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

Reading Time: 6 minutes

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.

***

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.

***

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.

***

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.

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How Description Leads to Understanding

Reading Time: 4 minutes

Describing something with accuracy forces you to learn more about it. In this way, description can be a tool for learning.

Accurate description requires the following:

  1. Observation
  2. Curiosity about what you are witnessing
  3. Suspending assumptions about cause and effect

It can be difficult to stick with describing something completely and accurately. It’s hard to overcome the tendency to draw conclusions based on partial information or to leave assumptions unexplored.

***

Some systems, like the ecosystem that is the ocean, are complex. They have many moving parts that have multiple dependencies and interact in complicated ways. Trying to figure them out is daunting, and it can seem more sane to not bother trying—except that complex systems are everywhere. We live our lives as part of many of them, and addressing any global challenges involves understanding their many dimensions.

One way to begin understanding complex systems is by describing them in detail: mapping out their parts, their multiple interactions, and how they change through time. Complex systems are often complicated—that is, they have many moving parts that can be hard to identify and define. But the overriding feature of complex systems is that they cannot be managed from the top down. Complex systems display emergent properties and unpredictable adaptations that we cannot identify in advance. But far from being inaccessible, we can learn a lot about such systems by describing what we observe.

For example, Jane Jacobs’s comprehensive description of the interactions along city sidewalks in The Death and Life of Great American Cities led to insight about how cities actually work. Her work also emphasized the multidimensionality of city systems by demonstrating via description that attempting to manage a city from the top down would stifle its adaptive capabilities and negatively impact the city itself.

Another book that uses description to illuminate complicated and intricate relationships is The Sea Around Us by Rachel Carson. In it she chronicles events in the oceans, from the cycles of plankton growth to the movement of waves, in accessible, evocative descriptions. It’s no trouble to conjure up vivid images based on her words. But as the book progresses, her descriptions of the parts coalesce into an appreciation for how multidimensional the sea system is.

Carson’s descriptions come through multiple lenses. She describes the sea through the behavior of animals and volcanoes. She explores the sea by describing its vertical integration from the surface to the depths and the bottom. She looks at the oceans through the lenses of their currents and relationships to wind. In total the book describes the same entity, the seas that cover the majority of the earth’s surface, through thirteen different descriptive lenses. Although the parts are broken down into their basics, the comprehensive view that Carson employs allows the reader to easily grasp how complicated the sea system is.

None of the lenses she uses impart complete information. Trying to appreciate how interconnected the parts of the system are by looking at just her description of tides or minerals is impossible. It’s only when the lenses are combined that a complete picture of the ecosystem emerges.

The book demonstrates the value in description, even if you cannot conclude causation in the specifics you’re describing.

One noticeable omission from the book is the role of plate tectonics in the movement of the ocean floor and associated phenomena like volcanoes. Plate tectonic theory is a scientific baby and was not yet widely accepted when Carson updated her original text in 1961. But not knowing plate tectonic theory doesn’t undermine her descriptions of life at the bottom of the oceans, or the impact of volcanoes, or the changing shape of the undersea shelves that attach to the continents. Although the reader is invited to contemplate the why behind what she is describing, we are also encouraged to be in the moment, observing the ocean through Carson’s words.

***

The book is not an argument for a particular way of interacting with the sea. It doesn’t need to make one. Carson’s descriptions offer their own evidence of how trying to change or manage the sea system would be extremely difficult because they reveal the multitude of connections between various sea phenomena.

Describing the whole from so many different angles illuminates the complex. By chronicling microinteractions, such as those between areas of hot and cold water or high and low pressure, we can see how changes in one aspect produce cascading change. We also get a sense of the adaptability of the living organisms that live in the oceans, like being able to live in depths that have no light (and therefore no plants that rely on photosynthesis) and adjusting biochemistry to take advantage of seasonal variations in temperature that affect water weight and salt contents.

The reader walks away from the book appreciating the challenge in describing in detail something as complicated as the ocean ecosystem. The book is full of observations and short on judgments, an approach that encourages us to develop our own curiosity about the sea around us.

The Sea Around Us is the Farnam Street book club’s summer selection. Members get additional resource materials to help get the most out of this fascinating book. Learn more or join in.

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Mirror Your Audience: Four Life Lessons From Performance Artist Marina Abramović

Reading Time: 6 minutes

Imagine it is a Saturday. You are in New York and decide to go to the Museum of Modern Art. There is a special exhibit on called The Artist is Present. Performance artist Marina Abramovic is sitting in one of the galleries. You wait in line to sit across from her, noticing that hundreds of people are lined up for the experience. Anticipation slowly builds as your time gets closer.

From your spot in line, you can see her seated in a red dress. Across the table from her, people sit; some for mere moments, others for long stretches. You later learn that there is no limit on the length of time you can sit at the table across from her, but you are not allowed to touch or talk to her. As you are waiting in line, the experience is like looking at a living painting.

