protect the cheater’s identity? I shared his name and other information with the AMC, of course. But I could also easily contact his high school through personal contacts. I could publish his name on the web. I could contact the Kean University contest manager, and other university based contact managers in the region to let them know. Should I take any or all of those steps? Or leave it to the AMC to take action.
    2. Is it important to protect the tutor’s identity? I have contact information for another of his students and that student’s father. What action on my part is appropriate? I was offended on many levels that this tutor placed no priority at all on honesty. In fact, he is a key figure in a cheating gang. If he were tutoring my child, I would want to know.
    3. How welcoming should I be of AIME students from outside the Bard Math Circle community? I want to promote and develop a culture of math enrichment in the Mid-Hudson Valley. I want to open up opportunities for students whose schools don’t have a math club, a math team or a math circle. But this cheating incident was very, very disappointing.
    4. What would you do with the cell phone? (I ended up shipping it to his parents by express mail, carefully bubble wrapped and insured. I included a brief letter explaining that the phone had been confiscated from their son during the AIME exam.)

     

     

  • Uniqueness of Factorization

    A few days ago I came across a proof of the Fundamental Theorem of Arithmetic (aka Unique Factorization) in Courant and Robbin’s What is Mathematics that I hadn’t seen it before. I liked it enough to learn it.

    Then another surprise – I saw it again yesterday in Primes and Programming by Peter Giblin, a book that Larry Zimmerman had recommended to a student from the summer high school program.

    The usual proof that I know is based on Euclid, and basically is a proof by Strong Induction. This new proof is by the Principle of Least Element. So the key is to suppose that unique factorization fails, and to reason about the least positive integer \(N\) that has more than one factorization into primes. Even though we’ll show this number doesn’t exist, we can deduce lots of information about it!

    First, some notation. Let’s say that two distinct prime factorizations of \(N\) are

    \[\text{(1)}\qquad N = p_1 p_2 \dots p_r \text{ and } N = q_1 q_2 \dots q_s\]

    Of course, we’ll arrange the primes in non-decreasing order, so that in particular, \(p_1\) and \(q_1\) are the smallest primes in those factorizations.

    The other primes don’t take a big role in what comes next, so let’s write \(P = p_2 \dots p_r\) and \(Q = q_2 \dots q_s\), so that we have

    \[N = p_1 P = q_1 Q.\]

    The first observation is that \(p_1\) and \(q_1\) are different primes, otherwise if they were equal, we could factor them off and then \(N/p_1 = P\) would be a smaller positive integer with two distinct prime factorizations.

    Now that that’s done, let’s assume without loss of generality that \(p_1 < q_1\), and we’ll form a new number:

    \[\text{(2)}\qquad M = (q_1 – p_1) Q\]

    By equation (2), it’s clear that \(M\) is a positive integer that is less than \(N\), and therefore does factor uniquely into primes. Now we rewrite \(M\) as follows:

    \[M = (q_1 – p_1) Q = q_1 Q – p_1 Q = N – p_1 Q = p_1 P – p_1 Q = p_1 (P – Q)\]

    That is, \[\text{(3)}\qquad M = p_1 (P – Q)\]

    We’re almost there. Note that because of equation (3), the prime \(p_1\) is a prime factor of \(M\). Now consider the factorization of \(M\) given in equation (2). Since we were careful to list the primes in non-decreasing order, \(p_1\) can’t be any of the primes in \(Q = q_2 \dots q_s\), and so it must be a factor of \((q_1 – p_1)\). Suppose that \((q_1 – p_1) = p_1 t\). Then solving for \(q_1\), we find that \(q_1 = p_1 (t+1)\). And this is a contradiction, since then \(q_1\) would not be a prime number!

  • Gathering 4 Gardner

    This year’s G4G, or Gathering for Gardner, Celebration of Mind II, falls on Friday, October 21. This is the second G4G since Martin Gardner passed away on May 22, 2010, and the G4G is intended to celebrate his life and work.

    Martin Gardner

    You can find a nearby celebration here: http://www.g4g-com.org/

  • MathFest! 2011 in Lexington, KY

    This year’s MathFest will be in Lexington, KY in early August. I’m going to present a talk about Math Circles in the Hudson Valley at the Fostering, Supporting and Propagating Math Circles for Students and Teachers session. I’ll also be presenting the teachers’ math circle session with my talk on the game of Nim. Both these are part of the SIGMAA on Circles.

