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This month's topics:

"Mathematics of the spheres"

The most efficient way to pack equal-sized spheres in three dimensions involves placing them in layers along a hexagonal tiling of the plane and fitting adjacent layers together so that each sphere in one layer fits into the dimple determined by 3 adjacent spheres in the layer below. In these arrangements each sphere touches 12 others, and the average density, or packing fraction (volume of spheres)/(volume of ambient space) is approximately 0.74. Thomas Hayes' 1998 proof that these packings are in fact optimal (the Kepler Conjecture) is now generally accepted. But suppose spheres are dumped into a container without being carefully stacked. What density can such a random packing expect to achieve? Experimentally it ranges between 55% (random loose packing) and 64% (random close packing). Clearly friction will play a role, but how? In the May 29 2008 Nature, a CCNY-Fortaleza team (Chaoming Song, Ping Wang, Hernán A. Makse) use statistical mechanics to describe a phase space for packings, and to give an intelligible model for these questions.

The phase space for sphere packings. For each value of the friction coefficient the dashed line represents the possible packing fractions that can be realized by a stable sphere packing. The mechanical coordination number, the average number of adjacent spheres that contribute to holding a given sphere in place, varies monotonically with friction. This image is from Francesco Zamponi's "News and Views" analysis of the Song-Wang-Makse paper in the same issue: Nature 453 606-607, and is used with permission.

"Mathematics of the spheres" is the way these items were characterized in the Nature "Editor's Summary."

E8 in The New Yorker

Geometry of the mental number line

It has been established that the number line, canonically orientated in a left-to-right manner, is a psychological reality. But what about distances? An report in Science, May 30 1998, is entitled "Log or Linear? Distinct Intuitions of the Number Scale in Western and Amazonian Indigene Cultures." The four-person France-USA team of researchers, led by Stanislas Dehaene, started from the observation that children, although they unhesitatingly place numbers on a line with the smaller to the left, "do not distribute the numbers evenly, [but] instead devote more space to small numbers, imposing a compressed logarithmic mapping." The shift from logarithmic to linear mapping "occurs later in development, between first and fourth grades, depending on experience and the range of numbers tested."

To determine to what extent this "mental revolution" is culturally determined by numbering systems, etc., the team "gathered evidence from psychological experimentation with the Mundurucu, an Amazonian indigene culture with little access to education." The Mundurucu are known to "entertain sophisticated concepts of number and space" although "their lexicon of number words is reduced and they have little or no access to rulers, measurement devices, graphs or maps." More specifically, the Mundurucu number lexicon, as reported elsewhere, has names for 1, 2, 3, 4, and 5 (but even those are used in a "casual manner"). In this study they are reported to also have ways of expressing 7 ("one handfull, two on the side" i.e. 5 + 2), 9 ("one handfull, four on the side"), and 10 ("two handfulls"), but only those. The psychological testing involved presenting subjects with a number line:

The Mundurucu number line, as used in tests of their conceptual spatial distribution of numbers. The segment is labeled at the left end with a "dot symbol" for 1, at the right end with the dot symbol for 10. A cursor can be moved to any intermediate position. After illustration in Science 320 1218.

Subjects were then presented with number stimuli; according to the test these could be

• "dot symbols" analogous to the limit markers, with 1, 2, ..., or 10 dots
• a sequence of 1, 2, ..., or 10 tones
• a spoken number-word in Munducuru, which could be pũn ma (1), xep xep (2), ebapũg (3), ebadipdip (4), pũg põgbi (5), or expressions for 7, 9, 10
• a spoken number-word in Portuguese: um, dois, ..., deis.

Some of the results:

• "Logarithmic thinking persists into adulthood for the Mundurucu, even for very small numbers in the range from 1 to 10, whether presented as dots, tones, or spoken Mundurucu words."
• "The most educated Mundurucu eventually understand that linear scaling, which allows measurement and invariance over addition and subtraction, is central to the Portuguese number word system. At the same time, they still do not extend this principle to the Mundurucu number words, where perceptual similarity between quantities is still seen as the most relevant property of numbers."

As they mention, "it is clear that the mental revolution in Western children's number line does not result from a simple maturation process."

"Girls = Boys at Math"

"Gender Similarities Characterize Math Performance" was the more scholarly title for an Education Forum piece in the July 25 2008 Science. The authors, a Berkeley-Wisconsin team of five led by Janet S. Hyde, analyzed the math performances of some 7 million students (grades 2 through 11) in a representative 10 states, assembled in the context of the No Child Left Behind (NCLB) legistation. Their main conclusion: "Our analysis shows that, for grades 2 to 11, the general population no longer shows a gender difference in math skills..."

The authors also focused on the upper tail of the distribution. Here it turns out that the large (and unexplained) difference in variance, (at least 10% larger for males, at every level), has a significant impact. For example in Minnesota, a boy is twice as likely as a girl to place in the 99th percentile on the tests. They remark: "Gender differences in math performance, even among high scorers, are insufficient to explain lopsided gender patterns in participation in some STEM fields." STEM = Science, Technology, Engineering, Mathematics; they calculate that a 2 to 1 ratio of males in the top percentile could predict a 67%-33% split in STEM careers, while the actual proportion in engineering PhD programs is 85%-15%.

When the authors attempted to measure different performances on more complex questions, they made a discouraging discovery: after sorting questions on a 4-point scale (1 = recall, ... , 4 = extended thinking) they reported: "For most states and most grade levels, none of the items were at levels 3 or 4. Therefore, it was impossible to determine whether there was a gender difference in performance at levels 3 and 4." Their comment: "With the increased emphasis on testing associated with NCLB, more teachers are gearing their instruction to the test. If the tests do not assess the sorts of reasoning that are crucial to careers in STEM disciplines, then these skills may be neglected in instruction, putting American students at a disadvantage relative to those in other countries..."

The authors also addressed (in the Supplemental Online Information) the discrepancy in SAT-M scores (male average 533, females 499), and explained it as a sampling artifact, since many more females take the test than males. In states (Colorado, Illinois, 2002) where the ACT test was administered to all high school students, "The gender gap in scores disappeared ... and, in fact, a slight gap favoring females emerged."

"Girls = Boys at Math" was the headline in ScienceNOW Daily News for July 24, 2008. Or, as they also put it, "Barbie was wrong."

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