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The mathematics of bee colony failurePesticide doses too weak to kill individual bees can nevertheless wipe out a colony. John Bryden, Richard Gill, Robert Mitton, Nigel Raine, and Vincent Jansen, a team of British biologists, report (Ecology Letters 16 14631469, December, 2013) their construction of a dynamical system which successfully models what happens when colonies of bumblebees (Bombus terrestris) are stressed by nonlethal doses of imidacloprid, a widely used nicotinederived insecticide. The most important feature of their model is the way it quantifies how the presence of a sufficient percentage of impaired individuals can reduce the collective health of the colony to the point where the colony dies (even though a colony can survive, and rebound, after many of its members are killed). Their dynamical system is a nonlinear variation on the predatorprey twopopulation model taught in elementary differential equations courses: $$\begin{array}{l} \frac{dS}{dt} = b(S+cI)  \frac{\mu}{(S+cI)+\phi}S  \beta S\\ \frac{dI}{dt} = \beta S  \frac{\mu}{(S+cI)+\phi}I \nu I, \end{array}$$ where $S$ is the number of healthy bees and $I$ the number of impaired ones; the paramater $c\leq 1$ "reflects the reduced contribution of the impaired bees to colony function, so that the effective operational size of the colony is $S + cI$." Other parameters: $b$ the eclosion (birth) rate, $\beta$ the rate at which bees become impaired, $\nu$ the mortality rate of infected bees. The overall per capita death rate $\mu/((S+cI)+\phi)$ varies inversely with the effective size of the colony. The authors explain: "This reflects that colonies containing more impaired bees are less good at maintaining essential colony functions (e.g. foraging, thermoregulation, defence and hygienic behaviour)." Vertical coordinate: colony size $S+I$; horizontal coordinate: the ratio $I/S$ of impaired to healthy bees. Twentyday trajectories for the $(S,I)$ dynamical system with parameter settings $b = 0.126,$ $c = 0.00332,$ $\beta = 0.12,$ $\nu = 0.00625, $ $\mu = 0.0209,$ $\phi = 0.402$ and various initial conditions. Black trajectories lead to colony survival, red trajectories to extinction. The central hyperbolic (unstable) stationary point is significant: "we predict that two colonies with similar stress levels can have divergent fates ... caused by the feedback through colony function. [R]andom events (such as other stresses or demographic noise) can put similar colonies on divergent trajectories. This will weaken the correlation between the level of a stressor and colony failure." The authors checked their model experimentally with 16 bumblebee colonies (8 fed a sugar solution laced with imidacloprid at 10 ppb, "which falls near the upper end of the field realistic range reported for nectar and pollen in agricultural crop species," and 8 controls.) Vertical coordinate: colony size; horizontal coordinate: time in days. Black data points are the mean for the eight pesticidetreated colonies; blue curve is the mean of eight runs of the $S,I$ system with initial data matching the eight colonies and parameters as above, except $\beta=0.202$. The dashed green line represents predictions generated by a different (and less satisfactory) model. Images courtesy of John Bryden. This report was picked up by Stéphane Foucart in the "Sciences" section of Le Monde online, November 7, 2013: Pesticides: les mathématiques au secours des abeilles. (Mathematics comes to help save the bees). After the Pisa reportThe Program for International Student Assessment (PISA) released their report for last year on December 3, 2013 ("Selected findings"). In particular, "The U.S. average mathematics, science, and reading literacy scores in 2012 were not measurably different from average scores in previous PISA assessment years with which comparisons can be made." The old news is that in mathematical literacy, US students are weaker than the OECD average: 9% of our 15yearolds score at proficiency level 5 or above (OECD average, 13%); 26% of them score below level 2 (OECD average, 23%). "American 15YearOlds Lag, Mainly in Math, on International Standardized Tests" was the headline for Motoko Rich's piece in the New York Times (December 3). "In the midst of increasingly polarized discussions about public education, the scores set off a familiar round of handwringing, blaming and credittaking." Rich gives several examples including quotes from teacher's union president Dennis Van Roekel: "..we as a nation haven't addressed the main cause of our mediocre PISA performancethe effects of poverty on students," and Marc S. Tucker (National Center on Education and the Economy) who blames recent efforts to reform public education by using measures like student test scores to evaluate teachers. " ... young people ... simply won't consider going into teaching." Rich also cites another familiar reaction: talk of the inevitable impact of low math (and other STEM) proficiency on the nation's economic competitiveness. He quotes Stanford economist Eric Hanushek: " ... increasingly, we have to rely on the skills of our work force, and if we don't improve that, we're going to be slipping." Is math boring, or what?Perhaps coincidentally, the New York Times ran an editorial on December 8, 2013, the Sunday after the release of the PISA data. "Who Says Math Has to Be Boring?" starts with the sentence: "American students are bored by math, science and engineering." More specifically, "Nearly 90 percent of high school graduates say they're not interested in a career or a college major involving science, technology, engineering or math ..." And essentially: "That's because the American system of teaching these subjects is broken. For all the reform campaigns over the years, most schools continue to teach math and science in an offputting way that appeals only to the most fervent students. The mathematical sequence has changed little since the Sputnik era: arithmetic, prealgebra, algebra, geometry, trigonometry and, for only 17 percent of students, calculus." "The system is alienating and is leaving behind millions of ... students, almost all of whom could benefit from realworld problem solving rather than traditional drills." The Times editorial board offers some possible remedies (numbered here for reference). 