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.

“But that was the thing that I was born for.”

marlinWhen I taught English as a Second Language at a middle school in Brooklyn (from 2005 to 2008), I had my students read The Old Man and the Sea, which they adored. One of our liveliest debates was about whether the old man enjoyed being alone; they found that a single textual passage could serve as evidence for either side. Moreover, they found it possible that he could like being alone and not like it at the same time.

For a side project, I had students select and illustrate a favorite quote. This illustration (pictured here) moved me; the student told me I could to keep it. The quote reads, “Perhaps I should not have been a fisherman, he thought. But that was the thing that I was born for.” Here, in the drawing, you see the skeleton of the marlin against a desolate beach, with driftwood and a restaurant table and chair. The scene looks desolate and broken, but there’s something grand about it too: the marlin’s skeleton looms much larger than the tiny furniture; there’s something here beyond what humans know and see. Another interesting thing here is the juxtaposition: the quote occurs well before this near-final scene. (The final scene, if one can call it that, is of the old man dreaming about the lions as the boy watches him.)

As I looked at this picture again, I began thinking about my students’ work over the years. They have made some remarkable things. I mention here the few that have links.

There was my students’ production of The Wizard of Oz in 2006.

One student wrote a terse, gorgeous poem that I quoted in full (with her permission and her mother’s) in my book, Republic of Noise.

When I began teaching philosophy at Columbia Secondary School, I found myself learning from (and sometimes roaring over) my students’ work. One line I recall often: “What have we here? It appears that I have arrived at exactly the perfect time. For the perfect time is always now.” (Context: the hermit from Tolstoy’s story “Three Questions” walks into a scene based on Gogol’s story “The Nose”; Epictetus and Erasmus’s Folly are also involved.)

Most recently, as readers of this blog know, my students created a philosophy journal, CONTRARIWISE, and had a great celebration in May. We look forward to an exciting second issue; in early fall, the editors-in-chief will call for submissions and announce contests.

These are all published things, known things, or soon-to-be-revealed things. Much more happens every day–in discussions, on homework assignments, on tests–that goes back into the mind, where it becomes part of other shapes and thoughts.

Why does the approaching new year bring up memories? I think a new year has a way of doing that–especially when it comes at this time of year. I remember my teachers too.

 

Note: I made an addition to this piece after posting it.

Don’t Criticize–Retire!

One of the most disturbing traits of our era is what I would call “age nationalism”–a belief that if you do not support the more recent innovations, whatever they may be, you are out of step with the times and should go away.

In The Chronicle of Higher Education, Steven Conn, a professor at Ohio State University, criticizes colleges’ current tendency to hold students’ hands and tell them exactly what they need to do to get that A. He cites the “writing rubric” and the endless “learning objectives” as examples of this trend.

I support his viewpoint. The pros and cons of rubrics aside, I was struck by the snide tone of many of the comments on Conn’s article. Their attitude was, “Rubrics are what we do now, and if you don’t like it, you shouldn’t be teaching.”

Here’s a quote from one such comment:

It’s not exactly clear why he went into teaching. –Sounds more like he wanted to get paid for reading his favorite books and discussing them with students who can process those books unassisted. The (educationally) rich just get richer.

Dear me. So a professor who expects students to come to class prepared–who expects them to be able to read and write and study–must be elitist and spoiled?

Here’s another comment (quoted in full):

I found myself, by the end of the article, hoping you would retire soon from teaching.

A rubric sets guidelines and documents expectations. It’s not an “outline” nor is it there to promote grade inflation. What you confuse as helicopter teaching is sound practices. A rubric provides the student with an assurance that you are organized.

If you were employed outside your safe ivory tower, and in the real world, you would see that the rubric you so disdainfully snub as making soft students is really management by objectives (MBO). It’s how people retain their employment.

What is this? A professor who doesn’t think in business terms (e.g., “management by objectives,” or MBO) is supposed to retire from teaching? Who will question the jargon, then? Apparently no one–for in this person’s view, the “real world,” or his version of it, has the final say.

There are many more in a similar spirit–and others that are more courteous, and still others that corroborate the author’s points. But what stands out is these commenters’ insistence that someone who questions the current trend should not be teaching at all. The reasoning, apparently, is as follows: “Teaching is X; Professor Conn does not seem to exemplify X; therefore, Professor Conn should not be teaching.” They do not stop to ask whether teaching really is uniformly X, and whether they can judge, on the basis of an op-ed, whether or not Professor Conn exemplifies X.

