Two Kinds of Writers

In 1920, the humorist and actor Robert Benchley wrote in Vanity Fair,

There may be said to be two classes of people in the world; those who constantly divide the people of the world into two classes, and those who do not. Both classes are extremely unpleasant to meet socially, leaving practically no one in the world whom one cares very much to know.

In the spirit of this quote, I hope there are not two kinds of writers: those who like to discuss the writing process and those who do not. Both kinds, in my view, would be rather irritating, though I’d be a little more receptive to the second. There’s a time and place for discussing the writing process, and an eternity for not doing so.

Problems with discussing the writing process? There’s so much variety that one cannot draw any conclusions about a “right” way. What’s more, the “process” discussions tend to ignore substance. There are writers who revise constantly and those whose first draft is almost always their last. There are those who adhere to a strict routine and those who write whenever the ideas strike them. There are those who suffer terribly from writer’s block and those who have never known it. There are those who insist on writing in pen, or with the trusty Remington, or through dictation. In the end, I don’t care what they do, if the writing is good.

Yet staying mum is problematic too. There are writers who hold themselves above describing what they actually do; they insinuate that their work is mystical and untouchable, and that any mention of process is the mark of a lesser talent. Or they refrain from discussing it lest they expose a weakness–an embarrassing first draft, for instance, or an abundance of unfinished work. Silence is golden, but gold can be the ornament of a snob.

The ideal would be to talk about it sometimes but not all the time. Just how much would depend on the person’s judgment and circumstances. If you have been invited to speak to young people about your writing process, and have agreed to do so, then a secretive attitude is out of place. However, if you are at a tea party where people are going on about how they love “workshopping” their work (and you don’t particularly love doing that), then you have every right to maintain a happy hush.

I revise a lot. One thing I enjoy about having a blog is that I can come back and change things later. (When I do, I indicate this in a note at the end of the post, unless the changes are too minor to mention.) I rethink things continually; months or years later, I may see a better way of putting them. This is true for my nonfiction, fiction, and poetry. This morning I made some revisions to an old poem, “Jackrabbit.” It’s one of my favorites (of my older pieces), but the original version and even a later version had some strained parts. The current version will rest as is.

Jackrabbit

This land has never been painted properly.
Mix clumps of juniper with moonbeam blue,
Throw in a bit of tooth, a bit of song,
to fill the silhouette with bite and tongue.

This is a real dirt road with imagined doubts,
senses, untasted dangers, destinations.
Headlights sweeping the long floor of the wild
pan a jackrabbit back and forth in time.

Caught in the blank emergency of beams,
he dodges his dilemma with a brisk
“what if, what if” that dances him to death.
He could not find a way out of the way.

Earlier that day I was on the phone,
missing all your relevant advice.
A wire had got caught up in my throat,
an answer-dodger. It distracted me.

It trembled so fast that it numbed my tongue.
It did this while you were trying to talk.
I couldn’t listen well because the dance
had blurred all trace of consonant and sense.

I think now that this may have been a crash
of my old givens against your offerings:
new junipers, or ways of seeing them,
new countries, or ways of getting there.

When I hung up, there was no wire or word.
The moon was gone, the road a long fur coat
on some unwitting wearer, blissed and hushed.
I forgot all about it until years later.

You had said: “You can go left or right.”
Take me straight! I shouted. Straight to the remedy.
Gallop like the nineteenth century
down to the police station or cemetery.

Striding answerless, a station incarnate,
a cop ticketed me for not listening.
Now I can bear the rabbits and the wires.
I inch through forks and roadkill, listening.

Note: I changed three words (and fixed a formatting glitch) after the initial posting.

Our Unwitting Teachers

Ninth grade was a tumultuous year for me. Too much to go into here–but many ups and downs. In the spring, I discovered that I enjoyed running track. Though bad at sports overall, I had good endurance; I was one of the few in the school who could run a full mile, at a good pace, without stopping. (Later I brought it up to three.) Through running, I got to know a few students outside of my usual circles. They chose to run with me because they trusted that I wouldn’t stop halfway through. As we were running, we had short-winded conversations about all kinds of things.

One day I was running with a girl had lost a family member a few years before. We were talking (inasmuch as we could) about life, and she said, “I want to experience everything–both the good and the bad.” Given what she had been through, this moved me and stayed on my mind afterward. I thought about how everything we go through can contribute to who we are; all of it, taken properly, can be a gift, no matter how difficult it seems at the time. I thought, also, of the ending of Thomas Hardy’s The Mayor of Casterbridge, which I had just read in class.

A month or two later, toward the end of the year, I decided to tell her that her words had helped me. I asked whether she had a few minutes to talk. She agreed and went with me to the “fountain room,” a luminous little meeting room with French windows and doors, in a corner of the hallway near the drinking fountain. I told her what I have said just here. She looked surprised and happy but didn’t say very much.

Later, when she signed my yearbook, she wrote: “You have been a source of knowledge about myself this year. Your talent and persistence has inspired me and helped me be calm and level headed. You told me that I helped you–well, telling me that made me cry that night in bed, because just earlier I had been chastising myself for being insensitive and unresponsive to so many people that were sensitive and yet very far away from me. I am so glad that I was able to do something for you. You deserve happiness an if I was able to help you attain that end, then I am happy.”

