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 Deep Problem with the School of One

This morning, Rachel Monahan reported in the New York Daily News that two of the three New York City schools that piloted the “School of One” decided to drop the program. After a great deal of expenditure and hype, the School of One didn’t show better results on the state math tests than regular math classes.

I am not surprised by this report. The School of One (which I discuss in the eighth chapter of my book) assumes that mathematics consists of a progression of skills. Its proprietary software program generates a daily “playlist” for each student and lesson plans for the teachers. Students enter the classroom, view their playlist, and go to their appointed station. On a given day, a student might play a video game, work in a small group, receive direct instruction from a teacher, or engage in some combination of these activities. Teachers might spend fifteen minutes with one group, a few minutes here and there with individuals, and another fifteen minutes with another group. The students take frequent multiple-choice quizzes, which help to determine their activities and grouping. Supposedly, by working at their own pace in their own preferred style, students will make great progress.

But mathematics is not an amusement park. It is about recognizing patterns and seeing problems in more than one way. It requires imagination as well as precision. In the best math classes, students learn to struggle with problems that at first seem daunting (but for which they are adequately prepared). They try this and that, seeming to get nowhere, and then suddenly they see it. In a flash, it is all clear—and the solution sheds light on problems from earlier lessons and problems still to come.

Students cannot rely on such flashes of insight, of course. After solving a difficult problem, they must practice solving similar problems until they come easily. Then they continue on to the next challenge, which often arises out of the problems they have solved. A good math curriculum has a clear, logical progression but also moves back and forth and outward. Over time, as students advance and gain knowledge and experience, they develop what Alfred North Whitehead called “that eye for the whole chess-board, for the bearing of one set of ideas on another.”

Personalized, computerized instruction doesn’t do justice to such a curriculum; even precocious students need guidance through the challenges. It is the teacher who knows how to pose a problem in different ways and to draw more than the obvious conclusions from it. It is the teacher who can glean where a student is going wrong and guide him back on track. Such teaching takes time. A class can easily spend an entire lesson on a single theorem or concept, and the students learn from each other’s efforts.

What happens when the lesson is fragmented, when students go off into their various groups and corners to play a game or work on an activity? Well, in many cases both the students and the mathematics itself are shortchanged. The students may make progress with problems of a basic sort (like those that appear in summer math workbooks) but will need the teacher for the trickier and subtler points. Also, flitting from activity to activity isn’t always helpful; mathematics requires focus and doggedness. (Yes, sometimes the solution comes to you after you walk away—but those hours of puzzling and pondering help to bring this about.)

So why has the the School of One enjoyed such hype? Not only are there powerful political and commercial entities behind it, but it appears to address a real problem. Today’s classrooms have a wide range of levels; the advanced and struggling students study together. Since tracking is not an option (especially at the elementary and middle school levels), the teacher is expected to accommodate all levels at once. Given that state of things, a personalized learning system (aided by software) sounds like a crystal palace of sorts. To some, it is the future.

But to paraphrase Fyodor Dostoevsky’s Underground Man, if it is raining and I crawl into a hen-house in order to stay dry, I will not call it a palace out of gratitude. It is still a hen-house. Something analogous holds true for the School of One. It is a makeshift solution, and an expensive one at that.

What can we do instead of expanding the School of One? We could adopt strong math curricula that give students a foundation in the early grades. We could allow for certain kinds of flexible tracking—so that, for instance, a fifth-grade student could take math  with sixth graders if she were prepared (but would take other classes with her fifth-grade classmates). We could have public lectures, seminars, and workshops on mathematics, so that parents, teachers, and others could grapple with math problems together. We could identify first-rate math textbooks, possibly translating a few from other languages, so that teachers did not have to scramble for appropriate resources. All of this would be far less expensive—and far truer to the purpose of teaching math—than the School of One.

