The Homework Conundrum

Unlike Alfie Kohn and others, I believe that homework has meaning and carries benefits. This is partly because I teach at the high school level. You can’t discuss Plato if you haven’t read Plato, and the only time to read Plato is outside of class. If you read Plato in class, then there’s no time to discuss it. It’s as simple as that. The subject matter in high school demands independent work.

It does in elementary school as well, though not quite as much. Children do need to read books at home, sometimes for school. There isn’t enough time in the school day for all of their reading. They need to practice math problems, a language, a musical instrument. If they are writing a report, they need to go to the library to look up information. At the same time, they need free time—time for play, exploration, thinking, and being with their family and friends.

Now, back to high school. The homework volume doubles and triples for an unfortunate reason: if you give students a reading assignment without a writing assignment, many will interpret this as no homework at all. So you pair reading assignments with questions. There’s the conundrum: to help ensure that the homework gets done, you end up doubling it.

This means more work for teachers as well as students. I have 259 students in all, and I carry large stacks of homework home. I find it important to read and comment on homework (and enjoy doing so) but sympathize with students’ complaint that it takes a lot of time. What can we do about this—in general, throughout our schools?

Students shouldn’t have to write for every subject every night. It makes sense that they should read on some days and write on others. But they have to treat the reading as a serious assignment, even though it doesn’t result in a concrete product right away.

For this to happen, we have to stop treating concrete products as the be-all and end-all of education. Yes, education should result in good work, but students should learn to hold things in their minds, to work without immediate results. They should develop integrity as students, pondering the material even when there’s nothing to turn in.

Students don’t develop these habits overnight. The best way to help them get there is to set a good example. That means showing them, in class, how to take interesting things into the mind, to make sense of them, to question them, to ponder them again. If I am introducing my students to Blaise Pascal, I expect them to remember what I tell them—and to bring it up in class discussion or on an exam. But I expect still more: they should be willing to enter Pascal’s Pensées (or the short selection we will be reading this week), puzzle through it, recognize its argument and its subtleties, and carry some of it with them.

It takes a long time to build such practice. An individual teacher can encourage it, but it is really the work of a school and of many schools. The life of the mind is almost a lost concept; we need to revive it. It begins with a strong foundation in elementary and middle school—not only in math and reading, but in literature, history, science, music, art, and drama. Students should memorize and recite poems from a young age, so that they develop an ear and a repertoire. They should learn to work through math problems that require skillful framing. In addition, they should learn to persist with things that they do not fully understand: sentence structures that bewilder them at first, terminology that seems out of reach, or melodies and harmonies that seem at first too complex to sing.

Last week a student told me that she had struggled with a passage from the Book of Job. She read it slowly, again and again, and started to glean its meaning. That’s what should happen on a larger scale. When students take the reading that seriously, there’s no need to check up on them every day—and they arrive at greater, not lesser, understanding.

But it is all too easy to cave in to the cultural demand for immediate rewards and punishments. Turn in your homework, and you get two points. Don’t turn it in, and you get a zero. Kids understand that language, and it makes sense that they would. It isn’t bad as a starting point; it can help them get on track. It should not be the end goal.

Homework should have meaning, but meaning does not arise in a vacuuum. It comes with the subject matter, with cultural values and habits, and with persistent teaching.

Elitism Versus Populism in Education

In a recent post (now deleted), I discussed what I saw as an anti-intellectual tendency in education. I gave only two examples and didn’t go into the complexities of the matter. (I later became dissatisfied with the piece.) In particular, I didn’t make a clear distinction between anti-intellectualism and anti-elitism. The two overlap and combine but are not identical.

Anti-elitism involves distrust of privilege and its distortions. In education, the elite are those who come with money or make a great deal of money; who hobnob with Bill Gates and Arne Duncan and take part in various wealthy organizations; who have strong media connections and can get op-eds in the big magazines; and who don’t teach day in, day out. They need some knowledge of education, or they lose credibility, like Cathie Black, who briefly served as NYC schools chancellor. Yet they don’t have to do the daily work of planning and conducting lessons, calling parents, correcting papers, setting up rooms, or rushing around to make photocopies and gather supplies. On the other hand, precisely because they don’t have to perform all these tasks or deal with so many youngsters, they have room to write, do research, think ideas through, and deliberate with others.