We were each other’s unwitting teachers. At the time, I wondered just how much people held back from each other, how many good things they could say to each other if they dared. Years later, it seemed to me that such inhibitions might have a place, that they might make room for the understood, the understated, and things that don’t go easily into words.

Now I see it in both ways. There are many who have taught and helped me and probably do not know it. Maybe it is a shame that they do not know it; maybe it is for the better. Sometimes the one seems true, sometimes the other, but I never know for certain which one holds.

I suspect that we tend to hold back from telling people what they taught us; thus, people have little idea how much they bring to others. Yet there’s no way to delimit “how much” these teachers bring; the students themselves do not know. So, speaking and holding back can both be wrong. But there is no wrong in perceiving one’s teachers: seeing a chance comment burst into an offering, or hearing a cadence that brings a change of heart. There must be good in witnessing these things without trying to capture them, without trying to say exactly how much or what they mean.

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

CONTRARIWISE Contests Galore

itsmyowninventionThroughout my teaching experience, there have been many surprises, many sources of wonder, but nothing quite like CONTRARIWISE, my students’ philosophy journal. It arose out of an assignment about Plato a year ago; the first issue, which came out in February, received a great review and many appreciative comments from readers. We had a glorious celebration in May; a week later, four students took part in an interview with Mark Balawender of PLATO. (If you aren’t sure what the fuss is all about, see the samples on the website.) But that was only the beginning. Now my students have announced an international contest and two national contests, as well as an open call.

Here is the international contest:

Your favorite cultural dish* is now its own nation. Who/what is its leader? Its citizens? What does each ingredient do for a living? You may refer to the ingredients, cooking utensils, eating utensils, human participants, or other aspects of the food’s preparation and consumption. Write about a philosophical problem this nation experiences—”anything from existential angst due to being eaten, to “okra should never have been chosen as ‘secretary of state.'” This can be a story, an essay, an epic poem written in the style of Beowulf, words set to a popular song (bonus points if it’s a song we don’t know and have to look up, and it becomes one of our favorite songs of all time), or anything, really.

Secondary school students are lucky to have these contests! When I bring up the international contest with adults, I often get the reaction: “What would I do with that?” followed by days of conversation about fondue, various pastas, etc., and what they could be as nations. (A recent comment: “We’re still thinking about the eggplant.”) Alas, we adults may muse to our hearts’ content but may not enter. That is just as well; I wrote a piece about the realm of flan, was proud of it at first, but then realized how contained it was and how much more possibility the contest held.

But that isn’t all. Here’s the first of the national contests:

Write a piece about how mathematics and philosophy are related. It could be a theorem with a variety of proofs, a comparison of a philosophical and a mathematical problem, a mathematical solution to an ethical issue such as adoption, or a poem about how to treat your x. You may use any format you wish, including pictures, and you may invoke higher dimensions.

Here is the second:

You are a knight or samurai (who strictly adheres to your society’s honor code) during the fall of feudalism in your nation. This time period can be any time after your chosen government begins to stop following the codes of chivalry or bushido. In 3,000 or fewer words, write a piece critiquing the government and explaining how you feel and what should be done about it. This could be in the form of a letter to be sent to your government, a poem to be nailed on the gates of a church…the format can be as creative as the piece itself. Just let us know what you intend the knight to do with his work at the top of your first page. Be sure to research your chosen nation!

The words of Khadijah McCarthy, a CONTRARIWISE contributor who participated in the PLATO interview, seem especially apt here:

There has to be a degree of eccentricity to the questions that we ask because we are not looking for your basic responses. We need philosophers who can transgress those boundaries and get people to come in and say I want to take a philosophy class and request it in schools around the world and around the nation. We do our best to really make people think. And the questions that they asked me, and I when I looked at them at face value, I thought, “I really don’t know how I am going to answer this.” … I think the best questions are the ones where you don’t know how you’re going to answer them. You’re going to have to formulate them and test them. So pretty much you’re a scientist, a philosopher…everything is wrapped up in one.

I can’t wait to see what comes in.

 

Image: Lewis Carroll, Through the Looking Glass, illus. John Tenniel, chapter 8, “It’s My Own Invention.”

Questions of Community

There are several related idols in contemporary culture: the group, the team, and the community. Each one has a different character, and each one has benefits and dangers.

I have discussed the pitfalls of group work on numerous occasions–most recently, in an interview with The Guardian (UK). I do not mean that group work is necessarily bad; it is just overemphasized. Thinking on one’s own–or participating in a whole-class lesson–gets short shrift.

In addition, I have discussed problems with the concept of a team. Teams have their place (many places, actually), but not every group or association is a team, nor should it be. Much important work is done by individuals and can be shortchanged by a team.

In relation to the above, I have also examined how collaboration differs from group work, and how belonging and apartness combine in education.

Today I will look at a somewhat touchier subject: community. Community, as I understand it, is an association of individuals with a loose common bond, be it geography, a common interest or attitude, or some other common characteristic. To many, community is an automatic good; what could possibly be wrong with having something in common with many others and, on account of this commonality, being part of a larger whole?