  • NYMC Summer Workshop 2011 at Bard College

    Join us at Bard this summer for a week-long residential program focused on the investigation of inequalities and optimization. Enjoy an environment of creative and insightful mathematical problem solving for middle school and high school math teachers who wish to deepen their mathematical understanding. No prior experience with inequalities required, just an interest in doing mathematics in a community of teachers. Workshops and activities led by NYMC instructors and Bard math professors.

    More information: http://nymathcircle.org/2011workshop

    Dates: July 25-29, 2011.
    Location: Bard College, Annandale-on-Hudson, NY (map).
    Accommodations will be provided.
    Audience: Middle and High School Mathematics Teachers.
    Theme: Optimization and Inequalities.
    Registration will open in February.

    Graduate credit is available, additional fees apply. All participants receive a certificate of participation and an official letter describing the 24 contact hours of the workshop.

  • Circle on the Road 2011: Houston

    March 18, 2011 to March 20, 2011

    The Math Circle on the Road is a series events around the country that brings together math circle organizers with people who plan to start math circles.

    Circle on the Road 2011: Houston

    Poster (pdf)

  • Bard Math Circle in Kingston

    View of the Kingston Library
    Kingston Library at 55 Franklin Street, Kingston, NY 12401

    Bard Math Circle meets in the Kingston Library

    The Bard Math Circle will host math activities in Kingston every second Saturday this fall, 1pm to 3pm.
    Each session will include time for puzzles and games, creative mathematical problem solving, and a hands-on math activity with a take-away.
    Meetings this fall: 1pm to 3pm on each second Saturday (September 11, October 9, November 13 and December 11) at the Kingston Library (55 Franklin Street, Kingston NY, 12401).
    First meeting: 1pm to 3pm on Saturday, September 11, 2010.
    Here is the library press release and posters:
    What is a Math Circle?

    According to the National Association of Math Circles,

    Mathematical Circles are a form of education enrichment and outreach that bring mathematicians and mathematical scientists into direct contact with pre-college students. These students, and sometimes their teachers, meet with mathematical professionals in an informal setting, after school or on weekends, to work on interesting problems or topics in mathematics. The goal is to get the students excited about the mathematics, giving them a setting that encourages them to become passionate about mathematics.

    This is certainly the case! The Bard Math Circle is hosted by Bard College mathematicians and math majors, and seeks to work directly with middle school students (although everyone is welcome) who already enjoy mathematics. Bring a friend along, and they may fall in love with math too.

  • Math Delights, a webpage.

    A wonderful website of Math Delights. Full of rich mathematical activities for 5-12 year olds. Collected by Nancy Blachman, founder of the Julia Robinson Math Festival.

  • Math Physics Explore

    Excerpt from website: http://www.mathphysicsexplore.org

    We develop a curiosity and interest in mathematics so students, parents and teachers can improve their math and science education. Activities focus on various topics in mathematics and their application to physical phenomenon.

    This website and our exploratorium:

    • Encourage each person to reach his/her potential
    • Provide guidance for children participating in science fairs
    • Help teachers to explain math concepts
    • Provides travel exhibits for schools

    We are located at 1054 Freedom Plains Road, Poughkeepsie, NY 12603.

  • Bard Math Circle

    The Bard Math Circle primarily targets middle school and elementary school students in the Mid-Hudson Valley region of New York State.
    In Kingston, NY (a small city in which 15% of residents live below the poverty line and 27% of children live below the poverty line) , the target audience for the past two years has been members of the Boys & Girls Club, whose members are primarily minority youth from low socio-economic households. The highlight of the activities in Kingston was a group visit to the Bard campus, in which the Boys & Girls Club members participated in math workshops led by Bard College math professors. In 2010, the primary location is the Kingston Library in Midtown Kingston and targets a wider middle school audience, a diverse socio-economic, racial and gender balanced group.
    In Tivoli, (a semi-rural community north of Bard College) the Bard Math Circle holds monthly sessions at the Tivoli Library, The audience consists of families with children in middle school and younger.
    In Red Hook, (a suburban and semi-rural rural community east and south of Bard College), we also target a middle school audience, in partnership with a math teacher at the Linden Avenue Middle School.

    Origins

    The Bard Math Circle was formed in 2007 by Bard College Professors Lauren Rose and Japheth Wood and then students Shelley Stahl and Ezra Winston. The Bard Math Circle is run jointly by students, under Bard’s Trustee Leader Scholar Program, and faculty at Bard College.