Long before rubrics entered higher education, there was a difference between small liberal arts colleges, which prided themselves on their nurturing atmosphere, and large universities, which emphasized scholarship. Many institutions sought and found a middle ground: a research institution with support systems for the students, or a college that fostered outstanding research.

When I was a high school student considering colleges, I wanted anything but a college that would coddle me. I applied to two universities, early action (Harvard and Yale); got into both; and chose Yale on the basis of visits, course syllabi, conversations, and instinct. I stumbled at various points in college–but that was part of growing up, intellectually and emotionally. Those were not grade-crazy days; getting a C on a college paper was considered a worthwhile experience.

Today we hear a lot about “grit” and the “importance of failure”–but students also hear that a B in high school–or any kind of lopsidedness–will limit their college prospects. They are told to take risks, but–as a recent fifth-grade test passage put it–to learn to be “smart” risk-takers, weighing the pros and cons of the risk in advance. One can try to avoid senseless, ill-conceived risks–but there’s really no such thing as smart risk-taking. It’s a contradiction in terms. A true risk involves the unknown, sometimes a lot of it. I remember, about 18 years ago, when a friend was going to Bulgaria for the summer. At a sendoff dinner, someone asked him what he hoped to get out of his trip. He replied, “I don’t know. That’s why I’m going.”

One has to have seen a different era to recognize that many students today are afraid of being on their own, afraid of anything less than an A, afraid of not knowing exactly what is expected of them. Not everyone is afraid; I see some students forge ahead with less concern about their grades than about what they learn. But they come under continual pressure to think and act on others’ terms.

The comments quoted above show hostility to intellectual independence–both that of the professor, who is putting forth a legitimate view, and that of the students. I do not mean that that any objection to his view is hostile. It is possible to defend rubrics without telling him to retire, or without insinuating that he is out of touch with the “real world” and clinging to some fading dream.

The “real world” is not what any particular group decides it is. It is continually tested and approximated. Those who put forth unpopular views, or who question current trends, are themselves affecting the real world by stretching the bounds of the possible.

 

Note: I previously referred to the professor as Steve Conn–but see that his byline is actually Steven Conn. The error is now fixed.

Homophonia and Parekbasiphobia

This is now a well-known story (and no hoax): Blogger Tim Torkildson was fired from his position at Nomen Global Language Center, Utah’s largest private English as a Second Language school, for posting a piece about homophones on the company’s website.

Homophones are words with like sounds and different meanings, such as where/wear, or/oar, and pair/pear. They may have the same spelling (for instance, rose/rose).

A post about homophones is entirely appropriate for the website of an ESL school. But Clarke Woodger, Nomen owner and boss, told the Salt Lake City Tribune that “people at this level of English … may see the ‘homo’ side and think it has something to do with gay sex.”

Well, and so what if they did? They are learning English, correct? They would soon learn what “homophones” actually were. In addition, they could learn the meaning of the prefix “homo-.” Part of the point of learning a language is learning what words and their parts actually mean–not staying stuck in what you think they mean.

If you avoid the very sounds of words because of their possible associations, you will end up in a verbal noose. But that’s only part of the story. Woodger’s greater concern–as reported to Torkildson and to the Salt Lake City Tribune–was that Torkildson was going off on too many tangents in his posts, and that he therefore couldn’t be trusted. This post on homophones–a wild digression, in Woodger’s view–was “the last straw.”

If we look at this story in terms of a fear of tangents and digressions (which I will call parekbasiphobia, as parekbasis is Greek for digression), then Woodger’s complaint is typical of a larger tendency in education.

Since my entry into public school teaching in 2005, I have seen widespread distrust of digressions. Teachers themselves understand the value of digressions–allowing a conversation to take an unexpected direction for the sake of larger understanding, or even for sheer fun. But policymakers and teacher trainers see it otherwise: to many of them, if you stray from the point for even a few seconds, you are wasting precious instructional time. You may be robbing children of the opportunity to meet the stated objective and thereby to achieve measurable progress.