Indeed, there is much to be said for it; many of us have longed to be part of a community of some kind and have rejoiced when we found one. But the word can be misused.

For one thing, as David Bromwich points out in Politics by Other Means (1992), it can be invoked manipulatively, for ideological ends. (Sometimes the “community” invoked might not even exist as such.)

Or the word might be invoked in reference to the most popular activities or views–and not in reference to the outliers. In my experience, “Support your community” rarely means, “Support the individuals within it.” Instead, it seems to mean, “Support those things that the majority supports, those things that draw a crowd.” I do not mean that the things that draw a crowd are unworthy–but a true community should have room for more. A genuine community, as I understand it, would honor its minorities, dissidents, independent thinkers, and others who don’t fit the group. There are circles within circles; the largest subcircle is not the whole (unless it is, of course).

I am likewise wary of communities where the members, because of the very nature of the bond, conceal important thoughts by choice or necessity–for instance, a “supportive community of writers” where everyone is supposed to praise everyone else. There must be room for genuine criticism; support should not be equated with applause.

Or take a workplace. Is that and can it be a community? It depends; at various jobs, I have become friends with my co-workers. Sometimes the entire staff has bonded. But no matter how warm the workplace, one must remember that at some level, it is a job. There is work to be done. Friendship and fellowship can form within it–but that should not be the expectation.

All of these pitfalls can be addressed with careful use of the word. There are different kinds of community, each with its offerings and restrictions. If one knows what one means by the word, one can avoid being deceived by it. But there is still another danger.

Belonging to a group is meaningful only if some true fellowship exists in it. Fellowship between two may be the best and strongest kind. As Emerson writes in his essay “Clubs” (the ninth chapter of Society and Solitude), “Discourse, when it rises highest and searches deepest, when it lifts us into that mood out of which thoughts come that remain as stars in our firmament, is between two.” Yet a community often interferes with the fellowship of two (or with solitude, for that matter); the individuals come under pressure to include others in their group, to level out their conversation, to accept the common denominator. If a community can make room for friendship and idiosyncrasy, if it does not try to smooth everyone down, if it recognizes that some affinities will run deeper than others, then it can be strong.

 

The First CONTRARIWISE Interview

Last May, Mark Balawender, communications director for PLATO (Philosophy Learning and Teaching Organization), interviewed the CONTRARIWISE co-editors-in-chief and two contributors. His wonderful piece was published today on the PLATO website.

CONTRARIWISE is my school’s philosophy journal. The inaugural issue, released last February, received a lovely review from Cynthia Haven. The second issue will feature an international contest!

What Should Teacher Education Be?

This is the closest I will come to reviewing Elizabeth Green’s Building a Better Teacher. (For earlier posts on specific parts of the book, see here and here; see also my response to the book excerpt “Why Do Americans Stink at Math” (New York Times, July 27). I find that the book raises important questions about teacher training but makes false oppositions between the “bad old days” and the promising present or future. In addition, I question its underlying assumption that we need a grand model for teacher training; as I see it, the best teacher education (and training) will be humble in scale and goal; it will give teachers the knowledge and skills they need to exercise independent thought, which will transcend existing models.

Elizabeth Green does us a great service by bringing the question of teacher education to the forefront and challenging the rhetoric and policy about “good” and “bad” teachers. She argues passionately that teachers can improve through deliberate study of the craft, yet she does not ignore the complexities of this proposition. The book is sure to meet with strong responses, because it deals with old (not new) controversies underlying pedagogy.

Unfortunately, she tries to resolve at least some of the complexities through a cosmic tale of slowly converging perspectives. We have Deborah Ball, Magdalene Lampert, and their TKOT group on the one hand, and Lemov and his “Taxonomy” group and “no-excuses schools” on the other. At first, it seems that Green is setting up a dialectic–but this does not seem to be the point. Slowly, through failures, revisions, and chance meetings, the two groups start to converge, or so it seems. Enter the Common Core, which (in Green’s depiction) seems to mesh well with both TKOT and the revised “Taxonomy.” It seems–though this may be incorrect–that Green is placing hope in the possibility that some great convergence will lead to a great master plan for teacher training.

Robert Pondiscio, who finds that Green comes “perilously close to undermining the case she sets out to build,” shares Green’s belief that any viable plan for teacher training must be scalable: “But if teachers are to be made, after all, rather than born, then good instructional practice must be something that can be identified, named, practiced, and mastered by millions.” (I wish I could attend the September 2 discussion, hosted by the Thomas B. Fordham Institute, between Pondiscio and Green; alas, I am at school until late afternoon and can’t possibly get to D.C.).

I argue the opposite: that both TKOT and the Taxonomy go wrong when they try to become comprehensive models. Scale them down a bit–make them into working principles for certain situations–and they can be of great use. The problem with an overarching model is that it comes from the minds of the few–so you have a few thinkers at the top, and many followers at the bottom. Teaching must allow for independence of thought, or education itself will be downgraded.

