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.

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.

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.

What Is Joy, and What Is Joy in Learning?

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

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

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

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

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

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

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

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

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

 

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

School Shocked by Non-Team-Playing Résumé

Lanham, MD—Last Saturday, nearly all of the teachers at Fernwood High school bustled around the building making photocopies, preparing lessons, or interviewing candidates for the many open positions. One applicant’s résumé became the subject of hallway gossip, frenzied tweets, and Facebook posts.

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

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

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

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

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

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

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

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

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

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

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

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

“Edification?

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

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

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

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

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

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

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

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

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

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

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

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

Daydreams, Lectures, and Helices

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Note: I corrected one term after the initial posting.

Squaring the Circle

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

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

 

 

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

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

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

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

the Love that moves the sun and the other stars.

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

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

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

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

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

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

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

You could consider the ratio in terms of this figure:

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

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

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

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

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

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

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

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

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