Mathematics education in the New York Times

"Math Under Common Core Has Even Parents Stumbling" (Mokoto Rich, June 30, 2014). "Across the country, parents who once conceded that their homework expertise petered out by high school trigonometry are now feeling helpless when confronted with firstgrade work sheets." Rich tries to give both sides of the picture:

"The new instructional approach in math seeks to help children understand and use it as a problemsolving tool instead of teaching them merely to repeat formulas over and over. They are also being asked to apply concepts to reallife situations and explain their reasoning. This is partly because employers are increasingly asking for workers who can think critically and partly because traditional ways of teaching math have yielded lackluster results. In global tests, American students lag behind children in several Asian countries and some European nations, and the proportion of students achieving advanced levels is low. Common Core slims down curriculums so that students can spend more time grasping specific mathematical concepts."

" ... textbooks and other materials have not yet caught up with the new standards, and educators unaccustomed to learning or teaching more conceptually are sometimes getting tonguetied when explaining new methodologies. ... Tensions over the Common Core have been heightened because the standards are tied to new standardized tests being introduced in many states. Teachers are fretting that their performance ratings will increasingly depend on how their students perform on these tests. ... Some educators said that with the Common Core's focus on questioning lines of reasoning and explaining answers, the new methods were particularly challenging for students with learning disabilities, or those who struggle orally or with writing.
Finally he quotes John White, Louisiana state superintendent of education: "This is a shift for an entire society. No one should be under any illusion that it's going to take just a year or two to rethink the way that we teach mathematics, because it is really challenging."

"Q: Why Does Everyone Hate the New Math? (A: Because no one understands it, not even the teachers)," by Elizabeth Greene, is the cover story in the Magazine, July 27, 2014. The actual title inside is "(New Math) $$ (New Teaching) = Failure," whereas the title online is "Why Do Americans Stink at Math?" This long and richly detailed article sets up the background: "As a nation we suffer from ... innumeracy the mathematical equivalent of not being able to read" (illustrated with gruesome anecdotes). It touches lightly on the history: "... in the 1960s, when schools gripped by a postSputnik inferiority complex unveiled an ambitious 'new math,' only to find, a few years later, that nothing actually changed. In fact, efforts to introduce a better way of teaching math stretch back to the 1800s. The story is the same every time: a big, excited push, followed by mass confusion and then a return to conventional practices." And then focuses on the main problem, as Greene sees it: we have now discovered better ways of teaching math, but we have not been able to train teachers to use them. "Most policies aimed at improving teaching conceive of the job not as a craft that needs to be taught but as a naturalborn talent that teachers either decide to muster or don't possess. Instead of acknowledging that changes like the new math are something teachers must learn over time, we mandate them as 'standards' that teachers are expected to simply 'adopt.' We shouldn't be surprised, then, that their students don't improve."
Greene suggests that we can learn from Japan, which "was able to shift a country full of teachers to a new approach." It took time and was not easy (parents complained), even though there "teachers teach for 600 or fewer hours each school year, leaving them ample time to prepare, revise and learn. By contrast, American teachers spend nearly 1,100 hours with little feedback."

[It is understandable but unfortunate that neither of these articles addresses the financial, attitudinal, institutional and political obstacles to improving the education and status of school teachers in the United States. TP]

A smaller scale, onefamilyatatime approach is advocated by Jordan Ellenberg (professor at math at Wisconsin) in an OpEd published on July 25, 2014: "Don't Teach Math, Coach It.". "[H]ow can we encourage kids in a difficult task like math without doing so in a way they'll come to resent?" Ellenberg, whose 8yearold son C.J. is a baseball fanatic, learned to imitate C.J.'s Little League coaches. "They don't get mad and they don't throw you off the team. They don't tell you that you stink at baseball, even if you do  they tell you what you need to do to get better, which everybody can do." For Ellenberg, coaching involves playing mathematical games with his children, and finding other mathrich games for them: chess, Monopoly, Rubik's Cube, Rush Hour, Set, Dragonbox. "Every one of these games shows kids mathematical ideas in a spirit of play, which is a big and often hidden part of the true spirit of math."
"These games are difficult, but also, for many kids, kind of addictive. Which means they also teach sitzfleisch, the ability to focus on a complicated skill for the length of time it takes to master it. Math needs that. (Baseball does, too.) It fits with the research of the psychologist Carol Dweck, which suggests that mentors should emphasize effort over native ability. We can't really teach kids to do things; we can only teach them to practice things."
New equilateral polyhedra
A paper in PNAS (February 25, 2014) by Stan Schein and James Gayed (UCLA) explains their discovery of a fourth class of convex equilateral polyhedra with polyhedral symmetry. (Equilateral means that all the edges have the same length; polyhedral symmetry means that each of them has the symmetries of one of the Platonic solids: tetrahedron, octahedron or icosahedron). The first three classes are the Platonic solids themselves, the 13 Archimedean solids and "the two rhombic polyhedra reported by Johannes Kepler in 1611" the rhombic dodecahedron and triacontahedron. As in Kepler's examples, the faces of the polyhedra in this fourth class (the convex, equilateral Goldberg polyhedra) are polygons that are planar, equilateral but not in general equiangular.
The authors start with Goldberg's 1937 method for generating combinatorial "cages" starting from the three Platonic solids with triangular faces: tetrahedron, octahedron and icosahedron. By overlaying in different ways an equilateral triangle on a regular hexagonal grid they obtain three families of hexagonal patterns in the triangle. One example is the pattern they call (3,0). They print this pattern on each (triangular) face of one of the solids; for example the isocahedron. Where segments of the pattern meet across a polyhedron edge, they are replaced by straightline segments in 3space.
Construction of a cage. The (3,0) pattern is printed on each face of an icosahedron; the broken segments crossing edges are replaced by straightline segments in 3space, shown thickened and in grey here. Images from PNAS 111 no. 8, 29202925, used with permission.
In this example pulling the icosahedron faces slightly apart will make the grey lines as long as the others, so the cage becomes equilateral (image below). But if we try to make each of the hexagons and pentagons equiangular as well, solid geometry forces many of those faces to be nonplanar, so the cage is not a convex polyhedron. The authors pinpoint and quantify the problem as follows.