Equals Sign Creates Jobs and Confusion

In their rush to implement Universal Design for Learning (UDL), a federally approved framework intended to maximize learning outcomes for all students, schools have hired gymnasts, carpenters, and political philosophers to provide multiple representations of the equals sign.

According to UDL, “An equals sign (=) might help some learners understand that the two sides of the equation need to be balanced, but might cause confusion to a student who does not understand what it means. … An important instructional strategy is to ensure that alternative representations are provided not only for accessibility, but for clarity and comprehensibility across all learners.”

“I thought the equals sign was pretty clear,” said John Knap, a high school mathematics teacher. “Not sure why we have to represent it in other ways. Yesterday two gymnasts came to my class to perform double flips, and the kids were supposed to grasp that the two routines were ‘equal.’ They weren’t equal. One was a little faster than the other. And of course the kids wanted to see more routines. We couldn’t get to the lesson.”

Kelly McEwen, a carpenter hired to provide alternative representations of the equals sign at elementary schools in San Diego, expressed misgivings over the project. “The pay’s great,” she said, “but I’m not sure I’m doing the right thing. I’m supposed to show them the spirit level and pretend it’s just like the equals sign. It isn’t just like the equals sign. I end up doing a lot of qualifying and explaining, and the teachers and kids get anxious. Plus, they’re waiting for me to take out the saw, which I never bring, for safety reasons.”

Political philosophers seem especially disgruntled with the project. “I was invited to come to Inspiration Academy to talk about political equality,” reported Andrew Ravny, author of numerous books on William Hazlitt and Thomas Jefferson. “I accepted gladly. When I arrived, I was told to draw stick figures and put a smiley face between them to show that they were equal. I did this and went on to say that two equals three in such a scheme, because both numbers have the same inherent dignity and rights. I don’t think I’ll be invited back.” He chuckled grimly. “Which is just as well, since I need to focus on my next book.”

We had the pleasure of interviewing the inventor of the equals sign, Robert Recorde, whom we heard stirring in his grave. We asked him why he had chosen to represent mathematical equality with two parallel lines. He replied that he did it “to auoide the tediouse repetition of these woordes : is equalle to.”

But why the two parallel lines? we asked.

“Bicause noe 2 thynges can be moare equalle,” was his reply.

We thought that Recorde would be pleased to learn that his equals sign was now inspiring multiple representations. As we told him about the reforms, we watched and listened closely for his reaction. But he replied in cryptic verse and then faded from our midst:

One thyng is nothyng, the prouerbe is,
Whiche in some cases doeth not misse.
Yet here by woorking with one thyng,
Soche knowledge doeth from one roote spryng,
That one thyng maie with right good skille,
Compare with all thyng: And you will
The practice learne, you shall sone see,
What thynges by one thyng knowen maie bee.*

“It’s a nice poem, but I’m not sure how it applies to classroom practice,” said Mercy Trout, director of instructional services in Boise, Idaho, who had accompanied us for the interview. “Is he saying kids should study math as math? Or is he saying all things are connected? What are the policy implications for school improvement?”

“Where are you, Recorde, and where’s the whetstone of witte when we need it?” cried another.

“I think he’s saying that if people do study math in a focused way, then they will see….” a third member of our party ventured. But it had grown dark and windy, and conversation turned to our flight back home and whether it would depart on time.

*The verse appears in Recorde’s preface to his Whetstone of Witte (1557).

Research Has Shown—Just What, Exactly?

In popular writing on psychology, science, and education, we often encounter the phrase “research has shown.” Beware of it. Even its milder cousin, “research suggests,” may sneak up and put magic juice in your eyes, so that, upon opening them, you fall in love with the first findings you see (until you catch on to the trick).

Research rarely “shows” much, for starters—especially research on that “giddy thing” known as humanity.* Users of the phrase “research has shown” often commit any of these distortions: (a) disregarding the flaws of the research; (b) misinterpreting it; (c) exaggerating its implications; or (d) cloaking it in vague language. Sometimes they do this without intent of distorting, but the distortions remain.

