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Explore Or Exploit? How To Choose New Opportunities

One big challenge we all face in life is knowing when to explore new opportunities, and when to double down on existing ones. Explore vs exploit algorithms – and poetry – teach us that it’s vital to consider how much time we have, how we can best avoid regrets, and what we can learn from failures.

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“Had we but world enough, and time,
This coyness, Lady, were no crime.
We would sit down and think which way
To walk and pass our long love’s day . . .

Let us roll all our strength and all
Our sweetness up into one ball,
And tear our pleasures with rough strife
Thorough the iron gates of life:
Thus, though we cannot make our sun
Stand still, yet we will make him run.”
—Andrew Marvell, To His Coy Mistress

Of all the questions life demands we answer, “To explore or to exploit?” is one we have to confront almost every day. Do we keep trying new restaurants? Do we keep learning new ideas? Do we keep making new friends? Or do we enjoy what we’ve come to find and love?

There is no doubt that humans are great at exploring, as most generalist species are. Not content to stay in that cave, hunt that animal, or keep doing it the way our grandmother taught us, humans owe at least part of our success due to our willingness to explore.

But when is what you’ve already explored enough? When can you finally settle down to enjoy the fruits of your exploration? When can you be content to exploit the knowledge you already have?

Turns out that there are algorithms for that.

In Algorithms to Live By, authors Brian Christian and Tom Griffiths devote an entire chapter to how computer algorithms deal with the explore/exploit conundrum and how you can apply those lessons to the same tension in your life.

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How much time do you have?

One of the most important factors in determining whether to continue exploring or to exploit what you’ve got is time. Christian and Griffiths explain that “seizing a day and seizing a lifetime are two entirely different endeavors. . . . When balancing favorite experiences and new ones, nothing matters as much as the interval over which we plan to enjoy them.

Time intervals can be a construct of your immediate circumstances, like the boundaries provided by a two-week vacation. For a lot of us, the last night in a lovely foreign place will see us eating at the best restaurant we have found so far. Time intervals can also be considered over the arc of your life in general. Children are consummate explorers, but as we grow up, the choice to exploit becomes more of a daily decision. How would your choices today be impacted if you knew you were going to live another five years? Twenty years? Forty years? Christian and Griffiths advise, “Explore when you will have time to use the resulting knowledge, exploit when you’re ready to cash in.”

“I have known days like that, of warm winds drowsing in the heat
of noon and all of summer spinning slowly on its reel,
days briefly lived, that leave long music in the mind
more sweet than truth: I play them and rewind.”
—Russell Hoban, Summer Recorded

Sometimes we are too quick to stop exploring. We have these amazing days and magical experiences, and we want to keep repeating them forever. However, changes in ourselves and the world around us are inevitable, and so committing to a path of exploitation too early leaves us unable to adapt. As much as it can be hard to walk away from that perfect day, Christian and Griffiths explain that “exploration in itself has value, since trying new things increases our chances of finding the best. So taking the future into account, rather than focusing just on the present, drives us toward novelty.

“Like as the waves make towards the pebbled shore,
So do our minutes hasten to their end;
Each changing place with that which goes before,
In sequent toil all forwards do contend.”
—William Shakespeare, Sonnet 60

There is no doubt that for many of us time is our most precious resource. We never seem to have enough, and we want to maximize the value we get from how we choose to use it. So when deciding between whether to enjoy what you have or search for something better, adding time to your decision-making process can help point the way.

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Minimizing the pain of regret

The threat of regret looms over many explore/exploit considerations. We can regret both not searching for something better and not taking the time to enjoy what we already have. The problem with regret is that we don’t have it in advance of a poor decision. Sometimes, second-order thinking can be used as a preventative tool. But often it is when you look back over a decision that regret comes out. Christian and Griffiths define regret as “the result of comparing what we actually did with what would have been best in hindsight.”

“Does the road wind uphill all the way?
Yes, to the very end.
Will the day’s journey take the whole long day?
From morn to night, my friend.
Shall I find comfort, travel-sore and weak?
Of labour you shall find the sum.
Will there be beds for me and all who seek?
Yea, beds for all who come.”
—Christina Rossetti, “Up-Hill”

If we want to minimize regret, especially in exploration, we can try to learn from those who have come before. As we choose to wander forth into new territory, however, it’s natural to wonder if we’ll regret our decision to try something new. According to Christian and Griffiths, the mathematics that underlie explore/exploit algorithms show that “you should assume the best about [new people and new things], in the absence of evidence to the contrary. In the long run, optimism is the best prevention for regret.” Why? Because by being optimistic about the possibilities that are out there, you’ll explore enough that the one thing you won’t regret is missed opportunity.

