# Tag: Math

## Learning Through Play

Play is an essential way of learning about the world. Doing things we enjoy without a goal in mind leads us to find new information, better understand our own capabilities, and find unexpected beauty around us. Arithmetic is one example of an area we can explore through play.

Every parent knows that children need space for unstructured play that helps them develop their creativity and problem-solving skills. Free-form experimentation leads to the rapid acquisition of information about the world. When children play together, they expand their social skills and strengthen the ability to regulate their emotions. Young animals, such as elephants, dogs, ravens, and crocodiles, also develop survival skills through play.

The benefits of play don’t disappear as soon as you become an adult. Even if we engage our curiosity in different ways as we grow up, a lot of learning and exploration still comes from analogous activities: things we do for the sheer fun of it.

When the pressure mounts to be productive every minute of the day, we have much to gain from doing all we can to carve out time to play. Take away prescriptions and obligations, and we gravitate towards whatever interests us the most. Just like children and baby elephants, we can learn important lessons through play. It can also give us a new perspective on topics we take for granted—such as the way we represent numbers.

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## Playing with symbols

The book Arithmetic, in addition to being a clear and engaging history of the subject, is a demonstration of how insights and understanding can be combined with enjoyment and fun. The best place to start the book is at the afterword, where author and mathematics professor Paul Lockhart writes, “I especially hope that I have managed to get across the idea of viewing your mind as a playground—a place to create beautiful things for your own pleasure and amusement and to marvel at what you’ve made and at what you have yet to understand.

Arithmetic, the branch of math dealing with the manipulation and properties of numbers, can be very playful. After all, there are many ways to add and multiply numbers that in themselves can be represented in various ways. When we see six cows in a field, we represent that amount with the symbol 6. The Romans used VI. And there are many other ways that unfortunately can’t be typed on a standard English keyboard. If two more cows wander into the field, the usual method of counting them is to add 2 to 6 and conclude there are now 8 cows. But we could just as easily add 2 + 3 + 3. Or turn everything into fractions with a base of 2 and go from there.

One of the most intriguing parts of the book is when Lockhart encourages us to step away from how we commonly label numbers so we can have fun experimenting with them. He says, “The problem with familiarity is not so much that it breeds contempt, but that it breeds loss of perspective.” So we don’t get too hung up on our symbols such as 4 and 5, Lockhart shows us how any symbols can be used to complete some of the main arithmetic tasks such as comparing and grouping. He shows how completely random symbols can represent amounts and gives insight into how they can be manipulated.

When we start to play with the representations, we connect to the underlying reasoning behind what we are doing. We could be counting for the purposes of comparison, and we could also be interested in learning the patterns produced by our actions. Lockhart explains that “every number can be represented in a variety of ways, and we want to choose a form that is as useful and convenient as possible.” We can thus choose our representations of numbers based on curiosity versus what is conventional. It’s easy to extrapolate this thinking to broader life situations. How often do we assume certain parameters are fixed just because that is what has always been done? What else could we accomplish if we let go of convention and focused instead on function?

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## Stepping away from requirements

We all use the Hindu-Arabic number system, which utilizes groups of tens. Ten singles are ten, ten tens are a hundred, and so on. It has a consistent logic to it, and it is a pervasive way of grouping numbers as they increase. But Lockhart explains that grouping numbers by ten is as arbitrary as the symbols we use to represent numbers. He explains how a society might group by fours or sevens. One of the most interesting ideas though, comes when he’s explaining the groupings:

“You might think there is no question about it; we chose four as our grouping size, so that’s that. Of course we will group our groups into fours—as opposed to what? Grouping things into fours and then grouping our groups into sixes? That would be insane! But it happens all the time. Inches are grouped into twelves to make feet, and then three feet make a yard. And the old British monetary system had twelve pence to the shilling and twenty shillings to the pound.”

By reminding us of the options available in such a simple, everyday activity as counting, Lockhart opens a mental door. What other ways might we go about our tasks and solve our problems? It’s a reminder that most of our so-called requirements are ones that we impose on ourselves.

If we think back to being children, we often played with things in ways that were different from what they were intended for. Pots became drums and tape strung around the house became lasers. A byproduct of this type of play is usually learning—we learn what things are normally used for by playing with them. But that’s not the intention behind a child’s play. The fun comes first, and thus they don’t restrain themselves to convention.

