“In a problem, the great thing is the challenge….”

In childhood I was given a book on probability, a subject that fascinated me. It had a series of intriguing problems, with humorous illustrations scattered throughout, and detailed solutions at the end. I loved the book, opened it up many times, but did not get far in it. I remember poring over the first few problems and browsing through the others. Then, after a series of moves and life changes, the book got misplaced.

Years later, I remembered it and wanted to find it, but I couldn’t remember the title or author. I asked people, searched in bookstores, searched online, and racked my memory, all to no avail. Then one day I read an interview with a dear friend of the family, George Cobb, who died last spring and whom I had not seen in many years. He mentioned using Frederick Mosteller’s Fifty Challenging Problems in Probability with Solutions (1965) in a probability course that he taught. Something told me that this might be the book; I looked it up, and sure enough, it was. He must have given me a copy as a gift. I ordered a Dover paperback (the original book was hardcover); it arrived the other day.

I opened it up and read the preface, which I probably hadn’t read before, since in childhood I didn’t bother much with prefaces, preferring instead to get right into the matter. It brought back a dim and beloved world. Mosteller writes:

Much of what I have learned, as well as much of my intellectual enjoyment, has come through problem solving. Through the years, I’ve found it more and more difficult to tell when I was working and when playing, for it has so often turned out that what I have learned playing with problems has been useful in my serious work.

In a problem, the great thing is the challenge. A problem can be challenging for many reasons: because the subject matter is intriguing, because the answer defies unsophisticated intuition, because of its difficulty, because of a clever intuition, or even because of the simplicity or beauty of the answer.

I turned to the first problem, which I now remembered clearly.

1. The Sock Drawer

A drawer contains red socks and black socks. When two socks are drawn at random, the probability that both are red is 1/2. (a) How small can the number of socks in the drawer be? (b) How small if the number of black socks is even?

The first part I figured out just by experimenting in my mind. The total number of possibilities for choices of two socks would be (t)(t-1), where t is the total number of socks. I would need r(r-1), the total number of possibilities for choosing two red socks, to be 1/2(t)(t-1). If the total number of socks were 4, and the number of red socks 3, this would work out.

The second part is much trickier–and the solution in the book involves setting up an inequality, using it to express the relation of r to b, and then trying out increasing even values of b until one of them works.

Last night I started thinking of a different solution, which I would execute with Perl. My underlying principle was this: if I could have Perl generate two tables, one of which held particular values for the total number of socks (t, t-1, t(t-1), and t’s even/odd value) and the other for the total number of red socks, and if I could write a program that iterated through the tables until it found a match where t(t-1) was twice r(r-1), then I could narrow down the list to those where t and r had the same even/odd value, which would make b even (since b = t-r). I worked on that for quite a while but couldn’t get Perl to do the iterations that I had in mind.

Then, when biking to the supermarket for last-minute groceries for dinner, I had a different idea.

use POSIX;

for ($redtotal = 1; $redtotal <= 1000000; $redtotal++) {
$redsocks[$redtotal][0] = $redtotal;
$redsocks[$redtotal][1] = $redsocks[$redtotal][0] – 1;
$redsocks[$redtotal][2] = $redsocks[$redtotal][0] * $redsocks[$redtotal][1];
$redsocks[$redtotal][3] = 0;
if ($redsocks[$redtotal-1][3] == 0) {
$redsocks[$redtotal][3] = 1;
}
else {
$redsocks[$redtotal][3] = 0
}
$redsocks[$redtotal][4] = 2 * $redsocks[$redtotal][2];
$product = $redsocks[$redtotal][4];
$square = sqrt($product);
$roundup = ceil($square);
$rounddown = floor($square);
if ($roundup != $rounddown) {
if (($roundup * $rounddown) == ($product)) {
if ((($roundup % 2) + ($redtotal % 2)) != 1) {
print (“$roundup”, ” total socks, “, “$redtotal”, ” red socks\n”);
}
}
}
}

The POSIX call just brings in some extra functions. The whole program consists of a “for” loop that iterates through values of $redtotal, the total number of red socks. First it established the elements of the array @redsocks. Then it assigns a few more variables.