It’s reasonable to be suspicious of elites, especially when they talk about the need for better teachers. Their degree of material comfort, compared to that of teachers, staggers and addles the mind. Some of them may work hard—I have no doubt that Wendy Kopp and Geoffrey Canada do—but they do not have to grade 200 homework assignments over a weekend. They don’t have to worry about where the chattering is coming from in a room, how to introduce students to Aristotle, or why a certain student isn’t handing in homework. Nor do they have to worry about being judged by students’ test scores—on tests that have little to do with their subjects. Working in the quiet of your office, or even giving talks around the country, carries nothing of this pressure or exhilaration. It has its own pressures and rewards. I am not diminishing the work of good education leaders—but put them in a classroom for a month, with all of the responsibilities, and many would find themselves overwhelmed.

On the other end of things, we have populism, which opposes elitism tooth and nail. Populism is essentially a belief in the virtue, authority, and wisdom of the people. Daniele Albertazzi and Duncan McDonnell characterize it as an ideology that “pits a virtuous and homogeneous people against a set of elites and dangerous ‘others’ who are together depicted as depriving (or attempting to deprive) the sovereign people of their rights, values, prosperity, identity, and voice.” Populists say (this is my paraphrase, not that of Albertazzi and McDonnell), “look at those people making all that money and enjoying all that power. What do they know about our concerns? Why should they be telling us what to do? Why aren’t we the ones setting policy?”

If you don’t sympathize at least a little with a populist outlook, then you are missing something. There’s every reason to be wary of the ultra-powerful, and to yearn for more popular influence over public affairs. But populism has its pitfalls, too. For one thing, it presumes to know who the people are and what they want; it assumes that they more or less agree, when in fact there may be deep divisions among them. Second, it values certain ideas because they (presumably) come from the people, not because they are good. Along these lines, it may dismiss good ideas merely because they appear to come from the elite. Third, it places high value on group thinking and majority rule; those who don’t fit in or who hold independent views are regarded with slight suspicion. (Granted, elite groups and policymaking bodies have plenty of their own groupthink; I highly recommend Irving L. Janis’s book on the subject.)

So, anti-populists, or skeptics of populism, champion independent thought and intellect; they remind us of the “tyranny of the majority.” They point out where popular and populist movements have gone wrong, how they have gotten swept up in an illusion of consensus and truth, when in reality they were deluded and divided. The anti-populists have a point, but they, too, can get carried away. They can distrust anything that looks like a popular movement, even if it’s well founded and badly needed.

How could we bring together the best of elitism and populism, so that we could evaluate ideas on their own merit, allow for individual voices and group efforts, and honor those who devote themselves to education, especially teachers? First, we would have to put an end to the education racket. In many circles, education reform has become lucrative, with consultants making more than a thousand dollars a day. This is obnoxious at best, crippling at worst. Second, when the New York Times and other publications have “panels” on education topics, they should not only include teachers in the discussion, but bring them to the forefront. Third, we should take ideas on their merits, instead of judging them by the speaker’s position and connections. Fourth, we should respect independent thought. No one should be spurned for differing from the group. We are more likely to respect and understand independent thought when we discuss something substantial—so let’s have more discussions of subject matter itself.

These are only preliminary thoughts; I intend to think and write more about this topic.


N0te: I made a few edits to this piece long after posting it.

A Sense of Tuning and Timing

In Book VIII of the Republic, Plato explains how the beautiful city, the kallipolis, succumbs to decay as anything else does. First, the leaders start having children at the wrong times; then the children, who are not raised properly, mature without a sense of poetry and music. Lacking this sense, they also lack a sense of proper governance.

Why might this be so? I asked my students. Why would good leaders need education in music and poetry?

The answers they offered said a lot about our times. “Music allows you to be creative,” said one.

“It’s self-expression,” said another.

“This is true, but is there more? What does it mean for Plato?” I asked.  They were momentarily stumped.

I directed them to a passage in Book III:

Aren’t these the reasons, Glaucon, that education in music and poetry is most important? First, because rhythm and harmony permeate the inner part of the soul more than anything else, affecting it most strongly and bringing it grace, so that if someone is properly educated in music and poetry, it makes him graceful, but if not, then the opposite. Second, because anyone who has been properly educated in music and poetry will sense it acutely when something has been omitted from a thing and when it hasn’t been finely crafted or finely made by nature. And since he has the right distastes, he’ll praise fine things, be pleased by them, receive them into his soul, and, being nurtured by them, become fine and good. He’ll rightly object to what is shameful, hating it while he’s still young and unable to grasp the reason, but, having been educated in this way, he will welcome the reason when it comes and recognize it easily because of the kinship with himself.

