Category: Numeracy

Why Math Class Is Boring—and What to Do About It

There are two types of people in the world: those who enjoyed mathematics class in school, and the other 98% of the population.

No other subject is associated with such widespread fear, confusion, and even outright hatred. No other subject is so often declared by children and adults alike to be something they “can’t do” because they lack an innate aptitude for it.

Math is portrayed as something you get or you don’t. Most of us sit in class feeling like we don’t.

But what if this weren’t the fault of the subject itself, but of the manner in which we teach it? What if the standard curriculum were a gross misrepresentation of the subject? What if it were possible to teach mathematics in a manner naturally incorporating the kinds of activities that appeal to children and learners of all ages?

All of those things are true, argues Paul Lockhart, a mathematician who chose to switch from teaching at top universities to inspiring grade-schoolers. In 2002, he penned “A Mathematician’s Lament,” a 25-page essay that was later expanded into a book.

In the essay, Lockhart declares that students who say their mathematics classes are stupid and boring are correct—though the subject itself is not. The problem is that our culture does not recognize that the true nature of math is art. So we teach it in a manner that would just as easily ruin any other art.

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To illustrate the harms of the typical mathematical curriculum, Lockhart envisions what it would look like if we treated music or painting in the same dreary, arbitrary way.

What if music education was all about notation and theory, with listening or playing only open to those who somehow persevered until college?

“Since musicians are known to set down their ideas in the form of sheet music, these curious black dots and lines must constitute the “language of music.” It is imperative that students become fluent in this language if they are to attain any degree of musical competence; indeed, it would be ludicrous to expect a child to sing a song or play an instrument without having a thorough grounding in music notation and theory.

Playing and listening to music, let alone composing an original piece, are considered very advanced topics and are generally put off until college, and more often graduate school.”

And what if art students spent years studying paints and brushes, without ever getting to unleash their imaginations on a blank canvas?

“After class I spoke with the teacher. ‘So your students don’t actually do any painting?’ I asked.

‘Well, next year they take Pre-Paint-by-Numbers. That prepares them for the main Paint-by-Numbers sequence in high school. So they’ll get to use what they’ve learned here and apply it to real-life painting situations—dipping the brush into paint, wiping it off, stuff like that. Of course we track our students by ability. The really excellent painters—the ones who know their colors and brushes backwards and forwards—they get to the actual painting a little sooner, and some of them even take the Advanced Placement classes for college credit. But mostly we’re just trying to give these kids a good foundation in what painting is all about, so when they get out there in the real world and paint their kitchen they don’t make a total mess of it.'”

As laughable as we may find these vignettes, Lockhart considers them analogous to how we teach mathematics as something devoid of expression, exploration, or discovery.

Few who have spent countless hours on the equivalent of paint-by-numbers in the typical math class could understand that “there is nothing as dreamy and poetic, nothing as radical, subversive, and psychedelic, as mathematics.” Like other arts, its objective is the creation of patterns. The material mathematical patterns are made from is not paint or musical notes, however, but ideas.

Though we may use components of mathematics in practical fields such as engineering, the objective of the field itself isn’t anything practical. Above all, mathematicians strive to present ideas in the simplest form possible, which means dwelling in the realm of the imaginary.

In mathematics, Lockhart explains, there is no reality to get in your way. You can imagine a geometric shape with perfect edges, even though such a thing could never exist in the physical, three-dimensional world. Then you can ask questions of it and discover new things through experimentation with the imaginary. That process—“asking simple and elegant questions about our imaginary creations, and crafting satisfying and beautiful explanations”—is mathematics itself. What we learn in school is merely the end product.

We don’t teach the process of creating math. We teach only the steps to repeat someone else’s creation, without exploring how they got there—or why.

Lockhart compares what we teach in math class to “saying that Michelangelo created a beautiful sculpture, without letting me see it.” It’s hard to imagine describing one of Michelangelo’s sculptures solely in terms of the technical steps he took to produce it. And it seems impossible that one could teach sculpture without revealing that there is an art to it. Yet that is what we do with math all the time.

