Math Digests August 2023

How Recommendation Algorithms Work—And Why They May Miss the Mark
Scientific American, August 17, 2023.

A top-tier recommendation algorithm can lure users into watching, buying, and scrolling more than they otherwise would. But recommendation is a tough problem. In this article for Scientific American, Manon Bischoff describes some of the tactics companies like Amazon and Netflix might use to optimize their recommendations. Whether or not they’re successful is another matter.

Classroom Activities: trigonometry, linear algebra, distance functions

  • (Mid level, Matrix algebra) Read the article by Manon Bischoff. Discuss: What’s your experience with recommendation algorithms? Do you have any ideas of how to create a recommendation algorithm that do not appear in the story?
  • To get students thinking critically about recommendation algorithms, try this lesson about algorithms from Canada’s Centre for Digital Media Literacy.
  • (Mid level) One technique Bischoff mentions, called collaborative filtering, predicts what you’ll like based on ratings from users similar to you. Users are represented as vectors of the ratings that each user has assigned to each product.
    • Suppose Alice, Bob, and Carol each ordered and rated three products from an online store. Alice gave the mop a rating of 5, the broom a 2, and the vacuum a 4. Bob gave the mop a 4, the broom a 4, and the vacuum a 4. Carol gave the mop a 1, the broom a 1, and the vacuum a 5.
      • Write down three vectors representing Alice, Bob, and Carol. Which users do you think are most and least similar to each other?
    • Calculate the Euclidean distance, Manhattan distance, and cosine similarity between Alice and Bob, and between Alice and Carol.
    • Bischoff mentions several ways to measure distance or similarity, including the Euclidean and Manhattan distance, but describes cosine similarity in particular. What advantages do you think this measurement has over other distance measurements? Do you think it’s more helpful in situations where ratings are missing, like in the table of user ratings from the story?

—Leila Sloman

Rhyme and reason – why a university professor uses poetry to teach math

The Conversation, August 8, 2023.

Ricardo Martinez, a math education professor at Pennsylvania State University, believes that math and poetry can open your eyes to the world. “Math and poetry create new metaphors that allow people to better understand societal issues,” Martinez wrote in an article for The Conversation. In this article, he introduces a new course that delves into the integration of poetry and mathematics to enhance learning, encouraging students to use poetry as a tool for personal expression and to explore real-world issues. His goal is to reveal the connection between mathematics and creative expression, make mathematics more engaging, and emphasize recognizing math in everyday life. “Math poetry becomes even more critical today as people need an outlet to communicate their truths about societal injustices,” he writes.

Classroom Activities: creative math, logic

  • (All levels) Fill out “I Am” math poems based on this template and share with each other.
    • Which questions were the hardest for you to answer. Why?
    • Read Martinez’s poem about the math of gender inequality and Yousef Kara’s poem relating trans identity to mathematical functions. Discuss what you learned from both.
  • (All levels) Listen to the poem “Mathematics” by Wayne Henry. (Content note: This poem includes brief use of profanity.) As you listen, list the different ways in which math appears in the poem (e.g. “17% chance of survival,” “22 weeks,” “They did the math”)
    • How does it contribute to the story? What role(s) does math play in the poem?
    • Explain how math can make a story more vivid.

—Max Levy

How to be successful as a research mathematician? Follow your gut

Nature, August 14, 2023.

According to a recent survey, 82% of American students in grades 7-10 fear math. Eugenia Cheng wants them to stop. Cheng teaches math to art students at SAIC, the School of the Art Institute of Chicago, and she has long believed that math is the least frustrating subject. “In every other subject, it seemed like you had to just believe somebody. There never seemed to be an explanation,” Cheng said in an interview with Nature. Cheng recently wrote a book called “Is Maths Real” that aims to satisfy curiosities, like the question of why $-1 \times -1$ equals 1. In this article, Cheng discusses her book and the life experiences that shaped her mathematical mind.

Classroom Activities: double negatives, logic

  • (All levels) Watch this Khan Academy video about multiplying negative numbers.
    • Compare this to a “double negative” in the English language (e.g. “She is not unafraid.”) Discuss other ways in which the logic of math matches or doesn’t match that of written language.
  • (Mid level) In the article, Cheng describes what she likes about math and writing. Think about what you like to do, personally or academically, now or in the future.
    • What draws you to these subjects or activities?
    • How does your math education help you perform better in these activities?
  • (High level) Cheng describes studying category theory in this podcast, where she describes it as “the mathematics of mathematics.”
    • Listen to the podcast (or read the transcript) and describe, in your words, what category theory is.
    • Explain Cheng’s argument for category theory in “higher dimensions.”
    • How does category theory compare to intersectionality and questions of social justice, according to Cheng?
      • Think of a social cause that you agree with. What logic or reasoning best explains why you support it?
      • Think of a social cause that you disagree with. What logic or reasoning do its most well-intentioned believers use to defend it?

