How I Stopped Worrying and Learned To Love Math

1001 Books To Read Before Your Kindle Battery Dies, Part 61
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?

Some of Lockhart's critics say that math should focus on memorized formulae, because knowledge is cumulative, and few students can savvy higher math without a comprehensive foundation. But how many want or need higher math? As a student of mine said, she'll never need to factor polynomials in real life. No, we don't study math for its perceived utility. But that's not to say that math isn't useful.

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.

Francis Bacon in the Empire of Indulgence

Janice Law, Moon over Tangier

Francis Bacon has grown bored and lovelorn in London. No, not that Francis Bacon, the other one: the Anglo-Irish figurative painter whose tortured, distorted human forms revolutionized postwar British art. So he follows his paramour and sometime torturer David to colonial Morocco, a paradise of anonymous sex and late imperial exoticism. Once in Tangier, however, he finds a warren of crime, including a mysterious murder tied to local art forgery.

Though this is American novelist Janice Law’s third, and reputedly final, “Francis Bacon Mystery,” the book itself is functionally freestanding. It doesn’t require any particular familiarity with prior novels, or Francis Bacon, or art generally, to grasp Law’s narrative of cruel romance, venal empire, and postwar chicanery. Though marketed as a mystery, Law essentially crafts a tale of character and circumstance, which just coincidentally involves fraud and murder.

Bacon simply wants to drink copiously and romance as many Moroccan “beach boys” as limited time and money allow. When one prominently flamboyant party host’s newly purchased Picasso proves forged, however, the local constable blackmails Francis into joining the investigation. Seems they have a blank warrant for David’s arrest, which they’ll sign if he doesn’t comply. Thus Francis becomes an unwilling, but entirely too proficient, art counterfeiter.

When it appears everybody lies about this deeply conflicted case, nobody appears particularly surprised, except perhaps Francis himself. The police have political motivations that don’t involve aestheticism or intellectual property. The party host proves remarkably duplicitous when the spotlight shines elsewhere. When Francis finds himself stalked by a suspiciously James Bond-like agent of the British Legation, he realizes he’s stumbled into a case with truly international implications.

Janice Law

Tangier, in Law’s telling, has a familiar yet discordant ring. As a colonial capital, it catches every expatriate the European homeland doesn’t want to see. But unlike Bogart’s Casablanca, this African melting pot doesn’t catch diverse wartime refugees fleeing Naziism and generalized homeland oppression; Tangier caters specifically to sexual outcasts, the bon vivants and sybarites forced into exile from Continental homes where homosexuality remains wholly illegal.

David, the man who induced Bacon’s newfound expat status, is the consummate charming abuser. A masterful musician and storyteller when sober, a raconteur of wartime extravagance, his fondness for gin transforms him into a knife-wielding psychopath. Francis admits his fondness for “rough trade,” but also opens his narrative by fleeing into a hog barn overnight to avoid David’s homicidal wrath. The romance apparently flees rough trade pretty quickly.

Yet Francis remains steadfastly willing to defend David, whose distant but amorous attentions combine mentor, lover, and big brother in equal, and equally terrifying, measures. The fact that neither Francis nor David remains particularly faithful reflects, not homosexuals’ innate moral corruption, but how pliable ethics become under circumstances of official oppression. As first-person narrator Francis explains, police often serve traumatic beat-downs to “beach boys” for sport.

One of Francis Bacon's iconic, and deeply
disturbing, Screaming Popes paintings

Readers selecting this book for gripping mystery may find this book awkwardly slow-moving compared to contemporary thrillers. Law doesn’t expend energy on creating complex mental puzzles, and though she does well in concealing her culprit, she doesn’t particularly care about red herrings and other familiar tropes. For Law’s money, historic Tangier and its milieu of Crumbled Empire decadence holds more interest. This is mainly a historical novel and costume drama.

And what a historical novel. More literate critics than I have extolled Law’s scrupulous attention to detail. Where less meticulous authors have recently frustrated me for their willingness to compromise accuracy to create a better-selling paperback, Law creates a narrative for audiences who desire the more difficult journey. Like visiting a foreign land, one gets the feeling nothing here is sanitized for modern Anglo-American sentiments. Bacon’s Tangier is very dangerous.

This painstaking accuracy sometimes results in moments that make modern readers distinctly uncomfortable. Watching the revolutionary independence movement evolve from their rooftop divans, Bacon and David are sometimes prone to observations we’d call racist. And although I balk at modern complaints about “cultural appropriation,” Bacon’s tendency to regard Moroccan culture as a profligate shopping trip uncomfortably reduces African culture to a shopping mall. As imperial conquerors often do.