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If school curriculums fundamentally misrepresent math, where does that misrepresentation come from? Lockhart views it as a self-perpetuating cultural deficiency.

Unlike other arts, we generally don’t celebrate the great works of mathematics and put them on display. Nor have they become all that integrated into our collective consciousness. It’s hard to change the feedback loops at play in education because “students learn about math from their teachers, and teachers learn about it from their teachers, so this lack of understanding and appreciation for mathematics in our culture replicates itself indefinitely.”

In schools, mathematics is treated as something absolute that needs no context, a fixed body of knowledge that ascends a defined ladder of complexity. There can be no criticism, experimentation, or further developments because everything is already known. Its ideas are presented without any indication that they might even be connected to a particular person or particular time. Lockhart writes:

“What other subject is routinely taught without any mention of its history, philosophy, thematic development, aesthetic criteria, and current status? What other subject shuns its primary sources—beautiful works of art by some of the most creative minds in history—in favor of third-rate textbook bastardizations?”

Efforts to engage students with mathematics often take the form of trying to make it relevant to their everyday lives or presenting problems as saccharine narratives. Once again, Lockhart doesn’t believe this would be a problem if students got to engage in the actual creative process: “We don’t need to bend over backwards to give mathematics relevance. It has relevance in the same way that any art does: that of being a meaningful human experience.” An escape from daily life is generally more appealing than an emphasis on it. Children would have as much fun playing with symbols as they have playing with paints.

Those whose mathematics teachers told them the subject was important because “you’re not going to have a calculator in your pocket at all times as an adult” have a good reason to feel like they wasted a lot of time learning arithmetic now that we all have smartphones. But we can imagine those who learn math because it’s entertaining would go out into the world seeing beautiful math patterns all over the place, and enjoying their lives more because of it.

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If the existing form of mathematics education is all backward, what can we do to improve it? How can we teach and learn it as an art?

Lockhart does acknowledge that the teaching methods he proposes are unrealistic within the current educational system, where teachers get little control over their work and students need to learn the same content at the same time to pass exams. However, his methods can give us ideas for exploring the topic ourselves.

An education in the art of mathematics is above all a personal process of discovery. It requires tackling the sort of problems that speak to us at that particular point in time, not according to a preordained curriculum. If a new direction seems of interest, so be it. It requires space to take our time with exploration and an openness to making judgments (why should mathematics be immune to criticism?) All of this is far from ticking boxes:

The trouble is that math, like painting or poetry, is hard creative work. That makes it very difficult to teach. Mathematics is a slow, contemplative process. It takes time to produce a work of art, and it takes a skilled teacher to recognize one. Of course it’s easier to post a set of rules than to guide aspiring young artists, and it’s easier to write a VCR manual than to write an actual book with a point of view.

We should probably let go of the idea that doing math is about getting the right answer. Being creative is never about getting to a destination.

Above all, mathematics should be something we engage with because we find it to be a fun, challenging process capable of teaching us new ways to think or allowing us to express ourselves. The less practical utility or relevance to the rest of our lives it has, the more we’re truly engaging with it as an art.

12 Life Lessons From Mathematician and Philosopher Gian-Carlo Rota

The mathematician and philosopher Gian-Carlo Rota spent much of his career at MIT, where students adored him for his engaging, passionate lectures. In 1996, Rota gave a talk entitled “Ten Lessons I Wish I Had Been Taught,” which contains valuable advice for making people pay attention to your ideas.

Many mathematicians regard Rota as single-handedly responsible for turning combinatorics into a significant field of study. He specialized in functional analysis, probability theory, phenomenology, and combinatorics. His 1996 talk, “Ten Lessons I Wish I Had Been Taught,” was later printed in his book, Indiscrete Thoughts.