(This is similar to one of the most effective strategies in game theory: tit for tat. Start out by being nice, then reciprocate whatever behavior you receive. It often works better paired with the occasional bout of forgiveness.)

“Tell me, tell me, smiling child,
What the past is like to thee?
‘An Autumn evening soft and mild
With a wind that sighs mournfully.’

Tell me, what is the present hour?
‘A green and flowery spray
Where a young bird sits gathering its power
To mount and fly away.’

And what is the future, happy one?
‘A sea beneath a cloudless sun;
A mighty, glorious, dazzling sea
Stretching into infinity.’”
—Emily Bronte, “Past, Present, Future”

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The accumulation of knowledge

Christian and Griffiths write that “it’s rare that we make an isolated decision, where the outcome doesn’t provide us with any information that we’ll use to make other decisions in the future.” Not all of our explorations are going to lead us to something better, but many of them are. Not all of our exploitations are going to be satisfying, but with enough exploration behind us, many of them will. Failures are, after all, just information we can use to make better explore or exploit decisions in the future.

“You know—at least you ought to know,
For I have often told you so—
That children are never allowed
To leave their nurses in a crowd.
Now this was Jim’s especial foible,
He ran away when he was able,
And on this inauspicious day
He slipped his hand and ran away!
He hadn’t gone a yard when—Bang!
With open jaws, a lion sprang,
And hungrily began to eat
The boy: beginning at his feet.”
—Hilaire Belloc, Jim Who Ran Away from His Nurse, and Was Eaten by a Lion

Most importantly, we shouldn’t let our early exploration mishaps prevent us from continuing to push our boundaries as we grow up. Exploration is necessary in order to exploit and enjoy the knowledge hard won along the way.

Mental Models for Career Changes

Career changes are some of the biggest moves we will ever make, but they don’t have to be daunting. Using mental models to make decisions we determine where we want to go and how to get there. The result is a change that aligns with the person we are, as well as the person we want to be.

We’ve all been there: you’re at a job, and you know it’s not for you anymore. You come in drained, you’re not excited on a Monday morning, and you feel like you could be using your time so much better. It’s not the people, and it’s not the organization. It’s the work. It’s become boring, unfulfilling, or redundant, and you know you want to do something different. But what?

Just deciding to change careers doesn’t get you very far because there are more areas to work in than you know about. A big change often involves some retraining. A career shift will impact your personal life. At the end of it all, you want to be happier but know there are no guarantees. How do you find a clear path forward?

No matter how ready you think you are to make a move, career changes are daunting. The stress of leaving what you’re comfortable with to venture into foreign territory stops many people from taking the first step toward something new.

It doesn’t have to be this way.

Using mental models can help you clarify the direction you want to go and plan for how to get there. They are tools that will give you more control over your career and more confidence in your decisions. When you do the work up front by examining your situation through the lens of a few mental models, you set yourself up for fewer regrets and more satisfaction down the road.

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Get in touch with yourself

Before you can decide which change to make, you need to get in touch with yourself. No change will be the right one if it doesn’t align with what you want to get out of life.

First, do you know where you want to go? Are you moving with direction or just moving? As a mental model, velocity reminds us there is a difference between speed and direction. It’s easy to move fast without getting anywhere. We can stay busy all day without achieving our goals. Without considering our velocity, we run a huge risk of getting sidetracked by things that make us move faster (more money, a title on a business card) without that movement actually leading us where we want to end up.

As the old saying goes, we want to run to something, not from something. When you start articulating your desired direction, you give yourself clear purpose in your career. It will be easier to play the long game because you know that everything you are doing is leading somewhere you want to be.

When it comes to changing careers, there are a lot of options. Using the mental model of velocity will help you focus on and identify the best opportunities.

Once you know where you want to end up, it’s often useful to work backward to where you are now. This is known as inversion. Start at the end and carefully consider the events that get you there in reverse order.

For example, it could be something as simple as waking up happy and excited to work every day. What needs to be true in order for that to be a reality? Are you working from home, having a quiet cup of coffee as you prepare to do some creative work? Are you working on projects aligned with your values? Are you contributing to making the world a better place? Are you in an intense, collaborative team environment?

Doing an inversion exercise helps you identify the elements needed for you to achieve success. Once you identify your requirements, you can use that list to evaluate opportunities that come up.

Inversion will help you recognize critical factors, like finances or the support of your family, that will be necessary to get to where you want to go. If your dream direction requires you to learn a new skill or work at a junior level while you ramp up on the knowledge you’ll need, you might need to live off some savings in the short term. Inversion, combined with velocity, will help you create the foundation you need now to take action when the right time comes.