When it’s your turn and you take your seat across from her, your experience transforms and becomes part of the art itself.

Marina Abramovic performed The Artist is Present, sitting eight hours a day for three months in 2010. She trained for months to build the physical stamina to perform the piece, and in her memoir Walk Through Walls she comments on how the performance demonstrated the profound need for people to connect.

Originally from Belgrade, Abramovic began her art career in the 1960s, starting with traditional painting, before moving into performance pieces in the early 1970s. Many of her pieces are iconic, and she continues to work all over the world.

Here are some of her insights from her memoir that transcend performance art and speak to how we can move through life to achieve our goals.

On fear

“It is incredible how fear is built into you, by your parents and others surrounding you.”

“Human beings are afraid of very simple things: we fear suffering, we fear mortality. What I was doing in Rhythm 0—as in all my other performances—was staging these fears for the audience: using their energy to push my body as far as possible. In the process, I liberated myself from my fears. And as this happened, I became a mirror for the audience—if I could do it, they could do it too.”

The relationship we have with fear can be one of the defining relationships of our life. Finding a way to accept and process our fears is an ongoing task that can be mastered by willingly exploring them.

“For the first three months, I place each student at a table with a thousand pieces of white paper and a trash can underneath. Every day they have to sit at the table for several hours and write ideas. They put the ideas they like on the right side of the table; the ones they don’t like, they put in the trash. But we don’t throw out the trash. After three months, I only take the ideas from the trash can. I don’t even look at the ideas they liked. Because the trash can is a treasure trove of things they’re afraid to do.”

Part of dealing with fear is acknowledging it. Hiding fear in the garbage bin does us no good. When we confront what we are afraid to do, we find immense opportunities to develop and grow because we increase our options and adaptability. We both remove limits and teach ourselves how to handle a wider spectrum of fear-inducing situations in the future.

On finding your place

“All at once it occurred to me—why paint? Why should I limit myself to two dimensions when I could make out from anything at all: fire, water, the human body? Anything! There was something like a click in my mind—I realized that being an artist meant having immense freedom.”

We need to explore until we find our “clicks.” Limiting ourselves to convention might not be satisfactory. As Abramovic began to develop her performance art pieces, her physical self was at the center of her work. In most of her pieces, the audience was confronted with the idea of art as a living thing: not a painting on a wall, but a physical body.

“The essence of performance is that the audience and the performer make the piece together.”

In one of her pieces, people entered the gallery while she stood completely still, dressed in a blouse and pants. On a table in front of her were dozens of objects—things like cards, lipstick, pins, and even a gun. Over the next hours, the audience was invited to use any object to do whatever they wanted to Abramovic. Slowly at first, but with increasing momentum, they began to interact with her body. They moved her arms, put lipstick on her, put cards in her hand. One person stuck a pin into her. Only at the end of the night did security intervene when someone picked up the gun and prepared to shoot her.

It must have been an incredible experience.

The notion that an audience can participate in the creation of art as it is happening has been an important aspect of many of Abramovic’s performances. In the dynamic nature of performance art, Abramovic has found her medium to explore the nature of art itself.

“I had experienced absolute freedom—I had felt that my body was without boundaries, limitless; that pain didn’t matter, that nothing mattered at all—and it intoxicated me. I was drunk from the overwhelming energy that I’d received. That was the moment I knew that I had found my medium.”

Paying attention to our moments as we experience them helps us find our place. Abramovic writes often in her memoir about the success that came with being fully committed to her performances while they were happening. Her ability to move through physical pain in pursuit of artistic expression, and to communicate with her audience, stems in part from her knowing she was creating from a place of authenticity.

On art

“What is art? I feel that if we see art as something isolated, something holy and separate from everything, that means it’s not life. Art must be a part of life. Art has to belong to everybody.”

Another famous piece of hers was done in a gallery space, where she lived in front of the audience in three open rooms. Each room was a square box with one side removed. One was a living room, another one was a bedroom, and the final one was a bathroom. Abramovic did everything in front of the crowds who came into the gallery—sleep, meditate, shower, and use the toilet. One interpretation of the work is that by inviting the audience to watch her doing some of the mundane daily tasks that we all do, she provided a way for them to find the art in their own lives.

“This is a rule of performance: once you enter into this mental-physical construct you’ve devised, the rules are set, and that’s that—you’re the last one who can change them.”

Part of Abramovic’s success has stemmed from her fearlessness in using performance art to explore basic questions of humanity, as well as from her total respect for the medium. Her pieces are thoughtful, and no doubt leave a lasting impression on those who are able to watch and engage with them.

On the business of art

“It’s interesting with art. Some people have the ability—and the energy—not just to make the work, but to make sure it’s put in exactly the right place, at the right moment. Some artists realize they have to spend as much time as it took them to get an idea in finding the way to show it, and the infrastructure to support it. And some artists just don’t have that energy, and have to be taken care of, by art lovers or collectors or the gallery system.”