Now they understood that Plato saw music education as a conduit to good taste and judgment—because, having learned to discern good craft in one sphere, one can recognize it elsewhere as well.

One can dispute this, of course. There are plenty of examples of people with musical prowess who show poor judgment in other areas of life. Nonetheless, there’s something to this idea of timing and tuning. When you learn to play or sing in tune and in rhythm, you do become more alert to form and detail. You come to sense the relationships between different parts of a work, whether it’s a sonnet, an opinion piece, or even a sentence. You may even notice when your mood is out of tune or out of step.

None of this transfer of sensibility is guaranteed. It’s possible to perform a sonata splendidly and then get into a needless argument. It’s possible to sense a flaw in a sestina but not in a policy proposal. Nonetheless, music and poetry can make a person more alert to tunings overall.

But of course music isn’t only tuning and timing. There’s tension between control and release, between discipline and abandon, between form and departure from form. You need both, but in what proportion? There’s no final formula. That’s where keen sense comes in.

Young people do not lack that sense. It’s just that many of them haven’t thought of music in that way. Why not? Much of it has to do with a popular belief in self-expression. It needs a counterbalance, and a strong one. Self-expression of a kind is important, but it’s the shaping that makes it interesting. It’s the shaping that allows works to speak to each other and to seep into the memory. It’s the shaping that allows us to carry a sensibility from one sphere into another.

This shaping, of course, requires knowledge; you must listen to many sonatas to understand what a sonata can be, or to depart from a sonata. Beethoven’s Opus 111 arises from the earlier sonatas; it could not have been composed in a void.

A good curriculum would include many works that help students understand form and shape. It would involve a great deal of listening to poetry, music, and speeches. It would not preclude self-expression, but it would lift that expression, enriching it with literature, history, mathematics, languages, and more.

Update: For more on self-expression and its pitfalls in the classroom, see Robert Pondiscio’s piece in the Atlantic.

Polla ta deina

Last night I went with a friend to see Philip Glass’s Einstein on the Beach at the Brooklyn Academy of Music. My friend was unable to get us seats together; one of us was to be in the first row of section A; the other, in the second row of section B. “There are no intermissions,” he warned, “and it’s about five hours long. So people are going to be getting up and moving around. We might want to take a break at some point and even trade seats.”

About an hour into the performance, people started shuffling around. I looked across the hall and saw my friend, who nodded at me. We got up and met each other outside.

“Do you need a break?” he asked.

“No,” I said. “How about you?”

“No, I’m fine,” he said. “So, what do you think?”

“It’s fantastic.”

“Yeah, it’s pretty awesome.”

We went back in and reclaimed our original seats. We took no more breaks after that.

When we returned, the dance sequence had begun–dancers leaping across the stage, dancing out of sight, appearing again, with grace and skill and cheer, over and over again, angle after angle, rotation upon rotation, leap after leap, in perfect synchronicity and pattern but also appearing by surprise, a dancer leaping, almost flying, and then more dancers and patterns, crossing each other, circling, pirouetting, leaving.

I thought of the “Ode to Man” from Sophocles’ Antigone, “Polla ta deina kouden anthropou deinoteron pelei…” (“Many are the wonders and terrors but none more wondrous than man”). I brought that ode to my eleventh graders on the first day of class and recited it for them in Greek. I explained to them the meanings of pantoporos (all-resourceful) and aporos (resourceless), of hupsipolis (great of city) and apolis (without city). But now it seemed I was seeing the ode before me, in a form I hadn’t before imagined.

It is a great thing when a performance puts you in awe, not only of the performers or of the piece, but of the possibilities in a day, in a crossing of the room. It’s easy to forget such awe or to let it get dusty.

I will never be able to dance across a stage like that, or play the way the Einstein violinist played, or make such a  tone of alto saxophone, or compose complex counterpoint that suddenly rises into something simple and pure. But I can lift myself in the things I do.

Tobias Wolff’s Old School: Truth, Tangent, and Return

After yesterday’s post on yearning and return, I realized I had omitted something that had been on my mind for a long time. Here it is.