In a geometric convex polyhedron (requires planar faces) suppose an edge, like the red edge in this illustration, joins two trivalent vertices where the face angles are $\alpha, \beta, \gamma$, and $\alpha', \beta', \gamma'$. Note that $\alpha$ and $\alpha'$ are on the faces opposite the edge.
The dihedral angle $A$ along the edge (the angle in space between the two planar faces it joins) can be calculated at either vertex from the equation $$\cos A = \frac{\cos\alpha\cos\beta\cdot\cos\gamma}{\sin\beta\cdot\sin\gamma} = \frac{\cos\alpha'\cos\beta'\cdot\cos\gamma'} {\sin\beta'\cdot\sin\gamma'}.$$ (Paul Kunkel has posted a derivation of this equation from the law of cosines).

Schein and Gayed define the dihedral angle discrepancy (DAD) along an edge of the cage to be the difference of the "dihedral" values calculated from the face angles at one end and at the other. If all the polygons in the icosahedral cage are equiangular, then for the edge joining a pentagon to a hexagon all the adjacent face angles are $120^{\circ}$ except for the one in the pentagon, which is $108^{\circ}$. The dihedral calculation along that edge gives $180^{\circ}$ at the 3hexagon vertex, and $\arccos(.7453..) = 138.2^{\circ}$ at the other. This shows that the two hexagons bordered by that edge cannot both be planar. Furthermore the DAD gives a measure of the nonplanarity, namely $41.8^{\circ}$ in this instance.
The authors finish their construction by proving that if the hexagon angles are allowed to vary (while preserving overall icosahedral symmetry) there are just enough degrees of freedom to make all of the DAD's zero and therefore to produce a unique symmetric, convex, equilateral polygon of the same combinatorial type as the cage. In the final count there is one such polyhedron built on the tetrahedral model, one on the octahedral, and a countable infinity of examples based on the icosahedron.
The equilateral cage is not convex: this is apparent along the edges. But there is a (unique) way to adjust the angles in the hexagons so as to make it a convex polyhedron while keeping the full icosahedral symmetry. Larger images: equilateral cage, convex polyhedron. Thanks to Stan Schein for these images.
This work was picked up by Dana Mackenzie in Science News, March 22, 2014, with the title: "Goldberg variations: New shapes for molecular cages."
MultimillionDollar Minds
"The MultimillionDollar Minds of 5 Mathematical Masters," by Kenneth Chang, ran in the New York Times on June 23, 2014. The piece reports the award of $3 million "Breakthrough" prizes to five mathematicians: Simon Donaldson (my colleague at Stony Brook), Maxim Kontsevich (IHÉS), Jacob Lurie (Harvard), Terence Tao (UCLA) and Richard Taylor (IAS). The prizes are financed by Yuri Milner, "a Russian who dropped out of graduate studies in physics and became a successful investor in Internet companies like Facebook," and Mark Zuckerberg, the founder of Facebook. As Chang tells us, "The Breakthrough Prize in Mathematics is the latest effort in Mr. Milner's crusade to make science lucrative and cool in a society that much more often celebrates athletes, entertainers, politicians and business tycoons." (There are also prizes in Physics and in the Life Sciences). Chang does his best to give some idea of the mathematics being rewarded.

"The accomplishments of Dr. Donaldson include using a mathematical theory originally developed for particle physics to study and classify possible shapes of fourdimensional space."

"The citation notes a wide swath of mathematical fields where Dr. Kontsevich repeatedly bumped into unexpected connections. For example, about 15 years ago, he collaborated on what looks like a simple procedure called interval exchange transformations, which is essentially like taking a piece of rope, cutting it into pieces and shuffling them together in a different order. The mathematics of cutting and reshuffling turns out to be complex, and recently reappeared in a new area of abstract algebra used in some theoretical physics models."

"Dr. Tao has worked on fundamental problems involving prime numbers and has examined the equations of fluid flow, seeing if there might be solutions with black holelike singularities where the fluid velocity turns infinite."

"Dr. Taylor, who first became known for helping fill a gap in the proof of Fermat's Last Theorem, has mapped out unexpected connections between algebra and symmetries in geometry."
But he tosses in the towel for Lurie: "Dr. Lurie was cited for cuttingedge advances in esoteric fields like 'higher category theory' and 'derived algebraic geometry'."
Tony Phillips
Stony Brook University
tony at math.sunysb.edu