One of the first “inservice trainings” I attended included a presentation about sticking to the point. “We want our lessons to go straight to the objective,” the presenter said, “not where our own imagination takes them. We want to be like this”–here she made a gesture of straight motion–“and not like this” (a gesture of a zigzag).

One of my greatest teachers, the poet John Hollander, showed us in lecture after lecture, seminar after seminar, what digressions could do. There was no imitating him–in no way could his teaching be a “model”–but I would not trade a single one of his lectures for something that stuck strictly to the point. For Hollander, the point itself was multifaceted; to understand it, one needed to take excursions into etymology, history, architecture, music, and more.

Now, how do I reconcile a defense of digression with my insistence that focus is essential for learning? On the surface, it seems that these two principles contradict each other, but they do not. There is a big difference between digression and all-out distraction. If one is attentive to the topic at hand, one can move this way and that within it. How and when one does so will depend largely on the situation. Not all digressions are helpful, but some may open up insights into the lesson’s central questions. You can miss the point by sticking too rigidly to the point.

By contrast, what doesn’t count as focus is a willful inattention to a lesson or topic–a preoccupation with one’s iPhone, or with the latest social gossip, or with the homework for the next class. Now, some would argue that such “distractions” should be made part of the lesson–that instead of battling them, teachers should welcome them and search for their inner meaning. On the whole, I disagree. There is a simple practice of setting aside one’s own immediate preoccupations for the sake of something else. If students (and teachers and schools) do not develop this discipline, they will be at the mercy of their urges and impulses.

But once the general focus is established, there’s room for a great deal of adventure. Just how much, and when–that’s a matter of judgment, and judgment is at the center of a teacher’s practice. Take away judgment, and you take it all away.

In fretting over Torkildson’s “tangents,” Woodger may seem ridiculous–but he represents a current of our time.

The School of Deep Understanding

Teresa Stanbury used to be a Common Core skeptic—until she stepped into a Common Core math classroom where deep learning was taking place. What she saw, struck her into Core dumbfoundedness.

The teacher, Gideon Pelous, buzzed about the room like a shimmering dragonfly while the children—second-graders from the deep inner city—discussed the essence of numerals in small groups.

Before the Core, students would be taught that two plus two equals four, but they would never know why. They would go through their lives not knowing how to explain this basic mystery. Now things were entirely different. The moribund learning of the ossified past had been exhumed and cast away.

“I just had a realization,” said Shelly Thomas, arranging four rectangular blocks in front of her. “I used to think that numerals were quantities. I was trying to figure out what the curve on the 2 meant, and what the double curve on the 3 meant. I even tried measuring them with my ruler. Then I had the insight that numerals aren’t quantities, but rather symbols that represent quantities.”

“You mean to say—“ sputtered Enrique Alarcón as he seized a crayon.

“Yep,” she continued. “This 1 here represents a unit of something. It can be a unit of anything. Now, when we say ‘unit,’ we have to be careful. That’s another thought that came to me, but I haven’t figured it–.”

“I have,” interrupted Stephanie Zill, banging on her Curious George lunchbox. “We use the word ‘unit’ in both a contextual and an absolute sense. That is, a unit is unchanging within the context of a problem, but it may change from problem to problem. Also, certain defined units, such as minutes and yards, have a predefined size that doesn’t change from one context to the next—until you consider relativity, that is.”

“Oh, I get it,” said Enrique. “So, this numeral 1 represents one unit, which could be a unit of anything, but within a given problem, the word “unit” does not change referent unless we are dealing with more than one kind of unit at once. Hey, what color crayon should we use: magenta or seaweed?”

“Magenta,” said Shelly. “So, moving on with this problem, let’s say the numeral 1 represents one of these blocks. The numeral two represents two blocks.” She set two blocks aside to emphasize her point.

“Fair enough,” answered Stephanie. “But how do you get from there to 2 + 2 = 4?”

“OK,” Shelly resumed, swinging her braids. “So, you have these two blocks, and you want to add another two blocks to them. But two blocks, you see, is actually two of one block. So when you add two blocks, you’re actually adding one block twice. Now if you put twenty single blocks together, you get twenty blocks, which isn’t the same as two blocks, but it can be, if you divide those twenty blocks into ten groups of two each. Just try it and you’ll see what I mean. But here we want four blocks, not twenty, so that means that instead of dividing the pile into ten groups of two each, we should divide it into five groups of four each. So we do that. Then we take one of those groups of four and line up the blocks, like this. Then we take our original two blocks and match them up to these four blocks. It turns out that we can do so twice. This means that we are taking two blocks and then two blocks again, which is the same as adding two plus two, and this turns out to be four, which once again, or maybe for the first time, because this is all super-new, is represented by the numeral 4.”

