Tony’s Take February 2023

This month’s topics:

The decolonization of mathematics.

Nature ran an editorial on January 31, 2023 with the title “Why we have nothing to fear from the decolonization of mathematics.” What might be feared? Through the international decolonizing movement, “university faculty members and students are exploring the contributions that people from many cultures have made to the story of different research fields,” they explain. The editors reference a backlash from people who insist that mathematics exists in a domain beyond culture. To give the example mentioned in the editorial, the roots of a quadratic equation do not depend on the particular identity of the person doing the extraction.

The editors remind us that universal truths may have very local origins. Their subhead here, “Maths made the modern world — and everyone stands to gain from the acknowledgment that the world made maths,” is actually illustrated by the history of the quadratic equation. That equation may exist in the world of platonic ideals, but our methods for solving it were developed here on Earth, starting in Mesopotamia about 4000 years ago. Similarly, zero could hardly be more disembodied, but as the editors mention, our notation for it dates from a certain time (“as early as the third or fourth century”) and a certain place on the Indian subcontinent.

Mathematicians know that whether math is discovered or invented, it takes people to do the discovering or the invention. Reminding students of this fact — and that those people lived in many different times, places, and cultures — may possibly dispel some of the forbidding aura that often cloaks the discipline. The Nature editors suggest that it can be “particularly empowering for people from historically marginalized groups.”

Some ongoing examples of more radical decolonization were spotlit by Rachel Crowell in a “Work” rubric article, also in Nature: “Maths Plots a Course to Cultural Equality.” We learn to distinguish between indigenous mathematics (teaching the mathematics native to a given culture) and indigenizing mathematics (teaching standard mathematics with reference to a given culture). Crowell gives a nice example of the latter: Kamuela Yong (University of Hawai’i-West O’ahu) indigenizes his pre-calculus course by working with Polynesian ocean-navigation techniques. Yong observed to Crowell that “although students might be especially interested in maths examples derived from their own cultural backgrounds, they also benefit from engaging examples that are rooted in other communities.” (Unfortunately this information seems only to be anecdotal: there is no reference to any systematic evidence showing that students learn more useful mathematics when their instruction has been indigenized.)

The pandemic’s impact on math education

When the National Assessment of Educational Progress released “the Nation’s Report Card” on Grades 4 and 8 mathematics and reading last fall, the picture they gave was initially reported as catastrophic. “The Pandemic Erased Two Decades of Progress in Math and Reading” was a typical headline (New York Times, September 1, 2022). On February 8, 2023 The 74 (“America’s Education News Source”) posted Kevin Mahnken’s interview getting the perspective of Sal Khan, the founder of Khan Academy. As Mahnken remarks, Khan Academy is “an internationally known learning tool reaching tens of millions students in over 100 countries”; it has become the go-to destination for online mathematics tutoring, particularly in the United States. So Khan is uniquely well-positioned to see where our students are. His own report: “It’s not as if scores went from decent to bad; they went from horrible to even-more-horrible.” (He gives the example of Detroit public schools, which went from 5% of eighth-graders being proficient in math to 4%.)

The interview goes beyond the pandemic and its effects. Khan emphasizes the cumulative aspect of math education. If you’re a little shaky on addition then you’ll never be comfortable with multiplication, and when you get to exponents and word problems, you will founder and sink. This explains why Algebra I is such a stumbling block for students in high school and college: their weakness in pre-algebraic and even arithmetic skills catches up with them. Khan does not believe in “direct instruction” — teachers should not lecture, but ask questions and make students think. When asked about memorization versus more conceptual learning, Khan says, “I absolutely think it’s got to be both.”

“A geometric poem”

… that doesn’t quite rhyme. Petals of Venus: The Fascinating Geometry Behind Planetary Motions was posted January 23, 2023 on the website The Earth and Venus both have near-circular orbits around the Sun. “The Petals of Venus” refers to the orbit of the planet Venus when the Sun-Venus-Earth system is tracked in an Earth-centered coordinate system. (The mental image we share now of a Solar System with planets orbiting the Sun is relatively modern, and less intuitive than a point of view setting us at the center).

Diagram showing Earth and Venus' circular orbits around the Sun.

Diagram showing the Sun's circular orbit around the Earth, and Venus' circular orbit around the Sun.
Two conceptions of the Sun-Venus-Earth system. (a.) Heliocentric: the Earth (blue-green) and Venus (white) follow near-circular orbits around the Sun (red). (b.) Earth-centric: the Sun follows a near-circular orbit around the Earth; Venus, as seen from the Earth, has a more complicated motion: the composition of its orbit about the Sun with the Sun’s perceived motion about the Earth.

In Earth-centered coordinates, the two quasi-circular motions (Sun around Earth and Venus around Sun) combine to approximate a kind of curve called a (generalized) trochioid.

A curve that travels around Earth in an intricate fashion, self-intersecting and looping several times, with its two ends almost but not quite meeting.
The generalized trochioid traced out by Venus in an Earth-centered perspective. In this view, Sun and Venus move counter-clockwise. This plot follows Venus for 3000 days (8.2 Earth years).

Note in this image that the path described by Venus almost closes up after 8 years: the planet has circled the Sun just a bit more than 13 times. (An Earth year is 365.256 days while a Venus “year” is 224.772 Earth days. The ratio is 1.6250067, slightly over 13/8 = 1.625). The Culturacolectiva posting ignores the slight discrepancy and displays an image like the one below, where the two ends of the 8-year curve match up exactly.

The “Petals of Venus.” A Maple-generated plot using the Culturacolectiva color scheme.

(They compound the error in Venus Pentagram, their animation of the orbit, which shows Venus tracing the same lovely, symmetrical curve over and over for more than 50 years). As they put it, “In a nutshell, if you plot the Venusian trajectory for eight years with the Earth as the center, you get a geometric poem that reminds us that everything in the cosmos is perfect.” Well, almost perfect.