Finally, the last step before you start evaluating the career environment is taking stock of the skills you already have. Why do you need to do this? So you know what you can repurpose. Here, you’re using the concept of exaptation, which is part of the broader adaptation model in biology. Exaptation refers to traits that evolved for one purpose and then, through natural selection, were used for completely unrelated capabilities. For instance, feathers probably evolved for insulation. It was only much later that they turned out to be useful for flying.

History is littered with examples of technologies or tools invented for one purpose that later became the foundation for something completely different. Did you know that Play-Doh was originally created to clean coal soot off walls? And bubble wrap was originally envisioned as material for shower curtains.

Using this model is partly about getting out of the “functional fixedness” mindset. You want to look at your skills, talents, and knowledge and ask of each one: what else could this be used for?

Too often we fail to realize just how versatile the experience we’ve built up over the years is. We’re great at using forks to eat, but they can also be used to brush hair, dig in a garden, and pin things to walls. Being great at presenting the monthly status update doesn’t mean you’re good at presenting monthly status updates. Rather, it means you can articulate yourself well, parse information for a diverse audience, and build networks to get the right information. Now, what else can those skills be used for?

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Evaluate the environment

Looking at different careers, we’re usually in a situation where the “map is not the territory.” It’s hard to know how great (or terrible) a job is until you actually do it. We often have two types of maps for the careers we wish we had: maps of the highlights, success stories, and opinions of people who love the work and maps based on how much we love the field or discipline ourselves.

The territory of the day-to-day work of these careers, however, is very different from what those two maps tell us.

In order to determine if a particular career will work for us, we need better maps. For example, the reality of being an actor isn’t just the movies and programs you see them in. It’s audition after audition, with more rejections than roles. It’s intense competition and job insecurity. Being a research scientist at a university isn’t just immersing yourself in a subject you love. It’s grant applications and teaching and navigating the bureaucracy of academia.

In order to build a more comprehensive map of your dream job, do your research on as large a sample size as possible. Talk to people doing the job you want. Talk to people who work in the organization. Talk to the ones that enjoy it. Talk to the ones who quit. Try to get an accurate picture of what the day-to-day is like.

Very few jobs are one-dimensional. They involve things like administrative tasks, networking, project management, and accountability. How much of your day will be spent doing paperwork or updating your coworkers? How much of a connection do you need to maintain with people outside the organization? How many people will you be dependent on? What are they like? And who will you be working for?

It’s not a good idea to become a writer just because you want to tell stories, open a restaurant just because you like to cook, or become a landscape designer just because you enjoy being outside. Those motivations are good places to start—because it’s equally terrible to become a lawyer just because your parents wanted you to. But you can’t stop with what you like. There isn’t a job in the world that’s pleasurable and fulfilling 100% of the time.

You give yourself a much higher chance of being satisfied with your career change if you take the time to learn as much as you can about the territory beforehand.

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Elements of planning

You know which direction you’re heading in, and you’ve identified a great new career possibility. Now what?

Planning for change is a crucial component of switching careers. Two models, global and local maxima and activation energy, can help us identify what we need to plan.

Global and local maxima refers to the high values in a mathematical function. On a graph, it’s a wavy curve with peaks and valleys. The highest peak in a section is a local maximum. The highest peak across the entire graph is the global maximum. Activation energy comes from chemistry, and is the amount of energy needed to see a reaction through to its conclusion.

One of the things global and local maxima teaches us is that sometimes you have to go down a hill in order to climb up a new one. To move from a local maximum to a higher peak you have to go through a local minimum, a valley. Too often we just want to go higher right away, or at the very least we want to make a lateral move. We perceive going down as taking a step backward.

A common problem is when we tie our self-worth to our salary and therefore reject any opportunities that won’t pay us as much as we’re currently making. The same goes for job titles; no one wants to be a junior anything in their mid-forties. But it’s impossible to get to the next peak if we won’t walk through the valley.

If you look at your career change through the lens of global and local maxima, you will see that steps down can also be steps forward.

Activation energy is another great model to use in the planning phase because it requires you to think about the real effort required for sustained change. You need to plan not just for making a change but also for seeing it through until the new thing has time to take hold.

Do you have enough in the bank to support yourself if you need to retrain or take a pay cut? Do you have the emotional support to help you through the challenges of taking on a brand-new career?

Just like fires don’t start with one match and a giant log, you have to plan for what you need between now and your desired result. What do you need to keep that reaction going so the flame from the match leads to the log catching fire? The same kind of thinking needs to inform your planning. After you’ve taken the first step, what will you need to keep you moving in the direction you want to go?

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After you’ve done all the work

After getting in touch with yourself, doing all your research, identifying possible paths, and planning for what you need to do to walk them to the end, it can still be hard to make a decision. You’ve uncovered so many nuances and encountered so many ideas that you feel overwhelmed. The reality is, when it comes to career change, there often is no perfect decision. You likely have more than one option, and whatever you choose, there’s going to be a lot of work involved.