The business of creating and the business of selling one’s creations require different skills and temperaments that are not always found in the same person. When we know our capabilities, we can try to set ourselves up for success by knowing how and where to ask for help. All of us want our outputs to be recognized in some way. What we don’t often realize is that we have a part to play in getting that recognition to happen. We can’t expect everything to magically fall into place.

“From the mid-1970s to the end of the decade, performance art caught on. . . . The originators of the medium were no longer young, and this work was very hard on the body. And the market, and art dealers especially, were putting increasing pressure on artists to make something to sell, because, after all, performance produced nothing marketable.”

Choices close some doors and open others. Having a personal definition of success is critical. We have to make sacrifices to achieve our goals, so we must be confident we are working toward our own idea of success and not someone else’s. Abramovic continues to do performance art, but she also became an art teacher and has partnered on numerous other art projects, including films. From her memoir, we learn that an insightful way of measuring our achievements is against what we believe gives life meaning.

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The Method of Loci: Build Your Memory Palace

Reading Time: 11 minutes

“When information goes ‘in one ear and out the other,’ it’s often because it doesn’t have anything to stick to.” —Joshua Foer

According to legend, the renowned Greek lyric poet Simonides of Ceos was dining at the home of a wealthy nobleman one night when he received word that two young men were waiting outside with a message for him. Simonides stepped outside—mere moments before the roof of the banquet hall caved in, killing everyone inside. The extent of the destruction was so great that the family and friends of the deceased guests despaired of being able to identify their bodies.

But Simonides found that if he pictured the spatial layout of the hall, he could mentally walk around it and recall the names of the guests seated in each place. In this way, he helped identify the unfortunate diners. From this experience, Simonides deduced that remembering factual information is easiest if we tie it to a physical location we are familiar with. As the great orator Cicero explained:

“He inferred that persons desiring to train this faculty must select localities and form mental images of the facts they wish to remember and store those images in the localities, with the result that the arrangement of the localities will preserve the order of the facts, and the images of the facts will designate the facts themselves, and we shall employ the localities and images respectively as a wax writing tablet and the letters written on it.

. . . It has been sagaciously discerned by Simonides or else discovered by some other person that the most complete pictures are formed in our minds of the things that have been conveyed to them and imprinted on them by the senses, but that the keenest of all our senses is the sense of sight, and that consequently perceptions received by the ears or by reflection can be most easily retained in the mind if they are also conveyed to our minds by the mediation of the eyes, with the result that things not seen and not lying in the field of visual discernment are earmarked by a sort of image and shape so that we keep hold of as it were by an act of sight things that we can scarcely embrace by an act of thought.”

Today we know this technique as the “method of loci” or “memory palace.” Here’s how constructing a memory palace can help you memorize information and recall it with ease.

Spatial memory

From the time we learn to walk, we start building up spatial memories—recollections of the layouts of physical spaces and their relationships to the objects in them. These memories tend to form fast and stick around for a long time.

The method of loci hijacks our innate aptitude for remembering physical spaces, using it to help us remember other kinds of information with greater ease.

Although it may seem as if people who manage unusual feats of memory have unusual innate capabilities, much of the available evidence suggests that isn’t the case. Instead, they tend to simply use innate capabilities in unusual, creative ways. In Moonwalking with Einstein, journalist Joshua Foer discovers this for himself while investigating memory championships. After contestants tell him anyone could do what they do with the right training, Foer sets his sights on the USA Memory Championship—and goes on to win.

Foer’s journey started by researching memory and its physical effects on the brain. Scientists had recently discovered that your brain is much like a muscle and that making it work could make it grow by creating new pathways at a cellular level.

Does that mean the brain of a “mental athlete” is different from that of a normal person? Not necessarily. Foer found research indicating memory specialists simply used different parts of their brains for recalling information. They converted it into a visual form and positioned it in a spatial location.

A memory palace is an ancient mnemonic device that leverages the way we find it easiest to recall information: spatially and visually.

You might have experienced the power of spatial memory if you’ve ever revisited a place where you experienced something unpleasant or sad. Minor details might have been enough to make you vividly remember the experience.

How to build a memory palace

Let’s look at an example to illustrate the concept.

Say your memory palace is your childhood home. Take a moment to conjure images and memories of that place. We are going to stick to the outside of the house. Mentally walk from the road to your front door, trying to remember as many details as possible.

Let’s imagine you want to remember to buy some steaks on the way home. Now put the steaks, exactly how they look in the grocery store, on your front porch.

Got it?

Okay, now let’s try to make the steaks into something more memorable. How about a cow sitting on your front porch—not like a cow would, but like a person would. Let’s make them exaggeratedly chewing, but we’ll make it bubblegum instead of grass. Now the cow is periodically blowing gigantic bubbles, so big that you’re worried they might pop. Maybe think about the strange smell of bubblegum and cow mixed together. What would the cow’s skin feel like? What would it feel like to have to pick bubblegum off of the cow’s face?

