Issue #10: December 22, 2024
Join us for an inspiring look at a mathematician’s path—
and his advice for you!
What’s Inside This Issue?
- Concepts and Terminology: Discover concepts and terminology introduced in the featured article.
- Featured Article: Embark on the inspiring journey of a mathematician, Joey Zou, from his early love for numbers to the challenges he overcame along the way. Discover his valuable advice for students navigating their own struggles with math.
- About the Interviewee: Read Joey Zou’s bio to learn about his upbringing, education, and hobbies.
- Hands-On Activities: Explore math in nature with three engaging, hands-on activities that reveal its fascinating patterns and connections.
- Environmental Equity and Sustainability: Explore the many ways math helps us understand and tackle challenges in environmental equity and sustainability.
- Setting the Stage for Your Future: A Public Health Path: Discover how math is the foundation for countless career paths. Find out which high school courses and college majors can help you unlock exciting opportunities!
- Glossary: Recap key concepts and terminology from the featured article.
1. Concepts and Terminology
- Number sense is a strong understanding of numbers, including their relationships, patterns, and how to use them to solve problems.
- Calculus is a branch of mathematics that studies rates of change and accumulation. It involves two primary concepts: differential calculus, which deals with instantaneous rates of change, and integral calculus, which deals with the accumulation of quantities.
- Number theory is the study of the properties of integers, including prime numbers, divisibility, and number sequences.
- Cryptography is the art of writing or solving codes. It’s like being a secret codebreaker, using math and science to protect information.
- Computed Tomography (CT) is a medical imaging technique that uses X-rays to create detailed cross-sectional images of the body. It’s like taking a 3D picture of your insides!
2. Featured Article
From Numbers to Real-World Wonders: A Chat with Mathematician
Dr. Joey Zou
Interviewed and Written by Dr. Sik Lee Dennig
Image Credit: Jen Christiansen; Source: “Large Number Discrimination in 6-Month-Old Infants,” by Fei Xua and Elizabeth S. Spelke, in Cognition, Vol. 74; January 2000 (reference); Source: News Medical: “What is Public Health, and Why is it Important?”
Introduction: Unleashing Your Inner Math Superpower
Did you know we’re born with a built-in number sense? Even as infants, we can distinguish between quantities, like 8 and 16 dots, without understanding numbers. Researchers test this using habituation: babies lose interest in repeated stimuli but re-engage with new quantities, proving early number sense. This ability sharpens with age and isn’t unique to humans—crows and dolphins also demonstrate it. In a 2005 study, dolphins correctly chose the board with fewer dots over 80% of the time.1 Scientists believe number sense aids survival, helping animals spot predators, track offspring, or find food.
For some humans, this natural ability with numbers leads to a lifelong fascination with math. Some pursue this passion professionally and become mathematicians. One such individual is Dr. Joey Zou, who currently teaches math at Northwestern University. He’ll take us on his journey, from his early fascination with numbers to his work today, showing how this foundation can evolve into exciting real-world discoveries in mathematics.
The Early Journey: Math as a Game
Joey’s journey to becoming a mathematician began simply because he was able to keep up with math class fairly easily as he was growing up.
“I just answered questions, and those questions led to more questions.
In math, there’s another world where there are rules, but you can see how far those rules can take you. As you delve deeper, you discover you can think about different things, and they have slightly different rules than what you’ve seen before. To me, that’s a lot of fun—trying to understand what the rules in math are, what you can say about them, and what you cannot. For a long time, this felt like just something to play with—almost like a game that I could explore and enjoy—the game of math.
It didn’t really become a passion until sometime between high school and college.”
Joey likened the game of math to video games. “One of my spare-time hobbies is playing video games,” he explains. “Something I find interesting about video games is that a lot of times they’re set in a fictional or even fantasy environment, where there are rules. They aren’t the kind of rules you see in everyday life, but the point is that you’re playing a game, and the game has its own rules.
You’re trying to see what’s the best you can do within those rules. Math is a lot like that.”