One final model you can use is probabilistic thinking. In this particular situation, it can be helpful to use a Bayesian casino.

A Bayesian casino is a thought experiment where you imagine walking up to a casino game, like roulette, and quantifying how much you would bet on any particular outcome.

Let’s say when investigating your career change, you’ve narrowed it down to two options. Which one would you bet on for being the better choice one year later? And how much would you part with? If you’d bet ten dollars on black, then you probably need to take a fresh look at the research you’ve done. Maybe go talk to more people, or broaden your thinking. If you’re willing to put down thousands of dollars on red, that’s very likely the right decision for you.

It’s important in this thought experiment to fully imagine yourself making the bet. Imagine the money in your bank account. Imagine withdrawing it and physically putting it down on the table. How much you’re willing to part with regarding a particular career choice says a lot about how good that choice is likely to be for you.

Probabilistic thinking isn’t a predictor of the future. With any big career move, there are inevitably a lot of unknowns. There are no guarantees that any choice is going to be the right one. The Bayesian casino just helps you quantify your thinking based on the knowledge you have at this moment in time.

As new information comes in, return to the casino and see if your bets change.

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Conclusion

Career changes are some of the biggest moves we will ever make, but they don’t have to be daunting. Using mental models helps us find both the direction we want to go and a path we can take to get there. The result is a change that aligns with the person we are, as well as the person we want to be.

How Julia Child Used First Principles Thinking

There’s a big difference between knowing how to follow a recipe and knowing how to cook. If you can master the first principles within a domain, you can see much further than those who are just following recipes. That’s what Julia Child, “The French Chef”, did throughout her career.

Following a recipe might get you the results you want, but it doesn’t teach you anything about how cooking works at the foundational level. Or what to do when something goes wrong. Or how to come up with your own recipes when you open the fridge on a Wednesday night and realize you forgot to go grocery shopping. Or how to adapt recipes for your own dietary needs.

Adhering to recipes will only get you so far, and it certainly won’t result in you coming up with anything new or creative.

People who know how to cook understand the basic principles that make food taste, look, and smell good. They have confidence in troubleshooting and solving problems as they go—or adjusting to unexpected outcomes. They can glance at an almost barren kitchen and devise something delicious. They know how to adapt to a guest with a gluten allergy or a child who doesn’t like green food. Sure, they might consult a recipe when it makes sense to do so. But they’re not dependent on it, and they can change it up based on their particular circumstances.

There’s a reason many cooking competition shows feature a segment where contestants need to design their own recipe from a limited assortment of ingredients. Effective improvisation shows the judges that someone can actually cook, not just follow recipes.

We can draw a strong parallel from cooking to thinking. If you want to learn how to think for yourself, you can’t just follow what someone else came up with. You need to understand first principles if you want to be able to solve complex problems or think in a unique, creative fashion. First principles are the building blocks of knowledge, the foundational understanding acquired from breaking something down into its most essential concepts.

One person who exemplifies first principles thinking is Julia Child, an American educator who charmed audiences with her classes, books, and TV shows. First principles thinking enabled Julia to both master her own struggles with cooking and then teach the world to do the same. In Something from the Oven, Laura Shapiro tells the charming story of how she did it. Here’s what we can learn about better thinking from the “French Chef.”

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Gustave Flaubert wrote that “talent is a long patience, ” something which was all too true for Julia. She wasn’t born with an innate skill for or even love of cooking. Her starting point was falling in love with her future husband, Paul Child, in Ceylon in 1944 when both were working for the Office of Strategic Services. Paul adored food, and his delight in it inspired Julia. When they each returned to their separate homes after the war, she decided she would learn to cook. Things got off to a bad start, as Shapiro explains:

“At first she tried to teach herself at home, but it was frustrating to bushwhack her way through one dish after another. She never knew whether she would find success or failure when she opened the oven door, and worst of all, she didn’t know why this recipe worked and that one didn’t.”

Seeking expert guidance, Julia started taking cooking classes three times a week at a Beverly Hills cooking school. Even that didn’t help much, however, and after she married Paul a year later, her experiments in their Washington, DC kitchen continued to go awry. Only when the couple moved to Paris did an epiphany strike. Julia’s encounters with French cooking instilled in her an understanding of the need for first principles thinking. Trying to follow recipes without comprehending their logic wasn’t going to produce delicious results. She needed to learn how food actually worked.