Green quotes Ball’s statement that the math she learned in school was “uninspiring at best, mentally and emotionally crushing at worst.” Her own pedagogical approach seems to repudiate and counter the “old style.” Yet one of her classmates might have been inspired by the lessons that she found so dull. I have seen math students–and have been a math student–who, listening to the teacher’s presentation, detected a pattern or corollary and jumped out of the seat with excitement. I have had teachers who expected this and who would pose questions along the way: “Where do you see this going? What would change if I did such-and-such instead?” In addition, my math teachers (in high school) were skilled at diagnosing my errors. They could quickly tell the difference between a careless error and a conceptual one; in addition, they recognized when I was solving a problem in a way they hadn’t considered. Good math pedagogy has been alive and well for a long time. (So has bad math pedagogy–but it often appeared in the guise of a new method.)

What about classroom discipline? In my book and in an op-ed, I criticize Lemov’s Taxonomy for its rigidity and excessive emphasis on external behavior. My main argument is that Lemov’s system promotes a “thinking gap” between those who depend on directives from moment to moment and those who have internal focus and direction. A classroom of students in the latter group–which you will find in top-level schools and colleges–do not need SLANT, nor do they get punished for minor aberrations (such as looking out the window). The focus–for them and for the teacher–is on the substance of the lesson; within that focus, they have great intellectual liberty. Helping students reach such self-possession is another matter–it takes some effort–but the Taxonomy, as a full model, is not the route. Yet certain techniques within the Taxonomy could be of help to teachers.

If, as a teacher, you have a mind of your own, you will find any model insufficient for your purposes. The challenge lies in recognizing those aspects that could be helpful. For example, I object strongly to an overemphasis on “reading strategies.” I find that generic strategies do little to illuminate specific texts–and that strategy instruction tends to bring down the intellectual level of a course. Granted, students need to learn strategies of various kinds, but they can do that in the context of the subject matter. Green describes Pam Grossman’s strategy emphasis with apparent enthusiasm that I do not share (see pp. 268 and 302, for instance). It is important to challenge such enthusiasms. Most principles of teaching can be taken too far; the challenge lies in recognizing when they do.

Likewise, a teacher should be willing to question advice. When Green prepares to give a guest lesson to a high school class, she accepts the regular teacher’s (Andy Snyder’s) judgment that the readings she initially selected would be “too boring” for the students. I do not blame her for deferring to his judgment here; this is a one-off occurrence, and he is a highly skilled and respected teacher. Yet in general it is important to question assumptions about what students will find “boring.” My students have gotten excited about John Stuart Mill, Hannah Arendt, and other writers that some would consider far beyond teenagers’ realm of interest. Much depends on what the teacher does with such works. That leads to the point of this piece.

Good teachers are knowledgeable, questioning, and self-questioning. They learn much from others–but also learn from the many hours of rumination over the course material, the lesson plans, and the students’ work. To insist on an opposition between the “bad old days” of teacher isolation and the “good new days” of collaboration is to set things up for a great error. Green writes, on p. 311: “The only way to get better teaching, [some teachers] argued, was lot leave teachers alone–‘liberate’ them, one columnist put it, and ‘let them be themselves.’ Yet leaving teachers alone was exactly what American schools had done for years, with no great success.”Here Green commits two fallacies: first, by quoting the columnist, she comes close to ridiculing the idea that teachers should be left alone–an idea that has great merit when not taken too far. Second, she implies that schools were uniformly leaving teachers alone for years–which is not true. Collaboration and professional development are not recent inventions.

In teaching, both solitude and collaboration have an essential place. If you never consult with others, you may develop blind spots; if you only consult with others, you may settle for the judgments of the group. Collaboration, at its best, is distinct from group work; it involves a great deal of solitary work. One goes off and thinks on one’s own; then one brings one’s insights to the table and listens to others. This allows for substantial discussion. When collaboration is reduced to group work, when it no longer has a solitary component, it becomes shallow. Although this varies widely from one situation to the next, I would say that the solitary work should take up about 80 percent of the time, and the remaining 20 percent should go to in–person collaboration. Instead, I see a widespread assumption that collaboration and meetings are one and the same.

What, then, should teacher education look like? First, teachers should have a liberal education–a background in math, literature, history, science, art, music, and preferably philosophy and a second language. They should have additional preparation in their own subject. This “preparation” should consist not simply of required courses and grades, but of intellectual discussion; “professional development” should often consist of literary and mathematical study.

Then what of the pedagogy? Teachers should be offered techniques and tools–with the emphasis on the underlying principles, and with the recognition that any given technique may be more appropriate for one setting than another. Beginning teachers–or teachers in an especially challenging setting–may need more structure at the outset, but ultimately they should be encouraged to find their way.

Finally, teachers must not be crushed with unreasonable duties. Too many teachers have to create their curricula on the fly, while teaching; this is  unreasonable and harmful. (Some aspects of a curriculum may well be spontaneous, and that’s good; but there’s more room for spontaneity when you know what it is you’re teaching.) Teachers should not be assigned to teach subjects that they don’t know; that, too, is a setup. Finally, teachers should have more time in the day for planning–both on their own and with colleagues.

These three facets of teacher preparation–liberal education, pedagogical techniques (to be used with judgment), and a restructuring of teachers’ responsibilities–would do a great deal to strengthen the teaching profession. Various pedagogical models could come into play, yet teachers would be expected to go beyond them. Is that not what we hope our students will do: learn, defy, and transcend the structure we have offered?