From Game to Game-changer: When Math Finds Real-World Applications
Initially, Joey saw math as a fun pastime and assumed it would take a backseat to “real life” as he grew older. “You know, people have to stop playing games when they grow up, right?” he says. But in college, he discovered the surprising depth and utility of math. “I learned that people actually study math for its own sake,” he shares. “Even things that seem weird or convoluted at first can turn out to be incredibly useful.”
Through three compelling examples, Joey demonstrates how math bridges theory and practice, addressing challenges across science, technology, and everyday life.
1. The Power of Calculus
In high school, Joey viewed calculus as a set of interesting rules without immediate practical significance. However, as he delved deeply in college, he realized its fundamental role in various scientific and engineering fields. Partial differential equations, an extension of calculus, are particularly fascinating to Joey. These equations help us understand complex phenomena like heat diffusion2, economic trends3, and stock market fluctuations.4 The unknown aspects of these equations continue to drive research and innovation.
2. From Pure Math to Cyber Security
Next, Joey illustrated the surprising connection between pure mathematics and modern technology. He shared the story of G.H. Hardy, a renowned mathematician who believed his work in number theory had no practical use. Ironically, his research on prime numbers laid the foundation for public-key cryptography, the backbone of internet security. This mathematical principle powers secure online transactions, protects sensitive data, and safeguards our digital world.
Image Credit: AZ Quotes
One of the most famous examples of public-key cryptography is the RSA algorithm.5 It relies on the difficulty of factoring large numbers into their prime factors (e.g., breaking down 30 into 2 × 3 × 5). This is where Hardy’s work comes in. By studying prime numbers, mathematicians like Hardy gave us the tools to create these incredibly strong codes that keep our online information safe.
3. Joey’s Work on X-Ray Imaging
Joey’s own research exemplifies how abstract math can solve practical challenges. He described an exciting project that combines math, science, and technology to improve X-ray imaging, a tool we encounter in medical settings and airport security.
To explain the mathematical foundation of X-ray imaging, Joey introduced the work of Johann Radon, a mathematician whose theoretical contributions laid the groundwork for modern imaging technology. The Radon Transform, central to this process, works by integrating (adding up) the values of a function along straight lines at various angles—essentially mimicking how X-ray beams pass through an object to collect data.
Radon Transform
Joey’s project tackled a specific problem with X-ray imaging: streak artifacts. These unwanted lines sometimes appear in X-ray images, obscuring the view and reducing clarity. “You’re supposed to see a clear white blob that shows the object, but instead, you get these weird streaks,” Joey explained. These artifacts occur due to the beam hardening effect, where X-rays interact unevenly with dense materials like metal or bone, distorting the image. His research aimed to mathematically describe and address this issue to improve image accuracy.
To illustrate, Joey referenced a dental X-ray image: “In the picture, there are some round objects (teeth), some of which are bright (metal implants), as well as many straight lines, which are artifacts (i.e. they’re not supposed to be in the picture). One possible cause of these artifacts is the ‘beam hardening effect’, where X-rays with different energies behave differently when passing through a material such as metal; one of my papers studies some mathematical aspects of this phenomenon.”
Image Credit: Kondo, A., Hayakawa, Y., Dong, J. et al.
Joey’s work highlights how math extends far beyond numbers and theoretical exercises—it’s a critical tool that drives innovation and improves technology we rely on daily. Whether it’s modeling economic trends, securing online transactions, or enhancing X-ray imaging, math plays a vital role in solving real-world challenges.
Challenges in Learning Math: Embracing Uncertainty
For Joey, the journey of learning math took a significant turn when he reached college. What had once seemed instinctive and straightforward quickly became more challenging. “In high school, math felt like a game of rules—who could calculate the fastest or crack a tricky problem first,” he reflects. But college math demanded more than just quick problem-solving; it required abstract thinking, a shift that proved both difficult and rewarding.
The real turning point came when Joey learned to embrace uncertainty. “In college, I was faced with abstract concepts that I didn’t immediately understand,” he admits. The hardest lesson he learned was accepting that he wouldn’t always have the answers right away. “Many of the concepts I encountered in undergraduate math are things I’m still not completely comfortable with,” he says. One example was topology, the study of shapes and their properties. The Möbius strip, a twisted loop that seems to defy our understanding of dimensions, exemplified this challenge. “It’s a two-dimensional object, but it doesn’t behave like one,” Joey explains. “It takes time to develop an intuition for how it works.”
