**I.**

*A Revolution in One Classroom: The Case of Mrs. Oublier *(link) is an oft-cited piece of education research by David K. Cohen. It’s a case study of just a single teacher (Mrs. O) and her math teaching, at a time (the ’80s) when California lawmakers sought to radically transform math teaching in the state.

Mrs. Oublier is a pseudonym, *oublier* meaning “forgotten” in French. She’s earned this pseudonym for thinking her teaching had undergone a revolution, though in the eyes of Cohen she hardly changed any of the important stuff. I guess the point is that she *oublier*-ed to make these changes? Or that reformers didn’t help her make them?

Anyway, a lot of the fun of the piece is seeing the funhouse-mirror ways in which Mrs. O interprets those cutting-edge ideas about manipulatives, small group work, and estimation. And Cohen has serious things to say about why policy-makers never quite reached Mrs. O in the way they intended to, though I might question some of his conclusions.

Another thing that’s interesting about this piece is what it’s not: a representative sample from the teaching population. It’s the story of one teacher. Cohen tells us that Mrs. O’s story matters, but why should we believe him?

There’s no denying that Cohen tells a good story. But isn’t research supposed to be more than a good story?

**II. **

Mrs. O has been teaching second grade math for four years. The kids like her; colleagues like her; administrators think she’s doing a great job.

As a student, Mrs. O hadn’t liked math much, and she didn’t do too well in school. When she got to college, though, she started doing better. What changed? “I found that if I just didn’t ask so many why’s about things that it all started fitting into place,” she tells Cohen. So, that’s not a great start.

And yet, Mrs. O tells Cohen that she’s interested in helping her students really understand math. She also tells him that she’s experienced a real revolution in her teaching, a departure from the traditional, worksheet+drill methods she used when she began. On the basis of his observations, Cohen is strongly inclined to agree with her on this.

In the centerpiece episode, Cohen catches Oublier in the midst of a fairly ridiculous lesson. Oublier wants to teach her students about place value (so far so good). To do this, she wants to introduce another base system (debatable, but not necessarily a disaster). So Oublier gives each kid a cup of beans and a half-white/half-blue board.

Mrs. O had “place value boards” given to each student. She held her board up [eight by eleven, roughly, one half blue and the other white], and said: “We call this a place value board. What do you notice about it?”

Cathy Jones, who turned out to be a steady infielder on Mrs. O’s team, said: “There’s a smiling face at the top.”

On a personal note, I have been teaching 3rd and 4th Graders for four years and the idea of giving kids those little cups of beans gives me minor terrors. What if the cups spills? How early do you have to get to school to set up the beans? What if a kid eats a bean?

Anyway, after Mrs. O has ensured that all the kids noticed that their boards are half-white and half-blue, she starts the game. The game is supposed to be about grouping and regrouping in place value systems, but it’s really entirely about beans. She calls out a command, and the kids add a bean. At no time does she connect the beans to numbers.

According to Cohen, this was no accident, as Mrs. O wasn’t really a fan of making numbers explicit in her activities:

This was a crucial point in the lesson. The class was moving from what might be regarded as a concrete representation of addition with regrouping, to a similar representation of subtraction with regrouping. Yet she did not comment on or explain this reversal of direction. It would have been an obvious moment for some such comment or discussion, at least if one saw the articulation of ideas as part of understanding mathematics. Mrs. O did not teach as though she took that view. Hers seemed to be an activity-based approach:

It was as though she thought that all the important ideas were implicit, and better that way.

Oublier is a *huge *believer in manipulatives — in fact, the transition from worksheets to manipulatives seems to be a big part of what her “revolution” entailed. For Mrs. O, kids learn *through *the physical manipulation of the objects. As in, learning is the direct result of touching beans:

Why did Mrs. O teach in this fashion? In an interview following the lesson I asked her what she thought the children learned from the exercise. She said that it helped them to understand what goes on in addition and subtraction with regrouping.

Manipulating the materials really helps kids to understand math, she said. Mrs. O seemed quite convinced that these physical experiences caused learning, that mathematical knowledge arose from the activities.