In 1949, at the age of 37, she enrolled in classes at the famous Cordon Bleu school of cooking. It changed her forever:

“Learning to cook at the Cordon Bleu meant breaking down every dish into its smallest individual steps and doing each laborious and exhausting procedure by hand. In time Child could bone a duck while leaving the skin intact, extract the guts of a chicken through a hole she made in the neck, make a ham mousse by pounding the ham to a pulp with a mortar and pestle, and turn out a swath of elaborate dishes from choucroute garnie to vol-au-vent financière. None of this came effortlessly but she could do it. She had the brains, the considerable physical strength it demanded, and her vast determination. Most important, she could understand for the first time the principles governing how and why a recipe worked as it did.”

Julia had found her calling. After six months of Cordon Bleu classes, she continued studying independently for a year. She immersed herself in French cooking, filled her home with equipment, and befriended two women who shared her passion, Simone Beck and Louisette Bertholle. In the early 1950s, they opened a tiny school together, with a couple of students working out of Julia’s kitchen. She was “adamant that the recipes used in class be absolutely reliable, and she tested every one of them for what she called ‘scientific workability.’” By this, Julia meant that the recipes needed to make sense per her understanding of the science of cooking. If they didn’t agree with the first principles she knew, they were out.

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When Paul transferred to Marseille, Julia was sad to leave her school. But she and her friends continued their collaboration, working at a distance on a French cookery book aimed at Americans. For what would become Mastering the Art of French Cooking, Julia focused on teaching first principles in a logical order, not copying down mere recipes.

She’d grown frustrated at opening recipe books to see instructions she knew couldn’t work because they contradicted the science of cooking—for example, recipes calling for temperatures she knew would burn a particular ingredient, or omitting key ingredients like baking soda, without which a particular effect would be impossible. It was clear no one had bothered to test anything before they wrote it down, and she was determined not to make the same mistake.

Mastering the Art of French Cooking came out in 1961. Shapiro writes, “The reviews were excellent, there was a gratifying burst of publicity all across the country, and the professional food world acknowledged a new star in Julia Child. What nobody knew for sure was whether everyday homemakers in the nation that invented the TV dinner would buy the book.” Though the book was far from a flop, it was the TV show it inspired that catapulted Julia and her approach to cooking to stardom.

The French Chef first aired in 1963 and was an enormous success from the start. Viewers adored how Julia explained why she did what she did and how it worked. They also loved her spontaneous capacity to adapt to unanticipated outcomes. It was usually only possible to shoot one take so Julia needed to keep going no matter what happened.

Her show appealed to every kind of person because it could make anyone a better cook—or at least help them understand the process better. Not only was Julia “a striking image of unaffected good nature,” the way she taught really worked. Viewers and readers who followed her guidance discovered a way of cooking that made them feel in control.

Julia “believed anybody could cook with distinction from scratch and that’s what she was out to prove.” Many of the people who watched The French Chef were women who needed a new way to think about cooking. As gender roles were being redefined and more women entered the workforce, it no longer seemed like something they were obligated by birth to do. At the same time, treating it as an undesirable chore was no more pleasant than treating it as a duty. Julia taught them another way. Cooking could be an intellectual, creative, enjoyable activity. Once you understood how it actually worked, you could learn from mistakes instead of repeating them again and again.

Shapiro explains that “Child was certainly not the first TV chef. The genre was almost as old as TV itself. But she was the first to make it her own and have an enduring societal impact.”

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If you can master the first principles within a domain, you can see much further than those who are just following recipes. That’s what Julia managed to do, and it’s part of why she stood out from the other TV chefs of her time—and still stands out today. By mastering first principles, you can find better ways of doing things, instead of having to stick to conventions. If Julia thought a modern piece of equipment worked better than a traditional one or that part of a technique was a pointless custom, she didn’t hesitate to make changes as she saw fit. Once you know the why of something, it is easy to modify the how to achieve your desired result.

The lessons of first principles in cooking are the same for the first principles in any domain. Looking for first principles is just a way of thinking. It’s a commitment to understanding the foundation that something is built on and giving yourself the freedom to adapt, develop, and create. Once you know the first principles, you can keep learning more advanced concepts as well as innovating for yourself.

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.

Common Probability Errors to Avoid

If you’re trying to gain a rapid understanding of a new area, one of the most important things you can do is to identify common mistakes people make, then avoid them. Here are some of the most predictable errors we tend to make when thinking about statistics.

Amateurs tend to focus on seeking brilliance. Professionals often know that it’s far more effective to avoid stupidity. Side-stepping typical blunders is the simplest way to get ahead of the crowd.

Gaining a better understanding of probability will give you a more accurate picture of the world and help you make better decisions. However, many people fall prey to the same handful of issues because aspects of probability go against what we think is intuitive. Even if you haven’t studied the topic since high-school, you likely use probability assessments every single day in your work and life.