Rota began by explaining that the advice we give others is always the advice we need to follow most. Seeing as it was too late for him to follow certain lessons, he decided he would share them with the audience. Here, we summarize twelve insights from Rota’s talk—which are fascinating and practical, even if you’re not a mathematician.

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Every lecture should make only one point

“Every lecture should state one main point and repeat it over and over, like a theme with variations. An audience is like a herd of cows, moving slowly in the direction they are being driven towards.”

When we wish to communicate with people—in an article, an email to a coworker, a presentation, a text to a partner, and so on—it’s often best to stick to making one point at a time. This matters all the more so if we’re trying to get our ideas across to a large audience.

If we make one point well enough, we can be optimistic about people understanding and remembering it. But if we try to fit too much in, “the cows will scatter all over the field. The audience will lose interest and everyone will go back to the thoughts they interrupted in order to come to our lecture.

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Never run over time

“After fifty minutes (one microcentury as von Neumann used to say), everybody’s attention will turn elsewhere even if we are trying to prove the Riemann hypothesis. One minute over time can destroy the best of lectures.”

Rota considered running over the allotted time slot to be the worst thing a lecturer could do. Our attention spans are finite. After a certain point, we stop taking in new information.

In your work, it’s important to respect the time and attention of others. Put in the extra work required for brevity and clarity. Don’t expect them to find what you have to say as interesting as you do. Condensing and compressing your ideas both ensures you truly understand them and makes them easier for others to remember.

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Relate to your audience

“As you enter the lecture hall, try to spot someone in the audience whose work you have some familiarity with. Quickly rearrange your presentation so as to manage to mention some of that person’s work.”

Reciprocity is remarkably persuasive. Sometimes, how people respond to your work has as much to do with how you respond to theirs as it does with the work itself. If you want people to pay attention to your work, always give before you take and pay attention to theirs first. Show that you see them and appreciate them. Rota explains that “everyone in the audience has come to listen to your lecture with the secret hope of hearing their work mentioned.

The less acknowledgment someone’s work has received, the more of an impact your attention is likely to have. A small act of encouragement can be enough to deter someone from quitting. With characteristic humor, Rota recounts:

“I have always felt miffed after reading a paper in which I felt I was not being given proper credit, and it is safe to conjecture that the same happens to everyone else. One day I tried an experiment. After writing a rather long paper, I began to draft a thorough bibliography. On the spur of the moment I decided to cite a few papers which had nothing whatsoever to do with the content of my paper to see what might happen.

Somewhat to my surprise, I received letters from two of the authors whose papers I believed were irrelevant to my article. Both letters were written in an emotionally charged tone. Each of the authors warmly congratulated me for being the first to acknowledge their contribution to the field.”

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Give people something to take home

“I often meet, in airports, in the street, and occasionally in embarrassing situations, MIT alumni who have taken one or more courses from me. Most of the time they admit that they have forgotten the subject of the course and all the mathematics I thought I had taught them. However, they will gladly recall some joke, some anecdote, some quirk, some side remark, or some mistake I made.”

When we have a conversation, read a book, or listen to a talk, the sad fact is that we are unlikely to remember much of it even a few hours later, let alone years after the event. Even if we enjoyed and valued it, only a small part will stick in our memory.

So when you’re communicating with people, try to be conscious about giving them something to take home. Choose a memorable line or idea, create a visual image, or use humor in your work.

For example, in The Righteous Mind, Jonathan Haidt repeats many times that the mind is like a tiny rider on a gigantic elephant. The rider represents controlled mental processes, while the elephant represents automatic ones. It’s a distinctive image, one readers are quite likely to take home with them.

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Make sure the blackboard is spotless

“By starting with a spotless blackboard, you will subtly convey the impression that the lecture they are about to hear is equally spotless.”

Presentation matters. The way our work looks influences how people perceive it. Taking the time to clean our equivalent of a blackboard signals that we care about what we’re doing and consider it important.

In “How To Spot Bad Science,” we noted that one possible sign of bad science is that the research is presented in a thoughtless, messy way. Most researchers who take their work seriously will put in the extra effort to ensure it’s well presented.

