The Folly of the Big Idea

America was made by and for big ideas. Insofar as big ideas have shaped it, it is ever on the verge of hyperbole and dream. "America is a land of wonders," wrote Alexis de Tocqueville, "in which everything is in constant motion, and every movement seems an improvement. The idea of novelty is there indissolubly connected with the idea of amelioration. No natural boundary seems to be set to the efforts of man; and what is not yet done is only what he has not yet attempted to do."¹ Our history abounds with vast spaces, ambitions, and concepts: the Declaration of Independence, the American West, Great Awakenings, Manifest Destiny, the silver screen, self-made millionaires, big business, superpower status, dreams of liberty, space exploration, Google, and more.

Of course, America isn't only big; the supersize comes with a counterpart. The small town, the ordinary person, the town hall meeting, the Girl Scout helping others every day—all of this figures in the American psyche as well. We may even distrust big ideas at times. According to the satirist P. J. O'Rourke, "distaste for grandiose notions is embedded in our language"—for instance, in expressions like "What's the big idea?"² What's missing from much of our discourse is the discipline of building from basic axioms to larger principles and creations (and breaking the principles down into their elements). Just as it takes patience to learn to play an instrument or lead an athletic team to victory, so it takes diligence to develop an idea or structure that can last.

It is such structure that allows a document like the Declaration of Independence to endure in our daily life and understanding. The declaration contains much more than a grand idea; drawing on centuries of philosophy, and resounding from phrase to phrase, it progresses from axioms to facts to conclusions. It suggests through its language and logic that one must know and grapple with the past in order to transform the present. It thus stands in contrast to many of the big ideas of today.

Today's big ideas come with an air of celebrity and accessibility; they glitter with glamour but demand little of us. While they have many manifestations, we see them epitomized in TEDTalks. TED (which stands for Technology, Entertainment, Design), a nonprofit that offers two annual conferences of short lectures on innovative ideas, mixes extreme elitism with extreme accessibility. Tickets to the annual Long Beach event cost $7,500 and upward, and are available by invitation or application only, yet the talks themselves may be viewed on the Internet by all, free of charge, and require minimal background knowledge. In a New Yorker article on TED, Nathan Heller writes, "By most measures, TED shapes its style against the mores of academia. Educational lectures are set at a podium; TED prizes theatrical movement. Academic work relies on communities of shared premises and interpretive habit; TED tries to communicate without those givens. Scholarship holds objectivity as a virtue; TED aims for the heart." Writing for Salon, Alex Pareene makes similar points with more of a sting: "What's most important is a sort of genial feel-good sense that everything will be OK, thanks in large part to the brilliance and beneficence of TED conference attendees. (Well, that and a bit of Vegas magician-with-PowerPoint stagecraft.)"³

The typical TEDTalk gives the impression that one need only feel and believe it to be part of it, like Peter Pan, whose wonderful thoughts allow him to fly. The TED viewer imagines himself an insider, capable of understanding the concepts because they excite him in the moment. Think big, dream big, he imagines, and he will be big too. Why, the lectures seem so simple, so relatable, they couldn't not be for him. Listen to Sir Ken Robinson and, aha, it's all clear! Our schools are locked in an industrial model, which stifles creativity and talent. Convert them to an agricultural model, and the possibilities will multiply. Listen to Salman Khan, and you learn that he plans to use his extensive instructional video library to "humanize" education on a "global scale." Listen to Susan Cain, and you might come to believe that we are "poised on the brink of dramatic change" regarding introversion, quiet, and solitude. If we heed Cain's call to "open up" our "suitcases" and show what's inside them, we may all be able to grace the world with our gifts.⁴ In each case, the TEDTalk casts a complex problem in grand, uplifting, and unchallenging terms. While many individual TEDTalks have merit, the conference as a whole has become the biggest forum for today's biggest fad: bigness itself.