Four hours from now when you leave work to head home, you’ll remember you had to pick something up from the grocery store. When you take a trip to your memory palace, walk up the drive and gaze at your front step. What do you think you are more likely to remember? The packaged steaks that you see all the time? Or the gum-chewing cow we created?

Now, imagine you also need to remember to buy carrots and olive oil, so you place those in your memory palace too. As you step onto the front step, past the bubblegum blowing cow, you see the row of muddy shoes you recall usually being next to the front door. You picture each shoe being filled with soil, the leaves and orange tops of carrots growing in them just visible. Before you can press the doorbell, the door is opened by the cartoon character Olive Oyl, wearing a crown of olive branches. And so on.

If you’re trying to learn something more complex and important than a shopping list, it’s possible to keep repeating this process until you’ve constructed an incredibly detailed memory palace with hundreds of items. Or you might use multiple different locations for various purposes. You could also use a journey, such as your walk to work.

No matter what, you want to make your mental image engage with the location as much as possible and ideally involve multiple senses. Avoid placing multiple different items in parts of your memory palace that look too similar, such as identical dining room chairs.

The beauty of the memory palace technique is that it’s highly adaptable for remembering all kinds of information in whatever way feels easiest to you.

Remembering numbers

The memory palace is a great way to recall a variety of things, but you will still hit a hard ceiling, and that ceiling conflicts with the Herculean amount of numbers some memory competitors can remember. What’s the trick? It turns out that there is a whole different tool just for recalling numbers.

PAO: Person – Action – Object
In this system, every two-digit number from 00 to 99 is processed into a single image of a person performing an action on an object. Joshua Foer writes:

“The number 34 might be Frank Sinatra (a person) crooning (an action) into a microphone (an object). Likewise, 13 might be David Beckham kicking a soccer ball. The number 79 could be Superman flying with a cape. Any six-digit number, like say 34-13-79, can then be turned into a single image by combining the person from the first number with the action from the second and the object for the third – in this case, it would be Frank Sinatra kicking a cape.”

As you can see, this is still about storing very vivid and memorable images. We don’t know about you, but Frank Sinatra kicking a cape hasn’t come to our minds before. It becomes a very powerful tool when you realize that you can use your “stock images” as a sort of algorithm to generate a unique image for every number between 0 and 999,999.

How a memory palace works

When we’re learning something new, it requires less effort if we connect it to something we already know, such as a physical place. This is known as elaborative encoding. Once we need to remember the information, we can “walk” around the palace and “see” the various pieces.

The idea is to give your memories something to hang on to. We are pretty terrible at remembering things, especially when these memories float freely in our heads. But our spatial memory is actually pretty decent, and when we give our memories some needed structure, we provide that missing order and context.

For example, if you struggle to remember names, it can be helpful to link people you meet to names you already know. If you meet someone called Fred and your grandmother had a cat called Fred, you could connect the two. Creating a multisensory experience in your head is the other part of the trick. In this case, you could imagine the sound of Fred meowing loudly.

To further aid in recall, the method of loci is most effective if we take advantage of the fact that it’s easiest to remember memorable things. Memory specialists typically recommend mentally placing information within a physical space in ways that are weird and unusual. The stranger the image, the better.

Returning to the name example, it will probably stick better if you imagine Fred the person being chased by a giant version of Fred the cat, compared to just imagining them next to each other.

Cicero further explains:

“But these forms and bodies, unlike all the things that come under our view, require an abode, inasmuch as a material object without a locality is inconceivable. Consequently (in order that I may not be tedious on a subject that is well known and familiar) one must employ a large number of localities which must be clear and defined and at moderate intervals apart, and images that are effective and sharply outlined and distinctive, with the capacity of encountering and speedily penetrating the mind; the ability to use these will be supplied by practice, which engenders habit, and by marking off similar words with an inversion and alteration of their cases or a transference from species to genus, and by representing a whole concept by the image of a single word, on the system and method of a consummate painter distinguishing the positions of objects by modifying their shapes.”

The Roman rhetorician Quintilian also recommended the memory palace technique:

“Some have thought memory to be a mere gift of nature, and to nature, doubtless, it is chiefly owing. But it is strengthened, like all our other faculties, by exercise, and all the study of the orator of which we have been speaking is ineffectual unless the other departments of it be held together by memory as by an animating principle. All knowledge depends on memory, and we shall be taught to no purpose if whatever we hear escapes from us.