Basically, we are trying to find out whether, for any particular r, 2r(r-1) can be expressed as the product of two consecutive integers, t(t-1). To find this t and t-1, take the square root of $product, and, if it is not an even integer, identify the integers immediately above and below it ($roundup and $rounddown). Then test them out by multiplying them with each other. If they equal $product, then you have a match. In that case, add the even/odd values of $roundup and $redtotal. If the sum does not equal 1, then they are either both even or both odd, in which case b will be even. Those are the matches that will be printed out.

Now have the program print out all the matches as specified above. For the purposes of the problem, we only need the lowest value (15 red socks, 21 socks in total), but it’s fun to see what happens after that. Here are the results (where $redtotal goes up to one million):

21 total socks, 15 red socks
697 total socks, 493 red socks
23661 total socks, 16731 red socks
803761 total socks, 568345 red socks


You can test them out by multiplying each number of total socks by the number one less than that, doing the same for the red socks, and then verifying that your second result is one-half of your first one. Let’s do this for the highest number here.

803,761 x 803,760 = 646,030,941,360
568,345 x 568,344 = 323,015,470,680

323,015,470,680 x 2 = 646,030,941,360

So, you see, it works!

There are probably ways to make the script more elegant. Instead of nesting the ifs, I could have used a series of ands, but I couldn’t get that to work correctly. I haven’t used Perl in years, so I’m a little rusty with the syntax. I was proud to be able to get this working.

The book was written long before Perl and more sophisticated programming languages came into use, long before it became possible to program from home. But the problems do just what they did before. They incite you to think, play, tinker, and solve. And this book is not only rejoining my collection but opening up to me in a new way after all these years.

If you try out this code, be sure to change the minus sign (in line 5) to a plain hyphen and the quotes near the end to plain quotes.

Song Series #8: Different Exiles

 

C37F8E25-F01E-4E04-AED2-9B789EA5ADEE

Exile: by its usual definition, the state of being banned from your own country. But exile can be internal too. Or even a fact of life, a condition of the things you need to do. Music demands a kind of exile; while it brings people together (intensely), it also demands truth, and truth gets you in trouble, whether obviously or not.

It’s a little more complicated than that. Musical truth is different from what we know as “telling the truth.” The stories in music don’t have to match point for point with the facts of your own life, but the shape will be true, the rhythm will be true, and the words will speak to you even if you don’t know what they mean. When this happens, you’re already cast out–in the best of ways, since exile can be joyous too–and you can’t take it back. You go about your life like everyone else, but as soon as a certain song starts playing in your head, you suddenly unbelong to your surroundings. The world will not bend to the music or vice versa.

Every good song, in that sense, is a song of exile. But a few stand out for me in this way. I’ll leave out the obvious exile ballads, such as Radiohead’s “Daydreaming,” Townes Van Zandt’s “Pancho and Lefty” or Leonard Cohen’s “Famous Blue Raincoat.” They are among my favorite songs, but their place in the “exile canon” is already clear. Instead, I’ll include Nick Drake’s “River Man,” Ferron’s “Shadows on a Dime,” Dávid Szesztay’s “2120,” Joni Mitchell’s “Hejira,” and Sonic Youth’s “The Diamond Sea.” (I had included “The Diamond Sea” in my previous post in this series, but I switched it over here.)

Nick Drake’s songs come back to me over the years; they are bare and raw and so perfectly formed and played. “River Man” seems to have to do with a world that has come to be too much, and a “river man” who knows a different way, but a way that may not be open.  The music creates a picture of it: the lingering vocals, the synthesizer against the acoustic guitar. As the song progresses, you sense the river more and more.

In the 1980s I listened to Ferron’s “Shadows on a Dime” endlessly (and heard her play it once in concert); I loved and love its syncopations, the lovely raspy vocals, the guitar sound, and the connecting stories, all leading up to the last verse:

And now a tired conductor passes by
He takes my ticket with a sigh
I don’t think he meant to catch my eye
But he doesn’t turn away.
He says “I have a daughter as old as you
And there’s really nothing anyone else can do
Do you play guitar…well good for you
Me I play the violin”
I imagine him with his hair jet black
Does he hide his fiddle in the back?
He gauged his words as the train went slack:
The New York train stops here

But I don’t forget the factory
I don’t expect this ride to always be
Can I give them what they want to see
Let me do it twice —
The second time for me.