If you have not yet read Tobias Wolff’s novel Old School (2003), please read it before reading this piece, which will reveal some of the ending. I also encourage you to put off reading reviews until you have read the book. Though widely praised, it has been strangely misunderstood by some, including Michiko Kakutani of the New York Times. Reviewing comes with pitfalls: the best reviews draw attention to good work (or warn against the mediocre), while the worst sacrifice the book to the reviewer’s own needs and frailties. Few reviewers are consistently insightful; they succumb to their own stuff, as we all do at times. That’s how I see Kakutani’s review. Enough of that.

I am writing about this book because, from the first reading in 2003 through the third and most recent one yesterday, I have been carrying it around in my mind. I pick it up (in my hands or in the imagination) and return to favorite passages. It says more about education than many an education book; it is part of my own education. It is the ending that stays with me, though everything else builds slowly to it—an ending that seems a tangent but becomes a return and vision. I will look at this return today.


Yearning and Return in Education

It’s already an old joke that the good old days of nostalgia are long gone: that once upon a time it was honorable to look back longingly at the past, but no longer. There’s truth in it; in education discussion I often hear people fault others for harking back to a golden age that never was. Bad, bad, they say; we must stay grounded in facts. Mr. Gradgrind (from Dickens’ Hard Times) works his way into many an argument.

It is dangerous, of course, to paint the past as golden, but there are reasons why we yearn for the past sometimes. We shouldn’t be so quick to push such yearning away. For me, the fall is usually a time of yearning. I find room and urge to take walks, watch the leaves leap and sweep over the sidewalk, and assemble past autumns in my mind. Details work their way in as well: a ribbon on the ground, a cat surveying the neighborhood, or the color of a coat.

As a teacher, I return to the classroom and see the students a little older and taller, excited to tackle books that I first read in high school, and I remember my own teachers and the way they spoke. The beginning of the school year comes with reminiscence. There’s a ceremonial feeling to it, even amidst the confusion of rooms and schedules; when you address each class for the first time, you remember layers of first days.

I remember a high school assembly at the start of my ninth-grade year. The teachers were seated on the stage. One of them, I knew, had gone through a divorce; I wrote in my diary (which I no longer have) that I saw a look of pathos on her face. In retrospect, I doubt it was pathos (I discovered later that she had irrepressible wit), but the word “pathos” was part of that day for me.

Part of the point of education is to learn to select what is good, to bring it into one’s life, and to pass it on; this requires knowledge, discernment, and feeling. Memory helps us make such selections. Those works that come back to us many times over the years, or that suddenly open up on the second or third reading, have a little more to them, in our minds, than the ones we read and forget. With the memory comes a bit of longing. I think back on the Southern Literature course I took in high school, and the advanced verse writing seminar in college; I have often wished to return to those rooms, and have carried a hint of them into my teaching.

By this I don’t mean that people should rely on their memories for guidance. What I hold dear from my high school years may not have been quite as I remember it, nor is it necessarily good for every student. Still, I carry something of the spirit of it, and must do so; it is precisely through holding my past that I can play with it in the present, even transform it.

Andrew Delbanco understands this well. His extraordinarily thoughtful book College: What It Was, Is, and Should Be (2012) looks back to earlier eras not to portray them as perfect, but to capture their meaning and wisdom. His book resists alarmism and paeans to good old days, but still looks back with nostalgia—wise, temperate nostalgia. I wouldn’t do his book justice with a short quotation here; I hope to write more about it another time.

The literary works that make their way into our memory, the ones that follow us around, contain this treasuring and pondering of the past. What would Job’s lamentations be without this treasuring and pondering? What would Allen Ginsberg’s “Howl” be without it? How can a student or a teacher approach this literature without understanding what it means to think back, sometimes with sadness or rage, sometimes with wistfulness or wit?  Why the cultural pressure to regard the past with a cold eye and move on?

Many young people understand the importance of looking back and yearning. They need adults who understand it too and who can help them make sense of the past. They need to find that promising terrain between sentimentality and dismissiveness. Through literature, they learn to store language in memory; through history, they learn to guard against memory’s distortions.

The point is not to live in the past, but rather to hold it, turn it, contemplate it, change one’s mind about it, reconsider it again, forgive it, and sometimes, when necessary, leave it behind.