. . . when we return to places, after an absence of some time, we not only recognize them, but recollect also what we did in them. Persons whom we saw there, and sometimes even thoughts that passed within our minds, recur to our memory. Hence, in this case, as in many others, art has had its origin in experiment. People fix in their minds places of the greatest possible extent, diversified by considerable variety, such as a large house, for example, divided into many apartments. Whatever is remarkable in it is carefully impressed on the mind, so that the thought may run over every part of it without hesitation or delay. Indeed, it is of the first importance to be at no loss in recurring to any part, for ideas which are meant to excite other ideas ought to be in the highest degree certain.

They then distinguish what they have written, or treasured in their mind, by some symbol by which they may be reminded of it, a symbol which may either have reference to the subject in general, as navigation or warfare, or to some particular word, for if they forget, they may, by a hint from a single word, find their recollection revived . . . they place, as it were, their first thought under its symbol, in the vestibule, and the second in the hall, and then proceed round the courts, locating thoughts in due order, not only in chambers and porticoes, but on statues and other like objects. This being done, when the memory is to be tried, they begin to pass in review all these places from the commencement, demanding from each what they have confided to it, according as they are reminded by the symbol. Thus, however numerous are the particulars which they have to remember, they can, as they are connected each to each like a company of dancers hand to hand, make no mistake in joining the following to the preceding, if they only take due trouble to fix the whole in their minds.

What I have specified as being done with regard to a dwelling house may also be done with regard to public buildings, or a long road, or the walls of a city, or pictures, or we may even conceive imaginary places for ourselves.”

An important caveat is that the method of loci only helps you recall the specific information you’ve used it to encode. While you might find it easier to use the technique the more times you repeat it, you’re unlikely to see an overall improvement in your general memory.

Despite its limitations, constructing a memory palace is a fun and creative way to learn things you don’t want to write down (such as passwords or security question answers) or want to be able to call to mind on the fly (such as decision-making checklists). And if you’re someone who enjoys a challenge, it can be satisfying to see just how far you can extend your innate abilities.

***

The memory palace reminds us of the importance of being mindful and paying attention to life. Foer takes it further, arguing that when we look at it critically, memory is a huge component of almost every aspect of our lives:

“How we perceive the world and how we act in it are products of how and what we remember. We’re all just a bundle of habits shaped by our memories. And to the extent that we control our lives, we do so by gradually altering those habits, which is to say the networks of our memory. . . . Our ability to find humor in the world, to make connections between previously unconnected notions, to create new ideas, to share in a common culture: all these essentially human acts depend on memory. Now more than ever, as the role of memory in our culture erodes at a faster pace than ever before, we need to cultivate our ability to remember. Our memories make us who we are.”

We are a culmination of our experiences. How we process this information and encode it into something meaningful is intrinsically tied to our memory. Understanding how it works and how to use tools or tricks to make it better is a worthy endeavor.

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The Precautionary Principle: Better Safe than Sorry?

Reading Time: 7 minutes

Also known as the Precautionary Approach or Precautionary Action, the Precautionary Principle is a concept best summed up by the proverb “better safe than sorry” or the medical maxim to “first do no harm.”

While there is no single definition, it typically refers to acting to prevent harm by not doing anything that could have negative consequences, even if the possibility of those consequences is uncertain.

In this article, we will explore how the Precautionary Principle works, its strengths and drawbacks, the best way to use it, and how we can apply it in our own lives.

Guilty until proven innocent

Whenever we make even the smallest change within a complex system, we risk dramatic unintended consequences.

The interconnections and dependencies within systems make it almost impossible to predict outcomes—and seeing as they often require a reasonably precise set of conditions to function, our interventions can wreak havoc.

The Precautionary Principle reflects the reality of working with and within complex systems. It shifts the burden of proof from proving something was dangerous after the fact to proving it is safe before taking chances. It emphasizes waiting for more complete information before risking causing damage, especially if some of the possible impacts would be irreversible, hard to contain, or would affect people who didn’t choose to be involved.

The possibility of harm does not need to be specific to that particular circumstance; sometimes we can judge a category of actions as one that always requires precaution because we know it has a high risk of unintended consequences.

For example, invasive species (plants or animals that cause harm after being introduced into a new environment by humans) have repeatedly caused native species to become extinct. So it’s reasonable to exercise precaution and not introduce living things into new places without strong evidence it will be harmless.

Preventing risks and protecting resources

Best known for its use as a regulatory guideline in environmental law and public health, the Precautionary Principle originated with the German term “Vorsorgeprinzip” applied to regulations for preventing air pollution. Konrad Von Moltke, director of the Institute for European Environmental Policy, later translated it into English.

Seeing as the natural world is a highly complex system we have repeatedly disrupted in serious, permanent ways, the Precautionary Principle has become a guiding part of environmental policy in many countries.

For example, the Umweltbundesamt (German Environmental Protection Agency) explains that the Precautionary Principle has two core components in German environmental law today: preventing risks and protecting resources.

