Category: Uncategorized

  • The Upstate New York Math Team

    Hey, I want to tell you about one of my recent projects, re-booting the Upstate New York Math Team. This was a regional math team, designed to help high school students from around Upstate New York to compete in ARML each year. The team was active and successful since almost the beginning of ARML (see below for what I could find of the historical record), but then fell apart, and seems to have last competed in 2015.

    The team was dormant in 2016, 2017, 2018, and 2019. With the encouragement of some colleagues, I started working on reviving the team in 2019 to participate in ARML 2020. Then COVID happened, and ARML was canceled. (Actually, the ARML organizers scrambled, and did offer an informal online contest in 2020 which we did participate in.)

    Over the past several years, I’ve been trying to network across Upstate New York to rebuild the team. The ARML (and NYSML) administration has been wonderful, and have connected me with math team coaches across the state, and with some alumni. With some other volunteers, we’ve built up an email list on Mailchimp, set up a shared Google drive, and opened up a Discord server. We’d like to set up a webpage. A small core group of volunteers has formed a Problem Squad, and publishes a monthly practice problem set. A small core group of high school students meets monthly to practice solving problems together.

    Besides sending Upstate New York high school students to ARML each year, we’d like to grow as an enrichment resource for students interested in mathematical problem solving. In the past, the Upstate New York Math Team was managed by a succession of high school math teachers, but then something broke. We’d like to re-organize as a 501(c)3 nonprofit. I think that’s the way to ensure the longevity of the effort.

    If you’re interested in helping, please get in touch!

    As part of my efforts to understand what the Upstate New York Math Team is, I searched for team history. Here’s how the team placed in past contests (but there are gaps and likely errors).

    YearABC
    201539
    2014
    201316
    201233
    20112545
    20102035
    2009
    200820
    200712
    2006
    200514
    200417
    2003104774
    200296783
    200120
    2000124255
    1999105048
    1998156(B)42(B)
    19972012(B)38(B)
    19961(B)21(B)36(B)
    19951623(B)33(B)
    1994
    1993
    199232(B)
    199112
    19901(B)
    198918
    198816
    19871(B)
    198615
    1985
    198410
    198316
    19823(B)
    Historical record of the Upstate New York Math Team

  • Hello world v2.0!

    Notice something different? I’ve changed servers (from Powweb to Dreamhost) and tried to move my WordPress site over. The text is here, but somehow the photos haven’t made it yet (but they are backed up), and the theme is changed. I tried to do this earlier this year, but didn’t have enough time to solve some issues, so I wiped that, and am now starting again.

  • 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!

  • How to Give a Good Math Talk

    There are a lot of sites with useful advice out there. This post is intended to collect several links for future reference.

    Technically Speaking – Videos of math research students presenting their findings. I love the examples of bad style along with good style. This comes out of an NSF-funded project.

    Joe Gallian’s Advice on Giving a Good PowerPoint Presentation from his article in Math Horizons.

    Giving a Conference Talk by Mike Dahlin.

    Oral Presentation Advice by Mark D. Hill, includes “How to Give a Bad Talk” by David A. Patterson.

    Giving an Academic Talk by Jonathan Shewchuk.

    I also have several resources on good mathematical writing that I’ll share later.

  • Math Conference photos

    Going through some old papers, I found the following conference photos from

    • Fall 1996 Fields Institute, Algebraic Model Theory Program
    • July 1-10, 1998, XI Simposio Latinoamericano Lógica Matemática, Mérida, Venezuela
    • 1998 Szeged Conference on Lattices and Universal Algebra

    1996 Fields Institute
    The Fall 1996 Algebraic Model Theory Program at the Fields Institute
    Photo key

    XI Simposio Latinoamericano Lógica Matemática

    1998 Szeged Conference on Lattices and Universal Algebra

  • Found Math

    I sent this photo in to the MAA Found Math series, and it was published! This looks very much like a geometric dissection proof of the Theorem of Pythagoras, but in the form of a discarded foldable card table.

    Pythagoras
    A proof without words found in the trash

  • What is Algebra good for?

    Algebra represents a shift into mathematical abstraction, providing tools necessary for further study, a focus on properties and principles that deepen our understanding of number, and also reasoning and insight that transfer to other domains of human knowledge.

  • The Emperor has no Clothes

    The NYC DOE recently released effectiveness ratings (TDRs for Teacher Data Reports) for all public school elementary and middle school teachers in NYC. Newspapers and websites immediately published them. The teacher with the lowest rating was identified and publicly ridiculed. The teachers with the highest ratings were lauded.

    Some important public figures also commented:

    Bill Gates came out against this in an opinion piece for the New York Times: Shame Is Not the Solution. He calls this move to release the data a big mistake, and notes that the feedback did not offer substantive information that would allow teachers to improve.

    Diane Ravitch predicted, in the New York Review of Books, a future in which No Student Left Untested. That is, the logical conclusion is for more grades and subjects to be tested, as more data is needed to humiliate more teachers. She also fears that the public humiliation will drive even the best teachers from the profession.

    The Math Babe (Cathy O’Neil) noted that Teaching Scores Released, and accurately calls this the “teaching to the test model”. She notes that open-sourcing this value-added model would make it somewhat more useful for teachers, and finally, echoing Ravitch, calls on Bloomberg to study whether the use of this model and publishing the results has “the intended effect of keeping good teachers and getting rid of bad ones”.

    Stuyvesant math teacher Gary Rubinstein did an interesting study, Analyzing Released NYC Value-Added Data, and presents some seeming extreme cases of teachers whose rating varies widely from one year to the next, and even in the same year. Very telling are his scatter plots, that make it clear that these extreme cases are not unusual at all.

    For those unfamiliar with the abuse of the Value-Added Model, please read the essay Mathematical Intimidation: Driven by the Data, written by Math for America president John Ewing. You’ll learn basic background and limitations of the Value-Added Model, and specific reasons why its application to teaching is misguided and a case of intellectual bullying. Ewing calls on mathematicians to “confront people who misuse their subject to intimidate others into accepting conclusions simply because they are based on some mathematics”.

    Closer to the field, I had the pleasure of reading a letter sent home to parents from Liz Phillips, the principal of PS 321 in Brooklyn. In this letter, principal Phillips comes out strongly against the TDRs and in full support of her teachers. She states that these TDRs will actually lead to a larger “achievement gap”, as only standout, high-performing schools like hers will be able to withstand the pressure to teach to the test. You can read a similar message from Ms. Phillips in the blogpost A principal at a high performing school explains why she is “absolutely sick” about the public release of the TDRs.

    Given all this, I believe that The Emperor has no Clothes. Showing off the results of this model strips away any pretense that this is useful, or even adds value to the public discussion on the teaching profession. Perhaps this moment is a hidden opportunity to embrace a different way to measure effective teaching that is based on classroom observation and coaching, and not on test scores or a value-added model.

    Experienced educators know that good teaching comes in many forms, and they can recognize it in the classroom. I imagine that the effort of developing and articulating a research model of effective teaching (which probably already exists), and organizing an extensive network of experienced educators who visit classrooms to observe and offer guidance to help teachers improve their practice will cost much less than of all this testing. And it just might make a difference.

  • Pi Day Resources

    My brother pointed me to some Pi Day resources today on the NCTM website: Celebrate Pi Day

    It’s nice to see this collection, which is a great resource for teachers to easily roll out their own Pi Day celebration. If any readers can point me to other Pi Day resources, please do: I hope to write a Pi Day page some day, with links to share.

    Further Pi Day pages:

    Exploratorium Pi Day

    MAA Celebrating Pi Day

    For those interested, please also see my post on Math Festival Resources.