Japheth Wood

Category: HCSSiM

  • AP Classes Are a Scam

    I heard about the following Atlantic article from @stevenstrogatz : October 13th, AP Classes Are a Scam which I found quite interesting.

    I thought much the same in those years when I taught a lot of freshman Calculus. My main observations were that

    1. Most students who had taken AP Calculus in High School had to take the Calculus sequence anyway, and resented that they had to essentially repeat a course.
    2. Students who had not taken AP Calculus in High School felt intimidated that they were in class with students who had, and felt completely inadequate.
    3. As a result, the AP students barely worked at all, since they had a superficial knowledge of Calculus, while the non-AP students worked very hard.
    4. Since AP Calculus is not college level Calculus, the effects were clear by midterms: the AP students had fallen too far behind, and the non-AP students were learning the material, and starting to enjoy it.
    5. I suspected that there were other, successful AP students, who weren’t in my class, and never took another math class in their lives. Thus, some of the most enthusiastic math students at the high school level were diverted out of the math major, since they saw Calculus as the final math class.

    So why did I come away with the impression that AP Calculus, presented as the highest level math class one could take in college, was essentially a terminal math class, serving to prevent bright and hard-working high school math kids from continuing in mathematics?

    I’ve come to understand this more in the contrast between acceleration and enrichment. Our educational system emphasizes acceleration, and works hard to move kids rapidly through material. There are a lot of incentives for this, like granting college credit. An alternative is to enrich the curriculum, and allow students to go deeper into the material.

    When I was in high school, I used math team to enrich my studies, as well as my own mathematical reading of fantastic authors like Martin Gardner. I don’t think I ever earned an academic credit for this enrichment, but it was profoundly enjoyable, and directed me into mathematics. I did benefit from acceleration as well, but my most memorable mathematical moments were from some inspirational math enrichment.

    So, why do we bother with AP courses? I think our students would benefit greatly if a similar amount of resources were invested into academic enrichment. I’d love to see after-school math circles, math clubs and math teams in every school, and I can imagine similar enrichment in other subjects.

  • An Interesting Prime Number Fact, Rubik’s Cube and the Gömböc

    In the summer of 2010 I traveled to Hungary for the 25th anniversary reunion of the Budapest Semesters in Mathematics program, and had the pleasure of seeing the inauguration of another study abroad program for computer science, the Aquincum Institute of Technology.

    The interesting part of the ceremony was a series of mathematics talks to celebrate the genius of Hungarian mathematics and technology. There were also talks by Balázs Bús, the mayor of Óbuda, and by János Kocsány, the CEO of Graphisoft Park.

    The Graphisoft Park Rubik's Cube
    The Graphisoft Park Rubik's Cube

    László Babai talked about Mathematical Generalizations of Rubik’s Cube, and mentioned the following.

    The diameter of the Rubik’s Cube graph is at least 20, but probably no more than 21 (Richard Korf, UCLA, 1997), and definitely no more than 26 (Gene Cooperman, Dan Kunkle, Northeastern, 2007).

    It was very interesting that just a month later, in July, 2010, the diameter was confirmed as 20. A team used 35 years of CPU time, donated by Google, to complete the computation. Even more interesting for me was to learn that the lower bound of 20 had been established in 1995 by Mike Reid, who identified the “superflip” position that required 20 moves to solve. Here’s an interesting website that documents progress on this problem: God’s Number is 20.

    I had met Mike in person about 25 years ago, when I attended the Hampshire College Summer Studies in Mathematics. Now that I am involved with the New York Math Circle, I’ve had the pleasure of meeting Mike’s old math teachers, who have wonderful stories to tell.

    Babai had brought up the diameter of the Rubik’s Cube graph because his talk was really about the connection between the size of a group at the diameter of its Cayley graph. For the Rubik’s cube, the group has about 34 quintillion elements (The exact number is 43,252,003,274,489,856,000. Remember: thousand, million, billion, trillion, quadrillion, quintillion, sextillion, septillion, …), but its diameter is just 20, which is on the order of the logarithm of the size of the group.

    Babai mentioned a recent result of Harald Helfgott and Akos Seres, On the diameter of permutation groups, which gives a “quasipolynomial upper bound” for the diameter.

    One beautiful formula that Babai presented was:

    $latex \prod_{p \leq x} p \approx e^x$.

    This seems related to the prime number theorem, that $latex \pi(x) \approx \frac{x}{\ln{x}}$, where $latex \pi(x)$ denotes how many primes are less than x. I leave it to the readers to find the connection.

    Another great talk was by Gábor Domokos, The Story of Gömböc. The gömböc is a solid object with just one stable point of equilibrium (and also one unstable point). If you place the gömböc on a flat surface, it rocks back and forth, and eventually stabilizes in the same position each time.

    Amazing invention–I want one!

  • Rubik’s Cube Conversation over Breakfast

    If you read my previous post then you know that I’m at HCSSiM for the second half of the second half, teaching a mini on the Fundamental Theorem of Algebra. I’m very impressed with everything, from the teaching faculty to the students, and am really enjoying soaking in interesting mathematics every day. There is really high level mathematics going on here, and the students are truly immersed in mathematical knowledge and culture.

    Conversation at breakfast with Lucas and Gabe.

    I walked into a conversation about Rubik’s Cube records this morning at breakfast. Talking about new cube records, Lucas complained about kids who ask him if he solves the cube by just doing the same sequence of moves over and over again. Of course not! This would only work in a cyclic group!