‘Cause these windows make a perfect frame
For New York buildings like upright trains
They hold me as I hold the rain
Fleeting shadows on a dime.

It is a song of exile because the narrator, the musician, is always on the road, as are others, like the train conductor who maybe “hides his fiddle in the back.”

Now for Dávid Szesztay‘s “2120,” one of my favorite songs on his album Dalok bentre. (I heard him play on Saturday night in Szeged; you can read my review here.) The video, directed by Pater Sparrow and starring Szesztay and his family, is brilliant, eerie, beautiful and sad, but I recommend listening to the song on its own first, since there are so many ways to hear and understand it. The refrain does so much and rhythmically with the simple words “Kinn meg fagy, kinn hagytak” (“Outside and freezing, they left you outside.”) And then, at the end, the repeated “mozogjál” (“get a move on,” “hurry up”) contains its opposite; it stays instead of moving on, or it does both at the same time; the word turns into something else, something beyond leaving and staying. I have been listening to this song and the whole album over and over.

I have included Joni Mitchell in this song series before–“Coyote,” from the same album as this–but it’s impossible to leave out “Hejira” here.

I know, no one’s going to show me everything
We all come and go unknown
Each so deep and superficial
Between the forceps and the stone

Now for Sonic Youth’s “The Diamond Sea.” I love the changes it goes through, the way the music creates the diamond sea. I also love the matter-of-factness of the main melody, and the way the lyrics build. As for its exile, it’s the passage of time and the sight of the diamond sea that make you unable to come back. “Time takes its crazy toll.” The two go together; not only will you eventually see the diamond sea, over the course of time, but over time it will also have its effect on you. The music takes you through this.

And that concludes the eighth installment of the song series.

I took the photo on Saturday night in Szeged.

Why I Like Robert Pondiscio’s Book (and Why “Main Idea” Is Duke, Not King)

how the other half learnsAfter reading many reviews and summaries of Robert Pondiscio’s outstanding book, How the Other Half Learns: Equality, Excellence, and the Battle Over School Choice, I worried that I already knew too much of the gist and wouldn’t have much left to enjoy or think about. The worry was unfounded. I read it this weekend in several sittings, unable to stop for long. I was drawn into the descriptions, the characters, the daily life of Bronx 1 classrooms, the pedagogical and curricular details, and Pondiscio’s subtle, surprising observations along the way. That very experience–of enjoying the content of the book–points to what I see as its main blind spot. In the book itself, the “Main Idea” is not king–so I am wary of pedagogical approaches that insist that yes, it is.

Granted, I am writing from a high school and college perspective, as I usually do. Elementary school and high school differ profoundly; when people do not acknowledge this, they often end up talking past each other. Some of the greatest misunderstandings in education discussion come from failures to specify what we are talking about. Pondiscio’s book comes to life, and to meaning, through its specificity. He is talking about elementary school–and not elementary school in general, but elementary school for very poor kids whose parents are determined to give them a foothold. Elementary school is where students should be learning certain basics–and the “main idea” is surely one of them.

The refrain “Main Idea is king” rings throughout the book. It’s what the teachers tell the students over and over, and exemplify in their classrooms, at Bronx 1 Elementary School, which Pondiscio visited for a year. Bronx 1 belongs to the Success Academy, a network of charter schools, founded by Eva Moskowitz, that has won both fame for its test score success and rebuke for perceived creaming and overhype. Pondiscio argues that the Success Academy schools don’t cream students; they cream parents. Is this fair? It depends on how you look at it. But for now, back to the Main Idea.

Revering the main idea will help you, up to a point, with reading comprehension. (For instance, if there is a main idea in a text, and if you can identify it, you can then figure out how the different parts of the text support it.) Such regal treatment will also help you with ELA standardized tests, which almost always include questions about the main idea. It will not help you with the kind of discussion that you find at private high schools and in college. In many texts (Pondiscio’s book included), the main idea is only the foundation, if even that; the really interesting stuff is to be found in the subordinate clauses, the observations, the connections, the hesitations, the contradictions. This is especially true with poetry and fiction, but it applies to nonfiction as well.

