The Elephant in the Reform

Elizabeth Green’s recent article and book excerpt “Why Do Americans Stink at Math?” has drawn keen responses from Dan Willingham, Robert Pondiscio, and others.Still, one problem needs more emphasis: the lack of focus in the classroom. Math, like most other subjects, requires not only knowledge, but concentrated and flexible thinking, on the part of teachers and students alike. With this in place, a number of pedagogical approaches may work well; without it, pedagogy after pedagogy will flail. The ongoing discussion has upheld a false opposition between old “rote” methods and (supposedly) new methods devoted to “understanding.” It is time to see beyond this opposition.

By “focus,” I mean concerted attention to the topic at hand. This is not the same as perfect behavior; I have known some “wiggly” students who were clearly thinking about the lesson. Nor does it mean passive intake; to the contrary, it can involve a great deal of questioning, comparison, imagination, and so forth. Such focus is largely internal; in this way it differs from what people commonly call “engagement.” A student may be highly focused while doing nothing physically; a student may be visibly active (in lesson activities) but not thinking in depth about the subject.

After leading into her discussion with a story, Green asserts that reforms such as the Common Core will fail if teachers have not been properly trained to implement them. “The new math of the ‘60s, the new new math of the ‘80s and today’s Common Core math all stem from the idea that the traditional way of teaching math simply does not work,” she writes. Improperly trained teachers will turn them into nonsense or, at best, a set of rote procedures:

Most American math classes follow … a ritualistic series of steps so ingrained that one researcher termed it a cultural script. Some teachers call the pattern “I, We, You.” After checking homework, teachers announce the day’s topic, demonstrating a new procedure: “Today, I’m going to show you how to divide a three-digit number by a two-digit number” (I). Then they lead the class in trying out a sample problem: “Let’s try out the steps for 242 ÷ 16” (We). Finally they let students work through similar problems on their own, usually by silently making their way through a work sheet: “Keep your eyes on your own paper!” (You).

Green contrasts this with a “sense-making” method used by the elementary school teacher and scholar Magdalene Lampert:

She knew there must be a way to tap into what students already understood and then build on it. In her classroom, she replaced “I, We, You” with a structure you might call “You, Y’all, We.” Rather than starting each lesson by introducing the main idea to be learned that day, she assigned a single “problem of the day,” designed to let students struggle toward it — first on their own (You), then in peer groups (Y’all) and finally as a whole class (We). The result was a process that replaced answer-getting with what Lampert called sense-making. By pushing students to talk about math, she invited them to share the misunderstandings most American students keep quiet until the test. In the process, she gave them an opportunity to realize, on their own, why their answers were wrong.

Like many others, Green confuses the outer trappings of the pedagogy with its internal intent and sense. A teacher at the front of the room, doing a great deal of the talking, could push the students’ thinking much more than a teacher who has them struggle on their own. Within each of these approaches, there can be variation. What makes the difference is the teachers’ and students’ knowledge of the subject, their willingness to put their mind to the topic at hand, and their flexibility of thought. (Willingham does address teachers’ knowledge and flexibility–but more needs to be said about the students’ own attitudes toward the lesson.)

The “elephant in the room” is our devotion to damage control in the name of something lofty. We are trying to repair situations where students are not doing all they can to master the material. Likewise, we are shaping the teaching profession to be more managerial, athletic, and social than intellectual. There’s a lot of mention of “collaboration”–but nothing about thinking about the subject on one’s own.

If students in a classroom are all putting their mind to the topic at hand (not because the teacher has “engaged” them but because this is what they do as a matter of course), and if the teacher knows the topic thoroughly and has considered it from many angles, then the learning will come easily–if there is a good curriculum, and if the students have the requisite background knowledge. That sounds like a lot of “ifs,”–but it comes down to something simple: when you enter the classroom, you have to be willing to set distractions aside and honor the subject matter. Honoring it does not mean treating it as dogma. It means being willing to make sense of it, ask questions about it, and carry it in your mind even when class is over.

If the above conditions are absent, then that is the problem, period. It is not a question of who is doing the talking, or how well or poorly the teachers have been trained.