Preventing risks means legislators shouldn’t take actions where our knowledge of the potential for environmental damage is incomplete or uncertain but there is cause for concern. The burden of proof is on proving lack of harm, not on proving harm. Protecting resources means preserving things like water and soil in a form future generations can use.

To give another example, some countries evoke versions of the Precautionary Principle to justify bans on genetically modified foods—in some cases for good, in others until evidence of their safety is considered stronger. It is left to legislators to interpret and apply the Precautionary Principle within specific situations.

The flexibility of the Precautionary Principle is both a source of strength and a source of weakness. We live in a fast-moving world where regulation does not always keep up with innovation, meaning guidelines (as opposed to rules) can often prove useful.

Another reason the Precautionary Principle can be a practical addition to legislation is that science doesn’t necessarily move fast enough to protect us from potential risks, especially ones that shift harm elsewhere or take a long time to show up. For example, thousands of human-made substances are present in the food we eat, ranging from medications given to livestock to materials used in packaging. Proving that a new additive has health risks once it’s in the food supply could take decades because it’s incredibly difficult to isolate causative factors. So some regulators, including the Food and Drug Administration in America, require manufacturers to prove something is safe before it goes to market. This approach isn’t perfect, but it’s far safer than waiting to discover harm after we start eating something.

The Precautionary Principle forces us to ask a lot of difficult questions about the nature of risk, uncertainty, probability, the role of government, and ethics. It can also prompt us to question our intuitions surrounding the right decisions to make in certain situations.

When and how to use the Precautionary Principle

When handling risks, it is important to be aware of what we don’t or can’t know for sure. The Precautionary Principle is not intended to be a stifling justification for banning things—it’s a tool for handling particular kinds of uncertainty. Heuristics can guide us in making important decisions, but we still need to be flexible and treat each case as unique.

So how should we use the Precautionary Principle? Sven Ove Hansson suggests two requirements in How Extreme Is the Precautionary Principle? First, if there are competing priorities (beyond avoidance of harm), it should be combined with other decision-making principles. For example, the idea of “explore versus exploit” teaches us that we need to balance doubling down on existing options with trying out new ones. Second, the decision to take precautionary action should be based on the most up-to-date science, and there should be plans in place for how to update that decision if the science changes. That includes planning how often to revaluate the evidence and how to assess its quality.

When is it a good idea to use the Precautionary Principle? There are a few types of situations where it’s better to be safe rather than sorry if things are uncertain.

When the costs of waiting are low. As we’ve already seen, the Precautionary Principle is intended as a tool for handling uncertainty, rather than a justification for arbitrary bans. This means that if the safety of something is uncertain but the costs of waiting to learn more are low, it’s a good idea to use precaution.

When preserving optionality is a priority. The Precautionary Principle is most often evoked for potential risks that would cause irreversible, far-reaching, uncontainable harm. Seeing as we don’t know what the future holds, keeping our options open by avoiding limiting choices gives us the most flexibility later on. The Precautionary Principle preserves optionality by ensuring we don’t restrict the resources we have available further down the line or leave messes for our future selves to clean up.

When the potential costs of a risk are far greater than the cost of preventative action. If a potential risk would be devastating or even ruinous, and it’s possible to protect against it, precautionary action is key. Sometimes winning is just staying in the game—and sometimes staying in the game boils down to not letting anything wipe you out.

For example, in 1963 the Swiss government pledged to provide bunker spaces to all citizens in the event of a nuclear attack or disaster. The country still maintains a national system of thousands of warning sirens and distributes potassium iodide tablets (used to reduce the effects of radiation) to people living near nuclear plants in case of an accident. Given the potential effects of an incident on Switzerland (regardless of how likely it is), these precautionary actions are considered worthwhile.

When alternatives are available. If there are alternative courses of action we know to be safe, it’s a good idea to wait for more information before adopting a new risky one.

When not to use the Precautionary Principle

As the third criteria for using the Precautionary Principle usefully, Sven Ove Hansson recommends it not be used when the likelihood or scale of a potential risk is too low for precautionary action to have any benefit. For example, if one person per year dies from an allergic reaction to a guinea pig bite, it’s probably not worth banning pet guinea pigs. We can add a few more examples of situations where it’s generally not a good idea to use the Precautionary Principle.

When the tradeoffs are substantial and known. The whole point of the Precautionary Principle is to avoid harm. If we know for sure that not taking an action will cause more damage than taking it possibly could, it’s not a good idea to use precaution.

For example, following a 2011 accident at Fukushima, Japan shut down all nuclear power plants. Seeing as nuclear power is cheaper than fossil fuels, this resulted in a sharp increase in electricity prices in parts of the country. According to the authors of the paper Be Cautious with the Precautionary Principle, the resulting increase in mortality from people being unable to spend as much on heating was higher than the fatalities from the actual accident.

When the risks are known and priced in. We all have different levels of risk appetite and we make judgments about whether certain activities are worth the risks involved. When a risk is priced in, that means people are aware of it and voluntarily decide it is worthwhile—or even desirable.