Oublier tells Cohen that she relies heavily on a textbook, *Mathematics Their Way*, and that this text was the major source of some of her new ideas about physical activities and teaching math. From poking around, it looks like the whole text has been posted online, including the lesson that Mrs. O was caught teaching. Here’s what the bean-counting activity looks like in the text:

OK, now the next page of that activity:

But you won’t believe what’s on the page after that:

This is sort of getting repetitive so I’ll just skip ahead five pages:

Cohen comes down pretty hard on this curriculum, and on Mrs. O for using it:

Math Their Wayfairly oozes the belief that physical representations are much more real than symbols. This fascinating idea is a recent mathematical mutation of the belief, at least as old as Rousseau, Pestalozzi, and James Fenimore Cooper, that experience is a better teacher than mere books. For experience is vivid, vital, and immediate, whereas books are all abstract ideas and dead formulations.

I’ve focused on the manipulative episode, but that’s just part of her teaching that’s detailed in the piece. According to Cohen, Oublier generally seems to adopt the exterior of cutting-edge math teaching while sort of missing their points. She asks kids to estimate, but doesn’t give them chances to think or share ideas. She uses manipulatives, but doesn’t really ask kids to think much with them. She puts kids into small groups, but basically uses this as a classroom management structure. She avoids numbers and abstraction wherever possible.

This was certainly not what California’s math reformers had in mind.

**III. **

The point, for Cohen, is that California’s math reformers let Mrs. O down. But how, exactly?

I found myself needing more context for the California reforms than Cohen provides. Fortunately, the journal issue in which Mrs. O originally appeared was entirely dedicated to the California math reforms. (In fact, every piece in that issue was a different in-depth case study like Mrs. O.)

Cohen actually leads off the issue with a helpful summary of the aims and methods of the 1985 math reforms (link). At their center was a document, the California Math Framework. The Framework called for a transformation of math teaching away from rote memorization and drill, and towards a focus on conceptual understanding, teaching kids to communicate about math, problem solve, work in groups, make sense of math, etc.

So far, nothing new. Reform groups like NCTM have been pumping out these documents for a century.

What *was* new was the muscle California chose to employ. The state education office said that they would only reimburse districts for textbooks that met the standards of the Framework. And then they actually followed through by rejecting *all *the texts that publishers initially submitted. Eventually, the state got what they wanted and created an approved list of textbooks for districts to choose from.

(As Alan Schoenfeld notes in his *Math Wars *piece, California — along with Texas and New York — determine what gets published nationally because of the size of their markets. The publishers basically design their books for the big states, and the rest of the country gets dragged along. So California’s reform muscle had national implications.)

This was half the plan. The other half was to change the state tests for kids so that they also reflected the vision of the Framework. The idea was that if textbooks and tests were in place, teachers would come around all on their own.

I missed this the first few times, but this is why Cohen dwells so much on Oublier’s textbook choice. Oublier’s favored *Math Their Way *text was *not *an accepted California text, and Oublier’s district had adopted something else. Oublier likes *Math Their Way*, though, so she just uses that in her classroom instead. None of her superiors seems to mind either.

In other words, that entire “change teaching by making a list of textbooks” plan was sort of stupid. It failed to account for the ability of teachers to *get other textbooks if they wanted to.*

The fundamental assumption of the policy seemed to be that teachers need permission, or perhaps incentives, to teach in new ways. As Cohen points out — over and over — this is not the case. Teaching in fundamentally different ways implies *believing *that you should teach differently as well as *knowing *how to do so.

It’s pretty simple, actually: if you want to change teaching, you can’t ignore the teachers.

**IV. **

Even as Cohen critiques the California reforms, he still seemed to me pretty cheery about the potential for policy to impact reform.

First, he really does seem to give a lot of agency to math textbooks. He keeps on talking about the influence of the *Math Their Way *book on Mrs. O. On the one hand, the book’s influence on her comes at the expense of the Framework’s reach. At the same time, if a textbook can really have such a strong impact on a teacher, then the premise of the California reforms has been upheld. If you’re a reformer reading Cohen, I imagine that your mind starts wandering: imagine what would’ve happened if we could’ve gotten the right book in her hands!