Suppose I am a math teacher. (I am not and never have been; I currently teach philosophy.) Suppose I am teaching students to solve a problem of the following kind: “A train travels an average of 90 miles per hour for the first half of its journey, and an average of 100 miles per hour for the entire trip. What was the train’s average speed for the second half of the journey?” First I must establish that by “half” I mean half of the distance traveled. Then I must start to anticipate errors and misunderstandings. (Someone will likely offer the answer “10 miles”; another might offer “110 miles.”) I must be able to get other students to explain why these are not correct.

Then how to proceed? I ask the students what information we have, and what we are trying to find out. We know that the journey consists of two equal parts. It doesn’t matter how long each one is, since we are looking at speed, not distance traveled. So, we will call it d, but we are not going to try to find out what d is. It does not matter here.

Let t1 designate the time taken (in hours) by the first half of the trip; t2, the time taken by the second half, and t the total time.

So, we know that d/t1 = 90 mph for the first half. Thus, t1= d/90.

We don’t know what d/t2 is for the second half, since we don’t know the train’s speed, or rate (r) for the second half. Thus, t2 = d/r.

We know that 2d/t (total distance divided by total time) = 100 mph. Thus, t = 2d/100.

We know that t = t1 + t2.

Thus, t = 2d/100 = d/90 + d/r. (One could call on a student to perform this step.)

Thus, 2d/100 = (d/90 + d/r).

Thus, 2/100 = 1/90 + 1/r. (Divide both sides by d.)

Thus, 1/50 = 1/90 + 1/r.

Thus, 1/r = 1/50 – 1/90.

Thus, 1/r = 4/450. (Some students might arrive at 4/45–important to be alert to this.)

Thus, r = 450/4 = 112.5 mph.

As I lay this out, I can see some of the misconceptions and confusion that might arise. Some students might remain convinced that we need to find out what d is. Some might assume that t1 and t2 are equal. Some might grasp the steps but not know how to go about doing this themselves. Some might not know how to check the answer at the end.

But if I go to class prepared to address these issues, and if the students continually ask themselves (internally) what they understand and what they don’t, then even this amateur lesson will get somewhere–unless the levels in the class are so disparate that some students don’t know what an equal sign is. Of course, doing this day after day is another matter; a teacher needs extensive practice in the subject matter in order to prepare lessons fluently.

I am not proposing a magic solution here. Attention is not easily come by, nor is flexible thinking. Nor is curriculum or background knowledge. (Math teachers will probably point out errors of presentation and terminology in my example above.)

But if we ignore students’ obligation to put their mind to the lesson (in class and outside), teachers’ obligation to think it through thoroughly, and schools’ obligation to honor and support such thinking, we will continue with confused jargon and hapless reforms. Moreover, classrooms that do have such qualities will be dismissed as irrelevant exceptions.

 

Note: I made a few revisions to this piece after posting it.

Update 8/23/2014: In response to a reader’s comment, I changed “elementary school teacher Magdalene Lampert” to “elementary school teacher and scholar Magdalene Lampert.” It was not my intention to understate her academic credentials–or to comment on her work.

What Is Joy, and What Is Joy in Learning?

This morning I read a piece by Annie Murphy Paul titled “Fostering Joy, at School and at Work.” She begins by describing the efforts of Menlo Innovations to create a joyous workplace (a great success, according to the CEO). Unsatisfied with the unscientific nature of this report, Paul then turns to research by the Finnish educators Taina Rantala and Kaarina Määttä on the subject of joy in schools. They conclude that (a) “teacher-centric” instruction does not foster joy (in their words, “the joy of learning does not include listening to prolonged speeches”), whereas student-centered instruction does; (b) students are more joyous when allowed to work at their own pace and make certain choices about how they learn; (c) play is a source of joy; and (d) so are collaboration and sharing. Before taking apart these findings (which hold some truth but are highly problematic), let us consider what joy is.

Joy is not the same as cheer, happiness, or even enjoyment. It does not always manifest itself in smiles and laughter. It is a happiness that goes beyond regular happiness; it has to do with a quality of perception—of seeing and being seen, of hearing and being heard. When you suddenly see the solution to a geometry problem, you are also seen, in a way, because your mind has come forward in a way that was not possible before. When you listen to a piece of music that moves you, it is as though the music heard you as well. Joy has a kind of limitlessness (as in “Zarathustra’s Roundelay” in Nietzsche’s Thus Spoke Zarathustra) and stricture (as in Marianne Moore’s poem “What Are Years?”). One thing is clear about joy: when it comes, it marks our lives. It is not to be dismissed.