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## Have fun with the unfamiliar

There are advantages and disadvantages to all counting systems. For Lockhart, the only way to discover what those are is to play around with them. And it is in the playing that we may learn more than arithmetic. For example, he says: “In fact, getting stuck (say on 7 +8 for instance) is one of the best things that can happen to you because it gives you an opportunity to reinvent and to appreciate exactly what it is that you are doing.” In the case of adding two numbers, we “are rearranging numerical information for comparison purposes.

The larger point is that getting stuck on anything can be incredibly useful. If forces you to stop and consider what it is you are really trying to achieve. Getting stuck can help you identify the first principles in your situation. In getting unstuck, we learn lessons that resonate and help us to grow.

Lockhart says of arithmetic that we need to “not let our familiarity with a particular system blind us to its arbitrariness.” We don’t have to use the symbol 2 to represent how many cows there are in a field, just as we don’t have to group sixty minutes into one hour. We may find those representations useful, but we also may not. There are some people in the world with so much money that the numbers that represent their wealth are almost nonsensical, and most people find the clock manipulation that is the annual flip to daylight savings time to be annoying and stressful.

Playing around with arithmetic can teach the broader lesson that we don’t have to keep using systems that no longer serve us well. Yet how many of us have a hard time letting go of the ineffective simply because it’s familiar?

Which brings us back to play. Play is often the exploration of the unfamiliar. After all, if you knew what the result would be, it likely wouldn’t be considered play. When we play we take chances, we experiment, and we try new combinations just to see what happens. We do all of this in the pursuit of fun because it is the novelty that brings us pleasure and makes play rewarding.

Lockhart makes a similar point about arithmetic:

“The point of studying arithmetic and its philosophy is not merely to get good at it but also to gain a larger perspective and to expand our worldview . . . Plus, it’s fun. Anyway, as connoisseurs of arithmetic, we should always be questioning and critiquing, examining and playing.”

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We suggest that playing need not be confined to arithmetic. If you happen to enjoy playing with numbers, then go for it. Lockhart’s book gives great inspiration on how to have fun with numbers. Playing is inherently valuable and doesn’t need to be productive. Children and animals have no purpose for play; they merely do what’s fun. It just so happens that unstructured, undirected play often has incredibly powerful byproducts.

Play can lead to new ideas and innovations. It can also lead to personal growth and development, not to mention a better understanding of the world. And, by its definition, play leads to fun. Which is the best part. Arithmetic is just one example of an unexpected area we can approach with the spirit of play.

## The Nerds Were Right. Math Makes Life Beautiful.

Math has long been the language of science, engineering, and finance, but can math help you feel calm on a turbulent flight? Get a date? Make better decisions? Here are some heroic ways math shows up in our everyday life.

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Sounds intellectually sophisticated, doesn’t it? Other than sounding really smart at after-work cocktails, what could be the benefit of understanding where math and physics permeate your life?

Well, what if I told you that math and physics can help you make better decisions by aligning with how the world works? What if I told you that math can help you get a date? Help you solve problems? What if I told you that knowing the basics of math and physics can help make you less afraid and confused? And, perhaps most important, they can help make life more beautiful. Seriously.

If you’ve ever been on a plane when turbulence has hit, you know how unnerving that can be. Most people get freaked out by it, and no matter how much we fly, most of us have a turbulence threshold. When the sides of the plane are shaking, noisily holding themselves together, and the people beside us are white with fear, hands clenched on their armrests, even the calmest of us will ponder the wisdom of jetting 38,000 feet above the ground in a metal tube moving at 1,000 km an hour.

Considering that most planes don’t fall from the sky on account of turbulence isn’t that comforting in the moment. Aren’t there always exceptions to the rule? But what if you understood why, or could explain the physics involved to the freaked-out person beside you? That might help.

In Storm in a Teacup: The Physics of Everyday Life, Helen Czerski spends a chapter describing the gas laws. Covering subjects from the making of popcorn to the deep dives of sperm whales, her amazingly accessible prose describes how the movement of gas is fundamental to the functioning of pretty much everything on earth, including our lungs. She reveals air to be not the static clear thing that we perceive when we bother to look, but rivers of molecules in constant collision, pushing and moving, giving us both storms and cloudless skies.