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Make it easy for people to take notes

“What we write on the blackboard should correspond to what we want an attentive listener to take down in his notebook. It is preferable to write slowly and in a large handwriting, with no abbreviations. Those members of the audience who are taking notes are doing us a favor, and it is up to us to help them with their copying.”

If a lecturer is using slides with writing on them instead of a blackboard, Rota adds that they should give people time to take notes. This might mean repeating themselves in a few different ways so each slide takes longer to explain (which ties in with the idea that every lecture should make only one point). Moving too fast with the expectation that people will look at the slides again later is “wishful thinking.”

When we present our work to people, we should make it simple for them to understand our ideas on the spot. We shouldn’t expect them to revisit it later. They might forget. And even if they don’t, we won’t be there to answer questions, take feedback, and clear up any misunderstandings.

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Share the same work multiple times

Rota learned this lesson when he bought Collected Papers, a volume compiling the publications of mathematician Frederic Riesz. He noted that “the editors had gone out of their way to publish every little scrap Riesz had ever published.” Putting them all in one place revealed that he had published the same ideas multiple times:

Riesz would publish the first rough version of an idea in some obscure Hungarian journal. A few years later, he would send a series of notes to the French Academy’s Comptes Rendus in which the same material was further elaborated. A few more years would pass, and he would publish the definitive paper, either in French or in English.

Riesz would also develop his ideas while lecturing. Explaining the same subject again and again for years allowed him to keep improving it until he was ready to publish. Rota notes, “No wonder the final version was perfect.

In our work, we might feel as if we need to have fresh ideas all of the time and that anything we share with others needs to be a finished product. But sometimes we can do our best work through an iterative process.

For example, a writer might start by sharing an idea as a tweet. This gets a good response, and the replies help them expand it into a blog post. From there they keep reworking the post over several years, making it longer and more definite each time. They give a talk on the topic. Eventually, it becomes a book.

Award-winning comedian Chris Rock prepares for global tours by performing dozens of times in small venues for a handful of people. Each performance is an experiment to see which jokes land, which ones don’t, and which need tweaking. By the time he’s performed a routine forty or fifty times, making it better and better, he’s ready to share it with huge audiences.

Another reason to share the same work multiple times is that different people will see it each time and understand it in different ways:

“The mathematical community is split into small groups, each one with its own customs, notation, and terminology. It may soon be indispensable to present the same result in several versions, each one accessible to a specific group; the price one might have to pay otherwise is to have our work rediscovered by someone who uses a different language and notation, and who will rightly claim it as his own.”

Sharing your work multiple times thus has two benefits. The first is that the feedback allows you to improve and refine your work. The second is that you increase the chance of your work being definitively associated with you. If the core ideas are strong enough, they’ll shine through even in the initial incomplete versions.

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You are more likely to be remembered for your expository work

“Allow me to digress with a personal reminiscence. I sometimes publish in a branch of philosophy called phenomenology. . . . It so happens that the fundamental treatises of phenomenology are written in thick, heavy philosophical German. Tradition demands that no examples ever be given of what one is talking about. One day I decided, not without serious misgivings, to publish a paper that was essentially an updating of some paragraphs from a book by Edmund Husserl, with a few examples added. While I was waiting for the worst at the next meeting of the Society for Phenomenology and Existential Philosophy, a prominent phenomenologist rushed towards me with a smile on his face. He was full of praise for my paper, and he strongly encouraged me to further develop the novel and original ideas presented in it.”

Rota realized that many of the mathematicians he admired the most were known more for their work explaining and building upon existing knowledge, as opposed to their entirely original work. Their extensive knowledge of their domain meant they could expand a little beyond their core specialization and synthesize charted territory.

For example, David Hilbert was best known for a textbook on integral equations which was “in large part expository, leaning on the work of Hellinger and several other mathematicians whose names are now forgotten.” William Feller was known for an influential treatise on probability, with few recalling his original work in convex geometry.