1. A More Flexible Curriculum. "While all students need a strong grasp of the fundamentals of critical thinking and problem solving, including algebra and geometry, they should be offered a greater choice between applied skills and the more typical abstract courses." For example: "Only 3 percent of graduates have taken an engineering course." 2. Very Early Exposure to Numbers. "A new study, by researchers at the University of Missouri, showed that ... [h]aving 'number system knowledge' in kindergarten or earlier  grasping that a numeral represents a quantity, and understanding the relationships among numbers  was a more important factor in math success by seventh grade than intelligence, race or income." 3. Better Teacher Preparation. "The most effective teachers have broad knowledge of their subjects. Too many lack that preparation." 78% of math teachers are certified "but that still leaves three million math students being taught by uncertified teachers." 4. Experience in the Real World. "A growing number of schools are helping students embrace STEM courses by linking them to potential employers and careers, taking math and science out of textbooks and into their lives. ... Though many of these efforts remain untested, they center around a practical and achievable goal: getting students excited about science and mathematics, the first step to improving their performance and helping them discover a career." At least a littleKonstantin Kakaes' rejoinder to the Times editorial, posted on Slate's "Future Tense" blog, December 10, 2013, has the title "Math Has to Be at Least a Little Boring." We read: "The crisis in mathematics education is, as the Times says, severe. It extends all the way to the editorial board of the newspaper, whose members do not appear to understand what mathematics is, how it is used in the sciences, or why it is important." In particular, "The Times' dichotomy between 'realworld problem solving' and 'traditional drills' does not exist. As in learning foreign languages, repetitive drills enable students to master techniques ... which can then be used to solve problems in the real world, and to develop more mathematical sophistication, which can then be bolstered by using new mathematical concepts in the real world. This is true in arithmetic, and also in algebra, geometry, calculus." And finally: "The Times' most misguided belief may be the insistence that the reason we need better math education is to train a new STEM workforce. The real reason we need better math education is to equip citizens to understand the world they live in." What the Times doesn't know about math instructionA rejoinder from a different direction is given by Daniel Willingham, Professor of Psychology at the University of Virginia, in his December 9 blog posting. "[T]he suggestions in the [Times] editorial showed a striking naiveté about what it will take to improve." Referring to the numbered items above, he says: "The editorial ignores the fact that 1 and 4 will be meaningless without 2 and 3. And it grossly underestimates the difficulty of implementing 2 and 3." Willingham makes the point that it's American students' weakness in conceptual understanding that makes math boring. "What could be more boring than executing algorithms you don't understand?" His remedy: "This conceptual understanding ought to start in preschool with ideas like cardinality and equality. 'Very early exposure to numbers' is not going to do it. ... That means putting activities into preK (e.g., games and puzzles that emphasize the use of space) that will provide a foundation for conceptual understanding so that firstgraders will be in a better position to understand what they are doing." And in general: "We need to pay much closer attention to preschool and to early elementary grades. That will entail developing methods of helping children understand the conceptual side of math methods we now lack. It will also entail professional development to train teachers in the conceptual side of math." Edward Frenkel on math and borschtAnother interview in connection with the publication of "Love and Math," this time in the Prospero (Books, arts and culture) blog on The Economist's website (December 5, 2013). The interviewer (identified only as "A.B.") asks, for example, "Does maths exist without human beings to observe it, like gravity? Or have we made it up in order to understand the physical world?" Frenkel: "I argue, as others have done before me, that mathematical concepts and ideas exist objectively, outside of the physical world and outside of the world of consciousness. We mathematicians discover them and are able to connect to this hidden reality through our consciousness. If Leo Tolstoy had not lived we would never have known Anna Karenina. There is no reason to believe that another author would have written that same novel. However, if Pythagoras had not lived, someone else would have discovered exactly the same Pythagoras' theorem. Moreover, that theorem means the same to us today as it meant to Pythagoras 2,500 years ago." Later on in the interview: "Mathematics is invading our world more and more and it communicates timeless, persistent and necessary truths which transcend time and space. The Langlands Programme should be as familiar to us as the theory of relativity." About the borscht: the interviewer then asks "What is the Langlands Programme?" In the course of his answer Frenkel is led to describe harmonics, and Langlands' discovery of "some secret patterns relating to numbers, which could be expressed using harmonics," which brings us to "similar and surprising patterns in geometry and quantum physics, in particular the electromagnetic duality," itself explained "in my book" using an analogy based on the recipe for borscht "which is a favourite soup in my home country, Russia." Would Math Exist Without Us?, drawn from this interview, was posted on "The Daily Dish" on December 9, 2013. Frenkel's epistemology turns out to have theological implications.
Tony Phillips 
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