The fad resembles the historical phenomenon of "high modernism" as described by James C. Scott in Seeing Like a State: "a particularly sweeping vision of how the benefits of technical and scientific progress might be applied—usually through the state—in every field of human activity." According to Scott, while we have become distrustful of high modernism, it persists in various forms to this day.⁵ One could apply Scott's description to phenomena that do not originate in the state: for instance, grand ideas propagated by entrepreneurs and philanthropists.

Something like high modernism persists with a vengeance in education reform. Many reformers insist that reform must be sweeping and replicable in order to count as reform at all. That expectation creates a conundrum. In order to be sweeping, a reform must standardize its language and methods; in doing so, it loses touch with the particulars of subject matter, school, and classroom. Granted, sometimes there is a need for sweeping systemic change of one kind or another. But when an idea must sound big in order to gain traction, when policymakers and reformers equate the thoughtful, modest initiative with the dreaded "status quo," the ideas themselves get shortchanged.⁶

Take, for instance, the New York City Department of Education's "Children First" initiative, launched in 2003, which mandated the Balanced Literacy, Everyday Mathematics, and Impact Mathematics curricula throughout the school system (except for some 200 high-performing schools). Teachers, parents, and education commentators criticized these curricula for their amorphousness and lack of content, but to no avail. A year or so later, teachers received word through their administrators that they were all required to follow the "workshop model," a generalized version of the Readers' and Writers' Workshops of Balanced Literacy.⁷ Principals conducting observations expected teachers to follow the model; teacher preparation programs reinforced it. A few years later, the Department of Education began loosening the mandate but did not acknowledge openly that it had made a mistake, or several. The mistake lay not only in the choice of curricula (or quasi-curricula) but in the insistence on a single model for teaching. A teacher needs the latitude to plan lessons that suit the topic. A workshop model may be suitable for some topics but not for all.

Another example can be found in recent special education reform. Many school districts around the country have adopted Universal Design for Learning (UDL), a framework developed in a joint project of the Center for Applied Special Technology and the US Department of Education. UDL enables teachers to design curricula for diverse learners in advance, instead of on the fly. According to UDL, current curricula are not only deficient but "disabled"; UDL claims to address these disabilities by providing multiple means of "representation, expression, and engagement." For instance, a teacher using an equal sign in mathematics class should consider in advance the possibility that some students don't know what it means and should therefore provide "alternative representations."⁸ While the intent of UDL is laudable, it errs in its wholesale disparagement of current curricula (some of which might be quite good) and in its insistence on multiple representations. (If students are having difficulty with the equal sign, they should learn to work with the equal sign itself, not with a substitute.) Nonetheless, UDL enjoys federal support—perhaps because it proposes drastic change and claims to improve outcomes for all students.

One could cite many more examples of big ideas in education—value-added assessment, differentiated instruction, discovery learning, small schools, online learning, and so forth—and find a similar pattern. Small schools have advantages (and disadvantages), but the size of a school is not in itself a predictor of its quality. Differentiated instruction has many meanings and manifestations and is not always appropriate for a lesson or course. Nonetheless, education reform sweeps up such concepts with enthusiasm, applies them broadly, and continues to champion them even when they start to founder. It seems too complicated, too unglamorous, to interpret ideas carefully and apply them where they belong. Yet this is the more rewarding practice.

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How did the "big idea" mindset take over education reform? Its recent ascent is due, at least in part, to the weakening of the middle class and the gradual loss of a liberal principle of education. By the latter I mean a principle that honors the liberal arts: the study of a range of subjects not only for their uses, but for their beauty, their fascination, and their role in cultivating the mind.

Over the past few decades, the middle class has been losing many of the attributes that once defined it (if it even exists at this point). In September 2012, the US Census Bureau reported that 48.5 million people in the United States, or 15.9 percent of the population, lived below the official poverty line in 2011; according to scholars, the middle quintile of the population, the "middle class," owned only about 4 percent of US wealth. Income disparities have widened to an extreme; while CEO compensation increased more than 725 percent between 1978 and 2011, worker pay increased only 5.7 percent. In addition, workers contend with job uncertainty. In 2009, there were 28,286 mass layoff events; while the mass layoff numbers have decreased since then, they remain considerably higher than they were in the 1990s. Moreover, whoever loses a job carries not only the burden of unemployment but also its stigma; employers routinely overlook applicants who are not employed.⁹ As workers devote energy to getting and keeping jobs, they lose not only the material aspects of middle-class existence, but some of its intellectual aspects as well. (The working class and middle class have never been identical—but as the latter shrinks, so does the overlap between the two.)