The Deep Problem with the School of One

This morning, Rachel Monahan reported in the New York Daily News that two of the three New York City schools that piloted the “School of One” decided to drop the program. After a great deal of expenditure and hype, the School of One didn’t show better results on the state math tests than regular math classes.

I am not surprised by this report. The School of One (which I discuss in the eighth chapter of my book) assumes that mathematics consists of a progression of skills. Its proprietary software program generates a daily “playlist” for each student and lesson plans for the teachers. Students enter the classroom, view their playlist, and go to their appointed station. On a given day, a student might play a video game, work in a small group, receive direct instruction from a teacher, or engage in some combination of these activities. Teachers might spend fifteen minutes with one group, a few minutes here and there with individuals, and another fifteen minutes with another group. The students take frequent multiple-choice quizzes, which help to determine their activities and grouping. Supposedly, by working at their own pace in their own preferred style, students will make great progress.

But mathematics is not an amusement park. It is about recognizing patterns and seeing problems in more than one way. It requires imagination as well as precision. In the best math classes, students learn to struggle with problems that at first seem daunting (but for which they are adequately prepared). They try this and that, seeming to get nowhere, and then suddenly they see it. In a flash, it is all clear—and the solution sheds light on problems from earlier lessons and problems still to come.

Students cannot rely on such flashes of insight, of course. After solving a difficult problem, they must practice solving similar problems until they come easily. Then they continue on to the next challenge, which often arises out of the problems they have solved. A good math curriculum has a clear, logical progression but also moves back and forth and outward. Over time, as students advance and gain knowledge and experience, they develop what Alfred North Whitehead called “that eye for the whole chess-board, for the bearing of one set of ideas on another.”

Personalized, computerized instruction doesn’t do justice to such a curriculum; even precocious students need guidance through the challenges. It is the teacher who knows how to pose a problem in different ways and to draw more than the obvious conclusions from it. It is the teacher who can glean where a student is going wrong and guide him back on track. Such teaching takes time. A class can easily spend an entire lesson on a single theorem or concept, and the students learn from each other’s efforts.

What happens when the lesson is fragmented, when students go off into their various groups and corners to play a game or work on an activity? Well, in many cases both the students and the mathematics itself are shortchanged. The students may make progress with problems of a basic sort (like those that appear in summer math workbooks) but will need the teacher for the trickier and subtler points. Also, flitting from activity to activity isn’t always helpful; mathematics requires focus and doggedness. (Yes, sometimes the solution comes to you after you walk away—but those hours of puzzling and pondering help to bring this about.)

So why has the the School of One enjoyed such hype? Not only are there powerful political and commercial entities behind it, but it appears to address a real problem. Today’s classrooms have a wide range of levels; the advanced and struggling students study together. Since tracking is not an option (especially at the elementary and middle school levels), the teacher is expected to accommodate all levels at once. Given that state of things, a personalized learning system (aided by software) sounds like a crystal palace of sorts. To some, it is the future.

But to paraphrase Fyodor Dostoevsky’s Underground Man, if it is raining and I crawl into a hen-house in order to stay dry, I will not call it a palace out of gratitude. It is still a hen-house. Something analogous holds true for the School of One. It is a makeshift solution, and an expensive one at that.

What can we do instead of expanding the School of One? We could adopt strong math curricula that give students a foundation in the early grades. We could allow for certain kinds of flexible tracking—so that, for instance, a fifth-grade student could take math  with sixth graders if she were prepared (but would take other classes with her fifth-grade classmates). We could have public lectures, seminars, and workshops on mathematics, so that parents, teachers, and others could grapple with math problems together. We could identify first-rate math textbooks, possibly translating a few from other languages, so that teachers did not have to scramble for appropriate resources. All of this would be far less expensive—and far truer to the purpose of teaching math—than the School of One.

Is It Possible to Care Only About Data?

This isn’t a satirical piece. (The last two posts were.)

I meant to say something about the comment I reposted the other day, the one that appeared on Eduwonkette’s blog in January 2009. The commenter warns against that sly sort of conformity that can take over a person in academia—the temptation to say those things that will lead to publications, speaking engagements, and grants—and urges Eduwonkette (and all of us, I suppose) to remain willing to admit to error.