So when you appreciate air this way, as a continually flowing and changing collection of particles, turbulence is suddenly less scary. Planes are moving through a substance that is far from uniform. Of course, there are going to be pockets of more or less dense air molecules. Of course, they will have minor impacts on the plane as it moves through these slightly different pressure areas. Given that the movement of air can create hurricanes, it’s amazing that most flights are as smooth as they are.

You know what else is really scary? Approaching someone for a date or a job. Rejection sucks. It makes us feel awful, and therefore the threat of it often stops us from taking risks. You know the scene. You’re out at a bar with some friends. A group of potential dates is across the way. Do you risk the cringingly icky feeling of rejection and approach the person you find most attractive, or do you just throw out a lot of eye contact and hope that person approaches you?

Most men go with the former, as difficult as it is. Women will often opt for the latter. We could discuss social conditioning, with the roles that our culture expects each of us to follow. But this post is about math and physics, which actually turn out to be a lot better in providing guidance to optimize our chances of success in the intimidating bar situation.

In The Mathematics of Love, Hannah Fry explains the Gale-Shapley matching algorithm, which essentially proves that “If you put yourself out there, start at the top of the list, and work your way down, you’ll always end up with the best possible person who’ll have you. If you sit around and wait for people to talk to you, you’ll end up with the least bad person who approaches you. Regardless of the type of relationship you’re after, it pays to take the initiative.”

The math may be complicated, but the principle isn’t. Your chances of ending up with what you want — say, the guy with the amazing smile or that lab director job in California — dramatically increase if you make the first move. Fry says, “aim high, and aim frequently. The math says so.” Why argue with that?

Understanding more physics can also free us from the panic-inducing, heart-pounding fear that we are making the wrong decisions. Not because physics always points out the right decision, but because it can lead us away from this unproductive, subjective, binary thinking. How? By giving us the tools to ask better questions.

Consider this illuminating passage from Czerski:

We live in the middle of the timescales, and sometimes it’s hard to take the rest of time seriously. It’s not just the difference between now and then, it’s the vertigo you get when you think about what “now” actually is. It could be a millionth of a second, or a year. Your perspective is completely different when you’re looking at incredibly fast events or glacially slow ones. But the difference hasn’t got anything to do with how things are changing; it’s just a question of how long they take to get there. And where is “there”? It is equilibrium, a state of balance. Left to itself, nothing will ever shift from this final position because it has no reason to do so. At the end, there are no forces to move anything, because they’re all balanced. They physical world, all of it, only ever has one destination: equilibrium.

How can this change your decision-making process?

You might start to consider whether you are speeding up the goal of equilibrium (working with force) or trying to prevent equilibrium (working against force).  One option isn’t necessarily worse than the other. But the second one is significantly more work.

So then you will understand how much effort is going to be required on your part. Love that house with the period Georgian windows? Great. But know that you will have to spend more money fighting to counteract the desire of the molecules on both sides of the window to achieve equilibrium in varying temperatures than you will if you go with the modern bungalow with the double-paned windows.

And finally, curiosity. Being curious about the world helps us find solutions to problems by bringing new knowledge to bear on old challenges. Math and physics are actually powerful tools for investigating the possibilities of what is out there.

Fry writes that “Mathematics is about abstracting away from reality, not replicating it. And it offers real value in the process. By allowing yourself to view the world from an abstract perspective, you create a language that is uniquely able to capture and describe the patterns and mechanisms that would otherwise remain hidden.”

Physics is very similar. Czerski says, “Seeing what makes the world tick changes your perspective. The world is a mosaic of physical patterns, and once you’re familiar with the basics, you start to see how those patterns fit together.”

Math and physics enhance your curiosity. These subjects allow us to dive into the unknown without being waylaid by charlatans or sidetracked by the impossible. They allow us to tackle the mysteries of life one at a time, opening up the possibilities of the universe.

As Czerski says, “Knowing about some basics bits of physics [and math!] turns the world into a toybox.” A toybox full of powerful and beautiful things.

## The Simple Problem Einstein Couldn’t Solve … At First

Albert Einstein and Max Wertheimer were close friends. Both found themselves in exile in the United States after fleeing the Nazis in the early 1930s, Einstein at Princeton and Wertheimer in New York.

They communicated by exchanging letters in which Wertheimer would entertain Einstein with thought problems.

In 1934 Wertheimer sent the following problem in a letter.