But these very uncomfortable moments contribute to the immediacy of Law’s narrative. As Bacon navigates Tangier’s difficult political byroads, we sympathize with him, while still realizing he’s a colonial asshole. We want him to win, but not too much. Where suit-clad Bogart represented a pre-war antihero model, Bacon epitomizes postwar moral collapse. His narrative is immersive, and perilous. And we emerge wondering: how much am I like these colonial taskmasters?

Marking a Trail Across the Digital Wilderness

Shlomo Benartzi with Jonah Lehrer, The Smarter Screen: Surprising Ways to Influence and Improve Online Behavior

If you’ve ever felt adrift in today’s technological world, helpless amid infinite options, unable to make meaningful decisions in buying or reading or even just watching five hundred TV channels, you’re not alone. UCLA behavioral economist Shlomo Benartzi has reliable scientific evidence demonstrating what perceptive people already know, that today’s online environment creates a panicked, lost-in-the-woods feeling in most people. Fortunately, wise engineering can reverse this paralyzing trend.

From its earliest origins, pro-Web enthusiasm has gushed over the Internet’s capacity to provide users more information, more choices, and more autonomy. It’s been the classic capitalist assertion, that simply having more options available flushes bad choices away and consolidates good choices. But anyone who’s shopped for consumer electronics online recently knows that simply having more choices isn’t good enough. Without guidance, increased options generate snap judgments, haste, and paralysis.

Benartzi himself has participated in new research demonstrating how screen users handle information overload. His conclusion, based on his own research and the published discoveries of his peers: human attention isn’t adapted for broad, undifferentiated knowledge dumps. (Anyone who’s read government reports recently knows this.) Wise information merchants going forward will need to make the digital screen conform to what Benartzi calls “the mental screen”—our attention capacity.

“It's not that screens are making us more superficial,” Benartzi writes. “Rather, the world of screens merely makes it easier for us to act on those superficial first impressions.” We gravitate toward first options, self-indulgence, and whatever’s placed in the center of the screen. When people order food digitally, Benartzi says (with generous scholarly evidence), we’re more likely to pick unhealthy options. When we get doused with a firehose of buying options, we pick familiar images, not good choices.

Too often, Benartzi and his sources claim, humans rely upon facile heuristics to comprehend digital resources. (Anyone who’s read Daniel Kahneman already knows this, but Benartzi proves how this applies specifically to screens.) We think better when we enjoy instant feedback… until we become overwhelmed. We think in purely present-tense terms, unless prompted. Simple design tweaks can transform not only our experiences, but our thinking about those experiences.

Shlomo Benartzi

Similar tweaks have the potential for abuse. Benartzi describes circumstances where unscrupulous operators have manipulated screen options to maximize unfair trades, and places where even worse abuse is possible. Some online vendors, for instance, have gotten caught spotlighting more expensive options on mobile devices than on computers, because apparently the smaller the screen, the smaller our attention to detail. Crooked vendors manipulate our impatience and forgetfulness for venal ends.

This tendency has remarkable ramifications. As America enters an election season, unprecedented numbers of voters will garner political opinions from articles read online. This seems innocent, except we don’t read well off screens. Benartzi writes: “screens allow us to read more than ever before, but they also encourage us to read poorly, and to remember less of what we read.” Screens encourage casual, unengaged, incomplete reading… and thinking.

Every fault Benartzi describes, though, has easy fixes. Some involve top-end design improvements. Counterintuitive as it seems, evidence demonstrates that we read better when confronted with clunky, inelegant typefaces. When confronted by endless buying choices (Amazon sells over 100,000 breakfast cereal varieties), categories help us prevent getting overwhelmed and numb. The disastrous Healthcare.gov rollout could’ve been salvaged with March Madness-style brackets.

Benartzi also suggests end-user improvements, though these are less numerous. In giving feedback, for instance, actually sign your name; this prevents the Disinhibition Effect. Seek carefully for resources which encourage without criticizing and which have meaningful action plans. Allocate attention carefully: it’s impossible to actually read those bottomless “terms and conditions” we all click on, but when somebody asks you a real question, read the whole thing before answering.

The answers aren’t always easy, and seldom one-size-fits-all. The problem with digital reading, for instance, has different solutions for different audiences. Younger readers will probably finish reading this review better on Amazon’s brightly colored, graphics-heavy layout, while my blog’s muted colors and text-driven format will attract older readers. Other problems have similarly nuanced solutions. My final takeaway from Benartzi’s explanation: effective design is more art than science.