One of our core goals at Farnam Street is to share the best of what other people have already figured out. We all want to make original and creative contributions to the world. But the best ideas that are already out there are quite often much more useful than what we can contribute from scratch.

We should never be afraid to stand on the shoulders of giants.

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Every mathematician has only a few tricks

“. . . mathematicians, even the very best, also rely on a few tricks which they use over and over.”

Upon reading the complete works of certain influential mathematicians, such as David Hilbert, Rota realized that they always used the same tricks again and again.

We don’t need to be amazing at everything to do high-quality work. The smartest and most successful people are often only good at a few things—or even one thing. Their secret is that they maximize those strengths and don’t get distracted. They define their circle of competence and don’t attempt things they’re not good at if there’s any room to double down further on what’s already going well.

It might seem as if this lesson contradicts the previous one (you are more likely to be remembered for your expository work), but there’s a key difference. If you’ve hit diminishing returns with improvements to what’s already inside your circle of competence, it makes sense to experiment with things you already have an aptitude for (or a strong suspicion you might) but you just haven’t made them your focus.

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Don’t worry about small mistakes

“Once more let me begin with Hilbert. When the Germans were planning to publish Hilbert’s collected papers and to present him with a set on the occasion of one of his later birthdays, they realized that they could not publish the papers in their original versions because they were full of errors, some of them quite serious. Thereupon they hired a young unemployed mathematician, Olga Taussky-Todd, to go over Hilbert’s papers and correct all mistakes. Olga labored for three years; it turned out that all mistakes could be corrected without any major changes in the statement of the theorems. . . . At last, on Hilbert’s birthday, a freshly printed set of Hilbert’s collected papers was presented to the Geheimrat. Hilbert leafed through them carefully and did not notice anything.”

Rota goes on to say: “There are two kinds of mistakes. There are fatal mistakes that destroy a theory; but there are also contingent ones, which are useful in testing the stability of a theory.

Mistakes are either contingent or fatal. Contingent mistakes don’t completely ruin what you’re working on; fatal ones do. Building in a margin of safety (such as having a bit more time or funding that you expect to need) turns many fatal mistakes into contingent ones.

Contingent mistakes can even be useful. When details change, but the underlying theory is still sound, you know which details not to sweat.

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Use Feynman’s method for solving problems

“Richard Feynman was fond of giving the following advice on how to be a genius. You have to keep a dozen of your favorite problems constantly present in your mind, although by and large they will lay in a dormant state. Every time you hear or read a new trick or a new result, test it against each of your twelve problems to see whether it helps. Every once in a while there will be a hit, and people will say: ‘How did he do it? He must be a genius!’”

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Write informative introductions

“Nowadays, reading a mathematics paper from top to bottom is a rare event. If we wish our paper to be read, we had better provide our prospective readers with strong motivation to do so. A lengthy introduction, summarizing the history of the subject, giving everybody his due, and perhaps enticingly outlining the content of the paper in a discursive manner, will go some of the way towards getting us a couple of readers.”

As with the lesson of don’t run over time, respect that people have limited time and attention. Introductions are all about explaining what a piece of work is going to be about, what its purpose is, and why someone should be interested in it.

A job posting is an introduction to a company. The description on a calendar invite to a meeting is an introduction to that meeting. An about page is an introduction to an author. The subject line on a cold email is an introduction to that message. A course curriculum is an introduction to a class.

Putting extra effort into our introductions will help other people make an accurate assessment of whether they want to engage with the full thing. It will prime their minds for what to expect and answer some of their questions.

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If you’re interested in learning more, check out Rota’s “10 Lessons of an MIT Education.

The Best-Case Outcomes Are Statistical Outliers

There’s nothing wrong with hoping for the best. But the best-case scenario is rarely the one that comes to pass. Being realistic about what is likely to happen positions you for a range of possible outcomes and gives you peace of mind.