You can make your own Möbius Strip by cutting a strip of paper, giving one end of it a half twist (one-half of a full rotation, or 180 degrees), then joining the ends with tape. Try drawing a line down the middle all the way along the strip – how many sides of the paper have the line you drew?
Möbius Strip
Image Credit: David Benbennick, CC BY-SA 2.0, via Wikimedia Commons
This constant feeling of uncertainty initially left Joey uneasy. But over time, he realized that discomfort wasn’t something to avoid—it was an essential part of the learning process. “The challenge was learning to be comfortable with being uncomfortable,” he says. Rather than expecting immediate mastery, he focused on the aspects of a concept he could grasp. Even if he couldn’t fully explain the Möbius strip, he could still use the limited information he had to say something meaningful about it: “Something I had to learn is that even if I don’t understand it completely, I can still say something useful about what I do understand. That’s how life works too, right? So, being comfortable with things I don’t fully understand, while having enough confidence to realize that it’s OK, and that I can still work with it—that was key.”
Joey’s shift in mindset became a cornerstone of his learning strategy. He began to embrace the idea that not everything needs to be fully understood right away. “It’s not about figuring everything out all at once,” he explains. It’s about recognizing what you can understand and using that as a foundation to move forward. This approach—acknowledging limitations while still progressing—became invaluable not just in math, but in life as well.
Advice for High School Students Struggling with Math
Building on the idea of embracing uncertainty, Joey offers advice to students who find math challenging, recognizing that the psychological barriers often play a significant role in their struggles.
1. Acknowledging the Challenge
“The biggest difficulty is just the psychology behind it,” Joey explains. He understands that math can feel overwhelming because of past experiences, and for many, it may seem like a subject full of roadblocks. “For a lot of people, math is traumatic. It’s something they’ve struggled with and had bad experiences with.” Joey doesn’t dismiss these feelings but instead encourages students to acknowledge them and take the pressure off themselves. He wants to reassure them that math can be difficult—even for those who excel at it. “It is difficult for a lot of people, and depending on how far you go, it’s difficult for everyone,” he says. This realization, he believes, helps ease some of the pressure by reminding students they’re not alone in their struggles.
2. Math is Learnable
Joey firmly believes that higher-level math ability, meaning beyond your survival number sense, isn’t necessarily something you’re born with; it’s a skill that anyone can develop with time and effort. “Being good at math isn’t an inherent personality trait. People may have different advantages in how they understand math, but it’s something that everyone can learn,” he says.
3. Taking Small Steps to Success
“The first step has to be getting over that fear,” Joey advises. “I think the next thing would be to just realize that there are specific rules, specific things in math. It’s not just an arbitrary set of rules to follow. It’s not just just an arbitrary game to play. And so, trying to get better at math or trying to just pass your math class, there are very specific things to try to focus on.”
Joey stresses the importance of breaking things down. “The fear of not understanding everything is natural, but the key is to identify what you really need to know. What’s the important thing here?” He adds, “The key is to take that small bit of understanding and see where you can go with it. By taking those small pieces and building on them, you can make progress.”
Finally, he suggests seeking help: “It’s useful to discuss, with a math teacher or friends or classmates who are also taking it, that there are specific parts of math to understand. If you understand that there are these specific things to look at then that should be the start of getting better at math.”
The road to learning math.
Mathematicians Who Defied the Odds: Stories of Diversity and Triumph
Mathematics has long been seen as a universal language that transcends borders, and throughout history, mathematicians from diverse backgrounds have made groundbreaking contributions that have shaped the field. Their stories not only highlight the power of mathematics but also the resilience and determination needed to overcome challenges in pursuit of knowledge. Here are the stories of three remarkable mathematicians who defied the odds and changed the world of math.
1. Maria Gaetana Agnesi (1718–1799), an Italian mathematician, overcame the barriers faced by women in education to become one of the first women recognized in the scientific community. She wrote Instituzioni analitiche ad uso della gioventù italiana (1748), one of the earliest and most influential calculus textbooks, which introduced differential and integral calculus. Agnesi also studied the curve known as the “Witch of Agnesi” in algebraic geometry. Her legacy continues to inspire women in STEM today.