 

Note: I made some edits to this piece (for style and clarity) after posting it. I made two more minor edits on September 1. Then, on September 8, I made a substantial addition to paragraph 10 and inserted a new paragraph after that.

The Role of Love in Teaching

This is not meant to be a spoiler, nor is it meant to be taken out of context. In the final chapter of Building a Better Teacher, Elizabeth Green remembers the advice–received separately from Doug Lemov and Andy Snyder–that good teachers must love their students. After making a hurtful comment to a student during a guest lesson, and seeing the expression on the girl’s face, Green writes, “Staring back at her, I thought about how she was a human, a person I cared about. I decided that I loved her.” (This has already been quoted in Charlie Tyson’s review of the book on Inside Higher Ed.)

Soon I will say something about the book as a whole. Right now, I want to consider the general questions: Should teachers love their students? Is it possible to love all of one’s students? What does it mean to love one’s students, or to love anyone?

I will take up the last question first, since I find that the word “love” is thrown about too carelessly. We live in a time when you can “like” something with just a click, and where “love” seems just a few clicks away from “like.” There’s also a widespread belief (rooted in various religious traditions) that if you have a loving heart, you can love everyone, especially children. I would say that love is much rarer and more difficult than that.

What does it mean to love someone? It is not easily pinpointed, because love is in motion, and it comes in different forms. If we are considering basic human love–of a nonfamilial and nonerotic kind, that is, love based on intellectual, spiritual, and emotional but not physical bonds–then it has perhaps three sides: first, a recognition of another person as human (that is, a recognition of the person’s dignity); second, an appreciation of the person’s particulars, the things that distinguish him or her from others; and third, a genuine wish for that person’s well-being–that is, the person’s movement toward the good. Each of these aspects contains still more: for instance, a recognition of what one doesn’t know about the person, and a recognition that he or she is not static but changing.

Given this definition of love, it seems, on the surface, that we can and should have this love for everyone. But it is one of the most difficult things in the world. Each of us is given certain insights and certain blindness, which may or may not change over time. The insights allow us to see another person’s beauty (or shortcomings, as the case may be); the blindness may prevent us from seeing the same. In addition, it is our very idiosyncrasies that give meaning to love in the first place. If everyone loved me, I don’t think I would feel loved at all. There is something important about being recognized in the crowd, of being singled out. If love were universal, we would have no names. Everyone might as well be called “X.”

Even dignity–the most basic element of love–is difficult to keep in view all the time. In I and Thou (1923), Martin Buber describes the fleeting nature of the true I-You encounter; it comes and goes and cannot be held, but once one has known it, one knows it is there: “You cannot come to an understanding about it with others; you are lonely with it; but it teaches you to encounter others and to stand your ground in such encounters; and through the grace of its advents and the melancholy of its departures it leads you to that You in which the lines of relation, though parallel, intersect. It does not help you to survive; it only helps you to have intimations of eternity.”

But if dignity, fully realized, is elusive, it is also the most stable of the elements; one can honor it in anyone, and one can always keep it in view. A teacher may not be able, all the time, to treat others (or even herself) with full dignity, but she can recognize when she does and doesn’t. (One of my poems from long ago, “Looking Glass,” has to do with this–though it isn’t about teaching.) I think Green may be talking primarily about dignity here, although she calls it love.

A teacher can keep dignity in view, strive to treat everyone with dignity, and recognize her own shortcomings in that regard. That, to me, is a worthy aspiration for all teachers. What about love, then?

Returning to the three sides of love–recognition of dignity, appreciation of particulars, and wish for the person’s well-being–I would say that it can never be mandated, in the classroom or anywhere else, and that any effort to enforce it will lead to betrayal of others and self. It is much too rare and too precious to be encoded. But then I am puzzled by Leviticus 19:18: “Thou shalt not take vengeance, nor bear any grudge against the children of thy people, but thou shalt love thy neighbour as thyself: I am the LORD” (In Hebrew: לֹא-תִקֹּם וְלֹא-תִטֹּר אֶת-בְּנֵי עַמֶּךָ, וְאָהַבְתָּ לְרֵעֲךָ כָּמוֹךָ:  אֲנִי, יְהוָה). If love of others is commanded here, what does it mean? It must be something different from the definition I gave above, yet it must also go beyond recognition of dignity.

In a short piece in The Jewish Magazine, Ahuva Bloomfield explains that the Hebrew ahava, “love,” has the same root as hav, “to give.” There is thus a connection between loving and giving–precisely because giving creates a connection with others. Bloomfield suggests that to give is, in fact, to love, because the act becomes the bond.

Yet giving, too, is a tricky thing. First, it’s challenging. Many of us fall short in generosity to ourselves, to others, or both. Also, giving must be tempered. Give too much, and you wear yourself out–and make yourself unable to listen or receive. Give the wrong things, in the wrong way, and you prevent others from showing what they have.

A parent comes to know these complexities well. You can wish to give comfort to your son or daughter who has gone through a disappointment–being turned down for the school play, for instance, or being rejected by a peer. The comforting has its place but can also get in the way. Young people (and older people) need to go through certain things in their raw form. So a parent comes to recognize when to give comfort and when not to do so. Not doing so is also a form of giving.