We dream about achieving the best-case outcomes, but they are rare. We can’t forget to acknowledge all the other possibilities of what may happen if we want to position ourselves for success.

“Hoping for the best, prepared for the worst, and unsurprised by anything in between.” —Maya Angelou

It’s okay to hope for the best—to look at whatever situation you’re in and say, “This time I have it figured out. This time it’s going to work.” First, having some degree of optimism is necessary for trying anything new. If we weren’t overconfident, we’d never have the guts to do something as risky and unlikely to succeed as starting a business, entering a new relationship, or sending that cold email. Anticipating that a new venture will work helps you overcome obstacles and make it work.

Second, sometimes we do have it figured out. Sometimes our solutions do make things better.

Even when the best-case scenario comes to pass, however, it rarely unfolds exactly as planned. Some choices create unanticipated consequences that we have to deal with. We may encounter unexpected roadblocks due to a lack of information. Or the full implementation of all our ideas and aspirations might take a lot longer than we planned for.

When you look back over history, we rarely find best-case outcomes.

Sure, sometimes they happen—maybe more than we think, given not every moment of the past is recorded. But let’s be honest: even historical wins, like developing the polio vaccine and figuring out how to produce clean drinking water, were not all smooth sailing. There are still people who are unable or unwilling to get the polio vaccine. And there are still many people in the world, even in developed countries like Canada, who don’t have access to clean drinking water.

The best-case outcomes in these situations—a world without polio and a world with globally available clean drinking water—have not happened, despite the existence of reliable, proven technology that can make these outcomes a reality.

There are a lot of reasons why, in these situations, we haven’t achieved the best-case outcomes. Furthermore, situations like these are not unusual. We rarely achieve the dream. The more complicated a situation, the more people it involves, the more variables and dependencies that exist, the more it’s unlikely that it’s all going to work out.

If we narrow our scope and say, for example, the best-case scenario for this Friday night is that we don’t burn the pizza, we can all agree on a movie, and the power doesn’t go out, it’s more likely we’ll achieve it. There are fewer variables, so there’s a greater chance that this specific scenario will come to pass.

The problem is that most of us plan as if we live in an easy-to-anticipate Friday night kind of world. We don’t.

There are no magic bullets for the complicated challenges facing society. There is only hard work, planning for the wide spectrum of human behavior, adjusting to changing conditions, and perseverance. There are many possible outcomes for any given endeavor and only one that we consider the best case.

That is why the best-case outcomes are statistical outliers—they are only one possibility in a sea of many. They might come to pass, but you’re much better off preparing for the likelihood that they won’t.

Our expectations matter. Anticipating a range of outcomes can make us feel better. If we expect the best and it happens, we’re merely satisfied. If we expect less and something better happens, we’re delighted.

Knowing that the future is probably not going to be all sunshine and roses allows you to prepare for a variety of more likely outcomes, including some of the bad ones. Sometimes, too, when the worst-case scenario happens, it’s actually a huge relief. We realize it’s not all bad, we didn’t die, and we can manage if it happens again. Preparation and knowing you can handle a wide spectrum of possible challenges is how you get the peace of mind to be unsurprised by anything in between the worst and the best.

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.

Alex Bellos: Every Number Tells a Story

“We depend on numbers to make sense of the world,
and have done so ever since we started to count.”

— Alex Bellos

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The earliest symbols used for numbers go back to about 5000 years ago to Sumer (modern-day Iraq). They didn’t really look far for names. Ges, the word for one, also meant man. Min, the word for two, also meant women. At first, numbers served a practical purpose, mostly things like counting sheep and determining taxes.

“Yet numbers also revealed abstract patterns,” writes Alex Bellos in his fascinating book The Grapes of Math: How Life Reflects Numbers and Numbers Reflect Life, which, he continues, “made them objects of deep contemplation. Perhaps the earliest mathematical discovery was that numbers come in two types, even and odd: those that can be halved cleanly, such as 2, 4 and 6, and those that cannot, such as 1, 3 and 5.