It struck me as a wise, thoughtful letter from someone who knew a thing or two. Two sentences, though, gave me pause, not because I disagreed with them, but because I saw room for qualification: “But you didn’t want to play the policy game. All you cared about was data.”

Now, the person was addressing the comment to Eduwonkette, but I want to consider it more generally. It is easy to see how the first sentence could be true for someone. Many of us don’t want to play the policy game. The second sentence—is it possible? Does anyone with a serious interest in education care only about the data? Would that even be a good thing?

I have noticed a common assumption on both (or multiple) sides of the education debates: that a person should “follow the data,” and that a reform or policy is viable only if it has “evidence” to support it. (Evidence and data are not identical—the word “data” is often misused—but that’s another matter.)

It makes sense that one should look at the evidence behind any initiative. Why should anyone pursue a reckless, unproven reform on a large scale? Why subject schools and districts to experiments that have little evidence in their support? Granted, one has to experiment now and then, but one should do so with caution.

But “evidence” has meaning only in relation to your goals and values. Evidence supports or does not support a conclusion—but that conclusion must have some bearing on your aims. For instance, a certain kind of instruction could be correlated with higher artichoke consumption, but unless you’re trying to get the kids to eat more (or fewer) artichokes, this won’t mean much. Often policymakers speak in terms of “achievement” and “success”–but achievement of what? Success at what? They are usually referring to test scores—a limiting goal, given the nature of the tests.

If your goal (or one of the goals) is to give students background in literature, mathematics, science, languages, history, and arts, then the evidence of the curriculum’s worth can be found in its content and the lessons. If students are reading Chaucer, if the teacher is leading lively discussions and directing students’ attention to Chaucer’s word-play and satire, then the good is self-evident (if you think learning Chaucer is a good thing). But this goal is fairly unpopular with with those who want to see quick, concrete results.

Another goal might be to prepare students for the challenges of adulthood, such as college or the workplace. Hence “college and career readiness.” Policymakers supporting this goal might ask employers and college admissions officers what they seek in their applicants. They would then judge curricula and instruction by these criteria. Such an approach makes sense on the surface but has flaws; employers and admissions officers may take certain kinds of knowledge for granted and not think to mention them. In addition, the point of getting into college or getting a job is to start building one’s life (not just to get in). Thus, one needs qualifications that transcend the immediate ones.

If the goal is to prepare people for civic life, then the evidence-gathering becomes trickier still. One might ask: who are the people whose editorials get published, who hold political office, or who speak at town hall meetings? What kind of rhetoric and knowledge do they show in their writing and speaking? What kind of education did they have? This is faulty, though, because there are many ways of participating in political and cultural life besides writing editorials, holding political office, or speaking at town hall meetings. If civic participation means engagement in constructive discourse on matters of public concern, then we’re in better shape, as we can identify the attributes of such discourse and the knowledge required for it. Even so, we might disagree over the priorities and emphases.

One could stick with the goals that carry the clearest evidence—graduating from high school, getting a job, etc.—but such goals justify only the most basic and pragmatic sort of education. Some would say that’s just plenty; if students want more than that, they should find it on their own. But that would put vast numbers of students at a disadvantage. Those who could afford a fuller education would study literature, history, languages, and so forth (in private schools or in wealthy public schools), while others would have little exposure to such things.

Thus it makes no sense to speak of “evidence” except in relation to what one is trying to do. Once one has established the goals, there are still more challenges, since the appropriate evidence can be difficult to define and gather. Much of it shows itself only over a long period of time. Much of it comes from individual experience. Many people dismiss individual experience as limited and biased, but it also has strengths, if we treat it responsibly.

My point here is not that we should disregard evidence or data. That would be irresponsible. Rather, we should ask, evidence of what? What are we trying to do, and why? What sort of information would help us see whether or not we are accomplishing it? How should we go about interpreting such information? How can we stay true to our goals but admit to mistakes and misunderstandings? How can we speak and work with those whose goals are different?

Equals Sign Creates Jobs and Confusion

In their rush to implement Universal Design for Learning (UDL), a federally approved framework intended to maximize learning outcomes for all students, schools have hired gymnasts, carpenters, and political philosophers to provide multiple representations of the equals sign.

According to UDL, “An equals sign (=) might help some learners understand that the two sides of the equation need to be balanced, but might cause confusion to a student who does not understand what it means. … An important instructional strategy is to ensure that alternative representations are provided not only for accessibility, but for clarity and comprehensibility across all learners.”