—Max Levy

Risky Giant Steps Can Solve Optimization Problems Faster

Quanta Magazine, August 11, 2023.

When you look for the fastest route through traffic, the most profitable price point for selling handmade necklaces, or the most delicious banana bread recipe, you’re trying to solve what is known as an optimization problem—to minimize or maximize some quantity (travel time, profit, deliciousness) in order to get the best possible outcome. When optimization problems are modeled with a computer, one popular algorithm for finding solutions is to choose a random starting point on a function and then move around in a series of steps to find the minimum value, a technique called gradient descent. At each step, you move in what seems to be the most promising direction—like finding your way to the bottom of a valley on a foggy day by walking in whatever direction points most steeply downhill. In an article for Quanta Magazine, Allison Parshall reports on a counterintuitive new result about gradient descent, posted online in July. Intuitively, taking small steps with gradient descent is the safest bet, because you’re likely to move steadily towards a minimum without jumping over it. The new paper shows, surprisingly, that gradient descent sometimes performs better with big steps, like giant leaps instead of small paces.

Classroom Activities: optimization, graphs, derivatives

  • (Precalculus) Sketch the graph of the function $f(x) = x^2$.
    • What is the minimum possible value of $f(x)$? What value of $x$ gives you this minimum value for $f(x)$?
    • Imagine that you started at the point $(2,4)$ on the graph of $f$, and plan to find the minimum using gradient descent, by taking a sequence of steps that each change the $x$-value by $1$. Which direction should you step—to the left (decreasing the $x$-value by $1$), or to the right (increasing the $x$-value by $1$)?
    • In the description of the new paper, Parshall writes, “Giant leaps are tempting but also risky, as they could overshoot the answer.” Starting at the point $(2,4)$, consider what would happen you took a step that changed the $x$-value by $7$ instead of $1$. What would be the new value of $f(x)$? Would you be closer or further away from finding the minimum value?
  • (Calculus) What is the derivative of $f(x) = x^2$ at the point $(2,4)$? Is the derivative positive, or negative? Does that mean the function is increasing at that point, or decreasing? How can the answer to these questions tell you what direction to choose for gradient descent?
  • (Precalculus, Calculus) For a function $f(x)$, the absolute minimum is the smallest value that the function reaches, while a local minimum is a value that is smaller than all of the points near it: see the first image here for an example.
    • Take a look at the graph of the function $f(x) = |x^3 – x^2 – 1|$. What is the absolute minimum of the function? What are the local minima?
    • Using gradient descent, starting at the point $(-2, 13)$ and taking steps that change the $x$-value by $1$, would you find the absolute minimum, or just a local minimum? Justify your answer.

—Tamar Lichter Blanks 

A brief illustrated guide to ‘scissors congruence’ − an ancient geometric idea that’s still fueling cutting-edge mathematical research

The Conversation, August 9, 2023.

Certain simple concepts, such as polynomial equations or prime numbers, turn out to mask enough difficult mathematics to fuel thousands of years’ worth of research. One such idea: Rearranging shapes by cutting them up and gluing them back together, as Maxine Calle and Mona Merling write in this piece for The Conversation. The ancient Greek mathematician Euclid defined area as the number conserved by that procedure. His definition rings strangely to modern ears. It might seem obvious that area as we currently think of it — as the amount of space inside a shape, perhaps measured by the number of 1 x 1 squares that you can squeeze in — is conserved by this rearranging procedure, but it’s not obvious that it’s the only measurement that’s conserved. Indeed, Euclid’s definition doesn’t scale up to three-dimensional shapes, and mathematicians are continuing to explore what happens in higher dimensions.

Classroom Activities: geometry

  • For students unfamiliar with area, go over this lesson plan. The plan mentions several applications of area — gardening, painting, or fitting a rug. Can you think of more reasons to study area?
  • (All levels) In a few sentences, say how you would define area. Why do you think Euclid might have defined area the way he did?
  • (Mid level) Calle and Merling write that if you start with a piece of paper of area 1, you can make any polygon with area 1 by cutting the paper up along straight lines and gluing the pieces back together along the edges. Suppose your original paper is a square with side lengths 1, and you cut and paste it into each of the following polygons. What will the side lengths of the new shapes be?
    • An equilateral triangle
    • A 45-45-90 triangle
    • A 30-60-90 triangle
    • A regular pentagon
    • A regular heptagon
  • (All levels) Modern mathematicians say two shapes are scissors congruent if you can turn one into the other by this cutting-and-pasting procedure. Prove the following pairs are scissors congruent by cutting and taping Shape 1 into Shape 2:
    • Shape 1: A square
      Shape 2: A 45-45-90 triangle
    • Shape 1: A square
      Shape 2: An equilateral triangle
    • Shape 1: A regular pentagon
      Shape 2: An equilateral triangle
    • (Mid level) Measure the side lengths of your shapes, and compare to your results from the previous problem.

—Leila Sloman 

Some more of this month’s math headlines