A middle-class existence used to offer free time, among many other things. Members of the middle class had room and time for "life, liberty, and the pursuit of happiness" (or could have it if they chose it). They went to college not only to find a job but to take interesting courses, form bonds with peers and professors, and participate in the college's cultural life. If they did poorly in a course or two, that wasn't the end of things; after all, one was expected to go through some trial and error in college. After college, they could find jobs that were challenging but not grueling, jobs that allowed them some time for their own pursuits. Some devoted themselves to their jobs and to advancement within their careers, but at least they had the option of claiming some time for themselves and for service to the community. This meant that they were at liberty to take on projects that might come to fruition slowly or not at all. Granted, such freedom (in college and afterward) carried the risk of confusion and extended adolescence, but for many it made room for intellectual play, meaningful pursuits, and patience.

Such conditions, in turn, allowed colleges and universities to emphasize the life of the mind.¹⁰ A few decades ago, despite shrinking humanities departments and growing economic anxiety, students were encouraged to take time to select a major; to explore different subjects and interests; to take challenging courses, even at the risk of lower grades; to pursue what interested them, not what would lead to the most lucrative jobs; and to take part in the cultural life of the college and the surrounding community. In his remarkable book College: What It Was, Is, and Should Be, Andrew Delbanco of Columbia University recalls how Judith Shapiro, former provost of Bryn Mawr and then president of Barnard, explained the meaning of college to a group of young people: "You want the inside of your head to be an interesting place to spend the rest of your life."¹¹

My own memories of college, though not all rosy by a long stretch, abound with illustrations of this principle. Students would crowd into a lecture hall to listen to lectures on art history, even if they weren't taking the course for credit. They would major in English even though people warned them (erroneously) that you couldn't do much with an English major. They would spend evenings discussing philosophical questions, even if they had a test the next day. They would perform community service because they believed in it and learned from it, not because it would look good on their resumes. (Of course many were thinking of their resumes, but it was possible not to do so obsessively.) Such intellectual liberty had pitfalls; many college students blundered without adequate guidance, or threw themselves into an extracurricular activity at the expense of their studies. Yet the very spirit of intellectual quest allowed students to transcend these troubles; a student who veered into excess could redirect herself and come out wiser. Wisdom, or the striving toward it, was part of the point of college.

The spirit of intellectual quest has not vanished, but it is shrouded in pragmatic language. Students are expected to set career goals as early as middle school and to shape their studies around these goals. ACT, which develops career-planning tools and assessments for students and employees, has conducted research on "major-interest congruence" (that is, the congruence between a student's major and his interests) and its relation to GPA and perseverance. The assumption is that student interest serves as a conduit to success (as measured by grades and graduation rates).¹² What happens to the slow struggle with difficult and compelling material? When students believe that they are supposed to succeed and only succeed, they look for the short route.

There are several seemingly short routes to success, if success means test scores and money. One is to set precise goals and take the safest route toward them—for instance, by avoiding the most challenging courses. Another is to make use of connections. Still another is to hit upon a lucrative big idea. The latter has gained appeal in the last few decades. People like Mark Zuckerberg, Malcolm Gladwell, and Steve Jobs tantalize the popular imagination, as they seem to have leapt to fame with little more than a concept. Such a prototype of success is by no means new, but it grows more seductive as people need it more.