“I thought the equals sign was pretty clear,” said John Knap, a high school mathematics teacher. “Not sure why we have to represent it in other ways. Yesterday two gymnasts came to my class to perform double flips, and the kids were supposed to grasp that the two routines were ‘equal.’ They weren’t equal. One was a little faster than the other. And of course the kids wanted to see more routines. We couldn’t get to the lesson.”

Kelly McEwen, a carpenter hired to provide alternative representations of the equals sign at elementary schools in San Diego, expressed misgivings over the project. “The pay’s great,” she said, “but I’m not sure I’m doing the right thing. I’m supposed to show them the spirit level and pretend it’s just like the equals sign. It isn’t just like the equals sign. I end up doing a lot of qualifying and explaining, and the teachers and kids get anxious. Plus, they’re waiting for me to take out the saw, which I never bring, for safety reasons.”

Political philosophers seem especially disgruntled with the project. “I was invited to come to Inspiration Academy to talk about political equality,” reported Andrew Ravny, author of numerous books on William Hazlitt and Thomas Jefferson. “I accepted gladly. When I arrived, I was told to draw stick figures and put a smiley face between them to show that they were equal. I did this and went on to say that two equals three in such a scheme, because both numbers have the same inherent dignity and rights. I don’t think I’ll be invited back.” He chuckled grimly. “Which is just as well, since I need to focus on my next book.”

We had the pleasure of interviewing the inventor of the equals sign, Robert Recorde, whom we heard stirring in his grave. We asked him why he had chosen to represent mathematical equality with two parallel lines. He replied that he did it “to auoide the tediouse repetition of these woordes : is equalle to.”

But why the two parallel lines? we asked.

“Bicause noe 2 thynges can be moare equalle,” was his reply.

We thought that Recorde would be pleased to learn that his equals sign was now inspiring multiple representations. As we told him about the reforms, we watched and listened closely for his reaction. But he replied in cryptic verse and then faded from our midst:

One thyng is nothyng, the prouerbe is,
Whiche in some cases doeth not misse.
Yet here by woorking with one thyng,
Soche knowledge doeth from one roote spryng,
That one thyng maie with right good skille,
Compare with all thyng: And you will
The practice learne, you shall sone see,
What thynges by one thyng knowen maie bee.*

“It’s a nice poem, but I’m not sure how it applies to classroom practice,” said Mercy Trout, director of instructional services in Boise, Idaho, who had accompanied us for the interview. “Is he saying kids should study math as math? Or is he saying all things are connected? What are the policy implications for school improvement?”

“Where are you, Recorde, and where’s the whetstone of witte when we need it?” cried another.

“I think he’s saying that if people do study math in a focused way, then they will see….” a third member of our party ventured. But it had grown dark and windy, and conversation turned to our flight back home and whether it would depart on time.

*The verse appears in Recorde’s preface to his Whetstone of Witte (1557).

Breakthrough Research Shows Role of Study in Learning

Education policymakers have long assumed that students who learn more have been taught more effectively. A startling new study conducted by an international consortium of scholars at Peutêtre University in Toronto has thrown this assumption into question. According to project director Pascal Feldspar, students who take a course in a subject—and study it—show more learning gains in the subject than students who do not. While gains varied slightly from one section of a course to the next, the starkest differences were between those who took a course and those who did not, and those who did the work and those who did not.

“It is too early to generalize the findings,” said Feldspar, “but we found, for instance, that students who took French for a year, and did their homework, showed significantly more learning gains in French by the end of the year than students who took no French.” The same applied to geometry, ancient history, piano, and Shakespeare. “We gave fifty students a test on Henry IV, Part 1,” he said, waving a copy of the play at us. “They were asked to identify a series of quotes, explain their meaning, and discuss their relation to the work as a whole. Before anyone had read Henry IV, performance on the test was uniformly poor. Then we split them up into an experimental group and a control group. At the end of the study, the members of the experimental group—the ones who studied the play—performed better than the control group by more than two standard deviations. Put simply, the more you study, the more you learn. Of course it’s more complicated than that, but that’s the idea.”

