## This month’s topics:

### Algebra in California.

A *Wall Street Journal* editorial “San Francisco Can’t Do the Math” (April 28, 2023) with subtitle “A battle over equity, achievement—and eighth-grade algebra” reports on a public petition, proposed by the *SF Guardians* (a group made up of “parents, teachers and concerned citizens”) asking the San Francisco Unified School District board to restore Algebra I to the eight-grade curriculum. This local piece of news earns mention on the editorial page of a national newspaper because Algebra I in 8th grade has become a symbol of a problem that transcends San Francisco and California: how to balance the need for well-prepared and well-supported students to receive appropriately challenging mathematics instruction with the need to close achievement gaps. In today’s United States this question has become yet another bitterly divisive issue. As the title suggests, the *WSJ* editorial board supports the petition.

How did this particular course become so important? Algebra I, Geometry, Algebra II, Pre-Calculus, Calculus is a 5-year sequence, so moving Algebra I to ninth grade jeopardizes the current ideal of high-school mathematics education, which culminates in a calculus course. (This is a fairly recent development. A kind of achievement inflation has made it seem necessary for a good student to take calculus in high school. In fact, building a solid foundation in algebra, geometry and trigonometry might well be a much better use of that student’s time.)

The *WSJ *‘s mention of “equity” alludes to the subtext that has made this question especially thorny. The SFUSD’s decision was in part motivated by statistics like the one graphed here which showed a large general failure rate for 8th-graders in Algebra I, and an even worse picture for Black and Latinx students. On the other hand, any change that diminished access to calculus would especially impact the 30 percent of SFUSD students identified as Asian, since nationwide 46 percent of that group take calculus in high school.

What actually happened? The SFUSD’s decision to move Algebra I from middle school to high school was made in 2014. One March 2023 preprint dives into the records of 23,309 district students and tabulates the consequences. As the authors (Elizabeth Huffaker, Sarah Novicoff, and Thomas S. Dee, all of Stanford University) explain, math classes the year after the policy was implemented were significantly more diverse, with almost all ninth graders now enrolled in Algebra I and the vast majority of tenth graders in Geometry. Enrollment in AP Calculus initially fell from 30% to 24%, with losses mostly coming from the group of Asian/Pacific-Islander students. These reductions, they report, eased off as students learned to take advantage of “acceleration options.” And the policy did not grant Black and Hispanic students more access to more AP classes, leaving large racial gaps in place.

### Integer sequences in *The New York Times*.

Siobhan Roberts has an article in the *Times* for May 21, 2023, about the 50th anniversary of OEIS, the On-Line Encyclopedia of Integer Sequences. The OEIS was the brainchild of Neil J. A. Sloane (for many years at Bell Labs); he described its genesis in My Favorite Integer Sequences (2002). Here’s how it can be useful. I was once counting the number of mazes of a certain type that had $n$ path segments stacked atop one another ($n$ levels). With one, two or three levels there is only one; there are 2 with four levels, 3 with five levels, 8 with six: I was generating an *integer sequence.* It starts $(1, 1, 1, 2, 3, 8)$ and continues $(14, 42, 81, 262, \dots )$. The numbers grew very rapidly, as did the length of my counting process: the calculation for $n$ levels took on the order of $n^2$ as many steps as the calculation for $n-1$. I soon got stuck. But an article in the *New York Times* about Neil Sloane and his Encyclopedia caught my eye. Part of Sloane’s collection, reportedly, was the sequence of ways of folding a strip of $n$ stamps. For my maze-obsessed mind, counting folded strips of stamps was clearly related to counting my type of mazes. But the stamp-folder (John E. Koehler) had found a shortcut that made the number of steps grow at a much lower rate (although still exponentially). This insight (details here) allowed me to continue my calculations up to $n=22$, where there are $73,424,650$ mazes. The maze sequence became a new entry in the encyclopedia (it is now sequence A005316 and has been extended up to level 44, a 19-digit number).