Stephanie and Enrique nodded, rapt. “That was deep,” said Enrique.

“Deep understanding,” Stephanie agreed.

“That’s just the beginning,” said Shelly. “In the old days, we would have left it like that and gone back to dealing with abstract representations of quantities. But thanks to the Common Core, we get to apply this equation to numerous real-life situations. So, say you have a pair of socks and another pair of socks. How many socks do you have?”

“Two pairs,” said Orlando, who had just wiggled his way over from another group that was taking too long to arrive at insights.

“Two pairs, but how many individual socks?”

“They aren’t individual. They’re pairs.”

“But let’s pretend that they’re still individual, even as pairs.”

“Does it matter if they don’t match?”

“No.”

“Wait,” interjected Stephanie. “I thought you said the units were supposed to be identical.”

“This leads us to question what identity really is,” rejoined Shelly. “Any object in the physical world has a set of attributes. If you consider only certain attributes, such as general shape and purpose, this object may be identical to other objects that otherwise don’t resemble it. However, if you focus on the attributes that differ, they you find yourself confronted with unalike and incomparable objects.”

“I see,” Orlando sighed. “So, if we’re just considering the sockiness of the sock—that is, the property that makes something a sock and not some other object—then we have four such socks.”

“That’s more or less on the right track,” said Shelly. “There are some subtleties that need to be taken into account, but since group time is up, we’ll have to leave that until tomorrow.”

Mr. Pelous called the class back to attention. “Mathematicians, what did we learn in our groups today about two plus two?”

“It equals four!” the students cried.

“Yes, and why?”

The room erupted in voices—all saying different things. Suddenly Orlando began waving his hand frantically.

“Orlando, do you have something to tell us about why 2 + 2 = 4?”

“Yes—if it didn’t equal four, then life would be absurd, or at least very, very strange!”

“And who’s to say it isn’t?” shouted a student from the corner.

“It can’t be that strange, or we wouldn’t be trying to explain it through math,” Orlando said. The bell rang.

Teresa Stanbury thanked Mr. Pelous and wandered dreamily out of the school, marveling at the Common Core and the wonders it had wrought.

The Elephant in the Reform

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

We know that t = t1 + t2.

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

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

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

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

Thus, 1/r = 1/50 – 1/90.

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

Thus, r = 450/4 = 112.5 mph.

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

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

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

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

 

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

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

The Forum (BBC World Service): Panel on Solitude

In April I took part in a panel discussion on solitude, along with authors Eleanor Catton and Yiyun Li and host Bridget Kendall, on BBC World Service’s program The Forum. (Update: I thought the podcast was going to expire on July 28, but it appears that it will be up for another year.)

Also, you may be interested Melvyn Bragg’s recent discussion on the philosophy of solitude, also on BBC. (Because my streaming is spotty, I haven’t listened to it yet, but I hope to do so soon. It looks promising.)

Finally, you may enjoy the recently posted samples from my students’ philosophy journal, CONTRARIWISE.

The Blessing of Slight Neglect

Almost a year ago, I commented on William Faulkner’s Nobel speech. I focused on how the sprawl of our lives–the pressure to be available around the clock, the leveling and spreading of our intentions–tends to break down our sense of sanctity (broadly defined),* and how, without a sense of sanctity, we lose touch with what he calls the “old verities.” Today I would like to comment on a different aspect of his speech: the “problems of the human heart in conflict with itself.” I begin, though, with a change of direction. Last week, I started a post along these lines and ended up dissatisfied with it. I realized that there was great danger in implementing Faulkner’s words in a literal way.

From Faulkner’s speech:

Our tragedy today is a general and universal physical fear so long sustained by now that we can even bear it. There are no longer problems of the spirit. There is only the question: When will I be blown up? Because of this, the young man or woman writing today has forgotten the problems of the human heart in conflict with itself which alone can make good writing because only that is worth writing about, worth the agony and the sweat.