For example, riskier investments tend to have higher potential returns. Although they might not make sense for someone who doesn’t want to risk losing any money, they do make sense for those who consider the potential gains worth the potential losses.

When only a zero-risk option would be satisfying. It’s impossible to completely avoid risks, so it doesn’t make much sense to exercise precaution with the expectation that a 100% safe option will appear.

When taking risks could strengthen us. As individuals, we can sometimes be overly risk averse and too cautious—to the point where it makes us fragile. Our ancestors had the best chance of surviving if they overreacted, rather than underreacted, to risks. But for many of us today, the biggest risk we face can be the stress caused by worrying too much about improbable dangers. We can end up fearing the kinds of risks, like social rejection, that are unavoidable and that tend to make us stronger if we embrace them as inevitable. Never taking any risks is generally a far worse idea than taking sensible ones.

***

We all face decisions every day that involve balancing risk. The Precautionary Principle is a tool that helps us determine when a particular choice is worth taking a gamble on, or when we need to sit tight and collect more information.

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Seizing The Middle: Chess Strategy in Business

Reading Time: 7 minutes

Chess can serve as an apt metaphor for other areas of our lives, especially business. That’s because the game is a microcosm of the ways we use strategic thinking. There are not many areas where we can quickly assess the quality of our decisions and whether they are likely to have the desired effects. Chess helps us develop strategic thinking because we get immediate feedback on our strategic decisions. It also shows the benefits of thinking ahead.

Perhaps its value for teaching strategic thinking has something to do with the game’s longstanding appeal. Chess has been around for an estimated fifteen centuries, and precursors go back at least 4,500 years; it both reflects and teaches important skills. Seizing the middle is a chess strategy embodying the value of forward thinking. It involves using pieces to commandeer the middle of the board. A player can then restrict their opponent’s movements by controlling the maximal number of pieces in the game.

Strategies akin to seizing the middle are also used in areas such as business, economics, and negotiation. Analogous strategies involve limiting an opponent’s options by asserting control over a resource or area, be it physical or conceptual. Some of the most profitable businesses throughout history employed this strategy and treated the world like a chessboard.

John D. Rockefeller infamously used the strategy of seizing the middle to control the oil industry throughout the nineteenth century. Before he turned forty (according to a Fortune estimate) Rockefeller had personal control over an estimated 90% of the US oil refining industry via the Standard Oil company, and by the time of his death he was the richest person alive. Depending on who you ask, he was either a callous figure who valued money above all else or a shrewd businessman who boosted employment and gave away most of his fortune. Unsurprisingly, every detail of his life and especially his business strategies have been analyzed at great length. While the opportunities Rockefeller capitalized on are unlikely to come about again, they show how chess strategies can translate into business acumen.

It’s hard to overstate just how important the oil industry is to any nation. Because he controlled the oil, Rockefeller could leverage his power to make almost any negotiation go his way. A power which he used on the railroad companies.

Rockefeller recognized early on that railroads were the lifeblood of the oil industry because oil had to be shipped, and thus he sought to gain control of them. Railroads were to the oil business what the middle of a chessboard is to a player—without dependable, controlled access to them, a company could make precious few moves. As he loathed competition, Rockefeller sought to eliminate it—and one of his maneuvers to reduce his competition in the oil business was making sure no one else could transport it around the country.

In the nineteenth century, it was customary for shipping companies to offer rebates (partial refunds) or generous discounts to their biggest customers. Once Standard Oil became the largest oil refining company in the United States, Rockefeller was in an excellent negotiating position with the railroads. In exchange for huge amounts of regular business, the shipping companies agreed to give him an unusually large rebate. Cutting the costs of transporting oil gave Rockefeller a robust competitive advantage. Combine this with his efficient manufacturing process and shrewd usage of byproducts, and Standard Oil’s prices were a fraction of the usual cost of oil. Unsurprisingly, other oil companies had no hope of offering lower or even equivalent rates and still making a profit. If any of them seemed like they might pose a threat, Rockefeller could use his influence over the shipping companies to restrict their ability to transport oil.

Although controlling access to the railroads was a key element in seizing the middle territory of the oil business, Rockefeller had many more pieces in play. Should restricting railroad access be unfeasible, he would cut off competitor’s access to equipment, undercut their prices or buy up all the available raw materials. The amount of control Rockefeller had allowed him incredible power over the entire industry. In The Politics of the Global Oil Industry, Toyin Falola and Ann Genova explain that “Standard Oil had extended its control not only over its competitors but also over oil transportation. Nearly every method of transport from the oil fields to the consumer was owned by Standard Oil, which allowed the company significant control over prices.”