Late in this book, Benartzi states what could arguably be his thesis: “We don't want endless possibility. What we really crave is effective curation.” The Internet has become a vast repository of information, much of it exciting, some terrifying; and new information gets created or repackaged daily. Top-level providers who follow Benartzi’s guidelines will more likely earn users’ attention, today’s real economic prize.

The Little Black Phone That Conquered China

Clay Shirky, Little Rice: Smartphones, Xiaomi, and the Chinese Dream

Unless you’ve lived or worked in China since 2010, you’ve probably never encountered Earth’s third-largest mobile phone company, Xiaomi. The company, whose name translates as “Little Rice” in Chinese, currently dominates China’s mid-range smartphone market. Despite having no market presence in North America or Europe, Xiaomi’s global aspirations have flourished, and in five years, it has progressed from geek-oriented software startup to punching above its weight class.

Clay Shirky, NYU professor and new media cheerleader, sees Xiaomi as emblematic, not of the mobile phone industry, but of China’s changing place in global technology leadership. China has a lengthy reputation as America’s favorite offshore dumping ground; Apple famously labels its products “Designed In California; Assembled In China.” But as Chinese manufacturers become increasingly comfortable with American designs, many have assumed the design role and created native products domestically.

If you’ve experienced the existential dread of realizing you’ve forgotten your phone, you understand how important networked mobile technology has become. This goes double for poor countries. As rising technology puts mobile phones within mass customer reach, villagers in Africa and Asia who cannot afford laptops, cars, or books, nevertheless have global access with affordable phones. Networked mobile access puts poor, distant villagers on (theoretically) equal footing with everyone else.

Shirky quotes technology maven Jan Chipchase that mobile phones have joined money and keys as the three things without which we cannot leave the house. This presents China unprecedented opportunity and unique obstacles. Mainland China remains among Earth’s last nominally Communist nations, though “Communist” now means more party control than economic principle. Mobile phones offer instant global access to information, but also permit instantaneous coordination, anathema to one-party states.

Xiaomi straddles this complicated line. The corporation got its start pitching an open-source Android variant operating system, crowdsourcing critical design elements. This democratic hybrid business model remains central to Xiaomi’s success, even as they’ve transitioned into hardware development. But the corporation must tapdance between the Party’s twin mandates of unprecedented personal liberty, and absolute political autocracy. Xiaomi’s technology must permit everyone to do everything… except challenge the state.

Clay Shirky

Chinese President Xi Jinping has recently pushed something he calls the Chinese Dream. Unlike Maoist slogans of expunging both ancient and Western influences, Xi’s vision has a distinctive monetary value. Like the American Dream, the Chinese Dream involves home ownership, wealth management, and upward mobility. However, it infuses distinct collectivist tenets into its language. Xi wants Chinese to become wealthy in ways that redound positively upon the state.

China actively restricts domestic access to global Internet sources. Chinese citizens can access Amazon, Gmail, and other international sites through demi-legal network loopholes, and enforcement runs two-tiered for elites versus plebeians. Whatever advances business and globalization, the Party deems good, or anyway tacitly permits; whatever permits citizens to organize, coordinate, or pitch any alternative to the state, the Party squashes. (State speech codes forbid puns, to excise double meanings.)

This creates market narrowing for Chinese firms outside China. Shirky describes how, when Xiaomi first marketed phones in South Asia, Indian media generated panic that Xiaomi was shipping user data into China. Well, all mobile service providers store metadata on native server farms; Apple and Samsung do it. Their servers merely dwell in non-totalitarian states. Xiaomi, by contrast, must simultaneously appease state officials and free-market buyers, a difficult global sell.

For Shirky, Xiaomi isn’t another corporation, despite its massive size (its market is huge enough to dwarf international firms while remaining primarily locked inside China’s borders). Rather, Xiaomi represents China’s strange relationship with modernity. China has become economically free but politically restricted; socially diverse but ethnically homogenous; global yet totalitarian. If it survives ten more years, the People’s Republic will become history’s longest-lived one-party state.

Thus China is already making history, even if that history feels muddled to technocratic Western observers. It’s yoked competing influences together, with apparent success, and seems to be winning the global three-legged race. But China’s aggregate wealth hasn’t translated into individual well-being; the typical Chinese citizen remains poor by world standards. How information firms like Xiaomi fare in coming years will define how China faces today’s global situation.

Shirky’s writing combines journalism with technical expertise to describe powerful conflicting influences which will define global market considerations within our lifetimes. Xiaomi’s rapid advance, from zygote to market dominance inside five years, and its odd, counterintuitive business model, are symbolic, for Shirky, of China’s changing role in global economics. To understand China through analogy, Shirty says, let’s understand Xiaomi. That isn’t easy, but it’s decidedly unnerving.