In teaching, giving takes many forms–and must often combine with abstinence from giving. A teacher gives to the students by showing a way into a subject–and also by letting them figure out certain things for themselves. She gives to the students by being alert to their ups and downs–but also respecting their privacy. In addition, to give well, a teacher must have integrity; she must know her own limits and be willing to stay true to them. In doing so, she allows the students to have limits as well.

Where does this leave us? It seems that a teacher should have, first and foremost, an active intellect and conscience–a willingness to seek and seek. At the root of this is a recognition that there is more to learn–that we are full of error, and that even the highest attainments are only hints.

Building a Better Definition

Here is what I like so far about Elizabeth Green’s Building a Better Teacher: It has a searching quality, as I mentioned yesterday.It has vivid descriptions of lessons in action. It discusses actual subject matter. It makes the important argument that teachers can improve their craft through deliberate study. It gives rich examples of such study. All of these qualities make the book a worthwhile read.

At the same time, I am puzzled by Green’s utter lack of skepticism over certain exemplars of pedagogy that she offers in the book. In saying this, I am not trying to disparage them. My point is only that they could use some critical questioning and examination–in the very spirit of the kind of lesson study that Green finds promising.

This is a preliminary review, with a focus on a particular passage (about a third-grade lesson) in the second chapter. I haven’t read the whole book yet (I read slowly and have been very busy), but I had so many thoughts about these few pages that I decided to start here.

The context: Deborah Loewenberg Ball, at the time a professor at Michigan State, a scholar of math pedagogy, and a teacher at the public school Spartan Village, was teaching her third-grade students about odd and even numbers. The lesson was one of many that she and her colleague Magdalene Lampert had filmed for close study and discussion. Just before this lesson, the fourth-graders had a conference with the third-graders in which they discussed their findings on the question: “Was zero even, odd, or, as some children argued, neither one?”

For this lesson, Ball intended to have the students move from conjectures to proofs about odd and even numbers. But something unexpected happens: a “tall boy named Sean” puts forth a surprising conjecture that six is both even and odd. His classmates then jump in to refute him. What follows is a lively but flawed discussion–flawed not because of the students’ insights, which are excellent, but because of the lack of attention to basic principles, such as the principle of identifying and building on one’s working definitions (or, in the absence of definitions, information leading up to them).

The problem throughout the entire passage is that we never learn whether the students have a working definition of odd numbers. This lack of information affects everything, as I will show. It seems that they have a working definition of even numbers–but at times they appear to confuse definitions with properties. Moreover, the working definition itself could be the cause of Sean’s confusion–but this possibility is not mentioned. More about all of this shortly.

Back to the conference: it is a brilliant idea to have fourth-graders present their findings to third-graders. This gives the fourth-graders a chance to teach others what they have learned, and it gives the third-graders a glimpse of knowledge and insights that lie ahead. In addition, a conference on zero is a great idea; there’s much to explore about zero. Yet I fail to see why the question of zero’s odd, even, or other status merits a conference (even a short one). If the students have a viable definition of odd and even numbers, they can immediately rule out the possibility that zero is odd. (If they do not have working definitions, then they have no way of discussing the question anyway.) Then, if the students have a viable definition of even numbers, they can see (without a great amount of trouble) that zero meets the criteria. One stumbling block might be the concept of dividing zero in two. Some students might think that can’t be done. So, that would be the meat of the discussion, but it’s easily digestible. There isn’t much gristle here.

The students themselves don’t seem to be clear about their working definitions, or whether or not they have them. After Sean has spoken, Cassandra goes up to the board to refute him. She says that six can’t be an odd number, because zero is even, one odd, two even, and so on up to six, which must be even.

Green comments on the reactions of the mathematician Hyman Bass as he watches the video.

Hy marveled as the video continued. These third-graders–not a gifted class, but average, public school third-graders from, Deborah said, a wide range of backgrounds and ability levels–were having a real mathematical debate. One of them had made a claim, and then the others were trying to prove him wrong. Cassandra’s proof followed a classic structure. First, she had invoked one definition of even and odd–the fact that integers alternate between the two types on a number line–to show that six could only be even. Then she had drawn out a counterargument. To be odd and still fit the alternating definition, she’d shown, zero would have to be odd too. But, she’d concluded with a flourish, they had just decided the other day that zero was even. QED: Sean’s conjecture was impossible.

The two descriptions of Cassandra’s words and actions don’t match–the second is much more sophisticated than the first–but that’s only a secondary problem. The bigger problem lies in the notion that “the fact that integers alternate between the two types on a number line” could be called a definition. To me, this appears as a property, not a definition. It makes sense that the students would be working from properties to definitions–but it’s essential to point out the difference.

The same confusion arises a couple of pages earlier, in a footnote regarding the evenness of zero: “Like all even numbers, zero can be divided evenly by 2, is surrounded on either side by odd numbers, and when it is subtracted from an even number, produces an even result.” Only the first of these qualifies as a definition, and it alone is necessary.