In Naked Statistics, Charles Wheelan takes the reader on a whistlestop tour of the basics of statistics. In one chapter, he offers pointers for avoiding some of the “most common probability-related errors, misunderstandings, and ethical dilemmas.” Whether you’re somewhat new to the topic or just want a refresher, here’s a summary of Wheelan’s lessons and how you can apply them.

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Assuming events are independent when they are not

“The probability of flipping heads with a fair coin is 1/2. The probability of flipping two heads in a row is (1/2)^2 or 1/4 since the likelihood of two independent events both happening is the product of their individual probabilities.”

When an event is interconnected with another event, the former happening increases or decreases the probability of the latter happening. Your car insurance gets more expensive after an accident because car accidents are not independent events. A person who gets in one is more likely to get into another in the future. Maybe they’re not such a good driver, maybe they tend to drive after a drink, or maybe their eyesight is imperfect. Whatever the explanation, insurance companies know to revise their risk assessment.

Sometimes though, an event happening might lead to changes that make it less probable in the future. If you spilled coffee on your shirt this morning, you might be less likely to do the same this afternoon because you’ll exercise more caution. If an airline had a crash last year, you may well be safer flying with them because they will have made extensive improvements to their safety procedures to prevent another disaster.

One place we should pay extra attention to the independence or dependence of events is when making plans. Most of our plans don’t go as we’d like. We get delayed, we have to backtrack, we have to make unexpected changes. Sometimes we think we can compensate for a delay in one part of a plan by moving faster later on. But the parts of a plan are not independent. A delay in one area makes delays elsewhere more likely as problems compound and accumulate.

Any time you think about the probability of sequences of events, be sure to identify whether they’re independent or not.

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Not understanding when events are independent

“A different kind of mistake occurs when events that are independent are not treated as such . . . If you flip a fair coin 1,000,000 times and get 1,000,000 heads in a row, the probability of getting heads on the next flip is still 1/2. The very definition of statistical independence between two events is that the outcome of one has no effect on the outcome of another.”

Imagine you’re grabbing a breakfast sandwich at a local cafe when someone rudely barges into line in front of you and ignores your protestations. Later that day, as you’re waiting your turn to order a latte in a different cafe, the same thing happens: a random stranger pushes in front of you. By the time you go to pick up some pastries for your kids at a different place before heading home that evening, you’re so annoyed by all the rudeness you’ve encountered that you angrily eye every person to enter the shop, on guard for any attempts to take your place. But of course, the two rude strangers were independent events. It’s unlikely they were working together to annoy you. The fact it happened twice in one day doesn’t make it happening a third time more probable.

The most important thing to remember here is that the probability of conjunctive events happening is never higher than the probability of each occurring.

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Clusters happen

“You’ve likely read the story in the newspaper or perhaps seen the news expose: Some statistically unlikely number of people in a particular area have contracted a rare form of cancer. It must be the water, or the local power plant, or the cell phone tower.

. . . But this cluster of cases may also be the product of pure chance, even when the number of cases appears highly improbable. Yes, the probability that five people in the same school or church or workplace will contract the same rare form of leukemia may be one in a million, but there are millions of schools and churches and workplaces. It’s not highly improbable that five people might get the same rare form of leukemia in one of those places.”

An important lesson of probability is that while particular improbable events are, well, improbable, the chance of any improbable event happening at all is highly probable. Your chances of winning the lottery are almost zero. But someone has to win it. Your chances of getting struck by lightning are almost zero. But with so many people walking around and so many storms, it has to happen to someone sooner or later.

The same is true for clusters of improbable events. The chance of any individual winning the lottery multiple times or getting struck by lightning more than once is even closer to zero than the chance of it happening once. Yet when we look at all the people in the world, it’s certain to happen to someone.

We’re all pattern-matching creatures. We find randomness hard to process and look for meaning in chaotic events. So it’s no surprise that clusters often fool us. If you encounter one, it’s wise to keep in mind the possibility that it’s a product of chance, not anything more meaningful. Sure, it might be jarring to be involved in three car crashes in a year or to run into two college roommates at the same conference. Is it all that improbable that it would happen to someone, though?

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The prosecutor’s fallacy

“The prosecutor’s fallacy occurs when the context surrounding statistical evidence is neglected . . . the chances of finding a coincidental one in a million match are relatively high if you run the same through a database with samples from a million people.”

It’s important to look at the context surrounding statistics. Let’s say you’re evaluating whether to take a medication your doctor suggests. A quick glance at the information leaflet tells you that it carries a 1 in 10,000 risk of blood clots. Should you be concerned? Well, that depends on context. The 1 in 10,000 figure takes into account the wide spectrum of people with different genes and different lifestyles who might take the medication. If you’re an overweight chain-smoker with a family history of blood clots who takes twelve-hour flights twice a month, you might want to have a more serious discussion with your doctor than an active non-smoker with no relevant family history.