So, let us look at the first of the research conclusions cited by Paul: that “teacher-centric” learning does not foster joy. My personal experience contradicts this flat out: some of my greatest joy in school (K-12, college, and grad school) happened when I was listening to a teacher or professor who had insights into the subject. The listening was not passive; to the contrary, it woke up my mind. Likewise, as a teacher, I have known those moments when students are listening raptly—not necessarily because of something I have done, but because the subject itself is so interesting.

Of course, students need a chance to engage in dialogue as well. I am not advocating for one-way discussion. Nor do I consider a lecture necessarily “teacher-centric”; it may be the most “student-centered” thing the students have encountered all day, in that it gives them something interesting to think about. Or rather, maybe it is subject-centered. Whatever it is, there is no need to rush to put it down. Take a closer look at it first. Consider the great freedom of listening–and the great gift of something to listen to.

Working at one’s own pace—yes, there may be joy in finding one’s own velocity and rhythm. But in the higher grades, this normally occupies the realm of homework. In the classroom, one is discussing the material—and such discussion can meet several levels at once. In a discussion of a literary work, for instance, some students may be figuring it out for the first time, whereas others may be rereading it and noticing new things. The class comes together in discussion—but outside of class the students may indeed work at their own speed and in their own manner (yet  are expected to complete assignments on time).

(I can already hear someone objecting that the researchers focused on early elementary school. Yes—and that is how they should present their findings. They should make clear that their research does not comment on “joy” in general—in school or anywhere else. Onward.)

As for play, it is immensely important—but play, like anything else, can be well or ill conceived. There is play that leads to amusement, and play that leads to joy. (Amusement is not a bad thing, but it is not joy.) Also, play does not always bear the obvious marks of a game, although it can. There is play in considering an untried possibility or taking an argument to its logical conclusion. There is play in questioning someone’s assumptions or taking apart an overused phrase. My students’ philosophy journal, CONTRARIWISE, is full of play of different kinds—and it’s also intellectually serious. An academic essay can be filled with play in that the author turns the subject this way and that. If you are immersed in a subject, it becomes difficult not to play with it. Play is the work of the intellect. So, I would say that when there is no play in a classroom, something is very wrong, and joy is probably absent—but this doesn’t mean that students should be playing “algebra badminton” (whatever that is—I just made that up) every day.

As for the researchers’ last point—about collaboration and sharing—yes, those can be rewarding. But did the researchers consider how much joy can also come from working alone, or, even better, having a combination of solitude and collaboration? As long as I can remember, I have loved to sing with others, but I don’t think that would have had meaning if I didn’t also sing alone, in private. It is there that one comes to know the song. If you have ever gone out into the woods to sing—or even sang quietly while walking to the subway—then you know what it is like. It seems sometimes that the song must be solitary in order to exist at all. I am only touching on this subject, which I have discussed at length elsewhere; in any case, sharing and collaboration are only a part of joy.

Joy is not always happy. The other day I experienced joy when reading “Winky” by George Saunders. The ending was so unsettling and perfect, so beautiful in its botching of a plan, that I cried “yes,” in not so many words. Maybe joy is a kind of wordless “yes.”

 

Note: I made a few minor edits after the initial posting.

School Shocked by Non-Team-Playing Résumé

Lanham, MD—Last Saturday, nearly all of the teachers at Fernwood High school bustled around the building making photocopies, preparing lessons, or interviewing prospective teachers. One applicant’s résumé quickly became the subject of gossip and tweets.

“Not a team player,” read the first item in the “Skills” section of Rebecca Seule’s résumé.

“I don’t see why anyone would list that,” commented Bruce Klop, a social studies teacher. “Obviously we want team players, so she must not want to be hired.”

“Either that, or she’s biting her thumb at us,” added English teacher Ophelia Obida. “It’s bad form, in any case.”

