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 t_{1 }designate the time taken (in hours) by the first half of the trip; t_{2, }the time taken by the second half, and t the total time.

So, we know that d/t_{1} = 90 mph for the first half. Thus, **t _{1}= d/90**.

We don’t know what d/t_{2} is for the second half, since we don’t know the train’s speed, or rate (r) for the second half. Thus, **t _{2} = d/r**.

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

We know that t = t_{1} + t_{2}.

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 t_{1} and t_{2} 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.*

## roma giudetti

/ July 31, 2014I love everything you write. My husband read me an excerpt from Green’s recent article and all I could think is that she doesn’t know much about actual teaching. I think reformers get focused too much on the process rather than the actual practice. Those who can teach, those who can’t reform education.

## Ze'ev Wurman

/ August 1, 2014Excellent review. And it goes directly to what I think was the biggest error of Green — the focus on “reform” techniques rather than on techniques appropriate to the subject matter at hand. Green also has a lot of factual inaccuracies about the evolution of the Japanese curriculum that she uses to prop her thesis, but that’s yet another story.

## Matthew

/ August 2, 2014always a pleasure to read your exegesis of things Diana.

“Putting our minds to the topic at hand;” often hard to do in this noisy world, as you have said quite eloquently. But either we do it or we don’t, and the results flow from there.

## andy

/ August 4, 2014Thanks for sharing this essay. I deeply appreciate your core point that the quality of the concentration students and teachers bring to a learning experience serves as the most important factor in the success of the experience. Crucial, obvious, and almost never discussed.

I understand that you’ve addressed elsewhere some of the factors that most affect concentration – give your book a plug! A first reader of this blog might be confused by the lacuna.

Your disparagement of Elizabeth Green’s analysis seemed too-quick, given the quotes that you helpfully provided. You wrote, “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.”

But Lampert’s innovation precisely addresses the overlap between the form of the lesson (which you disparage as “trappings”) and its “intent and sense”. She’s not arguing that a teacher lecturing COULDN’T push student thinking ever, she’s arguing that math lessons generally work better, for her most prioritized math education goals in those lessons, if done differently. She’s arguing that exactly because of the “intent and sense” which she’s bothered to actually think through. Her book’s also worth reading (http://www.amazon.com/Teaching-Problems-Magdalene-Lampert/dp/0300099479).

Appearance and essence often have much to do with each other.

## Diana Senechal

/ August 4, 2014Thank you for your comment. I appreciate your point about Lampert–but I was not commenting on her work itself. I was commenting on the false (and all too common) opposition between “traditional” and “new” methods, where the “traditional” supposedly promotes rote learning; the “new,” deeper understanding.

My argument is that we get a lot farther when thinking about actual subject matter than when putting forth a pedagogy such as “You, Y’all, We.” (Ironically, when I began teaching in NYC schools, the “new” method was “I, We, You.” THAT was supposed to promote understanding.) The pedagogy should form around the subject matter–and there’s no reason to assume it will be the same every day. Lecture and dialogue may be optimal for an introduction to a topic. After that, other methods and formats may be more suitable.

I see the “You” stage as something to be done at home. That’s what homework is for. It’s much more satisfying and enlightening to struggle with a problem when no one is around, when you can work with it in your mind. The buzz of the classroom makes this all but impossible, and group work restricts things still further. Yet I recognize that in the elementary grades, students have not developed these study habits yet. (Even by high school, many have not.)

A course based purely on “I” and “We” (in varying proportions) could go extremely well–and often does–not because it’s “the” preferred method, but because it works well for teachers who rejoice in presenting and discussing the subject, and for students who come prepared. This does not mean that it increases students’ understanding in and of itself. Students’ understanding of math will increase when they, along with teachers, spend time poring over problems and asking questions.

I agree with you that appearance and essence have a lot to do with each other. Still, to see their relation, one must go beyond the oppositions that pervade discussion today.

## Michael Paul Goldenberg

/ August 23, 2014Just wondering why you’ve chosen to refer to Magdalene Lampert as “the elementary school teacher Magdalene Lampert.” Surely you’re aware that Lampert has a Ph.D from Harvard and was, until recently, a professor at the University of Michigan. At the time she taught math to a class of fifth graders at Spartan Village Elementary School in East Lansing, she was a professor at Michigan State University.