Beyond Cohen’s implicit optimism about textbook reform, he also wonders aloud about the possibility that a bit of incentive-engineering could have steered someone like Mrs. O towards better teaching:

“The only apparent rewards were those that she might create for herself, or that her students might offer. Nor could I detect any penalties for non-improvement, offered either by the state or her school district.”

These two sources of optimism, when put in context, seemed a bit dated to me. Cohen published this article in 1990, just after NCTM published its *Principles and Standards for School Mathematics* in 1989. This was, in many ways, a higher-profile go at California’s Framework, and (surprisingly to all involved) it took off, becoming a blockbuster for NCTM.

In the 90s, NSF would fund the development of new math texts that were aligned with the NCTM standards. My sense is that they didn’t live up to the expectations of the textbook-optimists. The texts were just texts, tools that teachers could use well or poorly depending on their understanding of math and of teaching.

It turns out: textbooks can’t transform teachers.

(Textbooks, it also turns out, can become highly visible targets of controversy, and nearly all use of the reform textbooks became contentious in the 90s. So that seems like it needs to be part of the textbook-reform calculus.)

Cohen seems to think that *Math Their Way *transformed Mrs. O, but he also thinks that she didn’t really revolutionize her teaching. The changes were cosmetic. And there’s a huge difficulty determining how the text impacted because of the plain fact that she *chose *this curriculum. Presumably, she chose it because she was disposed to. It fit with her understanding of math and of teaching. It didn’t fundamentally challenge her, and I see no reason to think that a text has any such power of a teacher, even when imposed.

Cohen’s other musing — about incentives — has echoes in No Child Left Behind and performance pay reforms. These reforms have also failed to live up to the dreams of the reformers, as all reforms do, and teaching chugs along, mostly as it has.

At times, it seemed to me that Cohen believes that the fundamental problem, for Mrs. O, is that her views on the nature of math remain unchanged:

…however much mathematics she knew, Mrs. O knew it as a fixed body of truths, rather than as a particular way of framing and solving problems. Questioning, arguing, and explaining seemed quite foreign to her knowledge of this subject. Her assignment, she seemed to think, was to somehow make the fixed truths accessible to her students.

I’m not particularly sympathetic to this critique. Math, among other things, *is *a fixed body of truths (theorems, facts, relationships) that we ought to help students know.

But forget that for a moment. Cohen sometimes seems to think that this isn’t just a problem for Mrs. O, but the root* *problem. If we could just help Oublier see that math isn’t quite as she thinks it is — that it’s dynamic, a source of puzzles, it’s about thinking and not just about knowing — then her teaching really would undergo a real revolution.

This seems to be where we are, right now, in math education reform. We’re not trying to save the world with NSF-funded textbooks, and we’re not hoping to incentivise great teaching. We believe, like Cohen, that the fundamental problem is one of learning, and that the fundamental problem is a *fundamental problem, *some ambitiously big thing that, if we can help teachers attain, the rest of their teaching will fall into place.

Right now, one version of the “fundamental problem” is productive struggle. NCTM has included this in their latest set of reform standards, the *Principles to Actions* standards.* *And if you’re in Baltimore this July, you can attend a three-day summer institute focused on productive struggle. The workshop promises to show how productive struggle is tied to every dimension of effective math instruction, from planning to feedback to wider advocacy.

I don’t think I believe in this sort of reform either. Cohen keeps drawing comparisons in this piece between teacher and student learning — both are challenging, he says, both take time. And that’s true. But imagine if we treated students like teachers. In other words, imagine if instead of teaching math to kids we had a workshop a few times a year where we tried to *fundamentally alter their conceptions of math*, and then sort of hoped that the rest of their math learning would just fall into place.

I know the comparison isn’t exactly direct, or fair, but I don’t believe that any knowledge can be altered by changing one fundamental element. Knowledge isn’t really structured that way, it seems to me. It’s not built on a foundation. To alter teaching you’d have to alter it *broadly*, not centrally. And broad change just can’t happen in a three-day workshop.

The final source of optimism that Cohen raises is that maybe Mrs. O represents progress for math reform. Though she hasn’t seemed to internalize the message of the reform, this sort of messy progress is what progress actually looks like.