The New Queen of the Smoke-Eyed Dreamscape

1001 Books To Read Before Your Kindle Battery Dies, Part 60
Cat Rambo, Eyes Like Sky and Coal and Moonlight: Stories

Spirits attack a fort full of despair on the frontier of a mythic kingdom. A girl is cursed to carry flame sprites throughout the land, creating a magical massacre. Tourists on the dingy side of Bangkok meet a woman who may or may not be Andersen's Little Mermaid. An elemental sorceress gambles everything to save her nation and discovers that victory may be the key to her greatest loss.

Cat Rambo, hailed as a leading voice in fantastic fiction, collects twenty stories of the speculative, the bone-chilling, and the uncanny. The tales in this volume are so strange, so evocative, and so different from one another that it's hard to believe they were all written by the same person. Rambo has a remarkable talent for plunging readers into alien realities in only a few pages, a talent that's become lamentably rare in recent short fiction.

The title story features many themes common in Rambo’s writing. (No, she doesn’t use a pseudonym.) Describing a young woman’s encounters with the general of an all-woman army, Rambo delves into meanings of gender and role in worlds where post-industrial Western traditions don’t apply. The story arose from Rambo’s involvement with a multi-player online RPG, underlining the reciprocal relationship Rambo’s writing has with media technologies.

Sometimes Rambo wears her influences undisguised upon her sleeves. “I’ll Gnaw Your Bones, the Manticore Said” has a very Harlan Ellison-esque title, though the story feels more Poe-like in its themes, and Lovecraftian in its imagery. Other times, Rambo’s influences vanish subtly beneath her sophisticated storytelling. “The Towering Monarch of His Mighty Race,” told from an animal’s POV, seamlessly combines a slumgullion of fictional and non-fictional, mainstream and fringe sources.

Literary critics and writing teachers frequently pooh-pooh genre fiction as mere escapism—as though there’s anything “mere” about stepping outside our lives and circumstances to perceive the world through new eyes. Okay, sometimes mass-market paperback fiction lowballs its audience’s intelligence. And sometimes audiences willingly live down to whatever expectations well-crafted marketing lays upon them. Perhaps ours is an age of tragically lowered standards.

Cat Rambo

But Cat Rambo, and her shoestring publisher, Paper Golem, represent the other extreme of genre writing. Like gazing into a funhouse mirror, Rambo’s fiction presents us with ourselves from another viewpoint, making us consider our situations from outside our limited, self-important perspective. Though never overtly political, Rambo’s best stories challenge her readers’ status quo. Every story places, or traps, its readers in the spotlight. Ultimately, every story is about us.

Consider “Narrative of a Beast’s Life.” In both form and content, it mirrors 19th Century American slave narratives, like Olaudah Equiano and Frederick Douglass, while reminding us that the slaveholding mentality persists in post-industrial society. “Eagle-Haunted Lake Sammamish” questions what it means to “purchase” land forcibly seized from its original indigenous inhabitants. Can we ever truly own something stolen by our ancestors?

Also, Rambo continues a tradition in American fiction. From Sherwood Anderson’s Winesburg, Ohio, to Lovecraft’s Arkham, Massachusetts, American writers love creating fictional cities which come into existence, not whole and ready to visit, but distant, complex, glimpsed only through characters’ partial viewpoints. Rambo’s distant city of Tabat, a manifestation of the human id, isn’t a place we visit. It’s a place that ambushes, engulfs, and strands mere tourists.

These stories refuse to be limited to one or a few genres. Rambo freely mixes heroic fantasy with psychological horror, or steampunk with westerns. Hers is an innovative mind that will stop at nothing to tell the best possible story, and she writes for eager, curious readers. Every character she creates has a distinctive voice, and every story she tells expands her world, and the reader's as well.

I applaud Rambo for choosing a small press. However, the wing-and-a-prayer budget of Paper Golem apparently leaves Rambo without an editor, and her stories could intermittently use a little clean-up. Several sentences drop important words, and sometimes her punctuation could be called quirky. Though these are distracting, they never diminish my enjoyment. Some of Rambo's story notes, on the other hand, contain spoilers; read her notes only after the stories.

Rambo comes to the reading public with glowing recommendations from luminaries like Jeff VanderMeer and John Barth, and it's easy to see why. Her unconventional fantasy refuses to follow familiar paths, and her writing eclipses most genre fiction coming pell-mell from the major publishing houses. This debut short story collection, signalled the arrival of a bold voice in fantasy literature, promise Rambo, now SFWA president, has fulfilled with aplomb.