Paul Lockhart, A Mathematician's Lament: How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form
I find myself in the awkward position of arguing against a book I admire, because Paul Lockhart sells himself short in this remarkable book. First, Lockhart thinks he's identified a problem unique to current mathematics education, when he's actually identified schooling's primary flaw. Second, he thinks we shouldn't teach math as "merely useful," when it's profoundly useful, just not in the way math textbook authors commonly think.
Lockhart believes (and I agree) that the current focus on rote memorization, "skillz drillz," and repetitive exercises causes students to falsely believe math is a heap of formulae in a vacuum. He expertly dismantles how math is taught while demonstrating the discipline's true, dynamic nature. To Lockhart, mathematics should foster inquiry and a curious, question-driven mindset that then pursues answers. Math teaches how to face challenges without road maps.
But on page 40, Lockhart asks: "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?" Would Lockhart like me to start a list? I could do it alphabetically. Historian James Loewen and literature professor Gerald Graff have voiced the same complaints in their fields. Discouragement of inquiry is endemic throughout education today.
Reading Lockhart's demonstrations of popular mathematical concepts, I was struck by how seemingly complex concepts suddenly appeared both clear and welcoming. I remembered the difference between my undergraduate education, which favored memorization and regurgitation, and graduate school, which encouraged individual discovery and growth. But why must I or anyone wait for grad school before unlocking the truth for ourselves?
|A Renaissance woodcut depicting Euclid,|
the father of plane geometry
Lockhart's eloquent, graceful proofs demonstrate a mind which faces questions that lack explicit answers, and then pursues those answers. This process of investigation is necessary in a world where clear-cut options are few on the ground. Too many graduates, facing adult life, stumble when they find that work and family aren't closed cases. These habits of investigation and inquiry are useful inasmuch as life offers more questions than answers.
In the six years since I first read this book, American education reformers have addressed some of this book’s concern by widening the selection of learning heuristics children learn. Though old fogies like me have mocked Common Core’s alternate learning patterns for being different than what we learned, I’ve come to appreciate the mindset behind the differing patterns. Teaching children diverse ways to approach common problems with systematic rigor.
Sadly, Common Core functionally repeats math education’s underlying flaw, because no matter how many heuristics children learn, they still reduce learning to rote memorization. Lockhart, in this book, concedes that sometimes we must give struggling children correct answers on a silver platter. And arithmetic, often, is best learned by mimicry. But higher math, as Lockhart asserts, is more than a repetitive skill. It’s a true, and underappreciated, art form.
More than that, math is a simple joy. Edward de Bono writes that many people, confronted with new ideas that upset their preconceived notions, respond with laughter. Laughing is our human response to having our eyes opened, our minds widened. I never understood what that meant until I read this book. Lockhart incorporates several exercises from Euclidean geometry to open our thinking to math’s higher influences, and while reading, I repeatedly laughed like a madman.
Math is useful and desirable because it opens doors of thought. Too much of school appears closed to debate, but real life forces us to ask questions that don't have answers. People who aren't equipped to face such questions take on adult roles without their most valuable tools. Whether it's math or science or art or business, we must learn to face questions, weigh possibilities, and seek that "Eureka" moment.
Often, people who would dominate us seek to create the false impression that sophisticated questions can be solved as simply as textbook exercises. Appropriate education, including an introduction to math, should teach us not to plug memorized answers into dynamic, changing questions. Math teaches us that each question opens new answers, and, like all disciplines should, invites us to learn how to ask and investigate for ourselves.