Pigeon math in the New York Times
"Stumped by Math? Ask a Pigeon for Help" was the headline in the December 23 2011 issue of the paper. James Gorman reporting on a report published that day (Science 334 1664: "Pigeons on Par with Primates in Numerical Competence" by Damian Scarf, Harlene Hayne, and Michael Colombo, all at the University of Otago, New Zealand): "Pigeons have now shown that they can learn abstract rules about numbers, an ability that until now had been demonstrated only in primates." In the reported experiments, pigeons were trained to peck at images of one, two and three objects in increasing order. "It may have taken a year of training, with different shapes, sizes and colors of objects, ... , but all that work paid off when it was time for higher math." From the Science report: "Subjects were then tested on pairs of numerosities drawn from the range of one to nine. The pairs were one of three types: familiar-familiar (F-F) pairs contained two numerosities drawn from the training range, familiar-novel (F-N) pairs contained one trained numerosity and one novel numerosity drawn from the values four to nine, and novel-novel (N-N) pairs contained two novel numerosities." The pigeons performed above chance on all three tests. The team compares pigeons' performance with that previously recorded for monkeys, and speculates about the evolutionary history of this ability: "Our results suggest that, at least with respect to numerical competence, pigeons are on par with primates and are well perched [sic] to inform us about the selection pressures and neural structures required for abstract numerical cognition."
Of ants and basketball players
Science for November 11, 2011 picked up (under their "Random Sample" rubric) a press release from Arizona State University and ran the item as "When there are too many ants, try basketball." The reported work is a collaboration between Jennifer Fewell (area of research: the behavioral ecology and evolution of social insects) and Dieter Armbruster (dynamical ststems), both at ASU. Fewell "... is interested in how hundreds, even thousands of ant workers coordinate their activities to keep a nest thriving." Since the numbers were too big for useful modeling, she joined with Armbruster in the study of a sample of interactive populations of 5 individuals: basketball teams, "team dynamics as a substitute for ant dynamics;" they were able to tease out quantitative measures of distributed leadership that gave winning strategies for the teams and were akin to aspects of the collective behavior of ants. They recruited two undergraduates to watch videos of the first round of the NBA 2010 playoffs, and to record "who passed to whom and how often the ball traveled among various players." On the basis of this information they constructed passing networks, as above, for each team, and used them to generate scores for Passing Entropy and Centrality. The entropy presumably measures how unpredictable the next pass will be; centrality how star-like the passing diagram appears.
As Armbruster says in the press release, "In general the teams with high entropy win against teams with low entropy." In fact, in the 2010 series between the Lakers and the Thunder, the Lakers won 4 games to 2, and went on to win the championship. The Mavericks were eliminated in the first round.
Diagrams, left: Passing patterns for the Dallas Mavericks, the Oklahoma City Thunder and the Los Angeles Lakers; the nodes are players, labeled by position (PG = Point Guard, etc.); the links are weighted proportionally to the amount of traffic between their endpoints. The red links represent the "backbone" of the pattern: they include the highest weight links, up to 50-60% of the total weight. Note how the combinatorial richness of the Lakers' backbone shows up in the numerical scores: lower centrality, higher entropy. Images courtesy of Dieter Armbruster and Jennifer Fewell.
The Khan Academy in the math classroom
The front page of the New York Times business section for December 8, 2011 showed photos of a teacher and a student at the Summit school in San Jose: "Jesse Roe ... can use the teaching software to monitor the math progress of students like Cheyenne Grant, 14, right." The Times reporter, Somini Sengupta, explains that the software in question is an experimental hybrid: material from the Khan Academy's playlist of short, ultra-focused math lessons is blended with traditional classroom instruction. The students are all working at various Khan mini-lessons on computers networked to the teacher's; the teacher can check individual students' progress and intervene if a student seems to be stuck or to have become distracted. Sengupta also gives us a glimpse into the mind of Salman Khan, the 35-year old innovator who "has become something of an internet sensation" with his YouTube-housed lectures (which "got their start six years ago when Mr. Khan needed a way to help a cousin catch up on high school math." ) She tells us that, when Salman was in school, "Math became his passion. ... He came to see math as storytelling. 'Math is a language for thinking,' he said, 'as opposed to voodoo magical incantations where you have no idea where they're coming from.'"
Stony Brook University
tony at math.sunysb.edu