Schools, tests, and textbooks help promote the fantasy; everywhere we hear the buzzword of "success" (meaning high test scores, eventually a high salary, and possibly fame). Granted, schools emphasize the role of hard work in success, but they present success as a grand, noticeable achievement, usually the result of "thinking big." Quieter forms of success rarely enter the picture. Policymakers and education critics call for more "innovation" and "creativity" in the schools, as though one could skip over traditional subject matter and get on with the breakthroughs. At the same time, they present this "innovation" primarily as preparation for the workplace, not for an imaginative life. Students receive a mixed message: they hear, on the one hand, that they should take risks and think creatively; on the other, that they should follow directions exactly and choose their career paths early on. The "big idea" fantasy offers young people relief from this conundrum; if they think big enough, many imagine, they won't have to finish school or become anyone's employee.¹³

But the best kind of study consists neither of following directions exactly nor of rushing toward innovation. It has to do with building one's knowledge and understanding of a field, until insights start to come through. One need not wait years for insights, but they will deepen over time. One thinks creatively not as a result of trying to think creatively, but rather as a result of close study of a subject—or, in the K–12 years, a range of subjects. We find meaning in our learning as we start to relate the details to the larger parts, and the larger parts to the whole.

This slow progression remains important even at the highest levels of scholarship. Some of the most valuable ideas and creations do not come quickly, nor are they recognized immediately. They may not have vast ramifications; they may be of note mainly to those interested in the particular subject. This is no shame. To have the strength to work in relative obscurity, without quick and dramatic rewards, is to have a room of one's own, in Virginia Woolf's sense of the phrase: a place for untrammeled thought. When we scramble for quick results, we give up our quiet rooms.

Honoring the Liberal Arts

How can we tone down the "big idea" culture and make room for subtler, more interesting ideas? We could start by honoring the liberal arts in schools, colleges of education, and beyond. By "liberal arts" I mean, in addition to common definitions, those studies that hold intrinsic interest and beauty as well as practical applications. By "honoring the liberal arts" I mean not only implementing a liberal arts curriculum but also living it. This means having faculty meetings about historical documents, works of literature, or math problems (and not just how to teach them). It means bringing these topics into teacher preparation, so that prospective teachers will start thinking about them before they enter the classroom. It means looking at education not only in terms of its specific objectives but also in terms of its subtleties and surprises. It means scrutinizing reforms for their compatibility with these endeavors. If a reform is destructive of liberal arts curriculum and culture, then it should be adjusted, reconsidered, or discarded.

We have luminous examples of such practice. There is the Dallas Institute of Humanities and Culture, which offers courses in the humanities to teachers, principals, and superintendents. There is the Yale–New Haven Teachers Institute, where New Haven public school teachers work collegially with Yale scholars: the teachers study a subject in a seminar taught by a professor and then write curriculum units that incorporate what they have learned. There is the Bard Master of Arts in Teaching Program, which requires advanced study both in the elected discipline and in education courses (in about equal proportions). These and other programs could serve as guides.¹⁴

Honoring the liberal arts may sound like a big idea in itself, but it requires modesty, as its meaning comes clear only in the details. It may take decades to bring to full fruition, but the rewards will be apparent along the way. It is not an all-encompassing idea, nor can it be implemented in a uniform fashion. There will be variations from school to school and from student to student. Still, if we devote ourselves to the principles, we will enrich our practice and discourse.

Imagine, for instance, a classroom where students are reading Robert Frost's poem "Birches."¹⁵ The teacher reads it out loud and then takes them through it, posing questions and encouraging discussion as they go along. The poem takes them slowly from one place to another: from bent birches in the woods and the thought of a boy swinging them, to the trees after an ice storm:

Loaded with ice a sunny winter morning
After a rain. They click upon themselves
As the breeze rises, and turn many-colored
As the stir cracks and crazes their enamel.

It would be difficult not to pause over those lines; the phrase "As the stir cracks and crazes their enamel" could, in itself, change a student's sense of language. The poem then returns to the imagined boy (and what the narrator "was going to say when Truth broke in / With all her matter-of-fact about the ice-storm"), a boy whose play consists of swinging the birches, "Some boy too far from town to learn baseball, / Whose only play was what he found himself, / Summer or winter, and could play alone," and how he "subdued" every one of his father's trees. There is good fortune in this boy's solitude; because the baseball games are far away, he finds his own way of playing.

