"Hidden From History" is the headline for Cara Buckley's *New York Times* (September 6, 2016) article about the African-American female mathematicians who helped put men on the Moon. The occasion for the story is the publication of "Hidden Figures" by Margot Lee Shetterly, who grew up in Hampton, Virginia (her father worked at NASA) and who learned that among her neighbors were women who had "defied convention ... by having vibrant, and by most standards, unusual careers. Black and female, dozens had worked at the space agency as mathematicians, often under Jim Crow laws, calculating crucial trajectories for rockets while being segregated from their white counterparts." The arrangement started, Buckley tells us, "some 75 years ago [when] the hungry wartime machine needed manpower, and womanpower, to fill its depleted ranks. This helped open the door for black female mathematicians, who were recruited through job bulletin boards and newspaper ads. Their job title? 'Colored computers.'" Katherine Johnson, one of the principals of Ms. Shetterly's story, is still around at 98 and spoke with the reporter. ("Mrs. Johnson, a math savant, graduated summa cum laude from what is now West Virginia State University at 18, and heard about the job through a family connection." "She calculated rocket trajectories for the Mercury and Apollo missions, and last year President Obama personally awarded her the Presidential Medal of Freedom for her life's work.") No mathematical details, but when asked about a time "in the late 1950s, when she successfully pressed her supervisor into admitting her into traditionally all-male meetings," Ms. Johnson: "Well, I don't ever wait for something. I remember asking the question, 'Is there a law?' And he said, 'Let her go.' It was easier than arguing."

A quote from the trailer for the movie, due out next January: "I need a mathematician ... that can look beyond the numbers ... find math that doesn't yet exist ... before the Russians plant a flag on the damn Moon."

"How is it possible that some individuals struggle to calculate a tip, whereas others find solutions to complex, ancient mathematical problems? Although some have argued that language provides the basis for highlevel mathematical expertise, others have contended that such mathematical abilities are linked to nonverbal processes that underpin the processing of magnitude and space. In PNAS, Amalric and Dehaene report data that significantly advance our understanding of the origins of high-level mathematical abilities." That quote is from the commentary by Daniel Ansari (Psychology, University of Western Ontario) on a report by Marie Amalric and Stephane Dehaene in that same issue (May 3, 2016) of *Proc Natl Acad Sci USA*.

Amalric and Dehaene's paper is "Origins of the brain networks for advanced mathematics in expert mathematicians." They conducted extensive fMRI studies on 30 subjects, (fifteen of them were professional mathematicians, while the other fifteen "had the same education level but had specialized in humanities and had never received any mathematical courses since high school"), scanning them "as they evaluated the truth of advanced mathematical and nonmathematical statements." Their experiments lead them to three main conclusions:

- In professional mathematicians only, mathematical statements activated the same set of specific areas of the brain. This was independent of whether the statements involved algebra, analysis, topology or geometry.
- Those areas did not include areas related to language and to "general-knowledge semantics."
- "mathematical judgments were related to an amplification of brain activity at sites that are activated by numbers and formulas in nonmathematicians."

One of the images from Amalric and Dehaene, *PNAS* 113:4909-4917. Colored areas are those which showed greatest *contrast:* RED, among mathematicians, contrast between activation caused by mathematical versus non-mathematical statements. GREEN, in the entire population, contrast between activation caused by Arabic numerals versus all other visual stimuli. BLUE, again in the entire population, contrast between activation caused by single-digit calculation versus sentence processing. YELLOW, the intersection of the three activation maps.

Ansari predicts that this work, along with follow-up studies, will lead us to "better understand the complex mechanisms that allow some to understand a level of mathematical complexity that is elusive to the majority of humankind." But on a less rarefied level, this work has clear implications for the theory and practice of mathematical education.

Daniel S. Silver contributed "Mathematical Induction and the Nature of British Miracles" to *American Scientist*, September-October 2016. He distinguishes between the name "mathematical induction" (which seems to be due to Augustus de Morgan, 1838) and the practice, which as he explains, can already be detected in Euclid's proof that there are infinitely many primes. He also shows how the looser meaning of induction: a mental procedure that goes from a finite number of examples to a law, could potentially lead to mathematical disaster in the wrong hands.

Silver places the cristallization of the notion of mathematical induction in Britain in the context of the tension between the British and Continental schools (left over from the Newton-Leibniz controversy) and the tension between the traditional frame of mind and the new ideas that would find their epitome in the works of Darwin. The article has many delicious bits of British mathematical history, like Cayley's "proof" of what is known as the Cayley-Hamilton Theorem: he works it out for $2\times 2$ matrices and then says "I have verified the theorem, in the next simplest case, of a matrix of order 3... but I have not thought it necessary to undertake the labour of a formal proof of the theorem in the general case of a matrix of any degree." And the wonderful rejection letter sent in 1821 by Sir David Brewster, editor of the *Edinburgh Journal of Science*, to Charles Babbage, who had proposed a series of papers including one on induction: "The subjects you propose for a series of Mathematical and Metaphysical Essays are so very profound, that there is perhaps not a single subscriber to our Journal who could follow them."

Andrew Hacker continues to get high-level media coverage. He was Ira Flatow's guest on "Science Friday," August 28, 2016. The segment, "How Much Math Should Everyone Know?" along with a printed-out summary by NPR's Julia Franz, is available online. There were two other guests.

- Maria Droujkova ("is a founder of Natural Math, and has taught basic calculus concepts to 5-year-olds. ... For Droujkova, high-level math is important, and what it could use in American classrooms is an injection of childlike wonder.")
- Pamela Webster Harris (UT Austin) hates "rote memorization" and believes in teaching "chunking": if a student doesn't know $6\times 7$ "then I might ask them, 'Do you know three sevens?' And if a kid knows three sevens is 21 and I need six sevens, then I just need to double 21, which is 42." Back to Franz: "Harris argues that 'chunking' like this quickly leads children to higher math by showing that a complicated figure can be built from easier-to-digest composite figures. Students who learn these concepts early on, she says, may get more out of courses like algebra and calculus when they reach them."
- Hacker (who keeps talking about
*inflicting*math): "I'm going to leave it to those who are in mathematics to work out the ways to make their subject interesting and exciting so students want to take it. All that I ask is that alternatives, other options, be offered instead of putting all of us on the road to calculus."

Tony Phillips

Stony Brook University

tony at math.sunysb.edu