CONTRARIWISE and the Humanities

CONTRARIWISE appears in a video by the Dallas Institute of Humanities and Culture! The video–about the future of the humanities–features interviews with three Hiett Prize winners: Mark Oppenheimer, James E. McWilliams, and myself. A lovely segment is devoted to CONTRARIWISE. There are also some glimpses of the Summer Institute in action. Thanks to the Dallas Institute and the producer, Judy Kelly, and congratulations to all involved!

“I Contemplate a Tree”

buber tree 2On Wednesday I took two of my classes across the street to Morningside Park in order to look at trees. We had been reading the tree passage in Martin Buber’s I and Thou, which begins with the declaration, “I contemplate a tree.” The speaker first accepts the tree as a picture, “a rigid pillar in a flood of light,” then feels its movement, then observes it as a species, then perceives it as an expression of physical and chemical laws, and then “dissolves” it into a number. “Throughout all of this,” he writes, ” “the tree remains my object and has its place and its time span, its kind and condition.”

Then comes a shift: “But it can also happen, if will and grace are joined, that as I contemplate the tree I am drawn into a relation, and the tree ceases to be an It. The power of exclusiveness has seized me.”

I told my students that we could not replicate what Buber described in the passage–that the sheer effort to replicate it would defeat  the purpose. I asked them nonetheless to pay attention to what happened.

It was imperfect, of course, because we had little time and had to stay together. One or two students moved a little apart from the group; others clustered together and moved close to the tree of their choice. I saw them fingering the needles, observing the crinkles of the bark, noticing a long worm on the ground.

I had to keep an eye on everyone and everything, so I could not focus on a tree–but as I looked around, I was struck by each tree’s insistent form. Some were bare and gnarled; others showered you with color. Some had leaves falling from them as we watched; others stood warm and firm with their needles and cones. Some had berries or nuts; others, nothing but trunk and branches. Yet these had more than appeared at first glance. You could follow the lines in their bark as though listening to a story.

For my students, too, this was imperfect. Street noises and other distractions made it difficult for them to focus. All the same, they appreciated taking a few minutes to look at a tree; it was something they didn’t get to do very often.

On the way back to school, I thought of taking a picture of one of the trees. It seemed to go against the spirit of our outing, so I didn’t. Later in the day, I returned and took a shot. This set off a stream of thoughts about the nature of pictures and other mementos.

When you take a picture of something, you are turning it into a possession of sorts–something you “have” and can pull out at will. In one discussion of Buber, a student spoke of the satisfaction of Polaroid cameras–of seeing that tangible object emerge from the camera soon after the photo is taken. So, in the taking of a photo, there is some wish or effort to possess what is not really yours–to claim what cannot be claimed, to hold what cannot be held.

Yet it is also possible, when taking or looking at a photograph, to see it as a hint of something else–not as an object or possession, but as a reminder of something not possessed or contained. (Much of the early controversy over religious icons had to do with these different ways of regarding a picture.)

There is still a third possibility: the photograph can be a work of art and can take on its own life and limitlessness. It is then no longer merely a representation of something else. Buber writes about the creation of art:

The form that confronts me I cannot experience nor describe; I can only actualize it. And yet I see it, radiant in the splendor of the confrontation, far more clearly than all clarity of the experienced world. Not as a thing among the “internal” things, not as a figment of the “imagination,” but as what is present. Tested for its objectivity, the form is not “there” at all; but what can equal its presence? And it is an actual relation; it acts on me as I act on it.

To “actualize” a form, as Buber describes, one must allow oneself the confrontation–yet this cannot happen through effort of will alone. Is there a way, then, to make it possible, or does it just happen? In other words, can Buber’s words be “applied” to life and to ethics, or are they for contemplation only?

I believe that they can be applied, if one defines “applied” cautiously. Buber’s words cannot in themselves take us to the You–but they can make us aware of our tendency to claim and circumscribe things. (Buber stresses that we cannot survive without the It–but that the It cannot involve our whole being.)

So I take a picture, but with slight regret. First, my picture is far from a work of art, so it does not exist at that level. Second, it reminds me of the outing but leaves out almost everything. Third, while on the outing I resisted taking the picture, but later I caved in–so the picture is both removal and compromise. Yet it is pretty: the branches, leaves, and texture, the sense of something more.