    However, I argued that it is possible, if you look at it another way. Suppose that g is a long sequence of moves that traverses through all possible cube positions. Then you only have to do the sequence g once, and somewhere along the way, you’ll have solved the cube. Notice that the end result of performing the moves in sequence g is the identity permutation on the cube.

    We can improve this by finding a cube permutation g that generates a large cyclic subgroup of the cube group. Let G be the cyclic subgroup generated by g. If we can express g as a long sequence of cube moves that traverses through a complete set of coset representatives of G, then we have the cube neophyte’s dream: a sequence of cube moves, that it you do over and over again, will eventually solve the cube (of course, in the worst case scenario you’ll move through all possible configurations of the cube, but I’m not making any claim about the efficiency of this method!)

    We finished our breakfast conversation by posing a more reasonable problem: try this for a small group.

    1. Show that g = (1, 2, 3)(4, 5) is an element of S_5 of maximal order.
    2. Find a sequence of 120/6 = 20 permutations s_1, s_2, …, s_20 whose product is g, and whose partial products (s_1), (s_1 s_2), (s_1 s_2 s_3), …, (s_1 s_2 s_3 … s_20) is a set of coset representatives of the cyclic subgroup .
    3. Solve problem 2, where each s_i is a transposition.
    4. Solve problem 2, where each s_i is a transposition of adjacent elements.

    This seems like a good start to investigating the breakfast conversation problem. Let me know what you think!

  • HCSSiM 2010: Alive and Well!



    I visited the Hampshire College Summer Studies in Mathematics (    HCSSIM) program for the annual YP Day, and am happy to report the program is as vibrant as ever this summer.

    Yellow Pigs Day

    For those not familiar with HCSSiM lore and what YP day is, the initials YP stand for Yellow Pig, one of the two mascots of the program. David Kelly, the director of the program, has perhaps the largest collection of yellow pigs in the world.

    The other mascot of the program is 17, the favorite number of scads of summer studies students and alumni. YP Day (Yellow Pig’s Day) falls each year on July 17th, and is the traditional time for alumni to revisit the campus and reconnect with HCSSiM.

    This Year’s Faculty

    Arriving at HCSSiM on Friday in time for lunch, I met the faculty. In addition to Kelly, we have Rob Hochberg and Josh Greene also leading the workshops. Rob, a computer scientist, and I go way back to the 1980’s when we were both summer studies students, but this was my first time meeting Josh, a Columbia University mathematician, who attended HCSSiM in the 1990’s. To round out the crew we have the amazing:
    • Gabe, recent University of Chicago graduate
    • Misha, math graduate student at CUNY
    • Nadine, math graduate student at the University of Pittsburgh
    • Achyut, Hampshire College physics major
    • Peter, Harvey Mudd math major
    • Nate, U Mass math major
    • Julia, recent Hampshire College graduate
    • Emil, Harvey Mudd math major
    and of course the indefatigable Susan Goff, who has been the program assistant for over a decade.

    Prime Time Theorem

    I got my chance to meet the students later that afternoon, when I explained How to Solve a Cubic to them as Friday’s Prime Time Theorem speaker. The students are as sharp as ever, and an enthusiastic audience. I ended my talk with some open-ended questions.
    • Is there a quartic formula?
    • Is there a quintic formula?
    • Does every polynomial have a zero?
    Although most of the summer studies students know the general answer, I hope that these questions and my talk will pique their interest for my Mini later this summer on the Fundamental Theorem of Algebra. Apparently another Mini will focus on Galois Theory, so HCSSiM definitely has this topic covered!

    After dinner I had a chance to talk with the students during their problem session. They’re enjoying some Algebraic Number Theory, Cardinality Proofs, and Group Theory.

    Saturday, July 17th

    Yellow Pig’s Day fell on a Saturday this year, just perfect timing for the traditional Ultimate Frisbee Game, Kelly’s talk on The Mathematical and Social History of 17, the Yellow Pig Cake and the Singing of Traditional Yellow Pigs Day Carols.

    Saturdays start off with the morning workshops, though. I was able to wander between the three workshops and observe a wonderful proof of Wilson’s Theorem, a listing of Group Theory axioms (after two weeks of examples), and some interesting Cardinality Proofs.

    Mathcamp

    Surprisingly close this summer is Mathcamp, another wonderfully rich math environment for high school students. Math camp is just a 15 minute drive from Hampshire College at Mount Holyoke College, I visited Dan Zaharopol’s class on Real Analysis, and Alison Miller’s class on p-adic numbers.

    Among the more than 100 Mathcamp students, there are several who previously studied at HCSSiM, and I was intrigued discussing their positive experiences at both programs.

    Ultimate Frisbee Game, HCSSiM 2010 vs. HCSSiM <>

    My visit to HCSSiM 2010 ended with a grueling game of ultimate in excruciating heat. The alums were outnumbered three-to-one, but did well until the 2010 students developed a strategy that took advantage of their strength in numbers: they lined up and passed the frisbee down the field. Luckily, the aging alums still managed to come out ahead by dividing each team’s score by the number of players, and then multiplying by the average age.I drove back home soon after dinner, and enjoyed a beautiful double rainbow all the way through the Berkshires. I’m looking forward to returning in a few weeks to lead a Mini on The Fundamental Theorem of Algebra.