The main idea of How the Other Half Learns might run as follows: “While controversial in its approaches to admissions, instruction, and discipline, and perhaps impossible to scale, the Success Academy charter schools bring their students to academic success–in terms of test scores, college admissions, and more–and therefore deserve recognition and support.” I don’t need to read a whole book to get that point–but the book did much more than argue it. I was drawn into the description of specific lessons, walkthroughs, leaders’ and teachers’ meetings, hallway activity, Pondiscio’s meetings with families, and characters so vivid that I saw and heard them in my mind.

The Success Academy’s emphasis on the main idea–and other concepts important to the standardized tests–goes hand in hand, I think, with its avoidance of “teacher talk.” For if students are supposed to be doing most of the work, and teachers are to limit their talking, then students must have specific, recognizable tasks to perform.

Third-grade teacher Steven Madan has the children continually involved in tasks, continually (in Pondiscio’s words) “engaged and on their toes.” From p. 46:

“The best learning we get in the classroom comes from other scholars, because we learn from each other,” Madan tells his students, a notion that Success drills into teachers during the network’s summer Teacher School, or T School. The feedback new staffers hear most often is “too much teacher talk.” The standard remedy is to “put the lift on the scholars”: Don’t do the work for the kids. Don’t be afraid to let them struggle. That’s how they learn.

I have heard this many times before, in public schools: that if the teacher talks, she is “doing the work” for the kids. This does not have to be so. Students should have opportunities to work out some problems and puzzles on their own. But listening to the teacher is a demanding challenge in its own right: you must focus closely, figuring out what makes sense to you and what does not, formulating questions, and finding words for disagreements, hesitations, or extensions.

Yes, I am thinking in terms of high school and college, but elementary school students can do this too, and if they can’t, they should begin learning it gradually. This does not mean that teachers should talk all the time, talk needlessly, or strain their students’ attention beyond what they can handle. But “teacher talk” should not be deplored; not only does it have an important place in lessons, but students unused to it will have great difficulty later, not only in lectures, but also in seminars, where they also need to sustain their listening and deal with complex ideas.

I will come back to the “Main Idea” shortly–but want to comment on the “Math Lesson” chapter, if too briefly. Pondiscio states that the teacher’s (Kerri Lynch’s) math lesson, “with its push to get students, not teachers, to do the thinking, and its almost complete lack of direct instruction, bears the hallmark of Success Academy’s approach and a focus–nearly an obsession–of its teacher training” (p. 142). The actual lesson is lively and productive; students figure out, among other things, that 7/8 is greater than 3/4, and arrive at a clear explanation. But what happens with a student who understands, right off the bat, that this is so, and can explain why? What challenge is left? One way to challenge such a student–and others as well–is for the teacher to present an extended solution to a problem, ask the students to pay close attention to it, and then question them to see whether they understand it, can explain it, and can take it in new directions.

For instance, in a geometry class, you might ask students how they would bisect a segment, without using any numerical measurements. They may use the length of the segment itself and the lengths and angles on a right-triangle ruler. But they may not actually measure the lengths and angles.

If they can’t figure it out, give them a helping start: Create an isosceles triangle with the segment as the base. From there they can probably figure out that all they need to do is drop a line from the vertex opposite the base down to the base, at a perpendicular to the base. If they can’t, there are ways to offer hints without giving the solution away. When they finally get it, have a student explain it from start to finish; if he or she gets stuck, others may help out.

Once they have explained it, ask: So, how do you trisect a segment, using the same tools? Let them puzzle over it for homework, part of homework, or extra credit; welcome them to work on it together if they wish. The next day, see who has figured it out; if someone has, ask for a presentation, and ask questions along the way about the steps. If no one has figured it out, give a helping start again, and see whether they can take it from there.

This example still has the students doing the majority of the work–but it is possible for the teacher to present something without turning the lesson into a sequence of procedures or robbing the students of insight. To the contrary: they must learn how to make sense of what they see on the board–not only make sense of it, but take it farther.