One could nod vigorously and say, yes, we have forgotten the problems of the human heart in conflict with itself, and that’s part of the problem in education today. We look at social, political, economic problems–but not the problems each of us carries in our private minds: problems of love, loyalty, hate, betrayal, and their limitless combinations. Therefore, it seems that we should turn our attention again to these problems.

Yes, but how? The worst thing–and the thing likeliest to happen–if someone were to present this argument to education policymakers–would be for schools to mandate daily attention to the “old verities.” Teachers would be required to write an “old verity” on the board every day. When reading a work of literature, students would have to identify the “old verity” that it addressed. This is deadly and counter to Faulkner’s meaning–for he is speaking of fiction and poetry, not of dogma. (The links in the previous sentence point to the etymology of these words, which is interesting in terms of the “kneading” and “piling.”)

Something would likewise be forced and false about addressing “old verities” through so-called “informational text” (heavily touted in schools, even in English class). It cannot be done. Philosophy and history can tackle the central human problems–but to do so, they cannot rely on abstractions and information alone; they need insight and form as well. Insight and form belong to fiction and poetry, which in turn rely on a certain concealment, or a complex kind of revelation. That is, to see truths of this kind, you must also have room in your imagination for the unseen.

A passage from José Ortega y Gasset’s Meditations on Quixote sheds some light on this. (Note: These texts are among the shorter readings in the Epic course at the Dallas Institute’s Sue Rose Summer Institute for Teachers, where I am currently teaching. The past two weeks have been filled with the Iliad, the Odyssey, the Aeneid, the Divine Comedy, and other works–all of these read in their entirety–and we will finish next week with Moby-Dick.) Here’s Ortega y Gasset:

There are things which, when revealed openly, succumb or lose their value and, on the other hand, reach their fullness when they are hidden or overlooked. Some men refuse to recognize the depth of something because they demand that the profound should manifest itself in the same way as the superficial. Not accepting the fact that there may be several kinds of clarity, they pay exclusive attention to the clarity peculiar to surfaces. They do not realize that to be hidden beneath the surface, merely appearing through it, throbbing underneath it, is essential to depth.

Maybe there’s a blessing in having Faulkner’s “old verities,” the problems of the human heart, overlooked in schools. Maybe a certain kind of overlooking is the best kind of honor. I think we can honor them through curriculum and general bearing, without pounding themes and messages into the students’ heads. Give students room to hear the works, to wade through them, to feel their pull and counter-pull. Show, through selection, intonation and gesture, that something worthwhile is there.

I think of these lines from the beginning of James Merrill’s The Changing Light at Sandover:

The more I struggled to be plain, the more
Mannerism hobbled me. What for?
Since it had never truly fit, why wear
The shoe of prose? In verse the feet went bare.

One can go barefoot as a reader, too–and this means reading and interpreting in an unfettered way. Yes, one analyzes what one reads, often in great detail and depth, but one does not try to map everything onto a specific external meaning, method, or theory. One allows the literature its life, not all of which can be explained in external or technical terms. (Some of the most inspiring criticism is fiction of its own kind.) When one does this, when one enters literature with heart–in the Hebrew sense of “lev,” not our current sentimental sense–one will confront those verities willy-nilly.

There is a focus and clarity that comes from not fretting over what we are going to get from a given thing. Unfortunately, schools have been trained into a “customer service” mode. They are supposed to deliver a product to the students–who, for their part, are supposed to expect one. There is partial good to this; one does want students to learn and do concrete things. But one can accomplish this with recognition that it is not the whole.

Beyond this, I have started to think that certain kinds of “neglect”–not extreme or irresponsible kinds, but the kinds that let things hide just a little–may hold more good than we know. In the same way that a poem or essay revises itself when one steps away from it, so we, too, may take shape when others are not looking. We get to putter around and think things through. The neglect must be slight, though, and not self-justified. A world shrivels when it asserts that the things it ignores don’t exist.

 

*Sanctity: the quality or state of being holy, very important, or valuable (Merriam-Webster); I would add: the quality or state of being set apart from other things. (In the earlier post on Faulkner’s speech, I didn’t use the word at all; rather, I used terms that conveyed a similar meaning.)

Note: I made some edits to this piece after its initial posting.

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