Math Digests February 2025

The butterfly effect: this obscure mathematical concept has become an everyday idea, but do we have it all wrong?

The Conversation, February 5, 2025

In the 2004 movie The Butterfly Effect, protagonist Evan Treborn alters the entire course of his life by going back in time and changing minute decisions. The film’s title is an homage to a term coined by meteorologist Edward Lorenz, who famously asked: “Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?” The term ‘butterfly effect’ now refers to the potential for small changes to generate large effects. But according to Milad Haghani of the University of Melbourne, this overstates Lorenz’s original intent. “In reality, not all systems are chaotic, and for systems that aren’t, small changes usually result in small effects,” Haghani writes for The Conversation. In this article, Haghani describes what motivated Lorenz, and what this well-worn phrase really means.

Classroom Activities: chaos, randomness, prediction

  • (Mid level) Read the article and answer the following questions:
    • Why did Lorenz’s computer simulation of the weather give a different answer the second time he ran it?
    • Explain in your own words the meaning of the words random, deterministic, chaos, and the butterfly effect. Which of these words applies to the following systems? Justify your answer.
      • Flipping a coin
      • A presidential election
      • The daily temperature in your city
      • A ball tossed in the air
    • Haghani writes that there have been “oversimplifications and misconceptions” about the meaning of the butterfly effect. What oversimplifications and misconceptions might he be referring to?
  • (Mid level) Compare how easy it is to predict a random system versus a chaotic system.
    • Write down the daily high temperature predictions for your town for the next 10 days. As the days progress, write down the true temperature.
    • On Day 10, predict a random system. In pairs, assign one partner to be a random walker, and the other to be the predictor. The random walker will take 10 steps, each randomly chosen to be either to the left or the right. Before the walk begins, the predictor will try to predict the walker’s path. Then switch roles and repeat.
    • Calculate the prediction errors for both systems. Compare and contrast what’s involved in predicting the daily temperature versus a random walk.
  • (All levels) Check out our October 2021 and July 2023 digests for more activities on chaos theory.

—Leila Sloman

Kids’ real-world arithmetic skills don’t transfer to the classroom (timestamp 00:45)

Nature, February 5, 2025

It’s often hard to apply what you learn in school to the real world. With math class it’s no different. This podcast from Nature describes children in India who excel at arithmetic related to the markets where they work but struggle with traditional math problems taught in school. “I had been to markets and could see there were kids there, 8-year-olds, who could do the mathematics involved in selling,” an economics researcher told Nature. “When we gave them the nation-wide test, it was very clear that they were worse.” According to that researcher’s recent study, the opposite tended to be true as well: When students learned arithmetic with problems from school, they weren’t as proficient with market-related problems. “These findings highlight the importance of educational curricula that bridge the gap between intuitive and formal maths,” the researchers wrote in their abstract.

Classroom Activities: arithmetic, mental math

  • (All levels) With a partner, review this guide to mental math tricks and test each other.
  • (All levels) Play the online mental math game Math Heads (free, but registration required).
  • (Mid levels) Practice your ability to switch between real-world and school-style math with these examples.
    • You start an after-school job that earns \$20/hour working the 4-hour long dinner shift at a local restaurant. How many shifts do you need to work to save \$900?
    • Suppose you now have \$900 in your bank account, and you want to save a greater amount, $y$. Write an equation to calculate how many more hours you need to work ($x$) in order to save a total of $y$ dollars.
    • How does this equation change if you are now taxed at 20% on every dollar you earn?
    • Plot the equation with and without taxes.
    • How much tax revenue ($) do you generate for every 100 hours that you work?
    • How much would someone who earns $300/hour be taxed for 100 hours of work, if they find a loophole that allows them to pay just 1% in taxes?

—Max Levy

Oh, great: Rat populations are surging as cities heat up

Grist, January 31, 2025

As temperatures increase due to climate change, rat populations are up as well. Scientists recently analyzed the relationship between rat populations and the environment in American cities. “Females will reach sexual maturity faster. They’re able to breed more, and typically their litters are larger at warmer temperatures in the lab,” one researcher told Grist reporter Matt Simon. This article analyzes the new data.

