Math Digests April 2021

Trachette Jackson Fights Cancer with Math

Quanta Magazine, April 12, 2021

When Trachette Jackson was a student, mathematics and biology remained largely separate disciplines. Now, she is a pioneer in mathematical biology, with two decades of experience in using math to improve cancer treatments. In this podcast interview with host Steven Strogatz, Jackson discusses how she entered the field and how she creates mathematical models of tumors and treatments. A key drawback of chemotherapy, she explains, is that it kills healthy cells as well as cancer cells. With differential equations and computer simulations, Jackson is working to develop multi-drug therapies that more effectively target cancer cells while leaving healthy cells unaffected.

Classroom Activity: logistic functions

Use data from the American Cancer Society to introduce students to logistic functions. Have them use graphing calculators to determine the parameters for the logistic model, then estimate a woman’s risk of developing breast cancer at various ages. Conclude with a discussion of the model’s limitations and ways to improve it.

—Scott Hershberger

Is Soccer Wrong About Long Shots?

FiveThirtyEight, April 8, 2021

Conventional wisdom in the soccer world holds that players shouldn’t try to score when they’re far away from the goal. But does that advice hold up to scrutiny? Researchers at MIT studied this question using a Markov decision process. In this type of model, an agent in an environment (e.g., a soccer player on the field) makes decisions to maximize a reward (number of goals scored). The reward and the actions available are determined by the environment’s current state—in this case, the current location of the player. In this article, John Muller describes the conclusions of the paper and the implications for the game of soccer.

Classroom Activity: Markov decision processes, probability

• Use Van Roy et. al.’s interactive tool to explore how changing shooting behavior in a soccer game affects the number of goals.
• Discuss in class what the new paper means for soccer players. Should they change their shooting techniques? Why or why not?
• One example of a Markov process is the number of heads that have come up during a series of coin flips. Practice analyzing this process asking students to make bets (with candy, gambling chips, etc.) on the outcomes of a series of coin flips. How much are they willing to bet on a rare event, like ten heads in a row? How big does the payoff have to be? Analyze mathematically which bets are “worth it,” and discuss why. Discuss what this game has in common with the model in Van Roy et. al.’s paper.

—Leila Sloman

The Math of a Crumpled Piece of Paper Is Insanely Important. No, Seriously.

Popular Mechanics, April 2, 2021

A crumpled piece of paper may seem unremarkable. But when Harvard mathematicians looked closer, they found that the creases follow an elegant mathematical pattern. The research, which required hand-tracing thousands of creases, draws on the same mathematics that describes how rocks break down into smaller pieces. It provides a potential physical explanation for a surprising result: The total length of the creases increases logarithmically with the number of times the piece of paper has been crumpled. As writer Courtney Linder notes, understanding the “dynamics of squished paper” will also help researchers understand the folding of the Earth’s crust as well as engineer thin devices.

Classroom Activity: logarithms

Use this article as a fun example of where logarithms show up in everyday life. Discuss what it means that the total crease length increases logarithmically rather than linearly, quadratically, or exponentially with the number of crumples. Compare this with the total crease length that results from repeatedly folding a piece of paper in half or performing other simple folding patterns.

—Scott Hershberger

The Mathematics of How Connections Become Global

Scientific American, April 1, 2021

How well-connected does a community have to be before information flows easily among its members? Before an infectious disease spreads out of control? It turns out that if edges in an infinite network are randomly distributed, there is a precise level of connectedness that implies information or infectious disease will spread infinitely far. But real networks of human contacts, or networks that change over time, are far more complicated. In an article for Scientific American, Kelsey Houston-Edwards describes several real-world examples of these networks and the importance of understanding them.

Classroom activity: network theory

• Have students play with an interactive percolation simulator.
• Play an analog “percolation game”. Collaborate on drawing a network from real-world data. Discuss the network structure and whether or not it lends itself to percolation. Try asking questions like:
• How large are the clusters in this graph?
• Could we make the clusters disappear by adding or deleting a few nodes?
• Are the edges evenly distributed?
• How many edges are there, compared to the number of nodes?

—Leila Sloman