An old clattery auto is to drive a stretch of 2 miles, up and down a hill, /\. Because it is so old, it cannot drive the first mile— the ascent —faster than with an average speed of 15 miles per hour. Question: How fast does it have to drive the second mile— on going down, it can, of course, go faster—in order to obtain an average speed (for the whole distance) of 30 miles an hour?

Wertheimer’s thought problem suggests the answer might be 45 or even 60 miles an hour. But that is not the case. Even if the car broke the sound barrier on the way down, it would not achieve an average speed of 30 miles an hour. Don’t be worried if you were fooled, Einstein was at first too. Replying “Not until calculating did I notice that there is no time left for the way down!”

Gerd Gigerenzer explains the answer in his book Risk Savvy: How to Make Good Decisions:

Gestalt psychologists’ way to solve problems is to reformulate the question until the answer becomes clear. Here’s how it works. How long does it take the old car to reach the top of the hill? The road up is one mile long. The car travels fifteen miles per hour, so it takes four minutes (one hour divided by fifteen) to reach the top. How long does it take the car to drive up and down the hill, with an average speed of thirty miles per hour? The road up and down is two miles long. Thirty miles per hour translates into two miles per four minutes. Thus, the car needs four minutes to drive the entire distance. But these four minutes were already used up by the time the car reached the top.

## The Ability To Focus And Make The Best Move When There Are No Good Moves

“The indeterminate future is somehow one in which probability and statistics are the dominant modalities for making sense of the world.”

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Decisions, where outcomes (and therefore probabilities) are unknown, are often the hardest. The default method of problem-solving often falls short.

Sometimes you have to play the odds and sometimes you have to play the calculus.

There are several different frameworks one could use to get a handle on the indeterminate vs. determinate question. The math version is calculus vs. statistics. In a determinate world, calculus dominates. You can calculate specific things precisely and deterministically. When you send a rocket to the moon, you have to calculate precisely where it is at all times. It’s not like some iterative startup where you launch the rocket and figure things out step by step. Do you make it to the moon? To Jupiter? Do you just get lost in space? There were lots of companies in the ’90s that had launch parties but no landing parties.

But the indeterminate future is somehow one in which probability and statistics are the dominant modality for making sense of the world. Bell curves and random walks define what the future is going to look like. The standard pedagogical argument is that high schools should get rid of calculus and replace it with statistics, which is really important and actually useful. There has been a powerful shift toward the idea that statistical ways of thinking are going to drive the future.

With calculus, you can calculate things far into the future. You can even calculate planetary locations years or decades from now. But there are no specifics in probability and statistics—only distributions. In these domains, all you can know about the future is that you can’t know it. You cannot dominate the future; antitheories dominate instead. The Larry Summers line about the economy was something like, “I don’t know what’s going to happen, but anyone who says he knows what will happen doesn’t know what he’s talking about.” Today, all prophets are false prophets. That can only be true if people take a statistical view of the future.

— Peter Thiel

And this quote from The Hard Thing About Hard Things: Building a Business When There Are No Easy Answers by Ben Horowitz:

I learned one important lesson: Startup CEOs should not play the odds. When you are building a company, you must believe there is an answer and you cannot pay attention to your odds of finding it. You just have to find it. It matters not whether your chances are nine in ten or one in a thousand; your task is the same. … I don’t believe in statistics. I believe in calculus.

People always ask me, “What’s the secret to being a successful CEO?” Sadly, there is no secret, but if there is one skill that stands out, it’s the ability to focus and make the best move when there are no good moves. It’s the moments where you feel most like hiding or dying that you can make the biggest difference as a CEO. In the rest of this chapter, I offer some lessons on how to make it through the struggle without quitting or throwing up too much.

… I follow the first principle of the Bushido—the way of the warrior: keep death in mind at all times. If a warrior keeps death in mind at all times and lives as though each day might be his last, he will conduct himself properly in all his actions. Similarly, if a CEO keeps the following lessons in mind, she will maintain the proper focus when hiring, training , and building her culture.

It’s interesting to me that the skill that stands out to Horowitz is one that we can use to teach how to think and one Tyler Cowen feels is in short supply. Cowen says:

The more information that’s out there, the greater the returns to just being willing to sit down and apply yourself. Information isn’t what’s scarce; it’s the willingness to do something with it.