Tag: Paul Lockhart

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.

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.