The discussion goes on.Apparently the students do have a definition of even numbers: one girl, Jeannie, reminds them that an even number is “one that you can split up evenly without having to split one in half.” If this is indeed the working definition, then it seems possible (though it never gets mentioned as a possibility) that Sean’s confusion arises directly from this wording, particularly the word “evenly.” (His own explanation of his reasoning seems to proceed from such a misunderstanding.) He may have taken this definition to mean that a number is even if it can be divided into even numbers–a circular definition, but one that “evenly” seems to invite. In that case, there’s more to say about Sean’s conjecture. More about that in a minute.

Now another student, Mei, makes a great argument: by Sean’s reasoning, it could turn out that all numbers were both odd and even, in which case “we wouldn’t be even having this discussion!”

What Mei suggests here–but no one brings out–is that they have been working with the premise that a number is odd or even, but not both. If that is indeed one of their working premises, then it should be on the table. If it isn’t, then I wonder how they conceive of odd numbers in the first place.

I admire Mei’s energy and logic, but I feel bad for the student who has been sitting there quietly–who gets odd and even numbers and yearns to move on. I also feel bad for the student who has no idea at this point what has been established and what hasn’t.

To draw something helpful–and fascinating–out of this discussion, the teacher only had to remind the students to go back to their working definitions (and distinguish them from properties). This is important mathematical practice. One has to return to working definitions continually. Sometimes they come up for questioning. Sometimes a definition may prove flawed, or it may need better phrasing. But one must be clear about what the definitions are.

If, as I suspect, Sean thought that a number was even if it was divisible into even numbers, then the teacher could have clarified the meaning of “evenly” (and “even” elicited a rewording of the definition).

Then, to take up Sean’s idea (which is actually very interesting), she could have asked: Which numbers are divisible into even numbers only (assuming one does not treat 1 or -1 as a factor)? Students would notice that the positive integers in this set were 2, 4, 8, 16, …. in other words (though they wouldn’t have the vocabulary for this yet) exponentiation of 2 to the powers 1, 2, 3, 4, etc.

Many interesting things happen in the lesson–but the confusion over definitions and properties prevents the discussion from moving forward. For this reason, I do not share Green’s amazement, though I am grateful to the lesson (and to Green’s description) for stirring up some thoughts.

 

Note: I made some minor edits to this piece after posting it. Also, on 8/26/2014 I added one parenthetical sentence.

Prelude to a Preliminary Review

I have read the first four chapters of Elizabeth Green’s Building a Better Teacher: How Teaching Works (and How to Teach It to Everyone) (New York: Norton, 2014). I like it much better than I thought I would. I was initially thrown off by the title: I imagined teachers being “built” like Lego models by outside tinkerers who claimed to have “the answer.” The book is about nothing of the sort. It has a searching quality, which is a refreshing change from the packaged solutions that so often get bandied about.

But I read slowly and am preparing for the school year–so, instead of commenting on the book as a whole, I thought I would comment initially on the second chapter, which has been on my mind. First, though, I will explain why I am interested in the topic of math instruction.

I come to this as a complete layperson; I have never taught math, except as a tutor and as a summer intern. I have taken no math education courses, and am unfamiliar with the names in the field. (I actually was unaware of the work of Magdalene Lampert and Deborah Ball until very recently.) Yet I have strong opinions about this stuff, since my experience with math education is somewhat unusual.

Both of my parents (now retired) are math professors, so I grew up surrounded by math, math teachers, math students. When I was eight, my mother organized a symmetry festival that brought together mathematicians, scientists, artists, dancers, musicians, poets, and others. My father would give me complicated math problems to solve–completely in my head–on long road trips. He also showed me rudimentary computer programming (Fortran, I believe), with those big cards that would get fed into the machine. From their dual influences I learned that (a) math was beautiful and could be found anywhere; (b) math was easy to get wrong; and (c) math could occupy the mind for hours on end.

My education was unusual as well. I attended public and private schools in the U.S. and abroad and witnessed an array of curricula and pedagogical approaches. (I was quite aware of what I saw; even in childhood, I often critiqued what I saw in the classroom.) I was also very independent about my learning; I did my homework on my own (and did not show it to my parents), and from age 12 to 17, except for one year that we spent in Moscow, I lived away from home in order to attend a day school in Boston. These are the schools I attended:

Kindergarten and first grade: The Common School (private), Amherst, Mass.
Third grade (I skipped second): Center School (public), South Hadley, Mass.
Fourth and fifth grades: Smith College Campus School (private), Northampton, Mass.
Sixth grade: Paterswolde-Noord Openbare Basisschool (public), Paterswolde, Drente, The Netherlands.
Seventh grade: South Hadley Junior High School (public), South Hadley, Mass.
Eighth and ninth grades: The Winsor School (private), Boston, Mass.
Beginning of tenth grade: South Berwick High School (public), South Berwick, Maine.
Tenth grade (or the equivalent): School no. 75 (public), Moscow, U.S.S.R.
Eleventh and twelfth grades: The Winsor School (private), Boston, Mass.
College and graduate school: Yale University, New Haven, Conn.

describe the opening of the Soviet math textbook on Joanne Jacobs’s blog.

