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This month's topics:

Chicks and the number line

"Chickens Agree: Left Means Less; Right Means More" was the heading on a New York Times report by Erica Goode (Science section, January 30, 2015). As Goode reports: " ... a team of Italian researchers has found that newborn chicks, like humans, appear to map numbers spatially, associating smaller amounts with the left side and larger amounts with the right side." The researchers are Rosa Rugani, Giorgio Vallortigara, Konstantinos Priftis and Lucia Regolin from the universities of Padova and Trento; their work, "Number-space mapping in the newborn chick resembles humans' mental number line," appeared in the issue of Science dated that day. As the authors explain in their Abstract, "Humans represent numbers along a mental number line (MNL), where smaller values are located on the left and larger on the right. ... In our experiments, 3-day-old domestic chicks, once familiarized with a target number (5), spontaneously associated a smaller number (2) with the left space and a larger number (8) with the right space. The same number (8), though, was associated with the left space when the target number was 20. Similarly to humans, chicks associate smaller numbers with the left space and larger numbers with the right space." The experimental set-up, as described in their article:

chick with 5

chick with 2

chick with 8

"Three-day-old domestic chicks (Gallus gallus) learned to circumnavigate a panel to reach a food reward. At training [top], the panel depicted a target number of elements (5 in experiment 1). At testing, we presented each chick with two panels, one on its left side and one on its right side, both depicting the same number of elements, ... a different number from the target. Each chick underwent two tests: a small number test [center] in which the panels depicted a number of elements smaller than the target (2 in experiment 1) and a large number test [bottom] in which the number of elements was larger than the target (8 in experiment 1)." Images © Rosa Rugani, University of Padova, used with permission.

The results, as suggested by the images: when used to finding food behind "5 dots" and presented with two "2 dots" panels, chicks chose to look behind the left one 70.67% of the time; whereas between two "8 dots" panels, they chose the right one 71% of the time. The experiment was repeated with "20 dots" as the target number; then between "8 dots" panels they chose the left (70%) and between "32 dots" panels they chose the right (77.5%).

In their conclusion the authors remark: "... number-space mapping is implemented in humans through a topographical representation in the right posterior parietal cortex. ... we suggest that a similar predisposition to map numbers onto space is embodied in the architecture of [other] animal neural systems."

Long-term effects of gender bias in math education

In January, 2015 the National Bureau of Economic Research released a working paper entitled "On the origins of gender human capital gaps: short and long term consequences of teachers' stereotypical biases." The authors are Victor Lavy (Economics, Warwick) and Edith Sand (Economics, Tel-Aviv). The article was picked up by the New York Times: "How Teacher Biases Can Sway Girls From Math and Science" by Claire Cain Miller, February 7, and later in Slate: "Teachers Give Girls Better Grades on Math Tests When They Don't Know They Are Girls" by Amanda Marcotte on February 10.

Levy and Sand studied three cohorts of 6th grade students in Tel-Aviv, 2002-2004: 3011 students in all, in 112 classes, and then tracked them all the way through high school. They measured 6th grade teachers' gender-biased behavior by comparing their average marking of boys and girls in a "non-blind" classroom exam to the respective means in a "blind" national exam marked anonymously. "We show that there is large variation within schools in this measure, and that it has a significant effect on the academic achievements of both genders during middle school and high school in math, science and language and also on the choice of the level of math and science study programs in high school. These high stake choices determine practically whether a student will be able to meet requirements of admission to science and math studies at universities later in life."

For the Times, Miller got a quote from Lavy: "The most surprising and I think important finding in the paper is that a biasing teacher affects the work choices students make and whether to study math and science years later." In Slate, Marcotte reminds us of the so-called Pygmalion effect: " ... it's shown up in study after study: Just by believing that a student has an aptitude for something, the teacher makes it more likely to be true. And so it matters quite a bit that teachers have an unconscious belief that boys are better at math than girls, and we may be seeing the results in the many girls who might otherwise enjoy math getting discouraged and dropping out."

Picture$\rightarrow$text (and language) translation via linear algebra,

brought to you by Google. MIT's Technology Review has a rubric: "Emerging Technology from the arXiv" [arXiv is the site for posting one's latest research in preprint form]. On January 1, 2015 they ran "Best of 2014: How Google "Translates" Pictures into Words Using Vector Space Mathematics." The arXiv piece in question is "Show and Tell: A Neural Image Caption Generator," by Oriol Vinyals and three other Google engineers. Technology Review explains the new developments mainly in terms of an earlier arXiv item, "Exploiting Similarities among Languages for Machine Translation," by Tomas Mikolov, Quoc V. Le and Ilya Sutskever, all also at Google, which they had picked up in September 2013: "How Google Converted Language Translation Into a Problem of Vector Space Mathematics.". Here TR can give an idea of the process.

  • "The new approach ... relies on the notion that every language must describe a similar set of ideas, so the words that do this must also be similar. For example, most languages will have words for common animals such as cat, dog, cow and so on. And these words are probably used in the same way in sentences such as 'a cat is an animal that is smaller than a dog.' ...
  • This is an important clue. The new trick is to represent an entire language using the relationship between its words. The set of all the relationships, the so-called 'language space', can be thought of as a set of vectors that each point from one word to another. And in recent years, linguists have discovered that it is possible to handle these vectors mathematically. For example, the operation ${\tt king} - {\tt man} + {\tt woman}$ results in a vector that is similar to ${\tt queen}$.
  • It turns out that different languages share many similarities in this vector space. That means the process of converting one language into another is equivalent to finding the transformation that converts one vector space into the other."

TR continues: "Now [Google] is using the same machine learning technique to translate pictures into words. The result is a system that automatically generates picture captions that accurately describe the content of images."

Tony Phillips
Stony Brook University
tony at math.sunysb.edu

Math Digest Math Digest
On Media Coverage of Math

Math Digest includes posts throughout each mont, with summaries of math stories and unique insights (and occasionally videos, interviews and podcasts) on math-related topics recently covered by the media.

Recently posted:

On Cédric Villani, by Claudia Clark

Villani

In this article, writer and speaker Alex Bellos writes at length about mathematician Cédric Villani (above), director of the Institut Henri Poincaré. Villani is a 2010 Fields Medal winner for his work with the Boltzmann equation, a partial differential equation that “describes how a gas disseminates by considering the likelihood of any of its molecules being in any particular spot, with a particular speed, at a particular time.” However, this is not an article for mathematicians only: much of this article is an accessible introduction to some of the concepts and people instrumental in the development of calculus. For example, Bellos describes the method by which Archimedes calculated the area bounded by a line and a parabola, noting that “Archimedes was the earliest thinker to develop the apparatus of an infinite series with a finite limit.” Bellos also shows how the infinitesimal can be used to find the gradient of a tangent, or to calculate instantaneous speeds, in discussing the work of Isaac Newton. Bellos wraps up his article with a discussion of the importance of the Boltzmann equation and why Villani is interested in it: “Often in PDEs,” says Villani, “you have tension between various terms. The Boltzmann equation is the perfect case study because the terms represent completely different phenomena and also live in completely different mathematical worlds.” (Photo by Sandy Huffaker.)

See "Seduced by calculus," by Alex Bellos. Cosmos, 16 February 2015.

--- Claudia Clark

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