Or take poetry. So many poems have been ruined by lessons that insisted on a main idea or relied entirely on student discovery. What do you do with Robert Frost’s “Stopping by Woods on a Snowy Evening“? On the surface, the poem is about taking a few minutes of quiet–but in each stanza, the quiet is subtly disturbed. Even the title, “Stopping by Woods…” suggests a temporary stopping, not a permanent one. You do not have to summarize all of this in a single statement; you can instead look at and listen to the different pulls in the poem.

I have no way of knowing, but I suspect that the Success Academy high school’s initial difficulties had to do with the elementary and middle schools’ extreme focus on reading strategies and their stance against “teacher talk.” (Pondiscio states bluntly, when describing the high school’s beginnings, that it was “a disaster.”) Students’ difficulties with seminar may have stemmed, in part, from not knowing how to listen to others at length or how to explore a text or topic on its own terms. This might not have been a matter of classroom discipline alone. It might have had to do with intellectual practices.

This does not mean that teachers should abandon group work or paired discussion (I include both in my lessons). But there is a case for teaching something directly to the students. First, they may not know it; second, it can give them some ideas of how to think about the topic; and third, it can open up discussion at a higher level than would otherwise be possible. The main idea comes up in such instruction and discussion, but it is rarely the goal.  Rather, teacher and students focus on the text’s motion, details, digressions, and uncertainties. The students come to see more than they saw before. The main idea still matters, but it does not merit a crown. Let it be duke.

Pondiscio’s book demonstrates this unwittingly. It is a bracing pleasure; it raises memories, ideas, questions. It holds much more than a main idea. What’s more, it comes from an author with experience, insight, and a gift for writing. It could not have been achieved through turn-and-talk alone.

Note: Robert Pondiscio is a good acquaintance/friend (whom I have not seen in person for some time). I have been reading and enjoying his writing for years–and contributed many guest posts to the Core Knowledge Blog when he was its editor and lead author. 

Also, after posting this piece I realized I should refer to him, after first mention, by his last name rather than his first–since this is a review, not an informal comment–so I made the change.

Why Imagination Matters

poets walk park

Our schools have vacillated between adulating and dismissing imagination; neither attitude suffices. Imagination involves forming things in the mind; education cannot do without it. Yet to employ it well, one must understand it correctly and combine it with actual learning.

In his bracing book Why Knowledge Matters: Rescuing our Children from Failed Educational Theories, E. D. Hirsch Jr. explores the origins and consequences of our schools’ emphasis on “natural” creativity and imagination at the expense of concrete learning. He points to the destructive effects of this trend, both in the United States and in France (which moved from a common curriculum to a child-centered mode of instruction). In addition, he offers wise commentary on standardized tests, the teaching profession, and the Common Core initiative.

An admirer of Hirsch’s work and of Core Knowledge schools, I object to just one aspect of his argument: By opposing creativity and imagination to specific training and instruction, he limits both. Recognizing this possible pitfall, he acknowledges that a school with a strong curriculum can still encourage imagination—but he does not treat the latter as vital and endangered. Imagination, in his view, has been overemphasized; the necessary corrective lies in specific, sequenced instruction.

He writes (on p. 119): “I am not, of course, suggesting that it would be a good idea to adopt the in-Adam’s-fall-we-sinned-all point of view. Imagination can certainly be a positive virtue when directed to life-enhancing goals. But the idea that imagination is always positive and life-enhancing is an uncritical assumption that has crept into our discourse from the pantheistic effusions of the romantic period.” I dispute nothing in this statement but the emphasis (and the take on Romanticism–but that’s another subject). I would proclaim: “Imagination has been wrested apart from subject matter and thus distorted—but properly understood, it permeates all intellectual domains.”

What is imagination? It is not the same as total freedom of thought; it has strictures and structures. To imagine something is to form an image of it. Every subject requires imagination: To understand mathematics, you must be able to form the abstract principles in your mind and carry them in different directions; to understand a poem, you must perceive patterns, cadences, allusions, and subtleties. To interpret a work of literature, you must notice something essential about it (on your own, without any overt highlighting by the author or editor); to interpret a historical event, you may transport yourself temporarily to its setting.

