Japheth Wood

Author: japheth

  • Teaching Proofs

    Here’s a nice guide to what is a proof and how to write one:
    How to write proofs: a quick guide by Eugenia Cheng.

    This is brief, has a nice outline, and gives some good examples. I love the analogy made, that a good proof is like a good story: it has a beginning, middle and end. The point: a proof can be bad by having the parts in the wrong order.

    A shortcoming of the article, in my opinion, is the explanation of an induction proof. It’s a pet-peeve of mine, but I dislike seeing “n = k” and “n = k+1” in the same paragraph. Problems with teaching and learning induction is a greater issue that is partially addressed in the May/June 2008 article Some Observations on Teaching Induction by Mary E. Flahive and John W. Lee.

  • Bard MAT Spring 2012 Placements

    Here’s a map of our spring 2012 placements. The fall and winter quarters are focused on the Bronx, but in the spring, we really send them out around the city!

    View Bard MAT Spring 2012 Placements in a larger map
    Each of these schools is an amazing place to be an apprentice teacher, and they’re all extremely different learning environments and intellectual cultures.

  • 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.

  • Difficulties in Logic

    I gave my math students a Surprise Formative Assessment today. This was a quick check of their retention of a definition, and facility at forming the negation of a statement.

    Everyone did okay on the definition – it’s important to know precisely what things mean.

    The results of the negation exercise showed three areas that are slowing our progress through the IBL notes. I identify them here, and I’d love for my students to discuss them with each other in study groups, on the subway, in office hours, etc.

    First, expressing and identifying universal and existential quantifiers in informal but precise English usage. Many statements we use in mathematics state that some property is ALWAYS TRUE for EVERY object in a class of objects. These are universally quantified statements. Others state that THERE EXISTS at least one object from a class of objects for which some property holds. These are existentially quantified statements. It must be clear from your mathematical writing which sort of statement you mean!

    Second, the skill of correctly NEGATING a quantified statement. If a statement is NOT ALWAYS true, then THERE EXISTS an object for which the statement is false; a counterexample. Likewise, if THERE DOES NOT EXIST an object about which a statement is true, then the statement is false FOR EVERY OBJECT.

    Third, in a statement that involves multiple and nested quantifiers, there is something interesting and useful that happens when you negate it. We saw this today when negating the statement that “Sequence p_1, p_2, … converges to point x”, which is triply-quantified. Here it is, with parentheses to emphasize the scope of the quantifiers.

    FOR ALL open intervals S that include x, (THERE EXISTS a positive integer N, so that (FOR ALL integers n that are at least N, (p_n is in S.)))

    This has the format FOR ALL(THERE EXISTS(FOR ALL(statement))). When you negate it, it becomes: THERE EXISTS(FOR ALL(THERE EXISTS(negation of statement))).

    That is,

    THERE EXISTS an open interval S that includes x for which (THERE DOES NOT EXIST a positive integer N, for which (FOR ALL integers n that are at least N, (p_n is in S.)))

    We can move the negation further across the quantifiers:

    THERE EXISTS an open interval S that includes x so that (WHENEVER N is a positive integer, then it’s not true that (FOR ALL integers n that are at least N, (p_n is in S.)))

    pushing the negation along even further:

    THERE EXISTS an open interval S that includes x so that (WHENEVER N is a positive integer, THERE EXISTS some integer n that’s at least as big as N, for which (p_n is NOT in S.)))

    This last form is the most useful when crafting a proof by contradiction. If you assume that the sequence p_1, p_2, … does not converge to the point x, then you can immediately assume this special open interval S with some amazing special properties.

    For further background on negating statements, I recommend “The Nuts and Bolts of Proof” by Antonella Cupillari, which has a nice section entitled “How to Construct the Negation of a Statement”.

  • 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!

  • Poster Sessions

    After completing a research project or making a mathematical discovery of some sort, it’s important to communicate your results. One nice way to do this is with posters.

    If you’ve never made a poster before, it’s simply a collection of boxes that tells a story, in this case, your mathematical research. It should tell a story to someone reading it, but work even better if you’re standing in front of the poster conversing with your audience.

    Here are some photos I’ve taken of posters and poster sessions:

    There are a lot of useful websites out there about making a poster. Here is one of my favorites:
    Creating Effective Poster Presentations :: An Effective Poster

  • Creativity in Mathematics: Inquiry-Based Learning and the Moore Method

    I just viewed the recent video on the Modified Moore Method (MMO), also known as Inquiry Based Learning (IBL).

    Creativity in Mathematics: Inquiry-Based Learning and the Moore Method

    This video features interviews with MMO practitioners, researchers and students. I’ve had the pleasure of meeting many of them at various Legacy of R. L. Moore, which have been a lot of fun.

    I’ve been developing an approach to IBL in my own teaching, especially in combination with cooperative learning, and have at various times tried to write my own problem sequences. Not an easy task, but very worthwhile!

  • Project Euler

    My friend Aaron just pointed me to this site, but I think I’d seen it before, in a less-polished state.

    Project Euler (http://projecteuler.net/)

    This page has a sequence of 360 challenging math and computing problems. If you sign up for an account, you can track your progress in solving the problems. The problems are not trivial at all, so this looks like a great way to challenge yourself and grow both mathematically and in terms of programming.

    I’m curious who set up this site. Even more, I think it would be great to work as a team to solve these problems. Add this to my list of fun things to do, if I only had time…

  • 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)