The principal, Ariane Waarom, suspected there was more to the story. “No one would just do that on a lark,” she insisted. “She must have some unusual purpose.” She decided to give Seule a call, just to find out what she had in mind. “At the very least, it’ll prepare us against future onslaughts,” she told herself.

When asked why she had put such unreasonable words on her résumé, Ms. Seule had a lot to say.

“Not everything is a team,” she began.  “I love working with my colleagues. I go to them with an idea, or they come to me. Sometimes this leads to some kind of collaboration or other outcome, but it doesn’t have to. Most of the time, I just enjoy hearing what they’re doing with their classes.”

“Well, I think that counts as teamwork,” Principal Waarom ventured.

“But it’s not. You see, teams pursue concrete goals together. Each member’s role contributes to the whole in a somewhat predictable way. Take a sports team. Let’s start with the simplest kind, or rather, the most complicated kind: the duo. In doubles tennis, the two members of the team know each other’s strengths and weaknesses. They know who’s good with the long volleys and who’s good up at the net. They may work out strategies together, but they will also react instinctively to what comes at them. Still, they have one fairly simple goal: to beat the other team. A brilliant drop shot isn’t worth much, if their joint effort doesn’t hold up. Conversely, they may lack brilliant drop shots altogether yet win the game because they work well together. Bottom line: they’ve got to win repeatedly to be considered a good team.”

“That sounds an awful lot like what we’re trying to do here at Fernwood—win repeatedly,” Waarom replied. “In fact, I might bring up your analogy at a team development meeting.”

“You’re welcome to do so, but the analogy breaks down,” said Seule. “Yes, teachers have a common goal, which is to ‘win’ in some sense of the word. The problem—and this applies to many areas of education—lies in taking a part and pretending it’s the whole.”

“How would that not be the whole?” queried Waarom, intrigued.

“Well, for one thing, each subject has its particularities. Yes, we’re all trying to help our students advance intellectually, but this plays out in such different ways that we often don’t know or understand what others are doing. Let’s say a math teacher decides to teach students about the cosecant through this formula: ‘cos(θ) ∙ sin(θ) ∙ tan(θ) ∙ csc(θ) = sin(θ).’ Well, you can get students to figure out that csc(θ) is the reciprocal of sin(θ). But that’s not all. From there, they can figure out that cos(θ) ∙ tan(θ) = sin(θ), which of course makes sense. That in turn leads to the calculation that tan(θ) = sin(θ) / cos(θ). The more of these manipulations they do, the more they grasp out the trigonometric functions and their relations—all of them inherent in a right triangle. You can’t really convey this to teachers who don’t know trigonometry. Nor can they convey to you the complexity of a Donne poem you’ve never read.Of course, you could take time to read and think about the poem, or about the trigonometric functions. That’s a great thing to do, in fact. But that would be for your edification, not for the success of the team.”

“Edification?

“Edification. Similar to education, but based on a different metaphor.”

“I know what it is,” snapped Waarom, slightly piqued; “I’m just not sure it has a place in this picture. Scratch that,” she added. “It has a place. I’m just not sure it changes anything. You could still work as a team within the math department to find the best way of teaching those trigonometric functions. Don ‘t tell me some approaches aren’t better than others.”

“Sure, they are. But often you arrive at a good lesson by toying with the trigonometric functions in your head, not by conferring with a team.”

“Wouldn’t you want to share your findings with the team?” pressed Waarom.

“I wouldn’t mind doing so. But each teacher would still have to walk alone with these trig problems—and that’s not all.”

Waarom was getting urgent emails on the computer and throbs and flashes on her iPhone. “I’m sorry I can’t talk all day,” she said with genuine regret, “but is there some final takeaway here?”

“Only one thing: that education is only partly about the pursuit of goals. It’s also about the contemplation of interesting things. You cannot contemplate as a team. As a class, perhaps, or as a faculty. As an assembly or other gathering, perhaps. But not as a team.”

There was a knock on the door; someone had a complaint about a broken copier machine. “I have to go,” Waarom told Seule, “but I’d like to bring you in for an interview. I’ll transfer you over to the secretary.”