This game of swinging birches held many lessons for the boy and holds a few for us:

                                    He learned all there was
To learn about not launching out too soon
And so not carrying the tree away
Clear to the ground. He always kept his poise
To the top branches, climbing carefully
With the same pains you use to fill a cup
Up to the brim, and even above the brim.

But this boy, as we may have suspected, is not entirely imaginary: "So was I once myself a swinger of birches. / And so I dream of going back to be." The game of the birches starts to reveal itself as a game of excess and return, of gentle flirtation with the limits of life on Earth, and of understated wit and sadness: "Earth's the right place for love: / I don't know where it's likely to go better." I won't reveal the ending—but this enough suggests that you can read "Birches" and find yourself subtly recreated, swinging birches along with the boy and the older Frost, and years afterward still seeing the "trunks arching in the woods."

Someone might ask: "What will students be able to do as a result of this lesson? Shouldn't every lesson leave students able to do something that they couldn't do before?" Well, no, not every lesson has to teach students how to do something, and this lesson will teach them to do quite a bit. First of all, as they listen to the poem, they will hear how it plays with iambic pentameter, neither strictly following it nor pushing it away. They will hear lines bend from left to right just as the birches do. They may also pay attention to the three similes of the poem, all of them striking: the trunks trailing their leaves on the ground "Like girls on hands and knees that throw their hair / Before them over their heads to dry in the sun"; the boy climbing carefully to the top "With the same pains you use to fill a cup / Up to the brim, and even above the brim"; and life being "too much like a pathless wood / Where your face burns and tickles with the cobwebs / Broken across it, and one eye is weeping / From a twig's having lashed across it open." (There is a quasi-simile, too, early on: "You'd think the inner dome of heaven had fallen.") One could spend a great deal of time, likewise, discussing the metaphor of the swinging—which is only partly metaphor, after all, since the actual swinging plays a role here as well. Still, the point would be not to identify the similes and metaphors (a typical lesson objective) but to admire and ponder them, to live in them for a little while, and to notice their stirrings.

In other words, students studying such a poem would come away with specific knowledge of meter and figurative language, but their knowledge of the poem itself would be their greatest gain. To come to know the tone and wit of this poem, its melancholy and playful rumination, is to have a mentor, a memory, and a way of walking alone. It would be difficult to match these gifts.

When was the last time an education pundit or policymaker, a proponent of "creativity" and "innovation," suggested that students read "Birches"? I am willing to wager that it has never happened—and there lies the problem. This is the stuff that makes a difference in a school day and a life—but it slips from notice, since it isn't in line with big policy, at least not in an obvious way. Yet there is majesty in this poem. A student reading "Birches" learns about the quiet plunge from sky to earth, about play and yearning and love, about words and rhythms that offer us time and birches. "One could do worse than be a swinger of birches"—yes, indeed. (I gave away the last line, after all, but it's the poem as a whole that gives this line its meaning.) A liberal arts curriculum could be filled with works of this caliber, works that shape the way you see and hear the world.

A mathematics problem, too, might help to characterize a liberal arts curriculum. I chose a simple geometry problem, because it is intriguing, beautiful, and surprising.* I came upon it when reading Canto XIII of Dante's Paradiso. Here, St. Thomas tells Dante that King Solomon asked for wisdom, not for answers to vain questions, such as whether, within a semicircle, one can inscribe a triangle with no right angle. (The implication is that this is impossible.)¹⁶

The theorem asserts that any triangle inscribed within a semicircle must have a right angle at the vertex opposite the semicircle's diameter. (In precise terms, the triangle is "inscribed within a semicircle" in the sense that one of its sides coincides with the diameter of the semicircle and the vertex opposite this side lies on the semicircle.) In the figure below, points A, B, and C define the triangle and AC is the diameter of the semicircle.