Let’s take an example that shows all these distortions. Certain teaching methodologies emphasize a combination of gesture, speech, and listening. While such a combination makes sense, it is taken to extremes by Whole Brain Teaching, a rapid call-and-response pedagogical method that employs teacher-class and student-student dialogue in alternation. At the teacher’s command, students turn to their partners and “teach” a concept, speaking about it and making gestures that the partner mimics exactly. Watch the lesson on Aristotle’s “Four Causes,” and you may end up dizzy and bewildered; why would anyone choose to teach Aristotle in such a loud and frenzied manner?

The research page of the Whole Brain Teaching website had little research to offer a few months ago. Now it points to a few sources, including a Scientific American article that, according to the WBT website,  describes “research supporting Whole Brain Teaching’s view that gestures are central to learning.” Here’s an instance of vague language (distortion d). Few would deny that gestures are helpful in teaching and learning. This does not mean that we should embrace compulsory, frenetic gesturing in the classroom, or that research supports it.

What does the Scientific American article say, in fact? There’s too much to take apart here, but this passage caught my eye: “Previous research has shown”—eek, research has shown!— “that students who are asked to gesture while talking about math problems are better at learning how to do them. This is true whether the students are told what gestures to make, or whether the gestures are spontaneous.” This looks like an instance of exaggerating the implications of research (distortion c); let’s take a look.

The word “told” in that passage links to the article “Making Children Gesture Brings Out Implicit Knowledge and Leads to Learning” by Sara C. Broaders, Susan Wagner Cook, Zachary Mitchell, and Susan Goldin-Meadow,  published in the Journal of Experimental Psychology, vol. 136, no. 4 (2007), pp. 539–550. The abstract states that children become better at solving math problems when told to make gestures (relevant to the problems) during the process. Specifically, “children who were unable to solve the math problems often added new and correct problem-solving strategies, expressed only in gesture, to their repertoires.” Apparently, this progress persisted: “when these children were given instruction on the math problems later, they were more likely to succeed on the problems than children told not to gesture.” So, wait a second here. They didn’t have a control group? Let’s look at the article itself.

The experimenters conducted two studies. The first one involved 106 children in late third and early fourth grade, whom the experimenters tested individually. For the baseline set, children were asked to solve six problems of the type 6 + 3 + 7 = ___ + 7, without being given any instructions on gesturing. Children who solved any of the problems correctly were eliminated from the study at the outset. (Doesn’t this bias the study considerably? Shouldn’t this be mentioned in the abstract?)

From there, the students were assigned to groups for the “manipulation phase” of the study. Thirty-three students were told to gesture; 35 were told to keep their hands still; and 38 were told to explain how they solved the problems. The students who were told to gesture added significantly more “strategies” to their manipulation than did the students in the other two groups; however, nearly all of these strategies were expressed in gesture only and not in speech. Across the groups, students added a mean number of 0.34 strategies to their repertoire, 0.25 of which were correct (the strategies, that is, not the solutions).

It is not clear how many students actually gave correct answers to the problems during the manipulation phase. The study does not provide this information.

The second study involved 70 students in late third and early fourth grade; none had participated in the first study. After conducting the baseline experiment (where no students solved the problems correctly), the researchers divided the students into two groups for the manipulation phase. Children in one group were told to gesture; children in the other group were told not to gesture. The researchers chose these two groups because they were “maximally distinct in terms of strategies added.” (How did they know this in advance? This is not clear.)

Again, the students who had been told to gesture added more strategies to their repertoire; those told not to gesture added none.  The researchers state later, in the “discussion” section of the paper: “Note that producing a correct strategy in gesture did not mean that the child solved the problems correctly. In fact, the children who expressed correct problem-solving strategies uniquely in gesture were, at that moment, not solving the problems correctly. But producing a correct strategy in gesture did seem to make the children more receptive to the later math lesson.”