For the rest of the day, the principal thought about how the word “team” was overused. She brought it up at the next faculty meeting; many teachers heartily agreed. The school then decided not to call itself a team any more. Word leaked to the district; the superintendent announced that all schools had to rewrite their mission statements to exclude the word “team.” (He revered Fernwood for its test scores and reasoned that if the Fernwood team had abandoned the word ‘team,’ other schools should do the same.)

Panic set in across the district. They needed to call themselves something, soon. What would it be, if not a team?

No one thought of “school.” Instead, a well-paid consultant drafted spiffy mission statements that described schools as “success hubs.”

Now the challenge lay in finding résumés with “Success Hub Facilitator” in the “Skills” section. The task proved trivial; within fifteen minutes, they were streaming in.

Note: I made minor edits to this piece long after posting it.

Daydreams, Lectures, and Helices

What do daydreams, lectures, and helices have to do with each other? Quite a bit.

One of my favorite parts of Dante’s Purgatorio is at the end of Canto XVIII, when Dante starts dozing off. Here is Allen Mandelbaum’s translation of those lines:

aaaThen, when those shades were so far off from us
that seeing them became impossible,
a new thought rose inside of me and, from
aaathat thought, still others–many and diverse–
were born: I was so drawn from random thought
to thought, that, wandering in mind, I shut
aaamy eyes, transforming thought on thought to dream.

I read this as a tribute to daydreaming (though Dante is on the verge of sleep and a nightmare). To be “so drawn from random thought / to thought” (in the original: “e tanto d’uno in altro vaneggiai”) is to have an expanse and few restrictions; I love this kind of expanse, though of course I can’t have it all the time.

As I have said elsewhere, that is one thing I enjoy about lectures: they not only take my mind to unexpected places, but they send it wandering off to the side and back, or backwards and forwards. While listening to a lecture, I may do with my mind what I please; if the lecture is very good, then my mind is as though in a dance with it, sometimes spinning apart from it, sometimes drawing close. If the lecture is bad (or dreadfully dull, as lectures sometimes can be), then my mind can go off on its own. This, too, has its benefits.

Lecture or no lecture, I need time to let my mind go where it wishes. A few days ago I took out a textbook of three-dimensional calculus and started reading the chapter on vectors. The vector equation for a helix immediately made sense:

helixr(t) = cos t i + sin t j + t k

where i = <1, 0, 0>, j = <o, 1, 0>, and k = <o, 0, 1>. (These are unit vectors along the x-, y-, and z-axes, respectively.)

If you omit the z-axis, you can see that you have the vector equation for circular counterclockwise motion:

r(t) = cos t i + sin t j

Adding the component t k turns the circle into an upward spiral.

I toyed with this in my mind for a while. The next day, I encountered a helix again, when reading Taking the Back off the Watch: A Personal Memoir by the astrophysicist Thomas Gold (1920–2004). Before the helix passage, there was a wonderful comment on the possibilities for thought during a dull lecture:

A dull lecture is like an experiment in sensory deprivation. You are sitting in your seat, you can’t leave the room because that would be too rude, you are carefully shutting out the incoming information because you have decided you don’t want to hear it, and your mind is now completely free from external disturbances. It was during this lecture that I suddenly saw how all the facts of the case would fall together.

Yes, during this dull lecture he figured out why a sound entering the cochlea produces a “microphonic potential”–an electric potential that both amplifies the sound and mimics its waveform. He took his theory to Richard Pumphrey, with whom he had been investigating this matter; they published their papers in 1948. But that’s an aside here (though interesting in itself). I bring this up because his words about the lecture rang true, so to speak, in my mind. Then, a few pages later, I came upon his description of an experiment with a helix and an eel.

The eel can move forward along a sinusoidal curve, both horizontally and vertically. Thomas Gold and the zoologist Sir James Gray found that it could move swiftly and easily through a sinusoidal tube. Sir James Gray posited that the eel could therefore move through a helical tube; a helix, after all, is the addition of the vertical sinusoid to the horizontal sinusoid in three-dimensional space. Thomas Gold disagreed; he was convinced that the eel could not move through the helical tube. He was right.

