Japheth Wood

Category: NYMC

  • A brief 4,000 year history of Diophantine Equations

    I filled in for a NY Math Circle class over the weekend. Since the topic was Primitive Pythagorean Triples, I had a blast. I also shared the following outline with the students. Each item is full of wonderful mathematics and anecdotes!

    Plimpton 322, a Babylonian cuneiform tablet @ Columbia University. From 1900BCE – 1600BCE, and allegedly includes the Pythagorean triple (12709, 13500, 18541).

    Pythagoras was born ca 580BCE on the island of Samos. Famous quote: “All is Number”. His proof of the theorem that bears his name involved cutting up a square of side a+b and rearranging the pieces.

    Proclus (5th century CE) credits Pythagoras with the formula (2n+1, 2n^2 + 2n, 2n^2 + 2n + 1) of (necessarily) primitive pythagorean triples where the hypotenuse is 1 more than one of the sides. Proclus also credits Plato with the formula (2n, n^2 – 1, n^2 + 1) where the hypotenuse is 2 more than one of the sides. (For what n is this primitive?)

    Euclid of Alexandria (born 365 BCE) is famous for the 13-book Elements. The Theorem of Pythagoras is Book I, Proposition 47. Analysis on Primitive Pythagorean Triples appears as Lemma 1 “To find two square numbers such that their sum is also a square” in Book X, just before Proposition 29.

    By the way, Elisha Scott Loomis, an early 20th century mathematician, published The Pythagorean Proposition, which has 370 proofs of the theorem, and not a single one used trigonometry. This was republished in 1968 by the NCTM.

    Diophantus of Alexandria (200CE – 298CE) is famous for his book Arithmetica. He sought integer (and perhaps rational too?) solutions to algebraic equations. The term Diophantine Equation typically refers to equations where we seek positive integer solutions.

    Fermat (1608CE – 1665CE) wrote in the margin of his copy of Arithmetica that there are no integral values of x, y, z so that x^n + y^n = z^n if n > 2.

    Andrew Wiles of Princeton University announced a proof in 1993 of Fermat’s Last Theorem, after working in secret for 7 years. An error was found in his proof, which was salvaged in 1994. Wiles’ proof was published in 1995.

    Jumping back to 1900, David Hilbert asked mathematicians at the International Congress of Mathematicians to devise a method to determine whether a Diophantine equation has solutions. This is known as Hilbert’s 10th Problem.

    Julia Robinson (1919-1985) was a Californian mathematician who worked on Hilbert’s 10th problem for decades, from the 1940s until final achieving a solution (in joint work with Martin Davis, Hilary Putnam, Yuri Matiyasevich and others) in 1970. In general, there is no such algorithm!

    Mirroring the Hilbert Problems of 1900, the Clay Institute of Mathematics issued the Millennium Problems in 2000.

  • 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.

  • Remembering Mr. Geller

    Richard Geller at Stuyvesant, November 3, 2009
    Richard Geller at Stuyvesant, November 3, 2009

    I first met Mr. Geller in the mid-80s when I was a high school student. My Stuyvesant math team friends snuck me into the math team practice one morning, and Mr. Geller was the math team coach. Although he enjoyed a reputation for being quite strict, he made an exception in my case and I felt welcome.

    When I found myself working in New York starting in 2006, I met Richard again at the New York Math Circle. Richard attended every teachers class, and I pretty much did too.

    I took this photo of Richard at Stuyvesant High School just two years ago at the annual math team coaches workshop.  Mr. Geller was there early, and welcomed everyone. I was there to run a middle school math teachers’ circle with the New York Math Circle, and Richard was the one who had invited us in and arranged for the room.

    Richard was diagnosed with a melanoma cancer last year, and died on November 1, 2011 at about 1pm. (11/1/11 1pm). He will be missed. In a recent message, his widow, Barbara, wrote that

    In lieu of flowers, contributions can be made to a scholarship fund at Stuyvesant High School set up in Richard’s name. Checks can be made out to Richard B Geller Memorial Scholarship for Mathematics and sent to Barbara Geller, 50 West 97th Street #11T, New York, NY 10025.

    Here is further reading:

    New York Times article: Stuyvesant Students Mourn a Math Teacher Who Was No. 1

    The Stuyvesant Spectator article: Richard Geller: The Lifelong Mathematician

    Riverside Memorial Chapel: Richard Geller (August 20, 1946 – November 1, 2011)

  • Bard Math Circle in the News (bring your kids!)