Statistics give us a simple snapshot, but if we want a finer-grained picture, we need to think about context.

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Reversion to the mean (or regression to the mean)

“Probability tells us that any outlier—an observation that is particularly far from the mean in one direction or the other—is likely to be followed by outcomes that are most consistent with the long-term average.

. . . One way to think about this mean reversion is that performance—both mental and physical—consists of underlying talent-related effort plus an element of luck, good or bad. (Statisticians would call this random error.) In any case, those individuals who perform far above the mean for some stretch are likely to have had luck on their side; those who perform far below the mean are likely to have had bad luck. . . . When a spell of very good luck or very bad luck ends—as it inevitably will—the resulting performance will be closer to the mean.”

Moderate events tend to follow extreme ones. One area that regression to the mean often misleads us is when considering how people perform in areas like sports or management. We may think a single extraordinary success is predictive of future successes. Yet from one result, we can’t know if it’s an outcome of talent or luck—in which case the next result may be average. Failure or success is usually followed by an event closer to the mean, not the other extreme.

Regression to the mean teaches us that the way to differentiate between skill and luck is to look at someone’s track record. The more information you have, the better. Even if past performance is not always predictive of future performance, a track record of consistent high performance is a far better indicator than a single highlight.

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If you want an accessible tour of basic statistics, check out Naked Statistics by Charles Wheelan.

Descriptions Aren’t Prescriptions

When we look at a representation of reality, we can choose to either see it as descriptive, meaning it tells us what the world is currently like, or as prescriptive, meaning it tells us how the world should be. Descriptions teach us, but they also give us room to innovate. Prescriptions can get us stuck. One place this tension shows up is in language.

In one chapter of The Utopia of Rules: On Technology, Stupidity, and the Secret Joys of Bureaucracy, David Graeber describes his experience of learning Malagasy, the national language of Madagascar. While the language’s writing system came about in the fifteenth century, it wasn’t until the early nineteenth century that missionaries documented the rules of Malagasy grammar for the purpose of translating scripture.

Of course, the “rules” of Malagasy the missionaries recorded weren’t rules at all. They were reflections of how people spoke at that point in time, as far as outside observers could tell. Languages don’t usually come into existence when someone invents the rules for them. Instead, languages evolve and change over time as speakers make modifications or respond to new needs.

However, those early nineteenth-century records remained in place as the supposed “official” version of Malagasy. Children learned the old form of grammar in school, even as they spoke a somewhat different version of the language at home. For Graeber, learning to speak the version of Malagasy people actually understood in conversation was a challenge. Native speakers he hired would instruct him on the nineteenth-century grammatical principles, then turn and speak to each other in a whole other fashion.

When asked why they couldn’t teach him the version of the language they spoke, Graeber’s Malagasy teachers responded that they were just using slang. Asked why no one seemed to speak the official version, they said people were too lazy. Graeber writes, “Clearly the problem was that the entire population had failed to memorize their lessons properly. But what they were actually denying was the legitimacy of collective creativity, the free play of the system. ” While the official rules stayed the same over the decades, the language itself kept evolving. People assumed the fault of not speaking “proper” Malagasy lay with them, not with the outdated dictionary and grammar. They confused a description for a prescription. He writes:

It never seems to occur to anyone—until you point it out—that had the missionaries came and written their books two hundred years later, current usages would be considered the correct ones, and anyone speaking as they had two hundred years ago would themselves be assumed to be in error.

Graeber sees the same phenomenon playing out in other languages for which grammars and dictionaries only came into existence a century or two ago. Often, such languages were mostly spoken and, like Malagasy, no one made formal records until the need arose for people from elsewhere to make translations. Instead of treating those records as descriptive and outdated, those teaching the language treat them as prescriptive—despite knowing they’re not practical for everyday use.

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Why don’t people talk “proper”?

So why can’t people just speak a language per the official rules? If someone has gone to all the effort of identifying and recording the rules and people received instruction on them in school, why not follow them? Why keep changing things up?

If languages didn’t evolve, it would make life a lot easier for historians looking at texts from the past. It would also simplify matters for people learning the language, for those coming from different areas, and even for speakers across generations. Yet all languages change all the time.

Graeber suggests the reason for this is because people like to play. We find it dull to speak according to the official rules of our language. We seek out novelty in our everyday lives and do whatever it takes to avoid boredom. Even if each person only plays a little bit once in a while, the results compound. Graeber explains that “this playing around will have cumulative effects.”