Maria Gaetana Agnesi; Image Credit: Wikimedia Commons
2. Srinivasa Ramanujan (1887–1920), a self-taught Indian mathematician, made groundbreaking contributions to mathematics despite limited formal education and facing significant challenges. His passion for numbers led him to independently discover theorems and identities, compiling thousands of original formulas in notebooks that astonished the mathematical community. After sending a letter to British mathematician G.H. Hardy, Ramanujan was invited to Cambridge University, where they collaborated on numerous problems. Hardy called him his greatest discovery, and Ramanujan’s legacy continues to inspire mathematicians today.
Srinivasa Ramanujan; Image Credit: AshLin, CC BY-SA 3.0, via Wikimedia Commons
3. Katherine Johnson (1918–2020), an African American mathematician, overcame racial and gender barriers to play a crucial role in NASA’s early space missions. As one of the “human computers,” she calculated trajectories, launch windows, and emergency return paths, including for John Glenn’s historic 1962 orbital flight. Glenn trusted her expertise to verify the computer-generated calculations, ensuring mission success. Johnson also calculated the trajectory for Alan Shepard’s suborbital flight before that and contributed to the Apollo program after, helping get astronauts to the Moon and back safely. Despite facing segregation and gender bias, she earned the Presidential Medal of Freedom, and her legacy continues to inspire diversity and excellence in science and math.
Katherine Johnson; Image Source: NASA: History of NASA Langley
- Dolphin Research Center – Dolphins Can Do Math! (n.d.). Www.youtube.com. https://www.youtube.com/watch?v=A226SlCQQsA1
- Wikipedia Contributors. (2019, June 2). Heat equation. Wikipedia; Wikimedia Foundation. https://en.wikipedia.org/wiki/Heat_equation2
- Loo, A. (2023, March 4). Solow Growth Model. Corporate Finance Institute; Corporate Finance Institute. https://corporatefinanceinstitute.com/resources/economics/solow-growth-model/3
- Black-Scholes Model: What & How It Works Example & Benefits | Intrinio. (n.d.). Intrinio.com. https://intrinio.com/blog/how-the-black-scholes-option-pricing-model-works-its-benefits4
- The Mathematics behind RSA. (2022). Sjsu.edu. https://www.cs.sjsu.edu/~stamp/CS265/SecurityEngineering/chapter5_SE/RSAmath.html5
3. About the Interviewee
Yuzhou Joey Zou is currently a Boas Assistant Professor in the Department of Mathematics at Northwestern University. His research is in the topic of microlocal analysis (a subfield of mathematical analysis which generalizes calculus and studies operators which act somewhat like differential operators), with applications to inverse problems and imaging. He was born in China and moved to the U.S. at age 6, and grew up in Pittsburgh, Pennsylvania, before obtaining Bachelors degrees in Mathematics and Chemistry at the University of Chicago (with one quarter spent abroad in Hong Kong) and a Ph.D. in Mathematics at Stanford University. While a graduate student at Stanford, Joey took Cantonese courses with Dr. Sik Lee Dennig as a way of building upon the Cantonese he learned in Hong Kong and balancing a mathematics Ph.D. with other intellectual pursuits. In his spare time, Joey enjoys biking long distances, as well as watching American football and ice hockey.
4. Hands-on Activities
Activity 1: Discovering the Golden Ratio in Yourself
The Golden Ratio is a special number, about 1.618, that shows up when a line is divided into two parts. The cool part is that the ratio between the whole line and the bigger part is the same as the ratio between the bigger part and the smaller part. It’s like a perfect balance.
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- Forearm: Measure the length from your elbow to your wrist.
- Upper Arm: Measure the length from your shoulder to your elbow.
- Calculate the Ratio: Divide the length of your upper arm by the length of your forearm. This is the upper arm to forearm ratio
2. Measure Your Palm and Finger:
- Palm: Measure the length from the base of your fingers to your wrist.
- Finger: Measure the length from the base of your fingers to the tip of your longest finger.
- Calculate the Ratio: Divide the length of your palm by the length of your finger. This is the palm to finger ratio.