So, just an oversight, or a conscious attempt to make Lampert seem quite unexceptional and to further call into question Green’s judgment? It’s worth noting that Deborah Ball is the dean of the University of Michigan School of Education, prior to which she, too, was a professor at Michigan State. And the school year (’89-’90) when Lampert taught math to a 5th grade class at Spartan Village Elementary, Ball was teaching math to a 3rd grade class there. Having watched, analyzed, and taught with the video of the “Shea numbers” lesson that you speak of somewhat disparagingly in another blog post, I have to suggest that your own lack of background with mathematics, its teaching, and its learning is evident in your failure to see more than it being focused on student errors/misconceptions. If you see evidence of third graders in the US having the sort of dialogues and discussions about mathematics that took place in that lesson with any degree of frequency, I’d be very much surprised.

## Diana Senechal

/ August 23, 2014It did not occur to me that the designation “elementary school teacher” was an insult, nor didI intend it as a comprehensive epithet,any more than I take “high school teacher” as a full description of me.

And no, I was not familiar with Lampert’s work, but I find some of the lesson descriptions in Green’s book highly problematic (though I like and admire other aspects of the book). I will have more to say about that later.

As for being qualified or not, this is my blog; beyond that, the last I checked, freedom of speech still existed.

## Diana Senechal

/ August 23, 2014In addition, I have not yet discussed the “Shea numbers” lesson in any blog post. (I may do so soon.) I think you’re referring to a comment I made on Tom Loveless’s recent piece. In that comment, I made clear that I had read only excerpts from the book and that my impressions were incomplete.

## Michael Paul Goldenberg

/ August 23, 2014But of course, I didn’t say or suggest it was an insult. What it seems to be (and you’ve not denied it) is a conscious understatement of Lampert’s academic background.

Lampert and Ball are two of the most respected mathematics/teacher educators in the country. Interesting that you didn’t know that or care to find out.

Freedom of speech? I don’t recall the vaguest suggestion on my part that it doesn’t exist or that one needs to be qualified to write a blog. Heck, Sandra Stotsky has no qualifications whatsoever to serve on various committees dealing with mathematics, but magically gets appointed to some of the most influential ones out there. Politics, of course, plays no role whatsoever in those appointments, any more than does Walton Foundation money influence the sorts of research coming out of the University of Arkansas.

However, when someone who appears to know very little about mathematics teaching and learning weighs in so definitively, should she be surprised when others who see much that she does not point out one very likely reason for her blindness?

## Diana Senechal

/ August 23, 2014My focus in this piece was not on Lampert’s work–and no, I had no intention of understating her credentials. Nor was I at all familiar with her work or Ball’s.

Nor did I make any bones about my own qualifications or lack thereof; I stated outright that I had never been a math teacher. I explicitly, overtly made clear that I was not an expert on this subject. I intended my ideas to be taken on their merit.

I have strong opinions about math instruction because I grew up surrounded by mathematicians and attended many schools (private and public, in the U.S. and abroad) with markedly different curricula and pedagogical approaches. I saw that a highly “traditional” approach (such as those I encountered in the Netherlands, the Soviet Union, and in Boston) actually encouraged a great deal of flexible thinking and questioning. There was dialogue between teacher and student; the problems were carefully selected, and teachers dealt skillfully with questions and errors. We were taught procedures, yes–but that did not prevent us from understanding what we were doing. Because of these experiences (that differed greatly from each other but had traits in common), I question the oppositions that Green puts forth in the article.

Moreover, i did not intend this as the last word. Although my tone may have seemed “definitive,” I question myself continually, on this subject and many others.

## Marge

/ August 25, 2014Michael Paul Goldenberg: Come on man, be fair, is it really unreasonable to conclude you’re accusing Senechal of insulting Lampert? Trying to “make Lampert seem quite unexceptional” is insulting her. I’m not in your world–I’m a math-challenged mom looking for intel on math education. (My kid’s school is considering jettisoning Everyday Math, and I want to ask good questions.) I have never heard of any of these people. Anyway, fwiw, to this reader (enough caveats?) you come across as unhelpfully argumentative.

## Diana Senechal

/ August 25, 2014Thank you for your comments, Marge.