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There is no limit to the number of distinct triangles one can create in this manner.¹⁷

The proof takes few steps and requires only basic knowledge of geometry. Consider the semicircle and triangle in the figure below. M is the midpoint between A and C, and is therefore the center of the semicircle because AC is the diameter; r is the radius of the semicircle. Thus segments MA, MB, and MC are all of length r. That means that triangles AMB and BMC are both isosceles (MA=MB and MB=MC), and their base angles are therefore equal. Let us say that ∠ BAM is β degrees (where β is a positive number); then ∠ ABM is also β degrees. Likewise, let us say that ∠ BCM is α degrees; then ∠ CBM is also α degrees.

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Now, look at the triangle ABC, the triangle originally under consideration. We already know that ∠ BAM is β degrees and ∠ BCM is α degrees. We know, also, that ∠ ABC is (α + β) degrees, since it is the sum of ∠ ABM and ∠ CBM, which are β and α, respectively. Because the measures of the angles of any triangle add up to 180 degrees, we have α + (β + α) + β = 180 degrees. Therefore, 2(α + β) = 180 degrees; therefore, (α + β) = 90 degrees. Thus, you can see that ∠ ABC will always measure 90 degrees when triangle ABC is inscribed in a semicircle in the sense defined above.

If we continue to ponder the theorem and its proof, we start to see many extensions and implications (related here in brief, not in detail). We see, for instance, that when a right triangle is inscribed within a semicircle (in the sense above), the distance from the midpoint of the hypotenuse to the opposite vertex is half the hypotenuse's length. There's more to it: we could prove that every right triangle can be so inscribed.

The proof has still more implications. For instance, we could use it to define a circle once a diameter is given: consider all right triangles whose hypotenuse coincides with the diameter, then the collection of all the vertices opposite the hypotenuse is a circle with the given diameter. Thus, in addition to our usual definition of a circle, the set of points at equal distance from a given point on a two-dimensional plane, we would have two definitions to explore. In mathematics, if you have two distinct definitions of something, it is common to ask: Does either definition imply the other? In this case, the answer is yes, which we could determine with a little bit of effort.

Later on, when students learn about sine and cosine, they may return to the triangle inscribed in a semicircle (in the sense above) and see that they can now look at any chord (a line segment with endpoints on the circumference) on a given circle from different perspectives.^† This is just one example of a mathematical problem that can suggest fruitful problems at different levels of study.

This problem or series of problems has numerous corollaries, analogs, and applications. One can use it to find the center of a circle, to construct the tangent to a circle from a given point outside the circle, and much more. At the same time, its sheer elegance—which comes from its simplicity and details—will likely make an impression on students. Students who work with such problems start to perceive possibilities beyond the apparent, and thus enter further into mathematics, which requires the ability to see unexpected connections in seemingly unrelated settings, to be "maker[s] of patterns of ideas," to quote the mathematician G. H. Hardy.¹⁸ This geometry problem is not entirely removed from the swinging of birches.

*  *  *

These two examples—Robert Frost's "Birches" and the geometry problem—show how exciting and instructive a liberal arts education can be—and how the details lead to the greater meanings. No matter what "objective" is written on the board, lessons on such topics will go far beyond the objective. Students will learn skills—and many of them—but will also carry interesting things in their minds, discuss them with others, and think about them when alone. If we defend and strengthen this kind of education, if we bring subject matter into education discussions themselves, then our priorities will be clearer. We will be able to temper and tune (or reject) the reforms that come our way.

For instance, we can put standardized tests in their place. Educators have tried in vain to convey to policymakers how limited the standardized tests are. If there were a common understanding of the nature of a liberal arts education, if policymakers and pundits understood the discrepancy between the tests and the actual subject matter, they'd be less likely to treat standardized test scores as precise measures of teaching quality, school quality, or student achievement.

All the same, we can readily acknowledge that standardized tests tell us something. If students do especially poorly or especially well, it makes sense to look into the reasons. We can make better use of tests if we don't exaggerate their importance; we should take whatever important information they offer, leave the rest, and continue to teach poetry and geometry. The same holds true for numerous other reforms and ideas—online instruction, the "workshop model," personalized instruction, and even school choice. Each has its place, but none should interfere with our treasured work.