Rockefeller thought in terms of first principles, which often meant controlling his own means of production. For example, he cut the cost of barrels by manufacturing them himself and the cost of laying pipework by employing his own plumbers. As Standard Oil grew, Rockefeller’s power grew exponentially. At a certain point, no one could compete with him. With the lion’s share of the market and profits to match, he could get credit for almost unlimited loans, giving Standard Oil a further advantage over competitors and a dominance over the oil territory.

As Alfred Chandler explains in The Visible Hand, Rockefeller’s strategy was part of a wider transition to a new type of industry, beginning in the 1840s and ending with the crash of the 1920s. Businesses started “seizing the middle” and taking control of the resources they depended on. A single company could take charge of everything from the natural resources required to make a product to the transport systems necessary to deliver it to customers. The implications of this were dramatic. Chandler writes of Rockefeller:

“He and his associates then decided to obtain the cooperation of its rivals by relying on the economic power provided by their high-volume, low-cost operation. They began by asking the Lake Shore Railroad to reduce its rates from $2 to $1.35 a barrel on Standard Oil shipments between Cleveland and New York City if Standard provided sixty carloads a day, every day. The road’s general manager quickly accepted, for assured traffic in such high volume meant he could schedule the use of his equipment much more efficiently and so lose nothing by the reduced rate. Indeed, the general manager, somewhat gratuitously, offered the same rates to any other oil refiner shipping the same volume.”

Chandler describes how the change in business practices allowed managers to start thinking like chess players: a few moves ahead. Being able to anticipate and plan had the undeniably significant effect of allowing companies to invest more in research and development because they could forecast where current trends headed:

“In allocating resources for future production and distribution, the new methods extended the time horizon of the top managers. Entrepreneurs who personally managed large industrials tended, like the owners of smaller, traditional enterprises, to make their plans on the basis of current market and business conditions. . . . The central sales and purchasing offices provided forecasts of future demand and availability of resources.”

Seizing the middle didn’t just help create the energy industry as we know it today. The strategy also contributed to the creation of the modern film industry.

For four tumultuous decades, known as The Golden Age of Hollywood, eight studios all but governed the global film industry. Between the 1920s and 1960s, Fox, Loew’s Inc., Paramount, RKO Radio, Warner Bros., United Artists, Universal, and Columbia Pictures formed the studio system. Much like Standard Oil needed control over the railroads to ensure their success, the film oligarchy also prioritized power over distribution systems. In this case, that meant owning the cinemas that showed their films. For the most part, they also owned the production facilities, and held Hollywood staff and stars under strict long-term contracts.

For example, actor Cary Grant signed a five-year contract with Paramount in 1931. This gave the studio such control over him that they could literally loan him to other studios—in 1935, Paramount lent him to RKO so he could star alongside Katharine Hepburn. Having popular actors with the cache to draw audiences to anything they appeared in under contract restricted the movements of any other studios, dictating the number of pieces that could be on the board.

New anti-trust laws in the late 1940s and the rise of television in the 1950s contributed to the end of the Hollywood studio system. Both Grant and Hepburn escaped the grip of their respective studios and took control of their own careers. Grant refused to renew his contract once it expired and became possibly the first freelance Hollywood actor. Hepburn bought out her contract after being assigned to a string of unsuccessful films.

Hollywood nonetheless achieved a lot during the studio system days. This era, beginning with the rise of “talkies” (films with sound) shaped many of our expectations of films. Many major cinematic genres and conventions were devised during the Golden Age. The low cost of producing films with all aspects of production and distribution under tight control meant studios could take chances with unproven actors, directors, and scripts for films like Citizen Kane. Although the era produced a lot of formulaic, repetitive, or dull works, it also gave birth to many that remain popular and well-loved even now.

In The Hollywood Studio System: A History, Douglas Gomery describes how Adolph Zukor, founder of Paramount, devised the studio system:

“During the 1910s, Adolph Zukor through his Famous Players and then Paramount corporations developed a system by which to manufacture popular feature-length films, distribute them around the world, and present them in Paramount picture palaces. . . . Zukor taught the world how to make motion pictures popular and profitable in a global marketplace. He also laid down the principles of the studio system.

From his entry into the industry, Zukor wanted to take control of the new movie business . . . and began to develop a national distribution system which would hereafter serve as the basis for the studio system. . . . Zukor was smart and looked to see how other industries developed their corporate economic power . . . and made films in a factor like a system; and he developed a distribution division (Paramount) to sell his wares throughout the world.

. . . In the end, Zukor and his followers developed a set of operating principles. Their industry—symbolized by their Hollywood studios—would be made up of a small set of corporations that produced, distributed, and presented films in order to maximize the profits of their corporations. The number . . . would total eight.”

To control the game, one tries to control as much of the board as possible. At the outset, using your pieces to seize the middle of the playing field is a great strategy, because it gives you the widest possible vantage point from which to control the movement of the other pieces. Both Rockefeller and the studio system in Hollywood employed this strategy successfully, allowing them to anticipate change and maneuver effectively for decades.

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