Civic life, too, relies on imagination; to participate in dialogue, you must perceive possibility in others; to make informed decisions, you must not only know their history but anticipate their possible consequences. Imagination forms the private counterpart of public life; to participate in the world, you must be able to step back and think on your own, as David Bromwich argues in his essay “Lincoln and Whitman as Representative Americans” (and elsewhere).

Plato’s Allegory of the Cave describes the cultivation of the imagination. The uneducated mind, the prisoner in the cave, accepts the appearances of things (as manipulated by others); once embarked upon education, it slowly, painfully moves toward vision of true form. People are quick to dismiss Plato’s idealism as obsolete—but say what you will, it contains the idea of educating oneself into imagination, which could inform many a policy and school.

Schools and school systems have grievously misconstrued imagination; drawing on Romantic tendencies, as Hirsch explains, they have regarded it as “natural” and therefore good from the start. If imagination is best when unhampered and untouched, if it is indeed a process of nature, then, according to these schools, children should be encouraged to write about whatever pleases them, to read books of their own choice, and to create wonderful art (wonderful because it is theirs). Some years ago I taught at a school where we were told not to write on students’ work but instead to affix a post-it with two compliments and two suggestions–so as not to interfere with the students’ own voice.

This is silly, of course. Serious imaginative work—in music, mathematics, engineering, architecture, and elsewhere—requires knowledge, discipline, self-criticism, and guidance from others. You do not learn to play piano if your teacher keeps telling you, “Brilliant, Brilliant!” (or even, in growth-mindset parlance, “How hard you worked on that!”). To accomplish something significant, you need to know what you are doing; to know, you must learn. Mindset aside, you must be immersed in the material and striving for understanding and fluency. You must listen closely; you must acknowledge and correct errors.

Learning draws on imagination and vice versa; a strong curriculum is inherently imaginative if taught and studied properly. Students learn concrete things so that they can think about them, carry them in the mind, assemble them in interesting ways, and create new things from them. On their own, in class, and in faculty meetings, teachers probe and interpret the material they present. This intellectual life has both inherent and practical value; the student not only comes to see the possibilities of each subject but lives out such possibilities in the world.

Hirsch objects, commendably, to the trivialization of curriculum and imagination alike: for instance, the reduction of literature instruction to “balanced literacy” (where students practice reading strategies on an array of books that vary widely in quality). Conducted in the name of student interest, creativity, initiative, and so forth, such programs can end up glorifying a void.

Without strong curricula, creative and imaginative initiatives will lack meaning, especially for disadvantaged students who rely on school for fundamentals. You cannot learn subjects incidentally; while you may gain insights from a creative algebra project, it cannot replace a well-planned algebra course.

But imagination belongs at the forefront of education, not on the edges; it allows us to live and work for something more than surface appearance, hits, ratings, reactive tweets, and prefabricated success. Imagination reminds us that there is more to a person, subject, or problem than may appear at first. It enables public, social, private, economic, intellectual, and artistic life. Without it, we fall prey to shallow judgment (our own and others’); within it, we have room to learn and form.

 

Photo credit: I took this picture yesterday in Poets’ Walk Park in Red Hook, NY.

Some More Tinkering

After yesterday’s discussion of springs and sine graphs, I remembered a post from 2012, “Daydreams, Lectures, and Helices.” I decided to figure out how to graph a helix in R.

Well, after installing the rgl package and experimenting a bit, I got it to work.

helix

Here’s the code:

require(rgl)
x=seq(0.01,6.29,0.01)
y=seq(0.01,6.29,0.01)
z=seq(0.01,40,0.01)
plot3d(cos(x), sin(y), z, col=”red”, size=3)

I then rotated it into a good view. As my Latin teacher used to say, that’s all there is to it!

Next up (at some point): an animation of a spring.

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 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 takes the form of homework. In the classroom, one is discussing the material; such discussion can meet several levels at once. In a discussion of a literary work, for instance, some students may be puzzling through it 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 (while also completing 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 too 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, you end up playing with it. Thus, when there is no play in a classroom, something is 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, from 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; on February 6, 2017, I made a few more.

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 good, then my mind dances with it, sometimes spinning away, 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 = , j = , and k = . (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 and made a minor revision 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.