Those concerned about raising standards can take heart: a liberal arts education is far from fluffy. It gives students more knowledge and more opportunities for creative thinking than any of the known alternatives. Students read and discuss concrete works of literature; memorize poetry and come to know its rhythms and shapes; examine the fine points and implications of mathematical proofs; study the facts, questions, and ideas of history; learn rules and principles of grammar and rhetoric; write in many formats and styles, about many topics; take part in the arts and study works of art; and develop a sense of virtue and character. Such education requires intensive practice and perseverance, and the rewards are often inherent in the work. The very practice of struggling with problems, of building one's understanding, of devoting one's attention to something worthy and beautiful, makes for an interesting and even happy life (and interesting careers to boot). Over time, this practice could lead to an enrichment of the public imagination: a renewed tolerance, even appreciation, of the slow, murmuring labor of the mind. To make room for such education, policymakers will need to overcome their insistence on quick, concrete results. It will benefit them to do so. We live in an era that places a premium on "outcomes"—but sadly, the more we focus on outcomes, the less likely we are to attain them. To attain anything of value in education, you need a strong sense of what is valuable; you must see beyond the immediate goal. A student of a musical instrument must learn to play scales well, but she is not learning scales in order to play scales. She is learning them so that she can play those pieces that amaze and move her—and perhaps compose pieces of her own. It is precisely for the sake of these pieces that she will persevere, if she has the will and the proper instruction. If we devote ourselves to things of beauty, we will enjoy good outcomes along the way; if we devote ourselves narrowly to outcomes, we will lose our sense of beauty.

In Aesop's fable "The Frog and the Ox," a frog tries to puff himself up to the size of an ox and bursts in the process. We have seen many a reform burst, not because it was too big per se, but because it puffed itself beyond its actual worth. To gauge the worth of education reform, we must hold it up against our best conception of education. This conception must build slowly; it must be grounded in literature, mathematics, history, and other subjects. If we let these subjects guide us, if we make room to contemplate, absorb, and discuss what they hold, we will not get lost. Or, if we do, we can call up those things we have learned and, through the recalling and reviving, find our way again.

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Diana Senechal teaches philosophy and Russian at Columbia Secondary School for Math, Science & Engineering in New York City. She is the 2011 recipient of the Hiett Prize in the Humanities and the author of Republic of Noise: The Loss of Solitude in Schools and Culture. Her translations of the Lithuanian poetry of Tomas Venclova have been published in two books, Winter Dialogue and The Junction, as well as numerous literary journals. Her education writing has appeared in Education Week, The New Republic, Educational Leadership, American Educational History Journal, The Core Knowledge Blog, Joanne Jacobs, GothamSchools, and the Washington Post's The Answer Sheet.

*This theorem, generalized to a circle, is attributed to the ancient Greek philosopher Thales (ca. 624 BC– ca. 546 BC). According to Diogenes Laertius, Thales sacrificed an ox in honor of his discovery. (Diogenes Laertius, Lives of Eminent Philosophers, 1.1.24.) (back to article)

^†If the circle has radius r, then the length of any chord AB drawn from a point A on the circle can be expressed directly in terms of r and the angle β that AB makes with the diameter through A, namely, 2r cos β. One can also explore the opposite relationship using the length of the chord to determine the angle. (back to article)

Endnotes

1. Alexis de Tocqueville, Democracy in America, trans. Henry Reeve (New York: Colonial Press, 1900), 432.

2. P. J. O'Rourke, "Let's Cool It with the Big Ideas," Atlantic Magazine, July/August 2012.

3. Nathan Heller, "Listen and Learn," New Yorker, July 9 and 16, 2012, 76; and Alex Pareene, "Don't Mention Income Inequality Please, We're Entrepreneurs," Salon, May 21, 2012.

4. Sir Ken Robinson, "Sir Ken Robinson: Bring on the Learning Revolution!" TEDTalks video, filmed February 2010, posted May 2010; Salman Khan, "Salman Khan: Let's Use Video to Reinvent Education," TEDTalks video, filmed March 2011, posted March 2011; and Susan Cain, "Susan Cain: The Power of Introverts," TEDTalks video, filmed February 2012, posted March 2012.