Roberts gives some examples of sequences in the collection. The odd numbers $1, 3, 5, 7, \dots$, the evens $2, 4, 6, 8, \dots$, the prime numbers $2, 3, 5, 7, 11, \dots$ (no divisors except themselves and 1) and the Fibonacci sequence $1, 1, 2, 3, 5, 8, 13, \dots$ (each number is the sum of the two preceding ones). These sequences have purely mathematical definitions. The OEIS has other sequences, like the “eban” numbers $2, 4, 6, 30, 32, 34, 36, 40, \dots$ (numbers whose English names do not use the letter “e”), where the defining criterion may depend on the language we’re speaking or on the shape of the numerals we choose to represent numbers; these sequences are as much riddles or jokes as mathematics.

One sequence mentioned in the article looks like it might be of the latter type. It’s John Conway’s “Look and Say” sequence, which starts $1, 11, 21, 1211, 111221, 312211, \dots$. Here for example $312211$ is a verbal description of the preceding number $111221$, which reads as “three 1s, two 2s, one 1.” But this is not a joke: “Look and Say” turns out to have a mathematical life of its own (for example, no numeral bigger than $3$ ever appears). For more details, I recommend the YouTube video where the master himself talks about the sequence.

### Coccolith geometry.

Coccolithophores are tiny algae that play a large role in regulating the concentration of carbon dioxide (CO$_2$) in the oceans. *Emiliana huxleyi*, pictured above, is one of the most important members of the family. As plants, they participate in *photosynthesis*, using energy from sunlight to synthesize organic matter from water and CO$_2$. This has the result of taking dissolved CO$_2$ out of the ocean. On the other hand, the production of their calcium carbonate shells has the side effect of *releasing* CO$_2$ into the ocean. The notation used for the relative strength of these two effects is PIC:POC; it represents the ratio of the *particulate inorganic carbon* (present in the shell) and the *particulate organic carbon* (absorbed during photosynthesis) produced by these organisms. If PIC:POC $>1$, they are releasing more CO$_2$ than they absorb, and coccolithophore growth is a net CO$_2$ source; if PIC:POC $<1$, it is a net CO$_2$ sink.

“Estimating Coccolithophore PIC:POC Based on Coccosphere and Coccolith Geometry” was published in *JGR Biogeosciences*, April 18, 2023. The authors, Xiaobo Jin and Chuanlian Liu (Tongji University, Shanghai) collected samples of *E. huxleyi* and another coccolithophore at locations in the South China Sea. It had been suggested, based on work with laboratory cultures of these algae, that the PIC:POC ratio was correlated with a geometric property of the individual coccoliths. Namely, the ratio of its average thickness to its length, called the *lateral coccolith aspect ratio* $A_L$. To verify this suggestion for a large sample of wild-caught specimens, the authors developed a method for measuring both PIC:POC and $A_L$ from photographs taken with a conventional (light) microscope.

Jin and Liu exploit the light microscope’s focusing sensitivity: only the material in or very close to the plane of focus gets recorded. Since calcium carbonate is translucent, focus on the equatorial plane of a coccolithosphere (below, left) gives an image allowing the measurement of the inside and outside diameters of the shell. The inside diameter of the shell is the diameter of the included organic cell. From this number one can calculate the volume of the cell, and estimate the POC. The difference between the outside and inside radii is the thickness of the carbonate shell, so the volume of the inorganic part can be calculated, leading to an estimate of the PIC. On the other hand, focus on the top or the bottom will give useful images of individual coccoliths.

For the measurement of $A_L$, the ratio of a coccolith’s average thickness to its length, the authors used another property of light. The amount of light that gets through calcium carbonate depends on its thickness. Calibrating a grey scale with a precisely micro-machined calcium-carbonate wedge allowed them to assess the thickness of a coccolith at each pixel in the image. From there they could calculate the average thickness. Since the length of the coccolith can also be measured from the image, this procedure yielded $A_L$.

The correlation between PIC:POC and coccolith $A_L$ can be useful in the study of the long-term history of our climate, since in the fossil record intact coccolithospheres are rare (they do turn up, even in classroom chalk), whereas individual coccoliths are abundant. Margaret Xenopoulos surveyed this research in Eos, the American Geophysical Union’s news magazine, with the subtitle: “Math can be fun when reconstructing the ocean’s past and forecasting the future with algal geometry.”