3. Measure the Lengths of Your Fingers:
- Measure the length of each of your fingers: proximal phalanx (the bone closest to the hand), intermediate phalanx, and distal phalanx (the bone at the tip of the finger).
- Calculate the Ratios:
- Divide the length of the proximal phalanx by the length of the intermediate phalanx.
- Similarly, calculate the ratio between the intermediate phalanx and the distal phalanx.
4. Compare the ratios you get to the Golden Ratio and see if they are close.
Activity 2: Fibonacci in the World Around Us
1. The Fibonacci Sequence is a series of numbers that starts with 0 and 1, and each subsequent number is the sum of the two previous ones. It looks like this: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,…
Here’s how it works:
- Start with 0 and 1.
- The next number is the sum of 0 and 1, which is 1.
- The next number is the sum of 1 and 1, which is 2.
- The next number is the sum of 1 and 2, which is 3.
2. Take note of any patterns you see in nature that align with the Fibonacci Sequence. For example, observe the branching pattern of trees.
Image Credit: Botanica Mathematica: Fibonacci Tree
Fun fact: The Golden Ratio (approximately 1.618) and the Fibonacci sequence are closely related. As the Fibonacci numbers increase, the ratio between consecutive numbers gets closer and closer to the Golden Ratio. Example: 21/13=1.615.
Activity 3: The Golden Spiral
The Fibonacci sequence provides a way to approximate the Golden Spiral. Here are the steps to draw it:
- Start with Squares: Draw a series of squares, increasing in size according to the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, 21, etc.).
- Draw Quarter-Circles: In each square, draw a quarter-circle that connects opposite corners.
Interestingly, the Golden Spiral, with its characteristic growth pattern, is observed in various natural phenomena, such as the spiral patterns in nautilus shells and the arrangement of scales on pine cones.
5. Environmental Equity and Sustainability
Did you know math plays a key role in solving some of the planet’s biggest challenges? From combating climate change to ensuring clean air and water for everyone, math helps us understand and address problems in environmental equity and sustainability. Here are four examples of how mathematicians contribute:
1. Tracking Air Quality
Mathematicians develop models that analyze data from sensors measuring pollutants like PM2.5, a fine particulate matter deeply penetrating the lungs and posing significant health risks. These models help cities monitor air quality and improve public health. Ozone, a gas formed by atmospheric reactions, and PM10, another particulate matter, also contribute to air pollution, impacting respiratory health.
Image Credit: AirNow: Interactive Map of Air Quality
2. Solar Power Optimization
Using geometry and trigonometry, mathematicians calculate the optimal angles for solar panels to capture the most sunlight throughout the day.
3. Analyzing Climate Data
Mathematicians assist climate scientists in using statistical methods to track global temperature trends and understand the impact of human activities on the environment.
Image featured in What’s Going On in This Graph? | Global Temperature Change
4. Measuring Carbon Footprints
Through the creation of formulas and models, mathematicians calculate CO₂ emissions from activities like driving or electricity use, empowering individuals and communities to assess and reduce their environmental impact.
Image Credit: The Nature Conservancy
From data analysis to innovative problem-solving, math is at the heart of promoting a sustainable future—and you can be a part of it!
6. Setting the Stage for Your Future
Mathematics is the cornerstone of countless fields. From engineering marvels like bridges and skyscrapers to the algorithms powering artificial intelligence, math plays a vital role. Economists use it to model markets, physicists to understand the universe, and biologists to study complex systems.
In the digital age, math is essential for fields like computer science, data science, and finance. Whether you’re interested in technology, healthcare, or the arts, a strong foundation in mathematics will open doors to a wide range of exciting and satisfying career paths.
STEM DISCIPLINES:
- Math:
- Mathematics serves as the bedrock for all STEM disciplines, providing the essential tools and language for understanding and solving complex problems.
- Core concepts include calculus, linear algebra, discrete mathematics, statistics and probability, and logic.
- Engineering:
- Civil, Mechanical, Electrical, and other engineering fields heavily rely on calculus, differential equations, linear algebra, and statistics.
- Applications include structural analysis, control systems, and fluid dynamics.