I have no way of knowing if that’s true, but it certainly strikes me as possible. I haven’t read more recent work of Cohen’s. I wonder if, looking back on the last 30 years of reform, he’s still as optimistic.

**V.**

Hey, wait a second! This is just a single case study. We were swept along in this gripping tale (aptly summarized) and assumed she represented some larger trend, but that’s just the illusion of focus. Cohen’s fooled us, then, hasn’t he? Maybe Mrs. O means nothing at all. (Or, at least, nothing beyond her own case.)

There are two things that temper this sort of skepticism. First, the journal that published Mrs. O also published four other case studies in the same issue (open version). So in addition to the case of Mrs. O, you also get the case of Carol Turner, Cathy Swift, Joe Scott, and Mark Black.

(Unclear if the other pseudonyms are also supposed to be deeply meaningful. Mark Black, because policymakers treat him like a black box. Cathy Swift, because the reforms were too fast! The other two stump me. Maybe they’re anagrams? Joe Scott = COOT JEST.)

Five case studies are only a bit better than one, but these other four cases present a lot of the same mixed-success-at-best themes as Mrs. O’s case. That helps.

The other thing that tempers skepticism about Mrs. O’s relevance is that Cohen actually also identified the “forgotten teacher” problem in a very different piece of research.

That other piece is called *Instructional Policy and Classroom Performance: The Mathematics Reform in California. *This time around, Cohen and his team do pretty much the opposite of “sit in the back of a classroom and watch.” They survey 1,000 California elementary teachers. They ask teachers to rate how frequently they employ various instructional activities in class. Hey, they ask, wouldn’t it be nice if all these teacher responses really pointed to two types of teachers? We could call them “traditional” and “reform-friendly”…

Err, did I say “traditional”? I meant “conventional”:

Anyway, Cohen’s group also asked teachers what professional learning opportunities they had, in relation to the math reforms. (I love that ‘Marilyn Burns’ is an option.)

What they find basically supports Cohen’s take in his Mrs. O piece — reform is possible, but only when it focuses on professional development that targets teacher learning:

Our results suggest that one may expect such links when teachers’ opportunities to learn are:

- grounded in the curriculum that students study;
- connected to several elements of instruction (for example, not only

curriculum but also assessment);- and extended in time.
Such opportunities are quite unusual in American education, for professional

development rarely has been grounded either in the academic content of schooling or in knowledge of students’ performance. That is probably why so few studies of

professional development report connections with teachers’ practice, and why so many studies of instructional policy report weak implementation: teachers’ work as learners was not tied to the academic content of their work with students.

Some people love the Mrs. O piece, but hated the sort of study that we previously read here, the one about teacher-centered instruction for first graders. First, because they rely on teacher responses to survey questions, and how much can you really learn from that? Second, because the statistical work can hide researcher assumptions that then become tricky to dig out. Third, because with scale comes quality control issues. You really no longer know what you’re dealing with.

To which, we might ask, why did Cohen produce exactly this kind of study when it came to evaluate the success of California’s reforms?

I talk to just as many people, though, who hold the complete opposite view. To them, something like the Mrs. O study is useless, as it doesn’t help us identify the causal forces at work. Maybe the reform failed Mrs. O, but compared to what? There are no controls, and without some sort of random assignment to a treatment can we really be sure that a focus on teacher-learning would make the difference Cohen said it would?

Is it too soft of me to say that both critiques are right?

It’s not my job to study teaching, but it sure seems hard. Every research approach has trade-offs. The way I see things, it’s best to use multiple, incompatible approaches to study the same things in teaching from wildly different perspectives. Why? Because of how it’s possible to take wildly different incompatible perspectives on teaching.

At one point, Cohen points out that Mrs. Oublier seemed comfortable living in contradiction:

Elements in her teaching that seemed contradictory to an observer therefore

seemed entirely consistent to her, and could be handled with little trouble.

But there really isn’t anything strange here at all. Everyone is willing to live with some contradictions in their lives. Contradictions can be unlivable, but they can also be productive — in teaching, in life, but also in research. Intellectually incompatible perspectives can be desirable.

—

Anyway, enough about all this. What should we read next?