Math Digests May 2021

Instagram post misleads on vaccine efficacy by conflating two different measures

Politifact, May 28, 2021

Measuring the efficacy of a COVID-19 vaccine involves several different—and potentially confusing—statistics. When we say that the Pfizer vaccine is 95% effective, we’re talking about relative risk reduction. A recent Instagram post with thousands of likes conflated the concepts of relative risk reduction and absolute risk reduction, using numbers from a Lancet Microbe commentary to make the false claim that the vaccine is not effective. This Politifact article explains the difference between the two statistics, why they’re both important for policymaking, and why the Instagram post is misleading.

Classroom Activity: relative risk vs absolute risk

• Using the equations on page 4 of the appendix of the Lancet Microbe commentary, have students calculate the relative risk reduction and absolute risk reduction for different values of $a$, $b$, $c$, $d$, $n_1$, and $n_2$. In which situations does focusing on just one measure give a misleading impression of the impact of a treatment or vaccine?
• Have students look up the absolute risk and relative risk associated with other diseases and treatments, discussing how they are similar or different to the case of COVID-19 vaccines.

—Scott Hershberger

How to Solve Equations That Are Stubborn as a Goat

Quanta Magazine, May 6, 2021

For Quanta Magazine‘s series “Quantized Academy”, Patrick Honner introduces the grazing goat problem: If a goat is tied to the side of a barn by a length of rope, how much area can it graze? The answer depends on the shape of the barn, the length of rope, and whether the goat is inside or outside the barn. A more complex problem is to start with the size of the grazing area and deduce what length the rope is. Honner gives examples of goat grazing problems and shows how to solve them when possible. He also discusses situations in which the problem is not exactly solvable due to the appearance of transcendental equations.

Classroom Activity: circles, quadratic equations, trigonometric functions

• Complete the exercises at the end of the article.
• Consider solving for the length of rope $r$ which gives the goat access to half the area inside a square or circular barn. As discussed in the article, $r$ is very difficult or impossible to compute exactly. Have students estimate $r$ using their intuition, and check their answers with a calculator. Discuss techniques for making good estimates.
• In the case of the goat tied outside a square barn, the shape of the grazing area is a semicircle attached to two quarter-circles and we can compute the rope length $r$ If the goat is inside the barn, the shape is a sector of a circle attached to two triangles, and we cannot compute $r$ exactly. Discuss the differences between these situations, and what makes one more complicated than the other.

—Leila Sloman

Wide-Angle Oddity: Giant Bidens Meet Tiny Carters

The New York Times, May 5, 2021

As it turns out, it’s easy to shrink a president—all you need is the right camera lens. President Joe Biden and First Lady Jill Biden recently met with their predecessors from 40 years prior, Jimmy and Rosalynn Carter. A White House photographer snapped a photo of the four in a small room, and something immediately looked odd. The Carters looked miniscule compared to the Bidens, who knelt beside them on the edge of the photo. “It was as if the hosts had been turned into Hobbits,” writes Heather Murphy in The New York Times. Photographers interviewed by Murphy explain that the lens causes this distorted perspective. “Wide-angle” lenses capture a complete view of small spaces, but at a cost: Objects close to the camera or near the edge of a frame get exaggerated. But did the Bidens get bigger or did the Carters shrink? It’s just a matter of perspective.

Classroom Activity: focal length, projections

• Use smartphone apps (or online images) of fisheye lenses, which are an example of an ultrawide-angle lens, to play with this type of distortion. Notice how the distortion changes with position in the frame and with distance from the camera.
• Revisit the math of focal length in ellipses (or introduce the math of focal length for lenses with this site and interactive calculators). Supplement this with an experiment with reflective spoons. Explore where the image on the concave side of the spoon inverts—the image should only appear upright once the object passes inside its focal point.
• Discuss other distortions and projections, such as the Mercator Projection of the globe into a 2D map, which makes landmasses near the edge of a map such as Greenland appear much larger than they truly are.

—Max Levy

The Art of Mathematics in Chalk

Scientific American, May 2021

In the world of mathematical research, chalkboards provide a unique space for scratch work and collaboration. Jessica Wynne documents this in her recent book, Do Not Erase: Mathematicians and Their Chalkboards. The book includes photographs of mathematicians’ chalkboards along with essays by the mathematicians. The work shown in the photos spans many fields of math, from topology to statistics. For Scientific American, Clara Moskowitz gives a brief description of a few of the topics that show up, including branching waves, vertex models, symplectic dynamics, and more.

Classroom Activity: math art, visualization, collaboration

• Have students choose one of the topics described in the article to research and give a short presentation on. Can they connect the photo of the chalkboard shown to what they learned in their research?
• Introduce more collaborative exercises in class. Have students work at a chalkboard or whiteboard together rather than using paper or computers. Discuss how this affects the inquiry process. Was it easier to focus and collaborate at a chalkboard?

—Leila Sloman