Pythagoras and Number Gender

Pythagoras, the Greek teacher, who lived in the sixth century BC and is most famous for his theorem about triangles, agreed with the Sumerians on number gender. He believed that odd numbers were masculine, and even ones were feminine. This is where it gets interesting … why did he think that? It was, he believed, a resistance to splitting in two that embodied strength. The ability to be divisible by two, in his eyes was a weakness. He believed odd numbers were master over even. Christianity agrees with the gender theory, God created Adam first and Eve second. The latter being the sin.

Large Numbers

Numbers originally accounted for practical and countable things, such as sheep and teeth. Things get interesting as quantities increase because we don’t use numbers in the same way.

We approximate using a “round number” as a place mark. It is easier and more convenient. When I say, for example, that there were a hundred people at the market, I don’t mean that there were exactly one hundred people there. … Big numbers are understood approximately, small ones precisely, and these two systems interact uneasily. It is clearly nonsensical to say that next year the universe will be “13.7 billion and one” years old. It will remain 13.7 billion years old for the rest of our lives.

Round numbers usually end in zero.

The word round is used because a round number represents the completion of a full counting cycle, not because zero is a circle. There are ten digits in our number system, so any combination of cycles will always be divisible by ten. Because we are so used to using round numbers for big numbers, when we encounter a big number that is nonround— say, 754,156,293— it feels discrepant.

Manoj Thomas, a psychologist at Cornell University, argues that we are uneasy with large, non-round numbers, which causes us to see them as smaller than they are and carries with it practical implications when, say, selling a house. “We tend to think that small numbers are more precise,” he says, “so when we see a big number that is precise we instinctively assume it is less than it is.” If he’s right, the result is that you will pay more for expensive and non-round prices. Indeed his experiments seem to agree. In one, respondents viewed pictures of several houses and sales prices, some were round, and some were larger and non-round (e.g., $490,000 and $492,332). On average subjects judged the precise one to be lower. As Bellos concludes on large numbers, “if you want to make money, don’t end the price with a zero.”

Number Influence When Shopping

One of the ways to make a number seem more precise is by subtracting 1.

When we read a number, we are more influenced by the leftmost digit than by the rightmost, since that is the order in which we read, and process, them. The number 799 feels significantly less than 800 because we see the former as 7-something and the latter as 8-something, whereas 798 feels pretty much like 799. Since the nineteenth century, shopkeepers have taken advantage of this trick by choosing prices ending in a 9, to give the impression that a product is cheaper than it is. Surveys show that anything between a third and two-thirds of all retail prices now end in a 9.

Of course, we think that other people fall for this and surely not us, but that is not the case. Studies like this continue to be replicated over and over. Dropping the price one cent, say from $8 to $7.99 influences decisions dramatically.

Not only are prices ending in 9 harder to recall for price comparisons, but we’ve also been conditioned to believe they are discounted and cheap. The practical implications of this are that if you’re a high-end brand or selling an exclusive service, you want to avoid the bargain aspect. You don’t want a therapist who charges $99.99, any more than you want a high-end restaurant to list menu prices ending in $.99.

In fact, most of the time, it’s best to avoid the $ altogether. Our response to this stimulus is a pain.

The “$” reminds us of the pain of paying. Another clever menu strategy is to show the prices immediately after the description of each dish, rather than listing them in a column, since listing prices facilitates price comparison. You want to encourage diners to order what they want, whatever the price, rather than reminding them which dish is most expensive.

These are not the only nor most subtle ways that numbers influence us. The display of absurdly expensive items first creates an artificial benchmark. The real estate agent, who shows you a house way above your price range first, is really setting an artificial benchmark.

The $100,000 car in the showroom and the $10,000 pair of shoes in the shop window are there not because the manager thinks they will sell, but as decoys to make the also-expensive $50,000 car and $5,000 shoes look cheap. Supermarkets use similar strategies. We are surprisingly susceptible to number cues when it comes to making decisions, and not just when shopping.