Very well. But I was momentarily intrigued with the problem that would be elementary to mathematicians: is the vector equation

r(t) = cos t i + sin t j + t k

equivalent to the addition of two traveling sinusoidal waves, one horizontal, one vertical, in three-dimensional space? I grasped that it was but spent a little time explaining it to myself. Yes, and the two sinusoids must be a quarter-cycle out of phase with each other.

The first traveling sinusoidal wave has the equation r(t) = cos t i + t/2 k.

The second traveling sinusoidal wave has the equation r(t) = sin t j + t/2 k.

So, unless I’m missing something, these sinusoids are twice as scrunched as the resultant helix, their sum.

These have been my daydreams, or a fraction of them, over the past week or so. There were no lectures involved, but there were memories of lectures and the liberty I found in them.

Note: I corrected one term after the initial posting.

Squaring the Circle

Since the summer, I have been obsessed with the problem of squaring the circle—that is, finding the square whose area is equal to that of a given circle, with no tools other than a straight edge and compass.

I took interest in the problem when reading the ending of Dante’s Paradiso (in Allen Mandelbaum’s translation):

 

 

As the geometer intently seeks
to square the circle, but he cannot reach,
through thought on thought, the principle he needs,

so I searched that strange sight; I wished to see
the way in which our human effigy
suited the circle and found place in it—

and my own wings were far too weak for that.
But then my mind was struck by light that flashed
and, with this light, received what it had asked.

Here force failed my high fantasy; but my
desire and will were moved already—like
a wheel revolving uniformly—by

the Love that moves the sun and the other stars.

I, too, had difficulty reaching the principle—and it was precisely the principle that I needed to find. The difficulty lies in determining what the problem exactly is.

No one has found a way to square the circle with compass and straight edge alone; in fact, I believe it has been proved that it can’t be done. Yet I kept on trying, thinking that I would learn something from the attempt.

Yesterday morning, something close to a solution came to me—not a solution, mind you, but something that points in the direction of one. It’s probably old hat—or ancient hat—but it’s interesting, all the same.

We will consider the circle with radius r=1, since expanding or shrinking it (and the square) is then a trivial matter.

We know that if the circle has radius 1, its area is π. Thus, for a square to have that area, it must have sides of length √π. But how do you find length √π? There’s the puzzle.

When the square and circle have the same area, the ratio of the circle’s circumference to the square’s perimeter is a constant, 2π divided by 4√π ; that is, π/2 divided by √π; that is, √π/2.

In other words, the ratio of one-quarter of the circle’s circumference t0 one side of the square will always be √π/2 when the square and circle have the same area.

You could consider the ratio in terms of this figure:

|AO| is equal in length to the side of the square  (√π when the circle has radius 1); |OP| is one-quarter of the circle’s circumference (1/4 *  2π =  π/2), which you can measure with a string. (I know you’re only supposed to use a straight edge and compass, but this exception will prove helpful.)

Now, the ratio of |OP| to |AO| is equal to the ratio of|BC| to |AB|, where |BC| = √π = |AO|, and |AB| = 2. We already established this ratio a few paragraphs up.

(Again, the ratio of one-quarter of the circle’s circumference to the square’s side must be √π/2.)

So now the challenge is to tweak the figure until |AO|=|BC| (while keeping points A, P, and C on a single line and A, O, and B on a single line). When we get the two segments to equal length (without changing |OP| or |AB|), we have brought both |AO| and |BC| to √π. Now, you must do this by trial and error, by shifting the OP segment. If |BC| is greater than |AO|, then move OP closer to BC; if it is smaller than |AO|, then move OP closer to A. Your adjustments will be smaller and smaller until you either make |AO| and |BC| equal or come as close as you wish. The next step would be to find a mathematical way of doing this.

Once you have made |AO| and |BC| as close to equal as you wish, you then make AO the side of your square, which you superimpose on a circle of radius 1 (so that both have a common center). You will see that the square is mostly, but not entirely, contained within the circle. It will look like the first of the two figures above.

Although this is not a satisfactory solution, it results in a close approximation and seems to be on the right track.

Update (on a tangent, so to speak): The angle in the figure is the inverse tangent of √π/2, or 41.54822621257918513… Its common logarithm (1.6185524875…) is tantalizingly close to the golden ratio, but not close enough for the trumpets to sound.

Note: This is a reposting of a piece that originally appeared on October 19.