5. James C. Scott, Seeing Like a State: How Certain Schemes to Improve the Human Condition Have Failed (New Haven, CT: Yale University Press, 1998), 90. Scott credits the term "high modernism" to David Harvey (ibid., 377n3).

6. For a history of education reform of the past century, see Diane Ravitch, Left Back: A Century of Battles Over School Reform (New York: Simon & Schuster, 2001).

7. Sol Stern, "Bloomberg and Klein Rush In," City Journal, April 8, 2003; and David M. Herszenhorn, "Teachers Protest the Methods of a Regional Superintendent," New York Times, February 4, 2005.

8. National Center on Universal Design for Learning, "What Does It Mean to Say That Curricula Are Disabled?" www.udlcenter.org/aboutudl/udlcurriculum/disabledcurricula; National Center on Universal Design for Learning, "UDL Guidelines—Version 2.0: Principle 1. Provide Multiple Means of Representation," www.udlcenter.org/aboutudl/udlguidelines/principle1; and National Center on Universal Design for Learning, "FAQs from the National UDL Task Force," www.udlcenter.org/advocacy/faq_guides/general.

9. Sabrina Tavernise, "Soaring Poverty Casts Spotlight on 'Lost Decade,' " New York Times, September 13, 2011; Alemayehu Bishaw, "Poverty: 2010 and 2011" (Washington, DC: US Census Bureau, September 2012); Michael I. Norton and Dan Ariely, "Building a Better America—One Wealth Quintile at a Time," Perspectives on Psychological Science 6, no. 1 (January 2011): 9–12; Lawrence Mishel and Natalie Sabadish, CEO Pay and the Top 1%: How Executive Compensation and Financial-Sector Pay Have Fueled Income Inequality (Washington, DC: Economic Policy Institute, 2012), 2; Bureau of Labor Statistics, "Mass Layoffs—September 2012," news release, October 23, 2012, www.stats.bls.gov/news.release/archives/mmls_10232012.pdf; and Catherine Rampell, "The Help-Wanted Sign Comes with a Frustrating Asterisk," New York Times, July 25, 2011.

10. For the sake of simplicity, I treat the terms "college" and "university" as near-synonyms here, but they refer to institutions with different purposes and histories. For more on the distinction, see Andrew Delbanco, College: What It Was, Is, and Should Be (Princeton, NJ: Princeton University Press, 2012), 67–101.

11. Delbanco, College, 32–33.

12. Terence J. G. Tracey and Steven B. Robbins, "The Interest-Major Congruence and College Success Relation: A Longitudinal Study," Journal of Vocational Behavior 69, no. 1 (August 2006): 64–89.

13. Many districts have implemented career education programs in elementary and middle schools. In 2005, the South Carolina Legislature passed the Education and Economic Development Act, which includes the mandate that all middle school students must take a career inventory to help determine their career major in high school. See SC Commission on Higher Education, "South Carolina Education and Economic Development Act," www.che.sc.gov/AcademicAffairs/EEDA/EEDA.htm. For a critique of the "dropout billionaire," see Frank Donoghue, "Idolizing Dropouts: The Real Issues," Innovations (blog), Chronicle of Higher Education, February 23, 2012.

14. For more information on these programs, see their respective websites: www.dallasinstitute.org; www.yale.edu/ynhti; and www.bard.edu/mat. I am on the faculty of the Dallas Institute's Sue Rose Summer Institute for Teachers.

15. Robert Frost, "Birches," Mountain Interval (New York: Henry Holt, 1916), 37–40.

16. Dante Alighieri, Paradiso 13.94–102.

17. Jan Marthedal Rasmussen, "Arithmetic by Geometry," Kanooth (blog), April 24, 2010, www.kanooth.com/blog/2010/04/arithmetic-by-geometry.html.

18. G. H. Hardy, A Mathematician's Apology (Cambridge: Cambridge University Press, 1967), 98.

Reprinted from American Educator, Winter 2012–2013