When asked about indirect exposure to a subject, Feldspar pondered for a moment. “We haven’t conducted any experiments on this yet,” he replied, “but our hypothesis is that indirect exposure to a subject plays a role in learning it as well.” He added that students who took courses but did no work showed negligible progress. “You learn something from just being in the environment,” he said, “but it appears that you learn a great deal more if you study the material, day after day.”

Asked whether the researchers had used standardized tests for their experiment, Feldspar shook his head vigorously. “You see, the American standardized tests don’t contain much subject matter,” he explained. “Their emphasis is on skills. In our study, we used tests that were directly related to what students had learned in the course.” Even then, he said, the tests didn’t show everything. “We know well that the test can’t measure all that was taught, and that the students’ performance isn’t an exact indicator of how much they learned.” All the same, he said, the test results showed differences between students who had studied a subject and students who hadn’t. “There are always some who study and study and still don’t learn,” he said, “but on the whole, so far, it appears that students who study a subject have a certain learning edge in that subject.”

Other scholars were quick to dismiss the findings. “The Peutêtre findings rest on the long-disproven axiom that you can only learn subject matter through a transmission model,” said Lorraine Caulk, professor of educational neurobehavior at Instructors College in Iowa City. “In fact, research has shown that when we empower children as self-initiating scholars, they start to exhibit strategic thought behaviors without the hierarchical modeling of traditional task performance.”

“I’m not sure what that means,” retorted Feldspar. “If Dr. Caulk would care to clarify her statement, I’d be happy to try to respond to it.” Dr. Caulk replied that she was using standard research terminology and that Dr. Feldspar should acquaint himself with the literature.

At this time it is unclear whether the findings at Peutêtre will have any impact on the classroom. “I like the gist of it—that we should teach actual subject matter,” said Nellie Nekogda, principal of the High School for Innovative Thinking in New York City. “The problem is, we have no money for books, and we have to bring up our test scores in ELA. I’d love to do it, though, if we could get the resources and the support.” She leaned back in her chair. “My reformation, glittering o’er my fault,” she mused, quoting from Henry IV, Part 1. “It has been years since I thought about that play. How great it would be to teach it here. Some day,” she said, “some day.”

  • “To know that you can do better next time, unrecognizably better, and that there is no next time, and that it is a blessing there is not, there is a thought to be going on with.”

    —Samuel Beckett, Malone Dies

  • TEDx Talk

    Delivered at TEDx Upper West Side, April 26, 2016.



    Diana Senechal is the author of Republic of Noise: The Loss of Solitude in Schools and Culture and the 2011 winner of the Hiett Prize in the Humanities, awarded by the Dallas Institute of Humanities and Culture. Her second book, Mind over Memes: Passive Listening, Toxic Talk, and Other Modern Language Follies, was published by Rowman & Littlefield in October 2018. In February 2022, Deep Vellum will publish her translation of Gyula Jenei's 2018 poetry collection Mindig Más.

    Since November 2017, she has been teaching English, American civilization, and British civilization at the Varga Katalin Gimnázium in Szolnok, Hungary. From 2011 to 2016, she helped shape and teach the philosophy program at Columbia Secondary School for Math, Science & Engineering in New York City. In 2014, she and her students founded the philosophy journal CONTRARIWISE, which now has international participation and readership. In 2020, at the Varga Katalin Gimnázium, she and her students released the first issue of the online literary journal Folyosó.


    On April 26, 2016, Diana Senechal delivered her talk "Take Away the Takeaway (Including This One)" at TEDx Upper West Side.

    Here is a video from the Dallas Institute's 2015 Education Forum.  Also see the video "Hiett Prize Winners Discuss the Future of the Humanities." 

    On April 19–21, 2014, Diana Senechal took part in a discussion of solitude on BBC World Service's programme The Forum.  

    On February 22, 2013, Diana Senechal was interviewed by Leah Wescott, editor-in-chief of The Cronk of Higher Education. Here is the podcast.


    All blog contents are copyright © Diana Senechal. Anything on this blog may be quoted with proper attribution. Comments are welcome.

    On this blog, Take Away the Takeaway, I discuss literature, music, education, and other things. Some of the pieces are satirical and assigned (for clarity) to the satire category.

    When I revise a piece substantially after posting it, I note this at the end. Minor corrections (e.g., of punctuation and spelling) may go unannounced.

    Speaking of imperfection, my other blog, Megfogalmazások, abounds with imperfect Hungarian.

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