    Dear Friends,

    The Bard Math Circle made the front page of today’s Daily Freeman with a story about our math outreach activities at the Kingston Library. COUNTING ON FUN: Math Circle at Kingston Library takes middle school kids beyond boring basics (video) (Monday, January 17, 2011) http://dailyfreeman.com/articles/2011/01/17/news/doc4d33674c5f430298712442.txt 

    The Bard Math Circle meets twice per month at the Kingston Library with a program of math and logic games, problem solving and hands-on math activities, designed for middle school math students. Everyone is welcome! The next meeting will be on Saturday, February 12, 1-3pm.

    The Bard Math Circle was previously featured here:

    Las Noticias: Por el amor a las matemáticas (Wednesday, October 06, 2010)

    and in our La Voz: Círculo de matemáticas en Kingston (Agosto 2010)


    What is the Bard Math Circle?

    The Bard Math Circle, which targets middle school and elementary school students in the Mid-Hudson Valley, was formed in 2007 by Bard College Professors Lauren Rose and Japheth Wood. The Bard Math Circle is run jointly by students, under Bard’s Trustee Leader Scholar Program (http://inside.bard.edu/tls/), and math faculty at Bard College. Student leaders have included Shelley Stahl, Ezra Winston, Elias Halloran and currently Jackie Stone. Outreach activities have focused on Kingston and Tivoli.

    In the summer of 2010, Bard hosted the New York Math Circle’s Summer Workshop 2010 for teachers (http://nymathcircle.org/2010workshop). I brought (primarily NYC) middle school math teachers to Bard for a week-long residential math immersion experience. The workshop sessions were led by Bard math professors, and instructors from the New York Math Circle and the Albany Area Math Circle. The 2011 workshop, which is scheduled for the week of July 25-29, will welcome both middle school and high school math teachers to Bard.

    The New York Math Circle was featured recently in the news on NY1, “NYC’s 24-hour Newschannel on the web”:
    Bonus Curriculum Is Greatest Common Factor For Local Math Masters (12/26/2010)

    Please let me know if you would like to participate in any of these activities!

    Thanks,

    Japheth Wood, PhD
    Mathematics Faculty
    Bard MAT Program


    —
    “If you can’t solve a problem, then there is an easier problem you can solve: find it.”
    -George Polya

  • NY1 News Article about the New York Math Circle

    The New York Math Circle was recently featured on NY1 (“NYC’s 24-hour newschannel on the web”). Check out the video here:

    Bonus Curriculum Is Greatest Common Factor For Local Math Masters


    Many thanks to TV reporter Shazia Khan for pursuing this topic! She got in touch with the New York Math Circle (NYMC) and I helped coordinate the filming. What came out in the video was filmed in late October, and shows an amalgam of the middle school and high school classes that day.
    I learned a lot about how much effort the parents make to get their children to these high-quality math enrichment classes. Some devote their entire Saturdays to shuttling their children from activity to activity, and see it as an investment in the future.
    There was one notable exception that truly impressed me – one of the middle school students takes the subway train all the way from the Bronx for class on his own. He was clearly having a great time at the class, and it completely shows in his interview. Can you spot him in the video?
    The TV news reporter, Shazia, also had a lot of fun visiting the New York Math Circle, and is interested in covering further stories in STEM (Science, Technology, Engineering and Mathematics) Eduction. You can see more of her videos here: Shazia Khan news clips
    If you know of a good story for Shazia, please leave a comment, and I’ll make sure your idea gets passed on!

  • New York Math Circle Summer Workshop 2009

    The New York Math Circle had its first summer workshop last week, July 27-30, 2009 at Union College in Schenectady, New York. This workshop for middle school math teachers was organized by Japheth Wood (Bard College) and sponsored by the New York Math Circle. Many thanks to the Math Teachers’ Circle Network for help with initial planning last summer, and to the Albany Math Circle for local assistance and encouragement.

    View pictures in Japheth’s online Photo Album.

    Workshop Math Schedule:

    Monday, July 27th

    Informal Math Activities at College Park Hall

    • Mary O’Keeffe showed us hangmath and other intriguing games.

    Tuesday, July 28th

    Walking Tour with Mary O’Keeffe:

    Peg Smith workshop, “The Five Practices”

    David Hankin workshop

    Informal Math Activities

    Wednesday, July 29th

    Walking Tour with Mary O’Keeffe (see Tuesday for details)

    Larry Zimmerman’s workshop

    Mary O’Keeffe workshop, Playing Patterns to Infinity”

    Informal Math Activities (see Tuesday for details)

    Thursday, July 30th

    Sheila Krilov workshop, “How to Start and Run a Math Team”

    Blogs:

    Mary O’Keeffe is a professor at Union College, and an organizer of the Albany Math Circle. She contributed a wonderful collection of mathematics, games, activities and ideas to the week. Her blog documents much of this:

    Jesse Johnson is an awesome and enthusiastic math teacher in NYC, and shared her notes on her blog Math Be Brave.