The pedagogy at the Common School and Smith College Campus School was decidedly progressive (in terms of encouraging creativity and exploration). In some ways, this was great; I have vivid memories of making a Sarah Noble doll, making a hardcover book, learning change ringing, and much more. However, it was not there that I found myself intellectually challenged in math. My first intellectual challenge in math class was in the Netherlands, where we learned mental arithmetic. Although there were no projects and almost no student talk, I came to understand operations inside out by performing calculations rapidly in my head. (Yes, with problems of this sort, you do have to understand what you’re doing.) Later I found challenge in my math classes in Moscow and at the Winsor School. Both of these could be called “traditional”–but they involved a great deal of dialogue, puzzling through problems, diagnosing errors, and so on.

As a teacher, I have seen a variety of approaches to math instruction (and student responses). My overwhelming experience is that students are interested in math. At my first school, a high-poverty middle school in Brooklyn with a large immigrant population, I saw students eagerly discussing math problems during lunch, after school, and sometimes in my ESL class. The reasons were evident: math offered them, first of all, a universal language, and second, the satisfaction of an eventual solution. They tackled problems like conquerors. The teachers (many of them Russian) were well versed in their subject. Their pedagogy was straightforward (lecture combined with workshop), but they did subtle things within that. Later I taught at a Core Knowledge elementary school in an even poorer neighborhood (in East New York, Brooklyn). There, too, the students were excited about math. I saw some second-grade lessons in action–where the teachers combined direct instruction with questioning and experiential learning. At my current school, math is a favorite subject for many students; I have seen teaching approaches and styles that ranged from total lecture (that had the students intrigued and mentally involved) to dialogical teaching to an emphasis group work. Some teachers combined all of these.

In addition, throughout my life, I have enjoyed working on math problems for fun and enlightenment. (See here, here, and here.) I love logic problems and paradoxes, and enjoy figuring things out. At one point I taught myself Perl programming and worked for a year as a junior programmer at Macromedia.

When it comes to math education, I am not opposed to progressive approaches (a huge category in itself); I see much room for combination of the traditional (also a huge category) and the progressive. I protest when I see traditional methods dismissed offhand. That’s what I thought was going on in Elizabeth Green’s recent book excerpt in the New York Times. Her actual book shows a much subtler take, though. I stand by most of what I said in my response to the article–but would frame it a little differently now. More on that later.

In any case, I think this explains why I, a layperson in math, take interest in questions of math curriculum and instruction. Do my thoughts on the subject have merit? That I leave for others to judge.

Mourning: Together or Alone?

Over the past few weeks, I have been pondering two pieces: “Grief in the Digital Age” by Elise Italiano (Acculturated, August 1, 2014), and “The Problem with Collective Grief” by Arnon Grunberg (New York Times, June 21, 2014). I would not say that they contradict each other; they are on different tracks. Yet in combination they raise a question: are private and communal forms of mourning at odds with each other? (I separate mourning from grief; mourning includes ritual expressions of loss, whereas grief consists of the raw emotion.)

Elise Italiano explores how “status updates,” “selfies,” and other digital communications affect both private and communal grief—for instance, at Ground Zero, where one is surrounded by people sending tweets, talking on cell phones, and taking pictures of themselves. She finds this phenomenon profoundly isolating—as it separates people not only from each other, but also from solitude.

Arnon Grunberg describes the Dutch fervor over the downed Malaysian airplane (193 of those killed were Dutch). He perceives the calls for collective mourning as nationalist in essence and responds, “The sad thing about mourning is that it really is quite unshareable, that it involves an extremely individual emotion. People have the right not to show their emotions and not to share them, even when it comes to soccer and calamity.”

Both are right. Grief and mourning are highly personal, but there’s nothing intrusive about establishing a place or time for mourning. To the contrary: such places and times allow the private mourning its own stretch.

Take a place like Ground Zero. If cell phones and other digital devices were not permitted at all, then there would be fewer distractions—and both solitude and companionship would be possible in a way that they are not now. (Visitors are not supposed to make or receive cell phone calls inside the museum itself—but they are allowed to take pictures with their cell phones.) True, people would object to such a prohibition; many feel that they have the right to use their devices. But the loss of such a right would be outweighed by the increase of respect.

Something similar can be said for times of mourning. On one level, mourning cannot be timed. It comes when it comes, and goes when it goes. On the other, a person participating in ritual mourning need not display or force private emotion. The ritual mourning makes room for the private mourning, even if the two do not coincide.

Collective mourning can be constricting and oppressive when it lays claim to private emotions. But when it does not lay such claim, it dignifies the privacy. To mourn with others in a time and place—even if my mourning is out of sync with theirs—is to set aside the distractions and dishonorings, together, for a while.

Much of what we mourn is not recognized. I may mourn someone who is not a family member, or someone still alive but gone from my life, or something as seemingly mundane as a misunderstanding. All of these relate in some way to death, but they may not get a funeral, or I may not have an official place in them. The formal mourning makes a possibility for those (people and mournings) that have no place.

If I step into formal mourning, even clumsily, then I participate in something beyond my own impulsive sadness. I learn history; I temper my urges. If I accomplish this, the impulsive sadness takes its time and shape but also remembers others.

If I can mourn in an allotted room, on an allotted day, then I can carry such a room into other days, or such a day into other rooms.

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