- Computer Science:
- Algorithm design, data structures, artificial intelligence, and cryptography heavily rely on discrete mathematics, logic, and probability.
- Physics:
- Mathematical equations and models describe the fundamental laws of the universe, from classical mechanics to quantum mechanics.
- Calculus, differential equations, and linear algebra are indispensable.
- Chemistry:
- Quantum chemistry, chemical kinetics, and thermodynamics involve mathematical modeling and calculations.
- Biology:
- Mathematical models are used to study population dynamics, genetic inheritance, and the spread of diseases.
- Data Science:
- Techniques such as linear algebra, probability theory, and machine learning are used to extract insights from large datasets.
- Meteorology:
- Mathematical models predict weather patterns and climate change.
FINANCE AND ECONOMICS:
- Economics:
- Statistical analysis, econometrics, and game theory are used to analyze economic trends, make predictions, and understand market behavior.
- Finance:
- Mathematical models are applied to price financial instruments, manage risk, and optimize investment portfolios.
- Actuarial Science:
- Probability theory and statistical modeling assess risk and calculate insurance premiums.
- Operations Research:
- Mathematical optimization techniques are used to solve complex problems in logistics, supply chain management, and scheduling.
OTHER DISCIPLINES:
- Psychology:
- Statistical analysis is employed to analyze experimental data and understand human behavior.
- Architecture:
- Mathematical principles are applied to design structures that are both aesthetically pleasing and structurally sound.
- Art and Design:
- Geometry, perspective, and color theory are key mathematical concepts fundamental to many artistic disciplines.
How High School Students Can Prepare for Math-Intensive Careers.
1. Master Math Basics
- Take advanced math classes like Algebra, Geometry, Calculus, and Statistics.
- Focus on problem-solving skills through math challenges or puzzles.
2. Learn Coding
- Start learning programming languages like Python or Java. You can find free online courses!
- Get comfortable with tools like Excel or Google Sheets, which are great for data and math.
3. Join Clubs or Take Electives
- Take electives in subjects like Physics, Economics, or Computer Science to see how math is used in real life.
- Join a math club, robotics team, or science fair project to get hands-on experience.
4. Find a Mentor
- Look for someone who works in a field you’re interested in (like a scientist, engineer, or programmer) and ask them about their work.
- Ask for advice on how to study and what classes to take.
5. Take Online Courses
- Explore free courses on sites like Khan Academy, Coursera, or YouTube to learn more about subjects you’re curious about.
- Learn new math skills or explore fields like data science or economics.
6. Work on Personal Projects
- Try out math-based projects, like building something with coding or analyzing data from a hobby.
- Experiment with math in real-life situations (e.g., budgeting, planning, or even game design).
7. Prepare for College
- Get strong scores in math on the SAT or ACT to show colleges you’re ready. Find out how you can prepare for the test before you take it, and give yourself at least a couple of months to do that.
- Check out college programs that focus on fields like engineering, computer science, or finance to learn what you’ll need.
7. Glossary
- Calculus is a branch of mathematics that studies rates of change and accumulation. It involves two primary concepts: differential calculus, which deals with instantaneous rates of change, like finding the slope of a curve at a point, and integral calculus, which deals with the accumulation of quantities, like finding the area under a curve.
- Computed Tomography (CT) is a medical imaging technique that uses X-rays to create detailed cross-sectional images of the body. It involves taking multiple X-rays from different angles and then using computer processing to reconstruct a 3D image.
- Cryptography is the practice and study of techniques for secure communication in the presence of third parties called adversaries. It involves the design, implementation, and analysis of cryptographic algorithms and protocols.
- Fibonacci Sequence is a sequence of numbers where each number is the sum of the two preceding ones, starting from 0 and 1. It’s often found in nature, such as in the arrangement of leaves on a plant.
- Number sense is a strong understanding of numbers, including their relationships, patterns, and how to use them to solve problems.
- Number theory is the study of the properties of integers, including prime numbers, divisibility, and number sequences.
- The Golden Ratio is a special number, around 1.618, often found in nature and art. It’s like a magic number that creates beautiful proportions.
- The Golden Spiral is a spiral shape that grows outward based on the Golden Ratio. We can find it in many natural things, like seashells and galaxies.
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