We can all be swayed by irrelevant, random numbers, which is why it’s important to use a two-step framework when making decisions.

Numbers and Time

Time has always been counted.

We carved notches on sticks and daubed splotches on rocks to mark the passing of days. Our first calendars were tied to astronomical phenomena, such as the new moon, which meant that the number of days in each calendar cycle varied, in the case of the new moon between 29 and 30 days, since the exact length of a lunar cycle is 29.53 days. In the middle of the first millennium BCE, however, the Jews introduced a new system. They decreed that the Sabbath come every seven days ad infinitum, irrespective of planetary positions. The continuous seven-day cycle was a significant step forward for humanity. It emancipated us from consistent compliance with Nature, placing numerical regularity at the heart of religious practice and social organization, and since then the seven-day week has become the world’s longest-running uninterrupted calendrical tradition.

Why seven days in the week?

Seven was already the most mystical of numbers by the time the Jews declared that God took six days to make the world, and rested the day after. Earlier peoples had also used seven-day periods in their calendars, although never repeated in an endless loop. The most commonly accepted explanation for the predominance of seven in religious contexts is that the ancients observed seven planets in the sky: the Sun, the Moon, Venus, Mercury, Mars, Jupiter and Saturn. Indeed, the names Saturday, Sunday and Monday come from the planets, although the association of planets with days dates from Hellenic times, centuries after the seven-day week had been introduced.

The Egyptians used the human head to represent 7, which offers “another possible reason for the number’s symbolic importance.”

There are seven orifices in the head: the ears, eyes, nostrils and mouth. Human physiology provides other explanations too. Six days might be the optimal length of time to work before you need a day’s rest, or seven might be the most appropriate number for our working memory: the number of things the average person can hold in his or her head simultaneously is seven, plus or minus two.

Bellos isn’t convinced. He thinks seven is special, not for the reasons mentioned above, but rather because of arithmetic.

Seven is unique among the first ten numbers because it is the only number that cannot be multiplied or divided within the group. When 1, 2, 3, 4 and 5 are doubled the answer is less than or equal to ten. The numbers 6, 8 and 10 can be halved and 9 is divisible by three. Of the numbers we can count on our fingers, only 7 stands alone: it neither produces nor is produced. Of course the number feels special. It is!

Favorite Numbers and Number Personalities

When people are asked to think of a digit off the top of their head, they are most likely to think of 7. When choosing a number below 20, the most probable response is 17. We’ll come back to that in a second. But for now, let’s talk about the meaning of numbers.

Numbers express quantities, and we express qualities to them. Here are the results from a simple survey that paints a “coherent picture of number personalities.

From The Grapes of Math by Alex Borros
From The Grapes of Math by Alex Bellos

Interestingly, Bellos writes, “the association of one with male characteristics, and two with female ones, also remains deeply ingrained.”

When asked to pick favorite numbers, we follow clear patterns, as shown below in a heat map, in which the numbers from 1 to 100 are represented by squares. Bellos explains:

(The top row of each grid contains the numbers 1 to 10, the second row the numbers 11 to 20, and so on.) The numbers marked with black squares represent those that are “most liked” (the top twenty in the rankings), the white squares are the “least liked” (the bottom twenty) and the squares in shades of gray are the numbers ranked in between.

From the Grapes of Math by Alex Bellos
From the Grapes of Math by Alex Bellos

The heat map shows conspicuous patches of order. Black squares are mostly positioned at the top of the grid, showing on average that low numbers are liked best. The left-sloping diagonal through the center reveals that two-digit numbers where both digits are the same are also attractive. We like patterns. Most strikingly, however, four white columns display the unpopularity of numbers ending in 1, 3, 7 and 9.

Numbers are a part of our lives. We see them everywhere. They influence us, they guide us, and they help us solve problems. And yet, as The Grapes of Math: How Life Reflects Numbers and Numbers Reflect Life shows us, their history and patterns can also be a source of wonder.

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