After the children had solved and explained the problems in the manipulation phase, they were given a lesson on mathematical equivalence. (There was no such lesson in the first study.) The experimenter used a consistent gesture (moving a flat palm under the left side of the equation and then under the right side) for each of the problems presented. Then the students were given a post-test.

On the post-test, the students told not to gesture solved a mean of 2.2 problems correctly (out of six); those told to gesture solved a mean of 3.5 correctly. (I am estimating these figures from the bar graph.)

Why would anyone be impressed by the results? For some reason the researchers did not mention actual performance in the first study. In the second, it isn’t surprising that the students told not to gesture would fare worse on the test. A prohibition against gesturing could be highly distracting, as people tend to gesture naturally in one way or another. Again, there was no control group in the second study. Moreover, neither the overall mean performance on the test or the performance difference between the groups is particularly impressive, given that the problems all followed the same pattern and should have been easy for students who grasped the concept, provided they had their basic arithmetic down.

The researchers do not draw adequate attention to the two studies’ caveats or consider how these caveats might influence the conclusion (distortions a and b). In the “discussion” section of the paper, they state with confidence that “Children told to gesture were more likely to learn from instruction than were children told not to gesture.”

This is just one of myriad examples of research not showing what it claims to show or what others claim it shows. I have read research studies that gloss over their own gaps and weaknesses; popular articles that exaggerate the implications of this research; and practitioners who cite the  popular articles in support of their particular method. When I hear the phrase “research has shown,” I immediately suspect that it isn’t so.

*From Shakespeare’s Much Ado About Nothing; thanks to Jamie Lorentzen for reminding me of the phrase.

  • “To know that you can do better next time, unrecognizably better, and that there is no next time, and that it is a blessing there is not, there is a thought to be going on with.”

    —Samuel Beckett, Malone Dies

  • Always Different



    Diana Senechal is the author of Republic of Noise: The Loss of Solitude in Schools and Culture and the 2011 winner of the Hiett Prize in the Humanities, awarded by the Dallas Institute of Humanities and Culture. Her second book, Mind over Memes: Passive Listening, Toxic Talk, and Other Modern Language Follies, was published by Rowman & Littlefield in October 2018. In February 2022, Deep Vellum will publish her translation of Gyula Jenei's 2018 poetry collection Mindig Más.

    Since November 2017, she has been teaching English, American civilization, and British civilization at the Varga Katalin Gimnázium in Szolnok, Hungary. From 2011 to 2016, she helped shape and teach the philosophy program at Columbia Secondary School for Math, Science & Engineering in New York City. In 2014, she and her students founded the philosophy journal CONTRARIWISE, which now has international participation and readership. In 2020, at the Varga Katalin Gimnázium, she and her students released the first issue of the online literary journal Folyosó.


    On April 26, 2016, Diana Senechal delivered her talk "Take Away the Takeaway (Including This One)" at TEDx Upper West Side.

    Here is a video from the Dallas Institute's 2015 Education Forum.  Also see the video "Hiett Prize Winners Discuss the Future of the Humanities." 

    On April 19–21, 2014, Diana Senechal took part in a discussion of solitude on BBC World Service's programme The Forum.  

    On February 22, 2013, Diana Senechal was interviewed by Leah Wescott, editor-in-chief of The Cronk of Higher Education. Here is the podcast.


    All blog contents are copyright © Diana Senechal. Anything on this blog may be quoted with proper attribution. Comments are welcome.

    On this blog, Take Away the Takeaway, I discuss literature, music, education, and other things. Some of the pieces are satirical and assigned (for clarity) to the satire category.

    When I revise a piece substantially after posting it, I note this at the end. Minor corrections (e.g., of punctuation and spelling) may go unannounced.

    Speaking of imperfection, my other blog, Megfogalmazások, abounds with imperfect Hungarian.

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