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.