Classroom Activities:  data analysis, correlations, algebra

  • (High level) Use the original scientific article to answer the following questions:
    • How many cities did the scientists include in their analysis? List all of them in ascending order of climate change driven increase.
    • Figure 3 shows “Positive association between warming temperatures and rat numbers.” What does “positive association” mean, and what do the scientists conclude? (Hint: read the figure caption and any references to “Fig. 3” in the text)
    • Figure 4 shows “Negative association between vegetation cover and rat numbers.” What does “negative association” mean, and what do the scientists conclude? (Hint: read the figure caption and any references to “Fig. 4” in the text)
    • Explain the following conclusion in your own words: “In a relative weights analysis, 7% of the variation in trend strength was linked to the mean temperature increase a city had experienced relative to long-term temperature averages.” (Hint: refer to Figure 2.)
  • (Mid level) Suppose that you can model Big City’s future average temperature with the following equation: $$T_{2024+t} = 60 + 0.25t$$
    • Based on the description of the equation, how would you define the variables $T$ and $t$?
    • What will be Big City’s average temperature for 2025?
    • Which units does this equation most likely use? And by how much does temperature increase every 10 years?
    • If Big City’s rat population increases by 18% for every 1 degree of temperature increase, by what percent will the population increase after 20 years?

—Max Levy

How Common Are Plane Crashes? What Statistics Show

Newsweek, January 30, 2025

This year, the country has already witnessed several high-profile plane crashes. One recent collision near Washington D.C. between an American Airlines flight and a military helicopter killed every person involved. “Commercial aviation in the U.S. hasn’t had a major accident since 2009,” an aviation lawyer told Newsweek, asserting that flying remains safe. Still, it’s normal to worry when seeing news and tragedy. This Newsweek article shares statistics to analyze flight safety over the years.

Classroom Activities: data analysis, statistics

  • (Mid level) Answer the following questions based on the article
    • How many total fatal and nonfatal crashes occurred in 2023, according to the National Transportation Safety Board (NTSB)?
    • How many fatal and nonfatal crashes occurred in 2023 per million flight hours?
    • How many fatal and nonfatal crashes occurred in 2008 per million flight hours?
    • By what percent did the rates of fatal and non-fatal crashes change between 2008 and 2023?
  • (High level) Create a spreadsheet table using the data from 1982 through 2025 from this NTSB data of yearly and monthly totals of fatal accidents.
    • Which month of the year has the highest average over this period?
    • Which year has the highest average over this period?
    • What are the limitations of comparing data from the 1980s to data from the 2020s?

—Max Levy

Mathematics professor explains ranked choice voting

KPCW Local News Hour, February 21, 2025

In this segment on KPCW, a public radio station in Utah, mathematician Alan Parry gives the rundown on instant runoff voting—how it works, some of its mathematical quirks, and how it stacks up against traditional plurality voting.

Classroom Activities: voting, monotonicity

  • (Mid level) Listen to the segment.
    • What is the connection between the monotonicity property in voting, and the monotone functions you might have studied in math class?
  • (High level) In a 1996 paper, mathematician Douglas Woodall studied a hypothetical election in which 11 voters rank Candidate A first and Candidate B second; 7 voters vote only for Candidate B; and 12 voters vote only for Candidate C. He used this example to show that no voting method can simultaneously satisfy all three of the following properties:
      • Plurality: Suppose Cynthia receives $m$ votes in all (in some votes, she’s ranked first-choice votes and in others, she’s second- or third-choice), and Jane receives $n$ first-preference votes. If $n > m$, Jane should have a better chance of winning than Cynthia.
      • Condorcet: If Jane would beat each of her opponents in a head-to-head race, Jane should win the election.
      • Monotonicity: If Jane is set to win the election, and new voters arrive at the last minute with ballots that rank Jane first, Jane should still win the election when votes are recounted.
    • Who do you think should win Woodall’s hypothetical election? Why?
    • What if two voters are added who rank B first and A second?
    • What if five voters are added who rank C first and B second?
    • To see Woodall’s conclusions, navigate to Section 4 of the paper. Does this exercise change your thinking about fair voting methods?
  • Read more about instant runoff voting in this article by MIT’s Election Lab.
  • (All levels) In our November 2022 digests, students can analyze more hypothetical elections that do not respect monotonicity.

—Leila Sloman

More of this month’s math headlines