Languages still need conventions so people can understand each other. The higher the similarity between the versions of a language different people speak, the more they can communicate. At the same time, they cannot remain rigid. Trying to follow an unyielding set of strict rules will inevitably curtail the usefulness of a language and prevent it from developing in interesting and necessary ways. Languages need a balance: enough guidance to help everyone understand each other and provide an entry point for learners, and enough flexibility to keep updating the rules as actual usage changes.

As a result, languages call into question our idea of freedom: “It’s worth thinking about language for a moment, because one thing it reveals, probably better than any other example, is that there is a basic paradox in our very idea of freedom. On the one hand, rules are by their nature constraining. Speech codes, rules of etiquette, and grammatical rules, all have the effect of limiting what we can and cannot say. ” On the other hand, no rules whatsoever mean no one can understand each other.

Languages need frameworks, but no amount of grammar classes or official dictionaries will prevent people from playing and having fun with their speech.

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The dictionary is not the language

“The map is not the territory” means that any representation of reality has to be a simplification that may contain errors, become outdated, or reflect biases. Maps remove details that aren’t necessary for their intended use. Representations of complex systems may show expected behavior or ideal behavior. For example, the London Underground map doesn’t reflect the distances between stations because this information isn’t important to most commuters. If a map represented its territory without reducing anything, it would be identical to the territory and therefore would be useless. In fact, the simplest maps can be the most useful because they’re the easiest to understand and remember.

Sometimes maps are descriptive, and sometimes they’re prescriptive; often they’re a bit of both. We run into problems when we confuse one type for another and try to navigate an idealized territory or make the real territory fit an idealized image.

A language’s grammar and dictionary are a sort of map. They take a complex system—a language spoken by what could be tens of millions of people—and aim to represent it with something which is, by comparison, simple. The official rules are not the language itself, but they provide guidance for navigating it. Much like a map of a city needs periodic updates as parts are torn down, built up, renamed, destroyed, added, and so on, the official rules need updating as the language changes. Trying to learn Malagasy using grammar rules written two hundred years ago is like trying to navigate Antananarivo using a street map made two hundred years ago.

A map of a complex system, like a language, is meant to help us find our way by giving us a sense of how things looked at one point in time—it’s usually descriptive. It doesn’t necessarily tell us how that system should look, and we may run into problems if we try to make it conform to the map, ignoring the system’s own adaptive properties. Even if the cartographer never intended this, we can end up treating a map as a prescription. We try to make reality conform to the map. This is what occurs with languages. Graeber calls this the “grammar-book effect”:

People do not invent languages by writing grammars, they write grammars—at least, the first grammars to be written for any given language—by observing the tacit, largely unconscious rules that people seem to be employing when they speak. Yet once a book exists, and especially once it is employed in schoolrooms, people feel that the rules are not just descriptions of how people do talk, but prescriptions for how they should talk.

As we’ve seen, one reason the map is not the territory with language is because people feel compelled to play and experiment. When we encounter representations of systems involving people, we should keep in mind that while we may need rules for the sake of working together and understanding each other, we’re always pushing up against and reshaping those rules. We find it boring to follow a rigid prescription.

For instance, imagine some of the documents you might receive upon starting a role at a new company. Process documents showing step by step how to do the main tasks you’ll be expected to perform. But when the person you’re replacing shows you how to do those same tasks, you notice they don’t follow the listed steps at all. When you ask why, they explain that the process documents were written before they started actually carrying out those tasks, meaning they discovered more efficient ways afterward.

Why keep the process documents, then? Because for someone filling in or starting out, it might make sense to follow them. It’s the most defensible option. Once you truly know the territory and won’t change something without considering why it was there in the first place, you can play with the rules. Those documents might be useful as a description, but they’re unlikely to remain a prescription for long.

The same is true for laws. Sometimes aspects of them are just descriptive of how things are at one point in time, but we end up having to keep following them to the letter because they haven’t been updated. A law might have been written at a time when documents needed sending by letter, meaning certain delays for shipping. Now they can be sent by email. If the law hasn’t been updated, those delay allowances turn from descriptions into prescriptions. Or a law might reflect what people were permitted to do at the time, but now we assume people should have the right to do that thing even if we have new evidence it’s not the best idea. We are less likely to change laws if we persist in viewing them as prescriptive.

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Conclusion

Descriptions of reality are practical for helping us navigate it, while also giving us room to change things. Prescriptions are helpful for giving us ways of understanding each other and providing enough structure for shared conventions, but they can also become outdated or end up limiting flexibility. When you encounter a representation of something, it’s useful to consider which parts are descriptive and which parts are prescriptive. Remember that both prescriptions and descriptions can and should change over time.

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The FS team were saddened to hear of David Graeber’s passing, shortly after we completed this article. We hope his books will continue to inspire and educate new readers for many years to come.