Math Digests September 2022

Quantum Physics Titans Win Breakthrough Prize

Scientific American, September 22, 2022

When you shop online with a credit card or send an email to a friend, your information is packaged, encrypted, and sent through systems that were designed using math and physics—but not quantum physics. A fundamentally different kind of computer, called a quantum computer, could use properties of quantum mechanics to efficiently solve problems that are impractical for ordinary computers. Quantum computers, which exist today but are small and not yet powerful, could completely change how information is communicated and protected. In an article for Scientific American, Daniel Garisto reports on the latest Breakthrough Prize in Fundamental Physics, awarded to Charles H. Bennett, Gilles Brassard, David Deutsch, and Peter Shor for “foundational work in quantum information.” The prize was for trailblazing research on the ideas behind the machines: they developed much of the conceptual framework for quantum computing and quantum cryptography.

Classroom activities: quantum computing, quantum physics

  • (All levels) Play the Quantum Circuits card game from the American Physical Society PhysicsQuest 2021 kit, following the game’s pictorial rules for quantum gates. (There is also a teacher’s guide here.) Note that this game has cards that need to be printed out in advance!
  • (All levels) Play through the puzzles in the Quantum Chess game. 
    • (High level) Come up with your own example of a chess puzzle using superposition and measurement or entanglement.
    • (High level) Read this article about quantum superposition. Compare this to the way that superposition is represented in the chess game.

—Tamar Lichter Blanks

A Trip to Infinity

Netflix, September 26, 2022

Infinity: the question cosmology can’t answer

IAI News, September 23, 2022

No matter what level you’re at, infinity is indispensable to how we do mathematics today. Even elementary school students implicitly rely on it when they add and subtract using integer numbers—a fact duly acknowledged by any child who, in a contest to name the biggest number, has triumphantly yelled “infinity plus one!” But does infinity ever enter into the physical world? Both Peter Cameron’s article and this 80-minute Netflix documentary discuss the mathematics of infinity as well as its role in physics.

Classroom activity: infinity, integers, real numbers

  • (Mid level) If students are not already familiar with the concept of rational and irrational numbers, use this online lesson before doing the other activities.
  • (Middle school and up) Use this lesson plan, “Teaching the Mathematics of Infinity”, to study different kinds of infinity in class.
  • (High level) Watch this video from Numberphile explaining countability and Cantor’s diagonal argument.
    • Ask students if they think the following sets are countable: The points on an xy-plane, the set of prime numbers, the set of odd numbers, the set of line segments of any length.

—Leila Sloman

The Beautiful Game Theory – using mathematics to resolve human conflicts 

Horizon Magazine, September 1, 2022

Conflicts between humans are a lot like games. You can win, or you can lose. You can draw, or compromise to win some battles and accept defeat in others. If two people win the lottery, they simply split the winnings 50/50. But what’s the best compromise for conflicts that involve concepts more abstract than money? In math, “game theory” deals with finding the optimal strategies to resolve competing interests. Game theory can help win a game of poker, and it’s also used to understand global conflicts. Researchers recently created tools based on game theory to facilitate discussions about land use conflicts around the world, such as a conflict between farmers and geese conservation efforts in Scotland. In an article for Horizon, Gareth Willmer writes about this research and describes the role of math in messy human situations.

Classroom activities: game theory

  • (Low level) Can you guess how your classmates will behave? Given a range of integers between 0 and 100, guess the whole number that is closest to two-thirds of the average of all numbers guessed by your classmates. Collect all the individual guesses, calculate the average, then two-thirds of the average. Discuss how you did.
  • (Mid level) Play crops versus creatures, a game designed by the researchers to illustrate the math of settling complex conflicts. Note: Game play will be recorded and used in a study by researchers at the University of Stirling. Students must be 16 or older to fill out the consent form.
  • (Algebra II) For more information on game theory and helpful games and examples, use this NSF-sponsored teaching tool by Cornell University.

—Max Levy

How Scientists Massage Results With ‘P-Hacking’ 

Popular Mechanics, August 23 2022

Good science is not a story of what happened one time in one lab—it is repeatable and significant. Scientists strive to draw broad conclusions based on experiments that let them study the effects of an “experimental variable” on a sample. For instance, clinical trials let scientists conclude that vaccines protected people from Covid-19 better than a placebo did. To measure the statistical significance of these effects, scientists calculate a “p-value.” A p-value represents the likelihood that observed effects can be explained by chance, rather than the variable. (Lower p-values mean a result is less likely to be due to chance.) The p-value is standard across science, but it’s not foolproof: Researchers can cleverly analyze data to get better p-values, thereby overstating weak results. Though sometimes harmless, writes Sarah Wells, “when this practice is used in medical trials, it can have much deadlier results.” In this article, Wells explains how some scientists are exploring ways to close those loopholes.

Classroom activities: probability, statistics, p-hacking, fragility index

  • (All levels) Watch this brief TED-Ed video that explains how p-hacking works in more detail. 
  • (Mid level) According to the article, scientists hope to weed out significant yet weak results with a new measure called the fragility index. An experiment’s fragility corresponds to how sensitive it is to small perturbations. For example, if a drug’s clinical trial data turns from significant (p < 0.05) to insignificant (p > 0.05) if one single participant would have had a weaker outcome, then it would be considered fragile. Discuss why you think it’s important to make such a distinction.
  • (Mid level) Suppose you are testing whether an antibiotic works better than a placebo to treat an infection. Your clinical trial has 200 total participants, who are all infected at the start of the experiment. Half receive the placebo. Half receive the drug. By the end of the study, 10 placebo recipients and 90 drug-recipients are infection-free.
    • Calculate an experiment’s fragility index using this online calculator. 
    • Repeat this calculation assuming instead that 30 placebo recipients and 45 drug-recipients are infection-free. 
    • Discuss what each result means in terms of whether the antibiotic should be recommended as a treatment.

—Max Levy

Let’s learn about dealing with math anxiety

Science News Explores, September 6, 2022

As a mathematician, I can’t count the number of people who have told me they hated math as a kid. It’s hard not to wonder how many of those people would feel differently if they’d been able to engage with math without timed tests or the “genius myth” hanging over their heads. Maria Temming’s article for Science News Explores looks at ways to deal with math anxiety, sharing articles and resources to help students practice math in a stress-free way.

Classroom Activities: math anxiety, algebra, geometry, arithmetic

  • (All levels) Ask students to read some of the suggested articles under “Want to know more?”. After reading, ask students to reflect on the following discussion questions:
    • Do math assignments or tests make you more or less anxious than assignments in other subjects? Do you think you experience math anxiety? Why or why not?
    • Do you know anyone who you think experiences math anxiety? Write what you notice about this person’s relationship with math.
    • In this 2017 article, Evelyn Lamb writes about how a “fixed mindset” and the “genius myth” may contribute to math anxiety. Do you agree or disagree that these concepts make math anxiety worse? Why?
  • (All levels) Mathematical skills come up in many areas of our lives, sometimes when we’re not even aware of it. As a class, brainstorm some games and activities that use logical thinking or numbers. For homework, ask students to spend 20 minutes on an activity they enjoy that involves some mathematical skills, and answer the following questions:
    • Did you feel more or less anxious during the activity than you normally are while doing math?
    • How was the experience different or similar to your usual experience of math?
    • Did you use skills that might help you in math class? If so, what?

—Leila Sloman

Some more of this month’s math headlines: