Are you a math person or a language arts person? You probably have an answer to that question without thinking too much about it. In a world of binaries and polarizing opinions, this one is less contentious and seems pretty straightforward. But is it really that clear? If I were to answer that question solely based on my own biases and lived experiences, I would automatically answer that I’m a language arts person and that I don’t like math very much.
Growing up, I tended to gravitate toward adventure novels and fantasy books. I loved reading and my mom would buy me several new books a month at a cheap used book store by our apartment in Havana. However, I remember picking up a book once that was called something like “Mathemagic,” which was filled with math tricks. That book stayed behind when I left Cuba, so I don’t exactly remember what it was called, but it was truly fascinating. After reading each mind blowing trick, I would go and show it to my friends. Or really, I would go show off in front of my friends. Even though I’ve always been a language person, and not a numbers person, those math tricks were captivating. Maybe I was also a math person after all?
However, the math that I learned at school wasn’t fun. There were no tricks to learn. It was all about memorization. It was boring. There was nothing creative about it. It’s no wonder that I didn’t find it interesting. The way math is taught in schools is completely different than that of language. Imagine if all of your language classes were all about the rules without giving you any opportunity to write a creative assignment. It would be extremely boring and hard to get into. All that to say that I was bad at math growing up.
In high school, I even failed geometry. That’s embarrassing to admit, but it’s true. My teacher was old and tired and I didn’t particularly pay too much attention in class. Those things didn’t add up to a successful experience. When I went to summer school to re-take the class, I had a different teacher who made it more interesting. I passed with an A.
All of a sudden, I got it. It genuinely felt like a light bulb had gone off in my head. I remember the exact moment when it all clicked. I was at home, looking over the textbook and the way it was explained with fun and interesting examples made it come alive.
A big problem with this binary thinking of whether someone is a “numbers or words” person is that we’re actively contributing to that narrative. We repeatedly tell ourselves that we’re one or the other. If we do that long enough, we start to believe it.
Now, I don’t want deny that most people seem to have a pre-disposition to certain disciplines, but I don’t think that this makes us unqualified to learn another. While we may have an innate ability toward a certain field of study, it doesn’t mean that we can’t be competent in a different one Also, being competent is not the same as being an expert. I’m not going to drop everything tomorrow and become a mathematician, but I do want to learn more about math, in particular mathematical thinking. After all, this newsletter is called Constantly Curious. I love learning and I think being a well-rounded person makes you a more interesting person. While a lot of my essays lately are more personal in nature, I still enjoy writing and thinking about more “nerdy” subjects like this one. I hope you enjoy reading and thinking about these things as well.
Mathematical Thinking
Someone graciously granted me access to Masterclass, which is where our journey on this essay begins. I had never heard of Terrence Tao before, as I’m sure many of you haven’t either, but here’s some info on him.
Terence Tao is an Australian mathematician who is widely regarded as one of the finest mathematicians of his generation. He was born in South Australia and gained fame at a young age for his exceptional mathematical abilities. At age 8, he scored a 760 on the math portion of the SAT. At the age of 10, Tao became the youngest person in history to win a medal in the International Mathematical Olympiad. Throughout his career, he has received numerous awards and honors, including a MacArthur genius grant and the Fields Medal.
He has a class on Mathematical Thinking. This essay will cover what I learned from it. Hopefully some of these lessons will be useful to you as well. First of all, we must answer the question, what is mathematical thinking? Mathematical thinking is a particular approach to addressing problems and situations, characterized by logical reasoning, abstraction, pattern recognition, and problem-solving skills. Or as Tao puts it, “There is a certain way in which mathematicians approach problems, We abstract them. We break them up into pieces. We make analogies. We try to find connections with other problems.”
While I’m not to going to go over every detail of the class, which would be unfair to Tao, I will cover the parts that I found interesting or useful. Here are the lessons to hopefully get us on a more mathematical thinking trajectory:
Choose Well
When you're looking for a math problem to work on, Terence Tao suggests going for ones that are a bit challenging but still doable. According to Tao, the most exciting part of math is when you're working on problems that are just a little bit beyond what you already know how to do. Mathematicians can also change the details of a problem to make it easier, harder, more general, or more specific.
Simplify
When Tao tackles a math problem, he often uses a trick called abstraction to simplify it. This means he takes the important parts of the problem and represents them using math "objects" like numbers or shapes. To do this right, you've got to figure out which parts of the problem really matter. It’s a process of stripping the problem down to its basic elements. By getting rid of the parts that don't really matter, you can focus on what's really going on in the math problem.
Analogies
Analogies are key in understanding tricky problems, revealing their inner workings or even the big ideas behind them. Let's take the example of polling—asking people their opinions on something. It's a bit like dipping a test tube into the ocean to figure out how salty it is. Just as you need samples from different spots in the ocean to understand its overall saltiness, you need opinions from different people to gauge if a proposal is politically viable.
Add Meaning
This suggestion is one that I find fascinating. If you're not feeling motivated to tackle a problem, chances are you're not fully engaged in the whole process. Tao suggests shifting your mindset—rather than just solving for X or Y, think of it as embarking on a quest for clues or engaging in a strategic battle against a clever adversary. Since problem-solving can be demanding and often includes multiple failures, finding or attributing some sense of meaning in the journey becomes crucial. It's about turning the process into something more than just solving equations; it's about making it a meaningful endeavor.
Question Everything
When Tao was a young student, he often questioned the traditional ways of solving problems. There's this memory of him attempting to tackle an exam without relying on the quadratic formula, which was suggested by his teacher. He decided to give it a shot using his own method initially, only turning to the teacher's advice after struggling for an hour to solve the equation. He encourages others to challenge the status quo, even if it means being proven wrong eventually. It's all about questioning and experimenting with different approaches.
When You’re Stuck
What else can Tao teach us? Another section that jumped out of me was one where he talks about what to do when you’re stumped by a math problem, or really, any situation that needs concentration and creative problem solving. Here are Tao’s tips:
Jump Right In
Quit worrying and give it a shot. There's a possibility you won't nail the answer right away. However, the effort you put in might spark ideas that lead to success later on—or it could be the mental warm-up your brain needs for tackling future problems.
Simplify
In the realm of mathematics, this could mean working with smaller numbers, adjusting constraints, and consolidating like terms through addition or subtraction. In the broader scope of life, it might involve breaking down substantial problems into manageable pieces, making a list of what you find challenging, and attempting to see the bigger picture of your tasks from a bird's-eye view.
Be Confident
Think about the problems you've already cracked and see if the knowledge you gained can be used to tackle the issue in front of you. You've accomplished quite a bit in your life, no doubt. Chances are, you've likely solved problems similar to the one you're dealing with right now. Your past successes can be valuable tools for overcoming current challenges.
Enjoy the Process
The Danish physicist Niels Bohr once suggested that an expert is someone "who has found out by his own painful experience all the mistakes that one can make in a very narrow field." Tao finds value in this idea, using it to remind himself that time spent learning is never wasted. The objective isn't just about solving a problem rapidly or efficiently; it's about finding enjoyment in the process and extracting valuable lessons along the way.
Cheat (Ethically)
In this context, cheating refers to tackling a new problem using tools you're already familiar with. Similar to a video game's "cheat code," which lets you skip certain tasks or stages, math cheats can help you progress through a problem more quickly. Tao, as an example, might turn a curve into a straight line to simplify a problem using a mathematical tool called linearization. The idea is to use your existing knowledge strategically to navigate challenges more efficiently.
A Different Approach
When Terence feels stuck, it's often because he's fixated on solving a problem in a particular way. To break free from this mental roadblock, he consciously releases his preconceived notions. Terence believes that ideas born out of desperation can be beneficial in these instances. Letting go of rigid thinking allows for new, potentially more effective approaches to problem-solving.
Collaborate
The more people collaborating on a problem, the stronger the team. Perhaps one member is an expert in a unique math field, while another has insights from some obscure yet relevant literature. Tao enjoys teaming up to solve problems because it enhances the satisfaction of finding a solution. "When you and your collaborator are both standing at the blackboard... and everything checks out... that's a great experience [and] more fun than if you're working on your own." Teamwork not only brings diverse perspectives but also adds a layer of enjoyment to the problem-solving process.
Here are some articles to read that talk about the idea of math and language arts being more closely connected than we think:
How reading comprehension can boost math scores
Why Being Good at Language Arts Means That You Can Do Math
Tao also recommended a few books to learn more about mathematical thinking. I’m now reading one of those books. “How Not to Be Wrong: The Power of Mathematical Thinking.” This probably means that I’ll be talking about it soon in an upcoming essay, with a lot more to say for paid subscribers. (If you buy the book from the link above, I get a small percentage) Be on the look out for that!
I hope you enjoyed reading this essay. It took a lot of effort to put together. I’ll certainly be doing more masterclasses and breaking them down like this, so if you found it useful, please consider upgrading to a paid subscription, sharing this post and telling others in your feeds about Constantly Curious. It takes a lot of time and effort to put this newsletter together every week, so I’d appreciate your support.
What do you think of Terrence Tao’s tips for being a more mathematical thinker? Let me know in the comments.
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What an interesting and unexpectedly fun essay...