Whenever I take a picture, I have ambivalence of this kind; it is usually wound into a tight thought, but it is present all the same. Here, the thought unravels. To “apply” Buber, then, is not to encounter a tree fully, nor to stop taking pictures, but to come closer to knowing one’s intentions.

The Privacy of Teaching and the So-Called Status Quo

Today few people think of teaching in terms of the private thought it involves. They the very idea of privacy with distrust. Teachers’ work should be open to all observers at all times, according to the general sentiment; teachers should not object to having visitors walk in and out, having video cameras installed in classrooms, and so on. Yet even if we did all of these things–made the classroom a continual open house with the camera running–an aspect of teaching would remain firmly private, simply because there is no audience for it. Within this privacy, the teacher and the teaching may be going through great changes, yet on the surface, and in the judgment of most, they remain part of the “status quo.” The conception of the “status quo” is flawed in that it mistakes a superficial reality for the whole.

After any lesson, my mind streams with thoughts: was this a good way to present Kant? Did certain passages deserve more attention? What do I make of a particular student’s comments? How will I adjust tomorrow’s lesson?  Most important of all: how can I prepare my lessons with full mind and spirit, making the most of my intellect and judgment, but bringing out the students’ ideas? Some of these thoughts come up in conversation with others, but most do not. They do not fit into regular conversation, faculty or team meetings, education policy discussion, or anywhere else. They may get translated now and then into generic terms (student-centered teaching, teacher-centered teaching, etc.), but those terms are limiting and misleading. The important internal deliberation–over subject matter and the minute events of the day–resist facile terminology and quick summation.

There are also numerous situations where a teacher is torn between two goods and must privately make a decision, as it is impossible to consult someone about each of them. For example: we all want to give our students more resources. The Stanford-based talk show Philosophy Talk has a great website–with lots of informed and enjoyable discussions. Recently one of the show’s hosts posted a piece on the philosophy of humor. Good light reading material, except that it begins with a joke about a skeleton walking into a bar. “X walks into a bar” is a standard joke opening (and this joke is innocent enough), but all the same, mentioning a bar is an unspoken no-no, or at best an iffy matter, in K-12 teaching. So, a teacher might well decide, “Interesting post, but not for distribution.” In a given week, a teacher may have a dozen minor dilemmas of this sort. She will usually take the safer option, but not without questioning and occasional regret.

That in itself raises larger questions: How do I, as a teacher, present my subject matter in a way that is safe but not sterile? How do I show what it means to live without fear in the world–while taking all appropriate caution for my students’ sake? This leads to another great area of privacy: the teacher’s own life. A teacher can neglect her life for a while–many do, under the work pressure–but cannot keep that up indefinitely and still teach well. A teacher must have room and time to be with friends, form relationships, pursue interests, help others, clean the apartment, eat, exercise, read, and think. Those things do not come up in the classroom, yet they influence a teacher’s actions and bearing. A teacher who lives fully will show that fullness without divulging it. The students will pick up on that life. Similarly, students pick up on strain and trouble. Beyond that, a teacher does not live for the students or for teaching alone; a life has its own meaning and dignity.

Within each of these privacies, teachers and teaching can undergo great changes, often against a backdrop of a “status quo.” This year, I have been able to revise some of my lessons in ways that were not possible before; previously my energy was going into the rush and churn of each day. Because my teaching load is manageable now, and because I am teaching the Ethics course for the third consecutive year, I can refine it and make it more responsive to the students, without abandoning its substance. This is a source of joy, and I am grateful for the opportunity. Yet an outsider might look at the situation and perceive “status quo.” There are policymakers who believe in switching teachers around every few years so that they never teach the same subject or grade for very long. On the surface, such policy promotes change–but it prevents or ignores transformation. Transformation may happen slowly and may be difficult to perceive. (For more on this topic, you may read the talk I gave at the 2013 Annual Meeting of the National Association of Schools of Art and Design.)

This is part of the reason why I blog less frequently lately. My emphasis is changing. By definition, the private truths and struggles of teaching have no place in the regular discourse; unfortunately, the discourse disparages the very privacy. I cannot live without the privacy, yet I also yearn for a forum where I do not have to be quite so enclosed, where there’s more acceptance of internal life and its role in everything. Some of these thoughts will find their way into my second book, which is not autobiographical or primarily about education. (I will say more about it when the time is right.)

(Note: I made some minor edits to this piece after posting it.)

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.”

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.


Get every new post delivered to your Inbox.

Join 157 other followers